Stochastic ydrology of the reat Lakes A Systemic …...Stochastic ydrology of the reat Lakes -A...
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Stochastic Hydrology of the Great Lakes - A Systemic Analysis A Compilation of projects completed under the Hydroclimatic Work Group for the International Upper Great Lakes Study Revised December 2011
Transcript of Stochastic ydrology of the reat Lakes A Systemic …...Stochastic ydrology of the reat Lakes -A...
Stochastic Hydrology of the Great Lakes - A Systemic Analysis
A Compilation of projects completed under the Hydroclimatic Work Group for the International Upper Great Lakes Study
Revised December 2011
Context of Stochastic Hydrology Studies
Hydroclimate Working Group International Upper Great Lakes Study
Revised December 2011
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Context of Stochastic Hydrology Studies
1. Introduction The Hydrology and Climate Modelling Strategy (2008) is being undertaken as part of the International Joint Commission’s International Upper Great Lakes Study (IJC-IUGLS). The IJC, created by the Boundary Waters Treaty of 1909, has the responsibility to manage the shared waters between Canada and the United States. The IJC established the 5-year binational Study in February 2007 in response to concerns regarding declining lake levels. The study area includes lakes Superior, Michigan, Huron and Erie, and their interconnecting channels (St. Mary's River, St. Clair River, Lake St. Clair, Detroit River and Niagara River), up to Niagara Falls. More details on the mandate, study organization, Plan of Study, etc. are available at www.iugls.org. The Study completed Phase I in December 2009 investigating factors affecting water levels and flows including physical changes in the St. Clair River. Phase II is underway examining whether the regulation of Lake Superior outflows can be improved to address the evolving needs of the upper Great Lakes. In support of the Study, the Hydroclimatic Technical Work Group (HC TWG) was formed to assess changes to the contemporary hydrology affecting the levels of the lakes and to examine future climate variability and change and any future challenges in lake regulation that may be expected as a result of a changing climate regime. The first objective primarily related to Phase I of the Study; how much of the change in Lake Huron levels can be attributed to channel conveyance and how much to water supplies? The second objective primarily relates to Phase II Upper Lakes regulation; can we anticipate and adapt lake regulation to a changing climate? This peer review focuses on the work performed to address this second objective.
2. Science Question
The tasks selected here for peer review address the science question: “What potential impact could variations in the climate system have on future regulations of the Upper Great Lakes?” During the plan of Study development, the Study planners recognized early on that it would be hard to defend estimated probabilities of future lake level trends. Past studies that estimated the probabilities of wet or dry periods based on statistical extrapolation of the historic record overlooked climate variability and human induced climate change and there is ample evidence that those forecasts were often significantly misleading. The Study planners also recognized that the new evidence available on climate variability and change doesn’t improve confidence in forecasting the future – it informs one to be more respectful of the chance that collectively we will be wrong about the future. So the Study planners implemented an approach that would allow the development of test data for new regulation plans that recognize the hubris
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of assigning probabilities and would allow experts to discover the hydrologic situations that would be most difficult to manage.
3. Approach and Products This approach builds upon the stochastic and climate change hydrology from the Lake Ontario study (International Lake Ontario-St. Lawrence River Study Board, 2006) that established this precedent and an earlier study performed by Hydro Quebec to determine the Beauharnois Dam Probable Maximum Flood. These studies used stochastic generation of water supplies based upon the contemporary residual supplies to develop plausible sequences of water supplies not seen in the short historic record. This approach developed nearly 50,000 years of monthly and quarter-monthly water supplies with which to test proposed new regulation plans for robustness. This Study builds upon this approach through three tasks: 1) Updating the approach of Fagherazzi et al. 2005 with a new and longer record (1900-2008)
of the contemporary water supplies (Fagherazzi 2011). This modeling scheme includes the use of a Contemporaneous Shifting Mean – CARMA Model (CSM-CARMA) at the annual level and a temporal annual-monthly disaggregation scheme. One data set was generated:
A series of 55,000 annual and monthly net basin supplies generated using the CSM-CARMA model.
2) Development of a new approach which builds a stochastic model to simulate the climate
indices tele-connected with the Great Lakes water supplies (Lee and Ouarda 2011 and Seidou, Lee and Ouarda 2011). Climate indices generally contain non-stationary oscillations (NSO) and not many adequate stochastic models are available in the literature to reproduce the NSO processes. In this work, the stochastic simulation model based on the Empirical Mode Decomposition (EMD) is improved by decomposing the observed time series and then simulating the decomposed components with the NSO resampling (NSOR) technique. A multivariate version of NSOR (denoted as M-NSOR) is developed to manipulate the links between the climate indices themselves as well as to reproduce the NSO process. The proposed simulation model was tested with three climate indices (i.e. Arctic Oscillation, El Niño-Southern Oscillation, and Pacific Decadal Oscillation) for the annual and winter (January, February, and March – JFM) datasets. The dataset generated is:
A series of 50,000 annual net basin supplies generated using stacked observed NCEP variables as predictors using an NL-ARX model.
3) Development of a stochastic series representing a changed climate (Seidou, Lee and
Ouarda 2011). This approach links annual net basin supplies to a list of climate variables that could be obtained from both global climate mode runs and National Center for Environmental Prediction (NCEP) reanalysis. The advantage of this model is that these variables can be readily obtained from climate change experiments and used with the
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stochastic model to obtain climate change altered net basin supply time series. Three data sets were generated:
500 sequences of 100 annual net basin supply time series (corresponding to years 2001-2100) were generated using the outputs of the SRES A1B experiments of the third generation of the Canadian General Circulation Model (CGCM3),
500 sequences of 100 annual net basin supply time series (corresponding to years 2001-2100) were generated using the outputs of the SRES A2 experiments of the third generation of the Canadian General Circulation Model (CGCM3).
4. Conclusions The datasets have been provided to the Plan Formulation and Evaluation Group and the Adaptive Management Group. Both groups have found the datasets useful in plan formulation and in assessments of plausibility. The focus of this peer review is on the methodology presented. The conclusions of the Study to which this work is contributing will be peer reviewed at a later date when the Study report is drafted.
5. Revised Submission Components
Parts A & B Stochastic Modeling and Simulation of the Great Lakes System. A report
prepared by Dr. Laura Fagherazzi, Dr. José Salas and Dr. Oli Sveinsson. Revised December 2011
Part C Stochastic generation of synthetic residual Net Basin Supply for the Great Lakes System. A report prepared by Ousmane Seidou, Taesam Lee and Taha Ouarda.
Part D Multivariate Stochastic Simulation of Climate Indices with Empirical Mode Decomposition. A report prepared by Taesam Lee, Taha Ouarda and Ousmane Seidou.
6. References
Fagherazzi, L. (2011). Statistical Characteristics of the Historical Data Base. Hydro Quebec, Montreal, Québec, CA.
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Fagherazzi, L. (2011). Stochastic Modeling And Simulation of the Great Lakes System. Hydro Quebec, Montreal, Québec, CA. International Lake Ontario-St. Lawrence River Study Board, (2006). Options for Managing Lake Ontario and St. Lawrence River Water levels and Flows, Final Report. International Joint Commission, Washington, DC and Ottawa, ON. Lee, D.H., and Pietroniro, A. (2008). Hydrology and Climate Modelling Strategy. Prepared for the International Upper Great Lakes Study. Cincinnati, OH. Lee, T. and Ouarda, T. (2010). Multivariate Stochastic Simulation of Climate Indices with Empirical Mode Decomposition, Final Report. Institut National de la Recherche Scientifique, centre Eau et Environnement,Québec, CA. Seidou, O., Lee, T., and Ouarda, T. (2011). Stochastic generation of synthetic residual Net Basin Supply for the Great Lakes System. University of Ottawa, Ottawa (ON) K1N 6N5, Canada and Institut National de la Recherche Scientifique, centre Eau et Environnement,Québec, CA.
Parts A & B
Statistical Characteristics of the Historical Data Base
Stochastic Modeling and Simulation of the Great Lakes System
Combined under one title “Stochastic Modeling and Simulation of the Great Lakes System”
Laura Fagherazzi Water Resources Specialist Hydro Quebec Production
December 2011
STOCHASTIC MODELING AND SIMULATION OF THE GREAT LAKES SYSTEM
Final Report PREPARED BY:
Dr. Laura Fagherazzi Hydro Quebec
Dr. José Salas Colorado State University Dr. Oli Sveinsson The Icelandic Power Company
Revised December 2011
CONTENTS
Figure List .......................................................................................................................... 3
CHAPTER 1 Statistical Characteristics of the Historical Data Base ....................... 10
1.1 The Great Lakes Basin ...............................................................................................10
1.2 Great Lakes Hydrology ..............................................................................................11 1.2.1 Great Lakes Net Basin Supply ............................................................................................... 11 1.2.2 Great Lakes Components and Residual NBS Data set........................................................... 11
1.3 Great Lakes NBS Statistical Characteristics ............................................................13 1.3.1 Characteristics of the series of annual NBS – IULS 1900-2008 ............................................ 13 1.3.2 Seasonal Characteristics ........................................................................................................ 37 1.3.3 Spatial Characteristics ........................................................................................................... 46
1.4 Concluding remarks ...................................................................................................47
CHAPTER 2 Criteria for selecting and testing an adequate stochastic model ........ 48
2.1 Introduction .................................................................................................................48
2.2 Fundamental stochastic and statistical characteristics to be retained by an well
fitted model ...............................................................................................................................49 2.2.1 Temporal Characteristics ....................................................................................................... 49 2.2.2 Spatial Characteristics ........................................................................................................... 50
CHAPTER 3 CHAPTER 3 Proposed Modeling Strategies ...................................... 51
3.1 Introduction .................................................................................................................51
3.2 Contemporaneous Shifting Mean – CARMA (p,q) Models ....................................51 3.2.1 Univariate Shifting Mean Models .......................................................................................... 51 3.2.2 Fitting Multivariate CSM-CARMA models to the Great Lakes ............................................ 54 3.2.3 Annual-seasonal disaggregation approach ............................................................................. 55
3.3 Transformations ..........................................................................................................56
3.4 Upgrading the parameters of the stochastic model due to changes in the Great
Lakes historical NBS data set ..................................................................................................56
CHAPTER 4 Testing the selected Great Lakes CSM-CARMA Model ..................... 58
4.1 Statistics of the historical and generated annual NBS .............................................58
4.2 Monthly Statistics ........................................................................................................79
4.3 Spatial Characteristics ................................................................................................88
CHAPTER 5 Connecting Channel flow routed levels and outflows statistics ......... 94
5.1 Introduction .................................................................................................................94
5.2 Characteristics of the series of Annual Outflows .....................................................95
5.3 Characteristics of the series of Annual Lake levels ................................................116
CHAPTER 6 Drought and surplus related statistics on lake levels........................ 131
6.1 Introduction ...............................................................................................................131
6.2 Characterizing Lake Levels .....................................................................................131
CHAPTER 7 Concluding remarks ........................................................................... 139
References ...................................................................................................................... 140
Figure List
Figure 1 - The Great Lakes and the drainage area of the lakes (Clites and Quinn, 2003) 10 Figure 2 – Characteristics of the Annual NBS series - Lake Superior ................................ 17 Figure 3 – Characteristics of the Annual NBS series - Lake Michigan-Huron ................... 18 Figure 4 – Characteristics of the Annual NBS series - Lake St Clair ................................. 19 Figure 5 – Characteristics of the Annual NBS series - Lake Erie ....................................... 20 Figure 6 - Characteristics of the Annual NBS series - Lake Ontario .................................. 21 Figure 7 - Wavelet approach applied to Lake Michigan-Huron over land precipitation
series ......................................................................................................................... 23 Figure 8 - Wavelet approach applied to Lake Michigan-Huron Annual NBS series .......... 23 Figure 9 - Wavelet approach applied to Lake Erie over land precipitation series ........... 24 Figure 10- Wavelet approach applied to Lake Erie Annual NBS series ............................. 24 Figure 11– Retrospective Classification in two states - Lakes Michigan-Huron, Erie and
Ontario ...................................................................................................................... 26 Figure 12 – Lake Superior Run Statistics ........................................................................... 28 Figure 13 – Lake Michigan Huron Run Statistics .............................................................. 29 Figure 14 – Lake St Clair Run Statistics ............................................................................ 30 Figure 15 – Lake Erie Run Statistics .................................................................................. 31 Figure 16 – Lake Ontario Run Statistics ............................................................................ 32 Figure 17 – Lake Erie Run Statistics (Threshold 0.9 the historical mean) ....................... 33 Figure 18 – Lake Erie Run Statistics (Threshold 1.1 the historical mean) ....................... 34 Figure 19 – Lake Ontario Run Statistics (Threshold 0.9 the historical mean) ................. 35 Figure 20 – Lake Ontario Run Statistics (Threshold 1.1 the historical mean) ................. 36 Figure 21 - Lake Superior Monthly NBS Hydrographs – Julian Year ................................ 38 Figure 22 - Lake Michigan-Huron Monthly NBS Hydrographs – Julian Year ..................... 38 Figure 23 - Lake St-Clair Monthly NBS Hydrographs – Julian Year .................................. 39 Figure 24 - Lake Erie Monthly NBS Hydrographs – Julian Year ......................................... 39 Figure 25 - Lake Ontario Monthly NBS Hydrographs – Julian Year .................................. 40 Figure 26 – Lake Superior Lag-1, 2 and 3 Month to Month Correlation .......................... 41 Figure 27 – Lake Michigan-Huron Lag-1, 2 and 3 Month to Month Correlation .............. 42 Figure 28 – Lake St. Clair Lag-1, 2 and 3 Month to Month Correlation ............................ 43 Figure 29 – Lake Erie Lag-1, 2 and 3 Month to Month Correlation .................................. 44 Figure 30 – Lake Ontario Lag-1, 2 and 3 Month to Month Correlation ............................ 45 Figure 31 - A schematic representation of the different processes composing the Shifting
Mean Model (from Sveinsson et al., 2007)............................................................... 53 Figure 32 – Empirical Frequency Plots of 510 realizations of synthetic annual NBS - Lake
Superior ..................................................................................................................... 65 Figure 33 – Empirical Frequency Plots of 510 realizations of synthetic annual NBS - Lake
Michigan-Huron ........................................................................................................ 66 Figure 34 - Empirical Frequency Plots of 510 realizations of synthetic annual NBS - Lake
Erie ............................................................................................................................ 67
Figure 35 - Empirical Frequency Plots of 510 realizations of synthetic annual NBS - Lake Ontario ...................................................................................................................... 68
Figure 36 - Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 - Lake Superior annual NBS ............... 69
Figure 37 –Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 - Lake Michigan-Huron annual NBS ... 70
Figure 38 – Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 - Lake Erie annual NBS ....................... 71
Figure 39 – Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 - Lake Ontario annual NBS ................. 72
Figure 40 – Empirical Frequency Plots of annual maximum and minimum monthly NBS - Lake Superior ............................................................................................................ 74
Figure 41– Empirical Frequency Plots of annual maximum and minimum monthly NBS– Lake Michigan-Huron ................................................................................................ 75
Figure 42 - Empirical Frequency Plots of annual maximum and minimum monthly NBS - Lake Erie .................................................................................................................... 76
Figure 43 - Empirical Frequency Plots of annual maximum and minimum monthly NBS– Lake Ontario .............................................................................................................. 77
Figure 44 - Selected transformation for month 10 (October), Lake Erie .......................... 78 Figure 45 – Superposed observed and generated monthly hydrographs (m³/s) – Lake
Superior ..................................................................................................................... 80 Figure 46 – Superposed observed and generated monthly hydrographs (m³/s) – Lake
Michigan-Huron ........................................................................................................ 81 Figure 47 – Superposed observed and generated monthly hydrographs (m³/s) – Lake St
Clair ........................................................................................................................... 82 Figure 48 – Superposed observed and generated monthly hydrographs (m³/s) – Lake Erie
................................................................................................................................... 83 Figure 49 – Superposed observed and generated monthly hydrographs (m³/s) – Lake
Ontario ...................................................................................................................... 84 Figure 50 – Box-plots of the Lag-1 and Lag-2 Month to Month Correlation obtained from
510 samples – Star: historical value .......................................................................... 85 Figure 51– Box-plots of the Lag-1 and Lag-2 Month to Month Correlation obtained from
510 samples – Star: historical value .......................................................................... 86 Figure 52 – Box-plots of the Lag-1 and Lag-2 Month to Month Correlation obtained from
510 samples – Star: historical value .......................................................................... 87 Figure 53 – Box-plots of the Lag-1 and Lag-2 Month to Month Correlation obtained from
510 samples – Star: historical value .......................................................................... 88 Figure 54 – Box-Plot illustrating Lake Superior annual contemporaneous cross-
correlation statistics .................................................................................................. 89 Figure 55 – Box-Plot illustrating Lake Michigan-Huron annual contemporaneous cross-
correlation statistics .................................................................................................. 90 Figure 56 – Box-Plot illustrating Lake St. Clair annual contemporaneous cross-
correlation statistics .................................................................................................. 91
Figure 57 – Box-Plot illustrating Lake Erie annual contemporaneous cross-correlation statistics .................................................................................................................... 92
Figure 58 – Box-Plot illustrating Lake Ontario annual contemporaneous cross-correlation statistics .................................................................................................. 93
Figure 59 – Characteristics of the Annual outflow series (BOC) - St Mary River .............. 96 Figure 60 – Characteristics of the Annual outflow series (BOC) - St Clair River ............... 97 Figure 61 – Characteristics of the Annual outflow series (BOC) - Detroit River ............... 98 Figure 62 – Characteristics of the Annual outflow series (BOC) – Lake Erie outflows ..... 99 Figure 63- Empirical Frequency Plots of 510 realizations of synthetic annual outflows –
St Mary and St Clair River ....................................................................................... 105 Figure 64- Empirical Frequency Plots of 510 realizations of synthetic annual outflows –
Detroit river and Lake Erie outflows ....................................................................... 106 Figure 65 - Empirical Frequency Plots obtained from the historical series (shown in red)
and one synthetic series of length 510*109 – St Mary River outflows ................. 107 Figure 66 - Empirical Frequency Plots obtained from the historical series (shown in red)
and one synthetic series of length 510*109 – St Clair River outflows .................. 108 Figure 67 - Empirical Frequency Plots obtained from the historical series (shown in red)
and one synthetic series of length 510*109 – Detroit River outflows .................. 109 Figure 68 - Empirical Frequency Plots obtained from the historical series (shown in red)
and one synthetic series of length 510*109 – Lake Erie outflows ........................ 110 Figure 69 - Empirical Frequency Plots of 510 realizations of annual maximum outflow
and historical maximum outflow – St Mary and St Clair rivers .............................. 112 Figure 70 - Empirical Frequency Plots of 510 realizations of annual maximum outflow
and historical maximum outflow – Detroit river and Lake Erie outflows ............. 113 Figure 71 – Superposed observed and synthetic monthly outflows (m³/s) – St Mary and
St Clair Rivers .......................................................................................................... 114 Figure 72 – Superposed observed and synthetic monthly outflows (m³/s) – Detroit River
and Lake Erie outflows ............................................................................................ 115 Figure 73 – Lake Superior routed levels annual characteristics (BOC) ........................... 117 Figure 74 – Lake Michigan-Huron routed levels annual characteristics (BOC) .............. 118 Figure 75 – Lake Erie routed levels annual characteristics (BOC) .................................. 119 Figure 76 - Empirical Frequency Plots of 510 samples of annual synthetic levels vs.
