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Flood Studies Update

Technical Research Report

Volume I Rainfall Frequency

Volume II Flood Frequency Estimation

Volume III Hydrograph Analysis

Volume IV Physical Catchment Descriptors

Volume V River Basin Modelling

Volume VI Urbanised and Small Catchments

Volume II

Flood Frequency Estimation Conor Murphy, Conleth Cunnane, Samiran Das and Uzzal Mandal

Derived from Technical Research Reports by

NUI Galway and NUI Maynooth


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Abstract

Flood frequency estimation addresses the issue of flood risk assessment required in flood

zoning and spatial planning, and in the specification of flow values for the design of flood

alleviation and control works.

This volume describes flood frequency estimation research undertaken at NUI Galway and

NUI Maynooth and provides a statistical summary of flood flows in Ireland taken from the

data archives of the OPW, EPA and ESB. The research is based on analysis of annual

maximum flow records at some 200 river flow gauging sites. The records analysed range

from eight to 55 years in length. More detailed research has been carried out on subsets of

110 gauged sites for which the quality of recorded flows is judged the most reliable.

Most floods occur during the winter half of the year, with some notable floods also occurring

during summer, especially in August. Because of Ireland’s humid climate, the year-to-year

variation of flood flow values (indexed by the coefficient of variation) is typically quite small

by international standards. The so-called skewness of the flood series is also modest. While

no single statistical distribution can be considered “best” at all locations, it has been found

that the EV1 (Gumbel) and lognormal distributions provide a reasonable model for the

majority of stations.

Guidance is provided on the estimation of the design flood of required annual exceedance

probability at both gauged and ungauged locations and on how to express the uncertainty in

the resulting estimates. The use of growth curves or growth factors based on data pooled

from a group of sites is generally advocated, with the use of suitable 3-parameter

distributions recommended in many applications.

A procedure is presented for estimating the so-called index flood (QMED) at an ungauged

site. QMED estimation is especially important in Ireland because flood growth rates are

generally mild. The recommended procedure for QMED estimation at ungauged sites is to

transfer information from a nearby site, ideally one upstream or downstream of the site of

interest. The user must apply experience and technique to select the so-called pivotal gauged

catchment from which to make the transfer.

Examples presented show a range of difficult cases that can arise in practice. In some cases,

the user is advised to consider single-site estimation in addition to – or in combination with –

the generally recommended pooling method.

The volume concludes that flood estimation cannot be reduced to a formula-based procedure.

Individual analysts must make choices which reflect the circumstances of the problem, the

available flow data and their own knowledge and experience.

©Office of Public Works 2014


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Further information about the research

FSU Technical Research Reports (TRRs) are available in their original

form for researchers and practitioners who seek additional information

about a method. The original TRRs sometimes document exhaustive

application of a method to many catchments. In others, additional options

are reported.

Inevitably, the relevance of the original TRRs is influenced by OPW

decisions on which methods to implement, and how best to arrange and

support them. Readers who consult the original TRRs will notice editorial

re-arrangements and compressions, and occasional changes in notation and

terminology. These were judged necessary to enhance understanding and

use of the FSU methods amongst general practitioners. More significant

changes are labelled explicitly as editorial notes.


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Contents

Abstract ii

Contents iv

Notation xiv

Lower case symbols xiv

Upper case symbols xiv

Subscripts xv

Abbreviations and descriptor names xv

Glossary of terms xvii

1 Introduction and data 1

1.1 Why we estimate flood frequency 1

1.2 Flood data 2

1.2.1 Introduction 2

1.2.2 Annual maximum series 2

1.2.3 FSU flood datasets 2

1.2.4 Expressions of flood rarity 3

1.2.5 Index flood 3

1.3 Historical floods 4

1.4 Structure of volume 4

1.5 Material in common with rainfall frequency studies 5

1.5.1 Annual maximum and peaks-over-threshold series 5

1.5.2 Return period 6

1.5.3 Average recurrence interval 6

1.5.4 Langbein’s formula 7

2 Estimation of the index flood, QMED 8

2.1 Exploratory data analysis 8

2.1.1 Introduction to the datasets 8

2.1.2 Adjustments for period-of-record effects 10

2.1.3 Physical catchment descriptors 12

2.1.4 Rank correlations 14

2.1.5 Principal component analysis 15

2.1.6 Correlations and competing variables 16

2.1.7 Scatter-plots and summary information for selected PCDs 17

2.2 Rural-catchment model for estimating QMED from PCDs 18

2.2.1 Regression methods 18

2.2.2 Alternative methods 20

2.2.3 Selection of catchments for calibration and validation 20

2.2.4 Selecting PCDs 21

2.2.5 Choosing a model 22

2.2.6 Model performance 23

2.2.7 Checking for logical consistency 25

2.2.8 Checking and investigating the model residuals 25

2.2.9 Validation of model performance 28

2.3 Assessing model robustness 29


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2.3.1 Model robustness 29

2.3.2 Bootstrap resampling 29

2.3.3 Model robustness to influential stations 30

2.3.4 Robustness to spatial drift in model coefficients 31

2.4 Investigating the effect of arterial drainage 32

2.4.1 Approach 32

2.4.2 Analysis of stations for which both pre and post-drainage records held 32

2.4.3 Modelling QMED on drained and undrained catchments separately 34

2.5 Adjusting the selected model for urbanisation 34


2.5.2 Deviation of QMED on urbanised catchments from rural model 35

2.5.3 A special check 36

2.5.4 Approach to devising an urban adjustment 36

2.5.5 Exploratory data analysis 37

2.5.6 An urban adjustment model 37

2.5.7 Performance of urban adjustment model 38

2.5.8 Comparisons and contrasts with the FEH 39

2.6 Improving model performance by data transfer 39


2.6.2 Review of techniques 39

2.6.3 Geostatistical mapping of residuals 40

2.6.4 Possible disadvantages of automated methods 42

2.6.5 Recommended procedure for data transfer 43

2.7 Worked example of QMED estimation at an ungauged site 43

2.7.1 Illustrative example 43

2.7.2 Merging data transfers from two sites 46

2.7.3 Geostatistical mapping method 46

3 Trend and randomness 47

3.1 Tests 47

3.1.1 Methods 47

3.1.2 Formats 48

3.2 Findings 48

3.3 Pragmatism 49

4 Descriptive statistics – and inferences therefrom 50

4.1 Descriptive statistics 50


4.1.2 Summary statistics – the idea 50

4.1.3 Summary statistics based on moments and L-moments 50

4.1.4 Additional summary statistics 51

4.2 Summary statistics for Irish flood data 52

4.2.1 Summary statistics for 181 FSU stations 52

4.2.2 Variability and skewness of Irish flood data 58

4.2.3 Comparisons of CV with L-CV and of H-skew with L-skew 59

4.3 Geographical traits 61

4.4 Preliminary distribution choice from skewness v. record length plot 63

4.5 Preliminary distribution choice aided by L-moment ratio diagrams 66

4.5.1 L-moment ratio diagrams 66


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4.5.2 Implication for choice of 2-parameter distribution 66

4.5.3 Implication for choice of 3-parameter distribution 68

4.6 Exceptional recorded values and outliers 68

4.6.1 Qmax/Qmean ratios 68

4.6.2 High outliers 69

4.6.3 Low outliers 70

5 Probability plots … and inferences from them 71

5.1 Introduction 71

5.1.1 The idea of a probability plot 71

5.1.2 Synthetic examples 71

5.1.3 Standard plots for the FSU flood peak data 71

5.1.4 Approach taken to assessing plots 74

5.2 Linear patterns 74

5.3 Curve patterns 74

5.3.1 Assignment of patterns 74

5.3.2 Probability plots for 43 Grade A1 stations 75

5.3.3 Probability plots for 110 Grade A1+ A2 stations 76

5.3.4 Curve pattern in relation to skewness coefficient 77

5.4 Flood volumes associated with largest peaks on convex probability plots 79

5.4.1 Stations with a convex curve pattern 79

5.4.2 Hypothesis 79

5.4.3 Arrangement of study 79

5.4.4 Calculation of hydrograph volumes 80

5.4.5 Example 80

5.4.6 Summary of findings 82

5.4.7 A further check on the hypothesis 82

5.5 Flood seasonality 83

5.5.1 Seasonality of annual maximum floods 83

5.5.2 Seasonality of largest floods 84

5.5.3 Circular diagrams 84

5.6 Flood statistics on some rivers with multiple gauges 85

5.6.1 Down-river growth in QMED 85

5.6.2 Down-river variation in probability plots 85

6 Determining T-year flood magnitude QT by index flood method 87

6.1 Introduction 87

6.2 Regional flood frequency analysis and the index flood approach 88

6.2.1 Index flood approach 88

6.2.2 Type of region for pooling flood data 88

6.2.3 Choice of index flood 88

6.2.4 A 2-stage approach 89

7 Flood growth curve estimation 90

7.1 Introduction to xT 90

7.2 Single-site and pooled estimates of QT 90

7.2.1 Advantages and drawbacks of the two approaches 90

7.2.2 Choice of distribution 90

7.3 Pooling groups 91

7.3.1 The idea of pooling 91


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7.3.2 Construction of pooling groups 91

7.3.3 The 5T rule 92

7.3.4 Recommended metric for pooling-group construction 92

7.4 Growth curve estimation 93

7.4.1 Pooled L-moment ratios 93

7.4.2 Fitting growth curve distributions by the method of L-moments 93

7.4.3 Growth curves for particular distributions 94

8 Effect of catchment type and period of record on xT and QT 97

8.1 Data screening 97

8.1.1 Discordancy 97

8.1.2 Characteristics of the three discordant stations 99

8.1.3 Other heavily urbanised catchments 102

8.2 Effect of catchment type on pooled growth curve estimates 103

8.3 Temporal effect on pooled growth curve estimates 106

8.3.1 Periods of record considered 106

8.3.2 Summary results for 90 stations 106

8.3.3 Individual results for three decades (50 stations) 106

8.3.4 Individual results for five decades (26 stations) 108

8.3.5 Individual results for a different set of five decades 109

8.3.6 Outcome 109

8.4 Arterial drainage effect on pooled growth curve estimates 110

8.5 Implications for flood frequency estimation 111

8.5.1 Implications for pooling-group formation 111

8.5.2 Respecting recent flood data 112

9 Uncertainty estimation 112

9.1 Standard errors – an introduction 112

9.2 Standard error of QMED estimation from gauged flood data 113

9.3 A comparison with FEH methods 114

9.4 Standard error of QT in single-site estimation 115

9.4.1 Method 115

9.4.1 Relative standard error 115

9.4.2 Relative standard errors under the EV1 assumption 115

9.4.3 Relative standard errors under the GEV assumption 116

9.5 Standard error of pooled estimate of xT and of QT 117

9.5.1 Simulation method 117

9.5.2 Results based on EV1 simulations 118

9.5.3 Results based on GEV simulations with k = -0.1 119

9.5.4 Summary 120

9.6 Standard error of QT based on PCD estimate of QMED and pooled xT 121

10 Guidelines for determining QT 123

10.1 Introduction 123

10.1.1 Single-site or pooled analysis? 123

10.1.2 Probability plots 123

10.1.3 Factors to be borne in mind 123

10.2 Determining QT by single-site analysis 124

10.2.1 General guidance 124

10.2.2 Plotting positions 124


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10.2.3 Parameter estimation for EV1 distribution 125

10.2.4 Parameter estimation for LN2 distribution 126

10.3 Determining QT from pooled data 126

10.3.1 When flood data are available at the subject site 126

10.3.2 When the subject site is ungauged 127

10.4 Characteristic examples of probability plot behaviour 127

10.4.1 Good straight-line behaviour 127

10.4.2 Good straight-line behaviour but single-site and pooled disagree 128

10.4.3 Concave upwards behaviour with outlier 129

10.4.4 Convex behaviour 131

10.4.5 Unclear behaviour with extreme outlier 132

10.4.6 Irregular behaviour 133

10.5 Additional notes on the choice of distribution and method 134

10.5.1 Problems in the use of 3-parameter distributions for single-site analysis 134

10.5.2 Reconciling single-site and pooled analyses 135

10.5.3 Probability associated with a very large recorded flood 135

10.5.4 Flood growth curves with an upper bound 136

11 Data transfers revisited 137

11.1 Interim assessment of QMED data transfers 137

11.1.1 Subject sites used in the interim assessment 137

11.1.2 Methods used in the interim assessment 138

11.1.3 Results 139

11.1.4 Remarks 141

11.2 Further guidance on pivotal catchments and data transfers 141

12 Summary and conclusions 143

12.1 Data 143

12.2 Descriptive statistics 143

12.3 Seasonal analysis 143

12.4 Estimation of design flood 144

Acknowledgements 146

References 146

Appendices 151

Appendix A Review of stage-discharge relationships 151

A1 Terminology 151

A2 Review of stage-discharge relationships 151

A2.1 General form of stage-discharge relationships 151

A2.2 Analysis tools and background information 152

A2.3 Review procedure 152

A2.4 Gauging station surveys 153

A3 Gauging station classification 154

A3.1 Initial site classification 154

A3.2 FSU station classification 154

A3.3 Uncertainty analysis 155

A4 Production of annual maximum flood series 156

Appendix B Flood data exclusions 157

B1 Stations omitted from the QMED modelling research 157


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B2 Other notes on stations omitted from research 158

B3 General recommendation 158

Appendix C Notes on the regression methods used in Chapter 2 159

C1 Ordinary least-squares (OLS) regression 159

C2 Weighted least-squares (WLS) regression 159

C3 Generalised least-squares (GLS) regression 160

C4 Geographically-weighted regression (GWR) 161

Appendix D QMED models for drained/undrained catchments 164

D1 Partitioned approach 164

D2 Undrained catchments 164

D3 Drained catchments 165

D4 Choosing a general purpose model 166

Appendix E Trend analysis of AM flows in Irish rivers 168

E1 Introduction 168

E2 Purpose 168

E2.1 Importance of testing whether hydrological processes are stationary 168

E2.2 Types of change 169

E3 Procedure 169

E3.1 Steps in the analysis 169

E3.2 The idea of exploratory data analysis 170

E3.3 Hypothesis testing 170

E3.4 Use of p-values 171

E4 Statistical tests 171

E4.1 Parametric and non-parametric tests 171

E4.2 Null hypotheses 172

E4.3 Tests adopted 172

E4.4 Mann-Kendall test 173

E4.5 Spearman’s ρ test 174

E4.6 Mean-weighted linear regression test 175

E4.7 Mann-Whitney U test 175

E4.8 Turning points test (Kendall’s test) 176

E4.9 Rank difference test (Meacham test) 176

E5 Exploratory data analysis 176

E5.1 Selection of data 177

E5.2 Stations showing trend 177

E5.3 Stations with pre and post-drainage records 177

E6 Trend analysis and results 181

E6.1 Formats of flood data tested for trend 181

E6.2 Main test results 181

E6.3 Non-randomness 184

E6.4 Trend 184

E6.5 Discussion 186

E7 Summary 186

E8 Critical values for mean-weighted linear-regression test 187

Appendix F Additional summary statistics 189

Appendix G Sample probability plots and summary information 194


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Appendix H Probability-plot linear scores and curve patterns 195

Appendix J Flood volumes in relation to convex EV1 plots 199

J1 Flood peaks and volumes for Station 07006 Moynalty at Fyanstown 199

J2 Flood peaks and volumes for Station 15003 Dinan at Dinan Bridge 201

J3 Flood peaks and volumes for Station 16008 Suir at New Bridge 203

J4 Flood peaks and volumes for Station 16009 Suir at Cahir Park 205

J5 Flood peaks and volumes for Station 24013 Deel at Rathkeale 207

J6 Flood peaks and volumes for Station 24082 Maigue at Islandmore 209

J7 Flood peaks and volumes for Station 25017 Shannon at Banagher 211

J8 Flood peaks and volumes for Station 25021 Little Brosna at Croghan 213

Appendix K Seasonal distribution of annual maximum floods 215

Appendix L Distance metrics for pooling-group construction 217

L1 Introduction 217

L2 Notation 217

L3 Selecting variables to define the distance metric 217

L4 Statistics to help in choosing a good distance metric 218

L5 Pooled uncertainty measure, PUM 218

L6 Application to flood data at 90 A1 + A2 stations 218

L7 Discussion 220

L8 Alternative weightings of the recommended distance metric 220

Maps

Map 2.1: Spatial distribution of 205 catchments 11 Map 2.2: Factorial error in QMED estimated by selected rural model 26 Map 2.3: IDW-interpolated residuals from rural ℓnQMED model 41 Map 4.1: Specific QMED (m

3s

-1/km

2) for 176 A1 + A2 + B stations 61

Map 4.2: CV at 110 A1 + A2 stations 62 Map 4.3: Hazen skewness at 110 A1 + A2 stations 63 Map 8.1: Location of the 85 stations within four geographical regions 104

Map C.1: Spatial variation in the FARL coefficient as interpolated from GWR 163

Figures

Figure 2.1: Number of stations in each quality category (full dataset of 205 stations) 9 Figure 2.2: QMED adjusted for period-of-record effects 14 Figure 2.3: Annual maximum flow series for Station 09035 Cammock at Killeen Road 14 Figure 2.4: Relationship between PCDs summarising extent of arterial drainage 17 Figure 2.5: Association of ℓnQMED and selected PCDs for 205 catchments 18 Figure 2.6: Improvement of r

2 for a model size of one to nine variables 23

Figure 2.7: Observed and modelled QMED for the 145 calibration stations 24 Figure 2.8: Diagnostic plots of ℓnQMED model performance 25 Figure 2.9: Residuals versus individual PCDs for rural ℓnQMED model 27 Figure 2.10: Observed and modelled QMED for the 25 validation stations 28 Figure 2.11: Normal quantile-quantile plots of the bootstrapped model coefficients 30 Figure 2.12: Influence of individual stations in determining model coefficients 31 Figure 2.13: QMED for pre and post-drainage records at 15 stations 33 Figures 2.14: QMED change following drainage (against ARTDRAIN and ARTDRAIN2) 33 Figure 2.15: QMED change following arterial drainage (against BFIsoil) 34 Figure 2.16: Performance of rural QMED model on urbanised catchments 35 Figure 2.17: Correlations between ℓnUAF and selected PCDs 37 Figure 2.18: Performance of UAF-adjustment to QMED 38


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Figure 4.1: Relationship between L-CV and CV for 110 A1 + A2 stations 60 Figure 4.2: Relationship between L-skew and H-skew for 110 A1 + A2 stations 60 Figure 4.3: Skewness at 110 A1 + A2 stations versus record length 64 Figure 4.4: L-skewness against record length for 110 A1 + A2 gauging station 65 Figure 4.5: L-moment ratio diagram for annual maximum floods 67 Figure 4.6: Comparison with equivalent samples drawn from particular distributions 67 Figure 5.1: Relative frequency of curve patterns – 43 Grade A1 stations 76 Figure 5.2: Relative frequency of curve patterns – 110 Grade A1 + A2 stations 76 Figure 5.3: Hydrographs of four largest AM events, Station 24082 Maigue at Islandmore 81 Figure 5.4: Hydrograph volumes of four largest AM events at 24082 Maigue at Islandmore 82 Figure 5.5: FAI v. Hazen skewness, labelled by EV1 curve pattern (110 A1 + A2 stations) 83 Figure 5.6: Seasonality and magnitudes of 6969 AM floods (202 A1 + A2 + B stations) 84 Figure 5.7: Variation of QMED down the Barrow, Suir and Suck 85 Figure 6.1: Probability density function and cumulative distribution function 87 Figure 8.1: AM flow series for three discordant sites (Stations 08005, 09002 and 09010) 101 Figure 8.2: Decadal estimates for 50 stations (arranged in station-number order) 107 Figure 8.3: Decadal estimates for 26 stations (arranged in station-number order) 108 Figure 8.4: Comparison of pre and post-drainage flood statistics at 16 stations 111 Figure 11.1: Performance at Station 16002 Suir at Beakstown (512 km

2) 139

Figure 11.2: Performance at Station 16008 Suir at Newbridge (1120 km2) 139

Figure 11.3: Performance at Station 18050 Blackwater at Duarrigle (244.6 km2) 140

Figure 11.4: Performance at Station 24008 Maigue at Castleroberts (805 km2) 140

Figure 11.5: Performance at Station 26002 Suck at Rookwood (626 km2) 140

Figure C.1: Fitted model for inter-site correlation 160

Figure D.1: Performance achieved with undrained model (on 95 undrained catchments) 165 Figure D.2: Performance achieved with drained model (on 50 drained catchments) 166

Figure E.1: Stations for which AM flow series shows significant trend 179 Figure E.2: AM flow series for stations with arterial drainage works during record 180 Figure E.3: AM flow series showing likely influence of arterial drainage 185

Figure J.1: Hydrograph volumes for Station 07006 Rank 1, 2 and 4 AM flood peaks 199 Figure J.2: Hydrograph volumes for Station 15003 Rank 1, 2 and 4 AM flood peaks 201 Figure J.3: Hydrograph volumes for four largest Station 16008 AM flood peaks 203 Figure J.4: Hydrograph volumes for four of five largest Station 16009 AM flood peaks 205 Figure J.5: Hydrograph volumes for four largest Station 24013 AM flood peaks 207 Figure J.6: Hydrograph volumes for four largest Station 24082 AM flood peaks 209 Figure J.7: Hydrograph volumes for four of five largest Station 25017 AM flood peaks 211 Figure J.8: Hydrograph volumes for four largest Station 25021 AM flood peaks 213

Histograms

Histogram 2.1: Length of available record for QMED estimation (205 stations) 9 Histogram 2.2: QMED values across full dataset of 205 stations 10 Histograms 2.3: Catchment sizes for calibration and validation datasets 21 Histograms 4.1: CV and L-CV at Grade A1 and A2 stations 59 Histograms 4.2: Hazen skewness and L-skewness at Grade A1 and A2 stations 59 Histogram 5.1: Seasonal occurrence of annual maximum floods 83 Histogram 5.2: Month corresponding to series maximum flow (202 A1 + A2 + B stations) 84

EV1 Probability Plots

EV1 Probability Plot 5.1: Nine random samples of size 25 drawn from EV1 72 EV1 Probability Plot 5.2: Nine random samples of size 50 drawn from EV1 73 EV1 Probability Plot 5.3: Station 24082 Maigue at Islandmore 80 EV1 Probability Plot 5.4: Multiple stations on the Rivers Barrow, Suir and Suck 86


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EV1 Probability Plots 8.1: Three discordant sites (Stations 08005, 09002 and 09010) 100 EV1 Probability Plots 8.2: Two further heavily urbanised catchments 102 EV1 Probability Plot 10.1: Single-site and pooled estimates, Station 25016 128 EV1 Probability Plot 10.2: Single-site and pooled estimates, Station 09001 129 EV1 Probability Plot 10.3: Single-site and pooled estimates, Station 09010 129 EV1 Probability Plot 10.4: Single-site and pooled estimates, Station 25002 131 EV1 Probability Plot 10.5: Single-site and pooled estimates, Station 08009 132 EV1 Probability Plot 10.6: Single-site and pooled estimates, Station 09002 133 EV1 Probability Plot 10.7: Single-site and pooled estimates, Station 26008 134

EV1 Probability Plot J.1: Station 07006 Moynalty at Fyanstown 199 EV1 Probability Plot J.2: Station 15003 Dinan at Dinan Bridge 201 EV1 Probability Plot J.3: Station 16008 Suir at New Bridge 203 EV1 Probability Plot J.4: Station 16009 Suir at Cahir Park 205 EV1 Probability Plot J.5: Station 24013 Deel at Rathkeale (post-drainage) 207 EV1 Probability Plot J.6: Station 24082 Maigue at Islandmore 209 EV1 Probability Plot J.7: Station 25017 Shannon at Banagher 211 EV1 Probability Plot J.8: Station 25021 Little Brosna at Croghan 213

Box-plots Box-plots 8.1: Pooled GEV estimates of x100 showing effect of (a) PEAT and (b) FARL 105 Box-plots 8.2: Pooled GEV estimates of x100 showing effect of (a) location and (b) AREA 105 Box-plots 8.3: Period-of-record and decadal effects on pooled flood growth by GEV 106 Box-plots 9.1: Relative SE of single-site quantile estimates – EV1 assumed 116 Box-plots 9.2: As above but with lines superposed to show theoretical values of relative SE 116 Box-plots 9.3: Relative SE of single-site quantile estimates – GEV assumed 117 Box-plots 9.4: As above but with lines superposed to show theoretical values of relative SE 117 Box-plots 9.5: Relative standard error in xT – pooled EV1 simulations 119 Box-plots 9.6: Relative standard error in QT – pooled EV1 simulations 119 Box-plots 9.7: Relative standard error in xT – pooled GEV simulations for k = -0.1 120 Box-plots 9.8: Relative standard error in QT – pooled GEV simulations for k = -0.1 120

Box-plots L.1: 100-year PUM values for eight formulations of distance metric dij 220

Boxes

Box 1.1: The risk equation 6 Box 2.1: Collinearity 17 Box 2.2: Interpretation of urban adjustment models for QMED 39 Box 2.3: Data transfer procedure when one of the catchments is urbanised 43 Box 4.1: Calculation of moments and their dimensionless ratios 51 Box 4.2: Calculation of L-moments and their dimensionless ratios 52 Box 4.3: Station 06030 Big at Ballygoly 53 Box 4.4: Station 08009 Ward at Balheary 68 Box 7.1: FEH pooling scheme 92 Box 7.2: Qualitative outline of simulation results for flood quantile estimate 95 Box 8.1: Box-plots 103 Box 8.2: Incorporation of historical flood data 109 Box 9.1: Comparison with uncertainty of FEH methods 114 Box 10.1: Plotting positions 125 Box 10.2: Combined use of single-site and pooled estimates 135

Tables

Table 2.1: QMED adjustments for period-of-record effects 13 Table 2.2: Rank correlations amongst QMED and 19 PCDs (205 gauged sites) 15


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Table 2.3: Principal component loadings of PCDs on ℓnQMED for 190 rural stations 16 Table 2.4: Summary information for PCDs at 190 rural stations used in model-building 19 Table 2.5: PCDs in best three 1 to 9-variable models for ℓnQMED (with ℓnAREA forced) 22 Table 2.6: Performance diagnostics for the addition of each independent variable 22 Table 2.7: Coefficient and collinearity statistics for selected rural model for ℓnQMED 24 Table 2.8: Bootstrapped confidence intervals for model coefficients (BCa method) 29 Table 2.9: Factorial change in QMED following drainage 32 Table 2.10: Five catchments poorly predicted by the QMEDrural model 36 Table 2.11: PCDs for Suck at Rookwood worked example 44 Table 2.12: Data transfers for Suck at Rookwood worked example 45 Table 3.1: Number of cases (out of 94) in which the null hypothesis is rejected 48 Table 4.1: Principal summary statistics 53 Table 4.2: Average values of some statistics for gauging stations (by station grade) 58 Table 4.3: High outliers amongst 110 stations graded A1 + A2 69 Table 4.4: Low outliers amongst 110 stations graded A1 + A2 70 Table 5.1: Linear pattern statistics for 110 A1 + A2 stations 74 Table 5.2: EV1 plot curve patterns – 110 A1 + A2 stations ordered by Hazen skewness 77 Table 5.3: Stations for which flood volumes were specially investigated 80 Table 8.1: L-moment ratios, PCDs and station discordancy within set of 88 A1+A2 stations 97 Table 8.2: Stations showing large discordancy values (in pool of 88 A1 + A2 stations) 102 Table 9.1: Typical standard errors when estimating QMED from annual maxima 114 Table 9.2: Relative standard errors for growth factors xT and quantile estimates QT 121 Table 11.1: Rivers having three or more gauging stations for assessment 137

Table C.1: Test of significance of spatial variability in ℓnQMED model parameters 162

Table D.1: Coefficient and collinearity statistics for undrained ℓnQMED model 164 Table D.2: Coefficient and collinearity statistics for drained ℓnQMED model 166 Table D.3: Validation of partitioned and general models for ℓnQMED 167

Table E.1: Stations with significant trends 177 Table E.2: Stations with pre and post-drainage records 178 Table E.3: Test statistics for trend and change in AM flow series 181 Table E.4: Number of cases (out of 94) in which the null hypothesis is rejected 184 Table E.5: Stations for which the trend is judged most highly significant 185 Table E.6: Critical values of the test statistic bs 187

Table J.1: Basic information for Station 07006 Moynalty at Fyanstown 199 Table J.2: Basic information for Station 15003 Dinan at Dinan Bridge 201 Table J.3: Basic information for Station 16008 Suir at New Bridge 203 Table J.4: Basic information for Station 16009 Suir at Cahir Park 205 Table J.5: Basic information for Station 24013 Deel at Rathkeale (post-drainage) 207 Table J.6: Basic information for Station 24082 Maigue at Islandmore 209 Table J.7: Basic information for Station 25017 Shannon at Banagher 211 Table J.8: Basic information for Station 25021 Little Brosna at Croghan 213

Table K.1: Percentage of AM floods occurring in winter half-year (Oct-Mar) 215 Table K.2: Month of maximum recorded flood in AM series 216

Table L.1: Summary of AM flow dataset used in dij study 219 Table L.2: Mean values of PUM100, H1 and H2 for various pooling schemes 219


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Notation

Lower case symbols

d Inter-catchment distance in km (often measured centroid-to-centroid)

dij Dissimilarity (or other) distance/measure/metric between sites i and j

f(x) Probability density function

g Coefficient of skewness

h Hardness (of data transfer)

k Shape parameter of GEV (or GLO) distribution

ℓn, ln Natural logarithm

m Number of stations (e.g. in a pooling group)

n Number of years of record, sample size

p Number of PCDs/variables in regression model

r Rank

r2 Coefficient of determination

t Test statistic

t2 Sample L-CV value

t3 Sample L-skewness value

t4 Sample L-kurtosis value

2t , 3t and 4t Pooled values of t2, t3 and t4

u Location parameter in extreme value distributions

Vector of L-moment ratios for station i ui

w Weighting term

xT T-year growth factor

y Reduced variate (e.g. for use with EV1/GEV distribution)

yT Value of y corresponding to T-year event

yi Value of y at which ith smallest AM to be plotted

yL Reduced variate for use with Logistic (i.e. LO or GLO) distribution

yN Reduced variate for use with Normal distribution

z Variate used in statistical exposition, z = ℓnX

α Scale parameter in extreme value distributions

β Scale parameter (growth curve)

β Test statistic

γ Euler’s constant (≈0.5772)

λ1 1st L-moment

λ2 2nd

L-moment

λ3 3rd

L-moment

μ Mean

μ3 3rd

central moment

μz Mean of z = ℓnX

ρ Spearman’s rank correlation coefficient

σ Standard deviation

σz Standard deviation of z = ℓnX

ω Weighting term

Upper case symbols

A Matrix of sums of square and cross-products of pooled L-moment ratios

A1 Gauging station grade A1

A2 Gauging station grade A2

B Gauging station grade B


xv

CV Coefficient of variation

Di (Hosking and Wallis) discordancy measure for station i

E Expected value

F(Q), F Cumulative distribution function

Fi Frequency at which ith smallest AM is to be plotted

F Test statistic

H0 Null hypothesis (in a specific statistical test)

H1 Heterogeneity measure based on L-CV

H2 Heterogeneity measure based on L-CV and L-skewness

H-skew Hazen-corrected skewness coefficient

L-CV Coefficient of L-CV

L-kurt Coefficient of L-kurtosis

L-skew Coefficient of L-skewness

M Number of repetitions in simulation (i.e. sampling) experiments

M100 1st PWM

M110 2nd

PWM

M120 3rd

PWM

M(ρ) Correlation function (used to adjust QMED)

N, n Number of years of record, sample size

N(0, γ2) Normally distributed with mean of zero and a variance of γ

2

Q Flood peak discharge (m3s

-1)

Q1, …, QN Flood series arranged in chronological order

Q(1), …, Q(N) Flood series arranged in rank order (smallest first)

QI Index flood

Qmax Maximum flood on record

Qmean, Q Mean of annual maximum flow series

QMED, Qmed Median of annual maximum flow series

Qmin Minimum AM flood on record

QT T-year flood

R2 Coefficient of determination

T Return period (years)

TAM Return period on annual maximum scale (years)

TPOT Return period on peaks-over-threshold scale (years)

U Test statistic

Var Variance

Matrix of independent variables X

Γ Gamma function

Standardised Normal distribution function

Subscripts

adj Adjusted (estimate)

d Donor (site)

o Overlap (period)

s Subject (site)

Abbreviations and descriptor names

AD, AD2 Abbreviations for ARTDRAIN and ARTDRAIN2

AdjFac Adjustment factor (in data transfer)

AEP Annual exceedance probability

ALLUV Proportional extent of floodplain alluvial deposit

ALTBAR Mean elevation of catchment (m)

AM Annual maximum


xvi

ANN Artificial neural network

AREA Catchment area (km2)

ARI Average recurrence interval

ARTDRAIN Proportion of catchment area mapped as benefitting from arterial drainage schemes

ARTDRAIN2 Proportion of river network length included in Arterial Drainage Schemes

BCa Bias-Corrected and accelerated (see Section 2.3.2)

BFI Baseflow index

BFIHOST Baseflow index derived from HOST soils data (a UK descriptor)

BFIsoil Soil baseflow index (estimate of BFI derived from soils, geology and climate data)

CDF Cumulative distribution function

CFRAM Catchment flood risk assessment and management

C.I. Confidence interval

Cov Covariance

CV Coefficient of variation

D Downwards concave i.e. convex (a curve pattern)

DMF Daily mean flow

DRAIND Drainage density (km/km2)

EDA Exploratory data analysis

EPA Environmental Protection Agency

ESB Electricity Supply Board

EV1 Extreme Value Type 1 = Gumbel (a 2-parameter distribution)

FAI Flood attenuation index

FARL Index of flood attenuation by reservoirs and lakes

FEH Flood Estimation Handbook

FLATWET PCD summarising proportion of time soils expected to be typically quite wet

FOREST Proportional extent of forest cover

FSE Factorial standard error

FSR Flood Studies Report

FSU Flood Studies Update

GEV Generalised Extreme Value (a 3-parameter distribution)

GLO Generalised Logistic (a 3-parameter distribution)

GLS Generalised least-squares (a regression method)

GWR Geographically weighted regression

HA Hydrometric Area (e.g. HA10 is Hydrometric Area 10)

HGF Highest gauged flow

H-skew Hazen-corrected skewness

IDW Inverse distance weighting

iid Independently and identically distributed

L Linear (a curve pattern)

L-CV Coefficient of L-variation (Hosking and Wallis, 1997)

LH Left hand

L-kurt L-kurtosis

LN2, LN 2-parameter lognormal (a distribution)

LN3 3-parameter lognormal (a distribution)

LO Logistic (a 2-parameter distribution)

L-skew L-skewness

MSL Mainstream length (km)

N.D. Normal distribution

NERC (UK) Natural Environment Research Council

NETLEN Total length of river network above gauge (km)

NUI National University of Ireland

O/L Outlet

OLS Ordinary least-squares (a regression method)

OPW Office of Public Works

PASTURE Proportional extent of catchment area classed as grassland/pasture/agriculture


xvii

PCA Principal components analysis

PCD Physical catchment descriptor (see Volume IV)

PDF Probability density function

PEAT Proportional extent of catchment area classified as peat bog

POT Peaks-over-threshold

PUM Pooled uncertainty measure (see Section L5 of Appendix L)

PWM Probability-weighted moment

RDS Royal Dublin Society

RH Right hand

RMSE Root mean square error

ROI Region of influence

S S-shaped (a curve pattern)

S1085 Slope of main stream excluding the bottom 10% and top 15% of its length (m/km)

SAAPE Standard-period average annual potential evapotranspiration (mm)

SAAR Standard-period average annual rainfall (mm)

SD Standard deviation

SE, se Standard error (of an estimate)

STMFRQ Number of segments in river network above gauge (differs from FSR definition)

TAYLSO Taylor-Schwartz measure of mainstream slope (m/km)

U Upwards concave (a curve pattern)

UAF Urban adjustment factor

UK United Kingdom

URBEXT Proportional extent of catchment area mapped as urbanised

Var Variance

VIF Variance inflation factor

WLS Weighted least-squares (a regression method)

WMO World Meteorological Organization

Glossary of terms

Term Meaning

Adjusted r2

Effective coefficient of determination r2, after allowing for degrees of freedom

consumed by estimating model parameters

Analogue catchment Catchment that is hydrologically similar to subject catchment in terms of key

PCDs but too distant to make it a natural choice as the pivotal catchment

Annual exceedance

probability AEP Probability of one or more exceedances in a year of a given extreme value

Annual maximum

flow series Time series comprising the largest flow in each year or water-year of record

At-site estimation

The FSU refers to the at-site QMED but not to at-site estimation of the flood

frequency curve; the term single-site analysis is preferred because it provides a

clear contrast with pooled analysis

At-site QMED Estimate of QMED made directly from flood peak data at the particular site

Average recurrence

interval ARI Average interval (often measured in years) between successive exceedances of

a given extreme value

Bias The amount by which a procedure typically overestimates the true quantity; if

the bias is small, the estimator is said to be unbiased, irrespective of scatter

Calibration Comparison of a model’s predictions with actual data, and adjustment of its

parameters to achieve a better fit with reality


xviii

Term Meaning

Coefficient of

determination r2

Proportion of variation accounted for by (e.g.) a regression model

Confidence interval Bounds within which a population parameter is estimated to lie with a stated

(usually %) confidence; used to indicate the reliability of an estimate

Cumulative

distribution function

CDF Probability F(x) of a value of a random variable X being less than or equal to x

Curvature parameter Parameter controlling the shape of a distribution; chiefly referred to in this

volume as the shape parameter

Donor catchment

Gauged catchment (usually one nearby) whose data are relevant to flood

estimation at the subject site; the pivotal catchment is the donor catchment that

the user judges to be most relevant to the specific flood estimation problem

Easting and Northing Coordinates of a location expressed as distance eastwards and distance

northwards from a fixed reference point

Essentially rural

(catchment)

Catchment for which the proportional extent mapped as urbanised (URBEXT)

is less than 0.015

Exploratory data

analysis EDA

An exploratory approach to analysing (often large) datasets to summarise their

main characteristics; EDA typically provides extensive visual summaries

Factorial error Ratio of estimated value to true value

Factorial standard

error FSE

The factorial standard error of X is the exponential of the standard error of

ℓnX; under certain assumptions, about 68% of estimates of X are expected to

lie within the factorial range 1/FSE to FSE of the true value

Flood-poor period Period in which floods are few and/or small

Flood rating curve Long-term relationship between river flow and water level; the stage-discharge

relationship; used to infer high flows from water-level measurements

Flood-rich period Period in which floods are many and/or large

Generalised least-

squares (GLS)

regression

Variation of ordinary least-squares regression or weighted least-squares

regression that can take account of inter-site dependence in observations of the

dependent variable (e.g. QMED); see Section C3 of Appendix C

Geographically

weighted regression

(GWR)

Technique that expands ordinary regression for use with spatial data; see

Section C4 of Appendix C

Geometric mean nth root of product of a sample of n values of a positive variable

Geostatistical

mapping

Method of mapping a variable (such as the residual error from a model) by

recognising its spatial structure; often done by deriving a semivariogram to

summarise how the variance between pairs of points changes with their

separation

Growth curve Model specifying the proportional increase of peak flow with return period

Growth factor Factor by which the index flood is multiplied to estimate the T-year flood

Hardness of a data

transfer

A “hard” data transfer assumes that the factorial error that a model is seen to

make at the pivotal site will be replicated at the subject site; a “soft” transfer

respects the model’s performance at the pivotal site in part rather than in full

Heterogeneity

In the context of a proposed pooled analysis, the degree of unacceptable

dissimilarity in single-site growth curves at stations in the pooling group,

taking due account of their record lengths


xix

Term Meaning

Heteroscedacity Uneven error variance

Hydrometry The science, technology and practice of water measurement

Independent and

identically

distributed (iid)

Observations are often assumed to be independent (in time and in space) and

identically distributed (i.e. sharing a common behaviour) for the purpose of

statistical inference; the assumption may not always be realistic but simplifies

the statistical analysis necessary to make practical applications

Index flood

A reference flood that can be relatively reliably estimated from gauged data;

the index flood adopted in the FSU is the median annual flood QMED; this is

the median of the annual maximum (AM) flow series.

Interpolation Any method of computing new data points from a set of existing data points

Interquartile range Measure of dispersion or scale defined as the difference between the 3

rd

quartile and the 1st quartile

Kriging An interpolation method based on a distance-weighted average of data at

neighbouring locations

L-CV An L-moment ratio; provides a dimensionless measure of the spread of the

distribution or the sample data

L-kurtosis An L-moment ratio; provides a dimensionless measure of the peakiness of the


L-moments

Moments computed from linear combinations of the ordered sample values

that lead to summary statistics of (e.g.) variation and skewness; often more

efficient than ordinary moments in parameter estimation of distributions;

L-moments are intimately related to probability-weighted moments

L-moment ratio A dimensionless ratio of two L-moments; useful because values from different

catchments can be compared

L-skewness An L-moment ratio; provides a dimensionless measure of the asymmetry of the


Least-squares

regression A method of fitting a model based on minimising the sum of squared residuals

Location parameter Parameter representing value subtracted from or added to a variable x to

translate the graph of its probability distribution along the x-axis

Median annual flood

QMED

QMED is the median of the annual maximum (AM) series. Half of AM floods

are larger than QMED and half are smaller; thus, the annual exceedance

probability associated with QMED is precisely 0.5; QMED is said to have a

return period of two years on the AM scale of frequency

Met Éireann Irish National Meteorological Service

Method of

L-moments

A method of calibrating the parameters of a distribution so that the lower-order

L-moments respect their sample values

Method of moments A method of calibrating the parameters of a distribution so that the lower-order

moments respect their sample values

Metadata Information about information; catalogue information

Multiple linear

regression Linear regression using two or more independent variables

Non-parametric test

A test that does not require an assumption about the probability distribution

underlying the variable being studied (also known as a distribution-free

method); typically, non-parametric tests use ordinal information e.g. the

position of the values in an ordered sample


xx

Term Meaning

Non-stationarity Non-stationary effects; see stationary

Ordered sample Data sample {x(1), x(2), …, x(n)} in which elements have been reordered so that

x(1) ≤ x(2) ≤ … ≤ x(n).

Ordinary least-

squares (OLS)

regression

The classical least-squares regression approach in which observations are

treated as being equally reliable and mutually independent (see Section C1 of

Appendix C)

Outlier An observation that lies an abnormal distance from other values in a supposed

random sample from a population

Parametric test A test requiring an assumption about the probability distribution underlying

the variable being studied

Peaks-over-threshold

(POT) series

Time series of independent events exceeding a given threshold; series

comprises magnitudes (e.g. peak exceedance in m3s

-1) and their dates of

occurrence; abstraction requires criteria for judging whether successive peak

exceedances are mutually independent

Pivotal catchment Gauged catchment judged by the user to be most relevant to the specific flood

estimation problem

Pooled analysis

Combined analysis of standardised AM flow data from a group of gauged

catchments deemed hydrologically similar (or otherwise relevant) to growth

curve estimation at the subject site

Pooling group Set of gauged catchments thought to be hydrologically similar to the subject

catchment

Probability density

function PDF

For a continuous random variable x, the PDF specifies the relative frequency

or probability of occurrence of x over all subsets of its range of values

Probability-weighted

moments PWMs

Certain weighted linear functions of the ordered sample data that statistical

theory shows as useful and efficient for parameter estimation of probability

distributions; L-moments are a development of PWM theory

Quantiles

Values taken at regular intervals from the cumulative distribution function of a

continuous random variable; where the CDF is broken into four parts, the

quantiles are known as quartiles; if the CDF is broken into 100 parts, the

quantiles are known as percentiles

Quartiles

For an ordered sample, the quartiles are the three points that divide the dataset

into four equal groups, each group comprising a quarter of the data; the second

quartile is the middle observation i.e. the median of the data; the lower quartile

is the middle value between the smallest observation and the median, while the

third quartile is the middle value between the median and the highest

observation; if the quartile (or some other desired quantile) does not

correspond to an observation, it is usual to interpolate between successive

sample values (if n is odd, the median corresponds to the middle-ranking value

x(i) where i = (n+1)/2; if n is even, the median is taken as the average of x(j) and

x(j+1) where j = n/2)

Region of influence

ROI

The region of influence approach selects gauged catchments that are thought

hydrologically similar to the subject catchment; stations are recruited to the

pooling group according to their nearness to the subject catchment in a

measurement system or metric that represents catchment dissimilarity

Residual Observed value minus the value estimated by a model

Return period T

Average number of years between years with floods exceeding a certain value.

T is the inverse of the annual exceedance probability; thus, a 50-year return

period corresponds to an AEP of 0.02.


xxi

Term Meaning

Scale parameter Parameter controlling the spread of a distribution

Shape parameter Parameter controlling the shape of a distribution

Single-site analysis Analysis of flood peak data at a particular gauged site; sometimes referred to

as at-site estimation

Skewness A measure of the departure from symmetry of a distribution

Soft data transfer See hardness of a data transfer

Standard-period

average annual

rainfall SAAR

Standard-period average annual rainfall, i.e. annual average rainfall evaluated

across a WMO standard period; in FSU usage, SAAR relates to 1961-90.

Standard deviation Measure of spread of values about their mean

Standard error Estimated standard deviation of a sample statistic such as the mean, i.e. the

standard deviation of the sampling distribution of the statistic

Standard error of

estimate Standard error of the variable being estimated

Stationary

A stochastic process is said to be stationary if its probability distribution is

independent of time; if a series of AM flows can be assumed stationary, the

values can be analysed together without the need to consider their sequence;

the assumption is upset by non-stationarity such as long-term trend, step-

change or long-term cyclical variation

Step-change An abrupt change e.g. in the frequency or magnitude of flood occurrences

Stepwise regression A method of multiple regression which begins by selecting the best one-

variable model and then adds further variables one by one

Subject catchment Catchment for which the flood estimate is required

Summary statistic

A statistical measure (e.g. the mean or standard deviation) used to summarise a

set of observations or results; the measure may be relatively intricate or relate

to a formal statistical test of significance; a summary statistic summarises an

aspect of the data or analysis in one number

Trend A data series shows trend if, on average, the series is progressively increasing

or decreasing

Turning point A data point defined by a triple of consecutive values xi-1, xi, xi+1 such that

xi-1 < xi > xi+1 or xi-1 > xi < xi+1; corresponds to a local maximum or minimum

Variance inflation

factor (VIF)

Reciprocal of the tolerance, which in turn denotes the proportion of the

variance in a given regressor (i.e. one of the PCDs) that cannot be explained by

the other regressors

Water-year Hydrological year beginning 1 October and ending 30 September

Weighted least-

squares (WLS)

regression

A variation of ordinary least-squares regression that can take account of the

variable record lengths supporting observations of the dependent variable (e.g.

QMED); see Section C2 of Appendix C


xxii

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1

1 Introduction and data

1.1 Why we estimate flood frequency

Flood frequency analysis is concerned with the assessment of flood magnitudes of stated

frequency (or degree of rarity) for use as input into the process of flood risk assessment and

management. Flood risk assessment is needed in the design of flood relief and protection

works and in the assessment of the safety of existing and planned infrastructure. This

includes domestic properties, commercial and industrial buildings, bridges, roads and

railways, and critical infrastructure such as hospitals, electrical stations, gas stations and

water works.

No development can be guaranteed immune from flooding during its projected life. Flood-

proofing every structure would be prohibitively costly. As a result, developments for which

the consequences and costs are modest may be required to tolerate occasional flooding.

Assessment of the residual risk requires estimation of the probability of occurrence of the

flood magnitude that would inundate or damage the structure or infrastructure. It is these

probabilities that are estimated with the help of flood frequency analysis.

The Flood Studies Update (FSU) arises in the context of the multifaceted approach to flood

risk assessment and management outlined by the Government’s Flood Policy Review Group:

an expert group appointed by the Minister of State. Their report (OPW, 2004) recommended

that a two-pronged approach to flood management be pursued with a greater level of

importance attributed to non-structural flood relief measures supported, where necessary, by

traditional structural flood relief measures. In order to make this possible, the report

recommended that research be undertaken in various sectors to develop a strategic

information base. The update of the Flood Studies Report for Ireland was one such body of

research identified.

A pivotal element identified was the need for improved hydrological information to support

effective decision-making, and a proposal was included for the Flood Studies Update (FSU)

programme. The programme has components related to Rainfall Frequency (Volume I),

Flood Frequency Estimation (this volume), Hydrograph Analysis (Volume III), River Basin

Modelling (Volume V), and Urbanised and Small Catchments (Volume VI). Additional

important features are the application of Geographic Information Systems to automate the

calculation of Physical Catchment Descriptors from digitally mapped data (Volume IV) and

the use of Information Technology to deliver many of the FSU outputs to users via web-

based applications (the FSU Web Portal).

Components of the FSU programme were prescribed in Work Packages. This volume

describes research carried out at NUI Galway and NUI Maynooth between 2006 and 2009.


2

1.2 Flood data

1.2.1 Introduction

The flood data analysed here are river flows measured at formal gauging stations. The

standard unit of measurement is a cubic metre per second (i.e. m3s

-1). The unit is often

spoken as a “cumec”.

River flow measurement is a central element of hydrometry: the science, technology and

practice of water measurement. The measurement of flood flows is relatively specialised and

typically revolves around establishing a long-term relationship between river flow and water

level known as the flood rating curve. The rating curve is also known as the stage-discharge

relationship. Their review is discussed in Appendix A.

1.2.2 Annual maximum series

The flood frequency analyses reported here are chiefly based on annual maximum data. The

annual maximum flow is the largest instantaneous flow recorded in the hydrological year

beginning 1 October. This is also known as the water-year.

The annual maximum flow is sometimes referred to as the annual maximum flood. This

alternate name is a little loose. In some years, the annual maximum flow is too small to be

considered a flood event. However, the emphasis in analysing annual maxima is on flood

estimation, so the term annual maximum flood is often used.

The time series of annual maximum flows for a particular station is referred to as the annual

maximum (AM) series.

1.2.3 FSU flood datasets

Annual maximum series were provided for some 206 river-flow gauging stations. The

datasets are briefly introduced and some features noted.

It is to be expected that individual researchers favour slightly different interpretations and

groupings. The datasets introduced here are those underpinning the general research on flood

frequency and flood growth rates undertaken at NUI Galway. The variants used in the NUI

Maynooth research of Chapter 2 are discussed in Section 2.1.1.

Attention focused on stations whose quality gradings (see Section A3 of Appendix A) fall

into Categories A1, A2 or B. There are 199 gauged catchments made up of:

45 Grade A1 stations;

70 Grade A2 stations;

67 Grade B stations;

A further 17 stations with both pre and post-drainage records (see Section 2.1.1).

The FSU employs a standard flood known as the index flood. The definition adopted here

(see Section 1.2.5) is the median, QMED, of the annual maximum flows. Estimation of

QMED is relatively tolerant of imperfection in flood rating curves (see Appendix A). Hence,


3

the Grade B stations are included in the research on QMED estimation from physical

catchment descriptors reported in Chapter 2. However, the analysis of flood growth rates and

other aspects warrants use of only the best datasets. Most recommendations are therefore

based on results obtained from A1 and A2 graded stations only.

For consistency and completeness, results are sometimes presented or referred to for the

Grade B stations. However, other than in the Chapter 2 work, no weight is given to them

when making general deductions about Irish flood behaviour.

For various reasons, some of the studies described are based on subsets of the overall dataset

of A1 and A2 graded stations. For example, some analyses needed:

Annual maximum flow series that were strictly complete, i.e. without any gaps;

Multiple gauges on a river (i.e. sets of gauges up and down the river system);

Stations with gauged estimates of the baseflow index (BFI).

In addition, some studies were completed before data for all stations became available. Such

features or exceptions are generally noted in the relevant part of the report.

1.2.4 Expressions of flood rarity

Strictly, any measure of event rarity should refer to a precise feature such as the peak flood

level or the peak flow in m3s

-1. Except where expressly stated otherwise, the feature dealt

with in this report is the peak flow.

The recommended measure of flood rarity for communication to the public is the annual

exceedance probability (AEP) of the flood rather than the return period in years. For

example, a large and damaging flood may have an AEP of 0.01, meaning that there is a one

in a hundred chance of its magnitude being exceeded in any one year.

Despite this clear statement of preferred terminology, the principal measure of rarity adopted

here is the return period (T) in years. The return period is simply more convenient in

technical reports. The two measures of rarity are fully interchangeable, with:

AEP = 1/T 1.1

The return period is defined as the average number of years elapsing between successive

exceedances of that flood magnitude. The word average needs to be stressed as a reminder

that any flood magnitude can be exceeded at any time, regardless of the recent flood history.

Some further remarks on return period are made in Section 1.5.

1.2.5 Index flood

An index flood is a reference flood that can be relatively reliably estimated from gauged data.

The index flood adopted in the FSU is the median annual flood, QMED. This is the median

of the annual maximum (AM) flow series.

Half of AM floods are larger than QMED and half are smaller. Thus, the annual exceedance

probability associated with QMED is precisely 0.5. QMED is said to have a return period of

two years on the AM scale of frequency.


4

1.3 Historical floods

Remarkable floods have occurred in Ireland that do not feature in the systematically

measured data series. Most often this is because gauging stations did not exist on the relevant

river. One example is the notable flood on the River Dargle at Bray on 25/26 August 1986

occasioned by Hurricane Charley (Charlie in some sources). Even on this one river, there are

other known historical floods, notably in August 1905, September 1931 and November 1965.

More widely across Ireland, there are many late 19th

century and early 20th

century floods for

which there is photographic or documentary evidence but no formal gauged flow data. A

notable example is for the town of Mallow, for which documentary evidence suggests that

flooding from the Munster Blackwater in November 1853, November 1916 and December

1948 exceeded the largest flows in the formal gauged record.

A more recent example of a notable event that passed largely unmeasured by the river

gauging network is the Newcastle West flood of 1 August 2008. The town is on the River

Arra, an ungauged tributary of the River Deel.

There are gauging stations on some catchments known to have experienced significant floods

– e.g. the Deel at Crossmolina – but data have not been supplied. This is generally because

the station does not meet the required standard for inclusion in Grade A1, A2 or B.

Further examination of notable floods is warranted, especially where they may be

symptomatic of damaging events on small steep catchments of a kind not well represented in

the main database of Grade A1 and A2 stations.

The National Flood Hazard Mapping website (http://www.floodmaps.ie/) provides

information on these and other historically important floods. It is essential viewing when

estimating flood frequency on a specific river. Some notable events are summarised by Met

Éireann (e.g. http://www.met.ie/climate-ireland/weather-events/Aug1986_HurCharlie.pdf).

British Rainfall yearbooks – now available online – sometimes provide useful information in

earlier eras (e.g. http://www.metoffice.gov.uk/media/pdf/7/a/British_Rainfall_1905.pdf).

1.4 Structure of volume

Chapter 2 reports the NUI Maynooth study of the relationship between the index flood

(QMED) and Physical Catchment Descriptors (PCDs). Section 2.6 discusses the important

topic of data transfer methods, for which a detailed worked example follows in Section 2.7.

Later chapters report research undertaken at NUI Galway, with Chapters 3 to 5 describing

relatively general statistical analyses of the AM flow series:

Trend and randomness (Chapter 3);

Descriptive statistics (Chapter 4);

Probability plots (Chapter 5).

http://www.floodmaps.ie/

http://www.met.ie/climate-ireland/weather-events/Aug1986_HurCharlie.pdf

http://www.metoffice.gov.uk/media/pdf/7/a/British_Rainfall_1905.pdf


5

Later chapters review the general problem of determining the T-year return period flood

magnitude, QT. Topics include:

Introduction to QT estimation by the index flood method (Chapter 6);

Determining the flood growth factor, xT, by single-site and pooled analysis of the AM

flow data (Chapter 7), including the construction of pooling groups (Section 7.3);

Studying the effect of catchment type and period of record on pooled estimates of QT

(Chapter 8);

Assessment of uncertainty in estimates of QMED, xT and QT (Chapter 9).

Guidelines for determining the design flood QT are presented in Chapter 10, together with

some challenging examples where application of a single prescribed strategy may generate

QT estimates that are counter-intuitive. This leads on to a discussion in Section 10.5 of

additional topics such as:

Problems in the use of 3-parameter distributions in single-site analysis;

The choice between single-site and pooled estimates of QT;

Flood growth curves with an upper bound.

Data transfers and the selection of the pivotal catchment are further discussed in Chapter 11.

One of many conclusions drawn in Chapter 12 is that blind application of a rule-based

procedure for flood frequency estimation is both inappropriate and impractical. Factors

beyond codification in any determination of QT include:

The history of flooding at the site;

The trade-off between single-site and pooled estimates of flood growth;

The plausibility (or not) of a derived flood frequency curve that has an upper bound.

Supporting material is presented in appendices.

1.5 Material in common with rainfall frequency studies

Some topics encountered in flood frequency estimation overlap with the rainfall frequency

research presented in Volume I. They are therefore only briefly mentioned here.

1.5.1 Annual maximum and peaks-over-threshold series

By definition, the annual maximum (AM) series comprises the highest values in each year.

The second highest value in a year is ignored, whether or not it exceeds the highest values in

other years. In contrast, the peaks-over-threshold (POT) series consists of all extreme values

exceeding a certain threshold. Although not considered in the FSU flood frequency research,

POT methods can sometimes assist in the estimation of QMED when the length of record is

shorter than ten years (but see also Section 2.1.2).


6

The probability of the T-year extreme event being exceeded at least once in L years is:

L)

T

1(11r 1.2

Hydrologists typically refer to this as the risk equation. It can be reasoned by rewriting:

L)T

1(1r1

and noting that the probability of the T-year extreme not being exceeded in L years (i.e.

the LH side of the equation) is simply the probability of the T-year extreme not being

exceeded in any of the L individual years (i.e. the RH side of the equation).

Risks are not always as one imagines. For example, there is an even chance (r = 0.5) that

the 100-year rainfall is exceeded in any 69-year period:

0.500)100

1(11 69

The risk equation can be applied to flood peaks as well as to extreme rainfall depths of a

given duration. It should be noted that such applications assume that the system

producing the extremes is stationary, i.e. that annul maxima are statistically independent

and drawn from the same underlying distribution. This assumption is compromised by

climate change. In the case of flood risk, it may also be compromised by catchment

change.

1.5.2 Return period

The return period T is best thought of as the inverse of the annual exceedance probability.

For example, the peak flow corresponding to T = 50 has a probability of 0.02 of being

exceeded in any year. It can be helpful to refer to this return period as the return period on

the annual maximum scale, to avoid possible confusion with other measures of frequency

(see Sections 1.5.3 and 1.5.4).

Though it is often misunderstood by the public, return period is a useful concept to the

professional. Its importance is perhaps best conveyed in understanding the often appreciable

risk of an extreme event being experienced in the long run. The risk equation (see

Equation 1.2 in Box 1.1) expresses the chance of the T-year event being exceeded in a long

run or lifetime of L years.

Box 1.1: The risk equation

1.5.3 Average recurrence interval

The analysis of peaks-over-threshold series gives the average interval between events that

exceed a particular value. This is often termed the average recurrence interval (ARI). For

high values of T, values of ARI and T are nearly equal. But for T less than 20 years the

difference is large enough to be important. Some analysts refer to ARI as the return period

on the peaks-over-threshold scale, and denote it by TPOT.


7

1.5.4 Langbein’s formula

Langbein (1949) provides a formula relating the two measures of frequency, i.e. T and ARI:

ARI

1exp1

T

1 1.3

This yields pairs of values such as:

T = 1.16 years when ARI = 0.5 years (twice per year frequency);

T = 1.58 years when ARI = 1 year (once per year frequency);

T = 2.54 years when ARI = 2 years (one in two years frequency).

For return periods of five years or longer, the approximation T = ARI + 0.5 suffices.

Langbein’s formula converts any ARI of interest into an equivalent return period or vice

versa.

