ORI GIN AL PA PER

Quantifying uncertainties associated with depth durationfrequency curves

Majid Mirzaei • Yuk Feng Huang • Teang Shui Lee • Ahmed El-Shafie •

Abdul Halim Ghazali

Received: 10 April 2013 / Accepted: 28 July 2013 / Published online: 29 August 2013� Springer Science+Business Media Dordrecht 2013

Abstract Uncertainty in depth–duration–frequency (DDF) curves is usually disregarded

in the view of difficulties associated in assigning a value to it. In central Iran, precipitation

duration is often long and characterized with low intensity leading to a considerable

uncertainty in the parameters of the probabilistic distributions describing rainfall depth. In

this paper, the daily rainfall depths from 4 stations in the Zayanderood basin, Iran, were

analysed, and a generalized extreme value distribution was fitted to the maximum yearly

rainfall for durations of 1, 2, 3, 4 and 5 days. DDF curves were described as a function of

rainfall duration (D) and return period (T). Uncertainties of the rainfall depth in the DDF

curves were estimated with the bootstrap sampling method and were described by a normal

probability density function. Standard deviations were modeled as a function of rainfall

duration and rainfall depth using 104 bootstrap samples for all the durations and return

periods considered for each rainfall station.

Keywords Uncertainty analysis � Depth duration frequency curves �Generalized extreme value distribution � Bootstrap sampling

M. Mirzaei (&) � A. El-ShafieFaculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, Kuala Lumpur,Malaysiae-mail: mirzaeii@gmail.com

Y. F. HuangFaculty of Engineering and Science, Universiti Tuanku Abdul Rahman, Kuala Lumpur, Malaysia

T. S. Lee � A. H. GhazaliFaculty of Engineering, Universiti Putra Malaysia, Serdang, Selangor, Malaysia

123

Nat Hazards (2014) 71:1227–1239DOI 10.1007/s11069-013-0819-3

1 Introduction

Depth duration frequency (DDF) curves are generally used to assess the extreme character of

rainfall in hydraulic and hydrology in general. In the design of dams, for example, it is crucial to

have reliable estimates of extreme rainfall depths in the determination of the required discharge

capacity of the spillway in order to prevent overtopping or dam breaking. While underestimation

of these extreme values leads to an unacceptable risk, overestimation results in the construction

of costly infrastructures that will be underused most of the time. The first step in construction of

DDF curves is to fit some theoretical distribution to the extreme rainfall amounts for a number of

fixed durations. Although many common distributions can be used for fitting purposes, the

generalized extreme value (GEV) distribution is applied herein. The GEV distribution has been

used worldwide to model extreme rainfall (Fowler and Kilsby 2003; and Koutsoyiannis 2004). It

has been widely used in frequency modeling of heavy precipitation (Overeem et al. 2008; Fowler

and Ekstrom 2009), floods (Cunderlik and Ouarda 2007), and other variables. The main prob-

lems to be solved in the estimated GEV parameters are to be correlated for different durations and

are the determination of the unknown coefficients and uncertainty in this relationship. It is

important to assess the uncertainties related to extreme rainfall estimates and to propagate those

uncertainties into design decisions and risk assessment (Coles et al. 2003).

Burn (2003) represented resampling technique to calculate uncertainties and confidence

intervals for flood quantiles. The bootstrap sampling method is used both for the estimation of

correlation between estimated GEV parameters and for the confidence bands of the DDF curves.

The bootstrap method, first introduced and named by Efron (1979), is a resampling technique for

estimating the properties, such as the variance, of an estimator or statistic. The idea behind

bootstrap is that the sample values are the best guide to the underlying true distribution even

when the information about the true distribution is lacking. The advantage of this method over

analytical approximation in the classic method is its relative simplicity in implementation. It has

been widelyused in uncertainty analysis, especially when the analytical form is difficult to derive

or approximate (e.g., Abrahart 2001; Srinivas and Srinivasan 2005). Bootstrap sampling is a

technique for determining the accuracy of statistics in circumstances in which confidence

intervals cannot be obtained analytically or when an approximation based on the limit distri-