historical levels – ..................................................................................................... 123 Figure 77 - Empirical Frequency Plots of 510 samples of 109 years routed annual
generated levels vs. historical levels– Lake Erie ..................................................... 124 Figure 78 - Empirical Frequency Plots of 1 sample of 55 590 years of annual routed
levels- Lake Superior ............................................................................................... 125 Figure 79- Empirical Frequency Plots of 1 sample of 55 590 years of annual routed levels
- Lake Michigan-Huron ............................................................................................ 126 Figure 80 - Empirical Frequency Plots of 1 sample of 55 590 years of annual routed levels
- Lake Erie ................................................................................................................ 127 Figure 81– Superposed observed and synthetic monthly lake levels (m) – Lakes Superior
and Michigan-Huron ............................................................................................... 128
Figure 82 - Superposed observed and synthetic monthly lake levels (m) – Lakes St Clair and Erie ................................................................................................................... 129
Figure 83 ......................................................................................................................... 132 Figure 84 ......................................................................................................................... 133 Figure 85 ......................................................................................................................... 133 Figure 86 – Box-plots of stochastic monthly levels, with superposed monthly levels
sequences (synthetic and observed ) of maximum RI (surplus and deficits) ......... 134 Figure 87 – Box-plots of stochastic monthly levels, with superposed monthly levels
sequences (synthetic and observed ) of maximum RI (surplus and deficits) ......... 134 Figure 88 - Sequence of maximum deficit RL identified from Lake Erie synthetic series of
annual levels with concomitant levels on Lakes Superior and Michigan-Huron .... 135 Figure 89- Sequence of maximum surplus RL identified from Lake Erie synthetic series of
annual levels with concomitant levels on Lakes Superior and Michigan-Huron .... 136 Figure 90 - Sequence of maximum deficit RI identified from Lake Erie synthetic series of
annual levels with concomitant levels on Lakes Superior and Michigan-Huron .... 137 Figure 91 - Sequences of maximum surplus RI identified from Lake Erie synthetic series
of annual levels with concomitant levels on Lakes Superior and Michigan-Huron 138
Table List Table 1 - Great Lakes Data Set .......................................................................................... 12 Table 2 – Characteristics of the series of Annual NBS ...................................................... 16 Table 3 – Great Lakes Annual Contemporaneous Cross-correlation Matrix .................... 46 Table 4 – Characteristics of the series of historical annual NBS to be preserved by the
simulated series ........................................................................................................ 49 Table 5 - Characteristics of the series of historical seasonal NBS (monthlies and
quartermonthlies) to be preserved by the simulated series ................................... 49 Table 6 – Average sample statistics of 510 generated time series versus historical
statistics – Characteristics of Annual NBS - Lake Superior ....................................... 59 Table 7 - Average sample statistics of 510 generated time series versus historical
statistics – Characteristics of Annual NBS - Lake Michigan-Huron ........................... 60 Table 8 - Average sample statistics of 510 generated time series versus historical
statistics – Characteristics of Annual NBS - Lake St. Clair ......................................... 61 Table 9 - Average sample statistics of 510 generated time series versus historical
statistics – Characteristics of Annual NBS - Lake Erie ............................................... 62 Table 10 - Average sample statistics of 510 generated time series versus historical
statistics – Characteristics of Annual NBS - Lake Ontario ......................................... 63 Table 11 – Upper Lakes statistics of historical annual NBS and corresponding simulated
outflows using the Coordinated Great Lakes Regulation and Routing Model ......... 94 Table 12 – Upper Lakes annual statistics of synthetic NBS and corresponding simulated
outflows using the Coordinated Great Lakes Regulation and Routing Model ......... 95 Table 13 - Average sample statistics of 510 routed outflow series versus historical
statistics – Characteristics of Annual outflows series – St Mary River ................... 100
Table 14 - Average sample statistics of 510 routed outflow series versus historical statistics – Characteristics of Annual outflows series – St. Clair River ................... 101
Table 15 - Average sample statistics of 510 routed outflow series versus historical statistics – Characteristics of Annual outflows series – Detroit River .................... 102
Table 16 - Average sample statistics of 510 routed outflow series versus historical statistics – Characteristics of Annual outflows series – Lake Erie outflows ........... 103
Table 17 - Average sample statistics of 510 lake levels series versus historical statistics Characteristics of Annual lake levels series – Lake Superior .................................. 120
Table 18 - Average sample statistics of 510 lake levels series versus historical statistics Characteristics of Annual lake levels series – Lake Michigan-Huron ...................... 121
Table 19 - Average sample statistics of 510 lake levels series versus historical statistics Characteristics of Annual lake levels series – Lake Erie .......................................... 122
CONTEXT AND BACKGROUND The hydrological data used in the design, planning and operational studies of water resource schemes are often limited to historical records, which are usually short, incomplete, sparsely distributed in space, and poorly synchronized for multivariate analysis. Moreover, it is rare that a particular sequence of flow observations reoccurs in the same form in the future. As a result, it is important that models accurately capture key historical low and high flow sequences to avoid design and planning shortcomings due to improper estimations. To better grasp drought and flood variability, the International Joint Commission’s Lake Ontario – St. Lawrence River Study (LOSLRS) used both historical and synthetic hydrological scenarios. Using statistical properties from the historical series a set of stochastic hydrological Net Basin Supply (NBS) series for the Great Lakes and local inflows of the St. Lawrence River’s major tributaries was developed. This set constitutes a large number of potential hydrological scenarios that could occur in the future due to the natural variation in climate. The simulated series were then used along with the observed 1900-2000 data to design and evaluate the adequacy of newly proposed Lake Ontario multi-objective management strategies. A detailed report by Fagherazzi et al. (2005) and some published papers (Sveinsson and Salas, 2002; Sveinsson et al., 2007, Fortin et al., 2002 ) describe the multivariate modeling approach developed for generating the synthetic hydrological scenarios for the Great Lakes and local inflows of the St. Lawrence River major tributaries. The study showed that simulating annual NBS with models explicitly treating some special temporal features and disaggregating annual values into monthly or smaller quantities gave good results. The different models were calibrated with the 1900-2000 Residual NBS data set. The Upper Great Lakes Study Group follows the approach proposed by the LOSLRS for evaluating new proposed Lake Superior multi-objective strategies. A revised Great Lakes NBS data base (1900-2008) was available in June 2010. Consequently, the Upper Great Lakes Study Group demanded the upgrade of the stochastic models parameters to account for the revised NBS series' characteristics and the preparation of a set of 510 synthetic series of monthly NBS. This report summarizes the results of the stochastic modeling and simulation approach applied to updated net basin supplies of the Great Lakes System. First the special hydrological characteristics of the Great Lakes system and its statistical properties are described. Then a brief review is given of the technical details of the different procedures included in the stochastic modeling approach. Changes in the model parameters and generated synthetic series from the earlier application of the Lake Ontario – St. Lawrence River Study are analyzed and a summary of the obtained results is provided. The technical description of the models and modeling approach used has
already been published. The main goal of this report is to provide a complete description of the synthetic series (NBS, lake levels and outflows series) to be used by modellers, planners, managers and other interest groups.
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CHAPTER 1 Statistical Characteristics of the Historical Data Base
1.1 The Great Lakes Basin
The Great Lakes – St. Lawrence River system is a complex lake-river system with large amounts of over-year storage characterized by particular spatial and temporal properties of inflow series. The Great Lakes System is composed by a series of five lakes (Superior, Michigan-Huron, St. Clair, Erie, and Ontario) connected through four channels (St. Marys River, St. Clair River, Detroit River, and Niagara River). Although Lakes Superior and Ontario have been regulated in the past several decades modifications in the connecting channels have impacted the lakes’ outflows (Quinn, 1979). Figure 1 from Ouarda et al, 2010 illustrates the structure of the system, the drainage area for the Great Lakes covering an area of 770 000 km² and the regulation points.
Figure 1 - The Great Lakes and the drainage area of the lakes (Clites and Quinn, 2003)
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1.2 Great Lakes Hydrology
1.2.1 Great Lakes Net Basin Supply
The term Net Basin Supply (NBS) is used to describe the amount of water that is contributed to or lost from a lake within the confines of its natural drainage basin. NBS is the sum of water the lake receives through direct over-lake precipitation and runoff from its drainage basin minus the water the lake losses through evaporation: NBS = P + R – E (1.1) where P is the over-lake precipitation, R is the runoff into the lake, and E is lake evaporation . The NBS, however, can be computed indirectly as a residual of the water balance for a lake: NBS = ΔS + O – I ± D (1.2) where ΔS is the change in water storage (m), also referred to as CIS, is computed from the difference in lake levels over a time interval, such as the beginning and the end of a month, O is the outflow (m³/s), I is the inflow (m³/s), and D is the total diversion over the time interval (m3/s). Refer to Ouarda et al, 2010 (Reference 4) for the definition and a detailed description of each mentioned variable. There are some differences between the NBS derived from the estimated values of its components in Equation 1.1(component NBS) and the NBS computed as a residual in Equation 1.2(residual NBS) because of uncertainties in the different variables in both equations (Croley & Lee, 1993). The focus of this study is on the residual NBS and therefore the statistical analyses are only performed on the residual NBS.
1.2.2 Great Lakes Components and Residual NBS Data set
The data base has been coordinated between the Great Lakes Hydraulics and Hydrology Office, the Detroit District U.S. Army Corps of Engineers (USACE) and the Great Lakes-St. Lawrence Regulation Office of Environment Canada (EC) in Cornwall, Ontario. A coordinated 1900-2000 NBS and connecting channels flow data set was made available for the International Lake Ontario and St. Lawrence River Study (ILOSLRS) (Fagherazzi et al, 2005) and it will be referred to as the ILOSLRS Residual NBS data set in the following sections of this report. The analyzed hydrological series starts in January 1900 to December 2000. Julian year definition is used to prepare the tables and graphics.
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In June 2010, the data was updated and revised up to December 2008 taking into account new flow estimates of the St. Clair, Detroit and Niagara Rivers and the use of actual flow estimates rather than standard monthly averages in the computation of the monthly volumes of change-in-storage (on which the former set of residual NBS data were based). The changes to the St. Clair and Detroit Rivers flows date back to 1987 while the Niagara River flows were adjusted back to 1961. This updated and revised Residual NBS data set will be identified as the IULS data set. A list of the different data sets used in this report along with the period of available records is summarized in Table 1.
Table 1 - Great Lakes Data Set
Basin ID Record length (Years)
Period Temporal scale
Lake Superior NBS 101 (ILOSLRS)
109 (IULS)
1900-2000
1900-2008
Annual, monthly
Lake Michigan-Huron NBS 101 (ILOSLRS)
109 (IULS)
1900-2000
1900-2008
Annual, monthly
Lake Erie NBS 101 (ILOSLRS)
109 (IULS)
1900-2000
1900-2008
Annual, quarterly
Lake Ontario NBS 101 (ILOSLRS)
109 (IULS)
1900-2000
1900-2008
Annual, quarterly
Lake Superior Overland and Overlake Precipitation
52
61
1948-1999
1948-2008
Annual, monthly
Lake Michigan-Huron Overland and Overlake Precipitation
52
61
1948-1999
1948-2008
Annual, monthly
Lake Erie Overland and Overlake Precipitation
52
61
1948-1999
1948-2008
Annual, monthly
Lake Ontario Overland and Overlake Precipitation
52
61
1948-1999
1948-2008
Annual, monthly
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1.3 Great Lakes NBS Statistical Characteristics
The stochastic spatial and temporal characteristics of the Great Lakes Net Basin Supplies (NBS) are very complex. The NBS series temporal variability includes oscillatory behaviour (high and low frequency components) and sudden shifts in addition to the usual seasonality and variance-covariance properties. Many techniques are available for detecting and testing the statistical significance of these special temporal characteristics and were already applied to the Great Lakes data series by many authors (Fagherazzi et al, 2005, Fortin et al, 2004, Ouarda et al, 2010). However, it is difficult to interpret and elaborate over these results as not all the NBS series show similar temporal behaviour. A characteristic that had received a considerable attention in literature is the sudden pattern shift that is apparent in some but not all of the annual NBS series. Human intervention such as the construction of regulation structures, diversions, dredging and urbanization could induce a significant change in some important statistics such as the mean or variance of a hydrological series. Such changes in the mean could also be the result of low frequency climatic components and large-scale climate variability (Sveinsson and Salas, 2002). Stochastic models that explicitly account for shifts in the mean have been suggested in the literature (Potter, 1976; Boes and Salas, 1978). Once accepting the hypothesis that some series of annual NBS may exhibit sudden shifts in the statistical parameters, the shifting mean approach could be used for adequate modeling and simulation. Section 1.3.1 summarizes the particular characteristics of the series of annual NBS and some results from different approaches used for detecting changes in the NBS, while section 1.3.2 shows characteristics of the series of monthly NBS. Spatial characteristics are summarized in section 1.3.3.
1.3.1 Characteristics of the series of annual NBS – IULS 1900-2008
During the ILOSLRS (Fagherazzi et al., 2005) and IULS studies (Ouarda et al., 2010), a considerable effort was involved in assessing the particular characteristics of the annual NBS series of the Great Lakes. Only the main steps and results will be reported in this section. Temporal properties of the NBS such as time, frequency and spectral domains were analyzed. These include annual descriptive statistics like the mean, standard deviation, skewness, maximum and minimum values, autocorrelation coefficients and storage deficits and surplus related statistics (Runs statistics). A detailed Run Characteristics definition could be found in Sveinsson et al., 2007. Table 2 summarizes both descriptive and runs statistics for all the lakes using IULS 1900-2008 data set. A deficit or a surplus
14
can be defined by its duration RL (Run Length), its magnitude RS (Run Sum) and its intensity RI (Run Intensity) expressed as RI = RS/RL. Figures 2 to 6 summarizes the main statistics for each lakes annual NBS series (IULS 1900-2008 residual NBS data set). Six graphs (a to f) were created for each lake and represent the following statistics:
a) The time plot of annual NBS valus along with their corresponding mean; b) The annual values on normal probability plot; c) The serial correlations or ACF; d) The spectral analysis; and e) & f) The results of the procedure by Lee and Heghinian (1977) used to identify
the year of a shift in the mean of the annual series (under the hypothesis that a change has occurred with certainty).
The visual analysis of graphs "a" (annual NBS time series) for Lakes St. Clair, Erie and Ontario (figures 4, 5 and 6) suggest the presence of local non-stationarities in the annual NBS (Fagherazzi et al., 2005). Dry and wet sequences seem to have occurred simultaneously on all the three lakes, like the dry spells of 1930-1940 and 1960-1970, and the wet sequence of 1970-1987. Very similar behaviour was found in the precipitation series. On the other hand, the data for the other lakes shown in graphs “a” within figures 2 and 3 do not seem to exhibit any shift. Graphs "b" illustrate the annual NBS time series on normal probability plot and the Filliben test of normality. The Filliben correlation statistic measures the correlation between the sorted transformed NBS and the quantiles of normally distributed variates with the same mean and variance (Grygier et al, 1990). If the NBS fit a normal distribution, a plot of NBS value vs rank will lie close to a straight line and the correlation between these two variables (the Filliben’s correlation statistic) will be close to one. The Kolmogorov-Smirnov test provides the upper and lower bounds with a 95% confidence level. The data for Lakes Superior, Michigan-Huron, St. Clair and Erie were considered approximately normally distributed, however observing table 2 the Lake Ontario annual NBS observations are more positively skewed (0.23) suggesting the use of another distribution. The sample autocorrelation function (ACF) up to lag 15 is shown on graph "c". For Lakes Superior and Michigan-Huron (figures 2 and 3), the lag-1 and lag-2 correlation coefficients are low and non-significant at a 5% level of significance and as the lag increases the significance quickly decreases to zero. The plot of the spectrum (spectral density function) against the frequency displayed in graphs "d" illustrates the short memory seen in the ACF. On the contrary, the ACF for Lakes St. Clair, Erie and Ontario show some evidence of long term persistence in the sense that the correlations decay slowly as the time lag increases. From lag-1 up to lag-5 coefficients for Lake Erie
15
(Figure 5) and the lag-1 to lag-4 coefficients for Lake Ontario (Figure 6) are both significant with similar levels of significance. The lag-6 to lag-15 coefficients for Lake Erie are still positive and significant with the exception of lag-9 and lag-12, however, this is not the case for Lake Ontario even though their spectrum's shapes are quite similar. Lake St. Clair ACF (Figure 4) shows significant positive coefficients for all lags up to lag-11. Under the hypothesis that a change has occurred with certainty, the Lee and Heghinian procedure identifies a positive sudden shift in the mean NBS in the late sixties/early seventies, for Lakes Erie and Ontario (graphs "e", Figures 5 and.6) and late forties for Lake St. Clair (Figure 4). The mean values before and after the change-point are illustrated on graphs "f", however, the statistics of Lee and Heghinian only identify the most important shift in the series which could have many shifts.