For those favouring the notation TAM and TPOT, Langbein’s formula can be written:

POTAM T

1exp1

T

1 1.4


8

2 Estimation of the index flood, QMED

This chapter reports research undertaken by NUI Maynooth. The research is targeted at

relating the index flood to Physical Catchment Descriptors (PCDs). This is necessary to

allow QMED to be estimated at ungauged sites.

Even where flood data are available elsewhere on the same river, the model relating QMED

to PCDs is an intrinsic part of the procedure for transferring information from the gauged site

to the subject site. This topic is discussed in Sections 2.6 and 2.7, and revisited in

Chapter 11.

2.1 Exploratory data analysis

2.1.1 Introduction to the datasets

Estimation of the index flood for ungauged catchments is based on the construction of an

empirical model. This links the index flood (known only at gauged sites) to PCDs (known at

all sites). The PCDs are described in Volume IV.

The annual maximum flow series and values for QMED were provided for a total of 206

gauging stations. Not all stations provided were used. Some were discarded following an

exploratory data analysis. Other questionable stations came to light during model-building.

Appendix B identifies 16 stations omitted from QMED modelling, and justifies the decisions

taken. This left 190 stations for model-building.

Arterial drainage

Where arterial drainage has taken place within the record, the series in question was treated in

two parts and separate values of QMED extracted for the pre and post-drainage periods. In

total, 15 stations were divided into pre and post-drainage records.

The set of 190 catchments was thereby increased to 205 for model-building. These 205

comprise 190 catchments nominally in their current condition plus 15 in their pre-drainage

condition. The make-up of the 205 catchments can be further summarised as: 131 (i.e. 116 +

15) stations without arterial drainage, and 74 stations with arterial drainage (i.e. 59 + 15).

Flood data quality

Figure 2.1 shows the number of stations in each category of flood data quality. There are 58

stations graded A1, 78 stations graded A2 and 69 stations graded B.

Length of record

In total, the full dataset of 205 stations comprises 6350 annual maximum events. The stations

have a mean length of 31 years, with a maximum of 65 and a minimum of seven years. The

distribution of record-lengths is summarised in Histogram 2.1.


9

Grade A1

Grade A2

Grade B

69

78

58

Figure 2.1: Number of stations in each quality category (full dataset of 205 stations)

706050403020100

15

10

5

0

Number of annual maxima from which QMED estimated

Fre

quency

EPA stations

OPW stations

Histogram 2.1: Length of available record for QMED estimation (205 stations)

Most of the longer records are from OPW stations. The bimodality evident in the distribution

of record lengths likely reflects changing priorities and budgets. It may also reflect that

arterial drainage has divided some otherwise long flood series into shorter records. The FSU

treats pre-drainage and post-drainage flood series as separate stations.

Histogram 2.2 shows the spread of QMED values, which are seen to be distributed

approximately lognormally. The 205 stations are marked in Map 2.1. Because of the

collocation of pre-drainage and post-drainage stations, and the limited resolution of plotting,

fewer than 205 symbols are discernible on the map.


10

6.21

46

5.29

83

4.60

52

3.91

20

2.99

57

2.30

26

1.60

94

0.69

31

0.00

00

20

10

0

QMED m s

Fre

quency

1 2 5 10 20 50 100 200 5003 -1

[The 205 QMED values are seen to be approximately lognormally distributed]

[Logarithmic scale]

Histogram 2.2: QMED values across full dataset of 205 stations

2.1.2 Adjustments for period-of-record effects

Natural variability in climate can lead to flood series exhibiting so-called flood-rich and/or

flood-poor periods. Several major floods may occur in a flood-rich period only to be

succeeded by a flood-poor period in which no major flood occurs for many years. In

consequence, QMED estimates from short records can be unrepresentative of the longer term.

Estimates of QMED were therefore adjusted for period-of-record effects using a method

broadly based on Robson (1999c).

Stations having a flood series shorter than 20 years were taken to be short records in need of

adjustment. The adjustment is accomplished by transferring information from long-record

sites. To be considered for this role, the donor station had to have a Grade A1 or A2 flood

rating, have a record length of over 30 years and be free from arterial drainage.

Long-record catchments within a 50 km radius of the short-record (subject) site were initially

selected. Only donors that overlapped with at least three-quarters of the subject site’s annual

maximum data were considered.

The correlation between subject and donor sites was assessed using Spearman’s rank

correlation, ρ (e.g. Kendall, 1975). An observation has rank r if it is the rth

largest in a

dataset. Donors that fully overlap the period of record at the subject site were prioritised but

those revealing a weak or negative rank correlation were excluded from the adjustment

process. The number of donors used ranged from one (where the donor fully overlapped the

record at the subject site and ρ > 0.8) to five (where no outstanding donor was identified).


11

Map 2.1: Spatial distribution of 205 catchments

In order to transfer information from the donor site to the subject site, QMED at the donor

site was estimated using all available data and then recalculated using only the period of

record available at the subject site. The ratio of these two values provides the adjustment

factor at the subject site. The QMED estimate adjusted to the donor period was taken as:

ρM

od,

dsadjs,

QMED

QMEDQMEDQMED

2.1

Here, QMEDs,adj is the adjusted QMED at the subject site, QMEDs and QMEDd are estimates

at the subject site and the donor site based on their respective periods of record, and QMEDd,o

is QMED at the donor site evaluated for the period of overlap.

[Symbols placed at

catchment centroids]


12

The exponent M(ρ) is a moderating influence based on the strength of correlation between

donor and subject site. [Editorial note: The exponent is later termed the “hardness” of the

data transfer, and denoted by h. See Step 5 in Section 2.7.1.] M(ρ) is defined by:

1ρ4n

ρ3nρM

2

o

3

o

2.2

where no is the length of overlap between subject and donor sites and ρ is Spearman’s rank

correlation between annual maxima at subject and donor sites.

Where only one donor with a very strong correlation (ρ > 0.8) was found, the adjustment

process was deemed complete. However, where several donor sites were identified, a

combined adjustment was made by weighting each donor based on distance from subject site,

additional years of data provided by the donor, and the strength of correlation with the subject

site. The weighting factor is formulated as:

ρnnn100

d1w odo

2.3

where nd is the length of the donor series and d is distance in km.

Table 2.1 details the 28 stations at which the QMED value was adjusted for a period-of-

record effect. As confirmed in Figure 2.2, the procedure led to relatively minor adjustments

in QMED. The most notable exception was the Cammock at Killeen Road (Station 09035),

for which the adjustment raised QMED by 31% (from 11.70 to 15.28 m3s

-1). The Cammock

is a short-record station known to be affected by urbanisation. Five of the nine annual

maxima shown in Figure 2.3 were recorded in summer (Apr-Sep) rather than winter

(Oct-Mar), which is consistent with an urban flood regime.

2.1.3 Physical catchment descriptors

Some 20 physical catchment descriptors (PCDs) were considered. Their derivations are fully

documented in Volume IV. [Editorial note: The flood attenuation index FAI was

unavailable at the time of study. Because QMED represents a relatively frequent (2-year)

flood, the omission of a PCD reflecting floodplain attenuation is not thought to be too

important.] In initial studies, the FLATWET descriptor was found to enter the QMED model

in a counter-intuitive manner. Given the reservations about this PCD expressed by the FSU

Steering Group, FLATWET was omitted from later modelling of QMED.

The remaining 19 PCDs were logarithmically transformed prior to the main model-building.

Where the lower range of a particular descriptor can take a value of zero, 1.0 is added to the

value prior to the log transformation. This applied to the fractions URBEXT, FOREST,

PEAT, PASTURE, ALLUV, ARTDRAIN and ARTDRAIN2. In some tables and figures that

follow, ARTDRAIN and ARTDRAIN2 are abbreviated to AD and AD2 respectively.

PCDs were screened both by the non-parametric approach of calculating rank correlations

(see Section 2.1.4) and by plotting the ℓn-transformed PCDs against ℓnQMED (see

Section 2.1.6). Scatter-plots help to check for possible outliers and non-linear relationships,

as well as for cross-correlations between the PCDs.


13

Table 2.1: QMED adjustments for period-of-record effects

Station

number

Rating

grade River Station name

# annual

maxima QMED QMEDadj

Adjustt

factor

01055 B Mourne Beg M. Beg Weir 9 2.70 2.80 1.036

07006 A2 Moynalty Fyanstown 19 27.93 25.20 0.902

07041 A2 Boyne Ballinter Br. 7 165.00 165.28 1.002

08007 B Broadmeadow Ashbourne 17 8.24 8.12 0.986

08009 A1 Ward Balheary 14 5.00 5.09 1.018

08012 B Stream Ballyboghill 19 4.35 4.35 1.000

09010 A1 Dodder Waldron’s Br. 18 47.05 46.64 0.991

09035 B Cammock Killeen Road 9 11.70 15.28 1.306

10028 B Aughrim Knocknamohill 16 46.95 46.29 0.986

13002 B Corock Foulk’s Mills 19 7.01 6.98 0.996

14034 A2 Barrow Bestfield 17 117.00 117.07 1.001

15007 A2 Nore Kilbricken 13 53.45 53.58 1.002

15012 B Nore Ballyragget 16 77.11 76.18 0.988

16051 B Suir Clobanna 13 2.85 2.82 0.989

19046 B Martin Station Road 9 29.95 28.33 0.946

22003 B Maine Riverville 8 98.01 98.03 1.000

22035 B Laune Laune Bridge 14 116.40 110.42 0.949

23012 A2 Lee (Kerry) Ballymullen 18 15.66 15.83 1.011

25038 B Tyone Nenagh 17 39.30 37.68 0.959

25124 A2 Brosna Ballynagore 18 13.65 13.36 0.979

25158 A1 Bilboa Cappamore 18 43.88 37.06 0.845

26014 B Lung Banada Br. 16 42.82 42.18 0.985

26108 A2 Owenure Bellavahan Br. 15 57.32 55.92 0.976

30012 B Clare Claregalway 9 126.00 116.97 0.928

34010 B Moy Cloonacannana 12 95.42 99.21 1.040

34029 B Deel Knockadangan 9 110.00 110.00 1.000

36016 B Annalee Rathkenny 14 50.70 50.70 1.000

39001 B Swilly New Mills 17 47.80 47.05 0.984


14

2.3012.0001.6991.3011.0000.6990.301

2.301

2.000

1.699

1.301

1.000

0.699

0.301

QMED ( m s )

QM

ED

adj

2 5 10 20 50 100 200

1:1 line

2

20

200

50

100

5

10

3 -1

Station 09035

Figure 2.2: QMED adjusted for period-of-record effects

20042002200019981996

30

20

10

0

Water-year

Flo

od p

eak

Winter

Summer

Figure 2.3: Annual maximum flow series for Station 09035 Cammock at Killeen Road

2.1.4 Rank correlations

Table 2.2 shows rank correlations amongst QMED and the PCDs for the 205 catchments.

[Editorial note: The rank correlation is the correlation between the ranks of the variables

within the dataset. It is unaffected by whether or not the ℓn-transformation has been applied,

making it a convenient and relatively robust measure of the degree of association between

values of the PCDs.] The stronger correlations (and anti-correlations) are highlighted, with

values greater than 0.9 (or less than -0.9) shown in red. Values shown in orange and green

mark pairs of PCDs that are progressively less strongly correlated (or anti-correlated).

Flo

od p

eak (

m3 s

-1)

QM

ED

adj (

m3 s

-1)


15

Table 2.2: Rank correlations amongst QMED and 19 PCDs (205 gauged sites)

Competition is evident amongst the PCDs that reflect catchment size, namely: AREA, MSL,

NETLEN and STMFRQ. [Editorial note: The FSU definition of STMFRQ (see Volume

IV) is the number of streams in the catchment. This is one greater than the number of stream

junctions. STMFRQ is therefore heavily correlated with size variables. In contrast, the FSR

(NERC, 1975) defined STMFRQ as a standardised stream frequency i.e. the number of

junctions per unit area.]

Unsurprisingly, there is strong competition between the PCDs representing arterial drainage

(ARTDRAIN and ARTDRAIN2) and between the slope descriptors (S1085 and TAYLSO).

The marked associations between the land-use fractions FOREST, PEAT and PASTURE

reflect that afforestation tends to be more prevalent on catchments with peat-based soils,

whereas pasture is more typical of catchments with non-peat soils. These patterns are

consistent with the positive association of SAAR with FOREST and PEAT, and its negative

association with PASTURE. SAAR is the long-term average annual rainfall.

2.1.5 Principal component analysis

The transformed PCDs were subject to a principal component analysis to explore the

dominant factors in explaining the variation of ℓnQMED. Table 2.3 displays the results

following Varimax rotation. This special type of orthogonal rotation maximises the sum of

the variance of the loading vectors. Its goal is to simplify the structure of the components by

making the large loadings larger and the small loadings smaller.

[Editorial note: Because of the artificial and important influence of urbanisation on flood

response times and flood magnitudes, the exploratory work studied QMED on the

190 catchments that are largely rural. This explains why URBEXT does not appear in the

principal component analysis (PCA). The mean altitude (ALTBAR) was considered in the

PCA but did not prove influential.]


16

Table 2.3: Principal component loadings of PCDs on ℓnQMED for 190 rural stations

Component 1 2 3 4 5 6 7 8 9

ℓnAREA 0.961

ℓnMSL 0.965

ℓnNETLEN 0.986

ℓnSTMFRQ 0.957

ℓnDRAIND 0.937

ℓnS1085 -0.620 0.584

ℓnSAAR 0.700 0.307

ℓn(1+FOREST) 0.819

ℓn(1+PEAT) 0.841

ℓn(1+PASTURE) -0.877

ℓn(1+ALLUV) 0.851

ℓnSAAPE 0.902

ℓnFARL 0.870

ℓnBFIsoil 0.836

ℓnTAYSLO -0.603 0.629

ℓn(1+ARTDRAIN) 0.968

ℓn(1+ARTDRAIN2) 0.961

Unsurprisingly, the first component is dominated by terms (directly or indirectly) reflecting

catchment size. Large catchments tend to have small S1085 and TAYSLO slopes. The

second component reflects catchment wetness and land-use, with subsequent components

reflecting arterial drainage, storage attenuation (indexed by ℓnFARL) and drainage density.

2.1.6 Correlations and competing variables

Given the strong correlations between ℓnQMED and the PCDs reflecting catchment size –

ℓnAREA, ℓnMSL, ℓnNETLEN and ℓnSTMFRQ – only ℓnAREA was retained in the model-

building. AREA is the natural descriptor of catchment size.

Strong correlations were also noted between ℓnSAAR, ℓn(1+FOREST) and altitude (indexed

by ALTBAR). These reflect the influence of topography on rainfall and land-use. Forested

areas are chiefly located in upland areas of high rainfall. The traditional descriptor

(ℓnSAAR) was selected.

The slope descriptor ℓnS1085 showed a marginally stronger relationship with ℓnQMED than

did the alternate ℓnTAYLSO. More exhaustive studies in later modelling of ℓnQMED led to

ℓnS1085 being consistently selected in preference to ℓnTAYLSO. The latter was eventually

dropped from further analysis.


17

The decision to retain only one of the competing variables at an early stage of the analysis

avoids problems of collinearity (see Box 2.1). Although the descriptors of arterial drainage

are highly correlated (see Figure 2.4), it was unclear which variable to prefer. Both were

retained but ARTDRAIN2 ultimately proved the more useful.

Box 2.1: Collinearity

1.00.90.80.70.60.50.40.30.20.10.0

0.5

0.4

0.3

0.2

0.1

0.0

ARTDRAIN2

AR

TD

RA

IN

[Regression: ARTDRAIN = 0.276 ARTDRAIN2 (intercept not significant)]

Figure 2.4: Relationship between PCDs summarising extent of arterial drainage

2.1.7 Scatter-plots and summary information for selected PCDs

Figure 2.5 shows a matrix of scatter-plots and correlations for ℓnQMED and a selected subset

of the ℓn-transformed PCDs. Table 2.4 provides some summary information.

Collinearity refers in a strict sense to the presence of exact linear relationships within a set

of variables. Typically, these are a set of candidate explanatory (i.e. predictor) variables

in a regression-type model. In statistical usage, collinearity also refers to near-

collinearity, i.e. when variables are close to being linearly related.

In a multiple regression with collinearity, least-squares regression coefficients are highly

sensitive to very minor changes in the input data. The least-squares problem or the dataset

is said to be ill-conditioned. Some or all of the regression coefficients are likely to be

meaningless.

A typical approach to overcoming collinearity is to simplify the problem, e.g. by retaining

only one of the subset of variables that are highly correlated. Relatively arbitrary

decisions – as to which variables to retain and which to remove – are sometimes

unavoidable and inevitably influence the final model achieved.


18

Figure 2.5: Association of ℓnQMED and selected PCDs for 205 catchments

2.2 Rural-catchment model for estimating QMED from PCDs

2.2.1 Regression methods

Regression has long been used to relate a desired flood quantile to catchment physiographic,

geomorphologic and climatic characteristics (e.g. Nash and Shaw, 1965; NERC, 1975). The

outcome is an estimation equation. This is typically referred to as the catchment-

characteristic or catchment-descriptor equation.

The model typically takes the form of a power-law equation:

p321 β

p

β

3

β

2

β

1T .....xxxxaQ 2.4

where:

QT is the flood quantile of interest;

xi is the ith

physical catchment descriptor;

βi is the related model parameter;

a is a multiplier (which is a further parameter of the model);

p is the number of catchment descriptors.

Lower triangle shows scatter-plot. Upper triangle shows (Pearsonian) correlation coefficient. ℓnQMED 0.83 0.09 -0.39 0.25 -0.03 -0.10 -0.01 ℓnAREA -0.27 -0.71 -0.06 -0.13 0.29 0.05 ℓnDRAIND 0.45 0.49 -0.16 -0.49 -0.13 ℓnS1085 0.31 0.14 -0.44 -0.21 ℓnSAAR -0.40 -0.37 -0.24 ℓnFARL -0.24 0.03 ℓnBFI

soil 0.16

ℓn(1+AD2)


19

Table 2.4: Summary information for PCDs at 190 rural stations used in model-building

PCD

notation Unit Meaning Min

Geom

mean Max

Main form in which

variable is used

AREA km2

Catchment area from

DTM 5.46 218 7980 ℓnAREA

DRAIND km/km2 Drainage density 0.27 1.03 2.64 ℓnDRAIND

SAAR mm

Standard-period

average annual rainfall

(1961-90)

711 1101 2465 ℓnSAAR

URBEXT

fractions

Urban extent 0.00 0.019 0.683 ℓn(1+URBEXT)

PEAT Peat extent 0.00 0.126 0.802 ℓn(1+PEAT)

ALLUV Alluvium extent 0.00 0.034 0.108 ℓn(1+ALLUV)

PASTURE Pasture extent 0.00 0.697 1.000 ℓn(1+PASTURE)

SAAPE mm

Standard-period

average annual potential

evaporation (1961-90)

448 501 563 ℓnSAAPE

FARL –

Index of flood

attenuation from

reservoirs and lakes

0.632 0.948 1.000 ℓnFARL

BFIsoil –

Soil baseflow index

(estimate of BFI

derived from soils,

geology and climate

data)

0.294 0.574 0.814 ℓnBFIsoil

ARTDRAIN

(AD) –

Proportion of catchment

area mapped as

benefitting from arterial

drainage

0.00 0.047 0.367 ℓn(1+ARTDRAIN)

ARTDRAIN2

(AD2) –

Proportion of river

network length mapped

as included in Arterial

Drainage Schemes

0.00 0.150 0.846 ℓn(1+ARTDRAIN2)

S1085 m/km

Mainstream slope

(excluding top 10% and

bottom 15%)

0.24 2.61 30.8 ℓnS1085

The index flood adopted in the Flood Studies Update is the median annual flood, QMED.

This is the median of the annual maximum flood values. Equation 2.4 holds that changes in

physical catchment descriptors (PCDs) have a scaling effect on the index flood, with the

degree of scaling indicated by the exponent terms.

Several techniques are available for estimating the p+1 model parameters. The most common

is multiple linear regression. This is invoked by linearising Equation 2.4 through a

logarithmic transformation, leading to the form:

pp332211 xnβ...xnβxnβxnβanQMEDn 2.5


20

The basic principle of regression is to achieve a best fit to the data by minimising a measure

of the deviation of the estimated values (of ℓnQMED) from the observed values (of

ℓnQMED). The criterion typically applied is to minimise the sum of squares of the

deviations. This gives rise to the term least-squares regression.

Models of the form of Equation 2.5 can be fitted using regression techniques supported by

standard statistical packages. Some methods begin by selecting the best one-variable model

(i.e. p = 1) and then adding further variables one by one. This is stepwise regression.

However, for many datasets, computer power is typically sufficient to allow exhaustive

searching (across all possible linear models) for the best 1-variable model, the best 2-variable

model, etc. This is the technique generally employed here.

There are several versions of multiple linear regression. The formulation chiefly used is

ordinary least-squares (OLS). However, consideration is also given to weighted least-squares

(WLS) and generalised least-squares (GLS). Some details of linear least-squares regression

are given in Section C1 of Appendix C.

2.2.2 Alternative methods

McCuen et al. (1990) highlight that techniques such as OLS, while leading to an unbiased

estimate of ℓnQMED, can lead to a biased estimate of QMED. In addressing this problem, a

number of authors have applied more complicated procedures such as non-linear and non-

parametric regression (e.g. Pandey and Nguyen, 1999). The use of such techniques was not

attempted here. [Editorial note: In most flood-risk applications, proportional changes

and/or factorial errors in peak flow are of greater relevance than absolute changes/errors.

This makes estimates of ℓnQMED typically more relevant than estimates of QMED.]

As a precaution, and given the impressive results achieved by Dawson et al. (2006), the use

of Artificial Neural Networks (ANNs) was also examined. However, the selected approach

was found to outperform ANNs for the FSU dataset. The alternative method is not reported.

2.2.3 Selection of catchments for calibration and validation

The approach taken began by deriving a method of estimating QMED for use on essentially

rural catchments. Urban adjustments are considered in Section 2.5.

Thirty-five of the 205 catchments have an urban fraction (i.e. URBEXT value) greater than

0.015. Excluding these catchments left 170 stations for modelling QMED on essentially rural

catchments. These 170 catchments include 15 pairs of catchments for which pre-drainage

and post-drainage behaviour were both considered (see also Section 2.4).

Split-sampling allowed some testing of the derived models. The 170-catchment rural dataset

was split 85%/15% to provide 145 stations for calibration and 25 stations for validation. The

allocation of stations to the validation group was made at random. As seen in Histograms

2.3, this resulted in a fair distribution of catchment sizes across the two datasets.


21

15

10

5

0

5000200010005002001005020105

10

5

0

Num

ber

of

cat

chm

ents

in A

RE

A b

and

Calibration dataset (145 rural catchments)

Validation dataset (25 rural catchments)

Histograms 2.3: Catchment sizes for calibration and validation datasets

2.2.4 Selecting PCDs

Selecting the combination of PCDs to be included in the final QMED model was a lengthy

and iterative process. Not every stage is reported here.

An exhaustive search found the best five sets of variables by fitting every combination of

descriptors (up to a maximum of nine PCDs) to the 145-catchment calibration dataset.

The fitted models were assessed on size (i.e. number of PCDs included), the coefficient of

determination (r2), the root mean square error (RMSE) of prediction, their hydrological

realism and the behaviour of the model residuals.

The RMSE is also known as the standard error of estimate. Here, the RMSE applies to

ℓnQMED. The factorial standard error (FSE) of the index flood itself, i.e. of QMED, is

given by:

RMSEeFSE 2.6

Table 2.5 indicates the PCDs that feature in the three best-fitting OLS models of each model

size, i.e. using from one to nine PCDs as regressor variables. For example, the best-fitting

2-variable model uses ℓnAREA and ℓnBFIsoil and has a factorial standard error of

e0.465

= 1.59. The names shown in the table denote the relevant transformed variable, e.g.

AREA denotes ℓnAREA and ALLUV denotes ℓn(1+ALLUV). The descriptor PEAT was

included in the analysis but did not feature in any of the best models reported in Table 2.5.

AREA (km2)


22

Table 2.5: PCDs in best three 1 to 9-variable models for ℓnQMED (with ℓnAREA forced)

2.2.5 Choosing a model

Table 2.6 reports performance statistics for the best 1-variable, 2-variable, 3-variable, …,

9-variable models for estimating ℓnQMED on rural catchments. This was based on an

exhaustive search.

The r2 value increases inexorably as the model increases in size. But the r

2 value levels off at

the 7-variable model (see Figure 2.6). The adjusted r2 allows for the degrees of freedom

consumed by estimating model parameters. An F test (based on the ratio of variances)

confirms that the 8-variable model does not provide a significant improvement on the

7-variable model.

Table 2.6: Performance diagnostics for the addition of each independent variable

Number of PCDs r r2 r

2 change Adjusted r

2 RMSE F ratio

Significance

of F ratio

1 0.786 0.618 0.618 0.615 0.629 229.57 0.000

2 0.890 0.793 0.175 0.790 0.465 119.09 0.000

3 0.918 0.843 0.050 0.839 0.407 44.25 0.000

4 0.936 0.876 0.033 0.872 0.362 37.54 0.000

5 0.945 0.893 0.017 0.889 0.337 22.12 0.000

6 0.951 0.903 0.010 0.899 0.322 14.60 0.000

7 0.954 0.909 0.006 0.905 0.313 8.72 0.004

8 0.954 0.911 0.001 0.905 0.312 2.05 0.154

9 0.955 0.911 0.001 0.905 0.312 1.19 0.278


23

987654321

1.0

0.8

0.6

0.4

0.2

0.0

Number of PCDs in model

0.9

Figure 2.6: Improvement of r

2 for a model size of one to nine variables

Based on the 145 catchments in the calibration set, the following 7-variable model was

selected for use on essentially rural catchments:

ℓnQMED = –11.300 + 0.937ℓnAREA – 0.922ℓnBFIsoil + 1.306ℓnSAAR +

2.217ℓnFARL + 0.341ℓnDRAIND + 0.185ℓnS1085 + 0.408ℓn(1+ARTDRAIN2) 2.7

The model has an r2 of 0.909 and a root mean square error (RMSE) of 0.313. Exponentiat-

ing, the model can be written:

0.4080.1850.341

2.2171.3060.922

soil

0.9375

ARTDRAIN21S1085DRAIND

FARLSAARBFIAREA101.237QMED

2.8

The factorial standard error (FSE) in estimating QMED is therefore e0.313

= 1.37. This tells us

that, under standard assumptions, 68% of QMED estimates can be expected to lie within the

range 1/1.37 to 1.37 (i.e. 73% to 137%) of the true value. [Editorial note: This is a

strikingly good performance. While there is no directly comparable equation in the FSR

(NERC, 1975), the FEH QMEDrural model (for UK data) has a considerably wider error-band,

with FSE = 1.55 (Robson, 1999b). A revised FEH QMED model (Kjeldsen et al., 2008)

improves this to FSE = 1.43.]

2.2.6 Model performance

Figure 2.7 plots the modelled and observed QMED values and shows a good fit with little

evidence of heteroscedacity (uneven error variance). The labelled stations are discussed later.

Table 2.7 confirms that all coefficients are significant at the 0.05 level (│t statistic│> 1.96).

The standardised coefficients (β in the table) highlight the relative contribution of each

descriptor to explaining the variation in ℓnQMED. As expected, ℓnAREA is by far the most

important predictor.

Co

effi

cien

t o

f d

eter

min

atio

n,

r2


24

50020010050201052

500

200

100

50

20

10

5

2

Observed QMED (m s )

Pre

dic

ted Q

ME

D1:1 line

3 -1

Station 06030

Station 10004

Station 19046

Figure 2.7: Observed and modelled QMED for the 145 calibration stations

Table 2.7: Coefficient and collinearity statistics for selected rural model for ℓnQMED

Term/regressor Coefficient Standard

error

β

value

t

statistic

95% confidence

interval Variance

inflation

factor (VIF) Lower Upper

Constant -11.300 1.15 -9.82 -13.58 -9.02

ℓnAREA 0.937 0.03 1.02 29.46 0.87 1.00 1.80

ℓnSAAR 1.306 0.17 0.29 7.54 0.96 1.65 2.22

ℓnFARL 2.217 0.33 0.22 6.67 1.56 2.87 1.62

ℓnBFIsoil -0.922 0.17 -0.18 -5.46 -1.26 -0.59 1.70

ℓnS1085 0.185 0.04 0.18 4.41 0.10 0.27 2.45

ℓnDRAIND 0.341 0.07 0.14 4.85 0.20 0.48 1.34

ℓn(1+AD2) 0.408 0.14 0.08 2.95 0.13 0.68 1.11

In order to assess the possible impact of collinearity (see Box 2.1), an additional statistic was

studied: the variance inflation factor (VIF). This is the reciprocal of the tolerance, which in

turn denotes the proportion of the variance in a given catchment descriptor that cannot be

explained by the other regressors.

High VIF values (i.e. small tolerances) indicate that a large amount of the variance in one

regressor can be explained by the other regressors. VIF thus indexes the impact of

collinearity (amongst the regressors) on the stability of the multiple regression model. VIF

values are (by definition) greater than or equal to 1. Whilst only a guide, VIF values greater

than 10 are often regarded as indicating serious problems of collinearity. In weaker models,

values above 2.5 may sometimes be a cause for concern.

It is seen from the final column of Table 2.7 that VIF is less than two for all regressors

excepting ℓnSAAR and ℓnS1085. The modest correlation (rank correlation ρ = 0.30)


25

between these two PCDs leads them to compete in explaining the variation in ℓnQMED.

This is because higher values of SAAR and S1085 both tend to be associated with upland

areas, and lower values with lowland areas. That their VIF values are reasonably healthy (i.e.

less than 2.5) supports the retention of both descriptors in the model, and suggests that

ℓnS1085 is adding something useful.

2.2.7 Checking for logical consistency

Logical consistency – in the sense of being in accord with common reasoning – is often the

overriding factor in the final choice of a regression model. The 7-variable model for

ℓnQMED on rural catchments performs well in this respect:

QMED increases with AREA;

QMED increases with greater wetness (indexed by SAAR);

QMED increases with FARL, meaning that it decreases for increased attenuation;

QMED decreases with greater permeability (indexed by BFIsoil);

QMED increases with catchment steepness (indexed by S1085);

QMED increases with drainage density (DRAIND);

QMED increases with the extent of arterial drainage (indexed by ARTDRAIN2).

All these features are logically consistent with what is known about flood behaviour.

2.2.8 Checking and investigating the model residuals

The ordinary least-squares (OLS) approach to judging and testing the model requires that the

residuals are Normally-distributed with constant variance. The plots of Figure 2.8 suggest

that the residuals to the ℓnQMED model are well behaved. Visually, the fit to the assumed

Normal distribution is good, even in the tails of the distribution. There is little evidence of

changes in variance with increasing ℓnQMED.

Figure 2.8: Diagnostic plots of ℓnQMED model performance

0.0 0.2 0.4 0.6 0.8 1.0

Observed cumulative probability

Ex

pec

ted c

um

ula

tive

pro

bab

ilit

y

0.0

0.2

0.4

0

.6

0.8

1

.0

0 1 2 3 4 5 6

Fitted ℓnQMED

Res

idual

(on ℓ

n s

cale

) -0

.5 0

.0

0.5

1

.0

(a) Normal quantile-quantile (b) Residual versus fitted

1:1 line


26

Map 2.2 indicates the residuals from the 7-variable rural QMED model. The small grey

symbols mark the catchments on which the model performs particularly well. It is seen from

the graduated symbols that there is some semblance of a tendency for the model to over-

estimate ℓnQMED in the Midlands and West, and to underestimate it in the East and South.

Map 2.2: Factorial error in QMED estimated by selected rural model

In analysing the residuals further, Figure 2.9 shows scatter-plots of the selected rural model

against each of the seven selected catchment descriptors. Interest centres on:

Examining the relationship between residuals and PCDs to assess the success of the

model in capturing the range of catchment types represented;

Identifying the possible presence of a curved pattern in the residuals when plotted

against any descriptor (curvature would indicate non-linear relationships and suggest

the need to include additional transformations of the PCDs in the model-building).

● Model underestimates QMED ● Model overestimates QMED

Factorial error

in QMED


27

6.90784.60522.3026

1.0

0.5

0.0

-0.5

-1.0

-0.1054-0.5108-1.2040 7.60097.31326.9078

0.0000-0.2231-0.5108

1.0

0.5

0.0

-0.5

-1.0

0.69310.0000-0.6931-1.3863 3.21891.60940.0000-1.6094 0.69310.40550.0000

Figure 2.9: Residuals versus individual PCDs for rural ℓnQMED model

In terms of curvature, the residual plots are found to be well behaved, with little evidence of

non-linear relationships between the residuals and the PCDs. In examining model

performance for the range of catchment types, particular interest was directed at how well the

model performs for permeable catchments (with high BFIsoil) and how well the full range of

catchment sizes is represented.

Res

idual

(on ℓ

n s

cale

)

AREA (km2) BFIsoil SAAR (mm)

10 100 1000 0.3 0.6 0.9 1000 1500 2000

FARL DRAIND (km/km2) S1085 (m/km) ARTDRAIN2

0.6 0.8 1 .25 .5 1 2 0.2 1 5 25 0 0.5 1

Res

idual

(on ℓ

n s

cale

)


28

Catchments smaller than about 50 km2 are not well represented in either the calibration or

validation dataset (see Histograms 2.3 in Section 2.2.3). From the spread of residuals in the

AREA plot in Figure 2.9, the selected rural model performs somewhat better for larger

catchments. However, there is a hint (from the positive residuals) of a tendency towards

underestimation on the very largest catchments.

The weaker performance on smaller catchments is a typical finding of such studies e.g.

Kjeldsen et al. (2008). Inevitably this prompts concern, given that many applications of

PCD-based models are to smaller catchments.

2.2.9 Validation of model performance

The performance of the rural ℓnQMED model was validated by application to records held

back for the purpose (see Section 2.2.3). The 25 randomly selected stations provide a

relatively robust method of assessment.

The outcome of the test is fully satisfactory. Figure 2.10 shows observed and predicted

QMED values for the 25 validation stations. The coefficient of determination obtained is

r2 = 0.906. The absence of heteroscedacity (uneven error variance) in Figure 2.10 is

reassuring, and vindicates the approach taken to fitting the QMED model: namely, OLS

regression on ℓn-transformed variables.

The validation eases concern over possible weak performance on small catchments. There is

no evidence from Figure 2.10 that the QMED model performs worse on catchments smaller

than 100 km2 than on catchments larger than 300 km

2.

50020010050201052

500

200

100

50

20

10

5

2


Pre

dic

ted Q

ME

D

1:1 line

3 -1

Station 25017

Figure 2.10: Observed and modelled QMED for the 25 validation stations

[Editorial note: An incorrect value of AREA appears to have been used for Station 25017

and accounts for the disappointing performance on this catchment. Overall, the model

performance on the validation set is excellent.]

AREA < 100 km2

100 ≤ AREA < 300 km2

AREA ≥ 300 km2


29

2.3 Assessing model robustness

2.3.1 Model robustness

Model robustness is an inevitable concern when a 7-variable model has been fitted to a

dataset of 145 observations. Stringent tests were therefore performed to make sure that the

inclusion/omission of certain catchments did not impact significantly on model coefficients.

In general terms, the robustness of a prediction model can be thought of as its ability to

remain stable against external disturbances. One concern is that the presence of an unusual

catchment in the calibration dataset might exert undue influence on the model derived.

2.3.2 Bootstrap resampling

As a prelude to testing the robustness of the model to the inclusion/omission of particular

catchments, the coefficients of the rural ℓnQMED model were “bootstrapped”. In conducting

the resampling, 1000 new samples – each of the same size as the observed data – were drawn

with replacement from replications of the observed data.

The model coefficients were first calculated for the observed data and then recalculated for

each of the 1000 resamples. This yielded the confidence intervals (for the model

coefficients) shown in Table 2.8. The BCa method constructs confidence intervals using the

“Bias-Corrected and accelerated” bootstrap introduced by Efron (1987). This adjusts for bias

and for skewness in the bootstrap distribution.

Table 2.8: Bootstrapped confidence intervals for model coefficients (BCa method)

Boot-strapped

confidence intervals 2.5% 5%

OLS

result 95% 97.5%

Constant -13.70 -13.26 -11.300 -9.73 -9.41

ℓnAREA 0.86 0.87 0.937 0.98 0.99

ℓnBFIsoil -1.21 -1.17 -0.922 -0.45 -0.36

ℓnSAAR 1.05 1.09 1.306 1.63 1.70

ℓnFARL 1.65 1.77 2.217 2.76 2.86

ℓnDRAIND 0.26 0.29 0.341 0.65 0.70

ℓnS1085 0.08 0.09 0.185 0.23 0.24

ℓn(1+AD2) 0.12 0.18 0.408 0.63 0.68

Figure 2.11 shows the Normal quantile-quantile plots of the bootstrapped model coefficients.


30

Qu a n ti l e s o f Sta n d a rd No rm a l

Qu

an

tile

s o

f R

ep

lica

tes

-2 0 2

-14

-12

-10

-8(Intercept)


Qu

an

tile

s o

f R

ep

lica

tes

-2 0 2

0.8

50

.90

0.9

51

.00

1.0

5

LnArea


Qu

an

tile

s o

f R

ep

lica

tes

-2 0 2

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

LnBFIsoils


Qu

an

tile

s o

f R

ep

lica

tes

-2 0 2

0.8

1.0

1.2

1.4

1.6

1.8

LnSAAR


Qu

an

tile

s o

f R

ep

lica

tes

-2 0 2

1.5

2.0

2.5

3.0

3.5

LnFARL


Qu

an

tile

s o

f R

ep

lica

tes

-2 0 2

0.2

0.4

0.6

0.8

LnDRAIND


Qu

an

tile

s o

f R

ep

lica

tes

-2 0 2

0.0

0.0

50

.10

0.1

50

.20

0.2

5

LnS1085


Qu

an

tile

s o

f R

ep

lica

tes

-2 0 2

-0.2

0.0

0.2

0.4

0.6

0.8

Ln1ARTDRAIN2

Figure 2.11: Normal quantile-quantile plots of the bootstrapped model coefficients

2.3.3 Model robustness to influential stations

The influence of individual catchments on the final model coefficients was assessed using a

technique referred to as “Jackknife after bootstrap”. Jack-knifing recalculates model

coefficients exhaustively: omitting each data point (i.e. each catchment) in turn.

In testing the sensitivity of model coefficients to the data they were trained on, the model was

held to be overly sensitive if the removal of any individual catchment from the calibration

dataset resulted in coefficients becoming insignificant or falling outside the 95% confidence

intervals of the BCa percentiles. Groups of influential catchments were also considered.

Figure 2.12 depicts the absolute relative influence of individual observations on the model

formulation. Observations with absolute relative influence values greater than 2.0 were

selected for further testing. In testing the sensitivity of each parameter, the model was rerun

with influential observations omitted sequentially without replacement. The significance of

the resulting change in the model parameter was noted.

Intercept ℓnAREA ℓnBFIsoil ℓnSAAR V

erti

cal

scal

es m

ark

qu

anti

les

of

mo

del

coef

fici

ent

ℓnFARL ℓnDRAIND ℓnS1085 ℓn(1+AD2)

Horizontal scales mark quantiles of standard Normal distribution


31

(Intercept)

Ob s e rv a ti o n

Ab

solu

te R

ela

tive

In

flu

en

ce

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

02

46

5

2 4

9 19 5

1 0 1

1 2 8

LnArea

Ob s e rv a ti o n

Ab

solu

te R

ela

tive

In

flu

en

ce

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

02

46

5

2 4

5 5 7 5

8 5

9 1

1 0 7

1 0 91 3 9

LnBFIsoils

Ob s e rv a ti o n

Ab

solu

te R

ela

tive

In

flu

en

ce

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

02

46

5

2 44 6

4 9

5 4

5 5

9 2

LnSAAR

Ob s e rv a ti o n

Ab

solu

te R

ela

tive

In

flu

en

ce

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

02

46

2 4

9 1

9 5

1 0 1

1 2 8

1 3 8

LnFARL

Ob s e rv a ti o n

Ab

solu

te R

ela

tive

In

flu

en

ce

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

02

46

5

2 44 9

8 81 0 7

1 0 9

1 2 8

LnDRAIND

Ob s e rv a ti o n

Ab

solu

te R

ela

tive

In

flu

en

ce

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

02

46

5

2 4 7 5 9 09 39 5

1 0 7

1 2 21 3 7

LnS1085

Ob s e rv a ti o n

Ab

solu

te R

ela

tive

In

flu

en

ce

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

02

46

24 2 43 0 4 96 6

8 5

8 79 11 3 1

Ln1ARTDRAIN2

Ob s e rv a ti o n

Ab

solu

te R

ela

tive

In

flu

en

ce

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

02

46

2

4

1 76 26 66 7

9 01 0 71 0 9 1 4 4

Figure 2.12: Influence of individual stations in determining model coefficients

[Editorial note: Particular catchments (e.g. Sequential Numbers 5, 24, 49 and 109) are seen

to be rather influential. Sequential Numbers 5, 24 and 49 correspond respectively to

Stations 06030 Big at Ballygoly, 10004 Glenmacnass at Laragh and 19046 Martin at Station

Road. These are all stations for which the 7-variable model underestimates QMED

appreciably (see Figure 2.7). The AM flood data for Station 06030 Big at Ballygoly appear

suspect, although this may not compromise the gauged value of QMED. See Box 4.3.

Sequential Number 109 corresponds to Station 31002 Cashla at Cashla, for which QMED is

well-predicted by the model. This station is influential on the exponent of FARL because the

catchment has the greatest attenuation (FARL=0.632) of any within the calibration dataset.]

It was found that, even when all influential points were removed, the eight model coefficients

(i.e. the intercept term and the seven coefficients in Equation 2.7) remained significant and

within the BCa percentiles given in Table 2.8. It was therefore concluded that the QMED

model is not unduly sensitive to individual observations.

2.3.4 Robustness to spatial drift in model coefficients

An underlying assumption of the regression approach – undertaken here by ordinary least-

squares – is that the derived relationship is spatially constant, i.e. that the estimated

parameters remain constant over space. In hydrological modelling applications, this

assumption of spatial invariance can sometimes be unconvincing.

Geographically weighted regression (GWR) is a technique that expands standard regression

for use with spatial data (Fotheringham et al., 2002). GWR assesses local influences and

Labels (and horizontal scales) mark sequential number in calibration dataset of 145 catchments

Ver

tica

l sc

ales

mar

k “

Act

ual

rel

ativ

e in

flu

ence

”

Intercept ℓnAREA ℓnBFIsoil ℓnSAAR

ℓnFARL ℓnDRAIND ℓnS1085 ℓn(1+AD2)


32

allows a spatial shift in parameters. This can provide a more appropriate and accurate basis

for descriptive and predictive purposes. Some details are given in Section C4.

The GWR technique was employed to test if the coefficients of the ℓnQMED model are

spatially constant. The assumption of spatial invariance was found reasonable except in

regard of the FARL term. There is some limited evidence (see Map C.1 in Section C4) that a

higher coefficient of ℓnFARL may be appropriate in the East and North West, while a lower

coefficient may be apt around the Upper Shannon basin and in the West.

This test result was of borderline significance, and use of a spatially-varying model for

ℓnQMED is not recommended. However, the finding endorses the recommendation to

favour geographical closeness – as well as similarity in key PCDs such as FARL – when

selecting the pivotal catchment in a particular flood estimation problem (see Section 11.2).

2.4 Investigating the effect of arterial drainage

2.4.1 Approach

The effect of arterial drainage on QMED is represented by the appearance of ARTDRAIN2

in the 7-variable model of Equation 2.8 in Section 2.2.5. However, arterial drainage is a

sufficiently important feature of Irish flood hydrology to warrant more explicit investigation.

Its impact on the incidence of flooding downstream has long been a source of controversy,

with the opposing points of view well highlighted by Robinson (1990).

The further work took two main forms. Section 2.4.2 looks at stations for which both pre and

post-drainage records are held, while Section 2.4.3 considers the separate modelling of

QMED on drained and undrained catchments.

2.4.2 Analysis of stations for which both pre and post-drainage records held

Amongst the flood peak data studied in the FSU, 15 stations have both pre and post-drainage

records. Gauged values of QMED are shown in Table 2.9 and Figure 2.13.

Table 2.9: Factorial change in QMED following drainage

Station

#

QMED (m3s

-1)

Factorial

change

Station

#

QMED (m3s

-1)

Factorial

change Pre-

drainage

Post-

drainage

Pre-

drainage

Post-

drainage

03051 21.50 40.10 1.865 24004 39.27 62.41 1.589

07002 17.91 19.22 1.073 25017 391.74 414.17 1.057

07003 12.71 21.87 1.721 26012 29.61 47.68 1.610

07005 86.10 104.98 1.219 30004 42.30 90.34 2.136

07007 37.15 35.70 0.961 30005 22.88 36.78 1.608

07010 32.87 70.72 2.152 30061 247.97 250.07 1.008

07012 149.61 265.86 1.777 35011 86.71 132.23 1.525

24001 80.84 114.59 1.417 15 FSU catchments, pre and post-drainage


33

It is evident that, following arterial drainage, there are substantial increases in QMED at 11 of

the 15 stations. QMED is found to more than double at Stations 07010 and 30004. In

contrast, Station 07007 shows a slight reduction, while Stations 25017 and 30061 show little

or no change in QMED following drainage.

Figure 2.13: QMED for pre and post-drainage records at 15 stations

Disappointingly, there is no discernible association between the factorial change and either of

the PCDs summarising the extent of arterial drainage (see Figures 2.14).

0.50.40.30.20.10.0

2.5

2

1.5

1

0.5

ARTDRAIN

Facto

rial change

1.00.80.60.40.20.0

2.5

2

1.5

1

0.5

ARTDRAIN2

Facto

rial change

Figures 2.14: QMED change following drainage (against ARTDRAIN and ARTDRAIN2)

The one significant association found – the trend is just significant at the 95% level – was

with BFIsoil (see Figure 2.15). It is logically consistent that drainage is somewhat more

effective on naturally less permeable (i.e. low BFIsoil) catchments.

Pre-drainage QMED (m3s

-1)

Post

-dra

inag

e Q

ME

D (

m3s-1

)

- - - - 100% increase – – – 50% increase ––––– no change


34

1.00.80.60.40.20.0

2.5

2

1.5

1

0.5

BFI

Fac

tori

al c

han

ge

35011

30061

30005

30004

26012

25017

24004

24001

07012

07010

07007

07005

07003

07002

03051

soil Figure 2.15: QMED change following arterial drainage (against BFIsoil)

2.4.3 Modelling QMED on drained and undrained catchments separately

Although represented in the QMED model of Equation 2.8, the effect of arterial drainage was

explored further by partitioning the 145-catchment rural dataset into 50 drained catchments

and 95 undrained catchments. The aim was to assess whether explicit discrimination of

drained and undrained catchments could:

Enhance the overall modelling of QMED from PCDs;

More fully reveal the impact that arterial drainage has on QMED;

Identify the PCDs that best capture the drained and undrained responses.

The further analysis is reported in Appendix D, where it is concluded that there is little gain

in modelling QMED separately on drained and undrained catchments.

However, the analysis suggests that BFIsoil and ARTDRAIN2 are valued descriptors in

predicting QMED in drained catchments, whilst DRAIND and S1085 are more important in

predicting QMED in undrained catchments. Awareness of these sensitivities may help the

experienced user to judge the most appropriate choice of pivotal catchment in a given flood

estimation problem (see Section 11.2).

2.5 Adjusting the selected model for urbanisation

2.5.1 Introduction

The rainfall-runoff response of a catchment can be radically altered by urbanisation.

Impervious surfaces inhibit infiltration and reduce surface retention. Increases in surface

runoff are typically accompanied by a more intense and accelerated response.

In assessing the effect of urbanisation at the river-catchment scale, the approach taken was to

study the change from the natural (rural) condition. The outcome is an adjustment factor that


35

can be used to augment the performance of the rural model for use on catchments that have

undergone urbanisation.

The adjustment derived is strictly for catchments that have undergone urbanisation. For

reasons discussed by Reed and Robson (1999), such models are not suitable for anticipating

the effect of planned urban development.

The modelling of QMED in Section 2.2 treated catchments with a fractional urban extent

(URBEXT) less than 0.015 as essentially rural. The urban dataset thus comprises the 35

catchments in the FSU dataset having URBEXT≥0.015.

2.5.2 Deviation of QMED on urbanised catchments from rural model

Figure 2.16 summarises the outcome of applying the rural QMED model to the 35 urbanised

catchments. Estimation of QMED from PCDs is inevitably approximate, so it is difficult to

infer a definitive influence of urbanisation. As anticipated, the rural model underestimates

QMED on more catchments than it overestimates (20 as against 15). Regression confirms the

general tendency for the rural model to underestimate QMED when applied to urbanised

catchments.

200100502010521

200

100

50

20

10

5

2

1


QM

ED

est

imat

ed b

y r

ura

l m

odel

08005

09010

08009

3 -1

1100108008

Figure 2.16: Performance of rural QMED model on urbanised catchments

Five catchments (labelled in Figure 2.16 and listed in Table 2.10) have a factorial error

greater than 2, although only marginally. QMED at Stations 08005, 08008, 09010 and 11001

is underestimated by a factor of just over 2, while QMED at Station 08009 is overestimated

by a factor of just over 2 (see central column of Table 2.10). [Editorial note: Station 08009

Ward at Balheary attracts comment elsewhere, notably in Box 4.4 and Section 10.4.5.

Bhattarai and Baigent (2009) suggest that the surprisingly low gauged QMED may reflect

incomplete recording. While urbanisation may account for the underestimation at Stations

08005 and 09010, it is likely that some other factor (e.g. extensive arterial drainage) accounts

for the underestimation of QMED at Stations 08008 and 11001.]

Regression

line


36

Table 2.10: Five catchments poorly predicted by the QMEDrural model

Station

# QMEDobs QMEDrural

QMEDrural ∕ QMEDobs

URBEXT AREA ARTDRAIN2

08005 2.32 1.09 0.470 0.25 9.17 0.00

08008 40.90 19.97 0.488 0.04 107.92 0.44

08009 5.00 10.14 2.028 0.08 61.64 0.55

09010 47.05 22.33 0.475 0.24 94.26 0.00

11001 47.17 21.84 0.463 0.02 155.11 0.36

2.5.3 A special check

Catchments liable to urbanisation are not necessarily typical of the overall set of Irish gauged

catchments. A special check was made by seeking a set of 35 rural catchments that match

the 35 urbanised catchments in respect of catchment features other than urbanisation.

The experiment defined catchment similarity as a standardised distance in the 4-dimensional

space representing differences in values of ℓnAREA, ℓnSAAR, ℓnBFIsoil and ℓnFARL.

Because of the limited number of catchments available, some rural catchments were allowed

to appear more than once in the set of 35 rural catchments formed to match the 35 urbanised

catchments. The most problematic feature to match was catchment wetness (SAAR).

[Editorial note: On average, the matched catchments tend to be somewhat wetter than their

urbanised counterparts. This reflects that settlement patterns are biased towards drier parts of

Ireland.]

The special analysis (not reported) found that the rural QMED model of Section 2.2

represents QMED on the matched set of catchments relatively well, without notable bias. It

was therefore concluded that it was reasonable to use the 7-variable rural model as a vehicle

for estimating the rural element of QMED on urbanised catchments.

2.5.4 Approach to devising an urban adjustment

Despite the difficulty in detecting a clear fingerprint of urbanisation, effort was made to

derive an adjustment factor for urbanisation that could be used to scale up the rural model

results using the form:

ruralQMEDUAFQMED 2.9

UAF is an urban adjustment factor that describes the proportional increase in QMED induced

by urbanisation. QMEDrural denotes the rural estimate for QMED by the Section 2.2 model.

Values of UAF at the gauged sites are inferred by inverting Equation 2.9:

ruralQMED

QMEDUAF 2.10


37

2.5.5 Exploratory data analysis

UAF was ℓn-transformed and its relationship with catchment descriptors examined. Figure

2.17 shows scatter-plots and correlation coefficients across ℓnUAF and a selection of

catchment descriptors.

Figure 2.17: Correlations between ℓnUAF and selected PCDs

It is seen that ℓnUAF is not strongly correlated with the PCDs. The strongest relationships

found are with ℓn(1+URBEXT), ℓnAREA and ℓnSAAR. The marked negative correlations

between ℓnUAF and ℓnAREA, and between ℓn(1+URBEXT) and ℓnAREA, reflect that the

most heavily urbanised catchments tend to be small catchments.

2.5.6 An urban adjustment model

Bearing in mind the above relationships, several approaches to modelling UAF were

examined beginning with the basic model form of:

URBEXT)n(1gnUAF 2.11

The model was fitted by weighted least-squares (WLS) regression with the weights

proportional to the urban extent. Thus, greater weight is given to data from more heavily

urbanised catchments. This led to the model:

ℓnUAF = 1.482 ℓn(1+URBEXT) 2.12

and its multiplicative form:

1.482URBEXT)(1UAF 2.13

Lower triangle shows scatter-plot. Upper triangle shows correlation coefficient.

ℓnUAF -0.48 0.54 0.13 -0.10 -0.39

ℓnAREA -0.68 -0.03 0.21 0.16

ℓn(1+URBEXT) 0.09 0.07 -0.43

ℓnFARL -0.42 -0.44

ℓnBFIsoil -0.09

ℓnSAAR


38

The model returned an r2 (in ℓn space) of 0.300, a standard error of 0.735 and a factorial

standard error of 2.09. The coefficient 1.482 has a standard error of 0.139. The UAF model

has the merit of decreasing to 1.0 when URBEXT decreases to zero. The model returns a

value of 2.793 when URBEXT reaches a maximum of 1 (for a fully urbanised catchment).

The value of g in this model is very similar to the coefficient derived for the corresponding

model in the FEH, which gave a value of 1.49 (Table 18.1 of Reed and Robson, 1999). The

large factorial standard error highlights the large uncertainties involved in modelling UAF.

Nonetheless the model represents a theoretically plausible description of the impact of

urbanisation on the index flood and can be interpreted as:

Urban adjustment factor increases with urban extent;

Urban adjustment factor increases to a maximum of 2.793 when a catchment is fully

urbanised (URBEXT=1);

Urban adjustment factor decreases to 1 as URBEXT tends towards zero.

2.5.7 Performance of urban adjustment model

Figure 2.18 shows the performance of the urban-adjusted model on the 35 urbanised

catchments. In comparison with Figure 2.16 – which shows the performance of the

unadjusted model on these catchments – a small but just perceptible improvement is evident,

not least for Stations 08005 and 09010.

200100502010521

200

100

50

20

10

5

2

1


QM

ED

model

adju

sted

by U

AF

08005

3 -1

1100108008

08009

09010

Figure 2.18: Performance of UAF-adjustment to QMED

Urbanisation tends to have a greater effect on naturally permeable catchments. However, it

was not practical to consider a permeable-catchment adjustment given the very small number

of such stations in the available dataset. It is recommended that additional gauges be

established in such catchments.


39

2.5.8 Comparisons and contrasts with the FEH

The approach to modelling the urban adjustment mirrors some aspects of the methods of

Reed and Robson (1999). Important differences in the Irish study are that:

The smaller pool of gauged catchments and the narrower variation of catchment

permeability led to adoption of a simpler model structure in which UAF is determined

solely by the urban fraction URBEXT;

The basis of URBEXT values differs between Ireland and the UK, impeding

comparisons;

Values of URBEXT have not been backdated to represent the state of catchment

development at the mid-point of the period of record used to calculate QMED at the

gauged site;

Values of UAF inferred from Equation 2.10 were specifically constrained to be no

less than 1.0.

[Editorial note: Associating the long-term value of QMED with the current (higher) value

of urban extent will tend to underplay the effect of urbanisation on flood magnitudes.

However, the device of constraining “observed” UAF values to be no less than 1.0 may do

much to compensate.]

Box 2.2: Interpretation of urban adjustment models for QMED

2.6 Improving model performance by data transfer

2.6.1 Introduction

Even though the selected rural model marks a major improvement on previous approaches for

Ireland, the uncertainty ranges are still large. Every effort should be made to adjust the

QMED estimates made from PCDs by reference to flood data available from nearby or

similar catchments.

2.6.2 Review of techniques

In terms of adjusting model predictions based on the transfer of information from gauged

sites, there has been debate within the literature as to whether adjustments should be made

Editorial note: As with the UK work of Reed and Robson (1999), the worry lingers that

the urban adjustment to QMED might be mistaken for representing the total effect of

urbanisation. It should be recalled that the urban adjustment factor represents only the

residual effect of urbanisation on flood magnitudes. It does not represent the effect that

past development-control has successfully ameliorated.

The urban adjustment model has been fitted to the gauged flood data just as they are.

Thus, the model implies that typically this amount of urbanisation produces this amount of

an increase in flooding if development control continues to be as effective as it has been in

the past. It would be negligent to interpret the urban adjustment model as representing the

total effect of urbanisation on what would have hitherto been a greenfield catchment.


40

using information from catchments that are geographically close to the site of interest

(sometimes called donor catchments) or from catchments that are hydrologically similar in

terms of key descriptors but located anywhere within the study domain (sometimes called

analogue catchments). Previous work has tended to highlight the strong clustering of

residuals in regression models and used this to underscore the recommendation to favour

local catchments in QMED adjustments.

In a comprehensive assessment of the FEH statistical method for adjusting QMED values,

Morris (2003) found inappropriate adjustment of QMED to be a potentially important source

of error and suggested that the selection of gauges for the transfer of information be based on

catchment similarity judged from key PCDs. However, the selection and use of similar (as

opposed to local) catchments is subjective and can greatly affect the QMED estimate. Morris

also concluded that favouring donor sites that are located directly upstream/downstream of

the subject site could potentially help to reduce prediction errors further. In contrast Kjeldsen

et al. (2008) favour a method where the weight given to a particular data transfer is based on

geographical distance, rather than expressly in terms of catchment similarity.

This section summarises some work done on an automated method of data transfer before

explaining why more traditional methods were chosen for implementation. Section 2.7

provides a worked example of QMED adjustment by data transfer. The topic is taken up

further in Chapter 11.

2.6.3 Geostatistical mapping of residuals

Grover et al. (2002) indicate that the performance of global regression models can be

improved by mapping regression residuals using geostatistical methods and using these

mapped residuals to adjust QMED estimates at point locations. To complement the

traditional approaches reviewed above, geostatistical mapping of residuals was considered:

To explore the spatial pattern of errors in the estimation of ℓnQMED;

As a basis for adjusting QMED.

Geostatistical methods were used to interpolate and map the residuals of the rural 7-variable

model. The interpolation methods considered included Kriging, splines and inverse distance

weighting. Based on the assessment of a small validation set, the inverse distance weighting

(IDW) technique was found to be the most appropriate.

The IDW function determines interpolated values using a linear weighted combination of a

set of sample points. The weight assigned to each is a function of the distance of an input

point from an output cell location. The greater the distance, the smaller the influence the

point has on the output value.

A fixed radius of 55 km was adopted to select input stations for modelling ℓnResidual.

Interpolated residuals are shown in Map 2.3. Areas of underestimation of ℓnQMED are

indicated in red, while the blue-grey areas mark overestimation of ℓnQMED. Some

clustering of model errors is evident, with areas of overestimation of ℓnQMED in the South

East, much of the North West, the Mid-West and the South West, and areas of

underestimation in much of the East and South of the country.


41

Map 2.3: IDW-interpolated residuals from rural ℓnQMED model

[Editorial note: The extent to which the geostatistical mapping approach is driven by the

residual error for particular gauged catchments can be judged by comparing Map 2.3 with

Map 2.2 presented in Section 2.2.8. Regrettably, the colour schemes are transposed. The

overestimation of QMED around Wexford is largely based on the residual for Station 13002

Corock at Foulk’s Mill (QMEDobs = 7.01 and QMEDest = 10.74 m3s

-1). Here, QMEDest

denotes the estimate of QMED from the rural catchment model of Equation 2.8.]