bution is not satisfactory (Davison and Hinkley 1997). Bootstrap techniques have become very

popular in many areas of environmental sciences, including frequency analysis in climatology

and hydrology (Dunn 2001; Hall et al. 2004; Ames 2006; Kysely 2009; Twardosz 2009; Fowler

and Ekstrom 2009). There are two basic approaches to the bootstrap: While the nonparametric

bootstrap is based on resampling with replacement from a given sample and calculating the

required statistic from a large number of repeated samples (it is often termed simply ‘resam-

pling’), the idea of the parametric bootstrap is to randomly generate samples from a parametric

model (distribution) fitted to the data and to calculate the statistic from a large number of

randomly drawn samples. In both cases, one attempts to infer a distribution of the estimate of a

given statistic (e.g., model parameter, quantile of a distribution) from the available data.

The motivation of this study comes from a desire to obtain uncertainties in DDF curves

based on a bootstrap method analysis. The DDF curves and their uncertainties for four

stations in the western part of the Zayanderood basin in central Iran using the Bootstrap

method are to be estimated. The bootstrap method, a simple nonparametric technique

(Efron 1979) is proposed in this paper as it is simple to describe and easy to implement.

The paper is organized as follows: second section introduces the methods used in the

analysis, third section describes the analyzed area and the data sets used in the study. In

fourth and fifth sections, the results are presented and discussed. Sixth section draws up the

conclusion of the study.

1228 Nat Hazards (2014) 71:1227–1239

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2 Case study area

In this research, the upper part of the west of the Zayanderood River catchment area in

central Iran, between the longitudes of 32� 1701000N and 33� 1204900N and the latitudes of

50� 103600E and 50� 4602600E (see Fig. 1), is studied. This area was selected because it

covers the main source of the stream flow in the Zayanderood River and it has a reasonably

dense network of rain gauge stations. This semi-arid region has an annual average pre-

cipitation of 611 mm and an annual average temperature of 11 �C; there is a seasonal

distribution of precipitation with the wet season being in autumn, winter and spring, and

the dry season in summer. There are 16 rain gauge stations located in the study area, each

with more than 30-year daily rainfall records. The four selected stations have more than

50 years of daily rainfall data records. All the rainfall stations are depicted in Fig. 2.

Table 1 lists the rainfall stations used in this study.

3 Fitted model

The principle behind fitting several distributions to the data is to find the best type of

distribution that gives the highest probability of reproducing the observed data. As pre-

viously mentioned, the GEV distribution has been used worldwide to model extreme

rainfall events, and in this study, the GEV distribution was applied to fit the distribution of

24, 48, 72, 96 and 120 h maximum annual rainfall data.

Fig. 1 Location map of Zayanderood basin in Iran (a), location map of study area in Zayanderood basin(b) and map of study area (c)

Nat Hazards (2014) 71:1227–1239 1229

123

The GEV distribution incorporates the Gumbel’s type I (k = 0), Frechet’s type II

(k \ 0) and the Weibull or type III (k [ 0) distributions. The GEV distribution has the

cumulative distribution function:

F xð Þ ¼ exp � 1� kx� nð Þ

a

� �1=k( )

k 6¼ 0

F xð Þ ¼ exp � exp � x� nð Þa

� �� �k ¼ 0

where n ? a/k B x\ ? ? for k \ 0, -?\ x \ ? ? for k = 0, and -?\ x \ n ? a/k

for k [ 0. Here, n, a and k are the location, scale and shape parameters, respectively. The

quantiles of the GEV distribution are given in terms of the parameters and the cumulative

probability p by

Fig. 2 Locations of the rainfall-gauging stations within the study area

1230 Nat Hazards (2014) 71:1227–1239

123

xp ¼ nþ ak

1� � ln pð Þð Þkh i

k 6¼ 0

xp ¼ n� a ln � ln pð Þð Þk ¼ 0:

4 L moments

The parameters of each distribution were estimated using one of the following methods,

namely: moments, maximum likelihood, least squares or L-moments (Singh 1998). The

choice of the method for estimating the parameters, where applicable, has been based on

the least number of intensive computations required. For the small samples, the estimates

based on L-moments generally have low standard deviation and is not computationally

more difficult than maximum likelihood with a Bayesian prior distribution; therefore, the

method of L-moments is chosen for this study.

n ¼ k1 �a

k1� C 1þ k

� �� a ¼ k2k

1� 2�k �

C 1þ k� �

k ¼ 7:8590cþ 2:9554c2;

c ¼ 2= 3þ s3ð Þ � logð2Þ=log 3ð Þ:

The L moment estimators k1; k2; and s3 ¼ k3

.k2 (L skewness) were obtained by using an

unbiased estimator of the first three probability-weighted moments defined as

br ¼ nþ ak

1� r þ 1ð Þ�kC 1þ kð Þh i.

ðr þ 1Þ

The unbiased estimator of br (Landwehr et al. 1979; Hosking and Wallis 1995) is:

br ¼Xn

i¼1

i� 1ð Þ i� 2ð Þ i� 3ð Þ. . . i� rð Þn n� 1ð Þ n� 2ð Þ. . . n� rð Þ x ið Þ

� �

r ¼ 0; 1; 2; . . .;

where the x(i) are the ordered observations from a sample of size{x(1) B x(2) B …Bx(n)}

and k1 = b0, k2 = 2b1 - b0, and k1 = 6b2 - 6b2 ? b2 (Hosking 1990a, b; Wang 1996).

The L moment estimators for the GEV distribution were evaluated using the daily rainfall

data from 1954 to 2009; these series were derived and plotted for the four selected rainfall

stations (Table 1; Fig. 2). To estimate the less probable maximum annual rainfalls with

high return periods, the extreme data were fitted to the GEV distribution. In general, the

Table 1 Selected stations, their record length, latitude, longitude

Station name Nationalcode

Studycode

Data periodFrom-To

Latitude(N)

Longitude(E)

Elevation(m)

Chelgerd 42001 S1 1954–2009 50.13 32.45 2,324

Damaneh 42004 S2 1954–2009 50.48 33.02 2,300

Shahrukh Palace 42003 S3 1958–2009 50.47 32.65 2,098

Sade Zayanderood 42007 S4 1955–2009 50.78 32.72 1,990

Nat Hazards (2014) 71:1227–1239 1231

123

assessment indicated that the GEV distribution was best fitted to the annual maximum

daily rainfalls at all stations of the study area. Figure 3 shows the annual maximum daily

rainfalls against their GEV theoretical probability distributions.

In this paper, c = a/n is considered instead of a. The advantage of using c is that its

correlation with k is weak. For all the stations, a GEV distribution was fitted separately to

the running annual maximum for durations of 24, 48, 72, 96 and 120 h.

5 Bootstrap sampling

To investigate the uncertainty in DDF curves, the bootstrapping method is used to calculate

the confidence bands of the DDF curves. The bootstrap, introduced by Efron (1979), is a

technique for determining the accuracy of statistics in circumstances in which confidence

intervals cannot be obtained analytically or when an approximation based on the limit

distribution is not satisfactory (Efron and Tibshirani 1993; Davison and Hinkley 1997).

Zucchini and Adamson (1989) were the first who used the bootstrap to determine the

uncertainty of design storms. There are two basic approaches to the bootstrap: (1) Non-

parametric bootstrap, which is based on resampling with replacement from a given sample

and calculating the required statistic from a large number of repeated samples (it is often

termed ‘resampling’), (2) Parametric bootstrap that randomly generates samples from a

parametric model (distribution) fitted to the data and calculates the statistics from a large

0

50

100

150

200

250

300

350

0 100 200 300

S1

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

S2

0

20

40

60

80

100

0 20 40 60 80 100

S3

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

S4

Fig. 3 The annual maximum daily rainfalls against their GEV theoretical probability distribution; thex-axis indicates the calculated amount from the GEV (mm) and y-axis indicates observed annual maximarainfall (mm)