16
Table 2 – Characteristics of the series of Annual NBS – IULS 1900-2008
Statistics Lake Superior Lake Michigan-Huron Lake St. Clair Lake Erie Lake Ontario
μ (m³/s) 2013 3198 132 623 1005
σ (m³/s) 484 742 65 290 236
Min 949 1355 -3.33 -12.50 499
Max 3151 4865 292 1240 1558
Median 1067 3300 136 612 1000
Skewness 0.03 -0.05 0.09 0.12 0.23
ρ1 0.15 0.11 0.51 0.28 0.20
Run Length (Deficits)(years)
Mean (Rld) 2.15 2.04 2.89 2.76 2.39
S (Rld) 1.43 1.62 3.08 3.02 2.43
Max (Rld) 5 8 12 13 12
Run Sum (Deficits)(m³/s)
Mean (Rld) 806 1332 164 622 445
S (Rld) 852 1479 240 743 628
Max (Rld) 3714 6157 894 2465 3066
Run Length (Surplus)(years)
Mean (Rls) 2.04 2.32 3.17 2.43 2.35
S (Rls) 1.56 2.06 3.19 3.23 1.70
Max (Rls) 7 9 14 15 8
Run Sum (Surplus)(m³/s)
Mean (Rls) 806 1332 164 622 445
S (Rls) 653 1227 214 1115 401
Max (Rls) 2103 4396 888 5188 1752
Storage
Storage capacity 7676 12499 2011 8189 4466
Rescaled Range 15.86 16.84 31 28.20 19
Hurst 0.69 0.71 0.86 0.84 0.74
17
Figure 2 – Characteristics of the Annual NBS series - Lake Superior
18
Figure 3 – Characteristics of the Annual NBS series - Lake Michigan-Huron
19
Figure 4 – Characteristics of the Annual NBS series - Lake St. Clair
20
Figure 5 – Characteristics of the Annual NBS series - Lake Erie
21
Figure 6 - Characteristics of the Annual NBS series - Lake Ontario
22
The International Lake Ontario – St. Lawrence River Study, Fagherazzi et al.(2005) illustrated the application of two other methods to detect changes on the mean of the annual NBS and overland precipitation series. The first one used the Haar modified wavelet. Figures 7 and 8 illustrate the application of the Haar modified wavelet procedure to the annual series of overland precipitation and NBS for Lake Michigan-Huron. A normalized entropy variation function was developed to facilitate the localization and the sign of the shift of the mean. A positive sign was associated with wet periods and a negative sign represented dry periods. It should be noted that a high variation does not necessarily indicate a shift in the mean since no statistical test is used in the approach. Nevertheless, these figures give a good description of the time variations and indicate when the most important changes could occur.
The upper graph in figures 7 and 8 shows the annual time series, the middle graph displays the entropy function and the lower one illustrates a bar graph of the normalized entropy variation. For Lake Michigan-Huron, figure 7 shows that the most important change in the annual overland precipitation series from 1948 to1999 occurs around 1965. Figure 8 illustrates the approach applied to the annual NBS series (ILOSLRS, period 1900-2000) and it shows the important positive change of the mean around 1965 but also identifies a negative change around 1930. Lake Erie annual overland precipitation and annual NBS series (ILOSLRS, period 1900-2000) are shown on Figures 9 and 10. The entropy function identifies an important positive change in the mean in the early seventies but also one in the early forties. A negative change in the mean is detected in the early thirties and the sixties.
23
Figure 7 - Wavelet approach applied to Lake Michigan-Huron over land precipitation series
Figure 8 - Wavelet approach applied to Lake Michigan-Huron Annual NBS series
1940 1950 1960 1970 1980 1990 2000600
800
1000
1200Michigan Huron over land precipitation from 1948 to 1999
mm
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995
0
2
4
6
x 104
1950 1955 1960 1965 1970 1975 1980 1985 1990 19950
0.2
0.4
year
24
Figure 9 - Wavelet approach applied to Lake Erie over land precipitation series
Figure 10- Wavelet approach applied to Lake Erie Annual NBS series
1940 1950 1960 1970 1980 1990 2000600
800
1000
1200Erie over land precipitation from 1948 to 1999
mm
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995
-1
0
1
x 105
1950 1955 1960 1965 1970 1975 1980 1985 1990 19950
0.1
0.2
0.3
year
25
Perreault et al, (2002) proposed a two state multivariate hidden Markov (MHMM) to represent the Lakes Michigan-Huron, Erie and Ontario annual NBS series. Adopting a Bayesian perspective and using Gibbs sampling as a numerical tool to evaluate posterior distribution, it was possible to retrospectively identify wet and dry sequences. Figure 11 represents the posterior probability of being in a dry regime for each year. The upper left figure illustrates the series of annual NBS per unit area for the mentioned lakes. The two lower graphs seem to indicate a long wet regime in the seventies and early eighties. The upper right figure shows the annual NBS series per unit area for Lakes Erie and Ontario only. The wet sequence mentioned is more clearly defined in the preceding paragraph. Ouarda et al. (2010) analyzed changes in the annual and monthly NBS and hydro-climate variables of the Great Lakes. The time series was tested for trends using a nonparametric Mann-Kendall trend test, under independence, short term persistence (STP) and long term persistence (LTP) hypotheses. A shift detection analysis was performed on the NBS and water level time series as well.
26
Figure 11– Retrospective Classification in two states - Lakes Michigan-Huron, Erie and Ontario
27
The statistical properties of the series of annual NBS for Lakes St. Clair, Erie and Ontario summarized in Table 2 reinforce the wet and dry sequences behaviour. This is observed in the rather long run lengths and large run sums of deficits and surpluses with respect to the means and the high value of the Hurst statistics (> 0.7). It is interesting to note that Lake Erie has the longest maximum Surplus Run length of 15 years and the longest maximum Deficit Run length of 13 years.
Figures 12 to 16 illustrate Run Lengths (RL) and Run Sum (RS) histograms for surpluses and deficits for each of the five Great Lakes. The threshold level used for deficit and surplus run length analysis was the historical mean. Observing figures 15 and 16, the maximum deficit RL and RS histograms for Lakes Erie and Ontario are similar but it is not the case for the maximum surplus RL and RS as the Lake Erie maximum surplus (Run Length and Run Sum) are much higher than Lake Ontario. Two other threshold levels were analyzed. The two thresholds are 110 % and 90% of the historical mean which are used to analyze the sensitivity of the run statistics. For surplus maximum RL (Figures 17 and 18), it can be seen that Lake Erie is not affected by varying the threshold, but the Lake Ontario maximum RL histogram changes with a threshold decrease of 10% (Figures 19 and 20).
28
Figure 12 – Lake Superior Run Statistics
29
Figure 13 – Lake Michigan Huron Run Statistics
30
Figure 14 – Lake St. Clair Run Statistics
31
Figure 15 – Lake Erie Run Statistics
32
Figure 16 – Lake Ontario Run Statistics
33
Figure 17 – Lake Erie Run Statistics (Threshold 0.9 the historical mean)
34
Figure 18 – Lake Erie Run Statistics (Threshold 1.1 the historical mean)
35
Figure 19 – Lake Ontario Run Statistics (Threshold 0.9 the historical mean)
36
Figure 20 – Lake Ontario Run Statistics (Threshold 1.1 the historical mean)
37
1.3.2 Seasonal Characteristics
Figures 21 to 25 show the monthly variations in NBS. All 109 hydrographs were plotted against the monthly mean. The monthly mean pattern of Lake Superior NBS differs from the monthly mean patterns of both Lakes Erie and Ontario NBS. The hydrographs for Lake Superior show a triangular shape with higher NBS values during the month of May and the bulk of negatives values during January and December whereas Lakes Erie and Ontario hydrographs show a sinusoidal shape with high NBS values during March and April and lower values on September and October. Lake Michigan-Huron hydrographs show higher NBS values in April and May and lower values in October. Lake St. Clair hydrographs show a different pattern characterized by high NBS variations between December and June. Figure 26 shows the lag-1 to lag-3 month-to-month correlation of Lake Superior monthly NBS at a 5% level of significance. Although part of the observed fluctuations can be attributed to sampling variation, it is clear that the lagged correlations vary within the year. The lag-1 month-to-month correlation, that is the correlation between a specific month and the previous one, is significant in the late summer and fall months (August to November). Lag-3 month-to-month correlation is non-significant at the monthly timescale. Lake Michigan-Huron lag-1 month-to-month correlation (Figure 27) is significant for the spring and summer months (May to July) and the fall months. Lag-3 month-to-month correlation is non-significant for the monthly timescale. Lake St. Clair lag-1 month-to-month correlation (Figure 28) is significant and relatively important for all months except January. Lag-2 and Lag-3 month-to-month correlation are also significant for some months. Lake Erie (Figure 29) and Lake Ontario (Figure 30) lag-1 and lag-2 month-to-month correlations are relatively important for all months except the winter. These characteristics suggest that more than one lag month-to-month correlation are necessary to capture the temporal behaviour of both Lakes Erie and Ontario NBS series during the spring, summer and fall season.
38
Figure 21 - Lake Superior Monthly NBS Hydrographs – Julian Year
Figure 22 - Lake Michigan-Huron Monthly NBS Hydrographs – Julian Year
39
Figure 23 - Lake St. Clair Monthly NBS Hydrographs – Julian Year
Figure 24 - Lake Erie Monthly NBS Hydrographs – Julian Year
40
Figure 25 - Lake Ontario Monthly NBS Hydrographs – Julian Year
41
Figure 26 – Lake Superior Lag-1, 2 and 3 Month to Month Correlation
42
Figure 27 – Lake Michigan-Huron Lag-1, 2 and 3 Month to Month Correlation
43
Figure 28 – Lake St. Clair Lag-1, 2 and 3 Month to Month Correlation
44
Figure 29 – Lake Erie Lag-1, 2 and 3 Month to Month Correlation
45
Figure 30 – Lake Ontario Lag-1, 2 and 3 Month to Month Correlation
46
1.3.3 Spatial Characteristics
The Great Lakes annual contemporaneous cross-correlation matrix is illustrated in Table 3. Only contemporaneous cross-correlations (lag zero) were significant. The pattern between correlations is not straightforward because the correlations between the lakes do not always decrease as the distance between them increases.
Table 3 – Great Lakes Annual Contemporaneous Cross-correlation Matrix
Superior Michigan-Huron St. Clair Erie Ontario
Superior 1 0.5262 0.1661 0.2290 0.1801 Michigan-Huron 0.5262 1 0.4506 0.5076 0.6080 St. Clair 0.1661 0.4506 1 0.6256 0.5337
Erie 0.2290 0.5076 0.6256 1 0.6287
Ontario 0.1801 0.6080 0.5337 0.6287 1
47
1.4 Concluding remarks
Chapter 1 focused on the statistical and the temporal characteristics of the Great Lakes NBS. Annual and monthly statistics were elaborated on each site using the available record (1900-2008) and spatial correlation across sites was analyzed. The major findings were the following:
Annual NBS series for Lakes St. Clair, Erie and Ontario show evidence of shifts in the mean. The most important positive simultaneous shift on the lower lakes (Erie and Ontario) occurs in the seventies and spans many years.
Annual serial correlation coefficients for Lakes St. Clair, Erie and Ontario are significant for many lags. The ACF for Lakes St. Clair, Erie and Ontario shows evidence of long term persistence in the sense that the correlations decay slowly as the time lag increases.
Annual lag-1 and lag-2 serial correlation coefficients for Lake Superior are low and non-significant at a 5% level of significance. Annual lag-1 and lag-2 serial correlation coefficients for Lake Michigan-Huron are also low and non-significant. For Lakes Superior and Michigan-Huron the serial correlation coefficients decrease to zero quickly as the lag increases.
Only contemporaneous cross-correlations (lag zero) were significant at the annual scale. The pattern between correlations is not straightforward. The correlations between the lakes do not always decrease as the distance between them increases.
The seasonal characteristics are different for each lake. However, Lakes Erie and Ontario seasonal characteristics show some similarities.
48
CHAPTER 2 Criteria for selecting and testing an adequate stochastic model
2.1 Introduction
The statistical analysis presented in Chapter 1 showed that the stochastic characteristic of the Great Lakes NBS are quite complex. A number of approaches have been suggested in the literature for analyzing, modeling and simulating the Great Lakes NBS series. Time series models such as autoregressive with moving average terms (ARMA) and their multisite versions, alternative schemes including multisite modeling at seasonal (monthly) time scales and temporal disaggregating models have been used by different groups of experts in Canada and the United States with different degree of success. In 1991-1992 Hydro-Quebec conducted a major modeling and simulation study of the Great Lakes NBS in connection with the Beauharnois-Les Cedres extreme floods analysis (Rassam et al., 1992). A few years later, a joint effort under the supervision of the International Joint Commission (IJC) was undertaken with the participation of an international group of experts to simulate synthetic NBS series for the Great Lakes and local inflows of the St. Lawrence River's major tributaries. The simulated series was used to evaluate the adequacy of the newly proposed Lake Ontario multi-objective management strategies under a large number of potential hydrological scenarios. Both studies simulate annual NBS with models that incorporated sudden shifts in the mean and disaggregate such annual values into monthly and quarter monthly values with different disaggregation approaches. To verify how well the fitted models and the modeling approaches preserve not only the NBS statistics used in the fitting procedure but also other implicit properties important for the planning and management of the system being studied. The usual approach consists of generating the fitted model with a large number of synthetic time series (realizations) having the same length as the historical record. Each one of the statistics of interest is then estimated from each realization and their averages are compared with the corresponding historical sample statistics. A detailed set of characteristics of interest was defined with the participation of planners, managers and other users of the system. For example, the planners wanted the simulated series to reproduce the marginal distribution of annual and seasonal flows and the covariance and seasonal covariance as closely as possible because they describe the persistence of high and low flows, the variability of the multivariate supply and how much water may be available at different sites at varying times of the year.
49
2.2 Fundamental stochastic and statistical characteristics to be retained by an well fitted model
The next section summarizes what the modellers, planners, managers and other interest groups deem as important to be preserved by a well fitted stochastic model and modeling approach prepared for the Great Lakes system.
2.2.1 Temporal Characteristics
Table 4 summarizes the fundamental characteristics of the series of historical annual NBS to be preserved at the annual level. Some characteristics not specifically used in the model fitting such as the storage related characteristics (rescaled range, Hurst coefficient) and drought and surplus statistics were also included. Table 5 shows the seasonal (monthly and quarter monthly) NBS characteristics to be preserved by the simulated series. Table 4 – Characteristics of the series of historical annual NBS to be preserved by the simulated series
Basin Stochastic characteristics to be be explicitly modelled
Temporal statistics to be preserved
Superior Michigan-Huron
None Basic statistics (mean, standard deviation, skew, lag one and two serial correlation) Storage related characteristics Run statistics Frequency distribution
St. Clair Erie Ontario
Changes in the mean Persistence
Basic statistics (mean, standard deviation, skew, particular ACF characteristics) Storage related characteristics Run statistics Frequency distribution
Table 5 - Characteristics of the series of historical seasonal NBS (monthlies and quarter monthlies) to be preserved by the simulated series
Subbasin Statistics to be preserved
Superior Michigan-Huron St. Clair
Basic statistics (mean, standard deviation, skewness, lag-1 and lag-2 season-to season correlation) Annual-monthly correlation
50
Erie Ontario
Frequency distribution of monthly and quarter monthly maximum and minimum values.
2.2.2 Spatial Characteristics
Lag-zero (contemporaneous) cross-correlations for annual and seasonal (monthly and quarter monthly) NBS series also needs to be preserved. After verifying that the simulated samples reproduced the Great Lakes NBS historical characteristics, the realizations should be routed through the system using the adequate regulation plans (Lakes Superior and Ontario Regulation Plans) and the Great Lakes Coordinated Model to obtain simulated lake levels and outflows. Afterwards, further analysis is needed to compare the simulated lake levels and outflows with the routed historical levels and outflows.