ℓn(Residual)

0.9 – 1.1

0.7 – 0.9

0.5 – 0.7

0.3 – 0.5

0.1 – 0.3

0.0 – 0.1

-0.1 – 0.0

-0.3 – -0.1

-0.5 – -0.3

-0.7 – -0.5

Wexford


42

In order to extract a correction value from the interpolated map, the centroid of the subject

catchment is overlaid and the interpolated ℓnResidual extracted. The value is then added to

the prediction from the rural ℓnQMED model.

The method was assessed across the 25 validation stations. Whereas the basic model

achieved a coefficient of determination of 0.906 for ℓnQMED (see Section 2.2.9), applying

the IDW-adjustment increased r2 to 0.912. This suggests that the adjustment technique is

moderately effective.

It is recommended that additional monitoring stations are sought in districts where residuals

are uncomfortably large (i.e. in darker-shaded areas of Map 2.3), especially where Map 2.2

indicates the underlying data to be sparse.

2.6.4 Possible disadvantages of automated methods

Automated methods of adjusting estimates are becoming more commonplace in flood

hydrology. Here, the IDW method has been shown (by split-sample testing) to be moderately

effective in adjusting the PCD-based estimates of QMED.

However, the routine adoption of such automated methods comes at a price. It rules out the

use of experience and reasoning to make subjective but informed decisions about appropriate

adjustments. Factors that the human adjuster might consider include:

The perceived degree of similarity between donor and subject catchments;

The likely quality of the gauged estimate of QMED at the donor site;

Whether to involve more than one donor in the data transfer.

There is some support from Morris (2003) for the traditional view that data transfer from an

upstream or downstream site is often of greatest value. This is further supported by an

interim assessment reported in Section 11.1.

It should be noted that the automated method for adjusting QMED by geostatistical mapping

is executed directly in space by reference to distances between catchments. The method does

not map the residual error (in ℓnQMED) up and down a particular river system.

The distances used in the geostatistical mapping method are measured from catchment

centroid to catchment centroid rather than from gauged site to subject site. In cases where the

nearby gauges all lie in the same overall river basin, these inter-centroid distances reflect

something of the river structure, albeit imperfectly.

A particular weakness of an automated method can be a failure to consider “scale” effects.

Should data from a small tributary be used to adjust QMED values in the much larger river to

which it drains? And is a gauging station on the main river suitable for adjusting QMED on a

minor tributary? These are difficult questions to answer algorithmically, rather than by

individual judgement of the particular circumstances.

The FSU concludes that the geostatistical mapping approach embodied in Map 2.3 is useful

for investigating the spatial characteristics of model residuals. However, it does not

recommend its adoption. Instead, the user is exhorted to choose (and make) a data transfer


43

based on reasoned judgement of the situational context of their flood estimation problem in

relation to the available gauged data. The judgement should take account both of the

characteristics of the particular subject site and of the options available for data transfer.

2.6.5 Recommended procedure for data transfer

It is advocated that the user assesses the most appropriate data transfer by making a reasoned

selection of a pivotal catchment. This is the gauged catchment judged to be most relevant to

the specific flood estimation problem.

It is also a matter for the user to judge whether part or all of the observed ℓnResidual at the

pivotal site is to be added to the catchment-descriptor estimate of ℓnQMED when making the

adjustment at the subject site.

Data transfer methods are discussed further in Chapter 11. But many of the principal

elements are introduced in the worked example that now follows.

2.7 Worked example of QMED estimation at an ungauged site

The catchments in the worked example are all essentially rural; their URBEXT values range

from 0.0021 to 0.0055. The recommended procedure for data transfer is more complicated

when the donor catchment is appreciably urbanised (see Box 2.3).

Box 2.3: Data transfer procedure when one of the catchments is urbanised

While it is recommmended to incorporate urban adjustments in every case, the goal of the

worked example here is to focus on the fundamentals of data transfer. The occasional

references to QMEDrural remind the user that the data transfer is to be applied to the inferred

rural element of QMED only, as indicated in Box 2.3.

2.7.1 Illustrative example

The methodology for estimating the index flood at an ungauged site is illustrated for the Suck

at Rookwood. This corresponds to Station 26002 but is treated here as an ungauged site. The

Editorial note: The FSU recommendation is that adjustment factors are derived from,

and applied to, the inferred rural element of QMED, i.e. to QMEDrural. The full procedure

is therefore to:

1. Infer values of QMEDrural at gauged sites by inverse application of the urban

adjustment model (i.e. Equation 2.10);

2. Assess adjustment factors as the ratio of gauged to modelled values of QMEDrural;

3. Make the data transfer from gauged to subject site, thereby adjusting the estimate

of QMEDrural at the subject site;

4. Apply the urban adjustment factor in the normal manner (i.e. Equation 2.9) to

obtain the adjusted estimate of QMED at the subject site.

In all cases, the urban adjustment factor UAF is given by Equation 2.13:

1.482URBEXT)(1UAF


44

following provides a step-by-step guide to deriving an estimate of the index flood at this

location.

Step 1 Derive coordinates for ungauged location: In this case the catchment centroid has an

Easting of 172050 and a Northing of 270500.

Step 2 Derive catchment descriptor information: The relevant PCDs are given in Table 2.11.

They derive from the digital datasets made available with the FSU.

Table 2.11: PCDs for Suck at Rookwood worked example

PCD value unit PCD value unit

AREA 641.45 km2 URBEXT 0.0026 –

DRAIND 0.799 km/km2 FARL 0.979 –

S1085 0.500 m/km BFIsoil 0.6036 –

SAAR 1067.03 mm ARTDRAIN2 (AD2) 0.000 –

Step 3 Apply rural QMED model

From Equation 2.8:

0.4080.1850.341

2.2171.3060.922

soil

0.9375

ARTDRAIN21S1085DRAIND

FARLSAARBFIAREA101.237QMED

408.0185.0341.0217.2306.1922.0937.05 000.01500.0799.0979.003.10676036.045.641102373.1

18796.09263.09540.090125928.19.426102373.1 5

= 58.93 m3s

-1

Step 4 Apply urban adjustment factor

From Equation 2.13:

1.482URBEXT)(1UAF

1.482(1.0026)UAF

=1.0039

Thus, the PCD-based estimate of QMED is reached by Equation 2.9:

ruralQMEDUAFQMED

QMED = 1.0039 × 58.93

= 59.16 m3s

-1

The gauged QMED for the Suck at Rookwood (Station 26002) is 56.56 m3s

-1. So

QMED at this site is overestimated by a factorial error of 59.16/56.56 = 1.046 or

4.6%. This represents a very good performance. However, the user will not know

this for an ungauged site!


45

Step 5 Transfer data from gauged locations to improve model prediction at subject site:

The general procedure is to infer an adjustment factor, AdjFac, by reference to the

performance of the PCD-based model of QMEDrural at a nearby gauged site:

PCDrural,

gaugedrural,

QMED

QMEDAdjFac 2.14

The adjustment is then partially or fully transferred to the subject site:

PCDrural,

h

adjustedrural, QMEDAdjFacQMED 2.15

The typical procedure is to apply a full transfer by setting the exponent h to 1.0. If

QMEDrural is found to be 20% greater than the PCD-based estimate, it is assumed that

the model will be similarly in error at the subject site. Thus, the estimate of

QMEDrural at the subject site is adjusted by multiplying by 1.20.

The exponent h can be thought of as the hardness of the data transfer. h=1 denotes a

full (or “hard”) transfer. A partial (or “softer”) transfer might set h=0.5. In this case,

if QMEDrural is found (by examining QMEDrural,gauged) at the donor site to be 20%

greater than given by the PCD model (i.e. QMEDrural,PCD) the estimate of QMEDrural at

the subject site is adjusted by multiplying by a factor of 1.200.5

or 1.095. [Editorial

note: The relatively complex notation here reflects that data transfers are made to the

rural element of QMED only. See Box 2.3 above.]

Much skill attaches to deciding which of several possible donor catchments is pivotal

to improving QMED estimation at the subject site. With gauged sites both upstream

and downstream of the subject site, the choice is not clear-cut for the Suck at

Rookwood. Table 2.12 summarises data transfers from the gauged sites upstream and

downstream of the subject site. A clear head is required to track the adjustments.

Table 2.12: Data transfers for Suck at Rookwood worked example

Method

Data

transfer

from

AREA ℓnAREA Gauged

QMED

Modelled

QMED Implied

factorial

adjustment

QMED

at subject site

Gauged Adjusted

km2 – m

3s

-1 m

3s

-1

No data

transfer 641.5 6.46 59.16

Upstream

donor

Station

26006 184.8 5.22 24.23 28.05 0.864 51.1

Downstream

donor

Station

26005 1085.4 6.99 93.21 87.39 1.067 63.1

Compromise 0.960 56.8

IDW

interpolated several 59.8

Analysis of

gauged data

Station

26002 56.56


46

2.7.2 Merging data transfers from two sites

The adjustment factors implied by the model performance upstream and downstream of the

subject site are seen to be contradictory: the PCD model is overestimating QMED at Station

26006 Suck at Willsbrook but underestimating it at Station 26005 Suck at Derrycahill. A

reasonable approach is to seek a compromise by taking a weighted geometric mean of the

adjustment factors:

w1

Donor2

w

Donor1Compromise AdjFacAdjFacAdjFac

2.16

The weight, w, is chosen to reflect the relative quality or relevance of the donors to QMED

estimation at the subject site. The value chosen for w might reflect the quality or length of

record at the gauged sites, or their inter-centroid distances from the subject catchment.

It transpires that the QMED estimates at 26005 and 26006 both derive from very long

records. The flood rating at Station 26006 is rated the more highly (Grade A1 as against

Grade A2 for Station 26005). However, the inter-centroid distance to the subject catchment

is shorter for Station 26005 (73 km rather than 129 km for Station 26006). The pragmatist

might decide to give the donor sites equal weight, setting w = 0.5. Thus:

0.50.50.5

26006

0.5

26005Compromise 1.0670.864AdjFacAdjFacAdjFac 0.960

The final estimate of QMED at the subject site is therefore 0.96059.16 = 56.8 m3s

-1.

It transpires that this would be a very good call for the Suck at Rookwood, where QMED is

gauged to be 56.6 m3s

-1. The excellent outcome in this example is, however, of little comfort

to the user faced with making adjustments at genuinely ungauged sites.

2.7.3 Geostatistical mapping method

It transpires that the geostatistical mapping method of Section 2.6.3 also performs well on

this example. The IDW-technique of Map 2.3 yields an adjusted QMED estimate for the

Suck at Rookwood of 56.9 m3s

-1.

Were an automated method of QMED adjustment implemented in Ireland, it would be

important to determine an appropriate method of incorporating the urban adjustment. Most

likely, the geostatistical mapping would be applied only to the inferred rural element of

QMED, in keeping with the FSU recommendation of Box 2.3.


47

3 Trend and randomness

It is assumed in flood frequency analysis that successive values in an annual maximum (AM)

series have emerged randomly and independently from a common population of flood values.

In particular, this means that there should be no upward or downward trend in the series of

data nor should there be any strong clustering of groups of larger-than-average or smaller-

than-average values during the passage of time. Recognised tests of trend and randomness

are conducted to determine how well these assumptions are met.

The tests were applied to the AM flow series that lie at the heart of the FSU research. When

seeking to detect trend or non-randomness in hydrological variables, there can be merit in

also studying extremes abstracted in peaks-over-threshold (POT) format (e.g. Robson,

1999a). If such series are available, methods put forward by Buishand (1982) may be used.

The testing was limited to Grade A1 and A2 gauging stations. These include some post-

drainage records on catchments which had earlier experienced arterial drainage works.

3.1 Tests

3.1.1 Methods

Building on experience gained and reported in the UK Flood Studies Report (NERC, 1975)

and the WMO Report Detecting trend and other changes in hydrological data (Kundzewicz

and Robson, 2000), six tests were used to check for trends, shift and serial dependency in the

AM flood series:

Tests for trend

Mann-Kendall (non-parametric test for trend);

Spearman’s ρ (non-parametric test for trend);

Mean-weighted linear regression test (parametric test for trend);

Test for step-change

Mann-Whitney U test (non-parametric test for step-change);

Tests for serial dependency in time series

Turning points (non-parametric test for randomness);

Rank difference (non-parametric test for randomness).

Spearman’s ρ is the rank correlation. An observation has rank r if it is the rth

largest in a

dataset. Rank-based tests use the ranks of the data values rather than the values themselves.

Most rank-based tests assume that data are independent and identically distributed but

typically have the advantage of being robust and simple to apply.


48

3.1.2 Formats

Statistical trend analyses were carried out on the annual maxima in two formats. The main

analyses were undertaken for 94 AM flow series for stations graded A1 or A2. Their

selection is discussed in Sections E5.1 and E6.1 of Appendix E.

Additionally, a special analysis of the median of annual maxima taken in five-year blocks

was applied to 117 series. The technique of splitting the annual maximum series into (non-

overlapping) five-year segments – and retaining only the median (i.e. middle-ranking) values

– provides a macro view of the data that is unaffected by occasional unusually large or small

annual maxima. This format is reasonably tolerant of minor gaps in the AM series.

Results obtained for the two formats were generally in good agreement. Those for the

standard AM series are now summarised. Supporting material is presented in Appendix E.

3.2 Findings

The exploratory data analysis found 11 AM flow series that showed a significant trend with

time. Eight of the significant trends were upward and three downward. [Editorial note: Of

the three sites showing downward trend, Station 25002 Newport at Barrington’s Bridge and

Station 25003 Mulkear at Abington were later withdrawn from study for reasons given in

Appendix B. The downward trend for Station 25014 Silver at Millbrook might reflect rating

changes in 1971.] Testing for trend and step-change are a valuable element of quality

inspection. Close inspection is warranted before judging whether a statistically significant

change likely reflects a physical effect.

Overall, the results show an unexpected degree of non-randomness in the flood series.

Across all tests on the AM series (Table 3.1), approximately 10% of stations reject the null

hypothesis at the 1% level, whereas in a truly random situation only one station (from about

100 stations) might be expected to reject the null hypothesis. Further details of the testing are

reported in Appendix E.

Table 3.1: Number of cases (out of 94) in which the null hypothesis is rejected

Significance

level

Trend Step-

change Randomness Number

expected

(from 94

cases) by

chance

alone

Mann-

Kendall

test

Spearman’s

ρ

Mean-

weighted

linear

regression

Mann-

Whitney

U

test

Turning

point

test

Rank

difference

test

1% 12 13 8 24 5 9 ≈1

5%

(but not 1%) 14 13 8 15 4 6 ≈4

5%

(all cases) 26 26 16 39 9 15 ≈5


49

It is concluded that the Irish flood series exhibit more non-stationarity (i.e. trends and step-

changes) than is to be expected by chance. They also exhibit some tendency to non-

randomness. This casts doubt on the validity of the independent and identically distributed

(iid) assumption which underpins all standard methods of flood frequency analysis.

3.3 Pragmatism

International experience with flood frequency analysis nevertheless suggests that making the

iid assumption is the most suitable means of estimating probabilities within a national flood

frequency methodology. It is especially helpful that the iid assumption allows AM series

with gaps – e.g. due to recorder or processing malfunction – to be treated in the same way as

continuous records. For instance, if five annual maxima are missing from 40 years of data,

the series is taken to be equivalent to a record of 35 years, regardless of whether the five

unavailable years are contiguous or not.

While it is pragmatic to make the classical assumption that AM flows are independent and

identically distributed, it is important to recognise that the assumption is not strictly valid.

This highlights the importance of examining AM series closely in all cases: both in terms of

their summary statistics (Chapter 4) and by visual inspection of probability plots (Chapter 5).


50

4 Descriptive statistics – and inferences therefrom

4.1 Descriptive statistics

4.1.1 Introduction

The research reported in Chapters 4 and 5 represents a detailed exploratory data analysis.

Much of the thinking behind such analyses is to allow the data to “speak” without

prejudgement. Although complex to the non-statistician, the summary statistics employed in

Chapter 4 – and the conventions followed in Chapter 5 – are all relatively standard in the

analysis of extreme values such as annual maximum flows.

4.1.2 Summary statistics – the idea

The AM) flow values provide the basic information on which the entire study rests. The

distribution of values can be viewed as a histogram or in a probability plot (see Chapter 5).

However, for the purposes of applying statistical models – such as probability distributions

based on the iid assumption – it is helpful to summarise the data by a small number of

statistical measures. These are referred to as summary statistics.

Summary statistics expressed in dimensionless form are especially useful, not least for:

Comparing flood frequency behaviour on different catchments;

Drawing inferences about the suitability of probability distributions for describing

flood data.

4.1.3 Summary statistics based on moments and L-moments

In flood hydrology, the most useful statistics relate to:

The typical magnitude of flood flows, e.g. their mean or median value

The “scale” or spread of the data, e.g. indexed by their standard deviation

The skewness (i.e. asymmetry) of the data

The kurtosis or “peakedness” of the data.

Box 4.1 defines dimensionless forms of the scale and skewness statistics based on ordinary

moments. These are the coefficient of variation, CV, and the coefficient of skewness, g. As

well as defining the basic population quantities, Box 4.1 provides formulae for calculating the

statistics from sample data. It also introduces the basic notation {Q1, Q2, Q3, …, QN} to

denote the AM flow series for N years of data.

Skewness values tend to increase slightly with sample size N. Hazen skewness is ordinary

skewness g multiplied by the factor (1 + 8.5/N). This correction factor – introduced by

Hazen (1930) – was shown by Wallis et al. (1974) to provide an almost unbiased estimator of

skewness for lognormal and EV1 distributions and for (mildly) skewed distributions in

general. Hazen skewness is listed here as H-skew.


51

Notation

The AM flow series is denoted by: {Q1, Q2, Q3, …, QN}.

Population quantities Sample estimates

= mean

N

1i

iQN

1Q

= standard deviation

N

1i

2

i

1N

)Q(Qσ

3 = 3rd

central moment

N

1i

3

i3 )Q(Q2)1)(N(N

Nμ

μ

σCV = coefficient of variation

Q

σCV

3

3

σ

μg = coefficient of skewness

3

3

σ

μg

Hazen’s unbiased skewness H-skew

N

8.51g

Box 4.1: Calculation of moments and their dimensionless ratios

Box 4.2 provides corresponding information for summary statistics based on probability-

weighted moments (PWMs) and L-moments. Prominent amongst these are the L-moment

ratios: the L-CV, L-skewness and L-kurtosis (Hosking, 1990). Relatively comprehensive

presentations of these measures are given in texts such as Hosking and Wallis (1997) and

Robson and Jakob (1999).

4.1.4 Additional summary statistics

Dimensionless measures of scale and skewness are useful for detecting common behaviour

among several series in a dataset. Dimensionless measures of flood magnitude are not

feasible although the specific flood (i.e. the peak flow per unit area) can sometimes be useful

for comparison purposes.

Other dimensionless quantities such as Qmax/Qmed, Qmax/Qmean or Qmean/Qmed can also be

compared between catchments although it has to be realised that Qmax/Qmean and Qmax/Qmed

can be sensitive to the period of record available and have high sampling variability.

Fourth-order moment ratios tend to have very high sampling variability. Nevertheless, values

of L-kurtosis can sometimes be useful when judging the relative merit of particular

3-parameter distributions.


52

Box 4.2: Calculation of L-moments and their dimensionless ratios

4.2 Summary statistics for Irish flood data

4.2.1 Summary statistics for 181 FSU stations

Principal summary statistics are presented in Table 4.1 for 45 stations graded A1, 70 stations

graded A2, and 66 stations graded B. Columns headed H-skew denote values of Hazen

skewness. Columns headed L-kurt show L-kurtosis values. The notations Qmed and QMED

are interchangeable. Appendix F reports additional summary statistics for these 181 stations.

Notation

In some analyses and plots, data are ranked in ascending order. The ordered AM

series is denoted by: {Q(1), Q(2), Q(3), …, Q(N)} where Q(1) denotes the smallest AM

flow in the series. Thus, the largest flood in the N years of record is Q(N).

Population quantities Sample estimates

M100 = 1st PWM QM100 = Qmean = sample mean

M110 = 2nd

PWM

N

1i

(i)110 Q1)(N

1)(i

N

1M

M120 = 3rd

PWM

N

1i

(i)120 Q2)1)(N(N

2)1)(i(i

N

1M

M130 = 4th

PWM

(i)

N

1i

130 Q3N

3i

2N

2i

1N

1i

N

1M

1 = M100 = 1st L-moment

2 = 2 M110 – M100 = 2nd

L-moment

3 = 6 M120 – 6 M110 + M100 = 3rd

L-moment

4 = 20 M130 – 30 M120 + 12 M110 – M100 = 4th

L-moment

1

2

λ

λτ = L-CV

2

33

λ

λτ = L-skewness

2

44

λ

λτ = L-kurtosis

Above-listed sample values of

M100, M110, M120 and M130 are

inserted into these expressions to

yield the sample estimates of

L-CV, L-skewness and L-kurtosis.

L-moment

ratios

L-moments

Probability-

weighted moments

(PWMs)


53

Statistics for six stations asterisked in the tables are provided for information only. Stations

25001, 25002, 25003, 25004, 25005 and 25158 play no role in inferences later about the

suitability of particular distributions to describe annual maximum flows. This is because

embankments in the Mulkear catchment contain about four out of five AM flows but are

overtopped in larger floods. This leads to damping of the larger flood magnitudes and

extremely concave downwards probability plots (see Section 5.3). Further details of these

exclusions are given in Appendix B. Box 4.3 refers to a further exceptional station.

Box 4.3: Station 06030 Big at Ballygoly

Table 4.1: Principal summary statistics

Station number and name Grade N Qmean Qmed H-skew CV L-CV L-skew L-kurt

01041 Deele at Sandy Mills B 32 85.08 82.61 0.22 0.29 0.17 0.02 0.13

01055 Mourne Beg at Mourne Beg Weir B 9 2.92 2.70 1.03 0.38 0.23 0.16 0.02

06011 Fane at Moyles Mill A1 48 15.86 15.39 0.81 0.20 0.11 0.09 0.07

06013 Dee at Charleville Weir A1 30 27.81 27.37 0.10 0.27 0.16 0.02 0.01

06014 Glyde at Tallanstown A1 30 22.56 21.46 1.31 0.27 0.15 0.22 0.12

06021 Glyde at Mansfieldstown B 50 21.54 21.50 0.45 0.24 0.13 0.09 0.09

06025 Dee at Burley Bridge A1 30 18.32 18.69 -0.33 0.16 0.09 -0.07 0.19

06026 Glyde at Aclint Bridge A1 46 13.87 12.30 1.05 0.32 0.18 0.24 0.09

06030 Big at Ballygoly [see Box 4.3] B 30 20.58 10.03 3.42 1.61 0.61 0.68 0.48

06031 Flurry at Curralhir A2 18 13.58 11.70 3.12 0.52 0.26 0.39 0.32

06033 White Dee at Coneyburrow Bridge B 25 27.88 18.60 2.49 0.84 0.41 0.46 0.27

06070 Muckno L. at Muckno A1 24 13.32 13.19 0.78 0.25 0.14 0.14 0.14

07006 Moynalty at Fyanstown A2 19 26.73 27.93 -1.09 0.21 0.12 -0.20 0.07

07009 Boyne at Navan Weir A1 29 162.64 134.80 0.93 0.38 0.21 0.21 0.11

07033 Blackwater at Virginia Hatchery A2 25 14.93 14.62 1.74 0.24 0.13 0.16 0.27

08002 Delvin at Naul A1 20 5.62 5.32 1.98 0.21 0.11 0.27 0.17

08003 Broadmeadow at Fieldstown B 18 26.88 22.55 4.21 0.87 0.39 0.36 0.34

08005 Sluice at Kinsaley Hall A2 18 3.04 2.50 1.54 0.68 0.38 0.23 0.19

08007 Broadmeadow at Ashbourne B 15 9.88 8.24 0.89 0.49 0.29 0.18 -0.01

08008 Broadmeadow at Broadmeadow A2 25 44.55 40.90 1.74 0.63 0.34 0.28 0.16

08009 Ward at Balheary A1 11 10.38 6.59 5.43 1.41 0.56 0.68 0.65

08011 Nanny at Duleek Road Bridge B 23 31.00 32.22 -0.93 0.25 0.14 -0.15 0.20

08012 Stream at Ballyboghill B 19 4.21 4.35 -0.54 0.58 0.33 -0.09 0.11

09001 Ryewater at Leixlip A1 48 38.71 35.46 1.17 0.44 0.24 0.19 0.15

09002 Griffeen at Lucan A1 24 7.24 5.40 2.40 0.83 0.42 0.39 0.25

Editorial note: Station 06030 Big at Ballygoly is also marked in the tables because the

AM flood series analysed for this Grade B station is highly questionable. A radically

different rating curve has been applied to the first five years of data, leading to peak flows

that appear unrealistically large. The effect is not enough to perturb the estimate of

QMED greatly. However, the values of variability and skewness quoted in Table 4.1 (and

the maximum recorded flood quoted in Appendix F) are not to be relied on.


54


09010 Dodder at Waldron’s Bridge A1 19 70.15 48.00 3.24 0.86 0.42 0.42 0.30

09035 Cammock at Killeen Road B 9 12.04 11.70 2.75 0.60 0.33 0.33 0.23

10002 Avonmore at Rathdrum B 47 88.19 83.49 2.85 0.43 0.21 0.27 0.31

10021 Shanganagh at Common’s Road A1 24 7.87 7.36 0.95 0.37 0.21 0.19 0.07

10022 Cabinteely at Carrickmines A1 18 3.84 3.85 0.36 0.40 0.23 0.06 0.06

10028 Aughrim at Knocknamohill B 16 56.69 46.95 1.57 0.38 0.21 0.30 0.08

11001 Owenavorragh at Boleany B 33 49.85 47.17 3.03 0.33 0.16 0.24 0.30

12001 Slaney at Scarawalsh A2 50 169.50 157.00 1.59 0.36 0.19 0.18 0.17

12013 Slaney at Rathvilly B 30 45.16 43.55 -0.01 0.27 0.15 0.03 0.12

14005 Barrow at Portarlington A2 48 40.81 38.27 2.01 0.29 0.15 0.29 0.22

14006 Barrow at Pass Bridge A1 51 83.76 80.52 1.46 0.20 0.11 0.24 0.21

14007 Stradbally at Derrybrock A1 25 16.94 16.20 1.12 0.30 0.17 0.22 0.07

14009 Cushina at Cushina A2 25 6.69 6.79 1.42 0.23 0.13 0.14 0.24

14011 Slate at Rathangan A1 26 12.07 12.30 0.19 0.25 0.14 0.02 0.16

14013 Burrin at Ballinacarrig A2 50 16.54 16.05 0.37 0.26 0.15 0.07 0.06

14018 Barrow at Royal Oak A1 51 141.83 147.98 0.22 0.24 0.14 0.04 0.06

14019 Barrow at Levitstown A1 51 103.46 102.41 0.61 0.24 0.14 0.09 0.13

14029 Barrow at Graiguenamanagh A2 47 162.54 160.74 0.18 0.14 0.08 0.06 0.07

14033 Owenass at Mountmellick B 22 22.59 19.50 0.53 0.28 0.16 0.14 -0.07

14034 Barrow at Bestfield A2 14 137.30 125.00 2.23 0.32 0.17 0.33 0.17

15001 Kings at Annamult A2 42 89.39 88.75 0.14 0.28 0.16 0.00 0.08

15002 Nore at John’s Bridge A2 35 211.98 197.00 0.59 0.31 0.18 0.08 0.09

15003 Dinan at Dinan Bridge A2 50 143.58 150.76 -0.78 0.20 0.11 -0.15 0.09

15004 Nore at McMahons Bridge A2 51 38.96 37.28 0.96 0.31 0.17 0.13 0.15

15005 Erkina at Durrow Foot Bridge B 50 28.47 27.44 1.78 0.34 0.18 0.19 0.21

15012 Nore at Ballyragget B 16 77.16 77.11 0.90 0.30 0.17 0.08 0.21

16001 Drish at Athlummon A2 33 15.65 15.66 0.30 0.22 0.12 0.01 0.12

16002 Suir at Beakstown A2 51 55.40 52.66 1.79 0.30 0.16 0.17 0.19

16003 Clodiagh at Rathkennan A2 51 31.17 29.98 0.85 0.18 0.10 0.21 0.05

16004 Suir at Thurles A2 48 22.17 21.37 0.53 0.20 0.11 0.07 0.11

16005 Multeen at Aughnagross A2 30 23.11 21.79 1.33 0.18 0.10 0.20 0.13

16006 Multeen at Ballinclogh Bridge B 33 30.37 27.87 0.34 0.39 0.23 0.06 0.03

16007 Aherlow at Killardry B 51 79.18 75.84 0.46 0.32 0.18 0.09 0.06

16008 Suir at New Bridge A2 51 90.66 92.32 -0.32 0.13 0.07 -0.06 0.05

16009 Suir at Cahir Park A2 52 159.29 162.21 -0.41 0.17 0.10 -0.10 0.05

16011 Suir at Clonmel A1 52 234.52 223.00 0.42 0.30 0.17 0.09 0.05

16012 Tar at Tar Bridge B 36 55.20 54.57 0.39 0.28 0.16 0.06 0.08

16013 Nire at Fourmilewater B 33 101.69 93.21 0.86 0.42 0.24 0.16 0.08


55


16051 Rossestown at Clobanna B 13 2.95 2.85 2.53 0.36 0.19 0.33 0.26

18001 Bride at Mogeely Bridge B 48 71.07 71.49 -0.16 0.19 0.11 -0.03 0.10

18002 Ballyduff at Muns Blackwater B 49 353.65 344.00 0.24 0.16 0.09 0.06 0.10

18003 Blackwater at Killavullen B 49 282.76 266.15 1.10 0.23 0.13 0.16 0.10

18004 Ballynamona at Awbeg A2 46 30.96 31.20 1.25 0.17 0.09 0.04 0.33

18005 Funshion at Downing Bridge A2 50 56.69 53.05 1.63 0.27 0.14 0.22 0.18

18006 Blackwater at CSET Mallow B 27 291.30 286.00 1.10 0.14 0.08 0.19 0.12

18016 Blackwater at Duncannon B 24 80.99 79.65 0.49 0.23 0.14 0.12 0.00

18048 Blackwater at Dromcummer B 23 222.77 220.00 1.11 0.08 0.05 0.15 0.22

18050 Blackwater at Duarrigle B 24 121.96 124.50 0.12 0.20 0.11 -0.01 0.09

19001 Owenboy at Ballea Upper A2 48 15.87 15.42 0.48 0.17 0.09 0.09 0.12

19014 Lee at Dromcarra B 20 79.69 71.89 1.67 0.38 0.21 0.30 0.12

19016 Bride at Ovens Bridge B 8 28.74 29.58 -1.35 0.16 0.09 -0.14 0.22

19020 Owennacurra at Ballyedmond A2 28 24.63 22.40 0.12 0.34 0.20 0.03 0.04

19031 Sullane at Macroom B 9 131.09 135.90 1.46 0.27 0.16 0.16 0.17

19046 Martin at Station Road B 9 31.09 29.95 -0.34 0.26 0.16 -0.04 0.06

20002 Bandon at Curranure B 31 140.60 126.28 2.13 0.37 0.19 0.37 0.28

20006 Argideen at Clonakilty WW B 25 30.25 27.70 1.67 0.30 0.16 0.27 0.18

22006 Flesk at Flesk Bridge B 51 165.89 169.09 0.66 0.25 0.14 0.07 0.12

22009 Dreenagh at White Bridge B 24 11.91 11.47 1.17 0.16 0.09 0.18 0.19

22035 Laune at Laune Bridge B 14 112.81 116.40 -0.86 0.20 0.12 -0.15 0.03

23001 Galey at Inch Bridge A2 45 97.39 99.05 1.22 0.33 0.18 0.13 0.18

23012 Lee at Ballymullen A2 18 16.87 15.66 2.98 0.29 0.15 0.39 0.33

24002 Camogue at Gray’s Bridge A2 27 24.06 23.49 0.32 0.19 0.11 0.06 0.17

24004 Maigue at Bruree B 52 54.86 50.63 0.91 0.39 0.22 0.18 0.10

24008 Maigue at Castleroberts A2 30 120.96 119.13 0.28 0.26 0.15 0.05 0.09

24011 Deel at Deel Bridge B 33 103.01 104.55 0.42 0.22 0.12 -0.02 0.24

24012 Deel at Grange Bridge B 41 110.45 109.99 -0.02 0.16 0.09 0.01 0.13

24022 Mahore at Hospital A2 20 9.83 9.80 1.12 0.41 0.23 0.12 0.16

24030 Deel at Danganbeg B 25 52.89 52.00 0.88 0.15 0.08 0.11 0.10

24082 Maigue at Islandmore A2 28 135.47 140.01 -0.22 0.26 0.15 -0.04 0.13

25001 Mulkear at Annacotty* A2 49 133.95 132.88 -0.19 0.16 0.09 -0.02 0.16

25002 Newport at Barringtons Bridge* A2 51 61.15 62.64 -0.65 0.16 0.09 -0.14 0.06

25003 Mulkear at Abington* A1 51 69.45 68.98 0.05 0.15 0.08 0.00 0.11

25004 Bilboa at Newbridge* B 30 41.70 42.30 -0.16 0.23 0.13 -0.04 0.12

25005 Dead at Sunville* A2 46 28.73 29.63 -0.98 0.11 0.06 -0.19 0.10

25006 Brosna at Ferbane A1 52 86.77 81.91 0.71 0.25 0.14 0.14 0.18

25011 Brosna at Moystown B 51 85.64 82.02 1.31 0.34 0.18 0.14 0.23


56


25014 Silver at Millbrook Bridge A1 54 17.67 17.25 0.57 0.23 0.13 0.10 0.13

25016 Clodiagh at Rahan A2 42 23.04 22.57 0.62 0.22 0.13 0.08 0.16

25017 Shannon at Banagher A1 55 413.25 407.68 0.18 0.20 0.12 0.04 0.08

25020 Killimor at Killeen B 35 46.60 43.65 0.93 0.33 0.19 0.16 0.11

25021 Little Brosna at Croghan A2 44 28.03 28.58 -0.13 0.14 0.08 -0.03 0.07

25023 Little Brosna at Milltown A1 52 12.14 11.22 0.57 0.29 0.16 0.14 0.06

25025 Ballyfinboy at Ballyhooney A1 31 10.15 10.18 0.57 0.29 0.17 0.08 0.15

25027 Ollatrim at Gourdeen Bridge A1 43 23.32 22.10 0.27 0.28 0.16 0.05 0.13

25029 Nenagh at Clarianna A2 33 54.12 56.48 -0.09 0.24 0.14 -0.02 -0.02

25030 Graney at Scarriff Bridge A1 48 43.80 40.64 1.01 0.32 0.18 0.18 0.13

25034 L. Ennell Trib at Rochfort A2 24 1.50 1.48 -0.48 0.29 0.17 -0.08 0.12

25038 Nenagh at Tyone B 17 42.08 39.30 1.30 0.27 0.15 0.17 0.22

25040 Bunow at Roscrea A2 20 3.78 3.59 1.32 0.27 0.15 0.20 0.18

25044 Kilmastulla at Coole A2 33 25.38 22.70 1.23 0.34 0.19 0.25 0.15

25124 Brosna at Ballynagore A2 18 12.79 13.65 -0.31 0.36 0.20 -0.07 0.23

25158 Bilboa at Cappamore* A1 18 47.66 43.88 0.17 0.29 0.17 0.05 0.13

26002 Suck at Rookwood A2 53 56.98 56.56 2.29 0.22 0.11 0.22 0.29

26005 Suck at Derrycahill A2 51 92.80 93.21 0.27 0.18 0.10 0.03 0.14

26006 Suck at Willsbrook A1 53 26.57 24.23 3.87 0.37 0.15 0.40 0.41

26007 Suck at Bellagill Bridge A1 53 91.75 88.15 1.05 0.19 0.11 0.16 0.16

26008 Rinn at Johnston’s Bridge A1 49 23.68 22.94 1.49 0.19 0.10 0.19 0.19

26009 Black at Bellantra Bridge A2 35 13.66 13.22 0.93 0.16 0.09 0.20 0.11

26010 Cloone at Riverstown B 35 20.03 17.17 1.64 0.41 0.22 0.35 0.18

26014 Lung at Banada Bridge B 16 44.10 42.82 1.80 0.23 0.12 0.19 0.26

26018 Owenure at Bellavahan A2 49 9.19 8.95 0.77 0.20 0.11 0.13 0.11

26019 Camlin at Mullagh A1 51 22.34 21.18 1.17 0.25 0.14 0.23 0.12

26020 Camlin at Argar Bridge A1 32 11.21 11.27 0.16 0.19 0.11 0.03 0.07

26021 Inny at Ballymahon A2 30 65.88 66.34 -0.86 0.25 0.14 -0.11 0.19

26022 Fallan at Kilmore A2 33 6.64 6.49 0.47 0.30 0.17 0.09 0.05

26058 Inny Upper at Ballinrink Bridge B 24 5.98 5.35 1.60 0.38 0.21 0.28 0.18

26059 Inny at Finnea Bridge A1 17 12.98 12.20 0.31 0.18 0.10 0.11 0.17

26108 Owenure at Boyle Abbey Bridge B 15 56.29 57.32 0.28 0.18 0.11 0.06 -0.06

27001 Claureen at Inch Bridge A2 30 20.65 20.10 1.55 0.20 0.11 0.19 0.25

27002 Fergus at Ballycorey A1 51 34.22 32.60 1.37 0.23 0.12 0.18 0.22

27003 Fergus at Corofin A2 48 24.01 22.92 0.71 0.24 0.13 0.09 0.22

28001 Inagh at Ennistimon B 17 52.69 47.58 4.89 0.40 0.16 0.47 0.61

29001 Raford at Rathgorgin A1 40 14.17 13.46 0.34 0.19 0.11 0.09 0.10

29004 Clarinbridge at Clarinbridge A2 32 11.39 11.30 0.77 0.15 0.08 0.14 0.08


57


29007 L. Cullaun at Craughwell B 22 27.83 26.49 1.03 0.22 0.12 0.15 0.20

29011 Dunkellin at Kilcolgan Bridge A1 22 31.94 28.89 3.37 0.30 0.14 0.41 0.32

29071 L. Cutra at Cutra A2 26 16.00 15.70 0.74 0.23 0.13 0.11 0.19

30007 Clare at Ballygaddy A2 31 61.93 62.98 1.30 0.20 0.11 0.11 0.20

30012 Clare at Claregalway B 9 126.89 126.00 1.00 0.12 0.07 0.13 0.15

30021 Robe at Christina’s Bridge B 26 28.17 27.20 2.17 0.34 0.18 0.21 0.29

30031 Cong at Cong Weir B 24 94.35 93.88 -0.62 0.18 0.10 -0.02 0.13

30037 Robe at Clooncormick B 21 1.80 1.79 -0.42 0.37 0.21 -0.09 0.18

30061 Corrib Estuary at Wolfe Tone Bridge A2 33 274.97 247.97 3.04 0.32 0.15 0.41 0.38

31002 Cashla at Cashla A1 26 12.89 12.16 1.92 0.24 0.13 0.31 0.18

31072 Cong at Cong Weir B 26 49.08 43.20 2.04 0.38 0.20 0.31 0.20

32011 Bunowen at Louisberg Weir B 26 74.88 64.87 0.75 0.30 0.17 0.17 0.06

32012 Newport at Newport Weir A2 24 30.06 29.85 -0.12 0.12 0.07 0.00 0.18

33001 Glenamoy at Glenamoy B 25 62.11 59.30 1.65 0.28 0.15 0.20 0.19

33070 Carrowmore L. at Carrowmore A1 28 7.90 7.67 1.19 0.16 0.09 0.10 0.18

34001 Moy at Rahans A2 36 174.76 174.61 1.08 0.19 0.10 0.08 0.21

34003 Moy at Foxford A2 29 180.42 178.00 1.26 0.17 0.09 0.11 0.25

34007 Deel at Ballycarroon B 53 90.37 84.48 0.96 0.36 0.20 0.14 0.10

34009 Owengarve at Curraghbonaun A2 33 28.37 27.48 0.43 0.17 0.10 0.08 0.16

34010 Moy at Cloonacannana B 12 123.29 113.72 1.31 0.30 0.17 0.21 0.11

34011 Manulla at Gneeve Bridge A2 30 18.80 18.73 0.78 0.16 0.09 0.10 0.23

34018 Castlebar at Turlough A1 27 11.50 11.28 0.93 0.20 0.11 0.18 0.03

34024 Pollagh at Kiltimagh A2 28 20.70 20.80 -0.39 0.12 0.07 -0.05 0.14

35001 Owenmore at Ballynacarrow A2 29 30.52 31.16 0.10 0.21 0.12 -0.01 0.19

35002 Owenbeg at Billa Bridge A2 34 51.78 50.48 0.07 0.17 0.10 0.03 0.08

35005 Ballysadare at Ballysadare A2 55 77.78 75.42 1.04 0.26 0.14 0.20 0.14

35011 Bonet at Dromahair B 36 116.02 115.36 0.04 0.30 0.17 0.01 0.08

35071 L. Melvin at Lareen A2 30 26.95 26.29 0.76 0.18 0.10 0.12 0.20

35073 L. Gill at Lough Gill A2 30 54.81 54.05 0.34 0.22 0.12 0.08 0.14

36010 Annalee at Butlers Bridge A1 50 66.56 66.80 1.05 0.22 0.12 0.15 0.22

36011 Erne at Bellahillan B 49 17.91 18.23 -0.51 0.18 0.10 -0.10 0.11

36012 Erne at Sallaghan A1 47 14.22 14.12 0.21 0.22 0.13 0.03 0.10

36015 Finn at Anlore A1 33 23.14 22.08 2.62 0.32 0.16 0.32 0.30

36018 Dronmore at Ashfield Bridge A1 50 15.84 16.25 0.43 0.18 0.10 0.04 0.10

36019 Erne at Belturbet A2 47 89.60 89.95 -0.16 0.18 0.10 -0.03 0.06

36021 Yellow at Kiltybarden A2 27 24.96 23.37 1.85 0.22 0.12 0.20 0.22

36031 Cavan at Lisdarn A2 30 6.85 6.45 4.19 0.22 0.10 0.37 0.39

36071 L. Scur at Gowly B 20 6.36 6.49 -0.06 0.15 0.09 -0.04 0.01


58


38001 Owenea at Clonconwal B 33 70.02 70.63 1.78 0.16 0.09 0.08 0.26

39001 New Mills at Swilly B 30 44.88 44.25 0.06 0.20 0.12 0.01 0.08

39008 Leannan at Gartan Bridge A2 33 28.34 28.18 0.73 0.26 0.15 0.13 0.11

39009 Fern O/L at Aghawoney A2 33 45.91 45.72 1.03 0.26 0.14 0.16 0.15

4.2.2 Variability and skewness of Irish flood data

Table 4.2 reports average values of the summary statistics describing variability and

skewness. Inclusion of both measures based on moments (i.e. CV and Hazen skewness) and

measures based on L-moments (i.e. L-CV and L-skewness) facilitates comparisons with other

studies. Values of Qmax ∕ Qmean are of interest because they convey general information about

the flood regime of a region that is additional to the measures of variability and skewness.

[Editorial note: The averages presented in Table 4.2 have been calculated as arithmetic

means. Given that each of the summary statistics is defined as a ratio, use of a geometric

mean would have been more appropriate. The numbers of stations shown in each grade

reflect the six excluded stations in the Mulkear basin (see Section 4.2.1 and Appendix B).]

Table 4.2: Average values of some statistics for gauging stations (by station grade)

Station

grade

#

stations

Qmed ∕

Qmean CV

L-

CV

H-

skew

L-

skew

L-

kurt

Qmax ∕

Qmean

Qmax ∕

AREA

A1 43 0.943 0.311 0.165 1.153 0.172 0.158 1.852 0.278

A2 67 0.975 0.249 0.137 0.838 0.106 0.160 1.651 0.307

A1+A2 110 0.963 0.273 0.148 0.961 0.132 0.159 1.730 0.296

B 65 0.947 0.325 0.174 1.024 0.140 0.155 1.839 0.787

A1+A2+B 175 0.957 0.293 0.157 0.984 0.135 0.158 1.770 0.473

Here, attention is chiefly paid to the results for the Grade A1 and A2 stations. The average

CV of 0.273 is identical to that reported for Ireland in Table I.2.3 of the FSR (NERC, 1975)

based on 63 stations with records of 15 years or more. The average Hazen skewness of 0.963

is larger than the average weighted skewness of 0.662 reported in the FSR table, though the

FSR value is without Hazen’s correction. [Editorial note: For reasons discussed in Section

I.2.3.3 of the FSR, derivation of a pooled value of skewness warrants greater sophistication

than an arithmetic mean of individual values of skewness: regardless of whether the Hazen

correction is applied.]

The average value of Qmed/Qmean for the A1 + A2 stations is 0.963. The expected ratio for an

EV1 distribution with CV = 0.273 is 0.955.

In comparison to many countries, Ireland has a flood hydrology regime that can be

characterised as “low CV and low skewness”. This is typical of very humid conditions where

between-year variation in flood magnitudes is relatively small.


59

4.2.3 Comparisons of CV with L-CV and of H-skew with L-skew

The distributions of CV and L-CV are summarised in Histograms 4.1, and those of Hazen

skewness and L-skewness in Histograms 4.2. The distribution of AM values is positively

skewed at most sites (i.e. H-skew > 0 and L-skew > 0). This is particularly the case for the

Grade A1 stations, which are considered the most reliable at measuring across the full range

of flows.

30

20

10

0

30

20

10

0

1.61.41.21.00.80.60.40.20.0

30

20

10

0

CV of 43 A1 stations

Fre

quency CV of 67 A2 stations

CV of 110 A1 + A2 stations

30

20

10

0

30

20

10

0

0.60.50.40.30.20.10.0

30

20

10

0

L-CV of 43 A1 stations

Fre

quency

L-CV of 67 A2 stations

L-CV of 110 A1 + A2 stations

Histograms 4.1: CV and L-CV at Grade A1 and A2 stations

20

10

0

20

10

0

6420-2

20

10

0

H-skew of 43 A1 stations

Fre

quency

H-skew of 67 A2 stations

H-skew of 110 A1 + A2 stations

20

10

0

20

10

0

0.80.60.40.20.0-0.2-0.4

20

10

0

L-skew of 43 A1 stations

Fre

quen

cy

L-skew of 67 A2 stations

L-skew of 110 A1 + A2 stations

Histograms 4.2: Hazen skewness and L-skewness at Grade A1 and A2 stations


60

Figure 4.1 shows the relationship between L-CV and CV, and Figure 4.2 shows that between

L-skewness and Hazen skewness.

1.61.41.21.00.80.60.40.20.0

0.8

0.6

0.4

0.2

0.0

CV

L-C

V

Forcing intercept to zero yields: L-CV = 0.515 CV

Regression of y on x yields: L-CV = 0.441 CV + 0.027

Figure 4.1: Relationship between L-CV and CV for 110 A1 + A2 stations

6420-2

0.8

0.6

0.4

0.2

0.0

-0.2

Hazen skewness

L-s

kew

ness

Regression of y on x yields: L-skew = 0.122 H-skew + 0.014

Forcing intercept to zero yields: L-skew = 0.129 H-skew

Figure 4.2: Relationship between L-skew and H-skew for 110 A1 + A2 stations

While CV and skewness vary between gauging stations some of this variation is due to

sampling variability rather than true differences. [Editorial note: Das and Cunnane (2012)

explore the sensitivity of L-CV and L-skewness to record length. Exploring the same group

of 110 stations, they note that the largest values are typically associated with the shortest

records. Further work might explore whether the higher values are artefacts of the limited

period of record or a product of the nature of those catchments with shorter records.]


61

From Table 4.2 it can be seen that – for the 110 A1 and A2 stations – the ratio L-CV / CV =

0.148/0.273 = 0.542 and the ratio L-skew / H-skew = 0.132/0.961 = 0.137. These compare

reasonably well with the corresponding theoretical values of 0.540 and 0.149 for an EV1

distribution. [Editorial note: The first ratio is √6 ℓn2 / π. The second ratio is evaluated as

L-skewness/skewness = 0.1699/1.139 = 0.149.]

4.3 Geographical traits

Map 4.1 shows the pattern of specific QMED values, i.e. QMED/AREA. Unsurprisingly,

many of the higher values lie in the wetter regions (e.g. the North West and the South West).

Map 4.1: Specific QMED (m3s

-1/km

2) for 176 A1 + A2 + B stations

Legend

Sp.Qmed

!( 0.006829 - 0.151544

!( 0.151545 - 0.270321

!( 0.270322 - 0.418750

!( 0.418751 - 0.812329

!( 0.812330 - 1.823686

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[Symbols placed at catchment outlets]

Specific

QMED

0.007 to 0.152

0.152 to 0.270

0.270 to 0.419

0.419 to 0.812

0.812 to 1.824


62

The patterns of CV and Hazen skewness are illustrated in Map 4.2 and Map 4.3. It is

noticeable that many of the stations around Dublin have high variability in annual maximum

floods. The geographical pattern of Hazen skewness (see Map 4.3) is less clear. There may

be some weak tendency for higher values in the North Midlands and at stations nearer the

coast.

[Editorial note: Geographical traits are also discussed by Ahilan et al., 2012.]

Map 4.2: CV at 110 A1 + A2 stations

´Legend

CV

!( 0.110000 - 0.190000

!( 0.190001 - 0.270000

!( 0.270001 - 0.440000

!( 0.440001 - 0.860000

!( 0.860001 - 1.410000

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901090029001

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6070 6031

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35071

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26059

26022

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1002210021

CV

0.11 to 0.19

0.19 to 0.27

0.27 to 0.44

0.44 to 0.86

0.86 to 1.41



63

Map 4.3: Hazen skewness at 110 A1 + A2 stations

4.4 Preliminary distribution choice from skewness v. record length plot

Another way of viewing skewness values is explored in Figure 4.3 and Figure 4.4, which

respectively show skewness and L-skewness plotted against length of record. In the interest

of model parsimony, it is desirable to consider the adequacy of 2-parameter models such as

the Normal and EV1 distributions. The confidence intervals marked here are based on the

assumption of (a) the Normal distribution and (b) the EV1 distribution.

´Legend

H_SKEW

!( -1.090000 - -0.090000

!( -0.089999 - 0.620000

!( 0.620001 - 1.460000

!( 1.460001 - 2.620000

!( 2.620001 - 5.430000

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- 1.09 to - 0.09

- 0.09 to 0.62

0.62 to 1.46

1.46 to 2.62

2.62 to 5.43

Hazen

skewness



64

The Normal distribution confidence intervals for skewness are calculated using the standard

expression for samples of size N:

3)2)(N1)(N(N

1)6N(NskewnessSE

4.1

Other confidence intervals were obtained by simulation.

Figure 4.3: Skewness at 110 A1 + A2 stations versus record length

with confidence intervals for (a) Normal samples and (b) EV1 samples

Skewness vs RL (Normal Dist.)

-2

-1

0

1

2

3

4

0 10 20 30 40 50 60

Record Length

Skew

ness

Observed Theoretical Skew. Value for N.D. 95% C.I. 67% C.I.

Skewness vs RL (EV1)

-2

-1

0

1

2

3

4

0 10 20 30 40 50 60Record_Length

Skew

ness

Observed Theoretical Skew. value for EV1 67% C.I. 95% C.I.

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

L-S

kew

ness

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60L-S

kew

ness


-2

-1

0

1

2

3

4

0 10 20 30 40 50 60

Record Length

Skew

ness



-2

-1

0

1

2

3

4

0 10 20 30 40 50 60Record_Length

Skew

ness


-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

L-S

kew

ness

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

L-S

kew

ness


-2

-1

0

1

2

3

4

0 10 20 30 40 50 60

Record Length

Skew

ness



-2

-1

0

1

2

3

4

0 10 20 30 40 50 60Record_Length

Skew

ness


-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

L-S

kew

ness

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

L-S

kew

ness


-2

-1

0

1

2

3

4

0 10 20 30 40 50 60

Record Length

Skew

ness



-2

-1

0

1

2

3

4

0 10 20 30 40 50 60Record_Length

Skew

ness


-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

L-S

kew

ness

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

L-S

kew

ness

Sk

ewnes

s

Record length (years)

(a) Samples from Normal distribution

Skew

nes

s


(b) Samples from EV1 distribution


65

Figure 4.4: L-skewness against record length for 110 A1 + A2 gauging station

with confidence intervals for (a) Normal samples and (b) EV1 samples

Because of the relatively low values of observed skewness at many gauging stations, it is

appropriate to ask whether the data as a whole could be considered to have come from a

Normal distribution. Looking at the upper end of the skewness range in Figure 4.3a, it is seen

that 46 stations fall above the upper confidence interval for the Normal distribution. If the

data as a whole were Normal then only 2.5% of the 110 values (say three values) would be

expected to lie above the upper 95% confidence interval. Thus the Normal hypothesis cannot

be accepted.

L-Skewness vs RL (Normal Dist.)

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60Record_Length

L-S

kew

ness

Observed Theoretical L-Skew.value for N.D. 95% C.I. 67% C.I.

L-Skewness vs RL (EV1)

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

Record_Length

L-S

kew

ness

Observed Theoretical L-Skew value for EV1 67% C.I. 95% C.I.


-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60Record_Length

L-S

kew

ness



-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

Record_Length

L-S

kew

ness



-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60Record_Length

L-S

kew

ness



-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

Record_Length

L-S

kew

ness



-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60Record_Length

L-S

kew

ness



-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

Record_Length

L-S

kew

ness


L-s

kew

nes

s


(a) Samples from Normal distribution

L-s

kew

nes

s


(b) Samples from EV1 distribution


66

Looking at the lower end of the skewness range in Figure 4.3b, it is seen that 14 stations fall

below the lower confidence interval for the EV1 distribution. Although less divergent (than

for the Normal hypothesis in Figure 4.3a), the number is still many times too large (i.e.

14 >>3) for the hypothesis that all the samples come from the EV1 family to be accepted as a

whole.

The preliminary conclusion on the basis of skewness values is that the stations with the larger

values of skewness might be modelled by an EV1 distribution but with the reservation that

there is a serious number (19 out of 110 stations, or 17%) which are not in keeping with this

choice. Thus the choice of the EV1 distribution to represent all stations in Ireland is of

doubtful validity.

Skewness in the sample sizes typically found in hydrology is not renowned for its precision.

L-skewness offers some advantages. However, very similar findings are obtained from the

plots of L-skewness against record length shown above in Figure 4.4.

An alternative assessment approach is to use L-moment ratio diagrams.

4.5 Preliminary distribution choice aided by L-moment ratio diagrams

4.5.1 L-moment ratio diagrams

Another way to explore the suitability of different probability distributions is to use

L-moment ratio diagrams. These provide a useful diagnostic tool (Hosking and Wallis, 1997)

though cannot be regarded as entirely reliable in the context of the sample sizes typically

available. Diagrams based on conventional moment ratios are also possible but are regarded

as having weaker discriminating capability (see Appendix 3 of Cunnane, 1989).

The most useful L-moment ratio diagram is a graph of L-kurtosis against L-skewness.

Usually a 2-parameter distribution with a location and a scale parameter plots as a single

point on the diagram. A 3-parameter distribution with location, scale and shape parameters

typically appears as a line or curve. The distribution selection process begins with a scatter-

plot of the sample L-moment ratios. These are compared with the points/curves that

represent the theoretical L-moment ratios for the candidate distributions. Figure 4.5 shows

this for the AM flows at the 110 A1 + A2 sites.

4.5.2 Implication for choice of 2-parameter distribution

In the context of the 2-parameter distributions alone, the sample data for the 110 stations are

on average appreciably closer to the population L-moment ratios of an EV1 distribution (●)

than to those of the LO (♦) or LN (■) distribution. Here, LO denotes the Logistic distribution

and LN denotes the lognormal distribution.

Figure 4.6 shows sample L-moments of data simulated from EV1, LO and LN distributions:

each with the same record lengths and parameters as the actual records. In the case of the

EV1 simulated data, the scatter of points is narrower than in the observed data, especially on

the LH side. Many more of the observed values of L-skewness are negative than is the case

for the simulated data. The LO and LN simulated L-moments cover the range of observed

values in the negative L-skewness domain more adequately than do the EV1 simulated data.


67

However, the LO and LN simulated L-moments cover the range of observed values less well

in the positive L-skewness domain, where the EV1 simulated data perform better. Further

realisations of the experiment were made, for which the same general patterns were noted.

Figure 4.5: L-moment ratio diagram for annual maximum floods

Figure 4.6: Comparison with equivalent samples drawn from particular distributions

L-Moment Ratio( Observed data)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis

L-Moment Ratio (EV1 simulated data)

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis

L-Moment Ratio( LN simulated data)

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis

L-Moment Ratio (LO simulated data)

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis


-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7L-Skewness

L-K

urt

osis

L-Moment Ratio Diagram for 110 A1 & A2 Stations

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70L-Skewness

L-K

urt

osis

sample_data

sample_avg

EV1

LO

LN

GEV

LN3

GLO

Normal

110 A1 + A2 stations

L

-kurt

osi

s

L-skewness

L-k

urt

osi

s

L-skewness

(a) Observed data (b) EV1 simulated data

(c) LN simulated data (d) LO simulated data

Station 08009


68

On the basis of average values, EV1 looks a more suitable candidate than LO or LN.

However, the occurrence of so many negative values of L-skewness among the observed data

throws doubt on the suitability of EV1 as a universal 2-parameter model for Ireland.

[Editorial note: It is understood that Station 08009 Ward at Balheary was omitted from

these simulations (see Box 4.4 below).]

4.5.3 Implication for choice of 3-parameter distribution

From Figure 4.5 it is seen that the average of the 110 data points falls roughly half way

between the GEV and GLO curves. GLO denotes the Generalised Logistic distribution. The

preliminary conclusion from the L-moment ratio diagrams is that, in addition to the

2-parameter EV1 distribution, the 3-parameter GEV and GLO distributions are possible

candidates to describe Irish flood data.

4.6 Exceptional recorded values and outliers

The largest flow values, Qmax, in each record and the dimensionless value Qmax/Qmean are

amongst the additional summary statistics presented in Appendix F.

4.6.1 Qmax/Qmean ratios

The largest Qmax/Qmean among the 45 Grade A1 stations is 5.17 at Station 08009 Ward at

Balheary (see Box 4.4). There are only two other stations where this statistic exceeds 3.0,

namely: Station 09002 Griffeen at Lucan and Station 09010 Dodder at Waldron’s Bridge.

Box 4.4: Station 08009 Ward at Balheary

The large AM value at Waldron’s Bridge occurred in August 1986 as a result of Hurricane

Charlie. Of all the exceptional flows which occurred in South Dublin and Wicklow at that

time, it is the only flow to have been measured. Even allowing for the difficulties of

measurement, and extrapolation of the rating curve, it is truly a very large flood in the Irish

context.

At the 42 remaining Grade A1 stations, and at all 70 Grade A2 stations, the Qmax/Qmean ratio

is less than 3.0. These values are extremely low by European standards. It should be noted

that these quantities depend on record length and not all records in the dataset are of equal

length, varying from 15 to 55 years.

The Station 08009 Ward at Balheary record is relatively short and there are several years

with incomplete data. Regardless of whether the record is short or long, the 1992/93

annual maximum of 53.6 m3s

-1 on 12 June 1993 – even allowing for difficulties of

measurement and extrapolation of the rating curve – is a large outlier. As a precaution,

the station’s data were not included in the inferences and summaries of preceding sections.

[Editorial notes: Section 10.4.5 uses the AM flow series at this station to illustrate the

difficulties that a high outlier presents to the analyst. The series is also discussed by

Bhattarai and Baigent (2009) who assign a much lower peak flow to the 12 June 1993

event and suggest that the unusually small gauged QMED at this station may be because

other large floods are missing from the record. Classification of a station as Grade A1 is

not always a guarantee that flow records are acceptably complete.]