1232 Nat Hazards (2014) 71:1227–1239

123

number of randomly drawn samples. In this study, the nonparametric bootstrap was applied

in analysis of the observed datasets. The principle of the nonparametric bootstrap is to

provide a way to simulate repeated observations from the population (Fig 4). Table 2

shows the estimated GEV parameters, c and k, and their respective standard deviations. As

expected, n increases with increasing duration, whereas the parameter c seldom increase

with increasing duration. There seems to be no systematic variation of k with duration. In

the bootstrap method, Diaconis and Efron 1983 and Efron and Tibshirani 1993 emphasized

that new samples (bootstrap samples) are generated by sampling with replacement from the

original sample. The standard deviations in Table 2 were derived from 104 bootstrap

samples.

Fig. 4 The principle of the nonparametric bootstrap

Table 2 Estimated GEV parameters for D = 24, 48, 72, 96 and 120 h. In all stations, standard deviationsare estimated with the bootstrap and given between brackets

D (h) S1 S2

n c k n c k

24 34.90 (2.47) 0.552 (0.002) 0.019 (0.024) 8.45 (1.11) 1.049 (0.003) 0.140 (0.016)

48 56.43 (9.49) 0.601 (0.005) 0.015 (0.015) 13.21 (1.48) 1.060 (0.005) 0.005 (0.012)

72 81.52 (11.83) 0.555 (0.004) 0.014 (0.014) 22.37 (3.73) 1.074 (0.010) 0.120 (0.014)

96 92.28 (16.27) 0.543 (0.006) 0.021 (0.020) 24.72 (5.99) 1.070 (0.011) 0.080 (0.015)

120 118.29 (20.87) 0.600 (0.003) 0.016 (0.027) 30.83 (6.18) 1.061 (0.016) 0.100 (0.068)

D (h) S3 S4

n c k n c k

24 11.27 (1.16) 1.117 (0.002) 0.181 (0.018) 9.86 (1.20) 0.698 (0.002) 0.088 (0.018)

48 17.34 (2.21) 1.075 (0.004) 0.159 (0.022) 14.47 (2.42) 0.671 (0.004) 0.141 (0.022)

72 22.53 (2.40) 1.129 (0.011) 0.145 (0.042) 20.05 (2.23) 0.638 (0.011) 0.118 (0.042)

96 28.63 (3.88) 1.197 (0.003) 0.170 (0.045) 23.36 (3.88) 0.701 (0.003) 0.101 (0.045)

120 34.87 (4.46) 1.124 (0.020) 0.186 (0.045) 30.50 (4.46) 0.715 (020) 0.119 (0.045)

Nat Hazards (2014) 71:1227–1239 1233

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6 GEV parameters as a function of duration

One of the simpler and frequently used models that is popularly utilized in statistics is the

classical regression model

Y ¼ Xbþ 2

where Y = (Y1,…,Yn)t represents an observational vector of length n, X is a n 9 p known

matrix of explanatory variables and b is a vector of unknown regression coefficients of

length p that characterizes the relationship between observations and explanatory variables.

Classically, the vector 2 is assumed to be a zero-mean Gaussian vector. In this study, X

and b were defined as:

X ¼

1 ln D1

: :: :: :1 ln D5

0BBBB@

1CCCCA and b ¼ b0

b1

�

where D is rainfall duration for 24, 48, 72, 96 and 120 h and b0 and b1 are regression

coefficients of GEV parameters. The generalized least squares method was used to estimate

the regression coefficients b0 and b1.

Relations of the GEV parameters as a function of duration D (hours) were used to

construct rainfall DDF curves. In Fig. 5, the GEV parameters were plotted against D for

24, 48, 72, 96 and 120 h. It shows that c and the n have a linear relationship with the

logarithm of D for all the considered durations. There appears to be no systematic variation

of k with duration. According to the equations in the GEV regression model, the regression

coefficients were estimated for the 104 bootstrap samples. Table 3 shows the averages and

standard deviations of the regression coefficients. For k, the estimate of slope b1 was

approximately zero for most samples. This confirms that k may be considered to be

constant, and the X2-statistic in 5 % level was below the critical value.