51
CHAPTER 3 CHAPTER 3 Proposed Modeling Strategies
3.1 Introduction
Building a stochastic flow generation model for a large system is a long and complicated process involving many alternative tasks and choices, a large number of hypotheses, different assumptions and techniques for parameter estimation and analyses of simulated series before finding the adequate configuration of models and parameters maximizing the number of reproduced characteristics. A large number of alternative models can be constructed depending on which NBS characteristics have to be explicitly reproduced. The Great Lakes annual NBS special stochastic characteristics directed the choice to the Shifting Mean Models (SM) that explicitly account for sudden changes in the mean of the process under consideration and the time between shifts. The SM model was coupled with a temporal disaggregation model for disaggregating the annual NBS values into seasonal (monthly) values. This chapter briefly describes the SM model, the approach followed for selecting a contemporaneous model for all the Great Lakes system along with the parameter estimation difficulties encountered in the modeling process as well as the disaggregation strategies used to prepare synthetic monthly NBS from the generated annual NBS series. The first section of this chapter describes the models and modeling approach followed at the annual time step; section 2 summarizes the annual – monthly disaggregation approach; section 3 enumerates the probability distributions used to describe the annual and monthly NBS.
3.2 Contemporaneous Shifting Mean – CARMA (p,q) Models
Section 3.2.1 resumes the univariate formulation of the SM model, for modeling and simulating individual series of annual NBS. The Contemporaneous Shifting Mean-CARMA (p,q) model used to model and simulate the annual NBS is jointly described in section 3.2.2 and the annual to seasonal disaggregation approach is described in section 3.2.3. More detailed mathematical formulation is available in the references.
3.2.1 Univariate Shifting Mean Models
The Shifting Mean (SM) model is characterized by sudden shifts or jumps in the mean (Fagherazzi et al., 2005; Sveinsson et al., 2002, 2005). That is, the underlying process is characterized by multiple stationary states only differing from each other by having
52
different means that vary around the long term mean of the process. The process is auto-correlated, and the autocorrelation is assumed to arise from the sudden shifting pattern in the mean. A general definition of the SM model is given by Sveinsson et al., (2002, 2007). A general definition of the univariate shifting mean model is given by
ttt ZYX (3.1)
where { tX } is a sequence of variables representing the process under study; { tY } is a
stationary process with mean Y , variance 2
Y and autocorrelation function ACF Y ( ).
The ACF Y ( ) is zero at all lags in the SM model and Y (h) = h at all lags h=0,1, … for
the SM model with persistence in { tY }. tZ 's represent noise in the mean of the process
tX and is a sequence with mean zero and variance 2
Z . This noise is characterized by
levels with fixed values during each stationary state, that is 1Z = … = 1NZ ; 11NZ = … =
21 NNZ ; where iN is the span or the length of the noise i for i= 1,2,… . The noise level {
tZ } can be written as
)(],(
11
tIMZii SS
t
i
it
(3.2)
where iS = 1N + 2N + … + iN with 0S = 0, and ),( baI (t) is the indicator function equal to
one if t (a,b) and zero otherwise. 1N , 2N , … are positive geometric random variables
with parameter p, 0<p<1, such that pNE /1 . Thus the average length of each state
of the process is the inverse of the parameter of the positive Geometric distribution or
1/p.{ kM } is iid N(0, Var(M = Var(Z)). The sequences { iN }, { kM } and { tY } are assumed
to be mutually independent. A schematic representation of the different processes composing the Shifting Mean Model used for the Study is illustrated in figure 31. The model assumes that the observations are normally distributed with a constant variance. The mean, however, only remains constant during a particular state or shift with length distributed according to a geometric distribution. In the SM model formulation analyzed here, the sign of the noise level is random, that is
kM can take both positive and negative values. It is possible to build another SM model
where two consecutive shifts will always have opposite signs (Sveinsson et al., 2002).
53
Figure 31 - A schematic representation of the different processes composing the Shifting Mean Model (from Sveinsson et al., 2007)
It can be shown that the mean, the variance and the serial correlation of the process tX
are:
YtXE (3.3)
22
MYtXVar (3.4) h
MY
MX
ph
22
2 )1()(
, h = 1,2,… (3.5)
For the regular SM model described here, four parameters pMYY ,,, , the mean
and the standard deviation of the stationary process { tY }, the standard deviation of the
noise { kM } characterized by levels and the parameter p of the geometric distribution
need to be estimated.
The parameter estimates at each site in terms of the observed statistics ˆX , ˆ X , ˆ (1)X
and ˆ (2)X are given by:
54
ˆ (2)ˆ 1
ˆ (1)
X
X
p
(3.6)
ˆ ˆ
Y X (3.7)
222
MXY
(3.8)
2
2 2 ˆ (1)ˆ ˆ
ˆ (2)
XM X
X
(3.9)
If p is known, the last equation becomes:
2 2 ˆ (1)ˆ ˆ
(1 )
XM X
p
(3.10)
The estimation procedures can result in infeasible parameter estimates. The cause could be sample variability of the observed ACF. Sveinsson et al, 2002, proposed the use of a
smoothed ACF with a function h
X abh )( having the same form as the model ACF
described in the precedent paragraph for 0<b<1, 0<a<1 (Eq. 3.5).
3.2.2 Fitting Multivariate CSM-CARMA models to the Great Lakes
Figures 2 to 6 in Chapter 1 show the time series pattern (graph "a") and the sample ACF (graph "c") for each of the Great Lakes. Shifts in the mean can be observed in the series of annual NBS and the sample ACF for Lakes Erie, Ontario and St. Clair exhibit some long term persistence. Therefore, for these series, it may be appropriate to use a SM model. Lakes Superior and Michigan-Huron ACF do not have the shapes that are expected for a SM model. There is no evidence of persistence or shifting mean behaviour that could be observed in their time series pattern. As a result, mixing the Shifting Mean model with another time series model like the ARMA (p,q) model may be appropriate for the Lakes Superior and Michigan-Huron ACFs. To avoid the parameter estimation problems usually encountered when modeling the full dependence structure in space and time at multiple lags, contemporaneous versions of the full multivariate ARMA (p,q) models were developed. In the contemporaneous ARMA (p,q) model or CARMA (p,q), the AR and MA parameter matrixes are assumed to be diagonal, and their parameters can be estimated independently for each site. The equations needed to estimate the variance-covariance matrix of the noise is summarized in Sveinsson et al., 2007, section A.3.2. The CARMA model is capable of
55
preserving the lag-zero cross correlation between sites and the time dependence structure for each site. Similarly, the contemporaneous SM model or CSM is obtained by decoupling the multivariate SM model into univariate SM models and estimating the parameters at each site with the procedures described section 3.2.1. As mentioned in Sveinsson et al.,
2002, 2007, the decoupling is based on the hypothesis that 1N + 2N + … + iN is a
common sequence for all sites and this common parameter p could be obtained for all modeled sites. If the common p is not known, a p(i) parameter could be estimated at each site (i) and then a common p could be estimated as a weighted average of the estimated p(i) . The shifts observed in the means of Lakes St. Clair, Erie and Ontario annual NBS are thought to be mostly induced by changes in climate and they do seem to appear at the same time. As such, it would make sense to assume that in this particular geographic region, a common p could be found reflecting the common shifting pattern. A CSM-CARMA (p,q) model representing a mixture of one Contemporaneous Shifting Mean (CSM) model and one CARMA (p,q) was developed for the Great Lakes system. In the proposed multivariate model, Lakes Superior and Michigan-Huron were fitted by a CAR (2) model and the other three lakes (St. Clair, Erie and Ontario) were fitted by a CSM model. The model preserves the lag zero cross-correlation in space. For the CSM part of the model, a detailed description of the parameter estimation procedure can be found in Fagherazzi et al., 2005.
3.2.3 Annual-seasonal disaggregation approach
The Grygier and Stedinger disaggregation model (Grygier and Stedinger, 1990) is a contemporaneous model
,1,,, ΛDYCεBYAY (3.11)
where A , B , C , and D are diagonal N N parameter matrix (i.e.
contemporaneous), Y is the N 1 column vector of observations in year at the N
key sites, ,Y is the corresponding N 1 column vector of observations in the same
year season , and 1, Y is the previous season N 1 column vector. Grygier and
Stedinger's model relate each seasonal flow with the yearly flow explicitly preserving
the correlations of the annual data with same year seasonal data through the matrix A for each season. The lag 1 season to season correlation is preserved through the matrix
56
C for each season. The last term of the equation represents the sum of previously generated flows and is included to preserve the additivity property, i.e. monthly flows summing to annual flows. Since the parameter matrices in the Grygier and Stedinger's model are diagonal, cross-correlations are preserved only within each site. In addition, the model does not preserve the lag 1 correlation between the first season of a given year and the last season of the previous year.
3.3 Transformations
Most stochastic flow generating models work with normally distributed random variables. Because the models are linear and additive, they preserve normality. Unfortunately, annual NBS series are not always adequately described by the normal distribution. Seasonal NBS distributions are positively skewed and could have a negative bound other than zero because of lake evaporation. The most frequent approach to generate flows (or NBS) which have non-normal distributions is to transform the observed flows, select the adequate stochastic model using transformed data, generate normal random variables and then transform these variables back to flows (or NBS) with the desired marginal distribution. Specialized software like SAMS (CSU), SPIGOT (Cornell), PARADE (INRS-EAU and Hydro Quebec) include many options for doing transformations. These transformations include exponential, two and three parameters log normal, Pearson Type III (Gamma three parameter) and Box Cox distributions. The observed data at different time scales are then transformed into their normal equivalents. The shorter the time scale (i.e. weekly and quarter monthly) the harder it is to select the most adequate transformation. In fact, the choice of the right transformation for the critical month or quarter month (start of the snow melt, summer floods) is as important for the final results as the choice of the adequate statistical law for the classic flood frequency analysis. A detailed description of the employed transformation procedures (distributions, used tests for verifying the adequacy of a selected transformation, etc) can be found in Fagherazzi et al, 2005 and Sveinsson et al, 2007.
3.4 Upgrading the parameters of the stochastic model due to changes in the Great Lakes historical NBS data set
As mentioned before, a revised Great Lakes Residual Net Basin Supplies data base (1900-2008) was available in June 2010. The revised data set considers new flow estimates of the St. Clair, Detroit and Niagara rivers. Changes to the St. Clair River and
57
Detroit River flows go back to 1987 while the Niagara River flows were adjusted back to 1961. The upgrades of the parameters of the CSM-CARMA model used for the ILOSLRS (Fagherazzi et al, 2005) include:
A review of the transformations (probability distributions and parameters) for the annual and monthly NBS series;
Estimated new parameters for the CSM-CARMA model; and
Estimated new parameters for the Grygier and Stedinger disaggregation model (annual to monthly).
58
CHAPTER 4 Testing the selected Great Lakes CSM-CARMA Model
As explained in Chapter 2, the selected model must be tested to determine whether the model is able of reproducing the selected statistical properties of the historical series. For comparing statistical properties, a large number of realizations having the same length as the historical record (109 years) was generated with the fitted model. Each one of the statistics of interest was then estimated from each realization and their averages were compared with the corresponding historical sample statistics. To give a good idea of the distribution of the statistic of interest and preparing frequency plots, the number of realizations was set to 500, however, an extra ten realizations were prepared for a monthly-quarter monthly disaggregation "warm-up" procedure used later, for a total of 510 realizations. A set of figures summarizing the statistics of the Great Lakes generated annual and monthly NBS series versus the statistics of the historical NBS series is presented in the following sections of this report. To complete the set of required NBS characteristics, statistics on the annual and monthly series of routed lake levels and outflows were also prepared and presented in Chapter 5.
4.1 Statistics of the historical and generated annual NBS
Tables 6 to 10 show descriptive statistics of the historical series of annual NBS and the average statistics obtained from the 510 realizations (mean E, variance S, maximum and minimum value, median, skewness and autocorrelation coefficient lag-1). The comparison of the average statistics with their corresponding historical sample statistics shows that the descriptive statistics are very well preserved for all the lakes except a small underestimation of the Lake St. Clair standard deviation. The storage related statistics (Hurst, RAR) and both related and run (RL and RS) statistics for drought and surplus were generally well preserved considering that these characteristics were not used in the fitting of the stochastic model. As explained in section 3.2.1 describing univariate SM models, the average length of each state of the process (deficit or surplus) is the inverse of the parameter (local p) of the positive Geometric distribution or 1/p. Consequently, if the magnitude of the historical deficit and surplus run statistics are very different, one set could be better preserved than the other. Tables 7, 9 and 10 show that while the maximum deficit Run Length and Run Sum were underestimated for lakes Michigan-Huron and Ontario; these were well preserved for Lake
59
Erie. On the other hand, the maximum surplus Run Length and Run Sum were well preserved for Lakes Michigan-Huron and Ontario, but underestimated for Lake Erie. The storage and the drought and surplus related statistics are dependent on the sample size (Sveinsson et al., 2006). For further comparison, the maximum Run Length and Run Sum statistics were estimated from a single sample of 55,590 years of data (510 realizations *109 years). The obtained values are shown on the fifth column of Tables 7, 9 and 10 and are bigger as expected than those obtained from the short samples.
Table 6 – Average sample statistics of 510 generated time series versus historical statistics – Characteristics of Annual NBS - Lake Superior
60
Table 7 - Average sample statistics of 510 generated time series versus historical statistics – Characteristics of Annual NBS - Lake Michigan-Huron
61
Table 8 - Average sample statistics of 510 generated time series versus historical statistics – Characteristics of Annual NBS - Lake St. Clair
62
Table 9 - Average sample statistics of 510 generated time series versus historical statistics – Characteristics of Annual NBS - Lake Erie
63
Table 10 - Average sample statistics of 510 generated time series versus historical statistics – Characteristics of Annual NBS - Lake Ontario
64
The left panel of figures 32 to 35 show Empirical Frequency Plots describing 510 superposed realizations of annual NBS series for Lakes Superior, Michigan-Huron, Erie and Ontario (averaged from monthly generated data) versus the historical annual NBS series shown in red characters (averaged from historical monthly data). Each sample contains 109 years representing the historical NBS series. The probability of non-exceedance pi is calculated with the Weibull’s plotting position equations. Lake St. Clair’s Empirical Frequency Plots were not shown to limit the number of presented figures. The return period T is the average of inter-event times between flood events. If the observations are independent, the cumulative probability of non-exceedance F(QT), and the return period T are related by the following relationship:
F(QT) = P(QT q) = 1 - P(QT>q) = 1 – 1/T
Assuming independence, this relationship enables the ability to estimate the magnitude of an event QT given its return period T. Even though the Great Lakes annual NBS are not independent, the participants of the study were comfortable with the return period concept and requested graphics showing the relation log (1/1-p). As a result, the graph on the right side shows the sorted historical annual NBS and generated NBS versus the log (1/1-p) giving an enhanced view of the high tail. Both graphs in figures 34 to 37 show good correlation with the historical series and a lot of variability of the samples around the observed series. The Frequency Plots for Lake Erie shows the biggest variability around the historical series and "wet" and "dry" sequences could be easily identified. Figures 36 to 39 show Empirical Frequency Plots of the annual historical NBS series and one synthetic series of 55,590 annual NBS obtained by reorganising the 510 samples for 109 years. The graph on the left shows the high tail of the empirical distribution in a logarithmic scale whereas the graph on the right illustrates the lower tail. The generated values seem to adequately extrapolate the annual historical series.
65
Figure 32 – Empirical Frequency Plots of 510 realizations of synthetic annual NBS - Lake Superior
Each realization has the same length as the historical record shown in red (109 years)
66
Figure 33 – Empirical Frequency Plots of 510 realizations of synthetic annual NBS - Lake Michigan-Huron
Each realization has the same length as the historical record shown in red (109 years)
67
Figure 34 - Empirical Frequency Plots of 510 realizations of synthetic annual NBS - Lake Erie
Each realization has the same length as the historical record shown in red (109 years)
68
Figure 35 - Empirical Frequency Plots of 510 realizations of synthetic annual NBS - Lake Ontario
Each realization has the same length as the historical record shown in red (109 years)
69
Figure 36 - Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 - Lake Superior annual NBS
70
Figure 37 –Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 - Lake Michigan-Huron annual NBS
71
Figure 38 – Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 - Lake Erie annual NBS
72
Figure 39 – Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 - Lake Ontario annual NBS
73
Figures 41 to 43 are Empirical Frequency Plots that illustrate the maximum and minimum monthly NBS for a particular year. The left panel shows the sorted maximum monthly NBS for the historical NBS (red characters) and the 510 samples of 109 years. The right panel illustrates the sorted minimum monthly NBS for the historical NBS and the 510 samples of 109 years. Low frequency monthly maximums and minimums are well reproduced for Lakes Superior, Michigan Huron and Ontario. The lower extreme tail of the generated samples for Lake Erie shows somewhat higher values when compared with the historical minimum monthly NBS. As explained in Section 3.3, the extreme values result from the distribution selected when transforming the flows. Figure 44 shows the frequency distribution plot of the original and the transformed data using the selected transformation for month 10 on the normal probability paper. The maximum (and minimum) historical monthly NBS seems underestimated (or overestimated) for the same non-exceedance probability by the selected transformation. The consequence of using the selected transformation is that the distribution of extremes for the synthetic NBS may not match the historical NBS. Despite this fact, the overall fit shown in figure 41 was considered good.