69

4.6.2 High outliers

An outlier is an observation that lies an abnormal distance from other values in a random

sample from a population. In other words, an outlier lies significantly away from the general

range or pattern of the remaining data. The presence of outliers can create problems: both for

the summary statistics and in the interpretation of probability plots (see Chapter 5).

Most interest typically centres on high outliers, i.e. unusually large AM values. The 17

stations with the most obvious high outliers have been selected from the probability plots and

are listed in Table 4.3, where the large values are compared with Qmed rather than with Qmean.

Qmed is the index flood adopted in the FSU. [Editorial note: It is written QMED elsewhere

in the volume so that subscripts can be attached to it with greater clarity.]

It is seen from Table 4.3 that only three stations have a Qmax/Qmed ratio in excess of 3.0 and

only 12 stations have values in excess of 2.0. These numbers are low by comparison with

UK data, for instance.

Table 4.3: High outliers amongst 110 stations graded A1 + A2

Station

number

Station

grade

#

years

Largest

AM flow

[Qmax]

2nd

largest

[Qmax2]

3rd

largest

[Qmax3]

Qmed Qmax/

Qmed

Qmax/

Qmax2

Qmax2/

Qmax3

08009 A1 11 53.6 11.8 10.3 6.59 8.13 4.54 1.15

09010 A1 19 269.__ 156.__ 112.__ 48. 5.60 1.72 1.39

06031 A2 18 35.8 24.4 18.1 11.7 3.06 1.47 1.35

26006 A1 53 70.06 68.66 41.81 29.23 2.40 1.02 1.64

16002 A2 51 123.88 84.42 81.86 52.66 2.35 1.47 1.03

29011 A1 22 66.52 44.34 40.61 28.89 2.30 1.50 1.09

36015 A1 33 49.99 43.30 31.77 22.08 2.26 1.15 1.36

36031 A2 30 13.70 8.87 8.32 6.45 2.12 1.54 1.07

14005 A2 48 80.42 80.42 59.99 38.27 2.10 1.00 1.34

24022 A2 20 20.5 15.0 14.2 9.79 2.09 1.37 1.06

23012 A2 18 31.74 26.27 19.12 15.66 2.03 1.21 1.37

15004 A2 51 74.96 73.61 56.88 37.28 2.01 1.02 1.29

36021 A2 27 43.57 32.16 31.79 23.37 1.86 1.35 1.01

07033 A2 25 26.58 20.56 18.71 14.62 1.82 1.29 1.10

06011 A1 48 26.36 21.07 19.99 15.39 1.71 1.25 1.05

34001 A2 36 286.56 224.44 219.12 174.61 1.64 1.28 1.02

33070 A1 28 11.97 9.44 9.35 7.67 1.56 1.27 1.01


70

4.6.3 Low outliers

A low outlier is an AM value that is unusually small. This may arise from data error (e.g.

where the annual maximum derives from only a few months of gauged record) or a chance

absence of any high flow in that year (e.g. in 1976 in some permeable UK catchments). Low

outliers can influence skewness values unduly, leading to unsuitable model fits. The remedy

is to pay close attention to the probability plot (see Chapter 5).

Table 4.4 lists two stations with low outliers. Although not remarkably low, such outliers

may influence distributions fitted to the AM series. This highlights the importance of always

inspecting probability plots.

[Editorial note: Another series with a low outlier appears in EV1 Probability Plot 10.7 for

Station 26008 Rinn at Johnston's Bridge.]

Table 4.4: Low outliers amongst 110 stations graded A1 + A2

Station

number

Station

grade

#

years

Smallest

AM flow

[Qmin]

2nd

smallest

[Qmin2]

3rd

smallest

[Qmin3]

Qmed Qmed/

Qmin

Qmin2/

Qmin

Qmin3/

Qmin2

15003 A2 50 61.13 96.72 98.02 150.76 2.47 1.58 1.01

26059 A1 17 8.23 10.7_ 11.4_ 12.2 1.48 1.30 1.07


71

5 Probability plots … and inferences from them

5.1 Introduction

5.1.1 The idea of a probability plot

Probability plots are useful for the display and analysis of flood data, particularly to

determine whether or not a given sample is consistent with a particular population

distribution. The plots employ an inverse distribution scale so that a simple (usually

2-parameter) cumulative distribution function (CDF) plots as a straight line. On this scale,

points (xi, yi) for i = 1, 2, ..., n are expected to lie close to the line y = a + b x, where a and b

are respectively the location and scale parameters of the distribution. Conversely, strong

deviation from the line is evidence that the distribution did not produce the data.

What constitutes “strong deviation” from such a line often has to be judged subjectively.

Nevertheless, probability plots are widely used in flood hydrology for data display and

exploratory analysis. Experienced analysts use them to inform – rather than determine – the

final choice of distribution.

5.1.2 Synthetic examples

EV1 Probability Plot 5.1 provides a small selection of probability plots of samples of size 25

drawn randomly from an EV1 population. EV1 Probability Plot 5.2 does the same for

samples of size 50. In a proportion of cases, the plots reveal departures from a linear pattern

that might – on the basis of visual inspection – lead the analyst to reject the EV1 hypothesis.

This illustrates that probability plots are no guarantee of taking a good decision.

For real samples of annual maximum flood data of size 25 or 50, we do not know the

population from which they are drawn. Hence, inferences from probability plots have to be

considered carefully.

5.1.3 Standard plots for the FSU flood peak data

Three versions of probability plot were considered based on the Gumbel (EV1), the

2-parameter Logistic (LO) and the 2-parameter lognormal (LN) distributions. Plots were

drawn and studied for the AM flows of 186 Irish stations. The flood series range in length

from eight to 55 years and the associated catchment areas range from 10.1 to 2780 km2.

Appendix G shows a sample collection of EV1, LO and LN probability plots and summary

information for Station 14018 Barrow at Royal Oak. Further examples of EV1 probability

plots appear throughout the volume, most notably in Section 10.4 and Appendix J.


72

EV1 Probability Plot 5.1: Nine random samples of size 25 drawn from EV1

0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

350

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

350

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

350

-2 -1 0 1 2 3 4 5

EV1 y

Q

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue

EV1 reduced variate, y



EV1 reduced variate, y EV1 reduced variate, y



Random samples of size 25 drawn from

an EV1 population with

mean = 100 and SD = 30

(i.e. CV = 0.333)


73

EV1 Probability Plot 5.2: Nine random samples of size 50 drawn from EV1

0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

350

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

350

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

350

400

450

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5

EV1 y

Q

0

50

100

150

200

250

300

350

-2 -1 0 1 2 3 4 5

EV1 y

Q

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue

Val

ue







Random samples of size 50 drawn from

an EV1 population with

mean = 100 and SD = 30

(i.e. CV = 0.333)


74

5.1.4 Approach taken to assessing plots

It is acknowledged that inferences drawn from probability plots have low statistical power,

i.e. there is a high probability that a conclusion could be adopted even though it is not true.

Nevertheless probability plots have played a prominent role in hydrological frequency

analysis in the past and it is considered worthwhile to use them in this study as a way of

viewing the available data. Plotted data for the 110 A1 + A2 stations were therefore

investigated from the standpoint of both linearity (Section 5.2) and curvature (Section 5.3).In

the linearity investigation, a straight line is fitted to the plotted data by least-squares

regression. The perceived quality/adequacy of fit is scored by personal (visual) judgement,

with 1 denoting a very poor fit and 5 denoting a very good fit.

In the curvature investigation, an assessment of the curve pattern of the plotted data is made

by personal (visual) judgement and the pattern classified.

5.2 Linear patterns

For each (A1 and A2) station in turn, and for each type of plot (EV1, LO and LN), the

probability plot was examined from the point of view of linearity and subjectively scored

from 1 to 5 by visual judgement, where:

1 = very poor (fit)

2 = poor

3 = moderate

4 = good

5 = very good (fit)

The linear pattern scores for the 110 A1 + A2 stations are given in Appendix H and

summarised in Table 5.1.

Table 5.1: Linear pattern statistics for 110 A1 + A2 stations

A1 stations A2 stations A1 + A2 stations

Type of plot EV1 LO LN EV1 LO LN EV1 LO LN

Modal Score 4 2 3 4 3 4 4 3 4

Mean Score 3.3 2.6 3.3 3.4 3.1 3.4 3.3 2.9 3.3

Table 5.1 reveals that EV1 and LN give almost identical results and that either provides a

higher score (i.e. better fit) than by LO. It can be deduced from the linear patterns that –

amongst 2-parameter distributions – Irish flood data are more likely to be distributed as EV1

or LN than as LO.

5.3 Curve patterns

5.3.1 Assignment of patterns

For each (A1 and A2) station in turn, and for each type of plot (EV1, LO and LN), the

probability plot was judged from the point of view of curvature and a curve pattern assigned.


75

The assessment scheme was based on four curve patterns: linear, concave, convex and

S-curve (i.e. an elongated S shape) coded as:

L straight line

U concave upwards

D concave downwards (i.e. convex upwards)

S S-curve

To allow greater discrimination, each pattern is subdivided into 1 = mild and 2 = severe

categories. Finally, a further classification X (for extremes) is added if the patterns of the end

points of the plot differ from the pattern of the main body of the data. In summary, the

15 curve patterns allowed are:

L1 perfectly straight line

L2 little deviation from straight line

L2X body pattern is quite straight but with end disturbance

U1 mild concave upwards

U2 severe concave upwards

U1X mild concave but with end disturbance

U2X severe concave but with end disturbance

D1 mild convex upwards

D2 severe convex upwards

D1X mild convex upwards but with end disturbance

D2X severe convex upwards but with end disturbance

S1 mild S-curve

S2 severe S-curve

S1X mild S-curve but with end disturbance

S2X severe S-curve but with end disturbance

The curve patterns for the 110 A1 + A2 stations are presented in Appendix H.

5.3.2 Probability plots for 43 Grade A1 stations

The curve patterns of the probability plots for the 43 Grade A1 stations are summarised in

Figure 5.1. The pie charts indicate the relative frequency of the different curve patterns. A

different pie is shown for each of the three probability plots considered: EV1, LO and LN.

Just over half of the EV1 plots in Figure 5.1a show a linear trend (Prefix L) and eight a

convex pattern (Prefix D). Only six stations exhibit a concave pattern (Prefix U). The

scarcity of concave patterns reflects that in many cases the largest floods on record are close

together in magnitude. In very few cases are there outstandingly large floods in the series.

The influence of the choice of probability plot on visual assessment is seen in the contrasting

judgements for the Logistic (LO) plots in Figure 5.1b. Concave patterns (Prefix U) are the

most prevalent. Many linear and S-curve patterns are also found but hardly any convex

(Prefix D) patterns.

The curve patterns for the lognormal (LN) probability plots are summarised in Figure 5.1c.

These show some similarity to those for the EV1 plots. A linear trend (Prefix L) is again the


76

dominant pattern. However, considerably more S-curve patterns and fewer convex patterns

(Prefix D) are found than for the EV1 plots.

(a) EV1 plots (b) LO plots (c) LN plots

Figure 5.1: Relative frequency of curve patterns – 43 Grade A1 stations

5.3.3 Probability plots for 110 Grade A1+ A2 stations

When results for the 67 Grade A2 stations are included, half of the EV1 plots (see Figure

5.2a) continue to exhibit a linear trend (Prefix L). However, the proportion exhibiting a

concave pattern (Prefix U) is reduced and is now outstripped by S-curve patterns (Prefix S) as

well as by convex (Prefix D) patterns.

(a) EV1 plots (b) LO plots (c) LN plots

Figure 5.2: Relative frequency of curve patterns – 110 Grade A1 + A2 stations

Concave patterns (Prefix U) remain the most prevalent in the LO plots (Figure 5.2b), with

convex patterns (Prefix D) remaining scarce. Across the 110 A1 + A2 stations, not even one

of the LO probability plots was judged a perfect straight line (Code L1).

The curve patterns in the LN plots (Figure 5.2c) are largely as for the Grade A1 stations

alone. However, convex patterns (Prefix D) are somewhat more prevalent, and concave

patterns (Prefix U) somewhat less prevalent, than in the smaller dataset.

In many cases it was observed that those stations which fit quite straight on EV1 plots show a

concave (Prefix U) fit on Logistic paper. The reason might be due to the range of the

probability axis, because the probability axis of LO is relatively compressed compared to

EV1. That is why it was also noticed that those which fit as convex (Prefix D) on EV1 plots

are found to be broadly linear on LO.


77

5.3.4 Curve pattern in relation to skewness coefficient

Understandably, there is a link between the skewness of a data sample and the curve pattern

of the resulting probability plot. Stations are ordered in Table 5.2 according to Hazen

skewness. The curve patterns shown are those based on EV1 probability plots.

Fourteen of the 110 A1 + A2 stations (shown in bold) have negative Hazen skewness. It is

confirmed that:

Stations with high outliers tend to have high skewness;

Stations with low outliers tend to have low skewness;

EV1 plots for stations with H-skew < 0.2 predominantly have a convex (Prefix D)

curve pattern;

EV1 plots for stations with H-skew > 2.5 (shown in red) mostly have a concave

(Prefix U) curve pattern.

Table 5.2: EV1 plot curve patterns – 110 A1 + A2 stations ordered by Hazen skewness

H-s

kew

Station number

and river

# y

ears

Cu

rve

patt

ern

(EV

1 p

lot)

Ou

tlie

r

Sta

tion

gra

de

H-s

kew

Station number

and river

# y

ears

Cu

rve

patt

ern

(EV

1 p

lot)

Ou

tlie

r

Sta

tion

gra

de

-1.09 07006 Moynalty 19 D A2 0.78 34011 Manulla 30 S A2

-0.86 26021 Inny 30 D A2 0.81 06011 Fane 48 L High A1

-0.78 15003 Dinan 50 D Low A2 0.85 16003 Clodiagh 51 L A2

-0.48 25034 L Ennell trib 24 D A2 0.93 34018 Castlebar 27 L A1

-0.41 16009 Suir 52 D A2 0.93 07009 Boyne 29 L A1

-0.39 34024 Pollagh 28 D A2 0.93 26009 Black 35 S A2

-0.33 06025 Dee 30 D A1 0.95 10021 Shanganagh 24 L A1

-0.32 16008 Suir 51 D A2 0.96 15004 Nore 51 L High A2

-0.31 25124 Brosna 18 D A2 1.01 25030 Graney 48 S A1

-0.22 24082 Maigue 28 D A2 1.03 39009 Fern O/L 33 L A2

-0.16 36019 Erne 47 D A2 1.04 35005 Ballysadare 55 S A2

-0.13 25021 Brosna 44 D A2 1.05 26007 Suck 53 L A1

-0.12 32012 Newport 24 D A2 1.05 06026 Glyde 46 S A1

-0.09 25029 Nenagh 33 S A2 1.05 36010 Annalee 50 S A1

0.07 35002 Owenbeg 34 D A2 1.08 34001 Moy 36 L High A2

0.10 06013 Dee 30 D A1 1.12 14007 Derrybrock 25 L A1

0.10 35001 Owenmore 29 L A2 1.12 24022 Mahore 20 L High A2

0.12 19020 Owennacurra 28 D A2 1.17 26019 Camlin 51 L A1

0.14 15001 Kings 42 D A2 1.17 09001 Ryewater 48 L A1

0.16 26020 Camlin 32 L A1 1.19 33070 Carrowmore 28 L High A1

0.18 25017 Shannon 55 D A1 1.22 23001 Galey 45 L A2


78

H-s

kew

Station number

and river

# y

ears

Cu

rve

pa

tter

n

(EV

1 p

lot)

Ou

tlie

r

Sta

tio

n g

rad

e

H-s

kew

Station number

and river

# y

ears

Cu

rve

pa

tter

n

(EV

1 p

lot)

Ou

tlie

r

Sta

tio

n g

rad

e

0.18 14029 Barrow 47 D A2 1.23 25044 Kilmastulla 33 S A2

0.19 14011 Rathangan 26 L A1 1.25 18004 Ballynamona 46 S A2

0.21 36012 Erne 47 D A1 1.26 34003 Moy 29 S A2

0.22 14018 Barrow 51 L A1 1.30 30007 Clare 31 L A2

0.27 25027 Ollatrim 43 D A1 1.31 06014 Glyde 30 L A1

0.27 26005 Suck 51 L A2 1.32 25040 Bunow 20 L A2

0.28 24008 Maigue 30 L A2 1.33 16005 Multeen 30 L A2

0.30 16001 Drish 33 L A2 1.37 27002 Fergus 51 L A1

0.31 26059 Inny 17 D Low A1 1.42 14009 Cushina 25 S A2

0.32 24002 Camogue 27 L A2 1.46 14006 Barrow 51 L A1

0.34 29001 Raford 40 D A1 1.49 26008 Rinn 49 L A1

0.34 35073 L. Gill 30 D A2 1.54 08005 Sluice 18 L A2

0.36 10022 Carrickmines 18 L A1 1.55 27001 Claureen 30 L A2

0.37 14013 Burrin 50 S A2 1.59 12001 Slaney 50 L A2

0.42 16011 Suir 52 D A1 1.63 18005 Funshion 50 L A2

0.43 36018 Dronmore 50 L A1 1.74 08008 Broadmeadow 25 L A2

0.43 34009 Owengarve 33 L A2 1.74 07033 Blackwater 25 S High A2

0.47 26022 Fallan 33 L A2 1.79 16002 Suir 51 L High A2

0.48 19001 Owenboy 48 L A2 1.85 36021 Yellow 27 L High A2

0.53 16004 Suir 48 L A2 1.92 31002 Cashla 26 U A1

0.57 25014 Silver 54 L A1 1.98 08002 Delvin 20 L A1

0.57 25023 Little Brosna 52 L A1 2.01 14005 Barrow 48 L High A2

0.57 25025 Ballyfinboy 31 L A1 2.23 14034 Barrow 14 L A2

0.59 15002 Nore 35 L A2 2.29 26002 Suck 53 S A2

0.61 14019 Barrow 51 D A1 2.40 09002 Griffeen 24 S A1

0.62 25016 Clodiagh 42 L A2 2.62 36015 Finn 33 U High A1

0.71 25006 Brosna 52 L A1 2.98 23012 Lee 18 U High A2

0.71 27003 Fergus 48 L A2 3.04 30061 Corrib Estuary 33 U A2

0.73 39008 Gartan 33 S A2 3.12 06031 Flurry 18 U High A2

0.74 29071 L. Cutra 26 L A2 3.24 09010 Dodder 19 U High A1

0.76 35071 L. Melvin 30 L A2 3.37 29011 Dunkellin 22 U High A1

0.77 26018 Owenure 49 L A2 3.87 26006 Suck 53 S High A1

0.77 29004 Clarinbridge 32 L A2 4.19 36031 Cavan 30 S High A2

0.78 06070 Muckno 24 L A1 5.43 08009 Balheary 11 U High A1


79

5.4 Flood volumes associated with largest peaks on convex probability plots

5.4.1 Stations with a convex curve pattern

For many of the stations which display a convex appearance on EV1 probability plots, it is

noted that the largest and 2nd

largest floods are little greater than the 3rd

largest. This

contributes to the flattening out of the probability plot and the low skewness of the data

sample. Such stations are identified in Appendix H as having a D curve pattern (D for

concave downwards). There are 26 such cases (i.e. stations exhibiting a convex EV1 plot)

amongst the 110 A1 + A2 stations.

If one were to include a few of the stations for which the pattern is classed as S-curve, the

convex behaviour is characteristic of a quarter of the A1 + A2 gauged catchments. The

behaviour is important in itself but also because a statistical distribution fitted to such data

may imply an absolute upper bound to flood magnitudes that is little greater than the largest

flood already observed. Investigation of such controversial cases is clearly warranted.

5.4.2 Hypothesis

In the mildly graded rivers characteristic of much of Ireland, the tendency for the largest

flood peaks to be of similar magnitude is typically attributed to the influence of floodplain

storage (e.g. Mason 1992; Ahilan et al. 2012). The theory is that the magnitude of large

flood peaks is naturally limited by the wide floodplains that become available to water when

river flow exceeds the bankfull capacity of the river system.

A specific attempt was therefore made to test the hypothesis that large floods of similar

magnitude might differ volumetrically. Could flood volume (amongst the highest floods) be

increasing, even though the peak discharge is not?

5.4.3 Arrangement of study

To investigate this question, hydrograph volumes have been calculated for the three or four

largest AM flood peaks at seven stations. These are drawn from the 26 stations identified as

yielding a convex curve pattern (Prefix D) on EV1 probability paper (see Appendix H).

The availability of hydrograph data was a factor in the selection of the examples listed in

Table 5.3. Comparison of the relevant periods of record in the final columns of the table

indicates that hydrograph data are mainly available for the required periods. However, there

were some exceptions. The hydrographs found are presented in Appendix J.

[Editorial note: An eighth station – the Deel at Rathkeale (24013) – had been identified as

having a strongly convex probability plot. However, it transpired that pre- and post-drainage

records had been inadvertently combined. Ahilan et al. (2012) also fail to distinguish pre and

post-drainage records at this station. The nine lowest AM flows in their frequency analysis of

49 annual maxima for the Deel at Rathkeale derive from the ten years of pre-drainage record.

The post-drainage series is retained in the analysis of hydrograph volumes reported in

Appendix J, although EV1 Probability Plot J.5 is found to be only weakly convex.]


80

Table 5.3: Stations for which flood volumes were specially investigated

Station

number Station name

Station

grade

Hazen

skewness

Period for which:

Hydrograph

data available

AM series

analysed

07006 Moynalty at Fyanstown A2 -1.09 1956 – 2005 1986 – 2004

15003 Dinan at Dinan Bridge A2 -0.78 1972 – 2005 1954 – 2004*

16008 Suir at New Bridge A2 -0.32 1954 – 2005 1954 – 2004

16009 Suir at Cahir Park A2 -0.41 1940 – 2005 1953 – 2004

24013 Deel at Rathkeale A1 0.05 1972 – 2003 1969 – 2004

24082 Maigue at Islandmore A2 -0.22 1975 – 2001 1977 – 2004

25017 Shannon at Banagher A1 0.18 1989 – 2003 1950 – 2004

25021 Little Brosna at Croghan A2 -0.13 1961 – 2003 1961 – 2004 *AM flood for 2001 water-year missing at this station

5.4.4 Calculation of hydrograph volumes

On examination of hydrographs for the three or four largest AM floods, it is seen that some

exhibit a unimodal (i.e. one-peaked) hydrograph reflecting their likely origin in a single

period of heavy rainfall. However, many of them occur as a result of longer and more

fluctuating periods of rainfall which lead to a multimodal hydrograph. In consequence, it is

not practical to specify a unique time duration over which to evaluate the hydrograph volume.

Hydrograph volumes were therefore calculated across a number of time durations or

windows. It should be noted that these are set symmetrically about the time of the peak. The

windows correspond to ±12 hours, ±24 hours, ±84 hours, ±1 week and ±15 days, leading to

window sizes of 1, 2, 7, 14 and 30 days respectively.

5.4.5 Example

The method is illustrated for Station 24082 Maigue at Islandmore. This 764 km2 catchment

has 28 annual maxima (1977 to 2004); see EV1 Probability Plot 5.3. The hydrographs for the

four largest AM floods are shown in Figure 5.3. The Hazen skewness is -0.22.

EV1 Probability Plot 5.3: Station 24082 Maigue at Islandmore

2 5 10 25 50 100 500

0

50

100

150

200

250

-2 -1 0 1 2 3 4 5 6 7

AM

flo

w (

m3

s-1

)


winter peak

summer peak

Feb

'90 Nov

'00 Oct

'88 Dec

'98


81

Figure 5.3: Hydrographs of four largest AM events, Station 24082 Maigue at Islandmore

Hydrograph during Maximum Flood in the year 1989

0

50

100

150

200

250

01/02/1990 00:00 03/02/1990 00:00 05/02/1990 00:00 07/02/1990 00:00 09/02/1990 00:00 11/02/1990 00:00

Time

Dis

charg

e (

m3/s

)

Red zone= Vol. of 1day

Yellow zone= Vol. of 2day

Blue zone= Vol. of 1week


0

20

40

60

80

100

120

140

160

180

200

31/10/2000 00:00 02/11/2000 00:00 04/11/2000 00:00 06/11/2000 00:00 08/11/2000 00:00 10/11/2000 00:00 12/11/2000 00:00

Time

Dis

char

ge (m

3/s)





0

20

40

60

80

100

120

140

160

180

200

14/10/1988

00:00

16/10/1988

00:00

18/10/1988

00:00

20/10/1988

00:00

22/10/1988

00:00

24/10/1988

00:00

26/10/1988

00:00

28/10/1988

00:00

Time

Disc

harg

e (m

3/s)





0

20

40

60

80

100

120

140

160

180

200

23/12/1998 00:00 25/12/1998 00:00 27/12/1998 00:00 29/12/1998 00:00 31/12/1998 00:00 02/01/1999 00:00 04/01/1999 00:00

Time

Disc

harg

e (m

3/s)




Rank 1 AM event,

6 February 1990

Rank 2 AM event,

6 November 2000

Rank 3 AM event,

21 October 1988

Rank 4 AM event,

30 December 1998

Flo

w (

m3 s

-1)

Flo

w (

m3 s

-1)

Flo

w (

m3 s

-1)

Flo

w (

m3 s

-1)

±12 hr, , ±84 hr from peak


82

The flood volumes across 1, 2, 7, 14 and 30-day windows are illustrated in Figure 5.4. The

label Rank 1 AM flood denotes the event with the greatest peak flow. The 1-day and 2-day

flood volumes are noticeably similar across these four events. The Rank 1 AM flood event

yields the largest volume at all durations, but the difference is marked only at the longer

durations of 7, 14 and 30 days.

Figure 5.4: Hydrograph volumes of four largest AM events at 24082 Maigue at Islandmore

5.4.6 Summary of findings

Appendix J provides results for the eight stations studied. Their behaviours are little different

to the example above. No marked trend is noted in the 1 and 2-day flood volumes of the

hydrographs associated with the largest AM flood peaks.

It is concluded that flood volumes do not reveal any noticeable growth with rank among the

highest-ranking floods in those AM series which display convex patterns on EV1 (and LN)

probability plots. In other words, no explanation for the flattening out (of the upper part) of

the probability plots can be found among the flood volumes. The hypothesis put forward in

Section 5.4.2 is therefore not supported.

Because of the importance of such effects, it was decided to make a further check on the

hypothesis by examining whether the skewness of AM flood series might be related to the

Flood Attenuation Index developed in the FSU (see Chapter 3 of Volume IV).

5.4.7 A further check on the hypothesis

The Flood Attenuation Index, FAI, falls in the range 0 to 1. Values of the index indicate the

fraction of the catchment area potentially inundated in very large floods. Under the

Section 5.4.2 hypothesis that floodplain storage tends to lead to convex EV1 probability

plots, it is expected that catchments with high values of FAI might be associated with convex

curve patterns.

It was seen in Table 5.2 that skewness is a relatively good indicator of the general curve

pattern of the EV1 probability plots. Convex patterns predominate when the Hazen skewness

is less than 0.2, whilst concave patterns predominate when the Hazen skewness is greater

than 2.5.

Volume of hydrographs of different year during max peak

0

20

40

60

80

100

120

140

160

1 2 7 14 30Days

Mill

cu

. met

er

1998

1988

2000

1989

Flo

od v

olu

me

in 1

06 m

3

Duration across which flood volume evaluated

Rank 1 AM flood, Feb 1990

Rank 4 AM flood, Dec 1998

Rank 2 AM flood, Nov 2000

Rank 3 AM flood, Oct 1988


83

Extracting FAI values for the 110 A1 + A2 stations and plotting them against Hazen

skewness, it is demonstrated emphatically in Figure 5.5 that the Flood Attenuation Index

provides no explanation for the convex curve pattern found in 26 of the 110 EV1 probability

plots. The hypothesis put forward in Section 5.4.2 is finally rejected.

6543210-1-2

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Hazen skewness

FA

I

LS S

L

DL

U

DSD L

SDLD

LS

LSLL

D U

U

LL ULD

LL

L

DL

DL

LL SL

L SL SD

SL

D

SS D

LLD

D

LLLD L

LL U

L

DL L

SDDD L

L

L

LL L

D

LD

LDD

L

SL

S

L

L

L

L

L

LU

S

L U

L

LL

S

LD

L

U

SD LD

L

Mainly convex Mainly concave

Mainly linear or S-curve

Figure 5.5: FAI v. Hazen skewness, labelled by EV1 curve pattern (110 A1 + A2 stations)

5.5 Flood seasonality

5.5.1 Seasonality of annual maximum floods

Annual maximum flood data were available for a total of 203 Grade A1, A2 or B stations.

One station did not have dates attached to the annual maximum flood data. Although the

flood peak data for some stations were subsequently rejected from wider analysis (see

Section 4.2.1 and Appendix B), the worries did not in general compromise the date

information used to characterise flood seasonality. Consequently, the seasonality study

reported here is based on 202 stations. Collectively, the series provide a total of 6969 station-

years of data.

Although floods are seen to occur at all times of year, most rivers register their annual

maximum in the winter (October-March) half-year. The percentages of AM floods occurring

in the winter half-year at each station are listed in Table K.1 of Appendix K. The months

December and January are associated with the greatest number of AM flood events followed

by November and February. In all, 6094 of the 6969 annual maxima occurred in the winter

half-year. In the summer half-year, considerable numbers of flood peaks were observed in

August. July has the least number of AM floods. The general pattern of flood seasonality is

summarised in Histogram 5.1.

Histogram 5.1: Seasonal occurrence of annual maximum floods

Seasonal Flood Frequency

Total no. of station

years: 6969

0

200

400

600

800

1000

1200

1400

1600

1800

Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun JulMonth

No

. o

f O

cc

ura

nc

es

Num

ber

of

occ

urr

ence

s

Based on 6969 station-years

across 202 A1 + A2 + B stations

Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul


84

5.5.2 Seasonality of largest floods

The months of occurrence of the maximum flow in each flood series are given in Table K.2

and summarised in Histogram 5.2. Most of the series maxima occurred in the winter half-

year, with the single month of December provided the maximum at 91 of the 202 stations. At

only 20 stations was the maximum recorded flood in the summer half-year. However, the

single month of August supplied the series maximum at eight of the 202 stations examined.

Histogram 5.2: Month corresponding to series maximum flow (202 A1 + A2 + B stations)

5.5.3 Circular diagrams

The circular diagram in Figure 5.6 illustrates the seasonal distribution of AM flood peaks

together with their magnitudes in m3s

-1. The angular position indicates the flood month and

the radial distance shows the magnitude of the flood peak.

Figure 5.6: Seasonality and magnitudes of 6969 AM floods (202 A1 + A2 + B stations)

Month corresponding to Max. flow in an AM series

0

10

20

30

40

50

60

70

80

90

100

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth

No

. o

f o

cc

ure

nc

es

Num

ber

of

occ

urr

ence

s

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

800

Oct

Apr

JanJul

200

400

600

-800 -340 120 580 January

October

July

April

Radial distance indicates

AM flood peak in m3s

-1


85

5.6 Flood statistics on some rivers with multiple gauges

As part of exploratory data analysis it is valuable to examine flow data from cases where

several long-term gauging stations exist on the same river. This gives an opportunity to see

how flood magnitude increases in the downstream direction.

5.6.1 Down-river growth in QMED

Flood data for multiple stations on the Barrow, Suir, and Suck rivers are considered here. It

is seen from Figure 5.7 that QMED increases down-river in a generally systematic manner.

The variation with catchment size (Figure 5.7a) is confirmed to be more regular than the

variation with distance along the main channel (Figure 5.7b). MSL denotes the mainstream

length. The down-river growth in QMED is seen to be more pronounced on the Suir than on

the Barrow or Suck.

500020001000500200100

500

200

100

50

20

10

AREA (km )

QM

ED

Barrow

Suir

Suck

2 2001005020

500

200

100

50

20

10

MSL (km)

QM

ED

Figure 5.7: Variation of QMED down the Barrow, Suir and Suck

5.6.2 Down-river variation in probability plots

Down-river variation in probability plots is found to be relatively complex (see EV1

Probability Plot 5.4). This is especially the case for the Barrow, although the more marked

differences in pattern are in the less important LH part of the plot. The arrow indicates the

down-river sequence of stations.

The probability plots for the three most upstream stations on the Suir are similar in shape

through much of the range. [Editorial note: O’Sullivan et al. (2012) study the reach from

New Bridge to Cahir Park.] The plots for the other two stations are notably steeper,

especially for the most downstream station (16011 Suir at Clonmel). This highlights the

important role of tributaries in influencing flood frequency on the Lower Suir.

The probability plots for stations on the Suck are notable for their mild slope over much of

the range but with a kick in the RH part of the plot, reflecting the scope for occasional much

larger flood peaks. The disturbance is more marked at upstream stations than at downstream

stations.

QM

ED

(m

3s-1

)

AREA (km2) MSL (km)

● Barrow

● Suir

● Suck

● Barrow

● Suir

● Suck

(a) Variation with catchment size (b) Variation with distance along main channel


86

EV1 Probability Plot 5.4: Multiple stations on the Rivers Barrow, Suir and Suck

ST14005

ST14006

ST14019

ST14018

ST14029

Probability Plot Summary for stations on Barrow River

2 5 10 25 50 100 5000

50

100

150

200

250

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

)

14005

14006

14018

14019

14029

Qmed vs Catchment Area relationship for stations on Barrow River

0

20

40

60

80

100

120

140

160

180

0 500 1000 1500 2000 2500 3000

Catchment Area (km2)

Qm

ed

(m

3/s

)

16002

16008

16009

16011

Probability plot summary for stations on Suir River

50010050251052

0

50

100

150

200

250

300

350

400

450

500

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

)

16004

16002

16008

16009

16011

Qmed vs Catchment Area Relationship for stations on Suir River

0

50

100

150

200

250

0 500 1000 1500 2000 2500


Qm

ed

(m

3/s

)

26002

26005

26007

Probability Plot Summary for stations on Suck River

2 5 10 25 50 100 500

0

20

40

60

80

100

120

140

160

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

) 26006

26002

26005

26007

Qmed vs Catchment Area relationship for stations on Suck River

0

10

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200 1400


Qm

ed

(m

3/s

)

Suir

● 16011 Clonmel

16009 Cahir Park

× 16008 New Bridge

▲ 16002 Beakstown

+ 16004 Thurles




Annual

max

imum

flo

w (

m3s-1

) A

nnual

max

imum

flo

w (

m3s-1

) A

nnual

max

imum

flo

w (

m3s-1

)

Barrow

● 14029 Graiguenamanagh

× 14018 Royal Oak

14019 Levitstown

▲14006 Pass Bridge

+ 14005 Portarlington

Suck

26007 Bellagill Br

× 26005 Derrycahill

▲26002 Rookwood

+ 26006 Willsbrook

Return period in years




87

6 Determining T-year flood magnitude QT by index flood method

6.1 Introduction

Important developments in flood frequency analysis in the early 20th

Century were made by

Fuller (1914), Foster (1924) and Hazen (1930), amongst others. Gumbel (1941) brought the

basis of analysis to a new level by applying extreme value theory and the findings of Fisher

and Tippet (1928). Gumbel introduced the Extreme Value Type I distribution to flood

frequency analysis. This distribution is often known as the Gumbel distribution but is

denoted EV1 in this report. Powell (1943) introduced extreme value probability paper, which

in one form or another (ruled in terms of probability or reduced variate) is still widely in use

today. Then Langbein (1949) provided the link between the T-year flood estimated by

annual maximum analysis and that derived from partial duration (i.e. peaks-over-threshold)

models.

The concept of standard error and bias of statistical estimates of QT was introduced by

Kimball (1949) for the EV1 distribution and by Kaczmarek (1957) for EV1 and other

distributions although this information filtered only slowly into hydrological practice. The

first simulation-based studies of standard errors were reported by Nash and Amorocho (1966)

and by Lowery and Nash (1970). By the mid-1970s, extensive simulation studies on bias and

standard error were reported by e.g. Matalas et al. (1975). Many analogous studies followed,

up to including that by Lu and Stedinger (1992).

Since Fuller (1914), flood magnitudes have been estimated by fitting distributions, either

graphically or numerically, to series of annual maximum floods treated as if they are random

samples from known distributions. The T-year return period flood QT is the flood magnitude

which is exceeded on average once in every T years and has a probability of being exceeded

in any one year of p = 1/T. p is the so-called annual exceedance probability.

Let F(Q) be the cumulative distribution function of AM flood magnitudes. Then the non-

exceedance probability of magnitude Q in any one year is:

F(Q) = 1 – 1/T 6.1

so that

QT = F-1

(1-1/T) 6.2

The cumulative distribution function (CDF) is sometimes referred to as simply the

distribution function. Its connection with the probability density function is illustrated in

Figure 6.1.

Figure 6.1: Probability density function and cumulative distribution function

Probability Density Function

0

0.1

0.2

0.3

0.4

Pro

babili

ty D

ensity F

unction

area = 1/Tarea =

1 - 1/T

QT

Distribution Function

0

0.2

0.4

0.6

0.8

1

No

n E

xce

ed

an

ce

Pro

ba

bili

ty, F

(q)

F(q) = 1 - 1/T

1 - F(q) = 1/T

QT

Probability Density Function

0

0.1

0.2

0.3

0.4

Pro

babili

ty D

ensity F

unction

area = 1/Tarea =

1 - 1/T

QT

Distribution Function

0

0.2

0.4

0.6

0.8

1

No

n E

xce

ed

an

ce

Pro

ba

bili

ty, F

(q)

F(q) = 1 - 1/T

1 - F(q) = 1/T

QT

(b) Cumulative distribution function (a) Probability density function


88

6.2 Regional flood frequency analysis and the index flood approach

6.2.1 Index flood approach

Regional or geographical pooling of dimensionless flow values was introduced by Dalrymple

(1960) to provide regional growth factors, xT, for use at ungauged sites. The flood frequency

curve, QT, is estimated in the form:

QT = QI xT 6.3

where QI is the index flood and xT is the flood growth factor. The index flood is usually

taken as the mean or median of the AM flow series. The index flood adopted in the FSU is

the median annual flood, QMED.

The FSR (NERC, 1975) introduced the idea that the pooled approach embodied by

Equation 6.3 should be used for QT estimation at gauged sites if T > 2N, where N is the

number of years of gauged record. More stringently, the FEH recommended that the pooled

approach of Equation 6.3 should prevail if T > N and should play at least some role

(alongside single-site analysis) when N/2 ≤ T ≤ N (Reed, 1999).

In between publication of FSR in 1975 and FEH in 1999, considerable research was carried

out on the efficiency of the index flood approach, including: Greis and Wood (1981),

Hosking et al. (1985a and 1985b), Lettenmaier et al. (1987), Cunnane (1989) and Hosking

and Wallis (1997). The index flood approach is now widely accepted as a valid and well

understood method, both in its implementation and its properties.

6.2.2 Type of region for pooling flood data

Dalrymple (1960) and FSR (NERC, 1975) applied the index flood method in fixed

geographical regions. All sites in the region are deemed to share a common flood growth

curve, xT.

In contrast, the Region Of Influence (ROI) method developed by Burn (1990) defines a region

centred on the specific (subject) catchment. Gauged catchments are selected to join the

region according to their nearness to the subject catchment in geographic or catchment-

similarity space. The ROI method was favoured in the FEH project, where it was found to be

more effective than the use of fixed geographical regions (Jakob et al., 1999).

6.2.3 Choice of index flood

The index flood approach uses a typical flood such as the mean (Qmean) or median (Qmed) of

the annual maximum flows as the index variable. FSR chose Qmean (often written QBAR) for

flood frequency but adopted the 5-year return period rainfall depth, M5, as the index variable

for rainfall frequency. In the present study, Qmed is taken as the index flood and is generally

written QMED.

QMED has a fixed frequency. It is the 2-year return period flood on the annual maximum

scale. One in every two annual maxima exceeds QMED.


89

6.2.4 A 2-stage approach

Use of the index flood approach to estimate QT by Equation 6.3 involves

Estimation of the index flood for the site in question, i.e. the subject site;

Estimation of the growth factor xT.

The graph of xT against T is referred to as the growth curve.

These two steps are more or less independent and each is dealt with separately. QMED

estimation from physical catchment descriptors (PCDs) was dealt with in Chapter 2. Though

a crucial building block, the relevant model for estimating QMED from PCDs (i.e.

Equation 2.8) is rarely applied in isolation. Practical application to estimate QMED at an

ungauged subject site requires both estimation of QMED from PCDs and a data transfer from

the most relevant gauged site. The latter is known as the pivotal catchment.

Though simple in principle, data transfers from gauged to subject sites tend to be complex in

practice. Section 2.7 provides an example, with further discussion chiefly deferred until

Chapter 11. The second stage of QT estimation by the index flood approach is estimation of

the flood growth curve, xT.


90

7 Flood growth curve estimation

7.1 Introduction to xT

The growth factor, xT, is the factor which when multiplied by the index flood QI gives QT, the

flood magnitude of return period T. In this study QMED is used as the index flood. Thus:

QT = QMED × xT 7.1

The relationship between xT and T is generally referred to as the flood growth curve.

The chapter begins by outlining two methods of determining the growth factor xT and the

resulting QT. The effects of catchment type, geographical location and period of record on xT

and QT are then examined. The effectiveness of different combinations of PCDs in defining

the distance measure on which ROI pooling groups are formed is examined in Appendix L.

7.2 Single-site and pooled estimates of QT

7.2.1 Advantages and drawbacks of the two approaches

If a subject site coincides with a gauging station – and if a sufficiently long record of AM

flows exists at the station – a suitable distribution can be fitted to the data of that site for

direct estimation of QT. This means that QT can be estimated without going through the

intermediate steps of determining QMED and xT. The FSU refers to this direct estimation of

QT as single-site analysis. Some practitioners call it at-site analysis.

What constitutes a sufficiently long record is arguable. The FSR (NERC, 1975) adopted the

criterion N > 0.5T, where N is the number of years of data available (i.e. the record length)

and T is the required return period. In contrast, the FEH adopted the much more stringent

criterion N > 2T (Reed, 1999). Introductory remarks and background information about the

general topic of QT estimation were presented in Chapter 6.

If the record is insufficiently long, the standard error of estimate of QT i.e. SE(QT) will be

large. Adopting a pooled estimate of xT, and hence of QT, reduces SE(QT).

The choice between a single-site estimate and a pooled estimate also involves a trade-off

between the convenience of single-site estimation and the extra work involved in assembling

and analysing data for a pooled estimate. While a pooled estimate of xT has a smaller

standard error, the necessary assumption of pooling-group homogeneity is uncomfortable.

7.2.2 Choice of distribution

When undertaking single-site analysis, another question arises as to what constitutes a

suitable distribution to describe flood frequency. Generally, it is better to adopt a

2-parameter distribution than a 3-parameter distribution because SE(QT) is much smaller in

the former case even though the bias may be increased.

Some guidance on the choice of 2-parameter distributions is available from Chapter 5, where

EV1 and lognormal (LN) distributions are seen to be typically better descriptors of gauged

AM data in Ireland than the 2-parameter Logistic (LO). However, many samples of data do

not seem amenable to description by a 2-parameter distribution. Such cases include instances


91

where the probability plot exhibits marked concave, convex or S-shaped curvature. These

cases were denoted in Chapter 5 by the curve-pattern classifications U (for upwards concave),

D (for downwards concave) and S (for S-shaped). While it is possible that this is a period-

of-record effect that will be neutralised when much longer records are available, it seems

more likely that the cases represent a genuine and permanent departure from 2-parameter

behaviour.

One approach might be to avoid using a single-site estimate at a station where the sample

shows a strong departure from 2-parameter behaviour and to adopt a pooled estimate instead,

on the basis that this reflects an average type of Q~T behaviour for the type of catchment in

question. It should be recalled from EV1 Probability Plot 5.1 that random samples drawn

from a 2-parameter distribution can display non-linear behaviour in a small proportion of

cases.

7.3 Pooling groups

7.3.1 The idea of pooling

Although xT can be estimated at a gauged site by conventional flood frequency analysis of the

AM flood series there, it is generally accepted that an advantage can be gained – in terms of

reduction of standard error of estimate of QT – if the assumption is made that xT does not vary

between gauging stations that belong to a homogeneous pooling group. A pooling group of

gauging stations is said to be homogeneous if the same growth curve applies to all stations in

the group (or, even more strictly, if the same value of xT for stated T can be applied to all

stations in the group).

The regional approach of pooling data exploits a trade-off between space and time. If the

group of stations is fully homogeneous, n years of record at each of m stations can be taken to

be equivalent to n × m years of record at a single site. Hence, if xT is estimated from the

n × m AM values – after they have been suitably standardised (e.g. by division by their

respective QMED values) – it will have a smaller standard error of estimate than an xT value

obtained by single-site analysis.

For a perfect case, the reduction in the standard error is by a factor of the order of m0.5

. The

gain is more limited if the pooling group is only weakly homogeneous or if the period of

record available at the stations is not very typical of the longer term.

7.3.2 Construction of pooling groups

Pooling groups can be formed by using geographical regions but Jakob et al. (1999,

specifically Figure 16.5) found such pooling groups to be typically less homogeneous than

those formed by a region-of-influence (ROI) approach of the type proposed by Burn (1990).

The ROI approach forms a pooling group for a particular subject site by selecting the nearest

stations in catchment-descriptor space. The space is designed to reflect the hydrological

similarity of catchments.

In choosing a pooling scheme, a decision has to be made about the PCDs to be included in

the metric (i.e. the distance measure dij) and what weightings and transformations are to be

applied to them. The FEH pooling scheme provides an example (see Box 7.1).


92

Box 7.1: FEH pooling scheme

7.3.3 The 5T rule

In addition, a decision has to be taken about how many stations to include in the pooling

group. Jakob et al. (1999) investigated pooling groups of various sizes before adopting the

5T rule. This states that at least 5T station-years of data should be used when estimating the

T-year flood.

Adoption of such a rule is a compromise. If too few station-years of data are pooled,

precision of the xT estimate is sacrificed. If too many stations are included, the assumption of

homogeneity may be compromised. Hosking and Wallis (1997) show that a small departure

from homogeneity can be tolerated. Having slightly too many stations included is therefore

deemed preferable to pooling too few station-years of data.

The 5T rule is adopted here as a pragmatic means of determining how many stations to

include in the xT estimation. It is recognised that it may not be practical or appropriate to

apply the 5T rule in every situation.

7.3.4 Recommended metric for pooling-group construction

Research on pooling schemes is reported in Appendix L. The distance measure (or metric)

ultimately recommended is:

2

nBFI

ji

2

nSAAR

ji

2

nAREA

ji

ijσ

nBFInBFI

σ

nSAARnSAAR

σ

nAREAnAREAd

7.2

Here, σℓnAREA, σℓnSAAR and σℓnBFI are the standard deviations of ℓnAREA, ℓnSAAR and ℓnBFI.

Values to be used in implementation are 1.265, 0.173 and 0.219 respectively.

The use of AREA and SAAR mirrors the FEH scheme of Box 7.1, in full knowledge that

both feature strongly in the estimation of QMED from PCDs (see Section 2.2). The third

variable in the distance metric is gauged BFI. This differs in that values of the baseflow

index (BFI) derived from daily mean flow data are used in preference to estimates of BFI

from soil modelling. This is in keeping with the philosophy (where possible) of basing

In the original FEH pooling scheme (Jakob et al., 1999), the distance between

subject site i and gauged site j is taken to be:

2

BFIHOST

ji

2

lnSAAR

ji

2

lnAREA

ji

ijσ

BFIHOSTBFIHOST

σ

lnSAARlnSAAR

σ

lnAREAlnAREA

2

1d

It is seen that two of the FEH catchment descriptors (AREA and SAAR) are transformed

logarithmically. The three coordinates in size-wetness-permeability space are then

standardised by dividing by the standard deviation σ of the relevant PCD. Finally,

differences in catchment size (indexed by ℓnAREA) are down-weighted in importance

relative to differences in catchment wetness (indexed by ℓnSAAR) and in catchment

permeability (indexed by BFIHOST), by inclusion of the ½ multiplier.


93

estimates on measured rather than modelled data. Gauged BFI has also been found useful in

estimating some hydrograph shape parameters (see Chapter 6 of Volume III).

In cases where there is no gauged BFI value, BFIsoil is to be substituted. The physical

catchment descriptor BFIsoil is detailed in Chapter 5 of Volume IV.

7.4 Growth curve estimation

Growth curves are expressed as algebraic equations for xT in terms of a frequency variable,

typically the return period T. Examples of such equations follow later in Section 7.4.3.

Suppose there are m gauging stations in the pooling group. Following the method of

L-moments, the pooled growth curve is constructed from pooled values of the L-moment

ratios. L-moment ratios were introduced in Chapter 4; see Box 4.2 and Section 4.5 in

particular. Typically, two L-moment ratios – the L-CV and the L-skewness – are used in

fitting the pooled growth curve. Sample values of L-CV and L-skewness are denoted here by

t2 and t3. [Editorial note: Some authors omit the subscript and denote L-CV by t.]

7.4.1 Pooled L-moment ratios

Let t2(i)

and t3(i)

denote sample values of L-CV and L-skewness at the ith

site. Pooled values

of the L-moment ratios are calculated as weighted averages:

2t =

m

1i

i

m

1i

(i)

2i

w

tw

7.3

and

3t =

m

1i

i

m

1i

(i)

3i

w

tw

7.4

where m is the number of stations in the pooling group and wi is a weighting factor.

[Editorial note: Where required, pooled values of L-kurtosis (t4) are calculated by the same

weighting scheme and annotated in similar style.]

Following guidance by Hosking and Wallis (1997), weights are taken proportional to record

length, i.e. wi = ni where ni is the number of annual maxima at station i. wi is then the

number of station-years of data in the pooling group. This weighting scheme is

recommended when the group (or “region”) of stations being pooled is reasonably

homogeneous.

7.4.2 Fitting growth curve distributions by the method of L-moments

In the so-called method of L-moments, a statistical distribution is fitted to a dataset by

equating its theoretical L-moment ratios to the sample values of the L-moment ratios.

In growth curve analysis, AM flood series from different stations are made comparable by

dividing the AM values by the index flood. The FSU adopts QMED as the index variable.

QMED corresponds to the 2-year flood, Q2. In consequence, the flood growth curve is


94

anchored to take the value x2 = 1.0. This determines one parameter of the distribution used to

model the growth curve.

If a 2-parameter distribution such as the EV1 or LO is being used, the remaining parameter is

determined by equating theoretical and sample values of the L-CV. If a 3-parameter

distribution such as the GEV or GLO is being used, the remaining two parameters are

determined by equating theoretical and sample values of both the L-CV and the L-skewness.

If a 4-parameter distribution such as the kappa is used, theoretical and sample values of the

L-kurtosis are also equated (see Box 4.2).

In pooled growth curve estimation, the values 2t and 3t (and, where required, 4t ) are

equated to expressions for these quantities written in terms of the distribution’s unknown

parameters. The resulting equations are solved for the unknown parameters.

A 3-parameter distribution can be used when pooling data from a group of stations because

the extra information ensures that the resulting standard error is smaller than in the single-site

case. Further, the 3-parameter distribution avoids possible bias resulting from using a

2-parameter distribution when in fact a 3-parameter distribution is more appropriate. The

thinking is summarised in Box 7.2, adapted from Cunnane (1989).

7.4.3 Growth curves for particular distributions

The formulae below define growth curves for the four distributions:

Generalised Extreme Value (GEV) distribution;

Extreme Value Type 1 (EV1) distribution, also known as the Gumbel;

Generalised Logistic (GLO) distribution;

Logistic (LO) distribution

The equations for estimation of their parameters by the method of L-moments are taken from

Hosking and Wallis (1997).

GEV growth curve

In this study, the GEV (and its special case EV1) and GLO (and its special case LO) are the

forms of growth curve chiefly used. The dimensionless GEV growth curve is defined by a

shape factor k and a scale parameter as follows:

k

k

T1T

Tlnln2

k

β1x for k 0 7.5

The parameters k and are estimated from the sample values t2 (of L-CV) and t3 (of

L-skewness):

k = 7.8590c + 2.9554c2 7.6

where

ln3

ln2

t3

2c

3

7.7

and


95

)2k)(1Γ(1(ln2)k)Γ(1t

ktβ

kk

2

2

7.8

Box 7.2: Qualitative outline of simulation results for flood quantile estimate

FIXED SKEWNESS MODELS (usually with two parameters)

(a) Model Skewness < Parent Skewness Hence negative bias i.e. underestimation at large T

(b) Model Skewness > Parent Skewness Hence positive bias i.e. overestimation at large T

VARIABLE SKEWNESS MODELS (usually with > two parameters)

(c) At-site use (d) Pooled XT estimation + At-site Qmed

Small bias, large se

Small bias and small se

1

0

-1

T

1

0

-1

T

1

0

-1

T

1

0

-1

T

Bias/QT%

[ Bias ± 1.96 se ]/QT %

T

10 100 1000

T

10 100

Fixed skewness models (usually with two parameters)

Variable skewness models (usually with more than two parameters)

FIXED SKEWNESS MODELS (usually with two parameters)

(a) Model Skewness < Parent Skewness Hence negative bias i.e. underestimation at large T

(b) Model Skewness > Parent Skewness Hence positive bias i.e. overestimation at large T

VARIABLE SKEWNESS MODELS (usually with > two parameters)

(c) At-site use (d) Pooled XT estimation + At-site Qmed

Small bias, large se

Small bias and small se

1

0

-1

T

1

0

-1

T

1

0

-1

T

1

0

-1

T

Bias/QT%

[ Bias ± 1.96 se ]/QT %

T

10 100 1000

T

10 100

– – – – – – – Bias/QT

- - - - - - - - - (Bias 1.96 se)/QT

(b) Model skewness > Parent skewness

[leads to +ve bias, with overestimation

at long return period, T]

(c) Single-site analysis

[Small bias and large standard error]

(d) Gauged QMED + pooled xT estimation

[Small bias and small standard error]

1

0

-1

1

0

-1

1

0

-1

1

0

-1

(a) Model skewness < Parent skewness

[leads to -ve bias, with underestimation

at long return period, T]

Cases (a) and (b) illustrate the effect of choosing a distribution having fixed but

incorrect skewness. Cases (c) and (d) illustrate the effect of a more flexible

distribution: namely, low bias and large standard error (SE) when used in single-site

analysis but with much reduced SE (as well as low bias) when used in gauged QMED +

pooled xT estimation.

[Adapted from Figure 5.1 of Cunnane (1989)]

Key


96

EV1 growth curve

The EV1 growth curve is defined as:

1/T))ln(1ln(ln(ln2)β1xT 7.9

The parameter is estimated from the sample value t2 (of L-CV):

ln(ln2)γtln2

tβ

2

2

7.10

where γ is Euler’s constant = 0.5772.

GLO growth curve

The GLO growth curve is defined by a shape factor k and a scale parameter as follows:

kk

T 1T1k

β1/FF11

k

β1x

for k 0 7.11

The parameters k and are estimated from the sample values t2 (of L-CV) and t3 (of

L-skewness):

k = – t3 7.12

and

kπsinttkπk

kπsintkβ

22

2

7.13

LO growth curve

The LO growth curve is defined as:

1)βln(T1xT 7.14

The parameter is estimated from the sample value t2 (of L-CV):

2tβ 7.15

[Editorial note: Implementation of methods through the FSU Web Portal also provides for

use of the 2-parameter lognormal (LN2) and 3-parameter lognormal (LN3) distributions. For

somewhat obscure reasons, the recommendation is that single-site application of the LN2

distribution is fitted by a scheme not involving L-moments. This variant is shown later in

Section 10.2.4. However, in all applications of the LN3 distribution, and in pooled

applications of the LN2 distribution, the use of L-moment methods is recommended. It

should be noted that different researchers favour different formulations of the lognormal

distribution. The L-moment methods implemented for the LN2 and LN3 distributions are

based directly on Hosking (1990) and on no other source.]


97

8 Effect of catchment type and period of record on xT and QT

8.1 Data screening

8.1.1 Discordancy

The 88 A1 + A2 grade stations for which BFI values were available were used to explore the

effect of catchment type and period of record on xT and QT.

Very unusual (i.e. discordant) datasets in the set of 88 catchments were first identified using

the Hosking and Wallis discordancy measure:

3

NDi uuAuu ii 1T

8.1

The vector holds the L-moment ratios for station i, namely the L-CV (t2), L-skewness (t3), ui

and L-kurtosis (t4). The superscript T denotes the transpose of the vector. u is the

unweighted regional average and is the matrix of sums of squares and cross-products. A

Further details are given in Hosking and Wallis (1997).

Table 8.1 lists the L-moment ratios and some leading PCDs for the 88 stations. The final

column shows the discordancy value Di when the station is judged relative to the set of 88

stations as a whole. Hosking and Wallis (1997) recommend that any site having Di > 3 be

regarded as discordant.