7 Derivation of DDF curves

Now that the GEV parameters are described as a function of D, rainfall DDF curves are

constructed by substituting these relationships into below equation, so that the DDF curves

are given by:

x Tð Þ ¼ exp b0n þ b1n ln D �

� 1þ b0c þ b1c ln D � 1� � ln 1� T�1ð Þ½ �kGLS

n okGLS

0B@

1CA

Then, for station S1 is

x Tð Þ ¼ exp 3:2149þ 0:3125 ln Dð Þ

� 1þ 0:5699þ 0:0009 ln Dð Þ1� � ln 1� T�1ð Þ½ �0:0164n o

0:0164

0@

1A

1234 Nat Hazards (2014) 71:1227–1239

123

For station S2

x Tð Þ ¼ exp 1:1391þ 0:3031 ln Dð Þ

� 1þ 3:0658þ 0:00089 ln Dð Þ1� � ln 1� T�1ð Þ½ �0:0857n o

0:0857

0@

1A

0

40

80

120

160

0 24 48 72 96 120

0.30.350.4

0.450.5

0.550.6

0.65

0 24 48 72 96 120

0

0.005

0.01

0.015

0.02

0.025

0 24 48 72 96 120

D(hour)

05

101520253035

0 24 48 72 96 120

0.8

0.9

0.9

1.0

1.0

1.1

1.1

0 24 48 72 96 120

00.10.20.30.40.50.60.70.8

0 24 48 72 96 120

D(hour)

0

10

20

30

40

0 24 48 72 96 120

0.3

0.8

1.3

1.8

0 24 48 72 96 120

0

0.1

0.2

0.3

0.4

0 24 48 72 96 120

D(hour)

0

10

20

30

40

0 24 48 72 96 120

0

0.2

0.4

0.6

0.8

0 24 48 72 96 120

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 24 48 72 96 120

D(hour)

S1 S2 S3 S4)

(mm

ξγ

k

Fig. 5 GEV parameters plotted against duration D. The solid lines represent the generalized relationship atthe Chelgerd station

Table 3 Results of the regression of GEV parameters for the four stations

GEV parameter S1 S2

b0 r b0

�b1 r b1

�b0 r b0

�b1 r b1

�

lnn 3.2149 0.0065 0.3125 0.0039 1.1391 0.0175 0.3031 0.0425

c 0.5699 0.3862 0.0009 0.073 3.0658 1.0842 0.00089 0.00038

k 0.0164 0.0096 0.0857 0.0051

GEV parameter S3 S4

b0 r b0

�b1 r b1

�b0 r b0

�b1 r b1

�

lnn 1.625 0.1756 0.4045 0.0235 1.453 0.278 0.4042 0.054

c 1.1419 0.3869 0.0009 0.0849 0.672 0.239 0.0034 0.0486

k 0.1659 0.0068 0.1105 0.0129

Nat Hazards (2014) 71:1227–1239 1235

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For station S3

x Tð Þ ¼ exp 1:625þ 0:4045 ln Dð Þ

� 1þ 1:1419þ 0:0009 ln Dð Þ1� �ln 1� T�1ð Þ½ �0:1659n o

0:1659

0@

1A

For station S4

x Tð Þ ¼ exp 1:453þ 0:4042 ln Dð Þ

� 1þ 0:6725þ 0:0034 ln Dð Þ1� �ln 1� T�1ð Þ½ �0:1105n o

0:1105

0@

1A

By choosing a return period T, the rainfall depth x (mm) can be plotted as a function of

duration D using the above equations. Figure 6 presents the DDF curves for T = 5, 10, 20,

50 and 100 years. The curves show a strong increase in rainfall depth with D.

8 Modeling uncertainty in DDF curves

Uncertainty in DDF curves which is usually disregarded in view of the difficulties asso-

ciated in assigning a value to it should be considered in the design of hydraulic structures.