74
Figure 40 – Empirical Frequency Plots of annual maximum and minimum monthly NBS - Lake Superior
75
Figure 41– Empirical Frequency Plots of annual maximum and minimum monthly NBS– Lake Michigan-Huron
76
Figure 42 - Empirical Frequency Plots of annual maximum and minimum monthly NBS - Lake Erie
77
Figure 43 - Empirical Frequency Plots of annual maximum and minimum monthly NBS– Lake Ontario
78
Figure 44 - Selected transformation for month 10 (October), Lake Erie
79
4.2 Monthly Statistics
Figures 45 to 48 illustrate superposed observed (right panel) and synthetic (left panel) monthly hydrographs. Only 48 synthetic hydrographs are shown on the right panel. For each month, the indices of the maximum and minimum generated NBS were identified and the corresponding hydrographs were selected and plotted forming a "monthly envelope of synthetic hydrographs" that could be compared with the historical ones. Figures 50 to 53 show Box-Plots of periodic month to month correlations (lag-1 and lag-2), obtained from the 510 realizations. The star symbol indicates the values obtained from the historical NBS. The lag-1 month-to month correlation is very well reproduced for all the lakes. The Lakes Erie and Ontario lag-1 month to month correlation for the month of January is zero because the annual-monthly disaggregation model does not preserve the correlation between the first month of one year and the last month of the previous year. The lag-2 month to month correlations for Lakes Erie and Ontario are also well reproduced.
80
Figure 45 – Superposed observed and generated monthly hydrographs (m³/s) – Lake Superior
81
Figure 46 – Superposed observed and generated monthly hydrographs (m³/s) – Lake Michigan-Huron
82
Figure 47 – Superposed observed and generated monthly hydrographs (m³/s) – Lake St. Clair
83
Figure 48 – Superposed observed and generated monthly hydrographs (m³/s) – Lake Erie
84
Figure 49 – Superposed observed and generated monthly hydrographs (m³/s) – Lake Ontario
85
Figure 50 – Box-Plots of the Lag-1 and Lag-2 Month to Month Correlation obtained from 510 samples – Star: historical value
Lake Superior
86
Figure 51– Box-Plots of the Lag-1 and Lag-2 Month to Month Correlation obtained from 510 samples – Star: historical value
Lake Michigan-Huron
87
Figure 52 – Box-Plots of the Lag-1 and Lag-2 Month to Month Correlation obtained from 510 samples – Star: historical value
Lake Erie
88
Figure 53 – Box-Plots of the Lag-1 and Lag-2 Month to Month Correlation obtained from 510 samples – Star: historical value
Lake Ontario
4.3 Spatial Characteristics
Box-Plot illustrating the Great Lakes annual contemporaneous cross-correlation statistics calculated from the 510 NBS realizations are shown on figures 54 to 58. The star symbol indicates the values obtained from the historical NBS.
89
Figure 54 – Box-Plot illustrating Lake Superior annual contemporaneous cross-correlation statistics
90
Figure 55 – Box-Plot illustrating Lake Michigan-Huron annual contemporaneous cross-correlation statistics
91
Figure 56 – Box-Plot illustrating Lake St. Clair annual contemporaneous cross-correlation statistics
92
Figure 57 – Box-Plot illustrating Lake Erie annual contemporaneous cross-correlation statistics
93
Figure 58 – Box-Plot illustrating Lake Ontario annual contemporaneous cross-correlation statistics
94
CHAPTER 5 Connecting Channel flow routed levels and outflows statistics
5.1 Introduction
This section analyzes the statistics of the Great Lakes annual levels and outflows series prepared by Dr. Yin Fan and David Fay using the Coordinated Great Lakes Regulation and Routing Model of Environment Canada, Cornwall Section. (Fan Y. et al., 2005). The model inputs are the synthetic monthly NBS series for Lakes Superior, Michigan-Huron and Erie. The outputs are the monthly series of outflows for the interconnected channels and levels for Lakes Superior, Michigan-Huron and Erie. Table 11 summarizes statistics of historical annual NBS and simulated outflows at different intermediate points of the Great Lakes system. They were calculated using the historical NBS series of 109 years. This set of values constitutes the Base Case or reference basis for comparison analysis between the simulated outflows using historical NBS series and routed outflows from the generated NBS series. When simulating a 109 year period, the ending lake levels are different from the initial levels, thus there are changes in the volume of water in storage in each lake. Column 4 of Table 11 shows these changes in storage for each lake. When considering the ‘mass balance’ over the 109 year simulation period, these changes in storage have to be taken into account. Comparing column 5 and column 6, we see that the outflows are very close to the total NBS (inflows + net diversions) plus the changes in storage for each lake. The remaining differences are attributed to rounding error within the Coordinated Great Lakes Regulation and Routing Model.
Table 11 – Upper Lakes statistics of historical annual NBS and corresponding simulated outflows using the Coordinated Great Lakes Regulation and Routing Model
Average NBS
Diversion
Change in Storage over 109 years
NBS+Diversion+ Change in storage
Routed outflows (with coordinated model)
(m3/s) (m3/s) (m3/s) (m3/s) (m3/s)
Lake Superior 2013 141 5.76 2159.76 2164
Lake Michigan-Huron 3198 -91 13.32
St. Clair River 5280.08 5281
Lake St. Clair 132 0
Detroit River 5412.08 5413
Lake Erie 623 -0.6 6034.48 6034
TOTAL 5966 50 18.48 6034.48
95
The synthetic set of 55,590 monthly NBS series for Lakes Superior, Michigan Huron, St. Clair and Erie was routed with the coordinated model. The simulated lake levels and outflows series were then organised in 510 series of 109 years and annual average statistics were prepared. Table 12 shows annual average statistics of synthetic NBS and the corresponding simulated outflows at the same intermediate points of the Great Lakes system. When simulating a longer series, the effect of changes in storage is very small, as can be seen in Table 12 and the simulated annual average outflow of Lake Erie corresponds exactly to the sum of synthetic upper Lakes NBS plus diversions.
Table 12 – Upper Lakes annual statistics of synthetic NBS and corresponding simulated outflows using the Coordinated Great Lakes Regulation and Routing Model
Average NBS
Diversion
Change in Storage over 55,590 years
NBS + Diversion + Change in storage
Routed outflows (with coordinated model)
(m3/s) (m3/s) (m3/s) (m3/s) (m3/s)
Lake Superior 2014 141 -0.01 2154.99 2160
Lake Michigan-Huron 3197 -91 -0.03
St. Clair River 5260.96 5262
Lake St. Clair 133 0
Detroit River 5393.96 5395
Lake Erie 625 -0.01 6018.95 6017
TOTAL 5969 50 -0.05 6018.95
5.2 Characteristics of the series of Annual Outflows
Figures 59 to 62 illustrate the characteristics of the annual historical outflow series of the interconnecting channels. The top panel shows the time plot of annual NBS outflows along with their corresponding mean while the bottom panel illustrates the serial correlations or ACF up to lag 15. The visual analysis of the annual outflows for the St. Clair River, the Detroit River and Lake Erie (Figures 60 to 62) suggest the presence of local non-stationarity. As for the NBS series, dry and wet sequences seem to have occurred simultaneously on all three channels, like the dry spells of 1930-1940 and 1960-1970 and the wet sequence of 1970-2000. As expected, the outflows are highly correlated for many lags as the correlations decay slowly as the time lag increases. On the other hand, the data for the St. Marys River shown in figure 59 do not seem to exhibit any shift. Lag-1 and lag-2 correlation coefficients are significant at a 5% level of significance, but they decrease to zero quickly as the lag increases.
96
A complete trend and shift detection analysis could be found in Ouarda et al., 2010. Figure 59 – Characteristics of the Annual outflow series (BOC) – St. Marys River
97
Figure 60 – Characteristics of the Annual outflow series (BOC) – St. Clair River
98
Figure 61 – Characteristics of the Annual outflow series (BOC) - Detroit River
99
Figure 62 – Characteristics of the Annual outflow series (BOC) – Lake Erie outflows
100
Tables 13 to 16 compare annual statistics for the observed and synthetic routed outflows. The annual characteristics (descriptive statistics and run statistics) are very well reproduced. The standard deviation for the St. Clair River, the Detroit River and Lake Erie seems slightly underestimated; however, their deficits and surplus related statistics calculated from synthetic outflows are of the same magnitude. The reasons were explained in Chapter 4. The surplus related statistics appear consequently underestimated for the Detroit River and Lake Erie outflows. Nevertheless, the maximum RL and RS calculated from the overall sample of 55,590 years are higher than the historical ones.
Table 13 - Average sample statistics of 510 routed outflow series versus historical statistics – Characteristics of Annual outflows series – St. Marys River
101
Table 14 - Average sample statistics of 510 routed outflow series versus historical statistics – Characteristics of Annual outflows series – St. Clair River
102
Table 15 - Average sample statistics of 510 routed outflow series versus historical statistics – Characteristics of Annual outflows series – Detroit River
103
Table 16 - Average sample statistics of 510 routed outflow series versus historical statistics – Characteristics of Annual outflows series – Lake Erie outflows
104
Figures 63 and 64 show the Empirical Frequency Plots of 510 realizations of synthetic annual outflows for the St. Marys, St. Clair and Detroit rivers and Lake Erie outflows. As with previous figures, the red curve illustrates the historical annual outflows. All frequencies are very well reproduced. Figures 65 to 68 shows the same data, but reorganised into one sample of 55,590 years. The right panel shows sorted averaged annual values versus 1/(1-p) using a log scale (high tail). The left panel shows an enhanced view of the lower tail and the historical series versus 1/p. The synthetic series seem to extrapolate adequately the annual historical outflows series.
105
Figure 63- Empirical Frequency Plots of 510 realizations of synthetic annual outflows – St. Marys and St. Clair Rivers
Each realization has the same length as the historical record shown in red (109 years)
106
Figure 64- Empirical Frequency Plots of 510 realizations of synthetic annual outflows – Detroit River and Lake Erie outflows
Each realization has the same length as the historical record shown in red (109 years)
107
Figure 65 - Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 – St. Marys River outflows
108
Figure 66 - Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 – St. Clair River outflows
109
Figure 67 - Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 – Detroit River outflows
110
Figure 68 - Empirical Frequency Plots obtained from the historical series (shown in red) and one synthetic series of length 510*109 – Lake Erie outflows
111
Figures 69 and 70 display the Empirical Frequency Plots of 510 realizations of annual maximum outflows and historical maximum outflows for the St. Marys, St. Clair and Detroit Rivers and Lake Erie outflows. Again, the synthetic values seem to adequately extrapolate the annual historical series.
Figures 71 and 72 illustrate superposed observed (left panel) and synthetic (right panel) monthly outflow hydrographs. When plotting the generated outflow hydrographs, only the extreme generated hydrographs were presented.
112
Figure 69 - Empirical Frequency Plots of 510 realizations of annual maximum outflow and historical maximum outflow – St. Marys and St. Clair Rivers
113
Figure 70 - Empirical Frequency Plots of 510 realizations of annual maximum outflow and historical maximum outflow – Detroit River and Lake Erie outflows
114
Figure 71 – Superposed observed and synthetic monthly outflows (m³/s) – St. Marys and St. Clair Rivers
115
Figure 72 – Superposed observed and synthetic monthly outflows (m³/s) – Detroit River and Lake Erie outflows
116
5.3 Characteristics of the series of Annual Lake levels
Figures 73 to 75 illustrate the characteristics of the annual historical lake level series of Lakes Superior, Michigan Huron and Erie. The top panel shows the time plot of annual NBS levels along with their corresponding mean while the bottom panel illustrates the serial correlations or ACF up to lag 15. As for the outflows series, dry and wet sequences seem to be present simultaneously on Lakes Michigan-Huron and Erie, such as the dry spells of 1930-1940 and 1960-1970 and the wet sequence of 1970-2000. As expected, the lake levels are highly correlated for many lags as the correlations decay slowly as the time lag increases. On the other hand, the data for Lake Superior shown in figure 73 do not seem to exhibit any shift. Lag-1 to lag-3 correlation coefficients is significant at a 5% level of significance, but they decrease to zero quickly as the lag increases. Tables 13 to 16 compare annual statistics for the observed and synthetic routed lake levels. The annual characteristics (descriptive statistics) are very well reproduced. The runs statistics are fairly well reproduced. Figures 76 and 77 present the Empirical Frequency Plots of 510 realizations of synthetic annual levels from Lakes Superior, Michigan-Huron and Erie. As with previous figures, the red curve illustrates the observed annual levels. All frequencies are well reproduced. Figures 78 to 80 shows the same data, but reorganised into one sample of 55,590 years. The right panel shows sorted averaged annual values versus {1/(1-p)} using a log scale (high tail). The left panel shows an enhanced view of the lower tail and the historical series versus 1/p. The synthetic series seems to adequately extrapolate the annual historical lake levels series.
Figures 81 and 82 illustrate superposed observed (left panel) and generated (right panel) monthly levels hydrographs. When plotting the generated level hydrographs, only the extreme generated hydrographs were presented.
117
Figure 73 – Lake Superior routed levels annual characteristics (BOC)
118
Figure 74 – Lake Michigan-Huron routed levels annual characteristics (BOC)
119
Figure 75 – Lake Erie routed levels annual characteristics (BOC)
120
Table 17 - Average sample statistics of 510 lake levels series versus historical statistics Characteristics of Annual lake levels series – Lake Superior
121
Table 18 - Average sample statistics of 510 lake levels series versus historical statistics Characteristics of Annual lake levels series – Lake Michigan-Huron
122
Table 19 - Average sample statistics of 510 lake levels series versus historical statistics Characteristics of Annual lake levels series – Lake Erie
123
Figure 76 - Empirical Frequency Plots of 510 samples of annual synthetic levels vs. historical levels –
Lakes Superior and Michigan-Huron
124
Figure 77 - Empirical Frequency Plots of 510 samples of 109 years routed annual generated levels vs. historical levels– Lake Erie
125
Figure 78 - Empirical Frequency Plots of 1 sample of 55,590 years of annual routed levels- Lake Superior
126
Figure 79- Empirical Frequency Plots of 1 sample of 55,590 years of annual routed levels - Lake Michigan-Huron
127
Figure 80 - Empirical Frequency Plots of 1 sample of 55,590 years of annual routed levels - Lake Erie
128
Figure 81– Superposed observed and synthetic monthly lake levels (m) – Lakes Superior and Michigan-Huron
129
Figure 82 - Superposed observed and synthetic monthly lake levels (m) – Lakes St. Clair and Erie
130
Lake Erie simulated outflows and Lake Ontario NBS were then routed using the Lake Ontario pre-project outlet conditions (Caldwell and Fay, 2002). The procedure requires quarter-monthly NBS data for Lakes Erie and Ontario. The disaggregation approach applied to Lakes Erie and Ontario NBS for disaggregating monthly NBS to quarter-monthly NBS will be treated in another rapport. Nevertheless, the mentioned activities had to be done in order to assess the statistics of Lake Ontario outflows and levels.
131
CHAPTER 6 Drought and surplus related statistics on lake levels
6.1 Introduction
The implementation of exceptional measures and/or restrictions on water resources management plans is strongly related to the severity of a sequence of water supplies, surplus or deficits that could arrive to the system. In addition, the consequences of extended dry or wet Great Lakes NBS sequences and corresponding lake levels and outflows could vary according to the characteristics of the user’s need. In this context, the analysis of lake levels surplus and deficits related statistics could be useful to identify “challenging level subsamples” that could be helpful for assessing the long term global performance of the system. Stochastic analysis has been used in the past to prepare synthetic surplus and drought sequences that could be used for long term water resources planning. Although the notion of severity was often the main concern, the characterization of extreme events in terms of frequency and risk could also be analyzed (Salas et al., 2005). The following section analyzes some surplus and deficits sequences identified from the stochastic series of Lakes Superior and Erie routed lake levels described in Chapter 5. All maximum deficit and surplus related statistics were estimated from a single sample of 55,590 years of data. Planners and managers showed special interest on the Run Intensity RI characteristics of deficits.
6.2 Characterizing Lake Levels
Figure 83 shows run intensity RI characteristics of the sequences of deficits identified from the series of historical annual levels of Lakes Superior and Erie. It shows that Lake Superior RI characteristics are less important in magnitude than Lake Erie RI. Figures 84 and 85 illustrate run intensity RI characteristics for all deficits sequences identified from the synthetic series of annual levels for Lakes Superior and Erie. The order of magnitude between the deficit RI characteristics of both lakes seems well preserved. Figure 86 illustrates Box-Plots of Lake Superior stochastic monthly levels with superposed monthly levels sequences of four consecutive years of high and low levels, from the sequences of maximum RI. The figure shows that the monthly lake levels distributions are symmetric and illustrate the relative position of the monthly values of a sample of many consecutive months of high or low levels against the corresponding monthly marginal distribution.
132
Figure 86 also illustrates the relative position of the observed maximum RI sequences. The monthly levels for the period 1999-2008 corresponding to the second highest observed run deficit sequence (RI) as well as the surplus period in 1969-1977 (maximum surplus RI) were superposed in black. Figure 87 illustrates Box-Plots of Lake Erie stochastic monthly levels with superposed monthly levels sequences of six consecutive years of high and low levels from the sequences of maximum RI. The figure also illustrates the relative position of the monthly values of the period 1933-1945 corresponding to the maximum observed deficit RI whereas the monthly values of the surplus period 1969-1998 were plotted in magenta.