Table 8.1: L-moment ratios, PCDs and station discordancy within set of 88 A1+A2 stations

Station

#

#

annual

maxima

AREA SAAR BFI FARL URBEXT L-

CV

L-

skew

L-

kurt

Hosking+Wallis

discordancy, Di

06011 48 229.2 1029 0.71 0.87 0.011 0.11 0.09 0.07 0.44

06013 30 309.1 873 0.62 0.97 0.009 0.16 0.02 0.01 0.97

06014 30 270.4 927 0.63 0.93 0.012 0.15 0.22 0.12 0.57

06026 46 148.5 941 0.66 0.92 0.009 0.18 0.24 0.09 1.02

06031 18 46.2 931 0.56 1.00 0.015 0.26 0.39 0.32 1.98

06070 24 162.0 1046 0.73 0.83 0.013 0.14 0.14 0.14 0.02

07006 19 177.4 937 0.55 0.99 0.004 0.12 -0.20 0.07 2.46

07009 29 1658.2 869 0.71 0.99 0.008 0.21 0.21 0.11 0.74

07033 25 124.9 1032 0.44 0.89 0.006 0.13 0.16 0.27 0.72

08002 20 33.4 791 0.60 1.00 0.005 0.11 0.27 0.17 0.86

08005 18 9.2 711 0.52 1.00 0.250 0.38 0.23 0.19 4.71

09001 48 209.6 783 0.51 1.00 0.029 0.24 0.19 0.15 0.83

09002 24 35.0 755 0.67 1.00 0.210 0.42 0.39 0.25 6.26

09010 19 94.3 955 0.56 0.96 0.240 0.42 0.42 0.30 6.86

10021 24 32.5 799 0.65 1.00 0.242 0.21 0.19 0.07 1.10

10022 18 12.9 822 0.60 1.00 0.297 0.23 0.06 0.06 1.27

12001 50 1030.8 1167 0.72 1.00 0.006 0.19 0.18 0.17 0.19

14005 48 405.5 1015 0.50 1.00 0.024 0.15 0.29 0.22 0.54


98

Station

#

#

annual

maxima


CV

L-

skew

L-

kurt

Hosking+Wallis

discordancy, Di

14006 51 1063.6 899 0.57 1.00 0.017 0.11 0.24 0.21 0.55

14007 25 118.6 814 0.64 1.00 0.007 0.17 0.22 0.07 1.13

14009 25 68.3 831 0.67 1.00 0.001 0.13 0.14 0.24 0.47

14011 26 162.3 807 0.60 1.00 0.028 0.14 0.02 0.16 0.42

14018 51 2419.4 857 0.67 1.00 0.017 0.14 0.04 0.06 0.39

14019 51 1697.3 861 0.62 1.00 0.018 0.14 0.09 0.13 0.05

14029 47 2778.2 877 0.69 1.00 0.016 0.08 0.06 0.07 0.73

15001 42 444.4 935 0.51 1.00 0.002 0.16 0.00 0.07 0.49

15003 50 299.2 934 0.38 1.00 0.006 0.11 -0.15 0.09 1.62

16001 33 135.1 916 0.61 1.00 0.003 0.12 0.01 0.13 0.29

16002 51 485.7 932 0.63 1.00 0.011 0.16 0.17 0.19 0.05

16003 51 243.2 1192 0.55 1.00 0.002 0.10 0.21 0.05 1.85

16004 48 228.7 941 0.58 1.00 0.011 0.11 0.07 0.11 0.19

16005 30 84.0 1154 0.56 1.00 0.003 0.10 0.20 0.13 0.73

16008 51 1090.3 1030 0.64 1.00 0.007 0.07 -0.06 0.06 0.94

16009 52 1582.7 1079 0.63 1.00 0.008 0.10 -0.10 0.05 1.01

16011 52 2143.7 1125 0.67 1.00 0.007 0.17 0.09 0.05 0.62

18004 46 310.3 985 0.68 1.00 0.003 0.09 0.04 0.33 2.89

18005 50 378.5 1190 0.71 1.00 0.004 0.14 0.22 0.18 0.21

19001 48 103.3 1176 0.64 1.00 0.019 0.10 0.10 0.12 0.28

19020 28 74.0 1179 0.66 1.00 0.000 0.20 0.03 0.04 0.99

23001 45 191.7 1084 0.32 1.00 0.003 0.18 0.13 0.18 0.18

23012 18 61.6 1264 0.46 1.00 0.024 0.15 0.39 0.33 1.65

24008 30 806.0 939 0.54 1.00 0.007 0.15 0.06 0.09 0.22

24022 20 41.2 942 0.53 1.00 0.003 0.23 0.12 0.16 0.81

24082 28 762.8 942 0.52 1.00 0.006 0.15 -0.04 0.13 0.80

25006 52 1162.8 932 0.71 0.96 0.019 0.14 0.14 0.18 0.04

25014 54 164.4 1008 0.67 1.00 0.005 0.13 0.10 0.13 0.07

25016 42 275.2 947 0.61 1.00 0.028 0.13 0.08 0.16 0.08

25023 52 113.9 922 0.65 1.00 0.003 0.16 0.14 0.06 0.68

25025 31 161.2 905 0.73 1.00 0.009 0.17 0.08 0.15 0.14

25027 43 118.9 1021 0.65 1.00 0.006 0.16 0.05 0.13 0.22

25029 33 292.7 1109 0.58 1.00 0.013 0.14 -0.02 -0.02 1.37

25030 48 280.0 1184 0.54 0.85 0.001 0.18 0.19 0.13 0.23

25034 24 10.8 969 0.76 1.00 0.000 0.17 -0.08 0.12 1.36

25040 20 28.0 990 0.64 1.00 0.062 0.15 0.20 0.18 0.10

25044 33 92.5 1187 0.58 1.00 0.000 0.19 0.25 0.15 0.48

25124 18 215.4 955 0.87 0.78 0.035 0.20 -0.07 0.23 2.95

26002 53 641.5 1067 0.61 0.98 0.003 0.11 0.22 0.29 0.97


99

Station

#

#

annual

maxima


CV

L-

skew

L-

kurt

Hosking+Wallis

discordancy, Di

26005 51 1085.4 1054 0.56 0.98 0.002 0.10 0.04 0.14 0.28

26006 53 184.8 1121 0.54 0.97 0.006 0.15 0.40 0.41 2.83

26007 53 1207.2 1046 0.65 0.98 0.002 0.11 0.16 0.16 0.24

26008 49 280.3 1035 0.61 0.86 0.003 0.10 0.19 0.19 0.35

26018 49 119.5 1044 0.72 0.76 0.003 0.11 0.13 0.11 0.25

26019 51 253.0 980 0.54 0.99 0.013 0.14 0.23 0.12 0.71

26021 30 1098.8 945 0.83 0.81 0.004 0.14 -0.11 0.19 2.48

26022 33 61.9 916 0.58 1.00 0.006 0.17 0.09 0.05 0.69

26059 17 256.6 976 0.91 0.73 0.005 0.10 0.11 0.17 0.22

27001 30 46.7 1477 0.28 0.99 0.000 0.11 0.19 0.25 0.56

27002 51 564.3 1336 0.70 0.84 0.001 0.12 0.18 0.22 0.22

29004 32 121.4 1107 0.52 0.99 0.013 0.08 0.14 0.09 0.94

29011 22 354.1 1079 0.63 0.98 0.011 0.14 0.41 0.32 1.81

30007 31 469.9 1115 0.65 0.99 0.005 0.11 0.11 0.20 0.28

30061 33 3136.1 1422 0.78 0.66 0.007 0.15 0.41 0.38 2.38

31002 26 71.3 1530 0.53 0.63 0.000 0.13 0.31 0.18 1.03

32012 24 146.2 1784 0.59 0.84 0.000 0.07 0.00 0.18 0.90

34001 36 1974.8 1323 0.78 0.83 0.008 0.10 0.08 0.21 0.48

34003 29 1802.4 1340 0.80 0.82 0.008 0.09 0.11 0.25 0.85

34009 33 117.1 1257 0.40 1.00 0.011 0.10 0.08 0.16 0.22

34018 27 95.4 1555 0.66 0.73 0.055 0.11 0.18 0.03 1.73

34024 28 127.2 1177 0.52 0.92 0.007 0.07 -0.05 0.14 1.01

35001 29 299.4 1173 0.60 0.92 0.003 0.12 -0.01 0.19 0.92

35002 34 88.8 1381 0.42 0.99 0.000 0.10 0.03 0.08 0.41

35005 55 639.7 1198 0.61 0.90 0.002 0.14 0.20 0.14 0.27

35071 30 247.2 1364 0.77 1.00 0.001 0.10 0.12 0.20 0.28

36015 33 153.1 1091 0.42 0.96 0.000 0.16 0.33 0.30 0.99

36018 50 234.4 950 0.69 0.85 0.007 0.10 0.04 0.10 0.26

36019 47 1491.8 971 0.79 0.76 0.007 0.10 -0.03 0.06 0.58

36021 27 23.4 1570 0.27 1.00 0.000 0.12 0.20 0.22 0.30

36031 30 63.8 910 0.48 0.96 0.060 0.10 0.36 0.39 2.88

Three stations highlighted in red are seen to be discordant. These were excluded from further

analysis in this part of the study, even though it might be argued that this reduces the natural

variability within the dataset being studied. Before proceeding, it is instructive to consider

the characteristics that may explain why these stations are discordant.

8.1.2 Characteristics of the three discordant stations

The three discordant stations have several features in common. All three catchments lie close

to Dublin and are heavily urbanised (URBEXT > 0.20). They all have unusually high values


100

of L-CV. This is confirmed from the steepness of the EV1 Probability Plots 8.1, where it is

noted that summer events contribute widely to the AM flow series.

EV1 Probability Plots 8.1: Three discordant sites (Stations 08005, 09002 and 09010)

2 5 10 25 50 100 500

0

2

4

6

8

-2 -1 0 1 2 3 4 5 6 7

AM

flo

w (

m3

s-1

)


Station 08005 Sluice at Kinsaley Hall

winter peak

summer peakL-CV = 0.38

2 5 10 25 50 100 500 0

5

10

15

20

25

30

-2 -1 0 1 2 3 4 5 6 7

AM

flo

w (

m3 s

-1)


Station 09002 Griffeen at Lucan

winter peak


2 5 10 25 50 100 500 0

50

100

150

200

250

300

-2 -1 0 1 2 3 4 5 6 7

AM

flo

w (

m3 s

-1)


Station 09010 Dodder at Waldron's Bridge

winter peak

summer peakL-CV =0.42


101

One of the discordant stations is amongst those tested for trend and other non-randomness in

Chapter 3. Station 09002 Griffeen at Lucan showed nothing untoward in five out of the six

tests summarised in Table E.3. However, the parametric test based on samples drawn from

an EV1 distribution indicated a very highly significant trend. The sample CV for this station

is 0.83, making the particular test (which assumes a population CV of 0.30) inappropriate to

judging trend at this site.

No marked trend is apparent in Figure 8.1. The example confirms the value of graphical

display and of using a number of tests when exploring possible non-stationarity.

10

5

0

30

15

0

200520001995199019851980

300

200

100

0

Station 08005 Sluice at Kinsaley Hall

Station 09002 Griffeen at Lucan

Station 09010 Dodder at Waldron's Bridge

Figure 8.1: AM flow series for three discordant sites (Stations 08005, 09002 and 09010)

The discordant Stations 08005 and 09010 are amongst four catchments identified in

Section 2.5.2 for which the PCD model for QMEDrural seriously underestimates QMED. It

was noted there that urbanisation provides a plausible explanation for this.

It would therefore appear that flood characteristics of Station 08005 Sluice at Kinsaley Hall

and Station 09010 Dodder at Waldron’s Bridge are doubly severe:

QMED is unusually large (likely reflects urbanisation);

The growth curve is especially steep reflecting the high value of L-CV.

The high L-CV for Station 09010 is strongly influenced by the unusually large flood arising

from Hurricane Charlie in August 1986.

That greater variability (higher L-CV) is characteristic of some AM series in this part of

Ireland is consistent with the finding of Bruen et al. (2005) that flood growth curves around

Dublin tend to be steeper than prescribed for Ireland in NERC (1975). Table 8.2 provides a

summary of the three discordant stations.

AM

flo

w (

m3s-1

)

Hurricane

Charlie


102

Table 8.2: Stations showing large discordancy values (in pool of 88 A1 + A2 stations)

Station

number Station name

D

value L-CV CV Possible reason for discordancy

08005 Sluice at Kinsaley

Hall 6.16 0.38 0.68 Urbanised catchment

09002 Griffeen at Lucan 6.26 0.42 0.83 Urbanised catchment

09010 Dodder at

Waldron’s Bridge 8.94 0.42 0.86

Urbanised catchment; outlier arising from

Hurricane Charlie (August 1986)

8.1.3 Other heavily urbanised catchments

The dataset of 88 A1 + A2 stations includes two further heavily urbanised catchments:

Stations 10021 and 10022. These also lie in the area around Dublin. From EV1 Probability

Plots 8.2 it is seen that summer events (indicative of a faster flood response conditioned by

urbanisation) again feature in the AM flow series. However, the L-CV values and flood

growth rates are less exceptional than for the three discordant stations identified above.

EV1 Probability Plots 8.2: Two further heavily urbanised catchments

2 5 10 25 50 100 500

0

2

4

6

8

10

12

14

16

-2 -1 0 1 2 3 4 5 6 7

AM

flo

w (

m3 s

-1)


10021 River Shanganagh at Common's Road

winter peak

summer peak

L-CV = 0.21

2 5 10 25 50 100 500

0

1

2

3

4

5

6

7

8

-2 -1 0 1 2 3 4 5 6 7

AM

flo

w (

m3 s

-1)


10022 River Cabinteely at Carrickmines

winter peak



103

8.2 Effect of catchment type on pooled growth curve estimates

After exclusion of the three discordant stations, 85 catchments remain in the research dataset.

In various experiments, the 85 catchments have been divided into subgroups on the basis of:

Storage attenuation – as indexed by the FARL descriptor;

Peat content – as indexed by the PEAT descriptor;

Size – as measured by AREA;

Geographical location – within one of four regions defined in Map 8.1.

Rather than trying to compare entire growth curves simultaneously, assessments focused

chiefly on the 100-year growth factor x100, with some reference also to assessments of the

50-year growth factor x50.

The growth factor x100 has been estimated at each of the 85 gauging stations from its own

pooling group based on the size-wetness-permeability distance measure defined in

Equation 7.2 and using the 5T rule of Section 7.3.3 to determine the number of stations to be

pooled. Thus, each growth curve analysis pooled a minimum of 5 100 = 500 station-years.

Growth factor estimates were obtained by fitting GEV and GLO distributions to the pooled

data using the method of L-moments (see Section 7.4). Only the results for the GEV are

reported here. The effect of catchment type on pooled growth curve estimates did not differ

qualitatively between the two distributional assumptions.

The effect of catchment type on the 100-year flood growth factor are summarised in box-

plots. See Box 8.1 for an explanation of what these show.

Box 8.1: Box-plots

Box-plots 8.1a summarise the influence of the extent of peat cover (indexed by PEAT) on

values of x100, while Box-plots 8.1b summarise the influence from of the attenuating effect of

reservoirs and lakes within the catchment (indexed by FARL).

Box-plots provide a visual summary of data values

based on five numbers. The 50th

percentile is the

median. The box extends from the 25th

to 75th

percentiles. The part of the box tinted maroon

indicates the range of values between the 25th

and 50th

percentiles. The part tinted pale yellow indicates the

range from the 50th

to the 75th

percentile values.

[*Editorial note: The lines extending above the 75th

percentile and below the 25th

percentile are often

called whiskers. Conventions for drawing these vary.

The illustration is one interpretation which may not

be the one used in the box-plots below. It is possible

that the whiskers have been drawn down to the 9th

percentile and up to the 91st percentile.]

Var

iable

Median

Maximum*

Minimum*


104

Map 8.1: Location of the 85 stations within four geographical regions

In each case, the leftmost box-plot represents all 85 stations in the dataset. The median value

of x100 – represented by the separating line between the pale yellow and maroon boxes – is

seen to differ only slightly with the extent of peat cover, and without a definite pattern. The

median x100 values show a slight monotonically decreasing trend with increasing amount of

storage attenuation (i.e. for smaller FARL values). This is consistent with the storage action

of reservoirs and lakes typically attenuating flood growth rates.

[Editorial note: Box-plots 8.1 are cumulative. It is therefore puzzling that (e.g.) the x100

values for the 22 stations with PEAT > 0.15 do not span the range indicated for the x100

values for the 14 stations with PEAT > 0.20. As discussed in Box 8.1, it is possible that the

convention used to draw the whiskers was percentile-based. Even so, the diagrams appear

inconsistent. This is not a problem for Box-plots 8.2, which are non-cumulative except in

the LH column. Editing has sought to retain the most important findings, even when the

graphical support is not always clear.]

Shannon

West

South-West

East

Key___

Gauging station


105

Box-plots 8.1: Pooled GEV estimates of x100 showing effect of (a) PEAT and (b) FARL

The influences of geographical region and catchment size on values of x100 are summarised in

Box-plots 8.2a and b respectively. In terms of geographical region, the South-West has the

lowest median x100 (of just under 1.6) and the East has the highest value (of about 1.9).

Steeper flood growth curves in the East are commensurate with the finding of Bruen et al.

(2005). It should be noted that catchments of all types (e.g. size, degree of storage

attenuation and peat extent) are mixed together in these geographical groupings.

In terms of catchment size, the median x100 value shows a monotonic decrease with

increasing catchment area. In other words, there is some tendency for smaller catchments

(especially AREA < 100 km2) to exhibit steeper flood growth.

Box-plots 8.2: Pooled GEV estimates of x100 showing effect of (a) location and (b) AREA

Pooling Growth Curve(GEV)

1

1.2

1.4

1.6

1.8

2

2.2

2.4

All st(85) ≥5%

peat(44)

≥10%

peat(30)

≥15%

peat(22)

≥20%

peat(14)

farl

≤.98(30)

farl

≤.95(23)

farl

≤.90(19)

farl

≤.85(15)

X1

00


1

1.2

1.4

1.6

1.8

2

2.2

2.4

All st(85) West(17) South-

West(18)

East(24) Shannon

(26)

A≤100(19) A~101-

200(20)

A~201-

500(24)

A≥500(22)

X1

00


1

1.2

1.4

1.6

1.8

2

2.2

2.4


West(18)

East(24) Shannon

(26)

A≤100(19) A~101-

200(20)

A~201-

500(24)

A≥500(22)

X1

00

24 stns

in

East

26 stns

in

Shannon

All

85

stns

19 stns

with

AREA

≤100

20 stns

with

AREA

101-200

24 stns

with

AREA

201-500

22 stns

with

AREA

>500

Flo

od

gro

wth

fac

tor,

x1

00

All

85

stns

17 stns

in

West

18 stns

in

SW


1

1.2

1.4

1.6

1.8

2

2.2

2.4


West(18)

East(24) Shannon

(26)

A≤100(19) A~101-

200(20)

A~201-

500(24)

A≥500(22)

X1

00

All

85

stns

44 stns

with

PEAT

≥0.05

30 stns

with

PEAT

≥0.10

22 stns

with

PEAT

≥0.15

14 stns

with

PEAT

≥0.20

All

85

stns

30 stns

with

FARL

≤0.98

23 stns

with

FARL

≤0.95

19 stns

with

FARL

≤0.90

15 stns

with

FARL

≤0.85

Flo

od

gro

wth

fac

tor,

x100


106

8.3 Temporal effect on pooled growth curve estimates

8.3.1 Periods of record considered

Pooled growth curves have been derived from different periods of record in four formats:

Distinguishing early and late halves of each record (for 90 stations);

Distinguishing three decades: 1972-1981, 1982-1991 and 1992-2001 (for 50 stations);

Distinguishing five decades: 1952-1961, 1962-1971, 1972-1981, 1982-1991 and

1992-2001 (for 26 stations);

Distinguishing five offset decades 1957-66, 1967-76, 1977-86, 1987-96 and 1997-

2006 (34 stations for all five decades and 68 stations for the last three decades).

The periods detailed above refer to water-years, e.g. the decade 1972-1981 corresponds to the

period 1 Oct 1972 to 30 Sep 1982.

8.3.2 Summary results for 90 stations

Some 90 gauging stations were used in this part of the study, which was completed before the

data provider indicated that some stations were unsuitable for detailed analysis. It is believed

that the general nature of the findings would not be greatly altered had the study been

repeated with the five rejected stations excluded.

The effects of period of record (1st half of each record compared with 2

nd half) and of

successive decades are shown in Box-plots 8.3. Median x100 shows a decrease between the

earlier and later halves of records, while the decadal pattern of values of x50 is up and down.

Box-plots 8.3: Period-of-record and decadal effects on pooled flood growth by GEV

8.3.3 Individual results for three decades (50 stations)

There are 50 stations for which estimates are available for all three decades. The decadal

values of x50 for each station are displayed in Figure 8.2a. It can be seen that the largest x50

values occur in the middle decade (1982-91) in every case.


1

1.2

1.4

1.6

1.8

2

2.2

2.4

All record(90) 1st half

record(90)

2nd half

record(90)

Decade70(52) Decade80(83) Decade90(82)

X100 (

X50 in c

ase o

f decade a

naly

sis

)

90

records

in full

1st half of

90 records 2

nd half of

90 records


1

1.2

1.4

1.6

1.8

2

2.2

2.4


record(90)

2nd half

record(90)


X100 (

X50 in c

ase o

f decade a

naly

sis

)


1

1.2

1.4

1.6

1.8

2

2.2

2.4


record(90)

2nd half

record(90)


X100 (

X50 in c

ase o

f decade a

naly

sis

)

Flo

od

gro

wth

fac

tor,

x50

Flo

od

gro

wth

fac

tor,

x1

00

1972-

1981

52 stns

1982-

1991

83 stns

1992-

2001

82 stns

90

records

in full

[Values are pooled estimates at individual sites based on

the size-wetness-permeability pooling system of Equation 7.2 and the 5T rule]


107

The corresponding QMED values, standardised at each station by the 1972-81 value of

QMED, are shown in Figure 8.2b. In most cases, the QMED values for the 2nd

and 3rd

decade

exceed the 1st decade value. Although the 3

rd decade displays a number of remarkably high

QMED values, only the expected one third of stations have their largest QMED in that

decade. Figure 8.2c shows the estimated Q50 values, standardised at each station by the

1972-81 value. The majority of Q50 values in the 2nd

and 3rd

decades exceed those of the

1st decade, with more than half of these exceedances occurring during the 2

nd decade.

Figure 8.2: Decadal estimates for 50 stations (arranged in station-number order)

1972-81 1982-91 1992-2002 (a) Pooled growth factor, x50

(b) QMED (standardised to 1st decade)

(c) Q50 (standardised to 1st decade)

1972-81 1982-91 1992-2002

1972-81 1982-91 1992-2002


108

8.3.4 Individual results for five decades (26 stations)

There are 26 stations for which decadal estimates are possible for five decades. At some

stations, there are not a full ten years of data available, especially for the 1st decade. Only

stations with seven or more years of data available for the 1st decade are included. The

analyses were executed as for the 3-decade case above, with results presented in Figure 8.3.

Figure 8.3: Decadal estimates for 26 stations (arranged in station-number order)

(a) Pooled growth factor, x50

(b) QMED (standardised to 1st decade)

(c) Q50 (standardised to 1st decade)

1952-61 1962-71 1972-81 1982-91 1992-02

1952-61 1962-71 1972-81 1982-91 1992-02

1952-61 1962-71 1972-81 1982-91 1992-02


109

The 2nd

(1962-71) decadal value of x50 is larger than those of other decades at every station,

with an average value of approximately 1.7. This contrasts strongly with the results for the

5th

decade (1992-2001) where the average value is approximately 1.4. Figure 8.3b shows the

decadal standardised QMED values for which the largest values tend to be from the 2nd

(1962-71) and 5th

(1992-2001) decades. The Q50 values in Figure 8.3c also convey the

impression that the largest Q50 values occur for the 2nd

(1962-71) and 5th

(1992-2001)

decades.

8.3.5 Individual results for a different set of five decades

For 34 stations, it was possible to examine a different set of five decades, offset by five years.

Results for decades ending in 1966, 1976, 1986, 1996 and 2006 were found to be broadly

similar to those described in Section 8.3.4.

8.3.6 Outcome

Natural variability in climate can lead to flood series exhibiting so-called flood-rich and/or

flood-poor periods. The period over which flood data are available has been confirmed to

have an effect on flood frequency estimation. For example, there is some limited evidence –

by combination of Figure 8.3c and the equivalent result from offset decades (Section 8.3.5) –

that the period 1962-1976 was flood-rich.

This finding emphasises the importance of analysing the longest flood records available and,

of course, incorporating the most recent flood data. The incorporation of historical flood

information can also be important (see Box 8.2).

Box 8.2: Incorporation of historical flood data

8.4 Arterial drainage effect on pooled growth curve estimates

Pooled growth curves have been compared for pre and post-drainage records of 16 gauging

stations, eight of which are in the Boyne river basin (Hydrometric Area 07).

The recommended pooling procedure of Section 7.3.4 requires BFI, the baseflow index (see

Section 5.1 of Volume IV). For historical or hydrometric reason, some gauging stations are

unable to measure flow throughout the full range of low, medium, high and flood flows. At

the time of study, BFI values were not available for three of the 16 stations. Values of BFI

from nearby gauged stations were therefore substituted. [Editorial note: The recommended

procedure in such cases is to use the physical catchment descriptor BFIsoil in place of BFI.

However, the study reported was undertaken before completion of the BFIsoil model. It

Editorial note: Historical flood data was not central to the FSU project, in large measure

because of the extensive work done under other initiatives, most notably the National

Flood Hazard Mapping website (http://www.floodmaps.ie/). Cawley et al. (2005) discuss

historical floods in the Brosna, Corrib, Deel, Dodder, Griffeen and Lee catchments.

Guidance on procedures for incorporating historical flood information into the statistical

analysis of peak flows is nevertheless required. Bayliss and Reed (2001) advocate the use

of graphical methods. More theoretical methods are available in the literature, including:

Guo (1990), Guo and Cunnane (1991), Cohn et al. (1997) and Payrastre et al. (2011).

http://www.floodmaps.ie/


110

transpires that the BFI substitutions made for Stations 07011 and 30004 were somewhat less

appropriate than that made for Station 07007. BFI is itself influenced by arterial drainage.]

The results of the pooled estimation of x50, QMED and Q50 for pre-drainage and post-

drainage records at the 16 stations are summarised in Figure 8.4.

Figure 8.4: Comparison of pre and post-drainage flood statistics at 16 stations

The post-drainage growth factor (Figure 8.4a) is seen to be smaller than the pre-drainage

growth factor in every case. This reflects that the index flood QMED is itself greater after

Pre-drainage Post-drainage


(a) Pooled growth factor, x50

(b) QMED (standardised to value for pre-drainage series)

(c) Q50 (standardised to value for pre-drainage series)



111

drainage (see Figure 8.4b and also Section 2.4). The estimated Q50 values for the post-

drainage period exceed the pre-drainage values in ten of the 16 cases, although Station 24001

shows only a minor increase.

It is seen in Figure 8.4b and Figure 8.4c that one station (Station 30001 Aille at Cartonbower)

shows a contrary result, with QMED and Q50 appreciably reduced after drainage. This station

may warrant special investigation.

Wider effects of drainage on river flows have been extensively studied by (amongst others)

Lynn (1981), Robinson (1990) and Bhattarai and O’Connor (2004).

8.5 Implications for flood frequency estimation

8.5.1 Implications for pooling-group formation

Storage attenuation (indexed by FARL), catchment size and geographical location appear to

have a noticeable effect on the x100 growth factor, whereas the extent of peat cover does not.

It is therefore reassuring that differences in AREA appear in the recommended distance

metric (Equation 7.2) used to form pooling groups by the region-of-influence method (see

Section 7.3.2). The non-appearance of FARL and geographical location in the recommended

distance metric is less comforting.

Geographical location may be partially taken into account by SAAR but not uniquely so. It is

likely that users will want to take geographical location explicitly into account in some cases,

especially for locations in the East, not least around Dublin.

The lack of representation of FARL in the distance metric is a reminder of the weakness of

purely rule-based methods of flood frequency estimation. The practical recommendation is

that users should take particular heed of FARL values when diagnosing pooling groups that

appear heterogeneous.

Users need to be alert to the trade-off between tailoring the pooling-group constituency to the

particular characteristics of the subject catchment and retaining a sufficient number of

station-years in the pooling group. Adherence to the 5T rule of Section 7.3.3 is generally

recommended.

8.5.2 Respecting recent flood data

With regard to temporal effects, there is no clear evidence that respect for recent flood data

should be prioritised over respect for older flood data. This does not sit well with

engineering guidance that – with on-going global and land-use change – “The past is no

longer the key to the future, and the future is uncertain” (Irish Academy of Engineering,

2007). However, there is a one-sided nature to extreme events. Once they have been

observed, they are in the record forever. Until they have been observed, they are entirely

missing.

A theme from Ireland at Risk (Irish Academy of Engineering, 2007) is that we should not

wait for extreme events to be gauged before taking action. Although in part paradoxical, one

reaction could be to reinforce the practice of incorporating historical flood information (see

Section 1.3 and Box 8.2) into flood frequency estimation. “The past may not be the key to

the future. But neither is neglect of the past.”


112

9 Uncertainty estimation

9.1 Standard errors – an introduction

The standard error (SE) of an estimate of QT is an indication of how reliable that estimate is.

It is based on the assumption that the data upon which the estimate is based are randomly

drawn from a single population – an assumption that cannot easily be proven. If an infinite

number of similarly sized datasets were to be drawn from the same population and the value

of QT obtained from each set by the same procedure then the SE is defined as:

SE(QT) = Standard deviation of all the possible set of QT values.

This measure represents only the degree of scatter of the several estimates and does not refer

to whether the mean of these equals the true value in the population. If this equality holds,

the procedure whereby QT is calculated is said to be unbiased; otherwise it is said to be

biased.

In flood hydrology, randomness of annual maximum floods and lack of trend with time are

generally assumed. Likewise it is assumed that a single form of statistical distribution

describes all the AM flood series in a region or country. Such assumptions cannot be proven

and there is some evidence from Chapter 3 and Appendix E to suggest that Irish flood data

are not entirely trend free. The presence of low or high outliers (see Section 4.6) in some

datasets also makes interpretation difficult. If in reality the data are not drawn from a unique

homogeneous parent population then the SE cannot strictly be defined.

However, the SE concept is widely used in flood hydrology. Its value is determined on the

assumption of a unique homogeneous parent and the value of SE so obtained may be

considered as a lower bound on the true value, were it derivable. Hence the SE provides a

useful guide to the precision of the QT value obtained in any situation.

The aim in this chapter is to provide expressions or graphs which give an indication of the

order of magnitude of SE(QT) in single-site and pooled estimations of QT. The discussion is

restricted to the EV1 and GEV cases on the assumption that the LO and GLO (and LN2 and

LN3) cases would yield standard errors of the same order of magnitude.

9.2 Standard error of QMED estimation from gauged flood data

Let there be N annual maximum values at a gauged site. QMED is obtained as the median of

the annual maximum flow series:

QMED ≡ Median (Q1, Q2, … QN)

The Qi values are re-ordered from smallest to largest so that Q(1) < Q(2) < … < Q(N). If N is

odd, the single mid-value is taken as QMED. If N is even, QMED is taken as the average of

the two middle-most values.

The standard error (SE) of the mean in a random sample from a Normal distribution is:

Nσmean

XSE 9.1


113

The SE of the median in such a sample is about 25% greater:

Nσ1.253med

XSE

9.2

That the standard error of the median is 25% larger than that of the mean is less of a

drawback when applied to AM flood data, where it is helpful that QMED is unaffected by

low/high outliers in the data sample and has a return period precisely anchored at 2 years.

Since Irish flood data are slightly more skewed than the Normal distribution, SE(QMED) for

flood data will be slightly greater than by Equation 9.2. Simulation indicates that the

multiplier is increased from 1.253 to about 1.28 to 1.30, when N = 5.

Adopting the larger value of 1.30, Equation 9.2 can be written:

Nmean

QCV1.30med

QSE

9.3

and:

Nmed

Q

medQ

meanQ

CV1.30med

QSE

9.4

Taking average values of CV and of the ratio Qmed/Qmean for 110 Irish A1 + A2 stations into

account – from Table 4.2 these are seen to be 0.273 and 0.963 respectively – an approximate

value of the multiplier of QMED is reached as:

Nmed

Q963.0

1273.01.30

medQSE

9.5

Thus:

NQMED369.0QMEDSE 9.6

Here, Qmed has been written in the preferred FSU notation of QMED.

9.3 A comparison with FEH methods

The standard errors above are seen in Table 9.1 to be somewhat smaller than those quoted for

UK flood data in the FEH (Robson, 1999a) for records shorter than about 20 years. This

difference likely reflects the smaller CV and lower skewness of Irish data. But see Box 9.1.

Table 9.1: Typical standard errors when estimating QMED from annual maxima

Number of annual

maxima, N

Approx. standard error,

by Equation 9.6

Equivalent value inferred from AM

column of Table 12.3 of Robson (1999a)

5 0.165 QMED 0.218 QMED

10 0.117 QMED 0.138 QMED

15 0.095 QMED 0.109 QMED

20 0.083 QMED 0.079 QMED

50 0.052 QMED


114

Box 9.1: Comparison with uncertainty of FEH methods

One method of reducing the standard error of estimate is by “record extension” whereby a

short record at the subject site is extended by exploiting the correlation between the AM

series at the subject site and the AM series of one or more neighbouring sites (Fiering, 1963).

A variation on this approach to QMED refinement is used in Section 2.1.2.

9.4 Standard error of QT in single-site estimation

9.4.1 Method

Theoretical expressions for L-moment based estimates of QT in both EV1 and GEV cases

have been given by Lu and Stedinger (1992) as follows. For samples drawn from the EV1

distribution:

2

T 0.8046y0.4574y1.1128n

αQSE 9.7

where TQ is the estimate for the T-year flow event; α is the EV1 scale parameter;

1/T1lnlnyy T 9.8

is the EV1 (or Gumbel) reduced variate; and n is the number of observations in the sample.

For samples drawn from the GEV distribution:

1/23

3

2

210T kTakTakexpTaTaexpn

αQSE 9.9

where TQ is the estimate for the T-year flow event; a0(T), a1(T), a2(T) and a3(T) are

coefficients that depend on the return period T; α and k are scale and shape parameters

respectively; and n is the number of observations in the sample. The values for the

coefficients for different return periods are tabulated in Lu and Stedinger (1992). For

example:

For T=10, a0= -2.667, a1= 4.491, a2= -2.207, a3= 1.802;

For T=100, a0= -4.147, a1= 8.216, a2= -2.033, a3= 4.780.

Editorial notes: The comparison with the FEH is not straightforward because of the

different approaches taken to assessing the uncertainty of QMED. The values in the RH

column of Table 9.1 may not provide a fair reflection. The cited FEH table presents

estimates of the factorial standard error (FSE). To subtract 1.0 from these numbers and

present them as equivalent values of the relative SE appears inflationary. Use of

resampling methods – as in Robson (1999a) – would allow an authoritative comparison.

Robson found that, when N < 14, QMED can be estimated with slightly smaller standard

error by using the peak-over-threshold (POT) flood series. This gain may not transfer to

Ireland where – for historical and hydrometric reasons – AM flow data are curated to a

higher standard than POT data. Nevertheless, there may be merit in considering the POT

approach for short flood series (say N < 10) of high quality.


115

9.4.1 Relative standard error

The quantity chiefly discussed below is the relative standard error: SE(QT)/QT. In the EV1

case, this can be expressed as a function of the ratio /u and yT or equivalently as a function

of CV and yT or of L-CV and yT. Note that L-CV ≈ 0.5 CV in the EV1 case.

Equation 9.7 has been applied to the data of 85 A1 + A2 stations in which the values of u,

and QT were estimated from each site's data separately and the ratio SE(QT)/QT formed. The

true value of SE(QT) is not obtained by using estimated values of parameters in these

expressions. However, it is considered useful to examine the range of values so obtained

because part of their scatter is caused by inter-site heterogeneity, assuming such exists, as

well as by random sampling effects and unequal record lengths.

9.4.2 Relative standard errors under the EV1 assumption

Relative standard errors across the 85 sites are summarised in Box-plots 9.1 for six different

return periods. It is suggested that the interquartile range – i.e. the range of the middle 50%

of values – ought to give a good indication of the range within which the true SE values fall.

The extreme high and low values can be considered unrepresentative of the whole, and the

suggestion is therefore to focus on the box (rather than whisker) part of the box-plot (see Box

8.1 for an introduction to box-plots).

The results are presented a second time in Box-plots 9.2, along with theoretical EV1 relative

standard errors calculated from Equation 9.7, using the average sample size of 37 years and

for a range of L-CV values that span the values experienced among Irish AM flood data.

[Editorial note: The different style of the boxes in Box-plots 9.2 arises from a technical

difficulty in graphing.]

What these show – under the EV1 assumption – is that the relative standard error can be

taken to be between 5% and 7% for the 10-year quantile and between 6.5% and 10% for the

100-year quantile.

Box-plots 9.1: Relative SE of single-site quantile estimates – EV1 assumed

SE

(QT)/

QT a

s %

Based on 85 A1 + A2 stations

T = 5 T = 10 T = 25 T = 50 T = 100 T = 500

Relative Standard Error (At site/EV1)

0

2

4

6

8

10

12

14

16

18

T=5 T=10 T=25 T=50 T=100 T=500

se(Q

^T)/

QT %


116

Box-plots 9.2: As above but with lines superposed to show theoretical values of relative SE

(for N = 37 and a range of L-CV)

9.4.3 Relative standard errors under the GEV assumption

Corresponding results for the GEV case are presented in Box-plots 9.3 and Box-plots 9.4.

The theoretical values shown in Box-plots 9.4 are for a single value of the shape parameter

k = -0.1, which it is felt ought to cover the most extreme underlying population case arising

in Irish conditions. From the box-plots we can conclude, using the arguments as above, that

under the GEV assumption, the relative standard error can be assumed to be between 6% and

8% for the 10-year quantile and between 8% and 15% for the 100-year quantile. Although it

can be seen that the most extreme values are extremely large, these are not representative of

the true relative standard error.

Box-plots 9.3: Relative SE of single-site quantile estimates – GEV assumed

SE

(QT)/

QT a

s %

T = 5 T = 10 T = 25 T = 50 T = 100 T = 500

Relative Standard Error (Atsite/EV1)

LCV=0.15

LCV=0.3

LCV=0.25

LCV=0.2

LCV=0.1

0

2

4

6

8

10

12

14

16

18

T=5 T=10 T=25 T=50 T=100 T=500

se(Q

T)/

QT%

SE

(QT)/

QT a

s %


T = 5 T = 10 T = 25 T = 50 T = 100 T = 500

Relative Standard Error (At site/GEV)

0

10

20

30

40

50

60

70

80

90

100

T=5 T=10 T=25 T=50 T=100 T=500

se(Q

^T)/

QT %


117

Box-plots 9.4: As above but with lines superposed to show theoretical values of relative SE

(for k = -0.1, N = 37 and a range of L-CV)

9.5 Standard error of pooled estimate of xT and of QT

9.5.1 Simulation method

An estimate of the order of magnitude of the pooled estimate of xT, SE(xT), for selected

values of T, was obtained by simulation for each of the 85 stations. Each gauging station was

selected in turn as the subject site and the following procedure applied:

Step 1 Identify the gauging stations in the subject site’s pooling group using the

recommended distance metric (i.e. dij values given by Equation 7.2) to achieve a

minimum of 500 station-years of data in the pooling group (so as to comply with the

5T rule for the 100-year quantile).

Step 2 Random samples are drawn from EV1 populations for the subject site and for each

site in the pooling group. For each site, the sample size is taken as equal to the length

of the observed record at the site and the parameter values u and are those estimated

from the observed record by L-Moments.

Step 3 The sample QMED is obtained for the subject site.

Step 4 The L-CV value is obtained for each sample in the pooling group and the weighted

average of these is calculated (using Equation 7.3) to yield the pooled L-CV.

Step 5 The pooled L-CV is used to determine the pooling group’s EV1 growth curve

parameter β (using Equation 7.10)

Step 6 The subject site’s ))/11ln(ln()2ln(ln1 TxT is calculated for T = 5, 10,

25, …, 500 years

Step 7 The subject site’s QT = QMED × xT is calculated for T = 5, 10, 25, …, 500 years.

Step 8 Steps 2 to 7 are repeated 1000 times to provide 1000 values of xT and QT at the

subject site and the standard errors SE(xT) and SE(QT) are calculated for the subject

site by the following equations:

SE

(QT)/

QT a

s %

T = 5 T = 10 T = 25 T = 50 T = 100 T = 500


118

N

1i

M

1m

2

T

i

T

im

T

i,

TX

xx

M

1

N

1)SE(x 9.10

N

1i

M

1m

2

T

i

T

im

T

i,

TQ

QQ

M

1

N

1)SE(Q 9.11

where m

T

i,x and m

T

i,Q are the estimated T-year growth factor and T-year flood quantile

respectively at site i at the mth

repetition; T

ix and T

iQ are the mean of these estimated

measures; T

ix and T

iQ are the assumed true T-year growth factor and T-year quantile

at site i; N is the number of sites in the pooling group and M is the number of

repetitions. These expressions average the standard error over all the sites in the

pooling group to take heterogeneity in the pooling group into account. It should be

noted that the actual standard error for any individual site could be smaller or larger

than the calculated value depending on the values of L-CV and L-skewness at the

particular site; the extreme values in the box-plots below illustrate the range that

could occur.

9.5.2 Results based on EV1 simulations

Step 1 to Step 8 were repeated for each of the 85 stations. The resultant relative standard

errors for xT and QT are presented in Box-plots 9.5 Box-plots 9.6 respectively. The same

remarks apply to the interpretation of these plots as to those for the single-site quantile

estimates presented in Section 9.4.

The relative standard errors for xT range from 1.1% to 1.2% at T = 10 to 1.8% to 2.0% at

T = 100 years. The corresponding relative standard errors for QT differ little with return

period, ranging from 3.7% to 8.5% at T = 10 to 4.0% to 9.0% at T = 100 years.

Box-plots 9.5: Relative standard error in xT – pooled EV1 simulations

Relative Standard Error of Growth Curve (Pooled/EV1-Simulation)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

T=5 T=10 T=25 T=50 T=100 T=500

se(X

^T)/

XT %

SE

(xT)/

xT a

s %


T = 5 T = 10 T = 25 T = 50 T = 100 T = 500


119

Box-plots 9.6: Relative standard error in QT – pooled EV1 simulations

9.5.3 Results based on GEV simulations with k = -0.1

The simulation procedure described above was also applied with the GEV distribution used

instead of the EV1 in Step 2 to Step 6. Because the shape parameter k can be very variable –

and unreasonably large or small when estimated from single-site records – a single value of

k = -0.1 was adopted in all simulations. This is considered a conservative choice in the

context of Irish flood data.

In Step 4, L-skewness and average L-skewness are calculated as well as L-CV. In Step 5, the

GEV growth curve parameters and k are calculated by Equations 7.8 and 7.6. In Step 6, the

expression for xT given in Equation 7.5 is used.

The results are presented in Box-plots 9.7 and Box-plots 9.8. The standard error for xT

ranges from 1.4% to 1.8% at T = 10 to 3.7% to 5.0% at T = 100, while the corresponding

standard errors for QT show an increasing trend, from 3.8% to 9.0% at T = 10 to 4.6% to

10.6% at T = 100.

Box-plots 9.7: Relative standard error in xT – pooled GEV simulations for k = -0.1

Relative Standard Error of Quantile Estimate (Pooled/EV1-Simulation)

0

5

10

15

20

25

30

35

T=5 T=10 T=25 T=50 T=100 T=500

se(Q

^T)/

QT %

SE

(xT)/

xT a

s %

T = 5 T = 10 T = 25 T = 50 T = 100 T = 500

Based on 85 A1 + A2 stations S

E(x

T)/

xT a

s %


T = 5 T = 10 T = 25 T = 50 T = 100 T = 500


120

Box-plots 9.8: Relative standard error in QT – pooled GEV simulations for k = -0.1

9.5.4 Summary

The ranges for six return periods and both EV1 and GEV distributions are summarised in

Table 9.2. Since the 100-year flood is used for design purposes for many major projects in

Ireland, it is worth noting that its standard error can be considered to be approximately 10%

of Q100.

Table 9.2: Relative standard errors for growth factors xT and quantile estimates QT

Pooled EV1 case Pooled GEV case (k = -0.1)

Return

period, T

Relative SE (%)

in xT

Relative SE (%)

in QT

Relative SE (%)

in xT

Relative SE (%)

in QT

5 0.7 to 0.8 3.7 to 8.4 0.8 to 1.0 3.7 to 9.0

10 1.1 to 1.2 3.7 to 8.5 1.4 to 1.8 3.8 to 9.0

25 1.4 to 1.6 3.8 to 8.7 2.3 to 3.0 4.1 to 8.9

50 1.6 to 1.8 3.9 to 8.9 3.0 to 3.9 4.3 to 9.6

100 1.8 to 2.0 4.0 to 9.0 3.7 to 5.0 4.6 to 10.6

500 2.1 to 2.3 4.1 to 9.3 5.4 to 7.3 5.7 to 12.2

[Editorial note: These standard errors seem surprisingly small. Perhaps the assumptions

made are unduly restrictive. The scope for underestimation of flood quantiles is intrinsically

much greater than the scope for overestimation. This is evident in the lop-sided whiskers in

the standard errors of the quantile estimates shown in Box-plots 9.6Box-plots 9.8.]

9.6 Standard error of QT based on PCD estimate of QMED and pooled xT

Using the index flood method:

TmedT xQQ 9.12

SE

(QT)/

QT a

s %


T = 5 T = 10 T = 25 T = 50 T = 100 T = 500


121

The variance of the T-year event is approximated using Taylor series expansion as:

)xQVar()QVar( TmedT

9.13

)x,QCov()xE()QE(2)xVar()QE()QVar()xE( TmedTmedT

2

medmed

2

T

Ignoring effects arising from:

Bias in the sample median

Bias in the sample L-moment ratio

Covariance between the sample median and the estimated regional growth curve

leads to:

)xVar(|)(Q)QVar(|)(x)QVar( Tm

2

medmedm

2

TT 9.14

The expression for Var(QT) is dominated by the Var(QMED) term as Var(xT) affects only the

3rd

or 4th

decimal point in the value of the expression. Hence the expression effectively

reduces to:

)SE(Qx)SE(Q medTT

)Q(x)SE(QxQ)SE(Q medTmedTTT

medmedTT Q)SE(QQ)SE(Q

Applying the approximate SE in QMED from Equation 9.6 yields:

med

medTT

Q

Q

N

0.369Q)SE(Q

i.e. N

0.37Q)SE(Q TT 9.15

Therefore, SE(QT)/QT is independent of T but dependent on the value of N associated with

the worth of a PCD-based estimate of QMED.

Experience suggests that the worth of a PCD-based estimate of the index flood is typically

equivalent to about one year of AM data, e.g. Nash and Shaw (1965), p I.342 of NERC

(1975) and Hebson and Cunnane (1987). [Editorial note: Experience in the FEH was much

the same, with the FSE of 1.58 quoted for the recommended PCD model of QMEDrural

slightly inferior to the FSE of 1.52 quoted for estimation of QMED from one year of AM

flow data.] Inserting N=1 into Approximation 9.15, the relative standard error of QT is

estimated to be 0.37. This value is consistent with FSE = 1.37 for the PCD-based estimate of

QMED on rural catchments (see Equation 2.8 in Section 2.2.5). [Editorial note: As

discussed in Box 9.1, comparison of the relative standard error of X with the FSE of X minus

1.0 does not appear to be even-handed. The Equation 2.8 PCD-based model is especially

good of its kind. It is likely worth a little more than one year of AM flow data.]

The finding – in derivation of Approximation 9.15 – that the uncertainty in QT is dominated

by the uncertainty in QMED and is independent of return period may seem counterintuitive.

The result applies regardless of whether the growth curve is obtained using EV1 or GEV.


122

[Editorial note: This finding does not sit comfortably with the experience of many applied

hydrologists that growth-curve estimation is a skilled task that plays a crucial role in

estimating the T-year flood, QT. It is particularly helpful that Section 10.4 includes a number

of worked examples demonstrating the subtlety of flood estimation problems and their

resolution.]

[Editorial note: While it is possible to judge the relative uncertainty of different methods of

single-site analysis and the relative uncertainty of different methods of pooled analysis, the

inter-comparison of pooled and single-site methods is exceptionally challenging. A practical

reason why pooled methods are often to be preferred is the desire to promote consistency in

flood estimates obtained for different sites along a river and on different catchments that are

hydrologically similar.]


123

10 Guidelines for determining QT

10.1 Introduction

10.1.1 Single-site or pooled analysis?

QT is the T-year return period flood, i.e. the flood peak with an annual exceedance probability

of 1/T. If sufficient data exist and the required return period is short, QT may be determined

by single-site analysis. Although adequate AM flow data may exist, interpretation can still be

problematic if the probability plot is of unusual shape. In such a case, it may be necessary to

combine single-site and pooled approaches.

For estimating floods of long return period, a combined single-site and pooled approach is

recommended. Where the subject site is ungauged, QMED must be estimated from the

catchment descriptor model of Section 2.2.5 and enhanced where possible by transferring

information from a nearby or similar gauged site (see Sections 2.6 and 2.7, and Chapter 11).

10.1.2 Probability plots

It is essential to examine a probability plot when analysing AM flow data. When choosing

the statistical distribution or estimation procedure to adopt, practitioners are often greatly

influenced by the appearance of single-site data on probability plots. However, it must be

borne in mind that random samples from a particular statistical population show considerable

inter-sample variation when displayed on probability plots. Some samples do not display

convincing straight-line behaviour even when the parent distribution is known to be a straight

line on such a plot.

Plots in Section 5.1.2 show nine random samples from an EV1 population with CV=0.33, a

value slightly higher than the average CV for Irish Grade A1 station data. Sample sizes are

25 in EV1 Probability Plot 5.1 and 50 in EV1 Probability Plot 5.2. It can be seen that

departures from straight-line behaviour are more striking in the smaller samples than in the

larger samples. Some of the samples of size 25 would have been given a low linearity score

in the assessment of probability plots described in Chapter 5. However, relatively few of the

samples of size 50 depart too markedly from straight-line behaviour.

10.1.3 Factors to be borne in mind

It can be convenient to refer to the estimate of QMED made from a site’s own flood data as

the at-site QMED. Factors to be borne in mind when selecting a design flood magnitude

include:

i The site or pivotal station’s AM probability plot on EV1 and LN scales. [Editorial

note: There can be a trade-off for the practitioner between adopting a type of

probability plot in which they have gained experience and not pre-judging the most

appropriate probability distribution.]

ii The range of variation that can occur between random samples drawn from a single

population i.e. recognising that a single-site estimate may differ from the true value;

iii One or more pooled growth curves;

iv Preference in single-site analysis for a 2-parameter distribution (rather than a

3-parameter distribution);


124

v In homogenous pooling groups, the overall statistical superiority of “At-site QMED

plus pooled growth curve” estimation (see Box 7.2 in particular);

vi The robustness of “At-site QMED plus pooled growth curve” estimation to small

departures from pooling-group homogeneity;

vii Consideration of straight-line versus concave-downwards curves on probability plots,

and whether an implied upper bound on flood magnitude is physically meaningful;

viii Comparison of design water levels (corresponding to the proposed estimate of QT)

with “ground truth”, i.e. does the flood estimate make sense in terms of what has been

observed (or not observed) locally?

ix Consideration of the scope for underestimation when the “At-site QMED plus pooled

growth curve” estimate of QT is smaller than some of the observed floods at the

station;

x The credibility of outliers (i.e. unusually large or small flood values) and how they

might be verified, assessed and accommodated within the analysis.

10.2 Determining QT by single-site analysis

10.2.1 General guidance

QT may be estimated by single-site analysis provided that both:

The record length N is ten years or longer;

The required return period T is less than N, or not appreciably larger than N and

certainly no greater than 2N.

Ordinarily, a 2-parameter distribution should be used in single-site estimation: either the EV1

or LN2 distribution. In the case of EV1, the parameters should be estimated by L-moments

(see Section 10.2.3 below). In the case of LN2, the parameters can be estimated from the

mean and standard deviation of the logarithms of the AM data (see Section 10.2.4 below).

Use of L-moments on the untransformed AM data is also statistically efficient for fitting the

LN2 distribution, should the user prefer to use L-moment methods throughout.

It is assumed that the basic statistics of the station’s AM flow data have been computed and

that summary statistics (see Chapter 4) have been examined to see where they fall in the

range of observed values among all gauged catchments. In other words, it is assumed that the

user has carried out a thorough check of the station’s data: (i) as they stand, (ii) in the context

of data from similar types of catchment and (iii) in the context of the overall national dataset.

A starting point for consideration of catchment similarity is the distance metric dij used in

pooling-group construction, i.e. Equation 7.2. However, similarity in terms of other PCDs

may be important if the subject catchment has unusual features.

10.2.2 Plotting positions

The station’s AM flow values should be displayed on an EV1 based probability plot using

Gringorten plotting positions, and (using logged AM values) on a lognormal probability plot

using Blom plotting positions. These plotting positions are defined in Box 10.1.


125

Box 10.1: Plotting positions

10.2.3 Parameter estimation for EV1 distribution

EV1 parameters u and are estimated by the method of L-moments (see Box 4.2) as:

n2

λ

n2

M2Mα 2100110

10.1

and

α0.5772λα0.5772Mu 1100 10.2

Then:

TT yαuQ 10.3

where:

))T

1n(1n(yT 10.4

is the EV1 (or Gumbel) reduced variate.

EV1 (or Gumbel) plotting positions are defined by the EV1 reduced variate:

))n(Fn(y ii

where:

N,...2,1,i,0.12N

0.44iFi

is the Gringorten plotting-position formula.

These positions are suitable for use with the EV1 and GEV distributions.

Plotting positions for the Logistic (LO) and Generalised Logistic (GLO) distributions also

apply the Gringorten plotting-position formula but with the Logistic reduced variate:

i

iL

iF

F1ny

Plotting positions for the LN2 and LN3 distributions typically apply the Blom plotting-

position formula:

N...,2,1,i,1/4N

3/8iFi

with the standardised Normal variate:

))(FΦy i

1N

i

where:

Φ(y) is the standardised Normal distribution function.


126

10.2.4 Parameter estimation for LN2 distribution

Writing z = ℓnQ, LN2 parameters are estimated by the method of moments (see Box 4.1) as:

n

1i

iz zn

1μ 10.5

and

21

n

1i

2

iz zz1-n

1σ

10.6

Then

TZ

T eQ 10.7

where )T

1(1ΦZ 1

T and (F)Φ 1is the standardised inverse Normal distribution.

10.3 Determining QT from pooled data

10.3.1 When flood data are available at the subject site

Determining QMED

If a gauging station exists at the subject site then QMED is obtained directly from the AM

series of floods at that site. If the record is long, the standard error in QMED will be

relatively small (see Section 9.2). Since gauged data are so much better than information

obtained from a PCD-based formula, it is recommended to base the QMED value on the

observed data, even if the record is short.

An important proviso to this guidance is that flood data from upstream/downstream gauging

stations – as well as from other neighbouring stations – should be checked to see if the

QMED values obtained from the same period of record as those at the subject site are smaller

or larger than the long-term QMED at these sites. The ratio of long-term QMED to short-

term QMED can then be applied to the subject site’s QMED as an adjustment factor. Such

adjustments are discussed in Section 2.1.2.

Data transfer techniques (from gauged to subject site) are discussed in Chapter 11, in addition

to the detailed worked example of Section 2.7.

Determining xT

It is recommended that the growth factor be determined from pooled data in a pooling group

selected using the distance metric dij of Equation 7.2 and the 5T rule of Section 7.3.3. When

applied to estimation of the 100-year flood, the pooling group will be extended until a

minimum of 500 station-years of data are incorporated. It is suggested that such a pooling

group be used for all return periods rather than constructing a separate group for each return

period of interest.

If the hydrologist wishes to depart from this advice and to use geographical regions or

exclude certain types of catchment, the reasons for doing so will need to be documented.

Some diagnosis and amendment of the pooling group may be warranted if the pooled flood


127

data are strongly heterogeneous. However, it is important that a station with exceptional

flood data is excluded from the pooling group only if its physical characteristics are

fundamentally dissimilar to those of the subject catchment.

Consideration must also be given to the choice of a 2-parameter or 3-parameter distribution in

line with the discussion in Section 10.5 below. In the event that the single-site estimate of the

flood growth curve is steeper than the pooled one, consideration should be given to using a

combination of the single-site and pooled estimates of the flood growth curve or of the flood

frequency curve. While there is a possibility that this might lead to over-design it avoids (or

at least moderates) the less desirable outcome of under-design.

10.3.2 When the subject site is ungauged

The procedure for determining xT is as above. However, QMED has to be estimated from

PCDs using the methods set out in Chapter 2. Where appropriate, an urban adjustment factor

is to be applied. Indeed, it is recommended that the urban adjustment factor is habitually

applied, even when the catchment is essentially rural.

As described in Section 2.7, data transfers are a crucial part of QMED estimation at an

ungauged site. The recommended procedure requires the user to assess the most appropriate

data transfer by making a reasoned selection of the pivotal catchment. This is the gauged

catchment judged to be most relevant to the specific flood estimation problem.

Unless the subject and pivotal catchments are both rural, the recommended procedure

requires a clear head. It is the estimate of QMEDrural that is transferred from the pivotal site

to the subject site. Thus, the procedure is:

A value of QMEDrural is back-calculated from the gauged value of QMED at the

pivotal site by applying the urban adjustment factor of Equation 2.9 in reverse;

The data transfer is made from the pivotal site to estimate QMEDrural at the subject

site;

The relevant urban adjustment is made to obtain QMED at the subject site (i.e

applying Equation 2.9 in the normal manner).

The urban adjustment factor (UAF) is determined from the urban extent (URBEXT)

according to Equation 2.13. The procedure is further complicated if the user decides that

two or more QMED values at gauged sites are equally relevant, and a dual or multiple

transfer is required.

10.4 Characteristic examples of probability plot behaviour

10.4.1 Good straight-line behaviour

A first characteristic example is provided by Station 25016 Clodiagh at Rahan, which has 42

years of AM flow data. EV1 Probability Plot 10.1 shows near straight-line behaviour. In

Appendix H, it is assigned a score of 4 (meaning “good agreement with straight line”) and a

curve pattern of L2 (meaning “little deviation from straight line”).


128

EV1 Probability Plot 10.1: Single-site and pooled estimates, Station 25016

The Hazen skewness for this flood series is 0.62: considerably less than the theoretical value

of 1.14 in the EV1 distribution. Nevertheless, it seems reasonable to suggest that the 50-year

flood (or possibly even the 100-year flood) could be estimated by the EV1 distribution fitted

to the site AM flows in this case. From Box-plots 9.1, the standard error of Q100 in such a

case is expected to be approximately 6.5% to 10%, on the assumption that the underlying

parent distribution is indeed EV1.

The CV of the site AM flows is (at 0.22) rather lower than the national average of 0.27 for

110 A1 + A2 stations shown in Table 4.2. The pooled growth curves shown in EV1

Probability Plot 10.1 are nevertheless in general agreement with the site AM flows.

Despite the good behaviour of the single-site frequency analysis, it is reasonable to adhere to

the default recommendation to adopt the pooled estimate when estimating a design flood of

return period longer than the gauged record length of 42 years.

10.4.2 Good straight-line behaviour but single-site and pooled disagree

A second important example is provided by Station 09001 Ryewater at Leixlip which has 48

years of AM flow data. EV1 Probability Plot 10.2 shows excellent straight-line behaviour.

In Appendix H, it is assigned a score of 5 (meaning “very good agreement with straight line”)

and a curve pattern of L1 (meaning “perfect straight line”).

At 1.17, the Hazen skewness is practically equal to the theoretical value of 1.14 in the EV1

distribution, confirming the good fit of this distribution to the data. It seems reasonable to

suggest that the 50-year flood (or possibly even the 100-year flood) could be estimated by the

EV1 distribution fitted to the station’s AM data. From Table 9.2, the relative standard error

of Q100 in such a case is expected to be approximately 4% to 9%, on the assumption that the

underlying parent distribution is indeed EV1.

The CV of the site AM flows is (at 0.44) appreciably higher than the national average of 0.27

for 110 A1 + A2 stations shown in Table 4.2, and higher than that of other stations in the

hinterland. In consequence, the pooled growth curves shown in EV1 Probability Plot 10.2

are appreciably less steep than the single-site curve. Their adoption would lead to

underestimation of design floods.

25016 RIVER CLODIAGH @ RAHAN

EV1

2 5 10 25 50 100 5000

10

20

30

40

50

60

-2 -1 0 1 2 3 4 5 6 7 8EV1 y

AM

F(m

3/s

)AMF data

At-site EV1 fit

Pooled EV1 fit

pooled GEV fit

AM

flo

w (

m3 s

-1)

-2 -1 0 1 2 3 4 5 6 7


+ AM flow data

–– Single-site EV1 fit

–– Pooled EV1 fit

–– Pooled GEV fit


129


However much faith is placed in the superiority of the pooled approach in general, it is a fact

that the pooled EV1 estimate of Q50 in this case has been exceeded three times in 48 years.

Indeed, the pooled GEV estimate of Q50 has been exceeded six times in 48 years! This

evidence cannot be overlooked when a final design flood is selected, even for long return

periods. It could be that the observed sample displays a gradient on the probability plot that

is at the upper end of the scale of steepness, and that the sample has come from a population

that actually has a less steep growth curve, i.e. with lower CV. However, no practical design

project would ignore the single-site analysis in such a case.

10.4.3 Concave upwards behaviour with outlier

Another important case is exemplified by Station 09010 Dodder at Waldron’s Bridge, which

has a relatively short record with just 19 years of AM flow data. Even if the underlying

parent growth curve is a straight line, the short record makes it more likely to exhibit

departure from linearity on the probability plot (see examples in EV1 Probability Plot 5.1).

EV1 Probability Plot 10.3 shows marked concave upwards behaviour. In Appendix H, it is

assigned a score of 2 (meaning “poor agreement with straight line”) and a curve pattern of U2

(meaning “severe concave upwards”).