The bootstrap method was applied to assess these uncertainties. Only the uncertainty due to

the estimation of the GEV parameters and the associated sampling errors were evaluated in

this study. For each of the 104 bootstrap samples, the relationships between the GEV

parameters and duration were reestimated using the generalized least squares, so that 104

DDF curves could be constructed. For each DDF curve, the rainfall depths were derived for

100

150

200

250

300

350

400

20 30 40 50 60 70 80 90 100 110 120 130

Rai

nfa

ll D

epth

(m

m) 100

50

20

10

5

30

50

70

90

110

130

150

170

190

20 30 40 50 60 70 80 90 100 110 120 130

100

50

20

10

5

S2

40

60

80

100

120

140

160

180

20 30 40 50 60 70 80 90 100 110 120 130

Rai

nfa

ll D

epth

(m

m)

D (hr)

100

50

20

10

5

20

40

60

80

100

120

20 30 40 50 60 70 80 90 100 110 120 130

D (hr)

100

50

20

10

5

S4

S1

S3

Fig. 6 Rainfall DDF curves (solid lines) and 95 % confidence bands (dashed lines) for the return periods of5, 10, 20, 50 and 100 years at stations S1, S2, S3 and S4

1236 Nat Hazards (2014) 71:1227–1239

123

durations between 24 and 120 h in time steps of 1 h. Subsequently, for each of these

durations, the 104 depths were ranked in increasing order, and the 250th and 9750th values

were determined. These values were then plotted to form the 95 % confidence bands, as

shown in Fig. 6.

The relationship between standard deviation of rainfall depth and rainfall duration in

each return period was modeled as a function of D and T. For each station, standard

deviations were estimated using the 104 bootstrap samples for five durations and five return

periods as:

For station S1:

r ¼ 1:852þ 0:0335Dþ 0:0389T

For station S2:

r ¼ 0:5037þ 0:0119Dþ 0:0168T

For station S3:

r ¼ 0:0151þ 0:0196Dþ 0:1049T

For station S4:

r ¼ 0:0001þ 0:0116Dþ 0:0150T

The regression coefficients were estimated with the ordinary least squares method.

Figure 7 shows the DDF curves for return period (T) equal to 50 and 100 years for station

S1. The normal probability density functions which describe the uncertainties in these

curves are plotted for duration (D) corresponding to 48 and 96 h. According to the DDF

curves and their 95 % confidence bands, for longer return periods (T) uncertainty increases

substantially. Uncertainty can be described by a distribution. The bootstrap distribution of

estimated quantiles is described by the normal distribution. The parameters l and r of the

normal distribution that are the mean and standard deviation are modeled as a function of

D and T.

Fig. 7 Rainfall DDF curves (solid lines) for T = 50 and 100 years and normal probability density functions(dashed lines) which describe the uncertainties in the DDF curves for D 48 and 96 h

Nat Hazards (2014) 71:1227–1239 1237

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9 Conclusions

Extreme rainfall in the Zayanderood River basin in central Iran for durations between 24

and 120 h was studied. The GEV parameters of this time series were estimated using the

method of L-moments. Standard deviations among the estimated GEV parameters were

obtained using the bootstrap method. To calculate the correlation among estimated GEV

parameters at different durations, the generalized least squares method was used to

describe the variation of these parameters as a function of time. It was found that the shape

parameter k of the GEV distribution does not change with time, and the parameter c rarely

increases with increasing duration. Accordingly, the coefficient of variation increases with

decrease in duration, suggesting that the simple scale relation being ruled out. Uncer-

tainties in rainfall DDF curves are generally ignored resulting in the risks being under-

estimated or completely eliminated. The valuable aspect of this study is that the uncertainty

in DDF curves due to sampling variability was modeled and evaluated. However, it should

be noted that other sources of uncertainty such as measurement errors and uncertainty in

the choice of the distribution had not been taken into account in this study. The following

conclusions can be arrived at:

1. For any of the stations, the DDF curve can be represented by a specific equation

derived from the analyses where GEV parameters were described as a function of

duration

2. Uncertainty in DDF curves due to sampling variability was modeled and evaluated. It

was also quantified with the Bootstrapping method and described with a normal

probability density function.

3. For longer return periods, it was found that uncertainty increase substantially

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