Figure 83
Deficit Run Intensity characteristics from the historical sample of 109 annual levels -
Lakes Superior and Erie
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0 2 4 6 8 10 12 14 16Sequence ID
RI
RL/RS
Superior
RL/RS
Erie
133
Figure 84
Figure 85
Deficits Run Intensity characteristics from the synthetic sample of
55590 annual lake levels - Lake Superior
0,000
0,050
0,100
0,150
0,200
0,250
0,300
0,350
0 1000 2000 3000 4000 5000 6000
Sequence ID
RI
RI
Deficits Run Intensity characteristics from the synthetic sample of
55590 annual lake levels- Lake Erie
0,000
0,100
0,200
0,300
0,400
0,500
0,600
0 1000 2000 3000 4000 5000
Sequence ID
RI
RI
134
Figure 86 – Box-Plots of stochastic monthly levels, with superposed monthly levels sequences (synthetic and observed) of maximum RI (surplus and deficits) Lake Superior
Figure 87 – Box-Plots of stochastic monthly levels, with superposed monthly levels sequences (synthetic and observed) of maximum RI (surplus and deficits) Lake Erie
1 2 3 4 5 6 7 8 9 10 11 12
182.6
182.8
183
183.2
183.4
183.6
183.8
184
184.2
184.4
Leve
ls (
m)
Month
Lake Superior - Monthly Levels Distributions
1 2 3 4 5 6 7 8 9 10 11 12
173
173.5
174
174.5
175
175.5
Levels
(m
)
Month
Lake Erie - Monthly Levels Distributions
135
Figures 88 and 89 illustrate the sequences of maximum deficit and surplus RL identified from the Lake Erie synthetic series of annual levels. The figures also show the concomitant levels on Lakes Superior and Michigan-Huron. It can be seen that Lake Erie is strongly influenced by the hydrological characteristics of the lakes located upstream, as well as the local NBS characteristics. Figures 90 and 91 illustrate concomitant levels on the three lakes for the wettest and driest RI sequence of Lake Erie levels. Figure 88 - Sequence of maximum deficit RL identified from the Lake Erie synthetic series of annual levels with concomitant levels on Lakes Superior and Michigan-Huron
0 200 400 600 800 1000 1200182
183
184
Month
Superior
(m)
Simulated levels for a dry sequence of 86 years (6954 à 7041)
0 200 400 600 800 1000 1200175
176
177
Month
Mic
hig
an-H
uro
n
(m)
0 200 400 600 800 1000 1200172
174
176
Month
Erie (
m)
136
Figure 89- Sequence of maximum surplus RL identified from Lake Erie synthetic series of annual levels with concomitant levels on Lakes Superior and Michigan-Huron
0 100 200 300 400 500 600 700 800 900183
183.5
184
Month
Superior
(m)
Simulated levels for a wet sequence of 75 years (10967 à 12041)
0 100 200 300 400 500 600 700 800 900176
177
178
Month
Mic
hig
an-H
uro
n
(m)
0 100 200 300 400 500 600 700 800 900172
174
176
Month
Erie (
m)
137
Figure 90 - Sequence of maximum deficit RI identified from Lake Erie synthetic series of annual levels with concomitant levels on Lakes Superior and Michigan-Huron
0 20 40 60 80 100 120 140 160 180182
183
184
Month
Superior
(m)
Simulated levels for a dryest sequence (RI) of 15 years (23594 à 23609)
0 20 40 60 80 100 120 140 160 180174
176
178
Month
Mic
hig
an-H
uro
n
(m)
0 20 40 60 80 100 120 140 160 180172
174
176
Month
Erie (
m)
138
Figure 91 - Sequences of maximum surplus RI identified from Lake Erie synthetic series of annual levels with concomitant levels on Lakes Superior and Michigan-Huron
0 50 100 150 200 250 300183
183.5
184
Month
Superior
(m)
Simulated levels for a wet sequence of 21 years (19290 à 19310)
0 50 100 150 200 250 300176
177
178
Month
Mic
hig
an-H
uro
n
(m)
0 50 100 150 200 250 300172
174
176
Month
Erie (
m)
139
CHAPTER 7 Concluding remarks
As mentioned in the Introduction, the Great Lakes – St. Lawrence System is a vast and varying hydrological system whose stochastic spatial and temporal characteristics present particular properties. A characteristic that had received a considerable amount of attention in literature is the sudden shift pattern that is apparent in some but not all of the annual NBS series. The sudden changes in the annual mean of some of the NBS series had been successfully modeled by a mix of a Contemporaneous Shifting Mean CSM model and CARMA(p,q) model developed by Dr. Sveinsson and Dr. Salas of the Lake Ontario – St. Lawrence River Study Group (Sveinsson et al., 2002, 2006). Adequate software was added to the SAMS package (Sveinsson et al., 2007). The annual NBS series were disaggregated in monthly NBS series; however, choosing a multivariate temporal disaggregation strategy presented other challenges. Some strategies were tried and analyzed before finding the adequate compromise between the preservation of required explicit and implicit statistics and the number of parameters. The Grygier and Stedinger contemporaneous disaggregation model was finally chosen. The NBS series were routed with the Coordinated Great Lakes Regulation and Routing Model of Environment Canada, Cornwall Section. (Fan Y. et al., 2005). The model inputs are the synthetic monthly NBS series for Lakes Superior, Michigan-Huron and Erie. The outputs are the monthly series of outflows for the interconnected channels and lake levels for Lakes Superior, Michigan-Huron and Erie. This report described the temporal and spatial characteristics of the revised Great Lakes Residual NBS data base (1900-2008) that became available in June 2010. The CSM-CARMA model parameters were revised accordingly and a newly generated data set including all the lakes was prepared for the Upper Lakes Study Group. A complete set of figures showing the generated NBS sample’s statistics and the corresponding routed levels and outflows versus the observed characteristics was presented in the previous chapters. The synthetic NBS series and the routed lake levels and outflows series very well reproduced the characteristics of the corresponding historical series.
140
References
1. Boes, D. and Salas, J., Non-stationarity of the mean and the Hurst phenomenon, Water Resources Research, 14(1):135-143, 1978.
2. Caldwell, R. and D. Fay, Lake Ontario Pre-project Outlet Hydraulic Relationship, Environment Canada, Cornwall ON, 2002.
3. Fagherazzi L., Guay R., Sparks D., Salas J., Sveinsson O., Stochastic modeling and simulation of the Great Lakes – St. Lawrence River system, Report submitted to the IJC, 2005.
4. Fortin, V., Perreault, L., Salas, J.D., Retrospective analysis and forecasting of streamflows using a shifting level model, Journal of hydrology 296, pp.135-163, 2004.
5. Grygier, J.C., and Stedinger, J.R., “SPIGOT, A Synthetic Streamflow Generation Software Package”, Technical description, version 2.5, School of Civil and Environmental Engineering, Cornell University, Ithaca, N.Y.,1990.
6. Grygier, J.C., and Stedinger, J.R., Condensed Disaggregation Procedures And Conservation Corrections For Stochastic Hydrology, Water Resources Research, Vol 24, No 10, pp. 1584, 1988. Ouarda et al.
7. Lee, A. F. S., and M.S. Heghinian, A Shift of the Mean Level in a Sequence of Independent Normal Random Variables - A Bayesian Approach, Technometrics, 19(4), 1977, pp. 503-506
8. Ouarda T. et al, Analysis of changes in the Great Lakes Net Basin Supply (NBS) and explanatory variables, Report presented to IJC, 2010.
9. Potter, K., A stochastic model of the Hurst phenomenon: Non stationarity in hydrologic processes. PhD thesis, John Hopkins University, 1976
10. Perreault, L., Fortin,V., Latraverse, M., Modeling the Great lakes NBS using a Bayesian Multivariate Hidden State Markov Model, Consulting report for the IJC, 2002.
11. Salas, J.D., Delleur, J.W., Yevjevich, V., and Lane, W.L. , Applied Modeling of Hydrological Series, Water Resources Publications, 1980.
12. Salas, J.D., Fu C., Cancellieres A., Dustin D., Bode D., Pineda A. and Vicent E., Characterizing the Severity and Risk of Drought in the Poudre River, Colorado, Journal of Water Resources Planning and Management, ASCE, September/October 2005.
13. Sveinsson O.G.B., Salas, J.D., Lane, W.L. and Frevert, D.K., Stochastic Analysis, Modeling and Simulation (SAMS) Version 2007 - User’s Manual, Colorado State University, Water Resources Hydrologic and Environmental Sciences, Technical Report Number 11, Engineering and Research Center, Colorado State University, Fort Collins, Colorado, 2007.
14. Sveinsson O.G.B., Salas J.D. - Stochastic Modeling And Simulation Of The Great Lakes NBS Based On Univariate And Multivariate Shifting Mean – Dept. of Civil Engineering, Colorado State University, Fort Collins, January 2002.
141
15. Sveinsson, O.G.B., Salas, J.D., and Boes D., Prediction of Extreme Events in Hydrologic Processes that exhibit Abrupt Shifting Patterns – Journal of Hydrologic Engineering, ASCE, July/August 2005.
16. Sveinsson, O.G.B., and Salas, J.D., Multivariate Shifting Mean Plus Persistence Model for Simulating the Great Lakes Net Basin Supplies, Proceedings of the 26th AGU Hydrology Days, CSU, 173-184, 2006.
17. Sveinsson, O.G.B., Unequal Record lengths in SAMS, Unpublished manuscript. 2004.
18. Yin Fan and Fay D., Simulation of water levels and outflows from upper Lakes by CGLRRM with synthetic generated Great Lakes Net Basin Supply, Report presented to IJC, 2005.
Part C
Stochastic Generation of Synthetic Residual Net Basin Supply for the Great Lakes System
Dr. Ousmane Seidou University of Ottawa
Dr. Taesam Lee and Taha Ouarda Institut National de la Recherche Scientifique, centre Eau et Environnement
Revised December 2011
Stochastic Generation of Synthetic Residual Net
Basin Supply for the Great Lakes System
Prepared by
Dr. Ousmane Seidou
University of Ottawa
161 Louis Pasteur office A113, Ottawa (ON) K1N6N5, Canada
Dr. Taesam Lee and Taha Ouarda
Institut National de la Recherche Scientifique, centre Eau et Environnement
Chaire du Canada en Estimation des Variables Hydrométéorologiques
INRS-ETE 490, de la Couronne Québec (Québec) G1K 9A9, CANADA
Revised December 2011
ii
Table of content
Chapter I. Introduction........................................................................................................... 1
Chapter II. Contemporary Residual NBS Generation Methodology ....................................... 5
II.1 Historical data and statistical characteristics to be reproduced .............................. 5
II.2 General methodology for generating annual NBS sequences .................................. 7
II.3 Annual to monthly NBS disaggregation procedure .................................................. 9
Chapter III. Contemporary Residual Supplies using global climate indices ........................... 10
III.1 NL-ARX models ........................................................................................................ 10
III.2 Predictors refinement using wavelet transforms ................................................... 12
III.3 Available climate indices ......................................................................................... 15
III.4 Selection of the best NL-ARX model per lake ......................................................... 16
III.5 Synthetic NBS time series generation ..................................................................... 17
Chapter IV. Stochastic Sequence 2, 3 and 4: Contemporary and future Residual Supplies
Using Climate Models Outputs ..................................................................................................... 21
IV.1 Generation of the predictor variables for the 1948-2010 period .......................... 21
IV.2 Multiple regression Lake NBS models .................................................................... 23
IV.3 Synthetic NBS time series generation for the current period ................................ 25
IV.4 27
IV.5 Stochastic Sequences with Climate Change Emission Scenario A1B and A2 ......... 27
Chapter V. Annual to Monthly disaggregation of synthetic NBS series ................................ 28
V.1 Transformation ....................................................................................................... 28
iii
V.2 Application of the Disaggregation Method ............................................................ 29
V.3 Disaggregation results ............................................................................................ 30
Chapter VI. Verification of the statistical properties of the disaggregated NBS time series
(contemporary residual supplies using global climate indices) .................................................... 31
VI.1 Lakes NBS series cross-correlation ......................................................................... 32
VI.2 Verification of basic statistics in simulated annual and monthly NBS for Lake
Superior 33
VI.3 Verification of basic statistics in simulated annual and monthly NBS for Lake
Michigan-Huron with Georgian Bay .......................................................................................... 36
VI.4 Overall performance of the generation procedure ................................................ 40
Chapter VII. Verification of the statistical properties of the disaggregated NBS time series
(Contemporary Residual Supplies using NCEP variables) ............................................................. 41
VII.1 Lakes NBS series cross-correlation ......................................................................... 41
VII.2 Verification of basic statistics in simulated annual and monthly NBS for Lake
Superior 42
VII.3 Overall performance of the generation procedure ................................................ 46
Chapter VIII. Statistics of Lakes Levels Statistics simulated with Contemporary Residual
Supplies 47
VIII.1 Lake Superior levels statistics(contemporary supplies using ENSO) ...................... 48
VIII.2 Lake Superior levels statistics(contemporary supplies using NCEP variables) ....... 50
Chapter IX. Lake Levels Statistics (Sequences Accounting for Climate Change) ................... 54
IX.1 Lake Superior .......................................................................................................... 54
IX.1.1 Net Basin Supply .......................................................................................................... 54
IX.1.2. Lake Levels .................................................................................................................. 55
iv
IX.2 Lake Michigan Huron with Georgian Bay ............................................................... 55
IX.2.1 Net Basin Supply .......................................................................................................... 55
IX.2.2 Lake Levels ................................................................................................................... 56
IX.3 Lake Erie .................................................................................................................. 56
IX.3.1. Net Basin Supply ......................................................................................................... 56
IV.3.2 Lake Levels ................................................................................................................... 57
IX.4 Lake St-Clair............................................................................................................. 57
IX.4.1. Net Basin Supply ......................................................................................................... 57
IX.4.2 Lake Levels ................................................................................................................... 58
IX.5 Conclusion ............................................................................................................... 60
References 61
List of Tables
Table II-1. Basic statistical characteristics of 1900-2008 residual NBS supplies ........................... 5
Table II-2. Variance-Covariance matrix of 1900-2008 annual NBS series (×1 E+05 (m3/s)2) ......... 6
Table II-3: Predictors and class of function F for each generated synthetic NBS series ................ 7
Table III-1. Predictors considered in the study and their period of availability ........................... 16
Table III-2: Characteristics and performance of the best NL-ARX model for each lake ............. 16
Table III-3. Mean and autocorrelation of the residuals .................................................................. 17
Table III-4. Variance-covariance matrix of the residuals .............................................................. 17
v
Table III-5. Temporal correlation of the generated error term, the observed annual NBS time
series and the generated annual NBS time series ........................................................................ 19
Table III-6. Variance-Covariance matrix of observed annual NBS series (×(1E5 m3/s)2) .............. 19
Table III-7. Variance-Covariance matrix of the generated NBS time series (×(1E5 m3/s)2) .......... 19
Where i
tP is the original predictor, stdi
tP , is the standardized predictor, i
19901961 is the 1961-
1990 average of i
tP and iS 19901961 the 1961-1990 standard deviation of i
tP . This resulted in 25
series of 1948-2009 standardized NCEP monthly predictors for each Lake, presented in table V-
1. Table IV-1. List of NCEP predictors used in the study ............................................................. 21
Table IV-2. Variables included in the regression model, Lakes Erie, Michigan-Huron with
Georgian Bay and Ontario. M(Y) where M is a month refers to the month of the current year.
M(Y+1) refers to the month of the following year. M1(Y) to M2(Y+1) refers to the average of the
variable from M1(Y) to M2(Y+1) over the lake’s basin ................................................................. 23
Table IV-3. Variables included in the regression model, Lakes Superior and St-Clair. M(Y) where
M is a month refers to the month of the current year. M(Y+1) refers to the month of the
following year. M1(Y) to M2(Y+1) refers to the average of the variable from M1(Y) to M2(Y+1)
over the lake’s basin ..................................................................................................................... 24
Table IV-4. Calibration and validation performance of the multiple regression models ............. 24
Table IV-5. Mean and autocorrelation of the residuals (the number in the bracket represnets the
percentage of the bias to the mean of the NBS) ............................................................................ 25
Table IV-6. Variance-covariance matrix of the residuals .............................................................. 25
Table IV-7. Autocorrelation of the generated error term ............................................................... 25
Table IV-8. Variance-Covariance matrix of observed annual NBS series (×1E5 (m3/s)2) ............... 26
Table IV-9. Variance-Covariance matrix of the generated NBS time series (×1E5 (m3/s)2) .......... 26
Table VI-1. Transformation and Related Information (Ann=annual; 1, 2,…., 12=month;
Gam=gamma transformation; Log=Logarithmic transformation; None=no transformation;
A=normality hypothesis accepted; R=normality hypothesis rejected). ....................................... 28
1
Chapter I. Introduction
Stochastic simulation of multivariate hydrologic variables has a key role in evaluating alterna-tive designs and operation rules of hydrologic facilities. The performance of a given plan can be estimated by simulating the behavior of the water resources system using sequences of inputs that are sufficiently long enough to contain a large number of potential hydrological scenarios that could occur in the future, including rare and potentially catastrophic events. Historical ob-servations are often too short to be used for that purpose, and the only remaining one alterna-tive is to use a stochastic model that is able to reproduce key statistical characteristics of histor-ical observations. Even though the statistical characteristics are the same, the synthetic series will contain more extreme values than the historical data and therefore will allow testing the system in more unusual flow conditions.