9001 RIVER RYEWATER @ LEIXLIP

2 5 10 25 50 100 500

0

20

40

60

80

100

120

140

-2 -1 0 1 2 3 4 5 6 7 8EV1 y

AM

F(m

3/s

)AMF data

At-site EV1 fit

Pooled EV1 fit

Pooled GEV fit

9010 RIVER DODDER @ WALDRON'S BRIDGE

500100502510520

50

100

150

200

250

300

350

400

-2 -1 0 1 2 3 4 5 6 7 8EV1 y

AM

F(m

3/s

)

AMF data

Atsite GEV fit

Pooled GEV fit

Pooled EV1 fit

+ AM flow data




AM

flo

w (

m3 s

-1)

-2 -1 0 1 2 3 4 5 6 7


+ AM flow data

–– Single-site GEV fit



AM

flo

w (

m3 s

-1)

-2 -1 0 1 2 3 4 5 6 7


Hurricane Charlie

25/26 Aug 1986


130

This gauged record in South Dublin has attracted much comment already. In Section 2.5.2, it

is noted that the gauged value of QMED is much greater than indicated by the PCD model of

QMEDrural. Heavy urbanisation provides only a partial explanation of this and Bruen et al.

(2005) suggest that flood growth curves in this mid-Eastern region may be characteristically

steeper than in other parts of Ireland.

In Section 8.1, the station’s flood growth behaviour was noted to be the most discordant

amongst 88 A1 + A2 stations across Ireland. The variability and skewness of the AM flows

are both exceptionally high (CV = 0.86 and H-skew = 3.24; L-CV = 0.42 and L-skew = 0.42).

These are amongst the highest such summary statistics across the entire FSU dataset.

It was noted in Section 8.1.2 that the three most discordant stations (one of them Station

09010 Dodder at Waldron’s Bridge) are all close to Dublin and are all heavily urbanised.

Thus, flood behaviour at this site is exceptional both for the typical size of AM floods (high

value of QMED) and for their great variability (yielding an unusually steep growth curve). In

terms of flood risk, this might be viewed as the worst of all worlds. In mitigation, it is noted

that the incidence of Hurricane Charlie in August 1986 accounts for some of the high

curvature in EV1 Probability Plot 10.3.

While a 3-parameter GEV distribution provides a good graphical fit in this case, it has to be

borne in mind, from a theoretical point of view, that GEV single-site estimates have high

standard error especially with short datasets. That means that other similar sized later records

might not show the same pattern nor be so steep. Hence, notwithstanding the good fit of the

single-site GEV, it would normally be recommended that a pooled estimate be used for flood

estimation at this site except for estimates at short return period (T < 25 years). Because

other stations included in the pooling group inevitably have much smaller CV and skewness

than the Station 09010 data, the pooled growth curve is much lower than the single-site

growth curve. However, the pooled growth curve looks exceptionally low in comparison

with the observed data.

[Editorial note: This is an exceptionally difficult catchment to handle. It should be noted

that the three most discordant of 88 A1 + A2 catchments – all close to Dublin and all heavily

urbanised – were excluded from the simulation experiments reported in Chapter 9. This may

in part account for the relatively modest standard errors presented in Table 9.2. Whether the

three discordant catchments – Stations 08005 and 09002 and Station 09010 itself – were

made available for admission to the Station 09010 pooling group is unclear. The decision of

many analysts to exclude records from urbanised catchments when estimating flood growth

on a rural catchment is understandable. Important flood estimation problems in urbanised

catchments must, however, find a way of incorporating all available data.]

The hydrologist has to balance the weight of published evidence – indicating the advantages

of pooled over single-site methods – and their own beliefs about the representative nature of

the observed data and whether the behaviour shown could be repeated in another similar

length of record. In the present example for Station 09010, one has to ask whether the

single-site GEV estimate of Q100 ≈ 350 m3s

-1 is credible. The largest flood on record

(269 m3s

-1) was caused by Hurricane Charlie in August 1986. Its counterpart on the

neighbouring Dargle catchment had earlier been equalled twice in the previous 80 years.

Undoubtedly therefore, the flow of 269 m3s

-1 on the Dodder could occur again. But this does

not necessarily support the figure of Q100 = 350 m3s

-1. On the other hand, it is manifest that


131

the pooled curve is definitely too low. The pooled estimate of Q100 has been exceeded four

times in only 19 years!

This sort of dilemma presents itself whenever the subject site record has a CV and skewness

which are considerably larger than average (see also the example in Section 10.4.2). In such

cases, a prudent approach is to seek out and interpret historical information about flood

frequency on the particular river. In other words, it is necessary to reinforce the single-site

analysis in some way before crossing out the pooled analysis. Some compromise approach

that respects the single-site analysis at short return period but tempers the steepness and

curvature of its growth curve is another possibility. Regardless of the method adopted, it is

advisable in such cases not to place too much faith in estimates of QT for large T.

Somewhat similar behaviour is found for Station 29011 River Dunkellin at Kilcolgan Bridge

(N = 22 years, CV = 0.30, H-skew = 3.37, one notably large flood) and Station 36015 Finn at

Anlore (N = 33 years, CV = 0.32, H-skew = 2.62, two notably large floods). But the CV

values for these stations are not exceptionally large. It is the combination of high variability

and high skewness that makes Station 09010 particularly problematic. [Editorial note: The

FSU recommendation is to favour L-moment ratios (L-CV and L-skew) over conventional

moment ratios (CV and H-skew). Use of a dual currency risks confusion.]

10.4.4 Convex behaviour

Convex (or concave downwards) probability plots typically arise in cases where either there

are a number of high values which are almost equal or where there are one or more

exceptionally low annual maxima in the series. A fine example of a data series exhibiting

such behaviour is Station 25002 Newport at Barrington’s Bridge with 51 years of record (see

EV1 Probability Plot 10.4).


The CV of 0.16 and Hazen skewness of -0.65 are both amongst the lowest values recorded in

the entire FSU dataset. While a 3-parameter GEV distribution is seen to provide a very good

graphical fit, it has to be borne in mind from a theoretical point of view that GEV single-site

estimates have high standard error with short datasets (although 51 years is not considered

short). More particularly, the positive value of the curvature parameter (k = 0.55) of the

fitted GEV distribution implies an upper bound to floods on this catchment. The upper bound

+ AM flow data

–– Single-site GEV fit



AM

flo

w (

m3 s

-1)

-2 -1 0 1 2 3 4 5 6 7


140

120

100

80

60

40

20

0


132

of 78.39 m3s

-1 is only 5% greater than the three largest AM floods (each nominally assigned a

peak flow of 74.64 m3s

-1) on 3 December 1960, 12 December 1964 and 7 October 1967.

The closeness of the implied upper bound is not reasonable from a hydrological point of

view. While it might be satisfactory to use the fitted distribution up to a return period of

25 years, use of a pooled estimate is to be recommended for flood estimation at this site.

Because most other stations included in the pooling group will more than likely have

skewness greater than that of the site data, the pooled growth curve is less convex than the

single-site curve. Judgement of 50 and 100-year flood estimates on this catchment may

require that the pooled estimates are interrogated further in the context of historical

information about flooding on the river. It is known that this river – in common with the

neighbouring Mulkear – has been embanked since the 1920’s and that these embankments are

overtopped in about one out of every five years. This may go some way to explaining why

the three largest floods on record are exactly the same: each nominally assigned a peak flow

of 74.64 m3s

-1. [Editorial note: Station 25003 was amongst stations rejected from most

analyses in this volume, for reasons summarised in Appendix B. See also the further

discussion of upper bounds in Section 10.5.4.]

Other examples of this kind of behaviour are provided by Station 25021 Little Brosna at

Croghan (N = 44 years, CV = 0.14, H-skew = -0.13) and Station 34024 Pollagh at Kiltimagh

(N = 28 years, CV = 0.12, H-skew = -0.39). It is noted that two of these three examples are

based on records which are considered to be among the longest in the study.

10.4.5 Unclear behaviour with extreme outlier

An extreme example of a series with a high outlier is provided by Station 08009 Ward at

Balheary (N = 11 years, CV = 1.41, H-skew= 5.43) where ten of the 11 AM flows available

are less than 12 m3s

-1 but the largest is recorded as 53.6 m

3s

-1 on 12 June 1993. The single-

site and pooled estimates are shown in EV1 Probability Plot 10.5. The small sample-size

exacerbates the problem of trying to estimate a design flood at this location. This station has

by far the highest recorded skewness in the whole dataset, though it has to be acknowledged

that skewness measured from such a short record is unreliable.


8009 RIVER WARD @ BALHEARY

2 5 10 25 50 100 500

0

10

20

30

40

50

60

70

80

90

-2 -1 0 1 2 3 4 5 6 7 8EV1 y

AM

F(m

3/s

)

AMF data

Atsite EV1 fit

Pooled EV1 fit

Pooled GEV fit

AM

flo

w (

m3 s

-1)

-2 -1 0 1 2 3 4 5 6 7


+ AM flow data





133

[Editorial notes: EV1 Probability Plot 10.5 shows data for 15 rather than 11 years. Four

annual maxima (for the 1979, 1981, 1982 and 1983 water-years) were omitted in the main

FSU analysis because they derived from notably incomplete records. In any case-specific

application, it is good practice (as here) to investigate exclusions and consider their

incorporation. As noted in Box 4.4, a recent study has queried both the completeness of

records at Station 08009 and the validity of the peak flow recorded on 12 June 1993.]

The single-site and pooled estimates for this station are fundamentally incompatible, and no

unique recommendation can be made. Further work might consider the following points:

The meteorological conditions leading to the June 1993 storm should be examined

and the likelihood of these being repeated anywhere in the region (including the

subject site) should be assessed.

Even if the rating curve or other features of the measurement are less than

satisfactory, the water level achieved locally will be known. This could be used as the

basis of the design of important works and the floor levels of new dwellings.

The hydrologist has to balance the competing beliefs that (i) the June 1993 flood was

so large that it could never happen again and (ii) if it happened once, it could occur

again. It is relevant to note that the Hurricane Charlie (August 1986) flooding in Bray

was so large that many believed it unrepeatable until it was discovered that almost

exactly similar events had previously occurred in 1904 and 1932. This underscores

the importance of historical review.

Further examples of high outliers – but not of such an extreme nature – are provided by

Station 36021 Yellow at Kiltybarden, Station 36031 Cavan at Lisdarn and Station 26006

Suck at Willsbrook. See also Table 4.3 in Section 4.6. Low outliers (e.g. Table 4.4) may

also be unduly influential in single-site estimation.

10.4.6 Irregular behaviour

Other kinds of irregular behaviour on probability plots include an elongated S shape or a

noticeable change of slope between lower and upper segments of the graph. Examples

include Station 09002 Griffeen at Lucan (N = 25 years) and Station 26008 Rinn at Johnston's

Bridge (N = 50 years). The single-site and pooled estimates for these stations are displayed

in EV1 Probability Plot 10.6EV1 Probability Plot 10.7 respectively.


9002 RIVER GRIFFEEN @ LUCAN

2 5 10 25 50 100 5000

5

10

15

20

25

30

35

40

-2 -1 0 1 2 3 4 5 6 7 8EV1 y

AM

F(m

3/s

)

AMF data

Atsite EV1 fit

Pooled EV1 fit

Pooled GEV fit

+ AM flow data




AM

flo

w (

m3 s

-1)

-2 -1 0 1 2 3 4 5 6 7



134

Station 09002 is particularly problematical because there is a noticeable increase in slope

caused by three large floods in the record. This catchment has experienced an increase in its

urban fraction, especially in recent decades although no strong trend is evident in the time

series shown for this station in Figure 8.1. QMED is 5.25 m3s

-1. Adopting a relatively large

(for Ireland) value of x100 = 2.5 yields a Q100 of 13 m3s

-1. Yet this value has been amply

exceeded three times in 25 years. Additional local knowledge must be gained about the

meteorological and physical conditions leading to these floods. [Editorial note: Some

analyses of Station 09002 Griffeen at Lucan presented in this volume consider only 24 AM

flows, omitting the final modest AM value because it was preceded by two missing years.

See Figure 8.1.]

EV1 Probability Plot 10.7 shows an elongated S shape for Station 26008 Rinn at Johnston's

Bridge. While no single distribution could describe the probability plot, the upper end is not

too dissimilar to some of the random samples in EV1 Probability Plot 5.1. While application

of a pooled growth curve should provide satisfactory estimates of QT, it would be prudent to

check the sensitivity of analyses to inclusion/exclusion of the low outlier seen at this station.


10.5 Additional notes on the choice of distribution and method

10.5.1 Problems in the use of 3-parameter distributions for single-site analysis

In ordinary circumstances a 3-parameter distribution should not be used with single-site data.

An exception could be made if the data series is very long, say > 50 years, and the required

return period is short, say ≤ 25 years.

A 3-parameter distribution is more flexible and may give a better fit visually on a probability

plot. However, the estimation of a third parameter has the effect of increasing the standard

error of the estimated quantile. See diagram (c) of Box 7.2. The situation can also occur

where use of a 3-parameter distribution in single-site analysis leads to a Q-T relation that is

not intuitively acceptable because of extreme upwards or downwards curvature.

26008 RIVER RINN @ JOHNSTON'S BRIDGE

500100502510520

10

20

30

40

50

60

-2 -1 0 1 2 3 4 5 6 7 8EV1 y

AM

F(m

3/s

)

AMF data

At-site EV1 fit

Pooled EV1 fit

Pooled GEV fit

+ AM flow data




AM

flo

w (

m3 s

-1)

-2 -1 0 1 2 3 4 5 6 7



135

10.5.2 Reconciling single-site and pooled analyses

In the event that the probability plot shows a marked departure from straight-line behaviour –

either by way of upwards or downwards curvature or because of some form of an elongated

S shape – consideration must be given to applying a pooled growth curve xT to the gauged

value of QMED. Even this may leave doubt about the suitability of the method chosen if the

pooled growth curve differs substantially from whatever general pattern is shown by the site

data e.g. as in EV1 Probability Plot 10.2.

A decision may have to be made to trust the pooled growth curve as a matter of good

practice: based on the well-documented reduced standard error of estimate and robustness of

the pooling method. See diagram (d) of Box 7.2. When AM flows at the site display CV and

skewness greatly in excess of the pooled values – as in EV1 Probability Plot 10.3 – it can,

however, be difficult to trust the pooled growth curve. If a very large flood is observed

during the period of record the question arises as to whether it should over-ride any more

modest estimate of QT obtained by pooled analysis.

A progressive approach can be to use a weighted combination of the pooled and single-site

estimates (see Box 10.2). However, it should be noted that the relative weight given to each

component cannot presently be specified by any rule based on scientific evidence. The

weight must be chosen subjectively and supported by rational argument alone.

Box 10.2: Combined use of single-site and pooled estimates

10.5.3 Probability associated with a very large recorded flood

Where a very large observed flood has occurred, it is possible – under certain assumptions –

to calculate the probability that such a large flood could occur. For instance, if a sample of

50 floods are drawn randomly from an EV1 distribution with CV = 0.3, the probability that

the largest flood would exceed 1.25 Q100 – where Q100 is the population value of the 100-year

flood – is less than 7% and the probability that it would exceed 1.5 Q100 is less than 1%.

In practice, the true value of Q100 is unknown and has to be replaced by an estimate which

makes these percentage probabilities less reliable. Nevertheless, such calculations can be

adapted to provide an informal test of the hypothesis that the observed outlier is consistent

with the assumed parent population. The probabilities quoted above are based on the fact that

if Q is distributed as an EV1 with parameters u and , then the largest flood in N years (Qmax)

is distributed as an EV1 with parameters u + ℓnN and .

Editorial note: The FSU Web Portal supports the combined use of single-site and pooled

estimates. The site and pooled flood growth curves are combined by taking weighted-

averages of the site and pooled L-Moment ratios, specifically:

L-CVcomb = ω L-CVsite + (1 - ω) L-CVpooled 10.8

L-skewnesscomb = ω L-skewnesssite + (1 - ω) L-skewnesspooled 10.9

L-kurtosiscomb = ω L-kurtosissite + (1 - ω) L-kurtosispooled 10.10

where ω takes a value between 0 and 1.


136

10.5.4 Flood growth curves with an upper bound

If the AM flow series shows negative skewness, and the three or four largest floods differ

from one another by only a very small amount, the data series will usually exhibit convex

curvature (e.g. EV1 Probability Plot 10.4). In such situations, any 3-parameter distribution

fitted to the data will usually have an upper bound which is not very much larger than the

largest recorded flood. The GEV and GLO have an upper bound if the shape parameter k is

greater than zero.

While it is unwise to be prescriptive, it is seldom hydrologically realistic if the upper bound

to flooding (implied by the fitted distribution) is only a little greater than one or more floods

already experienced at the station. This objection applies regardless of whether the flood

frequency curve has been derived by single-site or pooled analysis. The experienced

hydrologist knows that some unprecedented rainfall could occur in the future which leads to a

runoff or routing mechanism different to all previous floods and which could therefore

produce a flood much in excess of the largest flood on record.

The general recommendation in such cases is to consider instead use of a 2-parameter

distribution growth curve that yields a straight-line growth curve on the relevant probability

plot. Nevertheless, judgement is required before a design flood can be specified. There is the

possibility that a straight-line 2-parameter distribution, when extrapolated to estimate very

rare floods, may produce flood estimates that are implausibly large in the context of the

relevant physical factors and the known water levels reached during previous floods. This is

an example where the skewness of the model is potentially greater than the skewness of the

parent data. See diagram (b) in Box 7.2. The hydrologist will need to apply judgement to the

estimation process in such a case.


137

11 Data transfers revisited

Many flood estimation problems arise at subject sites that are ungauged. The transfer of

information from gauged to subject site is a fundamental matter to be addressed by

practitioners.

The topic has been touched on elsewhere, not least in the detailed example of QMED

estimation at an ungauged site in Section 2.7. However, data transfer is of such importance

that some final remarks are made in Section 11.2, after reporting an interim assessment

undertaken as part of the FSU research.

11.1 Interim assessment of QMED data transfers

11.1.1 Subject sites used in the interim assessment

The assessment is made by treating gauged sites as if they were ungauged. Some 184 gauged

sites were identified as potentially suitable candidates on which to base the assessment.

It was decided to focus on rivers with at least three gauging stations, each with at least 20

years of data. The 38 gauging stations thus identified lie in ten different river systems. The

stations used are identified in Table 11.1.

Table 11.1: Rivers having three or more gauging stations for assessment

Hydrometric

Area (HA) River

name

#

gauges

Station numbers within HA

No. Name Most

u/s

Most

d/s

06 Glyde Glyde 3 26 14 21

14 Barrow Barrow 6 5 6 19 34 18 29

16 Suir Suir 5 4 2 8 9 11

18 Blackwater Blackwater 5 16 50 48 6 3

24 Shannon Estuary Deel 3 30 11 12

24 Shannon Estuary Maigue 3 4 82 8

25 Lower Shannon Brosna 3 124 6 11

26 Upper Shannon Suck 4 6 2 5 7

26 Upper Shannon Inny 3 58 59 21

36 Erne Erne 3 12 11 19

To the extent possible, the record-length criterion insulates the assessment from uncertainty

in observing QMED itself (see Section 9.2). The selection of rivers with three such gauges

allowed assessment of the relative merits of data transfer from upstream and downstream

sites. Comparisons are also made with data transfers from more distant (but possibly

hydrologically-similar) gauged catchments i.e. from gauged stations that do not lie either

upstream or downstream of the subject site.


138

11.1.2 Methods used in the interim assessment

The approach used in the interim assessment of data transfers loosely resembles the final data

transfer procedure recommended and illustrated in Section 2.7. The procedure for estimating

QMED at an ungauged site has three main steps:

Selection of the pivotal catchment;

Estimation of QMED from PCDs at the subject site and at the pivotal site;

Adjusting QMED at the subject site by reference to performance of the PCD model in

estimating QMED at the (pivotal) gauged site.

Because a number of different data-transfer strategies were being compared in the interim

assessment, use of the term donor catchment is appropriate here. [Editorial note: Use of the

term pivotal catchment would give the impression that a prior judgement had been made.]

The interim assessment compared a number of data transfers to each of the 38 subject sites

studied. These experimental subject sites were the 38 gauged sites identified in Table 11.1.

Each data transfer used only one donor catchment at a time.

Selection of the donor catchment

The donor site selection chose the nearest gauge on the same stream as the subject site. The

2nd

nearest, 3rd

nearest etc. in the upstream and downstream directions are also examined

where such sites exist. Use of a data transfer from an upstream or downstream site is good

practice and the donor site is often self-selecting. One would normally use the nearest such

gauge measured along the river network or the one whose AREA ratio (compared to the

AREA to the subject site) is closest to 1.0.

The interim assessment also tested the effectiveness of a data transfer from outside the

catchment. For historical reasons largely related to terminology introduced by the FEH, a

distant donor catchment is termed an analogue catchment. Whereas selection of an upstream

or downstream site as the pivotal catchment is relatively straightforward, selection of an

analogue catchment is inherently controversial. [Editorial note: For the transfer to be

justifiable, the analyst has to be convinced that the donor catchment is both (i) hydrologically

similar to the subject catchment, and (ii) the most hydrologically similar (or otherwise most

appropriate) of all such gauged catchments available to act as a donor.] The selection of an

analogue catchment is never straightforward.

For the interim assessment, the analogue catchment was selected as the “nearest” (i.e. least

dissimilar) catchment according to the dissimilarity metric of Equation 7.2. For reasons

explained in Section 2.6.4 (see also Section 11.2), the FSU recommendation is opposed to

any automated selection of the pivotal catchment. However, the strategy had the merit of

allowing the interim assessment to proceed in a straightforward manner.

Estimation of QMED from PCDs at the subject site and at the donor site

For logistical reasons, the assessment used a different model for estimating QMED from

PCDs than that developed and recommended in Chapter 2. The model uses just three PCDs:

-1.5390.8980.829 BFISAARAREA0.000302QMED 11.1


139

In terms of the particular dataset of 164 catchments used in its calibration, the model explains

82% of the total variance in ℓnQMED (i.e. r2 = 0.82). It yields a root mean standard error

(RMSE) of 0.472 in the ℓn domain, so that the factorial standard error is FSE = e 0.472

= 1.60.

As is to be expected, the model is inferior to the best models reported in Section 2.2. For

example, Table 2.6 reports a 3-variable model having r2 = 0.843 and RMSE = 0.407.

Use of the baseflow index in Equation 11.1 is somewhat controversial. BFI is formally

known only at gauged sites with daily mean flow data. The results that follow should

therefore be treated as interim rather than authoritative.

Adjusting QMED at subject site by reference to performance of PCD model at donor site

The method of data transfer used in the interim assessment was a full (or “hard”) data

transfer. Thus, if the PCD model is found to underestimate QMED by 33% at the donor site,

the PCD estimate of QMED at the subject site is increased by factor of 100/67, i.e. by 49%.

11.1.3 Results

The performances of various data transfer methods at five of the 38 experimental sites are

summarised in the figures below. The number above each adjusted estimate denotes the area

of the donor catchment in km2. A perfect data transfer would yield an adjusted estimate

equal to 100% of the gauged QMED at the site. In the data transfer, each of the five sites is

treated as if it were ungauged.

Figure 11.1: Performance at Station 16002 Suir at Beakstown (512 km

2)

Figure 11.2: Performance at Station 16008 Suir at Newbridge (1120 km2)

236 1120

1602 2173

309

0

100

236

512 1602 2173

1207

0

100

1st u/s 1

st d/s 2

nd d/s 3

rd d/s Analogue PCDs

% o

f obse

rved

QM

ED

%

of

ob

serv

ed Q

ME

D

2nd

u/s 1st u/s 1

st d/s 2

nd d/s Analogue PCDs


140

182 1050 1184

152

0

100

113 881

1058 233

293

0

100

Figure 11.3: Performance at Station 18050 Blackwater at Duarrigle (244.6 km2)

Figure 11.4: Performance at Station 24008 Maigue at Castleroberts (805 km2)

Figure 11.5: Performance at Station 26002 Suck at Rookwood (626 km2)

Consideration of these and other cases led to the interim conclusions that:

Data transfers from donor stations located upstream or downstream of the subject site

tend to perform better than a data transfer from an analogue site;

Some limited evidence was found to suggest that data transfer from a gauge sited

downstream of the subject site typically performs a little more strongly than data

transfer from a gauge sited upstream of the subject site.

Most data transfers (even from an analogue site) perform better than estimating

QMED from PCDs alone.

It must be recalled that the above assessment was made with an interim method of estimating

QMED from PCDs, not with the recommended method developed in Chapter 2.

246 764

1063

0

100

% o

f ob

serv

ed Q

ME

D

2nd

u/s 1st u/s 1

st d/s 2

nd d/s Analogue PCDs %

of

ob

serv

ed Q

ME

D

1st u/s 1

st d/s Analogue PCDs

% o

f obse

rved

QM

ED

1st u/s 1

st d/s 2

nd d/s Analogue PCDs


141

11.1.4 Remarks

The user is reminded that the estimation of QMED plays a particularly important role in flood

estimation in many Irish rivers, especially those of low gradient and a sluggish flood

response. Such rivers typically have flood growth curves of low gradient, making estimation

of the index flood especially influential on design flood magnitudes. This view is

underscored by the Chapter 9 discussion of uncertainty which ascribes – under certain rather

large assumptions – that most of the uncertainty in estimating QT in Irish rivers derives from

uncertainty in estimating QMED.

It is anticipated that the wider application of data transfer techniques will promote a healthy

discussion of many factors, including:

Methods of assessing hydrological similarity;

Strengths and weaknesses of the national hydrometric network in relation to sites at

which flood estimates are required;

The value of installing a flood gauge locally when flood frequency estimation at a

critical subject site is not well served by gauged networks.

11.2 Further guidance on pivotal catchments and data transfers

Implementation of the FSU research requires the user to make some important judgements.

The pivotal catchment is the user’s assessment of the most relevant catchment on which to

base a data transfer. Where flood data are available from a gauge sited upstream or

downstream of the subject site, this will often be readily selected as the pivotal catchment. In

other cases, the selection is likely to be more precarious and to hinge on the user’s judgement

of catchment similarity.

An automated judgement of catchment similarity is likely to give weight to differences in a

few leading factors – e.g. catchment size (represented by AREA), catchment wetness

(indexed by SAAR) and catchment permeability (indexed by BFI or BFIsoil) – and to neglect

all other factors. This is not a safe approach.

A particular feature present on one catchment and absent on another may lead to strong

differences in their flood behaviour. Arterial drainage (indexed by ARTDRAIN and

ARTDRAIN2) is perhaps the most notable such feature. Section D4 of Appendix D finds

evidence that BFIsoil and ARTDRAIN2 are important in characterising the post-drainage

flood response of a catchment, whilst the descriptors DRAIND and S1085 are more important

in characterising the response of undrained catchments. These findings may help the

experienced user to judge which PCDs to examine closely when judging hydrological

similarity as a means of selecting the pivotal catchment.

Other notable features to consider when assessing catchment similarity are the extent of

urbanisation (indexed by URBEXT) and the presence of large lakes (indexed by FARL).

Research reported in Section 2.3.4 endorses the recommendation to favour geographical

closeness – as well as similarity in key PCDs such as FARL – when selecting a pivotal

catchment for use in a particular flood estimation problem.


142

Selection of the pivotal catchment is a demanding task that calls for reasoned judgements.

Some judgements will be unsettling because of the impact of the data transfer on the final

flood estimates. In some cases – not least on small catchments – the pivotal catchment

selected will not be wholly convincing. This is not a reason to abandon making a data

transfer. However, it may be good reason to consider making only a partial transfer. An

approach to making a partial transfer is included in Step 5 of the procedure illustrated in

Section 2.7.1.

Two final remarks concern the treatment of urbanised catchments in data transfers. It is

recommended that data transfers are applied to the rural element of QMED. The relevant

procedure is explained in Box 2.3 in Section 2.7. Second, it is recommended that the urban

adjustment factor UAF is applied in all cases, even to subject catchments that are almost

entirely rural. With ever-increasing automation, and a persistent interest in making flood

estimates throughout a river basin, it is unhelpful to have a procedure in which an urban

adjustment is incorporated at some sites but not others.


143

12 Summary and conclusions

12.1 Data

Data from some 200 gauging stations in the Republic of Ireland were available from the

archives of OPW, EPA and ESB. Of these, 115 were of Grade A and 67 were Grade B.

Data series at 17 stations had both pre and post-drainage records available. Data from most

of these stations played an active role in the analysis and modelling of QMED reported in

Chapter 2.

Summary statistics (Chapter 4) and probability plots (Chapter 5) were prepared for all these

stations. However, no strong inferences were drawn from those of the Grade B stations.

12.2 Descriptive statistics

Examination of the data for trend and randomness (Chapter 3) found that about 10% of

records displayed some trend or lack of randomness. The usual assumption that AM flows

are statistically independent and identically distributed was nevertheless adopted, supported

by the fact that 90% of records did not show significant trend or lack of randomness.

The descriptive statistics indicate that – in comparison to international data and many UK

data – Irish AM flow data typically have low variability and low skewness, whether judged in

terms of traditional statistics (CV and skewness) or in terms of L-moment ratios (L-CV and

L-skewness). This possibly reflects the shallow longitudinal slopes of many Irish rivers.

Examination of probability plots and moment ratio diagrams suggests that – among

2-parameter distributions – the Extreme Value Type I (EV1) and lognormal (LN2) are

typically the most appropriate for use in Ireland. Some exceptions were found, not least for a

number of stations that display very low skewness. In some cases, this reflects that the three

or four largest values in the record are not appreciably different in magnitude from each

other.

Explanations for this latter phenomenon were sought in Section 5.4 by considering flood

volumes and by referring to the physical catchment descriptor FAI: an index of the

attenuation of flood magnitudes expected from floodplain storage effects. However, no

empirical evidence was found to explain the phenomenon.

12.3 Seasonal analysis

Seasonal analysis shows that two thirds of AM flows occur during the winter months of

October to March. At 11 stations, no AM floods occurred outside this winter season.

July is the least likely month in which AM floods occur while August and September are the

most likely summer months to supply AM floods.

At 20 (10%) of the 202 stations examined, the largest flow on record (i.e. the series

maximum, Qmax) occurred during summer, with eight of the occurrences in August. For 91

(45%) of the 202 stations, the series maximum occurred in the single month of December.


144

12.4 Estimation of design flood

Estimation of the T-year flood is considered for each of a wide range of circumstances that

can arise in practice: from gauged to ungauged sites and from short to long flow records. The

index flood method (see Sections 6.2 and 7.1) is recommended in which QT is expressed as:

QT = QMED × xT

Here QT is the flood of return period T years, QMED is the median of the AM flows at the

subject site and xT is the growth factor appropriate to the subject site. Unless there is a long

record at the subject site, xT is estimated by pooled analysis of AM flow data from an

appropriate homogeneous pooling group.

QMED is estimated from the gauged AM flow data, where available. If the data series is

long, the QMED value obtained is used directly. If the available flow record is short, or if

QMED is obtained (in the ungauged case) from a PCD-based equation, QMED is adjusted

with the assistance of data from a relevant gauged site judged by the user to be pivotal to

flood estimation at the subject site.

Tests conducted on data of 38 gauging stations located in ten different river systems were

used to assess the efficacy of different data transfer methods for improving the QMED

estimate. Data transfer from a downstream donor site was (on average) found to be slightly

superior to data transfer from an upstream donor site. Both were found to be superior to data

transfer from a catchment judged to be locally relevant or hydrologically similar by other

criteria (i.e. other than being directly upstream or downstream of the subject site).

In practical applications – at all ungauged sites and some gauged sites – the user must justify

their choice of the pivotal catchment. This is the user’s assessment of the most relevant

catchment on which to base a data transfer to the subject site. A worked example in

Section 2.7 illustrates the detailed mechanics of data transfer to refine the estimate of QMED

at an ungauged or short-record site.

Estimation of xT may sometimes be based on single-site analysis, if a sufficiently long data

record exists at the subject site. Otherwise, xT is estimated from the dimensionless L-CV and

L-skewness values obtained by averaging these quantities across a pooling group of gauged

catchments chosen to be hydrologically similar to the subject catchment. There can

sometimes be scope to combine single-site and pooled estimates of xT (see Box 10.2 in

Section 10.5.2).

Suitable members of the pooling group are chosen with the help of a dissimilarity metric dij.

Tests were carried out into the effectiveness of different combinations of PCDs in the

definition of dij. The most effective metric was based on differences in catchment size

(indexed by differences in ℓnAREA), differences in wetness (indexed by differences in

ℓnSAAR) and differences in permeability (indexed by differences in the baseflow index BFI).

Where a gauged value of BFI is unavailable, the physical catchment descriptor BFIsoil is

substituted.

Tests have been carried out into the effect on the estimated value of xT of catchment size

(measured by AREA), peat coverage (indexed by PEAT), “lakiness” (indexed by FARL),

geographical location and the available period of record varying from the 1950s to the 1990s.


145

None of these effects was judged to be sufficiently influential to make additional provision

for, other than through their influence on QMED.

Although some regard is to be paid to the homogeneity (or not) of pooled flood data,

catchments with discordant values of L-moment ratios should be rejected from a pooling

group only if they are found to be physically dissimilar to the subject catchment. Catchments

should not be rejected on the basis of their flood statistics alone.

The standard errors associated with estimates of xT and QT are investigated in Chapter 9. The

standard error of QT estimated by the index flood method is dominated by SE(QMED).

Consequently, when expressed as a percentage, SE(QT), varies only slightly with T. When

QMED is estimated from gauged data and xT is estimated from a pooling group containing

approximately 500 station-years of data, SE(QT) is of the order of 5 to 10% of QT regardless

of return period. If QMED is estimated from a PCD-based formula alone, and xT is estimated

from a pooling group containing approximately 500 station-years of data, SE(QT) is of the

order of 37% QT. [Editorial note: One route to developing greater certainty about

uncertainty may be to use resampling techniques in which the resampling is arranged by year

across all AM datasets implicated in the pooled and single-site analyses of a particular flood

estimation problem. Such resampling respects important spatial structure in AM flows and is

capable of revealing the degree of sensitivity of the final answers to the particular years of

record available.]

Chapter 10 provides guidelines for the estimation of QT, both from site and pooled data.

Several points that need to be taken into account in practical cases are outlined. It is

emphasised that blind use of a prescribed method can sometimes lead to a QT estimate which

is not always in accordance with the “ground truth” of gauged or historical flood data. This is

especially the case when single-site AM flow data have higher-than-average variability

and/or skewness. A number of examples are discussed, most of which throw up practical

problems of the type met in practice where difficult choices have to be weighed and decisions

made. Discussion is also provided on flood distributions with upper bounds, and on the

choice between 2 and 3-parameter distributions.

In some special cases, the user may need to consider single-site estimate in conjunction with

(and sometimes even in preference to) the pooled method of growth-curve estimation that lies

at the heart of the general FSU recommendations for estimating the T-year flood peak. When

competing estimates of QT are being compared or assessed at a gauged site, it is essential to

view probability plots that also show the AM flow data for the site. It is crucial to be alert to

information about historical floods experienced at the site or elsewhere on the river. It is

better to use such information subjectively than to ignore it entirely.

In conclusion, flood estimation cannot be reduced to a strict formula-based procedure.

Individual analysts must make choices which depend on the problem circumstances and

which take into account their own knowledge and experience. Users will be expected to

present and record a reasoned argument for the choices made. “Because it gives a smaller

number” is never an adequate reason.


146

Acknowledgements

The work reported was undertaken principally by NUI Galway and NUI Maynooth, with

contributions from OPW and Hydro-Logic Ltd. The help of many organisations and

individuals is gratefully acknowledged.

Particular thanks go to hydrometric staff in gauging authorities and to members of the FSU

Technical Steering Group.

Volume II was edited by Duncan Reed of DWRconsult, who added Sections 2.7 and 11.2.

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Appendices

Appendix A Review of stage-discharge relationships

A1 Terminology

The water level at a gauging station is generally known as the stage. The discharge is the

river flow. The stage-discharge relationship is commonly referred to as the rating curve or

simply the rating. All these terms are used interchangeably.

River flow is measured occasionally by field technicians. Historically, this was typically

undertaken using current meters to measure the flow velocity at different points across the

river cross-section. In more recent years, river flow is measured using Acoustic Doppler

Current Profilers (ADCPs). The resultant measured flows are typically known as gaugings.

The rating curve is constructed from the stage and discharge measurements. Because water

level can be recorded continuously, the rating curve allows river flow to be inferred

continuously.

Depths are generally in metres and flows in m3s

-1.

A2 Review of stage-discharge relationships

The review of stage-discharge relationships was necessary in order to provide a robust set of

annual maximum (AM) flows for the FSU research. Hydro-Logic Ltd reviewed all gauging

stations that had the potential for use within the FSU research. The work examined the stage-

discharge relationships held by the relevant hydrometric archiving authority, most notably the

OPW, the EPA and the ESB.

Rating curves are dynamic equations that require adjustment if there is a change in

instrumentation or a change in the channel control (e.g. due to debris/obstructions in the

channel, weed growth, tree growth etc.) Because of this, it was necessary to consider the full

history of the gauging station and to acquire all pre-existing stage-discharge relationships for

the gauging stations under consideration. These ratings were used as the starting point in the

review of stage-discharge relationships.

A2.1 General form of stage-discharge relationships

The stage-discharge relationships held by the OPW take the form:

pDG)(DCQ A.1

where:

Q is the discharge

D is the (gauged) water height or stage

DG is the datum shift (which can be positive or negative)

C is the discharge when the effective depth of flow (D + DG) is equal to 1

p is the slope of the rating curve (on logarithmic paper)


152

The C and p values sometimes derive from theory if the gauging station comprises a formal

structure. More typically, they derive from regression analysis of flow and depth gaugings.

Some hydrometric agencies use the alternate notation:

βahcQ A.2

where:

Q is the discharge

h is the (gauged) water height or stage

a is the gauge datum nominally corresponding to zero flow

c (constant) is the discharge when the effective depth of flow (i.e. h – a) is 1.0

(constant) is the slope of the rating curve (on logarithmic paper)

The stage-discharge relationship can be written in its equivalent logarithmic form. For

example, Equation A.2 can be transformed to:

ahlnβclnQln A.3

In the simplest case where the datum a is known – and where all gaugings are treated in a

single group – the parameters ℓn c and β can be obtained by linear regression analysis.

Appendix C provides an overview of regression methods.

A2.2 Analysis tools and background information

Analysis and evaluation tools in standard hydrometric software packages were used to review

any existing or proposed rating. Rating curves and gaugings were plotted using the Hydro-

Logic “Gauger Analysis” system (a system for rating development) or using the rating curve

editor utility “SKED” within the WISKI water management information system.

A key ingredient was to study the deviations between current-meter gauged flows (Qg) and

rating-curve calculated flows (Qc). These deviations were plotted both against stage and

against date. Such “deviation plots” assist in the identification and diagnosis of

discontinuities in the ratings.

Detailed reference was made to files of information about the station and its gauging history.

Metadata extracted included information on datum shifts, changes in flow control and other

technical matters. Contact was made with the hydrometric authority and, where appropriate,

with field technicians. This process helped to assess the validity of ratings and to judge

whether new work was needed to develop a better flood-flow rating.

A2.3 Review procedure

The following steps were undertaken in the review:

i A database was established and populated with readily available information. This

was refined and updated as the project progressed.

ii A prior estimate of the mean annual flood (Qmean) was generally available for each

gauging station. This was taken with the highest gauged flow (HGF) and the ratio

HGF/Qmean calculated. This was used in the initial classification of the gauging

station in terms of the likely quality of its high-flow measurements (see Section A3).


153

iii An exploratory analysis was undertaken by trawling through ratings classed as A, B

or C grade to gain a first impression of their quality and characteristics. Some sites

were identified for which some basic survey information could assist with the rating

classification. For example, simple surveys were arranged for a number of key sites

to establish the level of the river bank relative to the datum of the water level

recorder. Some further details are given in Section A2.4.

iv Periods of record for which a particular rating curve could be used were confirmed or

identified. Where arterial drainage works have affected the gauged site or its

catchment, the ratings and flow series were split into pre-drainage and post-drainage

elements.

v On completion of the exploratory analysis, a more detailed assessment of the ratings

began for stations initially graded A, B or C. During these more detailed

investigations it was ascertained whether the existing ratings could be used or whether

a refined or totally new rating was required. On the basis of this assessment, ratings

were selected for use in an uncertainty analysis (see Section A3.3). The final

recommended ratings were entered on the database.

vi Annual maximum (AM) water levels were provided by the OPW in Excel

spreadsheets. For the EPA sites, the required AM water levels were extracted from

the WISKI information management system, after filling or checking any gaps by

scrutiny of chart data.

vii AM flows were determined using the existing OPW/EPA rating. For sites where

there was scope to produce a better rating, AM flow series were also extracted using

the proposed new rating. If the differences between the two flow series were minor,

the existing OPW/EPA rating was generally adopted for use in the FSU. Differences

were normally considered acceptable if the QMED values calculated for the existing

and for the revised series agreed to within 10%. In other cases, a further review of the

ratings and station history files was undertaken to explore the differences.

viii Those ratings that were required to be changed for the purposes of the FSU research

were discussed with the OPW. If the proposed changes were deemed reasonable, the

revised ratings were adopted and used to produce the AM flow series for the FSU

research.

A2.4 Gauging station surveys

By examining the rating periods (see Step iv above), their respective gaugings and the

bankfull level it was possible in some cases to extrapolate the flood ratings above the HGF.

Initially, if the channel had a relatively regular cross-section and bankfull levels were known,

the bankfull levels could be used to set the limits of extrapolation.

For sites with a more complex stage-discharge relationship, a more detailed and thorough

topographic survey is required. However, it was not possible for the hydrometric authority to

undertake such detailed surveys within the timescale of the review.

Based on the exploratory analysis of ratings, a list of priority sites for bankfull-level surveys

was forwarded to the relevant OPW field technicians. This survey work was incorporated

into the routine visits to these sites. Within the timescale of the review, this was not possible

for EPA sites. Such survey information could be useful in further improvement and

refinement of the ratings.


154

For the sites identified by ESB for inclusion in the FSU, the bankfull levels were available

from previous reports undertaken for these sites.

A3 Gauging station classification

A3.1 Initial site classification

Prior to the review, the OPW had produced an initial classification of gauging stations. This

can be summarised as follows:

A sites Sites having stage-discharge ratings considered good for determining high and

flood flows.

B sites Sites with good high-flow ratings but for which there were some concerns as to the

quality of the flood-flow rating.

C sites Sites with reasonable medium to high-flow ratings but for which it was not

possible to determine flood flows with any confidence, due to the fact that at high

flows the site was either not rateable (e.g. due to hydraulic effects or bypassing) or

because there were insufficient gaugings to produce a rating.

P sites Sites classified as poor and not considered suitable for high and flood-flow

determinations. It is possible that some of these sites could be used in future if

sufficient gaugings and other information were available.

U sites Sites for which the data were unusable for determining high flows. These could

for example be “level-only” stations at which it is not possible to measure

discharges.

P and U sites were not considered in any detail during the project.

For sites archived by the EPA, the classification of ratings had been produced some years

earlier and was not complete for all sites. Values of the mean annual flood (Qmean) were

derived from the EPA's archive database for all sites considered for inclusion in the FSU.

The ESB also has a network of hydrometric gauging stations with an associated archive

dataset. Even though their network is small, the sites have extensive records going back for

many years and in some cases for up to 60 years. After exploration, six of the ESB stations

were identified for potential use in the FSU research.

From the list of sites produced, a simple ranking index was created, based on the ratio

HGF/Qmean where HGF denotes the highest gauged flow.

A3.2 FSU station classification

The basis for the first level of site categorisation for the FSU was as follows:

Grade A station – Suitable for flood frequency analysis: Site for which HGF is at

least 1.3 QMED and which has a stage-discharge relationship that the OPW judges

represents extreme floods reasonably.


155

Grade B station – Suitable for flows up to QMED: Site for which HGF is at least

0.95 QMED and for which the cross-section is thought to be without significant

change in channel geometry up to the stage corresponding to QMED.

Grade C station – Possibly suitable for extrapolation up to QMED: Site for

which the rating is considered well-defined up to about 0.8 QMED.

The categorisation was further refined to distinguish two classes of Grade A station:

Grade A1 station – Very good – Providing the very best flood data: Grade A

station with confirmed ratings that are considered good for flood flows well above

QMED and with some confidence of valid extrapolation up to 2.0 QMED using

suitable survey data and allowing for any flows across the floodplain.

Grade A2 station – Good – Providing the next best flood data: Other Grade A

stations.

A3.3 Uncertainty analysis

The FSU categorisation makes particular use of the ratio of HGF to QMED. Being defined

as the median of the AM flows, QMED is a relatively robust measure that is unaffected by

uncertainty in the measurement of the very highest flood flows. The HGF is taken as the

largest flow that is measured at the station with a fair degree of confidence. The ratio

HGF/QMED therefore provides a basic index of the likely quality of the flood-flow rating.

If, for example, HGF is 85 m3s

-1 and QMED is 53 m

3s

-1, HGF/QMED is 1.6. This suggests

that the rating relationship is potentially valid for relatively rare floods well in excess of

QMED. Unless there were specific doubts about the quality of the high-flow gaugings, the

station would be assigned to Grade A1.

The above classification system inevitably has a certain amount of subjectivity. Following

discussion with the OPW, a more analytical approach was explored in which the likely

uncertainty in the high-flow ratings was assessed from the scatter (in gaugings about the

stage-discharge relationship) at high flows, taking due account of the number of high-flow

gaugings used in its construction.

The uncertainty analysis constructed 95% confidence intervals about the linear regression

(e.g. Equation A.3) representing the stage-discharge relationship. This was done using the

Hydro-Logic Gauger Analysis software, in which it is possible to enter the stage value

corresponding to QMED, or any other stage value, and obtain 95% confidence limits on the

estimation of flow using the stage-discharge relationship.

During the review and analysis of the ratings, stage values corresponding to QMED for all

Grade A and B stations were entered and confidence limits obtained. This provided a

quantitative expression of the uncertainty in flow measurement using the stage-discharge

relationship. Stations were considered good (i.e. worthy of Grade A2) if 95% confidence

intervals for flows at the level of QMED were smaller than 30% and very good (i.e. worthy of

Grade A1) if they were smaller than 10%. This uncertainty analysis helped to confirm the

final classification of gauging stations.


156

A4 Production of annual maximum flood series

With the gauging station review complete, and in association with the EPA and Hydro-Logic

Ltd, OPW hydrometric staff prepared the AM flow datasets subsequently used in the research

reported in this volume. In the majority of cases, flow data supplied for use in the Volume III

research on hydrographs were also based on the rating curves confirmed or revised in the

above review of stage-discharge relationships.

The FSU research chiefly used flow data from Grade A1, Grade A2 and Grade B stations.

Data from other gauging stations should be considered in site-specific flood studies.


157

Appendix B Flood data exclusions

B1 Stations omitted from the QMED modelling research

Sixteen out of 206 stations were omitted from the QMED modelling research of Chapter 2 for

the reasons highlighted below:

15003

Station 15003 Dinin at Dinin Bridge was one of three stations questioned during model-

building because of their exaggerated influence on model coefficients. Further inspection

revealed a number of low outliers, i.e. exceptionally small annual maxima. This station is

known to be an extremely flashy catchment in a karst area (Castlecomber Plateau). QMED

may not provide an adequate index flood in these circumstances. [Editorial note: Quite

extensive use was made of this Grade A2 station in other chapters, most notably in the study

of flood volumes reported in Section 5.4 and Appendix J.]

19014, 19015, 19016 and 19031

Annual maximum flow series for Stations 19014, 19015, 19016 and 19031 in the Lee basin

arrived too late for inclusion in the Chapter 2 research. [Editorial note: Limited use was

made of Stations 19014, 19016 and 19031 in the research reported in other parts of Volume

II. These ESB stations are categorised Grade B.]

20006

Station 20006 Argideen at Clonakilty WTW was one of three stations questioned during

model-building because of their exaggerated influence on model coefficients. Further

inspection revealed criteria for omission. The station has an unusually small QMED.

Investigation of the flow series revealed a large number of years with missing months. This

raises the suspicion that the true AM flow may have been missed in a number of years.

[Editorial note: Some limited use of this Grade B station was made in other parts of Volume

II.]

25001, 25002, 25003 and 25005

Four stations in the Mulkear basin were omitted from analysis in line with recommendations

made by Joyce (2006, pers. comm.):

This river was subjected to a District Drainage Scheme in the late 1920s and early

1930s that protects large areas of land by extensive lengths of embankments that are

overtopped about once in five years. This means that the catchment responds almost

without storage attenuation for the smaller annual maxima, including QMED, and

with massive storage attenuation for the larger events.

[Editorial note: These stations were omitted from much of the research reported in this

volume. However, they are included in the Section 5.5 study of flood seasonality and in the

trend analyses reported in Chapter 3 and Appendix E.]


158

26010

The AM series for Station 26010 Cloone at Riverstown revealed suspicious outliers. On

investigation, it was found that a partially developed flood rating had been erroneously

applied. Neither the AM flow series based on the OPW rating, nor that based on the

Hydro-Logic Ltd rating, appeared to reflect changes in the stage-discharge relationship

adequately. [Editorial note: Some limited use was made of this Grade B station in other

parts of Volume II.]

30037

Station 30037 Robe at Clooncormick was one of three stations questioned during model-

building because of their exaggerated influence on model coefficients. The gauged QMED of

1.79 m3s

-1 is exceptionally small for a catchment area of 210 km

2. This is well below any

other catchment of a similar size. [Editorial note: Some limited use was made of this

Grade B station in other parts of Volume II.]

31075, 34005 and 36020

Discrepancies were found in the metadata descriptions of Stations 31075, 34005 and 36020.

QMED values were provided but without the AM values themselves. [Editorial note: None

of these stations was used in Volume II.]

36027

The AM series for Station 36027 Ballyconnel Canal East at Bellaheady revealed some

extremely low outliers. Given the station’s name, and the fact that it typically records an AM

flood of only 25 m3s

-1 from a nominal drainage area of 1501 km

2, it seems unlikely that this

represents a meaningful flood series for a natural river. [Editorial note: The only use of this

Grade A2 station in Volume II was in the Section 5.5 study of flood seasonality.]

B2 Other notes on stations omitted from research

The list in Section B1 is not comprehensive. Different researchers make different choices.

Part of the Volume II research considered data for Station 07004 (Kells) Blackwater at

Stramatt. However, for some unrecorded reason, the Chapter 2 research on QMED

modelling did not consider this Grade A2 station.

27070

The annual maximum series for Station 27070 Lough Inchiquinn at Baunkyle was excluded

from most parts of Volume II because of suspected extreme lake effects. Its retention in the

QMED modelling of Chapter 2 does not appear to have led to undue difficulty or influence.

B3 General recommendation

Practitioners are encouraged to study all relevant flood data carefully. A flood series rejected

for national study may include information that is valuable in a site-specific study.

Investigation or modelling may rehabilitate flood data considered suspect or unreliable.


159

Appendix C Notes on the regression methods used in Chapter 2

C1 Ordinary least-squares (OLS) regression

The catchment descriptor equation was fitted using multiple linear least-squares regression.

Under this approach, the relationship can be written in vector notation as:

= + y X β e C.1

where is the vector of dependent variables, is the matrix of independent variables, is the y X β

vector of regression coefficients and is the vector of random errors. The errors are assumed e

to be uncorrelated and N(0, γ2), meaning that they are Normally distributed with mean of zero

and a variance of γ2. This is referred to as the model error variance.

Grover et al. (2002) highlight that in hydrology the true value of (e.g. the vector holding y

QMED values observed at a set of sites) is typically unknown, and there is therefore an error

associated with its estimation. Adopting widely used notation (e.g. Stedinger and Tasker,

1985), if y is an unbiased estimate of the variable of interest then:

E[ ] = ŷ y C.2

and

Var[ ] = ŷ Σ C.3

where is the sampling covariance matrix associated with the estimate of . Therefore Σ ŷ

Equation C.1 is written:

= + y X β u C.4

where is a random vector of errors that are a combination of model and sampling errors u

defined as:

Var[ ] = = u Λ γ2 I + Σ C.5

where is defined as the full covariance residual matrix, is a vector of modelling errors, Λ γ2

I

is the identity matrix and is a matrix of sampling errors. The least-squares estimate for in Σ β

Equation C.4 is given by:

β = (T -1

)-1

T -1

X Λ X X Λ ŷ C.6

This is also known as the generalised least-squares estimator.

In this study, three least-squares methods – ordinary, weighted and generalised – were

applied to solve Equation C.6, thereby estimating the model parameters . Ordinary least-β

squares (OLS) is the simplest method and is suitable when the sampling error in the data is

small ( ≈ 0) and the error terms have equal variances and are uncorrelated. Σ

C2 Weighted least-squares (WLS) regression

The weighted least-squares (WLS) procedure for hydrologic regression introduced by Tasker

(1980) attempts to limit the uncertainty introduced by unequal record lengths. [Unequal

record lengths tend to lead to heteroscedacity, i.e. to observations that have unequal sampling

errors. For example, one will generally have a much better estimate of QMED (the median

annual maximum flood) where the median is taken from 30 rather than (say) eight annual


160

maxima. In consequence, the assumption of constant variance in OLS is no longer valid.] In

the WLS approach, a weighting term – proportional to the square root of the record length –

is used to represent the sampling error. The procedure used here follows Weisberg (1980).

C3 Generalised least-squares (GLS) regression

The generalised least-squares (GLS) procedure – introduced to hydrological application by

Stedinger and Tasker (1985) – is an extension of WLS which seeks to account for inter-site

correlation in flood data.

In applying the GLS procedure, it was assumed here that the inter-site correlations in AM

flow data provide a reasonable approximation to the correlations in the regression errors.

Inter-site correlation was assessed and represented by an exponential decay with distance,

with inter-site correlation falling to approximately 0.5 at a distance of 50 km (see Figure C.1).

An exponential spatial correlation was incorporated into the GLS approach. While it is

evident from Figure C.1 that this is an approximation only, its incorporation into the

estimation procedure provides some recognition of the effect of inter-site correlation.

Figure C.1: Fitted model for inter-site correlation

[Editorial note: The Figure C.1 plot of inter-station correlation in AM flows against inter-

catchment distance is a little puzzling. The perfect correlation of a small number of station-

pairs is explained by records that overlap in one year only. However, one would expect some

station-pairs – especially those having only a few years of record in common – to exhibit

negative correlation. It seems likely that this is a mistake in the plot rather than in the

analysis. The decay rate of the fitted exponential is a little slower than that reported by

Robson (1999b) for UK flood data.]

From the modelling conducted, it was found that the assumptions of the OLS approach – i.e.

Normally distributed residuals, equal variance and uncorrelated sampling errors in the data –

were broadly satisfied. The simpler approach was therefore adopted. The extension of the

methodology to WLS and GLS returned only very slight changes in model performance and

parameter values. The results are therefore not reported further.

2001751501251007550250

Distance (km)

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

Co

rrela

tio

n

0 50 100 150 200

Inter-centroid distance, d (km)

Corr

elat

ion,

r

1.0

0.8

0.6

0.4

0.2

0.0

r = exp(-0.0137 d)


161

C4 Geographically-weighted regression (GWR)

A few details of geographically weighted regression (GWR) are given. A global regression

model can be presented as:

εxvμ,β....xvμ,βvμ,βy nn110 C.7

where (μ, ν) denotes the coordinates of the samples in space. In GWR, the parameter

estimates are made using an approach in which the contribution of a sample to the analysis is

weighted based on its spatial proximity to the specific location under consideration. Thus,

the weighting of an observation is no longer constant in the calibration but varies with

location. Data from observations close to the location under consideration are weighted more

than data from observations far away.

The parameters are estimated from a variation of Equation C.6 (in Section C1) in which the

covariance matrix -1

is replaced by a special matrix of weights: Λ

β (μ, ν) = (T

(μ, ν) )-1

T

(μ, ν) X W X X W ŷ C.8

Here, β (μ, ν) is an estimator of , is the matrix of independent variables, and (μ, ν) is the β X W

weighting matrix. β is the vector of regression coefficients.

Several methods have been proposed to determine the weighting matrix. Let Wij denote the

weight of the specific point j in the space at which data are observed to any point i in the

space from which parameters are estimated. Under certain assumptions it is reasonable to

represent Wij – known as the kernel – by the continuous function:

2

/bdexpW

2

ij

ij C.9

where dij denotes the distance between i and j, and b is referred to as the bandwidth. An

alternative kernel utilises the bi-square or quartic function:

22

ijij /bd1W if dij < b C.10

0Wij otherwise

Fixed kernels in regions where data are dense may suffer from bias when the kernels are

larger than needed. When the kernels are smaller than needed, they may not estimate the

parameters reliably where data are scarce. Thus, spatially varying kernels have also been

proposed.

Parameter estimation in GWR is highly dependent on the weighting function and the

bandwidth of the kernel used. As the bandwidth increases, the parameter estimates will tend

to the estimate from a global model. The selection of the weighting function and bandwidth

can be determined using a cross-validation approach. In this work, GWR was deployed using

an adaptive bi-square kernel, and the selection of weighting functions and bandwidth was

based on cross-validation.


162

Table C.1 shows the results of the tests for spatial stability in model parameter coefficients

for each independent variable in the rural ℓnQMED model of Equation 2.7. It is evident that

the majority of parameter coefficients in the model are indeed spatially constant. The

exception is the coefficient of ℓnFARL. [Editorial note: The use of p-values is discussed in

Section E3.4.]

Table C.1: Test of significance of spatial variability in ℓnQMED model parameters

Parameter p-value Significance Parameter p-value Significance

Intercept 0.14

None

ℓnSAAR 0.13 None

ℓnAREA 0.84 ℓnFARL 0.05 Just significant

ℓnDRAIND 0.16 ℓnBFIsoil 0.36 None

ℓnS1085 0.24 ℓn(1+ARTDRAIN2) 0.86

Map C.1 suggests that the coefficient of ℓnFARL takes generally higher values in the East

and North West and lower values around the Upper Shannon basin and in the West.

Comparisons with Map 2.2 andMap 2.3 suggest that the mapped minimum just west of

Wexford is driven by results from a single Grade B site: Station 13002 Corock at Foulk’s

Mill. [Editorial note: The FARL value for this catchment is 1.0, implying no major

reservoir or lake on the river network. Thus, it appears that the isolated overestimation of

QMED (evident in Map 2.2) is leading to the minimum west of Wexford, rather than any

feature related to FARL. This illustrates that automated methods can lead to over-fitting of

observations and can bypass tests of physical reasoning. What is really needed is

intensification of the gauging network. It is understandable why the GWR approach was not

selected for implementation.]


163

Map C.1: Spatial variation in the FARL coefficient as interpolated from GWR

Coefficient of

ℓnFARL

-1.514 – -0.552

-0.552 – 0.411

0.411 – 1.374

1.374 – 2.337

2.337 – 3.299

3.299 – 2.337

1.374 – 4.262

4.262 – 5.225

5.225 – 6.188

6.188 – 7.151

Wexford


164

Appendix D QMED models for drained/undrained catchments

D1 Partitioned approach

A more detailed approach to modelling QMED was explored by deriving separate models for

drained and undrained catchments. The calibration set of 145 essentially rural catchments

comprises 95 undrained (i.e. without arterial drainage) and 50 drained (i.e. with arterial

drainage). Some of the undrained cases are pre-drainage records from catchments

subsequently drained.

D2 Undrained catchments

Following the methodology used to derive the rural model of Section 2.2, QMED values for

the 95 undrained stations were subject to an exhaustive search to derive the best combination

of catchment descriptors for model-building. Details of the 6-variable model resulting are

given in Table D.1. This is similar in make-up to the general rural model of Equation 2.7

except that the arterial drainage term is redundant. The similarity of the results (to those of

Table 2.7) increases confidence in the manner in which ℓn(1+ARTDRAIN2) indexes

drainage and in the meaningfulness of the other six variables.