In order to evaluate alternative management plans for the Great Lakes System, the Internation-al Joint Commission for the Great Lakes (IJC) uses the Coordinated Great Lakes Regulation and Routing Model, which requires monthly and quarter-monthly Net Basin Supply (NBS) as inputs. The model has been used in the past years with a synthetic 50000 years sequence generated using a multivariate contemporaneous mix of shifting mean SM and ARMA process developed (CSM-ARMA) by Fagherazzi et al., (2005). The CSM-ARMA preserves temporal and spatial corre-lation in the series but assumes that the mean of the process jumps to random values at ran-dom times with no physical explanation. Amadou et al. (2009) showed that there are nonlinear teleconnexions between most NBS components and climates indices like the North American Oscillation (NAO), the Pacific Decadal Oscillation PDO and the El-Nino Southern Oscillation (EN-SO). It was therefore decided to develop alternative stochastic models where the apparent non-stationary in the mean of annual NBS is explained using climate related variables. A preliminary analysis showed that ENSO was the best predictor for annual NBS among the three climate in-dices. A first model linking annual NBS components to the El Niño–Southern Oscillation (ENSO) indices using a non-linear autoregressive model with exogenous inputs (NL-ARX) was therefore developed. A 50000 years-long time series of synthetic annual ENSO values was afterward gen-erated using a combination of the empirical mode decomposition (EMD) and non-stationary oscillation resampling (NSOR). The synthetic yearly ENSO values were used with the NL-ARX model to generate the mean values of the NBS time series. Finally, a random noise was gener-ated and added to the mean using a stochastic process whose parameters were optimized to reproduce the key statistical characteristics (temporal and spatial variability, cross correlations and temporal correlation) of historical annual NBS time series. The generated annual NBS time series were was disaggregated into the monthly data employing a parametric disaggregation model (Grygier and Stedinger, 1988; Grygier and Stedinger, 1990). by Lee and Ouarda (2010). The monthly values were further disaggregated to the quarter-monthly time-scale using a non-parametric disaggregation approach based on Genetic Algorithms (Lee and Ouarda, 2010). The
2
resulting NBS sequences were finally routed with the Coordinated Great Lakes Regulation and Routing Model and resulting lakes levels were obtained. An extensive comparison of the statis-tical characteristics of the simulated and observed Lakes NBS and levels is presented in the re-port.
One of the requirements in the study is to include the effect of climate change (CC) in the gen-eration procedure. A straightforward way to integrate CC is to generate a new series of exoge-nous variables that contains the effects of global warming. One possibility is to extract or esti-mate ENSO values from General Circulation Models (GCMs) outputs and use them with the pre-viously generated model. However, after several discussions with the study board, a consensus emerged on the fact that the ability of current GCMs in reproducing ENSO is poor. It was there-fore decided to develop a second model which links annual NBS to a list of variables that could be obtained from climate models outputs for both present and future climate. The values of the predictors for current climate were obtained from NCEP reanalysis which are gridded data sets that are obtained by assimilating climatic observations all around the world. Future values of the predictors were obtained from climate change experiments (GCM runs using hypothetic emission scenarios). The link between the predictors and NBS components was described by a simple regression model and the best predictors were selected by stepwise regression. Only one GCM (the third generation of the Canadian general Circulation Model) and two emission scenarios (A1B and A2) were considered in this study because of the time constraints. Emission scenario A2 was selected to represent a pessimistic projection of future climate wile A1B repre-sents a more optimistic projection. The stochastic model second model was used to generate three additional data sets:
Using observed NCEP variables as predictors, a series of 50000 annual NBS was generat-ed (since NCEP variables were available for only 61 years, each 50000 years-long predic-tor vector was obtained by stacking the 61-year long NCEP vector one after the other until the desired length is achieved)
Using the outputs of the SRES A1B experiments of the third generation of the Canadian General Circulation Model (CGCM3), 500 sequences of 100 annual NBS time series (cor-responding to years 2001-2100) were generated.
Using the outputs of the SRES A2 experiments of the third generation of the Canadian General Circulation Model (CGCM3), 500 sequences of 100 annual NBS time series (cor-responding to years 2001-2100) were generated.
These last three time series were disaggregated and routed using the same procedure as for the first data set. The entire procedure is illustrated in Figure I.1.
3
Figure I-1. Monthly NBS generation procedure
The remainder of the report is organized as follows: chapter II presents the general methodolo-gy for generating of contemporary Net Basin Supplies, including model development for annual supplies and temporal disaggregation. In chapter III, the methodology is applied to generate long sequences of contemporary annual NBS supplies from global climate indices. In chapter IV, the same methodology is used to generate long sequences of contemporary annual NBS sup-plies using NCEP climate variables. The model developed in chapter IV is forced in chapter V with the outputs of the Canadian General Circulation Model (CGCM3) to obtain sequences of
Simulate or obtain PREDICTORS
Climate Indices (AO, ENSO, PDO) EMD-MNSOR
Climate variables
25 variables extracted from NCEP reanalyis and CGCM3 out-
puts (Current, A1B, and A2)
Simulation of the annual Great Lakes NBS
NLARX with the climate indices generated with EMD-MNSOR
model
Stepwise regression with climate variables:
Current, A1B, and A2
Temporal Disaggregation of the annual NBS to monthly NBS
Parametric temporal disaggregation model
4
synthetic NBS supplies that account for climate change (scenarios A1B and A2). The disaggrega-tion procedure is presented in chapter VI, while the statistical properties of the two generated contemporary NBS supplies are compared to those of observations in chapters VII and VIII. All the generated NBS sequences were routed in the coordinated Great Lakes Model and resulting lake levels statistics are presented in chapters IX, X, and XI (sequences accounting for climate change)
5
Chapter II. Contemporary Residual NBS Generation Method-ology
The general methodology for the generation of contemporary NBS supplies is presented in this chapter. It aims at reproducing long synthetic annual and monthly NBS time series for which some selected statistical properties are the same as the ones of the historical NBS series.
II.1 Historical data and statistical characteristics to be reproduced
Monthly residual NBS supplies of Lake Superior (SUP), Lake St-Clair (STC), Lake Michigan-Huron with Georgian Bay (MHG), Lake Erie (ERI), and Lake Ontario (ONT) were provided by the inter-national joint commission for the 1900-2009 periods. Annual residual NBS series are plotted on Figure II-1, while their basics statistical properties (mean, standard deviation, temporal correla-tion and variance-covariance matrix) of the series are presented in tables II-1 and II-2.
Figure II-1 1900-2009 annual residual NBS for Lake Superior (SUP), Lake St-Clair (STC), Lake Michigan-Huron
with Georgian Bay (MHG), Lake Erie (ERI), and Lake Ontario (ONT)
Table II-1. Basic statistical characteristics of 1900-2008 residual NBS supplies
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000-1000
0
1000
2000
3000
4000
5000
Year
An
nu
al N
BS
(m
3/s
)
SUP
MHG
ERI
ONT
STC
6
Lake Superior
Lake Michigan-Huron with Geargian Bay
Lake Erie Lake Ontario Lake St-Clair
Mean (m3/s) 2.0082e+003 3.1026e+003 742.0932 1.0784e+003 272.2272
Standard
deviation (m3/s) 446.7279 685.1876 269.4839 218.6631 60.9862
Temporal
correlation 0.1460 0.1117 0.2905 0.2052 0.5272
Table II-2. Variance-Covariance matrix of 1900-2008 annual NBS series (×1 E+05 (m3/s)2)
Lake Superi-or
Lake Michi-gan-Huron with Geor-gian Bay Lake Erie Lake Ontario Lake St-Clair
Lake Superior 2.343 1.890 0.319 0.206 0.052
Lake Michigan-Huron with Georgian Bay 1.890 5.508 1.088 1.067 0.216
Lake Erie 0.319 1.088 0.845 0.430 0.117
Lake Ontario 0.206 1.067 0.430 0.559 0.082
Lake St-Clair 0.052 0.216 0.117 0.082 0.042
For each of the five lakes, the following statistics of historical NBS sequences have to be repro-duced by the generation procedure at the yearly and monthly scales.
1. The historical mean residual NBS for each lake in order to be able to reproduce mean lake levels
2. The historical standard deviation for each lake in order to reproduce lake levels variability,
3. The historical lag one autocorrelation in order to reproduce flow persistence pat-terns
7
4. The variance-covariance matrix of the system has to be reproduced as well for a better reproduction of flow variability in downstream lakes and reaches.
The performance of the generation procedure will be assessed in chapters VII (1st stochastic model) and chapter VIII (2nd stochastic model) using box plots figures to visually compare the statistics of the generated series to the statistics of historical series. The comparison will there-fore be qualitative and no tolerance threshold is considered.
II.2 General methodology for generating annual NBS sequences
The general model used for generating synthetic annual NBS is the following:
ttt ξXNBS (1)
Where 54321
ttttt NBSNBSNBSNBSNBStNBS represents the vector of five simulated yearly
lake level NBS components at time t, 54321X ttttt XXXXXt is the part of tNBS which is
deterministically driven by known predictors, and 54321ξ ttttt t is the error vector.
Each of these vectors contains five elements corresponding to Lake Superior (subscript=1), Lake Michigan-Huron (subscript=2), Lake Erie (subscript=3), Lake Ontario (subscript=4) and Lake St-Clair (subscript=5).
The deterministic component of each lake’s NBS is linked to a set of climate related predictors.
i
t
i
t FX P (2)
Where i
tP is the vector of the ith lake predictor’s values at date t and F is a function describing
the deterministic relationship between annual NBS components and the set of predictors. The class of F and the predictors used for each of the generated synthetic NBS series is given in the table below:
Table II-3: Predictors and class of function F for each generated synthetic NBS se-ries
Data set Predictors Function F
Stochastic Sequence of Contemporary Residual Supplies using ENSO indicator
Simulated ENSO indices using the EMD-NSOR method
Non linear autoregressive model with exogeneous variables (NL-ARX)
Stochastic Sequence of Contemporary Residual Supplies using NCEP 500
NCEP reanalysis linear regression
8
Data set Predictors Function F
mb anomalies indicator
Stochastic Sequence with Climate Change emission scenario A2
CGCM3 Outputs linear regression
Stochastic Sequence with Climate Change emission scenario A1b
CGCM3 Outputs linear regression
The methodology used to choose the predictors and develop the deterministic relationship F for each dataset will be described in the next sections. The parameters of the model are esti-mated as follow:
Step 1: the parameters of F are estimated assuming that tξ is multivariate Gaussian with
noise. Since there is no temporal correlation intξ
, the temporal correlation in tNBS is
controlled by the serial correlation in tX , while the variance-covariance matrix of tNBS
depends of both the variance-covariance matrix of tX and the variance-covariance ma-
trix of tξ .
Step 2: the structure of tξ is modified to better reproduce the autocorrelation and the
variance-covariance matrix of the observed NBS. Each component of the noise vector tξ
is modelled as follow:
i
i
i
ti
i
t r 1 (3)
where 54321
tttttt follow a stationary multivariate normal distribution of
mean zero and variance-covariance matrix . 5,..,1, iri and the desire is that the ele-
ments of Ω reproduce the temporal correlation and the variance-covariance matrix of ob-served yearly NBS series.
Step 3. The stochastic sequences have to reproduce the characteristics of the 1900-2008 NBS sequences, while predictors are available on a shorter period (1948-2009 for NCEP
variables, and 1950-2008 for ENSO indices). Therefore, the mean of the generated se-
quences at the end of step 2 will correspond to the mean of historical NBS sequences
9
during the period of availability of the predictors. Therefore, a constant is added to each lake’s synthetic NBS sequences so that their mean match the historical mean. Therefore the mean of the generated sequences will always match the historical mean because of that correction.
II.3 Annual to monthly NBS disaggregation procedure
The employed disaggregation model is the Stedinger model (Grygier and Stedinger, 1988; Grygier and Stedinger, 1990). It will be described in detail in chapter VI;
10
Chapter III. Contemporary Residual Supplies using global cli-mate indices
Amadou et al. (2009) used a combination of wavelet transforms, inverse wavelet transform and non-linear ARX (NL-ARX) modeling is used to investigate the teleconnexion between 210 time series of Net Basin Supply (NBS) components of the Great Lakes System, and mean seasonal values of the AO, PDO and ENSO indices. A similar approach is used here to model the the
determisitic component of annual NBS tX . The only difference between the results presented
here and those of Amadou et al. (2009) is that the modelled variable is the residual annual NBS time series, and the predictors are the annual and winter values of three climate indices: Arctic Oscillation, El Niño-Southern Oscillation, and Pacific Decadal Oscillation. 50000 values of each of these climate indices were generated by Lee and Ourda (2010) using two different techniques: the Contemporaneous Shifting Mean and Contemporaneous Autoregressive Moving Average (CSM-CARMA) model and a multivariate version of non-stationary version of the Non Stationary Oscillation Resampling (M-NSOR) technique. They found that M-NSOR is superior to the CSM-CARMA model for reproducing the ENSO process while the other basic statistics are comparatively well preserved in both cases. Therefore long series of simulated ENSO could be used if models linking annual NBS to ENSO could be built as in Amadou et al. (2009). NL-ARX models were considered since they have been successfully used for component NBS in Amadou et al. (2009). The present chapter describes the methodology used to develop a NL-ARX model for each lake using ENSO as predictor. NL-ARX models are introduced in Section 2.1.1; Wavelet and Wavelet coherence presented in Section 2.1.2; Predictor refinement using wavelets is explained in Section 2.1.3; The input data sets for the NL-ARX models are described in section 2.1.4. The selection of the best NL-ARX model for each of the five Lakes is explained in Section 2.1.5. For more detailed explanations on the process, the reader is referred to Amadou et al. (2009).
III.1 NL-ARX models
An ARX model is a linear recurrence equation to relate the current value of a target variable x(s) with its past finite time series and the past finite time series of other y exogenous input var-iables. In the special case of a single exogenous variable y, the ARX model can be written as:
(4)
where s is a current time step, the contribution coefficient of an i-step past value of the ob-
jective function to its current value, the j-step past value of the exogenous input variable y,
and the time lag of the propagation of the exogenous input variable. , and are
called the orders of the ARX model and are usually found by comparing models of different or-ders with performance criteria such as the Akaike Information Criterion (AIC).
1 1
( ) ( ) ( ) ( )a bn n
i j k
i j
x s a x s i b y s j n e s
ia
jb
kn an bn kn
11
A nonlinear ARX model is obtained by transforming the output of the ARX model with a nonlin-ear function :
(5)
Various forms can be given to the function depending on the problem at hand. In this work, four possible forms were considered for .
No transformation:
Wavelet networks: where ψ is a mother wavelet, which
is a function satisfying the admissibility conditions described in section 1 (Zhang and Ben-
veniste, 1992). The number m of wavelet functions, parameters ,
and are all optimized to minimize det( '* )E E where E is the prediction error.
Minimizing det( '* )E E is the optimal choice in a statistical sense and leads to the maximum
likelihood estimates in case no information is known about the variance of the noise (Ljung, 2007). Function F is referred to as a network because it is the sum of several basis functions with different nodes and parameter. Each basis function can be seen as a node in a one lay-er network.
Sigmoid networks: where σ is a sigmoid function. Sigmoid
functions are non-linear parametric functions, often used in neural networks to introduce
nonlinearity in the model and/to clamp signals to within a specified range. A typical sigmoid
function is . The number m of sigmoid wavelet functions, parameters
, and are optimized using the same algorithm as for
the wavelet network.
Tree partition networks: where η is a piecewise linear
function of x and a partition of the x-space. The number of partitions as well as parame-
ters , and are optimized using the same algorithm as
for the wavelet network.
F
1 1
( ) ( ) ( ) ( )a bn n
i j k
i j
x s F a x s i b y s j n e s
F
F
( )F x x
1
( , ) ( ( ))m
k k k
k
F x w x
, 1,kw k m , 1,k k m
, 1,k k m
1
( , ) ( )m
T
k k k
k
F x w x
)exp(1
1)(
tt
, 1,kw k m , 1,k k m , 1,k k m
( , ) ( )T
k k k kF x w x if x D
kD
, 1,kw k m , 1,k k m , 1,k k m
12
III.2 Predictors refinement using wavelet transforms
Wavelets are families of basis functions that allow the decomposition of a signal in the time-frequency space. They have advantages over traditional Fourier methods in analyzing situations for which the weight of the various frequencies of the signal changes over time. By decompos-ing a time series in the time-frequency space, the dominant modes of variability can be deter-mined and their changes in time analyzed. The continuous wavelet transform of a discrete se-
quence is defined as the convolution with scaled and translated version of :
(6)
where the (*) indicates the complex conjugate , is the wavelet function with zero mean that must be localized in both time and frequency space (admissibility condition). An example of wavelet transform (annual runoff of Lake Erie) is presented on the figure below. The figure was generated with the wavelet toolbox developed by Torrence and Compo (1998). Values are only plotted within a cone of influence (COI) where there are no edge effects on the calculation of the coefficients.
Figure III-1. Wavelet transform of the annual runoff on Lake Erie watershed
Given two time series X and Y with wavelet transform WXn(s) and WY
n(s), the cross-wavelet spectrum is defined as:
nX
1
'
' 0
'*
NX
n n
n
n n tW s x
s
Year
Period
1950 1960 1970 1980 1990 2000
4
8
16
1/8
1/4
1/2
1
2
4
8
13
)()()( sWsWsW Y
n
X
n
XY
n (7)
)(sW XY
n is a complex number which’ modulus )(sW XY
n is called cross-wavelet power and per-
mits not only to depict the features common to both the climatic indices and the seasonal pre-cipitation, but also to highlight temporal variations in their relationship. An example of cross wavelet transform (PDO autumn and the autumn runoff on Lake Erie watershed) is presented on the Figure below.
Figure III-2. Cross waveletpower of PDO autumn and the autumn runoff on lake Erie watershed. In Figure III-2, blank regions on either end indicate the “cone of influence,” where edge effects become important while the
arrows represent the phase angle of )(sW XY
n .