Table D.1: Coefficient and collinearity statistics for undrained ℓnQMED model


error

β

value

t

statistic

95% confidence

interval Variance

inflation


Constant -11.145 1.44 -7.74 -14.01 -8.28

ℓnAREA 0.910 0.04 1.05 21.34 0.82 1.00 2.00

ℓnSAAR 1.328 0.22 0.35 6.03 0.89 1.77 2.79

ℓnFARL 2.762 0.41 0.31 6.78 1.95 3.57 1.71

ℓnS1085 0.214 0.05 0.24 4.07 0.11 0.32 2.76

ℓnDRAIND 0.477 0.12 0.18 4.03 0.24 0.71 1.69

ℓnBFIsoil -0.590 0.21 -0.13 -2.76 -1.02 -0.16 1.87

Nevertheless, there are substantial differences in some of the model coefficients. Of

particular note is the reduced importance of ℓnBFIsoil, indicated by the more modest

coefficient and β value in Table D.1. The coefficients of ℓnDRAIND (especially), ℓnFARL

and ℓnS1085 all increase relative to the general rural model (i.e. compared to the

corresponding values in Table 2.7).

Overall, the model provides good results with an r2

of 0.892 and a factorial standard error

(FSE) of e0.315

= 1.370. From the model diagnostics, the OLS approach is again judged

acceptable, with the logged residuals Normally distributed and showing little evidence of

heteroscedacity (i.e. uneven error variance).


165

Figure D.1 summarises the performance achieved for the undrained stations. Using the

subscript ud to denote undrained, the OLS model is:

ℓnQMEDud = –11.145 + 0.910 ℓnAREA + 1.328 ℓnSAAR + 2.762 ℓnFARL +

0.214 ℓnS1085 + 0.477 ℓnDRAIND – 0.590 ℓnBFIsoil D.1

50020010050201052

500

200

100

50

20

10

5

2


Pre

dic

ted Q

ME

D

1:1 line

3 -1

Station 06030 River Big at Ballygoly

Figure D.1: Performance achieved with undrained model (on 95 undrained catchments)

D3 Drained catchments

In order to fit a regression model to the drained catchments (i.e. catchments in their post-

drainage state), the exhaustive regression approach was again employed. Using the subscript

d to denote drained, the linear form of the selected 4-variable model is:

ℓnQMEDd = –11.214 + 0.976 ℓnAREA – 1.780 ℓnBFIsoil + 1.230 ℓnSAAR +

1.328 ℓnFARL D.2

Further details of the model and its performance are given in Table D.2 and Figure D.2.

Overall, the model performs well with an FSE of e0.318

= 1.374.

It is possible that the excellent degree of fit achieved (r2=0.936) reflects that catchments that

have undergone drainage tend to be of a similar ilk, so there is less variance for a model to

have to capture. The variance inflation factors in Table D.2 confirm the relative robustness

of the model.

Nevertheless, there are some interesting features. The coefficient of ℓnBFIsoil is about three

times stronger than in the undrained model, and the variable is second only to ℓnAREA in


166

helping to explain the variation in ℓnQMED. This suggests the particular usefulness of

BFIsoil in characterising the behaviour of drained catchments.

Table D.2: Coefficient and collinearity statistics for drained ℓnQMED model


error

β

value

t

statistic

95% confidence

interval Variance

inflation


Constant -11.213 2.09 -5.37 -15.42 -7.01

ℓnAREA 0.976 0.04 0.96 22.69 0.89 1.06 1.27

ℓnBFIsoil -1.780 0.28 -0.28 -6.30 -2.35 -1.21 1.41

ℓnSAAR 1.230 0.31 0.16 3.98 0.61 1.85 1.17

ℓnFARL 1.328 0.61 0.11 2.18 0.10 2.56 1.67

500200100502010521

500

200

100

50

20

10

5

2

1


QM

ED

pre

dic

ted b

y d

rain

ed m

odel 1:1 line

Station 24022

25017

3 -1

Figure D.2: Performance achieved with drained model (on 50 drained catchments)

D4 Choosing a general purpose model

The partitioned approach has derived separate models for predicting ℓnQMED on drained

and undrained catchments. To test the merit of the approach, its performance was assessed

on the 25 stations in the validation dataset, and against the performance achieved by the

general model of Section 2.2.

The validation set comprises 17 undrained stations and eight drained stations. Table D.3

compares the success of the partitioned and general models in explaining variation in

ℓnQMED across the validation stations.


167

The drained-catchment model of Equation D.2 is seen to provide only a modest improvement

over the general model when tested on the eight drained catchments in the validation set.

More notably, the undrained-catchment model of Equation D.1 is outperformed by the

general model when tested on the 17 undrained catchments in the validation set. This is

likely due to the fact that all six PCDs appearing in the partitioned model are no different to

those already in the general 7-variable model for ℓnQMED. The general model includes

ARTDRAIN2.

Table D.3: Validation of partitioned and general models for ℓnQMED

Validation stations r

2 by partitioned model r

2 by general (Section 2.2)

model Undrained Drained

17 undrained 0.898 0.915

8 drained 0.891 0.848

All 25 0.906

Given the mixed success of the partitioned approach in validation – and the fact that it has

fitted 12 parameters (seven for the undrained and five for the drained) as against eight

parameters for the general model – it was judged prudent to maintain the recommendation in

Section 2.2.5 to adopt the 7-variable general model to estimate the rural component of

QMED on all catchments: drained and undrained.

Nevertheless, the partitioned analysis has shown BFIsoil and ARTDRAIN2 to be particularly

important in characterising the post-drainage flood response of a catchment, and the

descriptors DRAIND and S1085 to be relatively important when estimating QMED on

undrained catchments. These findings may help the experienced user to judge the most

appropriate data transfer when estimating QMED at an ungauged subject site (see

Section 11.2).


168

Appendix E Trend analysis of AM flows in Irish rivers

E1 Introduction

A major assumption in flood frequency analysis is that all observations in the dataset are

independent (random) and identically distributed. The study reported here is by Mandal and

examines the validity of the independent and identically distributed (iid) assumption for

annual maximum (AM) flows in Irish rivers. See also Mandal (2011).

Randomness cannot be proved but it can be disproved by the presence of a non-random

feature such as trend. Statistical methods lie at the heart of testing for change in hydrological

data series. Before the tests are described, the purpose, importance and language of trend

analysis are first introduced.

E2 Purpose

The purpose of trend testing is to determine if the values of a variable generally increase (or

decrease) over some period of time (Helsel and Hirsch, 1992; Hirsch et al. 1993).

E2.1 Importance of testing whether hydrological processes are stationary

Detection of abrupt or gradual changes in hydrological records is of scientific and practical

importance, and fundamental to planning water resources and managing flood risks

effectively. Traditional “design rules” (e.g. for water resource or flood alleviation systems)

are based on the assumption that hydrological processes are stationary (see Box E.1) and the

principle that the past is the key to the future.

Box E.1: Stationarity

The assumption of stationarity is clearly questionable in the era of global change. If it is

incorrect, procedures for water-related structures such as dams and river embankments will

need to be reassessed to ensure that risks are properly appreciated. Systems might be under-

designed and therefore fail to perform adequately. On the other hand, if systems are over-

designed, they may not be cost-effective.

E2.2 Types of change

Changes in river flows can be caused directly by human activities such as urbanisation,

reservoir construction, drainage works, water abstraction and agricultural changes, or by

incidental factors such as changes in channel morphology. An apparent change in flood

behaviour may arise from measurement practices, e.g. a change in instrumentation, a change

in data processing or the failure to apply the flood rating curve appropriate to the particular

period of record.

A stochastic process is said to be strictly stationary if its probability distribution is

completely independent of time. It is said to be stationary in the wider sense if the mean

and variance are independent of time. The statistics of samples drawn from a stationary

process may vary due to sampling variability but not due to their size or their position in

the population.


169

Climate is the principal driver of the hydrological cycle. Since the climate system and water

cycle are intimately linked, any change in one induces change in the other. Widespread

effects therefore arise from climate variability and from human-induced climate change.

Change in a series can manifest itself in a number of ways. Attention typically focuses on

gradual progressive change – referred to as trend – and abrupt change – often referred to as

step-change. Cyclical behaviour makes for a third kind of non-stationarity. For example,

several major floods may occur in a so-called flood-rich period only to be succeeded by a

flood-poor period in which no major flood occurs for many years.

Attention usually focuses on testing for trend or step-change in the typical value of a data

series: for example, a change in the mean or median. Non-stationarity may also be revealed

in a change in variability or in autocorrelation, or in almost any aspect of data.

With a range of effects to be considered, exploratory data analysis and graphical display of

data series are essential.

E3 Procedure

E3.1 Steps in the analysis

Kundzewicz and Robson (2004) list the main stages in a statistical analysis of change:

Decide what type of data series (e.g. monthly averages, annual maxima, peaks-over-

threshold series, …) to test to meet the issue of interest;

Decide what types of change are of interest (e.g. gradual or step-change);

Check the assumptions in an exploratory data analysis or by applying a formal test;

Select the statistical methods:

A particular test (noting that it is good practice to use more than one);

A particular test statistic;

A method for evaluating significance levels;

Evaluate the significance levels;

Investigate and interpret the results.

E3.2 The idea of exploratory data analysis

Exploratory data analysis (EDA) is an advanced visual examination of the data and forms an

integral part of any study of change. Graphical presentation is pivotal. An advantage of a

large study such as that reported here is the ability to assess behaviour at many sites. Any

effect arising from climate is likely to influence more than one flood series.

The first stage of EDA is to examine the raw data to identify notable features such as:

Peculiarities or problems (e.g. unusually large/small values or gaps in the record);

Temporal patterns (e.g. trend, step-change or marked seasonality);

Regional/spatial patterns.

Exploratory data analysis also plays an important role in checking assumptions such as

independence or the presumed statistical distribution of data values. Types of graph that can


170

be useful for checking hydrological data series include histograms, probability plots,

autocorrelation plots, scatter-plots and smoothing curves (see Box E.2).

Box E.2: Smoothing

A well-conducted EDA is sometimes such a powerful tool that it eliminates the need for a

formal statistical analysis (Kundzewicz and Robson, 2004).

E3.3 Hypothesis testing

A statistical test requires a formal procedure that includes:

Stating a null hypothesis (the convention is to label this H0);

Declaring a test statistic and its distribution under the null hypothesis;

Stating a critical region for the test statistic in which – under the null hypothesis – the

value of the test statistic falls with probability α;

Computing the test statistic for the data sample;

Accepting/rejecting the null hypothesis according to whether the observed test

statistic value lies inside/outside the critical region.

When interpreting results it is necessary to remember that no statistical test is perfect, even if

all test assumptions are met. Adoption of a 5% significance level means that an error will be

made, on average, 5% of the time. [If the 5% significance level is adopted and the null

hypothesis is actually true, about one in 20 test results will (as a matter of chance) yield a test

statistic greater than the critical value. In consequence, the null hypothesis will be (wrongly)

rejected.]

It is relatively commonplace to term a result significant if the null hypothesis is rejected at the

5% level and to denote it highly significant if it is rejected at the 1% level. However, there is

no universal acceptance of terminology and it is widely recognised that the choice of significance level, α, is arbitrary.

[Editorial note: A statistical test determines the statistical significance of a result. This is

not to be confused with the practical significance of a feature. For example, data may reveal

a trend that is very highly significant statistically. Yet the scale of the trend may be too

minor to be of practical importance.] Where many tests are applied to a large number of data samples, there is considerable scope

for significant results to arise by chance. In consequence, the interpretation of results can

become complex. Even the presence of a highly significant result may provide only weak

evidence of change. However, if a batch or “basket” of significant results is obtained – e.g. if

Smoothing techniques are used to reduce irregularities (random fluctuations) in time series

data. They can provide a clearer view of the underlying behaviour of a series.

In time series where it is naturally strong, seasonal variation can impede the detection of

trends or cycles. Smoothing can remove seasonality and make long-term fluctuations in

the series stand out more clearly. A typical smoother is a moving average filter in which

each observation is replaced by the average of observations in a time window centred on

the time of the particular observation.


171

the null hypothesis is rejected by several tests or if a particular test rejects the null hypothesis

at a number of sites on a particular river or in a particular district – there can be greater

confidence that an important effect has been detected.

E3.4 Use of p-values

An alternative way to conclude a test of hypotheses is to compare the p-value of the sample

test statistic against the significance level (α). The p-value of the sample test statistic is the

smallest level of significance for which we can reject H0. In other words, the p-value of a

statistical hypothesis test is the probability of getting a value of the test statistic as extreme as

or more extreme than that observed by chance alone, if the null hypothesis H0 is true.

The p-value is compared with the actual significance level of our test and, if it is smaller, the

result is deemed significant. That is, if the null hypothesis were to be rejected at the 5%

significance level, this would be reported as “p < 0.05”. The smaller the p-value, the more

convincing is the rejection of the null hypothesis. It indicates the strength of evidence for

rejecting the null hypothesis H0, rather than merely concluding “Reject H0” or “Do not reject

H0”.

The p-value serves a valuable purpose in the evaluation and interpretation of research

findings. It enables the researcher to set their own level of significance and to reject or accept

the null hypothesis in accordance with their own criterion rather than that at a fixed level of

significance.

E4 Statistical tests

E4.1 Parametric and non-parametric tests

Various test procedures have been developed for detecting trend in hydrological time series.

An important classification is whether the test is parametric or non-parametric. A parametric

test is one that depends on the form of parent distribution from which the sample is assumed

to be drawn. Such tests are able to make full use of the available information but carry the

penalty that the distributional assumption may be incorrect. This may bias the results

obtained.

In contrast, non-parametric tests require few if any assumptions about the shape (e.g.

skewness) of the underlying population distributions. Consequently, they are also known as

distribution-free methods. Typically, non-parametric tests use ordinal information (often, the

ranks of the data values) rather than the values themselves.

Inevitably, non-parametric tests are less powerful than parametric tests. Thus, when the

relevant distributional assumption is met, it is preferable to use the parametric test.

Hydrological data are often strongly non-Normal. This means that tests which assume an

underlying Normal distribution are inadequate. In many real-life situations, distribution-free

methods are to be recommended because they require only minimal assumptions to be made

about the data series.

The Flood Studies Report (NERC, 1975) found that the EV1 distribution could adequately

describe Irish AM flow series. The use of tests based on the Normal distribution was

therefore not considered to be appropriate. Non-parametric tests are used in order to test the


172

underlying assumptions of independent and identically distributed (iid) time series.

However, in order to compare the power of the tests, two parametric tests are also used.

E4.2 Null hypotheses

The null hypothesis H0 for the tests for trend is that there is no trend in the data. The null

hypothesis H0 for the tests for changes or difference in means/medians is that there are no

changes or difference in the means/medians between two data periods. The null hypothesis

H0 for the tests for randomness (independence) is that the data come from a random process.

It is considered that the alternative hypotheses for all tests are non-directional i.e. all tests are

two-tailed tests.

E4.3 Tests adopted

Six tests were used in detecting trends, shift and serial dependency in the AM flood series:

Tests for trend

Mann-Kendall (non-parametric test for trend)

Spearman’s Rho (non-parametric test for trend)

Mean-weighted linear regression (parametric test for trend)

Test for step-change

Mann-Whitney U (non-parametric test for step-change in mean/median)

Tests for serial dependency of time series

Turning point (non-parametric test for randomness)

Rank difference (non-parametric test for randomness)

E4.4 Mann-Kendall test

The Mann-Kendall test is a non-parametric test for identifying trends in time-series data. The

data need not conform to any particular distribution.

The test compares the rankings of the sample data rather than the data values themselves.

The n time-series values {X1, X2, X3, …, Xn} are replaced by their relative ranks {R1, R2, R3,

…, Rn} where R1 marks the lowest value and Rn the highest. Where equal-ranking (i.e. tied)

values are found, each is assigned a mean rank. Thus, if the 4th

, 5th

and 6th

ranked values are

equal, each is assigned a rank of 5.0. See Box E.3 for a discussion of tied values.

Box E.3: Tied values

The Mann-Kendall test statistic S (known as Kendall’s sum) is given by:

1

1 1

sgnn

i

n

ij

ji RRS E.1

where sgn is the sign function defined by:

Editorial note: Exactly equal values are known as tied values or ties. For various

reasons – notably the rounding of water-level observations to a fixed number of decimal

places – ties are quite common in flood peak data. It can be important for methods to

explicitly recognise this feature, if best estimates are to be obtained.


173

sgn(x) = 1 for x > 0

sgn(x) = 0 for x = 0 E.2

sgn(x) = -1 for x < 0

A positive value of S indicates a possible upward trend; a negative value indicates a possible

downward trend.

Mann (1945) and Kendall (1975) document that if the null hypothesis H0 is true (i.e. if there

is no trend) then for n ≥ 8, the statistic S is approximately Normally distributed with mean

zero and variance:

18

5)1)(2ii(it5)1)(2nn(n

(S)Var

n

1i

i

E.3

where ti is the number of ties of extent i.

The standardised Mann-Kendall (MK) statistic, ZMK, is computed by:

SVar

1SZMK

for S > 0

0ZMK for S = 0 E.4

SVar

1SZMK

for S < 0

ZMK follows the N(0, 1), i.e. the standard Normal distribution with a mean of zero and a

variance of one.

The p-value of the MK statistic is found from the cumulative distribution function (CDF) of

the Normal distribution using:

ZΦ0.5p where dte2π

1ZΦ

Z

0

2t2

E.5

When the p-value is small enough, the trend is considered to be statistically significant. For

example, if p < 0.05 the trend is considered significant (if a 5% significance level is being

applied).

E4.5 Spearman’s ρ test

Spearman’s ρ (rho) is a rank-based test that determines whether the correlation between two

variables is significant. When applied to trend analysis of annual maxima, one variable is

taken as the year number and the other as the annual maximum value. Both variables are

replaced by their ranks. [Editorial note: This substitution is less suitable if there is a period

of record missing in the AM series.]

If the time series consists of n distinct values, the ranks will be the numbers 1 to n, with 1

corresponding to the lowest value in the series and n to the highest. If there are ties (see Box

E.3) in the series, each value in the tie group is assigned the same (mean) rank. Thus the

ranks can sometimes be non-integer.


174

The null hypothesis H0 of Spearman’s ρ test is that there is no relationship between the

variables. Given a time series of values {Xi, i=1, 2, … n}, rejection of the null hypothesis

implies that Xi increases or decreases with i (i.e. that a trend exists with time). The relevant

test statistic given by Sneyers (1990) is:

1nn

i-R6

1D2

n

1i

2

i

E.6

where Ri is the rank of the ith

observation, Xi.

Under the null hypothesis – following Lehmann (1975) and Sneyers (1990), and for large

sample size (n > 30) – the distribution of D can be taken to be Normal with mean zero and

variance 1/(n-1).

Dividing by the standard deviation, the standardised test statistic:

D1n1n1

D

DVar

DZSR

E.7

follows the standard Normal distribution Z ~ N(0, 1). The p-value of the statistic can then be

evaluated in the usual way. This test is particularly useful for the detection of gradual change

in time series.

Box E.4: Resampling methods

E4.6 Mean-weighted linear regression test

This is a parametric test that assumes that the data come from an EV1 distribution. It tests

whether there is a linear trend by examining the relationship between time (x) and the

variable of interest (y). The regression gradient is estimated by:

Editorial note: An alternate approach to testing whether an observed value of

Spearman’s rank correlation is significantly different from zero is by permutation

resampling. The time series of AM values is randomly re-ordered and the rank correlation

of the resample rs calculated. The procedure is repeated many times. If the actual rank

correlation lies in the top 2.5% or bottom 2.5% of the distribution of rs values obtained by

permutation resampling, the trend can be declared significant at the 5% level.

Critical values of test statistics in rank-based methods are typically complicated by the

presence of tied values (see Box E.3). A well-executed approach based on permutation

resampling will deal with this automatically, whereas a theoretical approach may require

specially designed adjustments such as those in Equation E.3.

Resampling methods can be used to evaluate test statistics or to construct confidence

intervals in a wide range of hydrological applications. The resampling method can be

designed to preserve important structure in the dataset, e.g. resampling AM series by year

number when estimating confidence intervals for a pooled flood growth curve.


175

n

1i

2

i

n

1i

ii

xx

yyxx

b E.8

The test statistic bs is:

y

bbs where

n

1i

iyn

1y E.9

Under the null hypothesis of bs = 0 (i.e. the hypothesis of independent and identically

distributed samples drawn from an EV1 distribution), the critical values of bs for various

significance levels were calculated by sampling experiments.

Ten thousand time series each of size n = 50 were generated from the EV1 population with a

mean μ = 100 and coefficient of variation CV = 0.3. The mean value of 100 was chosen

arbitrarily. The CV of 0.3 is chosen to be typical of Irish AM flood series. Critical values at

various significance levels for sample sizes from three to 60 were derived. They are

presented later in Table E.6 of Section E8.

E4.7 Mann-Whitney U test

The Mann-Whitney U test is a non-parametric test of the null hypothesis that two populations

are the same against the alternative hypothesis that one population tends to have larger values

than the other. When applied to test for step-change in a series, the full sample is broken

down into early and late samples of size n1 and n2.

For some flood series, the separation into subsamples is known a priori: the date of change

corresponding to drainage works or to a known change in flow measurement. In other cases,

the date of any step-change is speculative or unknown. [Editorial note: Kundzewicz and

Robson (2004) recommend use of the median change-point test (Pettitt, 1979) in these cases.]

The Mann-Whitney U test labels observations from the two subsamples before ranking them

jointly. If the subsamples differ substantially, their elements will be poorly mixed in the joint

ranking, with the elements of one subsample displaying typically low rank numbers and those

of the other displaying typically high rank numbers.

The test statistic U is calculated as the smaller of U1 and U2 where:

111

1 R2

1nnU

and 1212 UnnU E.10

Here, n1 and n2 are the sizes of the first and second subsamples and R1 is the sum of the ranks

attributed to members of the first subsample in the ranked total sample. An alternate name is

hence the rank-sum test.

The test statistic U is designed to take a low value when the subsamples are not well mixed.

If the observed U value is less than a certain critical value Ucr, the hypothesis that there is no

difference (in typical values) between subsamples is rejected, and a step-change declared

significant. The quantity Ucr depends on the significance level adopted and the subsample


176

sizes. It is tabulated by Siegel (1956). When both subsamples are larger than 10, U is

approximately Normally distributed with mean n1n2/2 and variance n1n2(n1+n2+1)/12.

E4.8 Turning points test (Kendall’s test)

This non-parametric test is based on counting turning points in the series. Turning points are

triples of consecutive values xi-1, xi, xi+1 such that xi-1 < xi > xi+1 or xi-1 > xi < xi+1. If N is the

number of turning points, then the test statistic is:

2916n

1042n3NS

E.11

Under the null hypothesis of iid values, S is N(0, 1) distributed (Srikanthan et al., 1983).

E4.9 Rank difference test (Meacham test)

This is a nonparametric test based on computing differences in ranks of consecutive values in

the time series. The test statistic is:

74n1n2n

101n3US

2

E.12

where U denotes the sum of absolute values of the rank differences. Under the null

hypothesis of iid values, S is N(0, 1) distributed (Srikanthan et al., 1983).

E5 Exploratory data analysis

E5.1 Selection of data

It is important to consider carefully the form and frequency of the data that should be

analysed. This usually depends on the focus of the study. For floods, the biggest flow is

often of interested; for droughts, it may be the duration of low flows. Selection of which

stations to use in a study is also important (Kundzewicz and Robson, 2004). For example, in

order to study the climate-change signature in river flow, data should ideally be taken from

baseline rivers and should be of high quality and extend over a long period. Data should be

quality-controlled before commencing an analysis of change.

For the present study, annual maximum flood series were selected for 117 stations graded A1

or A2. Some 79 of these have continuous records ranging from 17 to 58 years in length. The

remaining 38 stations have intermittent records with the number of consecutive missing

values in the series ranging from 1 to 16.

As part of the EDA, the AM flows for each station were plotted and a linear trend fitted using

Excel’s data analysis tool. Most of the AM series were found to be well behaved with only a

few series showing outliers.

E5.2 Stations showing trend

Twelve of the 117 series showed a notable upward or downward trend (see Table E.1). The

two stations marked in red are amongst those rejected from general study for reasons

discussed in Appendix B.


177

Table E.1: Stations with significant trends

Station

number Water-body and location

#

AM

Station


#

AM

07004 Blackwater (Kells) at Stramatt 48 25014 Silver at Millbrook 54

15004 Nore at Mcmahons Bridge 51 25023 Little Brosna at Milltown 52

16003 Clodiagh at Rathkennan 51 25029 Nenagh at Clarianna 33

24001 Maigue at Croom 51 26005 Suck at Derrycahill 51

25002 Newport at Barrington’s Bridge 51 34018 Castlebar at Turlough 27

25003 Mulkear at Abington 51 36011 Erne at Bellahillan 49

The data series and trend-lines for these 12 series are shown in Figure E.1. Note that each

diagram shows two stations. In addition to the rejected Mulkear at Abington (Station 25003),

the Nore at MacMahon’s Bridge and Nenagh at Clarianna also show significant downward

trend. The remaining nine series show upward trend. [Editorial note: Two missing annual

maxima – for 1981 at Station 07004 and for 1984 at Station 36011 – have been inadvertently

ascribed a value of 0.0 in the trend analysis. The gap at Station 07004 corresponds to when

arterial drainage was undertaken.]

E5.3 Stations with pre and post-drainage records

The likely impact of arterial drainage works on the AM flows in some of these catchments is

evident. Fourteen stations with pre and post-drainage records are listed in Table E.2 and

illustrated in Figure E.2. [Editorial note: Station 07004 shown in Figure E.1e is a 15th

such

station. It has been added to Table E.2.]

Table E.2: Stations with pre and post-drainage records

Station


#

AM

Station


#

AM

07002 Deel at Killyon 46 23002 Feale at Listowel 59

07003 Blackwater (Enfield) at Castlerickard 46 24001 Maigue at Croom 51

07004 Blackwater (Kells) at Stramatt 48 25001 Mulkear at Annacotty 49

07005 Boyne at Trim 47 26012 Boyle at Tinacarra 48

07007 Boyne at Boyne Aqueduct 45 30001 Aille at Cartronbower 48

07010 Blackwater (Kells) at Liscartan 46 30004 Clare at Corrofin 35

07011 Blackwater (Kells) at O’Daly’s Bridge 44 30005 Robe at Foxhill 49

07012 Boyne at Slane Castle 65

In most cases an upward step-change is evident in the AM flows following arterial drainage.

Station 25001 Mulkear at Annacotty is amongst those rejected from general study for reasons

discussed in Appendix B. It is seen from Figure E.2ℓ that Station 30001 Aille at

Cartronbower shows a downward step-change.


178

Figure E.1: Stations for which AM flow series shows significant trend

Annual Maximum Flow

y = 0.0979x - 181.51

y = -0.119x + 253.06

0

5

10

15

20

25

30

1951 1961 1971 1981 1991 2001

Year

Flo

w (

m3/

s)

25014 2523 Linear (2523) Linear (25014)

Annual Maximum Flow

y = 0.6966x - 1330.7

y = -0.2524x + 560.58

0

10

20

30

40

50

60

70

80

1972 1977 1982 1987 1992 1997 2002

Year

Flo

w (

m3/s

)

25029 25002 Linear (25029) Linear (25002)

Annual Maximum Flow

y = 0.4125x - 723.53

y = 0.8654x - 1605.7

0

50

100

150

200

250

1953 1963 1973 1983 1993 2003

Year

Flo

w (

m3/s

)

24001 26005 Linear (26005) Linear (24001)

Annual Maximum Flow

y = 0.2256x - 415.25

y = 0.1887x - 354.92

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

1953 1963 1973 1983 1993 2003

Year

Flo

w (

m3/

s)

7004 16003 Linear (16003) Linear (7004)

Annual Maximum Flow

y = 0.0763x - 133.58

y = 0.1354x - 257.85

0

5

10

15

20

25

1953 1963 1973 1983 1993 2003

Year

Flo

w (

m3/s

)

34018 36011 Linear (36011) Linear (34018)

(a)

(b)

(c)

(d)

(e)

(f)

15004

25014

25023

25029

25002

26005

24001

16003

36011

34018

07004

25003


179

Figure E.2: AM flow series for stations with arterial drainage works during record

Annual Maximum Flows - Deel at Killyon (7002)

0

5

10

15

20

25

30

35

40

1953 1963 1973 1983 1993 2003

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Blackwater at Castlerickard (7003)

0

5

10

15

20

25

30

35

40

1953 1963 1973 1983 1993 2003

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Boyne at Trim (7005)

0

50

100

150

200

250

1953 1963 1973 1983 1993 2003

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Boyne at Boyne Aqueduct (7007)

0

10

20

30

40

50

60

70

80

1953 1963 1973 1983 1993 2003

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Blackwater at Liscarton (7010)

0

10

20

30

40

50

60

70

80

90

100

1953 1963 1973 1983 1993 2003

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Blackwater at O'Daly's Bridge (7011)

0

10

20

30

40

50

60

70

1958 1963 1968 1973 1978 1983 1988 1993 1998 2003

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Boyne at Slane Castle (7012)

0

50

100

150

200

250

300

350

400

450

500

1940 1950 1960 1970 1980 1990 2000

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Feale at Listowel (23002)

0

100

200

300

400

500

600

700

800

900

1946 1956 1966 1976 1986 1996

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Maigue at Croom (24001)

0

50

100

150

200

250

1953 1958 1963 1968 1973 1978 1983 1988 1993 1998 2003

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Mulkear at Annacotty (25001)

0

20

40

60

80

100

120

140

160

180

200

1953 1963 1973 1983 1993 2003

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Boyle at Tinacarra (26012)

0

10

20

30

40

50

60

70

1957 1962 1967 1972 1977 1982 1987 1992 1997 2002

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Aille atCartonbower (30001)

0

5

10

15

20

25

30

35

1952 1957 1962 1967 1972 1977 1982 1987 1992 1997

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Clare at Corrofin (30004)

0

20

40

60

80

100

120

140

1951 1961 1971 1981 1991 2001

Time

Flo

w (

m3/s

)

Annual Maximum Flows - Robe at Foxhill (30005)

0

10

20

30

40

50

60

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

Time

Flo

w (

m3/s

)

07010 07011

07012

07005

23002

26012

07002 07003

07007

24001 25001

30005

30004

30001

(a)

(c)

(e)

(g)

(i)

(k)

(m)

(b)

(d)

(f)

(h)

(j)

(l)

(n)


180

E6 Trend analysis and results

E6.1 Formats of flood data tested for trend

Based on the availability of records and the evidence of drainage improvement works,

statistical tests were applied to the flood data in two formats:

Full AM flood series;

Median of 5-yearly blocks of the AM flood series.

The use of 5-year medians can be viewed as a method of smoothing. Moreover, it is tolerant

of data imperfection. Here, the median of four recorded AM values was accepted as a

reasonable estimate of the 5-year median in cases where a gap in measurement or processing

led to the AM value being unknown in isolated years.

Inferences from the two formats of flood data did not differ greatly. Consequently, only the

results for the conventional analysis of AM series are reported here. Results of applying tests

to AM series for 94 of the 117 A1 + A2 stations are now presented. The other 23 stations had

more than one consecutive AM value missing.

Fifteen of these 94 stations had isolated years where the AM value was missing. At these

sites, the missing years were disregarded and the set of available values assumed to have

occurred consecutively in the series.

E6.2 Main test results

Trends in both data formats were analysed using the tests described in Section E4. A Fortran

program was developed and used in all relevant calculations. Values of the various test

statistics are presented in Table E.3 for the annual maximum series from each of 94 stations.

Results significant at the 5% and 1% level are highlighted.

Table E.3: Test statistics for trend and change in AM flow series

Emboldened entries are significant at 5%; values in red are highly significant at 1% level

Station

#

#

AM

Mann-

Kendall

Spearman’s

ρ

Linear

regression

Mann-

Whitney

U

Turning

Point

Rank

difference

06011 48 0.248 0.281 0.558 0.117 0.415 0.327

06013 30 0.803 0.963 0.944 0.254 0.551 0.770

06014 30 0.301 0.365 0.234 0.852 0.234 0.267

06025 30 0.027 0.035 0.253 0.008 0.551 0.968

06026 46 0.130 0.125 0.038 0.860 0.812 0.319

07004 48 0.000 0.000 0.001 0.000 0.000 0.000

07006 19 0.972 0.913 0.983 0.221 0.127 0.853

07009 29 0.409 0.336 0.236 0.012 0.172 0.453

07033 25 0.176 0.184 0.133 0.064 0.870 0.171

08002 21 0.740 0.814 0.637 0.097 0.367 0.819

09001 48 0.644 0.601 0.699 0.877 0.642 0.498


181


Station

#

#

AM

Mann-

Kendall

Spearman’s

ρ

Linear

regression

Mann-

Whitney

U

Turning

Point

Rank

difference

09002 24 0.286 0.363 0.001 0.707 0.401 0.502

10021 24 0.172 0.105 0.186 0.006 0.737 0.587

11001 33 0.745 0.728 0.631 0.759 0.571 0.910

14006 51 0.153 0.144 0.144 0.777 0.143 0.643

14009 25 0.072 0.064 0.214 0.663 0.412 0.560

14013 50 0.300 0.255 0.300 0.509 0.733 0.934

14018 51 0.065 0.062 0.111 0.019 0.910 0.777

14019 51 0.929 0.852 0.984 0.492 0.215 0.595

14034 17 0.564 0.700 0.748 0.229 1.000 0.416

15003 51 0.968 0.977 0.763 0.095 0.910 0.477

15004 51 0.002 0.002 0.001 0.000 0.024 0.024

16002 51 0.121 0.131 0.091 0.865 0.215 0.874

16003 51 0.000 0.000 0.008 0.000 0.910 0.000

16004 48 0.319 0.348 0.429 0.734 0.816 0.310

16005 30 0.972 0.930 0.724 0.309 0.882 0.374

16008 51 0.024 0.018 0.317 0.002 0.215 0.440

16009 52 0.068 0.076 0.321 0.015 0.655 0.959

18005 50 0.118 0.129 0.151 0.014 0.733 0.352

19001 48 0.756 0.581 0.754 0.503 0.561 0.461

19020 28 0.086 0.069 0.030 0.223 0.757 0.787

23001 45 0.256 0.338 0.310 0.021 0.548 0.669

23002 59 0.099 0.087 0.047 0.010 0.060 0.780

23012 18 0.058 0.046 0.147 0.001 0.001 0.002

24001 51 0.008 0.003 0.003 0.000 0.573 0.000

24008 30 0.432 0.424 0.356 0.309 0.882 0.582

24013 36 0.099 0.081 0.474 0.028 0.787 0.897

24022 20 0.015 0.015 0.001 0.000 1.000 0.110

24082 28 0.040 0.029 0.042 0.085 0.757 0.162

25002 51 0.004 0.004 0.123 0.025 0.573 0.031

25003 51 0.008 0.005 0.176 0.076 0.024 0.149

25006 52 0.336 0.227 0.149 0.552 0.655 0.142

25014 54 0.001 0.001 0.007 0.017 0.584 0.009

25017 55 0.133 0.163 0.329 0.031 0.665 0.738

25021 44 0.015 0.032 0.354 0.001 0.144 0.220

25023 52 0.005 0.004 0.002 0.001 0.577 0.026

25027 43 0.121 0.116 0.219 0.002 0.805 0.808

25029 33 0.007 0.004 0.014 0.000 0.571 0.058

25030 48 0.290 0.282 0.655 0.228 0.561 0.996


182


Station

#

#

AM

Mann-

Kendall

Spearman’s

ρ

Linear

regression

Mann-

Whitney

U

Turning

Point

Rank

difference

25158 18 0.096 0.209 0.342 0.047 0.116 0.559

26002 53 0.759 0.810 0.579 0.346 0.185 0.925

26005 51 0.003 0.004 0.105 0.001 0.910 0.681

26006 53 0.203 0.277 0.279 0.057 0.008 0.272

26007 53 0.730 0.739 0.990 0.233 1.000 0.167

26008 50 0.044 0.052 0.391 0.012 0.088 0.071

26009 35 0.191 0.229 0.612 0.987 0.681 0.496

26017 49 0.241 0.417 0.817 0.059 0.066 0.167

26018 49 0.163 0.198 0.440 0.052 0.420 0.116

26019 51 0.038 0.051 0.180 0.246 0.367 0.550

26020 33 0.133 0.221 0.307 0.000 0.777 0.276

26021 30 0.830 0.960 0.676 0.361 0.766 0.419

26022 33 0.448 0.489 0.588 0.589 0.571 0.589

26059 23 0.023 0.028 0.227 0.829 0.303 0.359

27001 30 0.015 0.016 0.280 0.068 0.766 0.402

27002 51 0.266 0.292 0.418 0.078 0.215 0.770

27003 48 0.488 0.464 0.713 0.773 0.201 0.864

27070 29 0.378 0.371 0.033 0.879 0.006 0.001

29004 32 0.517 0.366 0.590 0.070 0.031 0.902

29011 22 0.085 0.065 0.030 0.045 0.159 0.606

29071 29 0.051 0.047 0.113 0.000 1.000 0.473

30007 31 0.586 0.453 0.968 0.594 0.770 0.314

31002 26 0.567 0.476 0.529 0.228 0.054 0.075

32012 24 0.655 0.755 0.923 0.977 0.737 0.977

33070 25 0.118 0.128 0.450 0.007 0.511 0.091

34001 36 0.017 0.014 0.164 0.004 0.787 0.166

34003 29 0.764 0.775 0.738 0.034 0.363 0.565

34009 33 0.938 0.726 0.919 0.368 0.024 0.077

34011 30 0.284 0.224 0.433 0.245 0.882 0.003

34018 27 0.007 0.006 0.097 0.001 0.753 0.128

34024 29 0.722 0.730 0.691 0.285 0.363 0.632

35001 29 0.311 0.281 0.373 0.432 0.006 0.038

35002 34 0.328 0.385 0.470 0.196 0.329 0.369

35071 30 0.261 0.209 0.277 0.018 0.457 0.361

35073 30 0.695 0.770 0.917 0.263 0.297 0.968

36010 50 0.122 0.175 0.330 0.029 0.494 0.013

36011 49 0.004 0.005 0.097 0.001 0.420 0.038

36012 47 0.011 0.009 0.029 0.009 0.078 0.003


183


Station

#

#

AM

Mann-

Kendall

Spearman’s

ρ

Linear

regression

Mann-

Whitney

U

Turning

Point

Rank

difference

36015 33 0.020 0.015 0.066 0.001 0.479 0.101

36018 50 0.019 0.020 0.132 0.014 1.000 0.798

36019 47 0.011 0.017 0.138 0.001 0.158 0.010

36021 27 0.662 0.596 0.521 0.789 0.270 0.287

36031 30 0.169 0.213 0.567 0.534 0.766 0.793

39008 33 0.780 0.763 0.758 0.113 0.571 0.651

39009 33 0.828 0.825 0.943 0.126 0.777 0.972

The numbers of times the null hypothesis is rejected in the records taking each test are

summarised in Table E.4.

Table E.4: Number of cases (out of 94) in which the null hypothesis is rejected

Significance

level

Trend tests

Step-

change

test

Randomness tests Number

expected

(from 94

cases) by

chance

alone

Mann-

Kendall

Spearman

ρ

Mean-

weighted

linear

regression

Mann-

Whitney

U

Turning

point

Rank

difference

1% 12 13 8 24 5 9 ≈1

5%

but not 1% 14 13 8 15 4 6 ≈4

5%

(all cases) 26 26 16 39 9 15 ≈5

E6.3 Non-randomness

The tests of randomness identified significant non-randomness in some 15 (or 16%) of the 94

AM series. This is considerably more than the five expected by chance alone. There is

therefore evidence of some non-randomness in Irish AM flow data.

E6.4 Trend

In the trend tests, 26 out of 94 cases show statistically significant trends at the 5% level

according to the non-parametric (Mann-Kendall and Spearman ρ) tests. There is therefore

strong evidence of trend in Irish AM flow data.

The AM series that show the most marked departures from the null hypothesis (of zero trend)

are listed in Table E.5, where the direction of trend is also noted. These are taken from the

results presented in Table E.3. [Editorial note: The stations have been informally ranked

according to the sum of the p-values indicated by the three tests of trend.]


184

It is the nature of applying such tests widely that some of the stations exhibiting trend have

their own peculiarity. Possible explanations for trend are indicated in the comment column.

Arterial drainage is seen to feature in four of the seven series most clearly showing trend.

Table E.5: Stations for which the trend is judged most highly significant

Σ p-

values

Station

number Station name

Trend

noted Comment AD

* AD2

*

0.001 07004 Blackwater (Kells)

at Stramatt Increase

Arterial drainage completed 1981;

more step-change than trend 0.380 0.138

0.005 15004 Nore at Mcmahons

Bridge Increase

Arterial drainage completed 1981;

more step-change than trend 0.000 0.000

0.008 16003 Clodiagh at

Rathkennan Increase

Different rating before and after

1978 0.000 0.000

0.009 25014 Silver at Millbrook Decrease


1971; temporarily different rating

used 1984-1988

0.258 0.371

0.011 25023 Little Brosna at

Milltown Increase


1971 0.000 0.000

0.014 24001 Maigue at Croom Increase Maigue drainage works spanned

1973 to 1986; rating change in 1989 0.124 0.513

0.025 25029 Nenagh at

Clarianna Increase

Unexplained; drainage works pre-

dates record 0.076 0.411

*AD and AD2 denote the PCDs of arterial drainage: ARTDRAIN and ARTDRAIN2

The AM flow series for these stations were shown in Figure E.1. Station 24001 also appears

in Figure E.2i. An example of the likely influence of arterial drainage is shown in Figure E.3.

2010200019901980197019601950

30

20

10

0

AM

peak

flo

w

Pre-drainage

During drainage

Post-drainage

07004 Blackwater (Kells) at Stramatt

Figure E.3: AM flow series showing likely influence of arterial drainage


185

E6.5 Discussion

The possibility of trend in the Irish AM flow series cannot be dismissed. Besides

hydrometric reasons such as rating curve uncertainties and weir changes, there are of course

possible hydrological reasons for trend based on land-use changes such as hydroelectric

development, channel improvements and land drainage – and, of course, long-term variation

or change in climate.

It is important to examine the test results alongside graphs of the data, and with as much

historical knowledge about the data as possible (Kundzewicz and Robson, 2004). The impact

of drainage works on the AM flows at 12 stations was identified through EDA, as discussed

previously and illustrated in Figure E.2. An abrupt change in AM flows after drainage is

noted at most of these stations. In cases where drainage works are absent or pre-date the

record, further examination is recommended of series exhibiting trends in the AM flow.

Given effects from arterial drainage, changed hydrometry and other factors, it is barely

practical to explore the possible effect of climate change on the AM flows by empirical

analysis alone. Exploration of a small number of records proved inconclusive.

E7 Summary

In order to test for trend and other non-randomness in the flood series of Irish rivers,

six statistical tests were applied to AM flow series of 94 stations.

Non-parametric (i.e. distribution-free) statistical tests are considered appropriate to

detecting trend and other non-randomness.

However, a mean-weighted linear regression trend test method has been developed

based on a distribution known to be relatively suitable for AM flow series on Irish

rivers, namely: an EV1 distribution with CV = 0.3. Critical values of the test statistic

were obtained from sampling experiments and are presented in Section E8.

Tests for randomness identified non-randomness in 15 (or 16%) of the 94 AM series

examined. This is appreciably more than the five expected by chance alone. There is

therefore evidence of some non-randomness in Irish AM flow data.

Trend tests revealed some notable effects. Out of 94 records, 26 (28%) show

statistically significant trends (≈13 at the 1% level and a further 13 at the 5% level).

This is very much greater than the number expected by chance alone. Most trends

were upward rather than downward.

Tests for step-change revealed notable jumps in many AM flow series. Out of 94

records, 39 (41%) show statistically significant jumps (24 at the 1% level and a

further 15 at the 5% level). This is very much greater than the number expected by

chance alone. Most jumps were upward rather than downward.

The tests applied do not discriminate well between trend and step-change.

Arterial drainage is implicated in a considerable number of the significant step-

changes and trends reported. Hydrometric changes may account for some effects.

In cases where drainage works are absent or pre-date the record, further examination

of records is recommended for stations exhibiting trends in the AM flow.

Exploration of the possible effect of climate change on the AM flood flows proved

inconclusive.


186

E8 Critical values for mean-weighted linear-regression test

Critical values of the test statistic bs for the mean-weighted linear regression are given in

Table E.6. These are based on 10,000 simulated random samples from an EV1 distribution

with CV = 0.3. This is a typical value of CV for Irish AM flow data.

Table E.6: Critical values of the test statistic bs

Sample size

n

Lower tail Upper tail

0.005 0.01 0.025 0.05 0.05 0.025 0.01 0.005

3 -0.5247 -0.4742 -0.3993 -0.3388 0.3250 0.3949 0.4724 0.5221

4 -0.3360 -0.3018 -0.2585 -0.2104 0.2083 0.2489 0.2995 0.3364

5 -0.2323 -0.2109 -0.1794 -0.1482 0.1497 0.1793 0.2171 0.2422

6 -0.1822 -0.1662 -0.1387 -0.1182 0.1142 0.1367 0.1631 0.1815

7 -0.1434 -0.1321 -0.1106 -0.0921 0.0921 0.1094 0.1298 0.1417

8 -0.1184 -0.1053 -0.0885 -0.0744 0.0752 0.0884 0.1047 0.1186

9 -0.1004 -0.0910 -0.0756 -0.0633 0.0612 0.0738 0.0886 0.0986

10 -0.0832 -0.0765 -0.0655 -0.0540 0.0529 0.0632 0.0761 0.0872

11 -0.0737 -0.0666 -0.0558 -0.0460 0.0471 0.0565 0.0672 0.0741

12 -0.0639 -0.0584 -0.0488 -0.0410 0.0398 0.0474 0.0561 0.0623

13 -0.0578 -0.0513 -0.0427 -0.0352 0.0352 0.0433 0.0504 0.0571

14 -0.0514 -0.0454 -0.0377 -0.0319 0.0325 0.0384 0.0470 0.0516

15 -0.0456 -0.0403 -0.0344 -0.0290 0.0292 0.0342 0.0404 0.0459

16 -0.0404 -0.0365 -0.0308 -0.0255 0.0256 0.0304 0.0355 0.0388

17 -0.0378 -0.0344 -0.0281 -0.0237 0.0230 0.0274 0.0328 0.0362

18 -0.0328 -0.0308 -0.0257 -0.0216 0.0219 0.0262 0.0311 0.0348

19 -0.0310 -0.0280 -0.0236 -0.0197 0.0200 0.0237 0.0279 0.0311

20 -0.0302 -0.0269 -0.0219 -0.0182 0.0186 0.0217 0.0257 0.0287

21 -0.0270 -0.0244 -0.0203 -0.0174 0.0173 0.0204 0.0239 0.0267

22 -0.0255 -0.0235 -0.0189 -0.0158 0.0162 0.0196 0.0232 0.0256

23 -0.0235 -0.0213 -0.0180 -0.0151 0.0149 0.0176 0.0209 0.0232

24 -0.0226 -0.0203 -0.0170 -0.0144 0.0140 0.0167 0.0199 0.0220

25 -0.0214 -0.0193 -0.0158 -0.0132 0.0133 0.0160 0.0190 0.0215

26 -0.0192 -0.0171 -0.0144 -0.0122 0.0123 0.0149 0.0179 0.0201

27 -0.0184 -0.0167 -0.0142 -0.0119 0.0117 0.0138 0.0165 0.0186

28 -0.0177 -0.0157 -0.0134 -0.0112 0.0111 0.0135 0.0162 0.0175

29 -0.0163 -0.0146 -0.0127 -0.0105 0.0106 0.0126 0.0149 0.0166

30 -0.0161 -0.0146 -0.0121 -0.0101 0.0096 0.0116 0.0136 0.0154


187

Sample size

n

Lower tail Upper tail

0.005 0.01 0.025 0.05 0.05 0.025 0.01 0.005

31 -0.0147 -0.0134 -0.0113 -0.0094 0.0094 0.0112 0.0133 0.0150

32 -0.0147 -0.0135 -0.0110 -0.0092 0.0091 0.0108 0.0132 0.0146

33 -0.0137 -0.0123 -0.0105 -0.0089 0.0085 0.0103 0.0124 0.0138

34 -0.0133 -0.0122 -0.0100 -0.0084 0.0083 0.0100 0.0118 0.0130

35 -0.0124 -0.0113 -0.0095 -0.0081 0.0082 0.0098 0.0114 0.0124

36 -0.0124 -0.0108 -0.0091 -0.0077 0.0078 0.0092 0.0110 0.0125

37 -0.0124 -0.0103 -0.0086 -0.0073 0.0073 0.0087 0.0104 0.0114

38 -0.0124 -0.0101 -0.0086 -0.0071 0.0069 0.0083 0.0098 0.0110

39 -0.0124 -0.0095 -0.0081 -0.0070 0.0068 0.0080 0.0097 0.0110

40 -0.0124 -0.0090 -0.0078 -0.0065 0.0065 0.0077 0.0092 0.0102

41 -0.0124 -0.0090 -0.0075 -0.0063 0.0064 0.0076 0.0089 0.0099

42 -0.0124 -0.0086 -0.0072 -0.0060 0.0061 0.0073 0.0088 0.0097

43 -0.0124 -0.0085 -0.0072 -0.0059 0.0059 0.0070 0.0083 0.0092

44 -0.0124 -0.0078 -0.0067 -0.0056 0.0056 0.0067 0.0080 0.0092

45 -0.0124 -0.0078 -0.0067 -0.0055 0.0055 0.0066 0.0080 0.0089

46 -0.0124 -0.0075 -0.0063 -0.0054 0.0053 0.0063 0.0074 0.0081

47 -0.0124 -0.0072 -0.0062 -0.0053 0.0051 0.0061 0.0073 0.0081

48 -0.0124 -0.0070 -0.0059 -0.0049 0.0050 0.0060 0.0071 0.0077

49 -0.0124 -0.0069 -0.0057 -0.0048 0.0049 0.0058 0.0071 0.0077

50 -0.0124 -0.0065 -0.0055 -0.0046 0.0047 0.0056 0.0068 0.0075

51 -0.0124 -0.0063 -0.0054 -0.0045 0.0046 0.0054 0.0064 0.0071

52 -0.0124 -0.0061 -0.0052 -0.0044 0.0044 0.0053 0.0062 0.0070

53 -0.0124 -0.0060 -0.0051 -0.0043 0.0041 0.0049 0.0059 0.0065

54 -0.0124 -0.0059 -0.0050 -0.0042 0.0041 0.0049 0.0058 0.0064

55 -0.0124 -0.0059 -0.0049 -0.0041 0.0040 0.0049 0.0058 0.0063

56 -0.0124 -0.0057 -0.0048 -0.0041 0.0039 0.0047 0.0056 0.0062

57 -0.0124 -0.0054 -0.0046 -0.0038 0.0038 0.0046 0.0054 0.0061

58 -0.0124 -0.0053 -0.0045 -0.0037 0.0038 0.0045 0.0054 0.0059

59 -0.0124 -0.0051 -0.0043 -0.0036 0.0037 0.0044 0.0053 0.0058

60 -0.0124 -0.0051 -0.0042 -0.0035 0.0036 0.0043 0.0050 0.0055


188

Appendix F Additional summary statistics

Station number and name Grade AREA

km2

N Qmean Qmed Qmax Qmax/

Qmean

Qmax/

AREA Qmed/

Qmean

01041 Deele at Sandy Mills B 116.20 32 85.08 82.61 147.16 1.73 1.27 0.97

01055 Mourne Beg at Mourne Beg Weir B 10.80 9 2.92 2.70 4.72 1.61 0.44 0.92

06011 Fane at Moyles Mill A1 234.00 48 15.86 15.39 26.36 1.66 0.11 0.97

06013 Dee at Charleville Weir A1 307.00 30 27.81 27.37 41.84 1.50 0.14 0.98

06014 Glyde at Tallanstown A1 270.00 30 22.56 21.46 39.40 1.75 0.15 0.95

06021 Glyde at Mansfieldstown B 321.00 50 21.54 21.50 33.00 1.53 0.10 1.00

06025 Dee at Burley Bridge A1 176.00 30 18.32 18.69 23.57 1.29 0.13 1.02

06026 Glyde at Aclint Bridge A1 144.00 46 13.87 12.30 24.12 1.74 0.17 0.89

06030 Big at Ballygoly See Box 4.3 B 10.20 30 20.58 10.03 122.00 5.93 11.96 0.49

06031 Flurry at Curralhir A2 45.30 18 13.58 11.70 35.80 2.64 0.79 0.86

06033 White Dee at Coneyburrow Bridge B 57.40 25 27.88 18.60 92.30 3.31 1.61 0.67

06070 Muckno L. at Muckno A1 153.50 24 13.32 13.19 20.93 1.57 0.14 0.99

07006 Moynalty at Fyanstown A2 176.00 19 26.73 27.93 34.05 1.27 0.19 1.05

07009 Boyne at Navan Weir A1 1610.00 29 162.64 134.80 297.60 1.83 0.18 0.83

07033 Blackwater at Virginia Hatchery A2 129.00 25 14.93 14.62 26.58 1.78 0.21 0.98

08002 Delvin at Naul A1 37.00 20 5.62 5.32 8.96 1.59 0.24 0.95

08003 Broadmeadow at Fieldstown B 76.20 18 26.88 22.55 110.00 4.09 1.44 0.84

08005 Sluice at Kinsaley Hall A2 10.10 18 3.04 2.50 7.81 2.57 0.77 0.82

08007 Broadmeadow at Ashbourne B 34.00 15 9.88 8.24 18.79 1.90 0.55 0.83

08008 Broadmeadow at Broadmeadow A2 110.00 25 44.55 40.90 123.69 2.78 1.12 0.92

08009 Ward at Balheary A1 62.00 11 10.38 6.59 53.60 5.17 0.86 0.64

08011 Nanny at Duleek Road Bridge B 181.00 23 31.00 32.22 45.41 1.46 0.25 1.04

08012 Stream at Ballyboghill B 22.10 19 4.21 4.35 8.15 1.93 0.37 1.03

09001 Ryewater at Leixlip A1 215.00 48 38.71 35.46 91.50 2.36 0.43 0.92

09002 Griffeen at Lucan A1 37.00 24 7.24 5.40 23.70 3.28 0.64 0.75

09010 Dodder at Waldron’s Bridge A1 95.20 19 70.15 48.00 269.00 3.83 2.83 0.68

09035 Cammock at Killeen Road B 54.70 9 12.04 11.70 27.80 2.31 0.51 0.97

10002 Avonmore at Rathdrum B 233.00 47 88.19 83.49 266.64 3.02 1.14 0.95

10021 Shanganagh at Common’s Road A1 30.90 24 7.87 7.36 14.30 1.82 0.46 0.94

10022 Cabinteely at Carrickmines A1 10.40 18 3.84 3.85 6.89 1.79 0.66 1.00

10028 Aughrim at Knocknamohill B 204.10 16 56.69 46.95 102.00 1.80 0.50 0.83

11001 Owenavorragh at Boleany B 148.00 33 49.85 47.17 120.70 2.42 0.82 0.95

12001 Slaney at Scarawalsh A2 1036.00 50 169.50 157.00 399.00 2.35 0.39 0.93

12013 Slaney at Rathvilly B 185.00 30 45.16 43.55 72.30 1.60 0.39 0.96

14005 Barrow at Portarlington A2 398.00 48 40.81 38.27 80.42 1.97 0.20 0.94

14006 Barrow at Pass Bridge A1 1096.00 51 83.76 80.52 137.38 1.64 0.13 0.96


189


km2


Qmean

Qmax/

AREA Qmed/

Qmean

14007 Stradbally at Derrybrock A1 115.00 25 16.94 16.20 29.30 1.73 0.25 0.96

14009 Cushina at Cushina A2 68.00 25 6.69 6.79 11.19 1.67 0.16 1.02

14011 Slate at Rathangan A1 163.00 26 12.07 12.30 18.70 1.55 0.11 1.02

14013 Burrin at Ballinacarrig A2 154.00 50 16.54 16.05 26.32 1.59 0.17 0.97

14018 Barrow at Royal Oak A1 2415.00 51 141.83 147.98 216.07 1.52 0.09 1.04

14019 Barrow at Levitstown A1 1660.00 51 103.46 102.41 162.86 1.57 0.10 0.99

14029 Barrow at Graiguenamanagh A2 2762.00 47 162.54 160.74 206.21 1.27 0.07 0.99

14033 Owenass at Mountmellick B 185.00 22 22.59 19.50 33.00 1.46 0.18 0.86

14034 Barrow at Bestfield A2 2060.00 14 137.30 125.00 247.00 1.80 0.12 0.91

15001 Kings at Annamult A2 443.00 42 89.39 88.75 151.00 1.69 0.34 0.99

15002 Nore at John’s Bridge A2 1605.00 35 211.98 197.00 393.00 1.85 0.24 0.93

15003 Dinan at Dinan Bridge A2 298.00 50 143.58 150.76 187.52 1.31 0.63 1.05

15004 Nore at McMahons Bridge A2 491.00 51 38.96 37.28 74.96 1.92 0.15 0.96

15005 Erkina at Durrow Foot Bridge B 387.00 50 28.47 27.44 61.85 2.17 0.16 0.96

15012 Nore at Ballyragget B 945.00 16 77.16 77.11 133.00 1.72 0.14 1.00

16001 Drish at Athlummon A2 140.00 33 15.65 15.66 24.49 1.56 0.17 1.00

16002 Suir at Beakstown A2 512.00 51 55.40 52.66 123.88 2.24 0.24 0.95

16003 Clodiagh at Rathkennan A2 246.00 51 31.17 29.98 45.72 1.47 0.19 0.96

16004 Suir at Thurles A2 236.00 48 22.17 21.37 34.89 1.57 0.15 0.96

16005 Multeen at Aughnagross A2 87.00 30 23.11 21.79 34.31 1.48 0.39 0.94

16006 Multeen at Ballinclogh Bridge B 75.00 33 30.37 27.87 58.07 1.91 0.77 0.92

16007 Aherlow at Killardry B 273.00 51 79.18 75.84 138.03 1.74 0.51 0.96

16008 Suir at New Bridge A2 1120.00 51 90.66 92.32 110.91 1.22 0.10 1.02

16009 Suir at Cahir Park A2 1602.00 52 159.29 162.21 206.00 1.29 0.13 1.02

16011 Suir at Clonmel A1 2173.00 52 234.52 223.00 389.00 1.66 0.18 0.95

16012 Tar at Tar Bridge B 228.00 36 55.20 54.57 92.20 1.67 0.40 0.99

16013 Nire at Fourmilewater B 91.00 33 101.69 93.21 207.02 2.04 2.27 0.92

16051 Rossestown at Clobanna B 34.18 13 2.95 2.85 5.67 1.92 0.17 0.96

18001 Bride at Mogeely Bridge B 335.00 48 71.07 71.49 96.93 1.36 0.29 1.01

18002 Ballyduff at Muns Blackwater B 2338.00 49 353.65 344.00 479.00 1.35 0.20 0.97

18003 Blackwater at Killavullen B 1258.00 49 282.76 266.15 502.74 1.78 0.40 0.94

18004 Ballynamona at Awbeg A2 324.00 46 30.96 31.20 52.70 1.70 0.16 1.01

18005 Funshion at Downing Bridge A2 363.00 50 56.69 53.05 109.73 1.94 0.30 0.94

18006 Blackwater at CSET Mallow B 1058.00 27 291.30 286.00 397.00 1.36 0.38 0.98

18016 Blackwater at Duncannon B 113.00 24 80.99 79.65 114.82 1.42 1.02 0.98

18048 Blackwater at Dromcummer B 881.00 23 222.77 220.00 269.00 1.21 0.31 0.99

18050 Blackwater at Duarrigle B 244.60 24 121.96 124.50 175.00 1.43 0.72 1.02

19001 Owenboy at Ballea Upper A2 106.00 48 15.87 15.42 22.03 1.39 0.21 0.97


190


km2


Qmean

Qmax/

AREA Qmed/

Qmean

19014 Lee at Dromcarra B 168.00 20 79.69 71.89 157.21 1.97 0.69 0.90

19016 Bride at Ovens Bridge B 123.00 8 28.74 29.58 34.87 1.21 0.60 1.03

19020 Owennacurra at Ballyedmond A2 75.00 28 24.63 22.40 38.70 1.57 0.52 0.91

19031 Sullane at Macroom B 210.00 9 131.09 135.90 201.18 1.53 0.67 1.04

19046 Martin at Station Road B 60.40 9 31.09 29.95 41.95 1.35 0.70 0.96

20002 Bandon at Curranure B 431.00 31 140.60 126.28 287.11 2.04 0.44 0.90

20006 Argideen at Clonakilty WW B 79.30 25 30.25 27.70 55.60 1.84 0.25 0.92

22006 Flesk at Flesk Bridge B 325.00 51 165.89 169.09 282.83 1.70 0.43 1.02

22009 Dreenagh at White Bridge B 37.00 24 11.91 11.47 16.35 1.37 0.63 0.96

22035 Laune at Laune Bridge B 559.65 14 112.81 116.40 141.48 1.25 0.40 1.03

23001 Galey at Inch Bridge A2 196.00 45 97.39 99.05 210.07 2.16 1.07 1.02

23012 Lee at Ballymullen A2 60.00 18 16.87 15.66 31.74 1.88 0.53 0.93

24002 Camogue at Gray’s Bridge A2 231.00 27 24.06 23.49 35.21 1.46 0.15 0.98

24004 Maigue at Bruree B 246.00 52 54.86 50.63 104.55 1.91 0.29 0.92

24008 Maigue at Castleroberts A2 805.00 30 120.96 119.13 194.86 1.61 0.24 0.98

24011 Deel at Deel Bridge B 273.00 33 103.01 104.55 171.74 1.67 0.48 1.01

24012 Deel at Grange Bridge B 359.00 41 110.45 109.99 141.95 1.29 0.16 1.00

24022 Mahore at Hospital A2 39.70 20 9.83 9.80 20.50 2.09 0.52 1.00

24030 Deel at Danganbeg B 248.00 25 52.89 52.00 72.20 1.37 0.45 0.98

24082 Maigue at Islandmore A2 764.00 28 135.47 140.01 206.35 1.52 0.27 1.03

25001 Mulkear at Annacotty See §4.2.1 A2 646.00 49 133.95 132.88 178.58 1.33 0.28 0.99

25002 Newport at Barringtons Bridge §4.2.1 A2 223.00 51 61.15 62.64 74.64 1.22 0.33 1.02

25003 Mulkear at Abington See §4.2.1 A1 397.00 51 69.45 68.98 92.79 1.34 0.23 0.99

25004 Bilboa at Newbridge See §4.2.1 B 125.00 30 41.70 42.30 59.50 1.43 0.49 1.01

25005 Dead at Sunville See §4.2.1 A2 190.00 46 28.73 29.63 33.42 1.16 0.18 1.03

25006 Brosna at Ferbane A1 1207.00 52 86.77 81.91 147.21 1.70 0.12 0.94

25011 Brosna at Moystown B 1227.00 51 85.64 82.02 194.25 2.27 0.41 0.96

25014 Silver at Millbrook Bridge A1 165.00 54 17.67 17.25 27.03 1.53 0.16 0.98

25016 Clodiagh at Rahan A2 274.00 42 23.04 22.57 36.14 1.57 0.13 0.98

25017 Shannon at Banagher A1 7980.00 55 413.25 407.68 596.51 1.44 0.07 0.99

25020 Killimor at Killeen B 197.00 35 46.60 43.65 89.55 1.92 0.32 0.94

25021 Little Brosna at Croghan A2 493.00 44 28.03 28.58 35.80 1.28 0.07 1.02

25023 Little Brosna at Milltown A1 116.00 52 12.14 11.22 20.05 1.65 0.17 0.92

25025 Ballyfinboy at Ballyhooney A1 160.00 31 10.15 10.18 17.40 1.71 0.11 1.00

25027 Ollatrim at Gourdeen Bridge A1 118.00 43 23.32 22.10 40.46 1.73 0.34 0.95

25029 Nenagh at Clarianna A2 301.00 33 54.12 56.48 74.06 1.37 0.25 1.04

25030 Graney at Scarriff Bridge A1 279.00 48 43.80 40.64 87.04 1.99 0.31 0.93

25034 L. Ennell Trib at Rochfort A2 12.00 24 1.50 1.48 2.21 1.47 0.18 0.99


191


km2


Qmean

Qmax/

AREA Qmed/

Qmean

25038 Nenagh at Tyone B 139.00 17 42.08 39.30 67.90 1.61 0.21 0.93

25040 Bunow at Roscrea A2 30.00 20 3.78 3.59 6.29 1.67 0.21 0.95

25044 Kilmastulla at Coole A2 98.90 33 25.38 22.70 45.75 1.80 0.46 0.89

25124 Brosna at Ballynagore A2 254.00 18 12.79 13.65 22.50 1.76 0.09 1.07

25158 Bilboa at Cappamore See §4.2.1 A1 116.00 18 47.66 43.88 75.09 1.58 0.65 0.92

26002 Suck at Rookwood A2 626.00 53 56.98 56.56 104.87 1.84 0.17 0.99

26005 Suck at Derrycahill A2 1050.00 51 92.80 93.21 135.94 1.46 0.13 1.00

26006 Suck at Willsbrook A1 182.00 53 26.57 24.23 70.06 2.64 0.38 0.91

26007 Suck at Bellagill Bridge A1 1184.00 53 91.75 88.15 147.84 1.61 0.12 0.96

26008 Rinn at Johnston’s Bridge A1 292.00 49 23.68 22.94 41.02 1.73 0.14 0.97

26009 Black at Bellantra Bridge A2 97.00 35 13.66 13.22 18.76 1.37 0.19 0.97

26010 Cloone at Riverstown B 100.00 35 20.03 17.17 40.80 2.04 0.14 0.86

26014 Lung at Banada Bridge B 222.00 16 44.10 42.82 70.90 1.61 0.77 0.97

26018 Owenure at Bellavahan A2 118.00 49 9.19 8.95 13.98 1.52 0.12 0.97

26019 Camlin at Mullagh A1 260.00 51 22.34 21.18 37.03 1.66 0.14 0.95

26020 Camlin at Argar Bridge A1 128.00 32 11.21 11.27 15.59 1.39 0.12 1.01

26021 Inny at Ballymahon A2 1071.00 30 65.88 66.34 92.51 1.40 0.09 1.01

26022 Fallan at Kilmore A2 950.00 33 6.64 6.49 11.06 1.67 0.01 0.98

26058 Inny Upper at Ballinrink Bridge B 59.00 24 5.98 5.35 12.20 2.04 0.15 0.89

26059 Inny at Finnea Bridge A1 249.00 17 12.98 12.20 16.70 1.29 0.07 0.94

26108 Owenure at Boyle Abbey Bridge B 533.00 15 56.29 57.32 73.07 1.30 0.14 1.02

27001 Claureen at Inch Bridge A2 48.00 30 20.65 20.10 31.70 1.53 0.66 0.97

27002 Fergus at Ballycorey A1 562.00 51 34.22 32.60 59.76 1.75 0.11 0.95

27003 Fergus at Corofin A2 168.00 48 24.01 22.92 40.50 1.69 0.24 0.95

28001 Inagh at Ennistimon B 168.00 17 52.69 47.58 129.28 2.45 0.44 0.90

29001 Raford at Rathgorgin A1 119.00 40 14.17 13.46 19.66 1.39 0.17 0.95

29004 Clarinbridge at Clarinbridge A2 123.00 32 11.39 11.30 14.77 1.30 0.12 0.99

29007 L. Cullaun at Craughwell B 278.00 22 27.83 26.49 42.93 1.54 0.14 0.95

29011 Dunkellin at Kilcolgan Bridge A1 354.00 22 31.94 28.89 66.52 2.08 0.19 0.90

29071 L. Cutra at Cutra A2 123.50 26 16.00 15.70 24.30 1.52 0.20 0.98

30007 Clare at Ballygaddy A2 458.00 31 61.93 62.98 95.98 1.55 0.21 1.02

30012 Clare at Claregalway B 1075.40 9 126.89 126.00 155.00 1.22 0.02 0.99

30021 Robe at Christina’s Bridge B 138.00 26 28.17 27.20 60.70 2.15 0.12 0.97

30031 Cong at Cong Weir B 891.00 24 94.35 93.88 122.28 1.30 1.82 0.99

30037 Robe at Clooncormick B 210.00 21 1.80 1.79 3.19 1.77 1.59 0.99

30061 Corrib Estuary at Wolfe Tone Bridge A2 3111.00 33 274.97 247.97 601.59 2.19 0.19 0.90

31002 Cashla at Cashla A1 72.00 26 12.89 12.16 21.10 1.64 0.29 0.94

31072 Cong at Cong Weir B 891.00 26 49.08 43.20 103.00 2.10 0.57 0.88


192


km2


Qmean

Qmax/

AREA Qmed/

Qmean

32011 Bunowen at Louisberg Weir B 68.60 26 74.88 64.87 125.00 1.67 1.28 0.87

32012 Newport at Newport Weir A2 138.30 24 30.06 29.85 36.60 1.22 0.26 0.99

33001 Glenamoy at Glenamoy B 73.00 25 62.11 59.30 116.00 1.87 0.53 0.95

33070 Carrowmore L. at Carrowmore A1 90.00 28 7.90 7.67 11.97 1.52 0.13 0.97

34001 Moy at Rahans A2 1911.00 36 174.76 174.61 286.56 1.64 0.15 1.00

34003 Moy at Foxford A2 1737.00 29 180.42 178.00 282.00 1.56 0.16 0.99

34007 Deel at Ballycarroon B 156.00 53 90.37 84.48 198.91 2.20 0.12 0.93

34009 Owengarve at Curraghbonaun A2 113.00 33 28.37 27.48 38.58 1.36 0.34 0.97

34010 Moy at Cloonacannana B 471.00 12 123.29 113.72 193.07 1.57 0.41 0.92

34011 Manulla at Gneeve Bridge A2 144.00 30 18.80 18.73 26.05 1.39 0.18 1.00

34018 Castlebar at Turlough A1 93.00 27 11.50 11.28 17.33 1.51 0.19 0.98

34024 Pollagh at Kiltimagh A2 128.00 28 20.70 20.80 24.70 1.19 0.19 1.00

35001 Owenmore at Ballynacarrow A2 299.00 29 30.52 31.16 46.04 1.51 0.15 1.02

35002 Owenbeg at Billa Bridge A2 90.00 34 51.78 50.48 69.37 1.34 0.77 0.97

35005 Ballysadare at Ballysadare A2 642.00 55 77.78 75.42 132.71 1.71 0.21 0.97

35011 Bonet at Dromahair B 294.00 36 116.02 115.36 188.01 1.62 1.26 0.99

35071 L. Melvin at Lareen A2 247.20 30 26.95 26.29 37.91 1.41 0.15 0.98

35073 L. Gill at Lough Gill A2 384.00 30 54.81 54.05 78.40 1.43 0.20 0.99

36010 Annalee at Butlers Bridge A1 774.00 50 66.56 66.80 106.62 1.60 0.14 1.00

36011 Erne at Bellahillan B 318.00 49 17.91 18.23 23.60 1.32 0.07 1.02

36012 Erne at Sallaghan A1 263.00 47 14.22 14.12 21.63 1.52 0.08 0.99

36015 Finn at Anlore A1 175.00 33 23.14 22.08 49.99 2.16 0.29 0.95

36018 Dronmore at Ashfield Bridge A1 233.00 50 15.84 16.25 24.43 1.54 0.10 1.03

36019 Erne at Belturbet A2 1501.00 47 89.60 89.95 119.43 1.33 0.08 1.00

36021 Yellow at Kiltybarden A2 23.00 27 24.96 23.37 43.57 1.75 1.89 0.94

36031 Cavan at Lisdarn A2 52.00 30 6.85 6.45 13.70 2.00 0.26 0.94

36071 L. Scur at Gowly B 66.00 20 6.36 6.49 8.13 1.28 3.25 1.02

38001 Owenea at Clonconwal B 109.00 33 70.02 70.63 113.38 1.62 0.44 1.01

39001 New Mills at Swilly B 49.00 30 44.88 44.25 61.50 1.37 0.10 0.99

39008 Leannan at Gartan Bridge A2 78.00 33 28.34 28.18 43.88 1.55 0.56 0.99

39009 Fern O/L at Aghawoney A2 207.00 33 45.91 45.72 76.67 1.67 0.37 1.00


193

Appendix G Sample probability plots and summary information

Water

-year

AM flow

(m3 s

-1)

Date Station 14018 River Barrow at Royal Oak (2415 km2)

1954 193.75 12/12/1954 Summary statistics

1955 99.88 28/01/1956 N = 51 QMED = 147.98

1956 148.07 31/12/1956 Qmean = 141.83 L-CV = 0.138

1957 114.83 12/02/1958 CV = 0.240 L-skew = 0.036

1958 98.02 21/12/1958 H-skew = 0.218 L-kurt = 0.064

1959 111.04 30/12/1959

1960 161.76 05/12/1960 Summer peaks are tabulated in red and (for EV1 case) plotted in red

1961 83.08 17/01/1962

1962 118.69 07/02/1963

1963 95.83 26/03/1964

1964 166.52 13/12/1964

1965 177.97 19/11/1965

1966 160.19 23/02/1967

1967 147.98 25/12/1968

1968 173.01 10/01/1968

1969 121.31 23/02/1970

1970 113.56 25/11/1970

1971 124.64 03/02/1972

1972 102.09 29/12/1972

1973 183.86 02/02/1974

1974 150.23 28/01/1975

1975 92.58 03/12/1975

1976 138.49 23/02/1977

1977 145.76 05/02/1978

1978 145.76 28/12/1978

1979 162.55 28/12/1979

1980 107.94 27/10/1980

1981 106.72 05/01/1982

1982 135.65 09/11/1982

1983 139.93 26/03/1984

1984 151.73 15/12/1984

1985 102.84 28/08/1986

1986 148.73 11/12/1986

1987 145.76 04/02/1988

1988 90.44 26/10/1988

1989 216.07 08/02/1990

1990 150.23 29/12/1990

1991 97.66 07/01/1992

1992 199.57 15/06/1993

1993 151.73 05/02/1994

1994 212.34 28/01/1995

1995 150.23 09/01/1996

1996 99.86 20/02/1997

1997 178.80 05/01/1998

1998 183.86 31/12/1998

1999 165.72 27/12/1999

2000 192.49 07/11/2000

2001 151.73 26/02/2002

2002 164.13 16/11/2002

2003 109.18 17/01/2004

2004 148.73 29/10/2004

EV1 plot

2 5 10 25 50 100 500

0

50

100

150

200

250

-2 0 2 4 6 8

AM

flo

w (

m3s-1

)


winter peak

summer peak

Return period (years)

2 5 10 25 50 100 500

0

50

100

150

200

250

-6 -4 -2 0 2 4 6

Logistic reduced variate, yL

LO plot


AM

flo

w (

m3s-1

)

2 5 10 25 50 100 500

1.4

1.6

1.8

2

2.2

2.4

-2.5 -1.5 -0.5 0.5 1.5 2.5

log

10(A

M f

low

)

Standardised Normal variate, yN

LN plot



194

Appendix H Probability-plot linear scores and curve patterns

Station number and name Grade #

years

Linear score Curve pattern

EV1 LO LN EV1 LO LN

06011 Fane at Moyles Mill A1 48 4 2 3 L2 U1 L2

06013 Dee at Charleville Weir A1 30 2 3 4 D1 S1 L2

06014 Glyde at Tallanstown A1 30 4 2 3 L1 U1 L2

06025 Dee at Burley Bridge A1 30 2 4 3 D1 L2 S1

06026 Glyde at Aclint Bridge A1 46 3 1 2 S1 S2 S2

06031 Flurry at Curralhir A2 18 2 2 3 U2 U2 U2

06070 Muckno L. at Muckno A1 24 4 4 5 L2 L2 L1

07006 Moynalty at Fyanstown A2 19 2 3 2 D2 D1 D2

07009 Boyne at Navan Weir A1 29 4 1 3 L2 D2X L2

07033 Blackwater at Virginia Hatchery A2 25 4 3 3 S1 U1 S1

08002 Delvin at Naul A1 20 4 1 2 L2 U1 U1

08005 Sluice at Kinsaley Hall A2 18 4 3 5 L2 U1 L1

08008 Broadmeadow at Broadmeadow A2 25 4 2 4 L2 U1 L2

08009 Ward at Balheary A1 11 1 1 2 U2 U2 U1

09001 Ryewater at Leixlip A1 48 5 2 5 L1 U1 L1

09002 Griffeen at Lucan A1 24 1 1 4 S2X U2 L1

09010 Dodder at Waldron’s Bridge A1 19 2 2 4 U2 U2 L2

10021 Shanganagh at Common’s Rd A1 24 4 2 3 L2 S1 L2

10022 Cabinteely at Carrickmines A1 18 4 3 4 L2 L2X L2

12001 Slaney at Scarawalsh A2 50 4 2 3 L2 U1 L2

14005 Barrow at Portarlington A2 48 3 2 2 L2 U1 U1

14006 Barrow at Pass Bridge A1 51 5 2 3 L1 U1 U1

14007 Stradbally at Derrybrock A1 25 4 2 3 L2 U1 S1

14009 Cushina at Cushina A2 25 4 3 4 S1 U1 S1

14011 Slate at Rathangan A1 26 3 3 3 L2 L2 L2

14013 Burrin at Ballinacarrig A2 50 4 3 4 S1 S2 S1

14018 Barrow at Royal Oak A1 51 3 3 4 L2 S1 S1

14019 Barrow at Levitstown A1 51 3 3 4 D1 S2 S1

14029 Barrow at Graiguenamanagh A2 47 3 4 5 D1 S1 L1

14034 Barrow at Bestfield A2 14 4 2 2 L2X U2X U2X

15001 Kings at Annamult A2 42 2 2 3 D2 S2 S1

15002 Nore at John’s Bridge A2 35 4 3 4 L2 U1 L2

15003 Dinan at Dinan Bridge A2 50 2 3 2 D2 S1 D2

15004 Nore at McMahons Bridge A2 51 5 3 5 L2 U1 L2

16001 Drish at Athlummon A2 33 5 4 5 L1 U1 L1

16002 Suir at Beakstown A2 51 3 2 4 L2X U1X L2X


195


years


EV1 LO LN EV1 LO LN

16003 Clodiagh at Rathkennan A2 51 4 3 3 L2 S1 S1

16004 Suir at Thurles A2 48 5 4 5 L1 U1 L1

16005 Multeen at Aughnagross A2 30 5 2 2 L2 U1 U1

16008 Suir at New Bridge A2 51 2 4 3 D2 L2 D1

16009 Suir at Cahir Park A2 52 2 3 3 D2X D2X D1X

16011 Suir at Clonmel A1 52 3 2 3 D1 S1 S1

18004 Ballynamona at Awbeg A2 46 2 2 2 S2 S2 S2

18005 Funshion at Downing Bridge A2 50 4 2 3 L2X U2 L2X

19001 Owenboy at Ballea Upper A2 48 4 4 4 L2 L2 S1X

19020 Owennacurra at Ballyedmond A2 28 3 4 3 D2 S1 D1

23001 Galey at Inch Bridge A2 45 4 2 3 L2 U1 L2

23012 Lee at Ballymullen A2 18 2 1 1 U2 U2 U2

24002 Camogue at Gray’s Bridge A2 27 4 5 4 L2 L2 L2X

24008 Maigue at Castleroberts A2 30 3 5 4 L2 L2 L2X

24022 Mahore at Hospital A2 20 5 3 5 L1 U1 L2

24082 Maigue at Islandmore A2 28 2 5 3 D1 L2 D1

25006 Brosna at Ferbane A1 52 4 3 4 L2 L2 S1

25014 Silver at Millbrook Bridge A1 54 4 3 4 L2 U1X L2

25016 Clodiagh at Rahan A2 42 4 4 5 L2 L2 L2

25017 Shannon at Banagher A1 55 3 4 5 D1 L2 L1

25021 Little Brosna at Croghan A2 44 3 4 5 D1 L2 L2

25023 Little Brosna at Milltown A1 52 2 2 2 S2 S2 S1X

25025 Ballyfinboy at Ballyhooney A1 31 5 4 5 L1 L2 L2

25027 Ollatrim at Gourdeen Bridge A1 43 4 5 3 D1 L2 D1

25029 Nenagh at Clarianna A2 33 3 3 4 S2 S2 S1

25030 Graney at Scarriff Bridge A1 48 4 2 4 S1X U1 L2

25034 L. Ennell Trib at Rochfort A2 24 2 4 3 D2 L2 D1

25040 Bunow at Roscrea A2 20 4 3 4 L2 U1 L2

25044 Kilmastulla at Coole A2 33 3 3 3 S1 S1 S2

25124 Brosna at Ballynagore A2 18 3 4 2 D1 L2 D1

26002 Suck at Rookwood A2 53 2 1 2 S2X S2X L2X

26005 Suck at Derrycahill A2 51 4 4 5 L1 L2 L1

26006 Suck at Willsbrook A1 53 1 1 1 S2X S2X S2X

26007 Suck at Bellagill Bridge A1 53 5 3 5 L1 U1 L1

26008 Rinn at Johnston’s Bridge A1 49 4 3 3 L2 U1 L2

26009 Black at Bellantra Bridge A2 35 4 3 2 S1 S1 S2

26018 Owenure at Bellavahan A2 49 4 3 4 L2 S1 L2

26019 Camlin at Mullagh A1 51 4 3 3 L2 S1 L2


196


years


EV1 LO LN EV1 LO LN

26020 Camlin at Argar Bridge A1 32 3 3 2 L2 S1 L2

26021 Inny at Ballymahon A2 30 1 2 1 D2 D1 D1

26022 Fallan at Kilmore A2 33 4 3 4 L2 S1 S1

26059 Inny at Finnea Bridge A1 17 2 2 1 D1 D1 D2X

27001 Claureen at Inch Bridge A2 30 4 3 3 L2 U1 L2

27002 Fergus at Ballycorey A1 51 4 2 4 L2 U1 L2

27003 Fergus at Corofin A2 48 4 4 4 L2 L2 L2

29001 Raford at Rathgorgin A1 40 3 4 4 D1 S1 S1

29004 Clarinbridge at Clarinbridge A2 32 4 3 3 L2 L2 S1

29011 Dunkellin at Kilcolgan Bridge A1 22 2 2 2 U1 S1 S1

29071 L. Cutra at Cutra A2 26 4 3 4 L2 S1 L2

30007 Clare at Ballygaddy A2 31 4 3 4 L2X L2X L2X

30061 Corrib Estuary at Wolfe Tone Bridge A2 33 1 1 2 U2 U2 U1

31002 Cashla at Cashla A1 26 2 1 1 U2X U2X U2X

32012 Newport at Newport Weir A2 24 3 5 3 D1 L2 L2

33070 Carrowmore L. at Carrowmore A1 28 4 3 4 L2X L2X L2X

34001 Moy at Rahans A2 36 4 3 4 L2X L2X L2

34003 Moy at Foxford A2 29 3 3 3 S1 S1 S2

34009 Owengarve at Curraghbonaun A2 33 4 4 5 L2 L2 L1

34011 Manulla at Gneeve Bridge A2 30 4 4 4 S1 L2 L2

34018 Castlebar at Turlough A1 27 4 3 3 L2 U1 U1

34024 Pollagh at Kiltimagh A2 28 3 4 3 D1 L2 L2

35001 Owenmore at Ballynacarrow A2 29 4 5 4 L2 L2 L2

35002 Owenbeg at Billa Bridge A2 34 3 5 4 D1 L2 L2

35005 Ballysadare at Ballysadare A2 55 4 2 4 S1 S1 L2

35071 L. Melvin at Lareen A2 30 4 3 4 L2 U1 L2

35073 L. Gill at Lough Gill A2 30 3 4 4 D1 L2 D1

36010 Annalee at Butlers Bridge A1 50 4 3 4 S1X S1X S1X

36012 Erne at Sallaghan A1 47 3 4 3 D1 L2 L2

36015 Finn at Anlore A1 33 2 2 2 U2 U2 U1

36018 Dronmore at Ashfield Bridge A1 50 4 4 4 L2 L2 L2

36019 Erne at Belturbet A2 47 2 3 3 D1 S1 S1

36021 Yellow at Kiltybarden A2 27 3 3 3 L2 U2 S1

36031 Cavan at Lisdarn A2 30 2 1 1 S2 S2 S2

39008 Leannan at Gartan Bridge A2 33 3 2 3 S1 S1 S1

39009 Fern O/L at Aghawoney A2 33 5 3 5 L1 U1 L1


197

Appendix I [This page and appendix are intentionally blank]


198

Appendix J Flood volumes in relation to convex EV1 plots

J1 Flood peaks and volumes for Station 07006 Moynalty at Fyanstown

Table J.1: Basic information for Station 07006 Moynalty at Fyanstown

Nominal area Period of AM

flow data # years

Median Mean CV Hazen skewness

km2 m

3s

-1 m

3s

-1

2 176 1986 – 2004 19 27.93 26.73 0.206 -1.09

EV1 Probability Plot J.1: Station 07006 Moynalty at Fyanstown

Figure J.1: Hydrograph volumes for Station 07006 Rank 1, 2 and 4 AM flood peaks

Notes

Hydrographs available for three of four largest AM flood peaks (figures opposite );

1-day flood volumes are noticeably similar for these three events; at longer duration,

the Rank 2 AM event shows higher volume than the Rank 4 event, with the Rank 1

event giving the least volume among the three; the volume of the Rank 1 event is

only 50% of that of the Rank 2 event across 7, 14 and 30-day durations;

The Rank 2 and Rank 4 events display multiple flood peaks, explaining why their

hydrographs have greater volume than the unimodal Rank 1 event.

EV1

'93'95'04'00

2 5 10 25 50 100 500

0

5

10

15

20

25

30

35

40

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

)

w inter peak

summer peak


0

5

10

15

20

25

1 2 7 14 30Days

Mil

l. c

u.

mete

r

2000

1995

1993

Flo

od v

olu

me

in 1

06 m

3






199


0

5

10

15

20

25

30

35

40

02/10/1993 00:00 04/10/1993 00:00 06/10/1993 00:00 08/10/1993 00:00 10/10/1993 00:00 12/10/1993 00:00

Time

Dis

ch

arg

e (

m3/s

)Red zone= Vol. of 1day




0

5

10

15

20

25

30

35

22/11/1995 00:00 24/11/1995 00:00 26/11/1995 00:00 28/11/1995 00:00 30/11/1995 00:00 02/12/1995 00:00 04/12/1995 00:00 06/12/1995 00:00

Time

Dis

ch

arg

e (

m3/s

)





0

5

10

15

20

25

30

35

31/10/2000 00:00 02/11/2000 00:00 04/11/2000 00:00 06/11/2000 00:00 08/11/2000 00:00 10/11/2000 00:00

Time

Dis

ch

arg

e (

m3/s

)




Rank 1 AM event,

6 October 1993

Rank 2 AM event,

29 November 1995

Rank 4 AM event,

6 November 2000


200

J2 Flood peaks and volumes for Station 15003 Dinan at Dinan Bridge

Table J.2: Basic information for Station 15003 Dinan at Dinan Bridge


flow data # years


km2 m

3s

-1 m

3s

-1

2 298

1954 – 2004

(2001 missing) 50 150.76 143.57 0.196 -0.78

EV1 Probability Plot J.2: Station 15003 Dinan at Dinan Bridge

Figure J.2: Hydrograph volumes for Station 15003 Rank 1, 2 and 4 AM flood peaks

Notes

Hydrographs available for two largest AM flood peaks (figures opposite );

With only two data samples, it is speculative to infer how hydrograph volumes differ

with flood magnitude. The Rank 1 AM event corresponds to the August 1986 storm,

which was noted for its intensity and short duration.

Flo

od v

olu

me

in 1

06 m

3


Rank 1 AM flood, Aug 1986


'98 '68 '97 '85


201


0

50

100

150

200

22/08/1986

00:00

23/08/1986

00:00

24/08/1986

00:00

25/08/1986

00:00

26/08/1986

00:00

27/08/1986

00:00

28/08/1986

00:00

29/08/1986

00:00

30/08/1986

00:00

Time

Dis

char

ge

(m3/

s)



Blue zone= Vol. of 1w eek


0

50

100

150

200

14/11/1997

00:00

15/11/1997

00:00

16/11/1997

00:00

17/11/1997

00:00

18/11/1997

00:00

19/11/1997

00:00

20/11/1997

00:00

21/11/1997

00:00

22/11/1997

00:00

Time

Dis

ch

arg

e (

m3/s

)




Rank 1 AM event,

26 August 1986

Rank 2 AM event,

18 November 1997


202

J3 Flood peaks and volumes for Station 16008 Suir at New Bridge

Table J.3: Basic information for Station 16008 Suir at New Bridge


flow data # years


km2 m

3s

-1 m

3s

-1

2 1120 1954 – 2004 51 92.32 90.66 0.125 -0.32

EV1 Probability Plot J.3: Station 16008 Suir at New Bridge

Figure J.3: Hydrograph volumes for four largest Station 16008 AM flood peaks

Notes

Hydrographs available for four largest AM flood peaks (figures opposite );

The flood hydrographs display similar shape and translate to generally similar flood

volumes;

There is very little difference among flood volumes for these four events, particularly

at the 1, 2 and 7-day durations.

See Section J4 below and analysis of Suir hydrographs in Section 5.8 of Volume III.

'87'83 '68 '60

EV1 Plot

2 5 10 25 50 100 500

0

20

40

60

80

100

120

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

)

w inter peak

summer peak

Volume of hydrographs of different years during max peak

0

50

100

150

200

250

1 2 7 14 30Days

Mil

l. c

u.

mete

r

1987

1983

1968

1960

Flo

od v

olu

me

in 1

06 m

3







203


0

20

40

60

80

100

120

26/11/1960

00:00

28/11/1960

00:00

30/11/1960

00:00

02/12/1960

00:00

04/12/1960

00:00

06/12/1960

00:00

08/12/1960

00:00

10/12/1960

00:00

12/12/1960

00:00

14/12/1960

00:00

Time

Disc

harg

e (m

3/s)





0

20

40

60

80

100

120

140

16/12/1968

00:00

18/12/1968

00:00

20/12/1968

00:00

22/12/1968

00:00

24/12/1968

00:00

26/12/1968

00:00

28/12/1968

00:00

30/12/1968

00:00

01/01/1969

00:00

Time

Dis

ch

arg

e (

m3/s

)





0

20

40

60

80

100

120

29/01/1984

00:00

31/01/1984

00:00

02/02/1984

00:00

04/02/1984

00:00

06/02/1984

00:00

08/02/1984

00:00

10/02/1984

00:00

12/02/1984

00:00

14/02/1984

00:00

16/02/1984

00:00

Time

Dis

char

ge (m

3/s)





0

20

40

60

80

100

120

26/01/1988

00:00

28/01/1988

00:00

30/01/1988

00:00

01/02/1988

00:00

03/02/1988

00:00

05/02/1988

00:00

07/02/1988

00:00

09/02/1988

00:00

11/02/1988

00:00

Time

Dis

char

ge (m

3/s)




Rank 1 AM event,

4 December 1960

Rank 2 AM event,

24 December 1968

Rank 3 AM event,

6 February 1984

Rank 4 AM event,

4 February 1988


204

J4 Flood peaks and volumes for Station 16009 Suir at Cahir Park

Table J.4: Basic information for Station 16009 Suir at Cahir Park


flow data # years


km2 m

3s

-1 m

3s

-1

2 1602 1953 – 2004 52 162.21 159.29 0.172 -0.41

EV1 Probability Plot J.4: Station 16009 Suir at Cahir Park

Figure J.4: Hydrograph volumes for four of five largest Station 16009 AM flood peaks

Notes

Hydrographs available for four of five largest AM flood peaks (figures opposite );

The hydrograph shapes are relatively similar, especially in their rising limbs;

It is possible to compare the hydrograph shapes with those typically seen at Station

16008 upstream (see Section J3); three flood events (Dec 1960, Dec 1968 and Feb

1990) are among the five largest floods at both stations;

The flood volumes are relatively similar in magnitude, particularly at the 1, 2 and 7-

day durations; at 30-day duration, the Rank 1 flood peak gives the largest flood

volume; the Rank 5 flood in Oct 2004 followed a notably dry autumn.

See Section J3 above and analysis of Suir hydrographs in Section 5.8 of Volume III.

'04'68 '00 '60 '89

EV1 Plot

2 5 10 25 50 100 500

0

50

100

150

200

250

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

)


0

50

100

150

200

250

300

350

1 2 7 14 30Days

Mill cu

. m

ete

r

2004

2000

1960

1989

Flo

od v

olu

me

in 1

06 m

3






'87'83 '68 '60

EV1 Plot

2 5 10 25 50 100 500

0

20

40

60

80

100

120

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

)

w inter peak

summer peak


205


0

50

100

150

200

250

30/01/1990

00:00

01/02/1990

00:00

03/02/1990

00:00

05/02/1990

00:00

07/02/1990

00:00

09/02/1990

00:00

11/02/1990

00:00

13/02/1990

00:00

15/02/1990

00:00

Time

Dis

char

ge

(m3/

s)





0

50

100

150

200

250

25/11/1960

00:00

27/11/1960

00:00

29/11/1960

00:00

01/12/1960

00:00

03/12/1960

00:00

05/12/1960

00:00

07/12/1960

00:00

09/12/1960

00:00

11/12/1960

00:00

13/12/1960

00:00

15/12/1960

00:00

Time

Disc

harg

e (m

3/s)





0

50

100

150

200

250

29/10/2000

00:00

31/10/2000

00:00

02/11/2000

00:00

04/11/2000

00:00

06/11/2000

00:00

08/11/2000

00:00

10/11/2000

00:00

12/11/2000

00:00

Time

Dis

char

ge

(m3/

s)





0

50

100

150

200

250

20/10/2004

00:00

22/10/2004

00:00

24/10/2004

00:00

26/10/2004

00:00

28/10/2004

00:00

30/10/2004

00:00

01/11/2004

00:00

03/11/2004

00:00

05/11/2004

00:00

Time

Dis

char

ge

(m3/

s)




Rank 1 AM event,

7 February 1990

Rank 2 AM event,

4 December 1960

Rank 3 AM event,

6 November 2000

Rank 5 AM event,

29 October 2004


206

J5 Flood peaks and volumes for Station 24013 Deel at Rathkeale

Table J.5: Basic information for Station 24013 Deel at Rathkeale (post-drainage)


flow data # years


km2 m

3s

-1 m

3s

-1

2 426 1969 – 2004 36 109.60 108.31 0.152 0.05

EV1 Probability Plot J.5: Station 24013 Deel at Rathkeale (post-drainage)


Notes


Although the crest segments of the hydrographs have a characteristic shape, their

overall structure differs according to the complexity (i.e. multimodality) of the event;

The Rank 1 and Rank 2 AM floods have greater volume than the Rank 3 and Rank 4

AM floods;

At 1 and 2-day duration, the difference is minor; however, at 7-day duration, the

difference in volume is considerable [reflecting the multimodality of the hydrographs

in the Rank 1 and Rank 2 AM floods].

Volume of hydrograph of different years during max peak

0

10

20

30

40

50

60

70

80

1 2 7 14 30Days

Milli

on c

u.m

1988

1980

1998

1973

Flo

od v

olu

me

in 1

06 m

3






'73 '98 '80

'88 '94


207


0

20

40

60

80

100

120

140

160

26/11/1973

00:00

27/11/1973

00:00

28/11/1973

00:00

29/11/1973

00:00

30/11/1973

00:00

01/12/1973

00:00

02/12/1973

00:00

03/12/1973

00:00

04/12/1973

00:00

05/12/1973

00:00

06/12/1973

00:00

Time

Dis

ch

arg

e (

m3/s





0

20

40

60

80

100

120

140

160

25/12/1998 00:00 27/12/1998 00:00 29/12/1998 00:00 31/12/1998 00:00 02/01/1999 00:00 04/01/1999 00:00

Time

Dis

ch

arg

e (

m3/s

)





0

20

40

60

80

100

120

140

160

28/10/1980 00:00 30/10/1980 00:00 01/11/1980 00:00 03/11/1980 00:00 05/11/1980 00:00 07/11/1980 00:00 09/11/1980 00:00

Time

Dis

char

ge

(m3/

s)





0

20

40

60

80

100

120

140

15/10/1988 00:00 17/10/1988 00:00 19/10/1988 00:00 21/10/1988 00:00 23/10/1988 00:00 25/10/1988 00:00 27/10/1988 00:00 29/10/1988 00:00

Time

Dis

char

ge (m

3/s)




Rank 1 AM event,

1 December 1973

Rank 2 AM event,

30 December 1998

Rank 3 AM event,

2 November 1980

Rank 4 AM event,

22 October 1988


208

J6 Flood peaks and volumes for Station 24082 Maigue at Islandmore

Table J.6: Basic information for Station 24082 Maigue at Islandmore


flow data # years


km2 m

3s

-1 m

3s

-1

2 764 1977 – 2004 28 140.01 135.47 0.264 -0.22

EV1 Probability Plot J.6: Station 24082 Maigue at Islandmore


Notes


Not much difference is shown among the volumes of hydrographs of these four

events for the 1 and 2-day volume;

The Rank 1 AM flood event yields the largest volume at all durations, but especially

at 7, 14 and 30-day duration.

'98 '88 '00

'89

EV1 Plot

2 5 10 25 50 100 500

0

50

100

150

200

250

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

)


0

20

40

60

80

100

120

140

160

1 2 7 14 30Days

Mill

cu

. met

er

1998

1988

2000

1989

Flo

od v

olu

me

in 1

06 m

3







209


0

50

100

150

200

250

01/02/1990 00:00 03/02/1990 00:00 05/02/1990 00:00 07/02/1990 00:00 09/02/1990 00:00 11/02/1990 00:00

Time

Dis

charg

e (

m3/s

)





0

20

40

60

80

100

120

140

160

180

200

31/10/2000 00:00 02/11/2000 00:00 04/11/2000 00:00 06/11/2000 00:00 08/11/2000 00:00 10/11/2000 00:00 12/11/2000 00:00

Time

Dis

char

ge (m

3/s)





0

20

40

60

80

100

120

140

160

180

200

14/10/1988

00:00

16/10/1988

00:00

18/10/1988

00:00

20/10/1988

00:00

22/10/1988

00:00

24/10/1988

00:00

26/10/1988

00:00

28/10/1988

00:00

Time

Disc

harg

e (m

3/s)





0

20

40

60

80

100

120

140

160

180

200

23/12/1998 00:00 25/12/1998 00:00 27/12/1998 00:00 29/12/1998 00:00 31/12/1998 00:00 02/01/1999 00:00 04/01/1999 00:00

Time

Disc

harg

e (m

3/s)




Rank 1 AM event,

6 February 1990

Rank 2 AM event,

6 November 2000

Rank 3 AM event,

21 October 1988

Rank 4 AM event,

30 December 1998


210

J7 Flood peaks and volumes for Station 25017 Shannon at Banagher

Table J.7: Basic information for Station 25017 Shannon at Banagher


flow data # years


km2 m

3s

-1 m

3s

-1

2 7989 1950 – 2004 55 407.68 413.25 0.203 0.18

EV1 Probability Plot J.7: Station 25017 Shannon at Banagher

Figure J.7: Hydrograph volumes for four of five largest Station 25017 AM flood peaks

Notes

Hydrograph unavailable for Rank 1 AM flood peak;

Hydrographs available for next four largest AM flood peaks (figures opposite );

note that Rank 5 AM flood event on 2 Feb 1995 is equalled by one on 28 Dec 1959;

Hydrographs are severely attenuated for this large catchment; there is little difference

in the volumes of the four events across any of the durations.

'94 '01 '89 '99'54

EV1 Plot

2 5 10 25 50 100 500

0

100

200

300

400

500

600

700

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

)

Volume of hydrograph of different years during max peak

0

200

400

600

800

1000

1200

1400

1 2 7 14 30Days

Mill. c

u. m

ete

r

1994

2001

1989

1999

Flo

od v

olu

me

in 1

06 m

3



Rank 5= AM flood, Feb 1995




211


0

100

200

300

400

500

600

16/12/1999 00:00 21/12/1999 00:00 26/12/1999 00:00 31/12/1999 00:00 05/01/2000 00:00 10/01/2000 00:00

Time

Dis

ch

arg

e (

m3/s





0

100

200

300

400

500

600

22/01/1990 00:00 27/01/1990 00:00 01/02/1990 00:00 06/02/1990 00:00 11/02/1990 00:00 16/02/1990 00:00 21/02/1990 00:00 26/02/1990 00:00

Time

Dis

ch

arg

e (

m3/s

)





0

100

200

300

400

500

600

03/02/2002

00:00

05/02/2002

00:00

07/02/2002

00:00

09/02/2002

00:00

11/02/2002

00:00

13/02/2002

00:00

15/02/2002

00:00

17/02/2002

00:00

19/02/2002

00:00

21/02/2002

00:00

23/02/2002

00:00

Time

Dis

ch

arg

e (

m3/s

)





0

100

200

300

400

500

600

21/01/1995 00:00 26/01/1995 00:00 31/01/1995 00:00 05/02/1995 00:00 10/02/1995 00:00 15/02/1995 00:00

Time

Dis

ch

arg

e (

m3/s

)



Blue zone= Vol. of 1week Rank 5= AM event,

2 February 1995

Rank 3 AM event,

9 February 1990

Rank 2 AM event,

27 December 1999

Rank 4 AM event,

13 February 2002


212

J8 Flood peaks and volumes for Station 25021 Little Brosna at Croghan

Table J.8: Basic information for Station 25021 Little Brosna at Croghan


flow data # years


km2 m

3s

-1 m

3s

-1

2 493 1961 – 2004 44 28.58 28.03 0.141 -0.13

EV1 Probability Plot J.8: Station 25021 Little Brosna at Croghan


Notes


The hydrographs are rather intricate for this station, with shorter-term fluctuations

superposed on the main body of the hydrograph;

Although the Rank 1 AM flood has the largest peak discharge, it gives a considerably

smaller flood volume than the other high-ranking AM floods, most notably at 7, 14

and 30-day durations. [Editorial note: It appears that a rating curve adopted in the

FSU has been wrongly applied to the 1961 and 1962 AM water levels: ignoring a

rating change on 14 June 1963. It is likely that the FSU flood peak of 35.8 m3s

-1 on

5 Nov 1962 is greatly exaggerated. This would account for the anomalous shape of

the crest segment of the hydrograph shown opposite for the Rank 1 AM event.]

'99'00 '62

EV1 Plot

2 5 10 25 50 100 500

'94

0

5

10

15

20

25

30

35

40

45

-2 -1 0 1 2 3 4 5 6 7EV1 y

AM

F(m

3/s

)

w inter peak

summer peak


0

10

20

30

40

50

60

70

1 2 7 14 30Days

Mill cu

. m

ete

r

1994

1999

2000

1962

Flo

od v

olu

me

in 1

06 m

3







213


0

5

10

15

20

25

30

35

40

31/10/1962 00:00 02/11/1962 00:00 04/11/1962 00:00 06/11/1962 00:00 08/11/1962 00:00 10/11/1962 00:00 12/11/1962 00:00

Time

Dis

char

ge (

m3/

s)Red zone= Vol. of 1day




0

5

10

15

20

25

30

35

40

29/10/2000

00:00

31/10/2000

00:00

02/11/2000

00:00

04/11/2000

00:00

06/11/2000

00:00

08/11/2000

00:00

10/11/2000

00:00

12/11/2000

00:00

14/11/2000

00:00Time

Dis

char

ge (m

3/s)





0

5

10

15

20

25

30

35

40

16/12/1999

00:00

18/12/1999

00:00

20/12/1999

00:00

22/12/1999

00:00

24/12/1999

00:00

26/12/1999

00:00

28/12/1999

00:00

30/12/1999

00:00

01/01/2000

00:00

03/01/2000

00:00

Time

Disc

harg

e (m

3/s)





0

5

10

15

20

25

30

35

40

21/01/1995

00:00

23/01/1995

00:00

25/01/1995

00:00

27/01/1995

00:00

29/01/1995

00:00

31/01/1995

00:00

02/02/1995

00:00

04/02/1995

00:00

Time

Dis

char

ge (m

3/s)




Rank 1 AM event,

5 November 1962

Rank 2 AM event,

6 November 2000

Rank 3 AM event,

25 December 1999

Rank 4 AM event,

28 January 1995

[This may not be Rank 1 AM flood.

See notes on previous page.]


214

Appendix K Seasonal distribution of annual maximum floods

Table K.1: Percentage of AM floods occurring in winter half-year (Oct-Mar)

Station

number

%

AMs

Oct–

Mar

Station

number

%

AMs

Oct–

Mar

Station

number

%

AMs

Oct–

Mar

Station

number

%

AMs

Oct–

Mar

Station

number

%

AMs

Oct–

Mar

Station

number

%

AMs

Oct–

Mar

01041 91 09002 68 16009 90 24012 88 26010 91 32012 88

01055 89 09010 63 16011 90 24013 87 26012 98 33001 68

03051 100 09035 44 16012 92 24022 100 26014 100 33070 89

06011 100 10002 83 16013 88 24030 88 26017 94 34001 97

06013 90 10021 63 16051 85 24082 82 26018 96 34003 97

06014 97 10022 60 18001 90 25001 73 26019 88 34004 92

06021 94 10028 69 18002 94 25002 76 26020 88 34007 85

06025 80 11001 91 18003 96 25003 78 26021 90 34009 91

06026 91 12001 90 18004 85 25004 87 26022 88 34010 58

06030 63 12013 73 18005 84 25005 87 26058 83 34011 97

06031 94 14005 90 18006 93 25006 92 26059 96 34018 100

06033 80 14006 88 18016 83 25011 78 26108 100 34024 90

06070 85 14007 76 18048 83 25014 81 27001 73 34029 86

07002 96 14009 80 18050 92 25016 93 27002 96 35001 90

07003 83 14011 81 19001 92 25017 98 27003 88 35005 93

07004 98 14013 82 19014 90 25020 91 27070 79 35011 89

07005 91 14018 96 19016 100 25021 91 28001 82 35071 90

07006 84 14019 92 19020 89 25023 71 29001 85 35073 87

07007 91 14029 98 19031 89 25025 90 29004 81 36010 96

07009 90 14033 82 19046 67 25027 88 29007 82 36011 96

07010 93 14034 82 20001 93 25029 85 29011 86 36012 98

07011 98 15001 88 20002 94 25030 92 29071 83 36015 94

07012 94 15003 84 20006 84 25034 83 30001 94 36018 92

07033 84 15004 92 22006 86 25038 82 30004 97 36019 96

07041 100 15005 90 22009 79 25040 75 30005 86 36021 73

08002 76 15012 75 22035 100 25044 83 30007 94 36027 93

08003 78 16001 88 23001 87 25124 94 30012 100 36031 90

08005 67 16002 90 23002 80 25158 72 30021 92 36071 75

08007 73 16003 82 23012 83 26002 91 30031 100 38001 73

08008 84 16004 90 24001 88 26005 94 30037 57 39001 83

08009 67 16005 87 24002 81 26006 94 30061 91 39008 82

08011 87 16006 82 24004 87 26007 94 31002 81 39009 88

08012 68 16007 84 24008 87 26008 92 31072 69 202 FSU

catchments 09001 77 16008 90 24011 85 26009 89 32011 65


215

Table K.2: Month of maximum recorded flood in AM series

Station

number

Month

of

series

max

Station

number

Month

of

series

max

Station

number

Month

of

series

max

Station

number

Month

of

series

max

Station

number

Month

of

series

max

Station

number

Month

of

series

max

01041 Dec 09002 Dec 16009 Dec 24012 Dec 26010 Nov 32012 Dec

01055 Dec 09010 Dec 16011 Nov 24013 Dec 26012 Feb 33001 Sep

03051 Dec 09035 Oct 16012 Nov 24022 Dec 26014 Nov 33070 Oct

06011 Dec 10002 Nov 16013 Nov 24030 Aug 26017 Dec 34001 Oct

06013 Dec 10021 May 16051 Jan 24082 Dec 26018 Dec 34003 Oct

06014 Dec 10022 May 18001 Oct 25001 Dec 26019 Oct 34004 Oct

06021 Dec 10028 Oct 18002 Oct 25002 Oct 26020 Dec 34007 Oct

06025 Nov 11001 Aug 18003 Dec 25003 Dec 26021 Dec 34009 Nov

06026 Dec 12001 Nov 18004 Jan 25004 Dec 26022 Dec 34010 Jun

06030 Nov 12013 Nov 18005 Dec 25005 Nov 26058 Jan 34011 Oct

06031 Dec 14005 Dec 18006 Nov 25006 Dec 26059 Jan 34018 Dec

06033 Dec 14006 Dec 18016 Jan 25011 Dec 26108 Jan 34024 Nov

06070 Nov 14007 Feb 18048 Dec 25014 Dec 27001 Jan 34029 Jan

07002 Dec 14009 Feb 18050 Dec 25016 Dec 27002 Dec 35001 Oct

07003 Nov 14011 Feb 19001 Jan 25017 Dec 27003 Dec 35005 Oct

07004 Dec 14013 Dec 19014 Dec 25020 Jan 27070 Dec 35011 Oct

07005 Oct 14018 Feb 19016 Jan 25021 Dec 28001 Dec 35071 Jan

07006 Dec 14019 Feb 19020 Dec 25023 Nov 29001 Jan 35073 Dec

07007 Dec 14029 Dec 19031 Feb 25025 Jan 29004 Jan 36010 Dec

07009 Nov 14033 Dec 19046 Mar 25027 Dec 29007 Jan 36011 Dec

07010 Nov 14034 Feb 20001 Oct 25029 Feb 29011 Jan 36012 Dec

07011 Dec 15001 Dec 20002 Dec 25030 Dec 29071 Dec 36015 Oct

07012 Jan 15003 Aug 20006 Dec 25034 Dec 30001 Dec 36018 Dec

07033 Jan 15004 Feb 22006 Dec 25038 Feb 30004 Nov 36019 Dec

07041 Nov 15005 Dec 22009 Dec 25040 Oct 30005 Dec 36021 Oct

08002 Aug 15012 Feb 22035 Jan 25044 Dec 30007 Feb 36027 Jan

08003 Nov 16001 Dec 23001 Dec 25124 Feb 30012 Dec 36031 Oct

08005 Aug 16002 Dec 23002 Aug 25158 Jan 30021 Jan 36071 Apr

08007 Nov 16003 Dec 23012 Aug 26002 Oct 30031 Dec 38001 Sep

08008 Jun 16004 Dec 24001 Oct 26005 Dec 30037 Nov 39001 Sep

08009 Dec 16005 Sep 24002 Feb 26006 Dec 30061 Jan 39008 Dec

08011 Aug 16006 Dec 24004 Dec 26007 Nov 31002 Oct 39009 Dec

08012 Dec 16007 Sep 24008 Dec 26008 Nov 31072 Jul 202 FSU

catchments 09001 Dec 16008 Dec 24011 Dec 26009 Oct 32011 Sep


216

Appendix L Distance metrics for pooling-group construction

L1 Introduction

This appendix describes work underlying the distance metric recommended in Section 7.3 for

pooling-group formation:

2

lnBFI

ji

2

lnSAAR

ji

2

lnAREA

ji

ijσ

lnBFIlnBFI

σ

lnSAARlnSAAR

σ

lnAREAlnAREAd

[Equation 7.2]

Under the Region Of Influence (ROI) approach (Burn, 1990), the aim is to select catchments

that are hydrologically similar to the subject catchment. Catchments are recruited to the

pooling group (Reed et al., 1999) using a distance metric which represents their closeness to

the subject catchment in “catchment descriptor” space. There are many possible

arrangements and variations when constructing the catchment-descriptor space in which to do

the pooling, e.g.

Descriptors other than AREA, SAAR and BFI might have been used;

A different number of descriptors could have been used (e.g. two or four descriptors

rather than three);

Different transformations might have been applied;

Components in the distance metric could have been assigned different weights.

L2 Notation

The general form of the distance metric used for selecting members of a pooling group is:

n

1k

2

jk,ik,kij XXWd L.1

where n is the number of catchment descriptors, Xk,i is the normalised value of the

kth

catchment descriptor at the ith

site and Wk is the weight applied to descriptor k reflecting

the relative importance of that catchment descriptor. The subscript j applies to the subject

site and the subscript i applies to the available gauged sites.

In choosing a distance measure dij, a decision has to be made about which catchment

descriptors are to be included in the distance measure, whether logarithms or other

transformations are to be used, and what weightings are to be applied.

Following Jakob et al. (1999), catchment descriptors are normalised by dividing by the

sample standard deviation after any transformation. Thus, the term used to represent

catchment size is Xk,i = ℓnAREAi/σℓnAREA.

L3 Selecting variables to define the distance metric

The research considered use of the physical catchment descriptors AREA, SAAR, BFI and

FARL. The first three were found useful by Jakob et al. (1999). The fourth descriptor

responds to a specific finding in Section 8.2 that pooled growth factors in Ireland appear

sensitive to FARL. Some UK work (Kjeldsen et al., 2008) has also found FARL relevant in

this context.


217

L4 Statistics to help in choosing a good distance metric

The objective is to find a distance metric which leads to pooling groups which are most

homogeneous (i.e. least heterogeneous), thereby exploiting the national resource of AM flow

data effectively. The search is supported by three statistics:

H1, heterogeneity measure based on dissimilarity of L-CVs across pooling group;

H2, heterogeneity measure based on dissimilarity of L-CVs and L-skewnesses;

PUM, the pooled uncertainty measure (see next section).

The heterogeneity measures H1 and H2 are those put forward by Hosking and Wallis (1997).

L5 Pooled uncertainty measure, PUM

The pooled uncertainty measure (PUM) is a weighted average of the differences between site

and pooled growth factors measured on a logarithmic scale (Jakob et al., 1999):

2

M

1i

i

M

1i

Tp

Ti

Tlong

long

ii

n

lnxlnxn

PUM

L.2

where Mlong is the number of long-record sites and (for the ith

site): ni denotes record length,

iTx is the T-year single-site growth factor and iTpx is the T-year pooled growth factor.

PUM is a measure of how effective a pooling method is at identifying a homogeneous region.

A good pooling method will yield low values of PUM.

L6 Application to flood data at 90 A1 + A2 stations

Ninety stations graded A1 or A2 were used in the analysis. PUM has been evaluated at

100-year return period for those 85 (of the 90) stations having a record of 20 years or longer.

[Editorial note: The pooling group for each of the 85 stations used the distance metric under

test to select from the set of 90 catchments.] Stations were added to the pooling group

according to the 5T rule of Section 7.3.3, i.e. until there are at least 5 100 = 500

station-years of AM flow data in the pooling group.

The Generalised Extreme Value (GEV) distribution was used to calculate the pooled and

single-site growth factors. In making such experiments, Jakob et al. (1999) exclude the

subject site from its own pooling group when evaluating PUM. Here, PUM was evaluated

both with and without the subject site. However, only results for the latter case are reported.

Eight combinations of the four variables AREA, SAAR, BFI and FARL have been tested for

use in Equation L.1. The eight pooling schemes are:

Scheme 1: ℓnAREA

Scheme 2: ℓnAREA, ℓnSAAR

Scheme 3: ℓnAREA, ℓnSAAR, BFI

Scheme 4: ℓnAREA, ℓnSAAR, BFI, FARL


218

Scheme 5: ℓnSAAR

Scheme 6: BFI

Scheme 7: ℓnAREA, BFI

Scheme 8: ℓnAREA, ℓnSAAR, ℓnBFI

Initially, all weights Wk in Equation L.1 were set to unity. The dataset is summarised in

Table L.1. [Editorial note: It would appear that the dataset is rather similar to that used in

Chapter 8 and includes five relatively heavily urbanised catchments (URBEXT > 0.20 and

highlighted in Table 8.1) that might have warranted omission.]

Table L.1: Summary of AM flow dataset used in dij study

#

stations

Shortest record

length (years)

Longest record

length (years)

Mean record

length (years)

# station-years of

AM flow data

90 18 55 37.47 3372

Table L.2 summarises the mean variation in PUM100, H1 and H2 values achieved with the

eight different pooling schemes. Small values of PUM100, H1 and H2 indicate superior

performance. The numerical measures vary relatively little between methods except that

Scheme 6 (using BFI alone) is consistently the least-effective.

Table L.2: Mean values of PUM100, H1 and H2 for various pooling schemes

Sch

eme

Variables in distance metric dij

Mean value of

PUM100 H1 H2

1 ℓnAREA 0.1956 6.167 2.770

2 ℓnAREA, ℓnSAAR 0.1929 5.228 2.768

3 ℓnAREA, ℓnSAAR, BFI 0.1958 5.438 2.823

4 ℓnAREA, ℓnSAAR, BFI, FARL 0.1966 5.016 2.958

5 ℓnSAAR 0.1913 5.528 2.726

6 BFI 0.2053 7.141 3.214

7 ℓnAREA, BFI 0.2044 6.135 2.950

8 ℓnAREA, ℓnSAAR, ℓnBFI 0.1958 5.558 2.877

[Editorial note: Box-plots L.1 show the 100-year PUM values. Comparison with the results

presented in Table L.2 indicates that the ranking of methods would be appreciably altered

were this based on median rather than mean values of PUM100. Corresponding box-plots (not

included) indicate that rankings in terms of H1 and H2 would also change. Were median

values of the three measures used to rank the methods overall, the performance of Scheme 3

– which is recommended below – would in fact be comparable with or better than the other

seven schemes.]


219

Box-plots L.1: 100-year PUM values for eight formulations of distance metric dij

L7 Discussion

The use of ℓnAREA and ℓnSAAR (Scheme 2) might be warranted in Irish conditions.

However, any margin by which it outperforms (e.g.) Scheme 3 is minor and may not be

statistically significant. From a hydrological standpoint there is merit in including a third

component – i.e. additional to ℓnAREA and ℓnSAAR – that reflects a catchment feature

known to influence flood growth rates. Candidates include catchment permeability (indexed

by BFI) and storage attenuation (indexed by FARL). An extended investigation explored this

and sought to optimise the terms and weights (Wk) in Equation L.1.

L8 Alternative weightings of the recommended distance metric

It was found that weights of 1.7, 1.0 and 0.2 applied respectively to ℓnAREA, ℓnSAAR and

BFI in Scheme 3 offered a small but useful improvement over the unweighted schemes

reported in Table L.2. An alternative distance metric is therefore:

2

BFI

ji

2

SAAR

ji

2

lnAREA

ji

ijσ

BFIBFI0.2

σ

lnSAARlnSAAR

σ

lnAREAlnAREA1.7d

L.3

This led to values of mean PUM100, H1 and H2 of 0.1889, 5.20 and 2.62 respectively.

[Editorial note: Comparison with the values in Table L.2 confirms that introduction of the

additional weights leads to improved performance. It should be noted that this has been

achieved at the expense of an additional two parameters. Further analysis might confirm the

significance of this improvement. Any new recommendation on the choice of distance metric

is likely to be influenced by feedback from practitioners on the use of Equation 7.2 in

practical cases.]

Further details and additional research are presented by Das and Cunnane (2011 and 2012),

who favour the same distance metric but with a further-altered set of weights:

2

BFI

ji

2

SAAR

ji

2

lnAREA

ji

ijσ

BFIBFI0.1

σ

lnSAARlnSAAR

σ

lnAREAlnAREA1.5d

L.4

Pooled Uncertainty Measure(PUM)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

lnA lnA+lnS lnA+lnS+B lnA+lnS+B+F lnS B lnA+B lnA+lnS+lnB

PU

M100

ℓnAREA ℓnAREA ℓnAREA ℓnAREA ℓnSAAR BFI ℓnAREA ℓnAREA

ℓnSAAR ℓnSAAR ℓnSAAR BFI ℓnSAAR

BFI BFI BFI

FARL

Scheme Scheme Scheme Scheme Scheme Scheme Scheme Scheme

1 2 3 4 5 6 7 8

PU

M10

0


220

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