A set of predictors was obtained from original climate indices by selecting only the frequencies for which the estimated coherence between the NBS component and the climate index is above a given threshold. The explanatory variables for use in the modeling process were derived as follows:
1. A cross wavelet analysis of the candidate NBS component and candidate climate index is performed to obtain a wavelet coefficient matrix A
Year
Period
1950 1960 1970 1980 1990 2000
4
8
16
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
14
2. A threshold value is selected (each value in the set will be
picked once and will lead to one distinct predictor)
3. The mean wavelet coherence is calculated for all frequencies and frequencies for which the mean wavelet coherence is above the threshold are selected.
4. The lines of the wavelet coefficients matrix in A are set to zero for all the frequencies not selected at step 3 to ignore components for which the mean wavelet coherence is below the threshold. A new wavelet coefficients matrix A’ is obtained.
5. Inverse wavelet transform is applied A’ to obtain the derived predictor.
6. In order to avoid edge effect on the derived predictors, values that are not within the cone of influence (COI) are discarded. Therefore the length of the derived series de-pends on the frequencies for which the mean wavelet coherence is above the threshold. Therefore the lower the selected frequencies, the shorter the predictor length. Fre-quencies that lead to a derived predictor shorter than 60% of the original series length are not considered.
The process of frequency selection and inverse wavelet transformation is illustrated in Figure III-3. The histogram of the wavelet coefficients (averaged over time for each frequency) is pre-sented in the upper panel. The periods (in years) for which the mean wavelet coherence is higher than the threshold are {2.0661 9.2764 9.8280 10.4124 11.0316 11.6876 12.3825 13.1188}. Inverse wavelet transform is applied to the wavelet coefficient matrix after the coef-ficients for all other frequencies are set to zero, to obtain the derived predictor presented on the lower panel. Depending of the level of coherence between the NBS component and the climate index, up to 10 predictors can be derived from a given climate index.
0,0.1,0.2,0.3,...0.9
15
Figure III-3. Frequency selection and inverse wavelet transform for =5 (Mean autumn PDO and mean Runoff on Lake Erie watershed)
III.3 Available climate indices
Six time series of climate indices (ENSO winter, ENSO annual, PDO winter, PDO annual, NAO winter and NAO annual) were initially considered in this study. At the time they were chosen, there was an ongoing study on their predictability using Non-stationary Oscillatory Resampling (Lee and Ourada, 2000) and it was expected that synthetic values of these indices would be available later for use in the stochastic generation models.. The historical observations of these indices were used to calibrate candidate NL-ARX models for each of the five Lakes. The length of the historical observations of these climate indices are presented in the table below.
16
Table III-1. Predictors considered in the study and their period of availability
CLIMATE INDICE PERIOD OF AVAILABILITY
ENSO winter 1950-2009
ENSO annual 1950-2009
PDO winter 1900-2009
PDO annual 1900-2009
NAO winter 1900-2009
NAO annual 1900-2009
III.4 Selection of the best NL-ARX model per lake
The following procedure was used for each of the five Lakes:
Several candidate NL-ARX models were developped using different combinations of predictor variables, wavelet coherence threshold and non-linearity.
The predictorvector is then split into a 60% calibration set and a 40% validation set.
The parameters of the NL-ARX model are estimated using the maximum likelihood method (assuming that the error is a white noise)
The model having the best Nash-sutcliffe coefficient in the calibration period is selected. A Nash-sutcliffe coefficient equal to 1 corresponds to a perfect fit while a Nash coefficient of 0 indicates that the model predictions are as accurate as the mean of the observed data. A Nash coefficient less than zero occurs when the observed mean is a better predictor than the model.
The characteristics of the selected NL-ARX model for each of the five Lakes are presented in Table 3-2.
Table III-2: Characteristics and performance of the best NL-ARX model for each lake
LAKE PREDICTOR DELTA NONLINEARITY an bn kn Nash (calibration) Nash (validation)
SUP ENSO Winter 0.3000 Wavenet 0 1 1 0.6275 0.4075
STC ENSO Winter 0.3000 Wavenet 0 2 0 0.2548 0.2233
MHG ENSO Winter 0 Sigmoidnet 0 1 0 0.0784 0.2845
ERI ENSO Winter 0.3000 Wavenet 0 2 0 0.6214 0.3893
ONT ENSO Annual 0.5000 Wavenet 0 1 1 0.4186 0.3119
17
The fit of the NL-ARX models is generally good, with a validation Nash coefficient ranging from 0.2233 (Lake St-Clair) to 0.4075 (Lake Superior)
III.5 Synthetic NBS time series generation
The NL-ARX models developped in the previous sections were were to generate the 50000 values of mean NBS values using the synthetic climate indices generated by Lee and Ouarda (2010). In order to obtain NBS stochastic sequences, 50000 errors values should be generated as well with the adequate statistical properties. The table below presents the statistical properties of the residual time series obtained by substracting the outputs of the NL-ARX models from observed annual NBS series.
Table III-3. Mean and autocorrelation of the residuals
Lake SUP MHG ERI ONT STC
Mean (cms) -19.285 (-.96%) 5.982 (.19%)
-21.108 (-2.84%)
-5.590 (-.52%)
26.028 (9.56%)
Lag 1
autocorrelation
-0.1484 0.4643 -0.1653 0.0753 0.3522
Table III-4. Variance-covariance matrix of the residuals
SUP MHG ERI ONT STC
SUP 1.1143e5 142.2695 6.6507e4 6.2396e3 1.4831e4
MHG 142.2695 2.1865e3 1.1506e4 5.6802e3 5.1765e3
ERI 6.6507e4 1.1506e4 3.4306e5 5.2030e4 3.0060e4
ONT 6.2396e3 5.6802e3 5.2030e4 3.4834e4 2.3538e4
STC 1.4831e4 5.1765e3 3.0060e4 2.3538e4 4.5060e4
18
It appears in table 3 that the residuals displays a certain level of autocorrelation, which can be significant (0.46 for Lake Michigan-Huron and 0.35 for Lake St-Clair). Therefore, the error model was altered from white noise (with no serial correlation) to a correlated multivariate normal process as described in chapter 2:
i
t
i
ti
i
t r 1 (8)
where 54321
tttttt follow a stationary multivariate normal distribution of mean zero
and variance-covariance matrix ji, . For lake number I, we have:
iii
i
ti
i
t rr ,
221var1var (9)
i
tvar can be approximated with the variance of the residuals for Lake i. Therefore, if ir is
known, ii, can be estimated from equation (8). The values of ir and ii, are obtained using
the following algorithm:
1. For ir =-0.99 to 0.99 (increment=0.01)
a. Calculate ii, from equation (9)
b. Draw 50000 values of it from a normal distribution with mean 0 and variance
ii,
c. Calculate it from equation (8)
d. Calculate itNBS from equation (1)
e. Calculate the serial correlation of itNBS
2. End For
3. Select the value of ir for which the serial correlation of itNBS is the closest to the
historical temporal correlation of annual NBS at Lake i.
The elements of are afterward estimated by minimising the sum of squarre errors between
the variance-covariance matrix of itNBS and the observed variance covariance matrix of
annual NBS on the five lakes. A first guess for is the variance-covariance matrix of the
realisations of it , calculated from equation (7) and observed residuals. A computer code
which minimizes the objective function while maintaining the positive-definite property of was used to obtain the final values of the coefficients of . The optimized values of
5,..,1, iri are presented in Table III-5, along with the temporal correlatiob coefficients of the
observed and simulated NBS series. It can be seen from the values that the generation procedure reproduces the temporal correlation of observed NBS series very well.
19
Table III-5. Temporal correlation of the generated error term, the observed annual NBS time series and the generated annual NBS time series
Lake SUP MHG ERI ONT STC
ir 0.6490 -0.1580 0.1730 0.5290 0.5530
Observed NBS autocorrelation 0.1463 0.1115 0.2842 0.2023 0.5070
Generated NBS autocorrelation 0.1461 0.1155 0.2804 0.1974 0.5168
Table III-6 and III-7 present the variance covariance matrix of observed and simulated NBS se-ries. It can be seen that the diagonal elements of the two tables are very close, while the non-diagonal elements are of the same order of magnitude. The percent difference between the coefficients of the Variance-Covariance matrix of the generated NBS time series to the coeffi-cients of the Variance-Covariance matrix of the observed NBS series is presented in table III-7. While the difference in the diagonal is at most 0.71%, it is up to 71% elsewhere (covariance be-tween Lake Erie and Lake Ontario NBS series). The generation procedure presented in this chapter was therefore able to reproduce the variance of individual Lakes NBS but not the cross-correlation.
Table III-6. Variance-Covariance matrix of observed annual NBS series (×(1E5 m3/s)
2)
SUP MHG ERI ONT STC
SUP 2.34 1.89 0.32 0.21 0.05
MHG 1.89 5.51 1.09 1.07 0.22
ERI 0.32 1.09 0.84 0.43 0.12
ONT 0.21 1.07 0.43 0.56 0.08
STC 0.05 0.22 0.12 0.08 0.04
Table III-7. Variance-Covariance matrix of the generated NBS time series (×(1E5 m3/s)
2)
SUP MHG ERI ONT STC
SUP 2.33 0.67 0.24 0.19 0.03
MHG 0.67 5.52 0.55 0.54 0.12
ERI 0.24 0.55 0.85 0.12 0.11
ONT 0.19 0.54 0.12 0.56 0.05
STC 0.03 0.12 0.11 0.05 0.04
20
Table III-8. relative difference in percent between the coefficients of the Variance-Covariance matrix of the generated NBS time series to the coefficients of the Variance-Covariance matrix of the observed NBS series
SUP MHG ERI ONT STC
SUP 0.41 64.45 24.28 5.59 48.85
MHG 64.45 0.20 49.24 48.99 43.77
ERI 24.28 49.24 0.17 71.70 4.75
ONT 5.59 48.99 71.70 0.21 43.61
STC 48.85 43.77 4.75 43.61 0.72
21
Chapter IV. Stochastic Sequence 2, 3 and 4: Contemporary and future Residual Supplies Using Climate Models Outputs
In chapter III, we were able to generate synthetic contemporary NBS supplies with acceptable statistical properties. The next task would be the inclusion of climate change (CC) in the genera-tion procedure. A straightforward way to integrate CC is to generate a new series of exogenous variables that contains the effects of global warming. The model developed in chapter III (NL-ARX models with ENSO predictors) were not considered because of concerns about the ability of GCM in reproducing ENSO. These concerns were discussed at the beginning of chapter I.
A second model which links annual NBS to a list of climate variables that could be obtained from both GCM runs and NCEP reanalysis was used instead. The two only differences between the model presented here and the model presented in chapter III are the following:
Climate indices are replaced by time series of NCEP variables averaged over the Great Lakes’ watersheds
Stepwise regression replaces the NL-ARX models.
The model performance is assessed using the Nash-Sutcliffe efficiency coefficient on the cali-bration and validation period. Once the model deemed satisfactory, it was forced with project-ed predictors to simulate annual NBS for the 2010-2100 period.
IV.1 Generation of the predictor variables for the 1948-2010 period
The monthly values of 25 variables (listed in the table below) covering the 1948-2009 period were extracted and/or calculated from the NOAA Earth System Research Laboratory website (ftp.cdc.noaa.gov). Vorticity and divergence were calculated following DAI CGCM3 Predictors (2008) while other variables were calculated. The 25 variables are exactly the ones used in DAI CGCM3 Predictors (2008) because they have been specifically developed for downscaling appli-cation. The variables were first spatially interpolated and averaged over the watersheds of each lake, and then standardized using the mean and the standard variation of the 1961-1990 period over the lake’s watershed:
i
ii
tstdi
tS
PP
19901961
19901961,
(9)
Where i
tP is the original predictor, stdi
tP , is the standardized predictor,
i
19901961 is the 1961-1990 average of i
tP and
iS 19901961 the 1961-1990 standard deviation ofi
tP . This resulted in 25 series of 1948-2009 standardized NCEP monthly pre-
dictors for each Lake, presented in table V-1.
22
Table IV-1. List of NCEP predictors used in the study
Variable No. Variable Pressure level
1 Mean sea level pressure -
2 Wind Speed 1000hPa
3 U-component 1000hPa
4 V-component 1000hPa
5 Vorticity 1000hPa
6 Wind Direction 1000hPa
7 Divergence 1000hPa
8 Wind Speed 500hPa
9 U-component 500hPa
10 V-component 500hPa
11 Vorticity 500hPa
12 Geopotential 500hPa
13 Wind Direction 500hPa
14 Divergence 500hPa
15 Wind Speed 850hPa
16 U-component 850hPa
17 V-component 850hPa
18 Vorticity 850hPa
19 Geopotential 850hPa
20 Wind Direction 850hPa
21 Divergence 850hPa
22 Specific Humidity 500hPa
23 Specific Humidity 850hPa
24 Specific Humidity 1000hPa
25 Temperature at 2m
23
Apart from the temperatures, all these variables are pressure, wind and humidity related
variables. They all affect precipitation either directly either by moisture redistribution.
In order to generate the maximum number of potential predictors for the annual NBS of a given Lake, the 25 standardized variables were further averaged over periods of 1, 3, 6 and 12 Months. The time lags between the end of the period over which a predictor is averaged and the end of the year range from -16 Months to +8 months. This resulted in a total of 1920 different predictors (and non-independent) predictors.
IV.2 Multiple regression Lake NBS models
For each of the five lakes, predictors and annual NBS were available for a period of 53 years (1957-2009). The 53 years were divided into a 60% calibration period (1957-1988) and a 40% validation period (1989-2009). Stepwise regression was applied to select the best explanatory variables for each lake’s NBS. The use of stepwise regression insures that only meaningful pre-dictors are used.
Table IV-2. Variables included in the regression model, Lakes Erie, Michigan-Huron with Georgian Bay and Ontario. M(Y) where M is a month refers to the month of the current year. M(Y+1) refers to the month of the following year. M1(Y) to M2(Y+1) refers to the average of the variable from M1(Y) to M2(Y+1) over the lake’s basin
ERI MHG ONT
1 850hPa Wind Speedfor APR(Y) to
MAR(Y+1)
500hPa V-component500hPa V-
component for JUN(Y+1) to
JUN(Y+1)
850hPa Wind Speedfor APR(Y) to
MAR(Y+1)
2 500hPa Specific Humidity500hPa
Specific Humidity for FEB(Y) to
JUL(Y)
850hPa Wind Direction500hPa
Wind Direction for for JUN(Y) to
MAY(Y+1)
500hPa Wind Direction
for Jan(Y) to DEC(Y)
3 500hPa Wind Direction500hPa Wind
Direction for OCT(Y) to DEC(Y)
850hPa Wind Speedfor OCT(Y) to
MAR(Y+1)
1000hPa Wind Direction
for SEP(Y-1) to AUG(Y)
4
500hPa Wind Direction500hPa
Wind Directionfor OCT(Y) to
DEC(Y)
500hPa Specific Humidity
for JUL(Y) to JUL(Y)
5
500hPa Specific Humidity500hPa
Specific Humidity for JUL(Y) to
JUL(Y)
500hPa Geopotential for
SEP(Y+1) to SEP(Y+1)
6
500hPa Specific Humidity500hPa
Specific Humidity for MAY(Y) to
JUL(Y)
500hPa U-component for
FEB(Y+1) to JUL(Y+1)
7
500hPa Specific Humidity500hPa
Specific Humidity for AUG(Y-1) to
JUL(Y)
500hPa Wind Direction
for OCT(Y) to DEC(Y)
24
Table IV-3. Variables included in the regression model, Lakes Superior and St-Clair. M(Y)
where M is a month refers to the month of the current year. M(Y+1) refers to the month of the following year. M1(Y) to M2(Y+1) refers to the average of the variable from M1(Y) to M2(Y+1) over the lake’s basin
STC SUP
1 500hPa V-component for
JUN(Y+1) to JUN(Y+1)
850hPa U-component for SEP(Y)
to FEB(Y+1)
2 500hPa Wind Direction for JUN(Y)
to MAY(Y+1)
850hPa V-component for NOV(Y)
to JAN(Y+1)
3 Mean sea level pressure for
OCT(Y+1) to OCT(Y+1)
1000hPa Wind Speed for NOV(Y)
to NOV(Y)
4 500hPa Wind Speed for
MAR(Y+1) to AUG(Y+1)
1000hPa U-component for AUG(Y)
to OCT(Y)
5 500hPa U-component for
MAY(Y+1) to JUL(Y+1)
1000hPa V-component for JUL(Y)
to SEP(Y)
6 850hPa Geopotential for NOV(Y)
to APR(Y+1)
1000hPa Wind DirectionJUN(Y) to
AUG(Y)
7 500hPa Wind DirectionJan(Y) to
DEC(Y)
1000hPa Wind DirectionMAR(Y)
to AUG(Y)
8 1000hPa Wind Speed for SEP(Y) to
NOV(Y)
SUP_c3a1tempna APR(Y) to
APR(Y)
9
500hPa U-component for AUG(Y)
to JUL(Y+1)
10
500hPa Wind Direction for
MAY(Y+1) to MAY(Y+1)
Table IV-4. Calibration and validation performance of the multiple regression models
Lake Nash (calibration) Nash (Validation)
ERI 0.3530 0.4347
MHG 0.7375 0.6030
ONT 0.7229 0.6542
STC 0.6153 0.7660
SUP 0.7657 0.6828
25
IV.3 Synthetic NBS time series generation for the current period
Synthetic NBS generation using NCEP variables follows exactly the same procedure as the one used with climate indices. The mean and autocorrelation of the residuals are presented in table IV-5, while their variance-covariance matrix is presented in Table IV-6.
Table IV-5. Mean and autocorrelation of the residuals (the number in the bracket represnets the
percentage of the bias to the mean of the NBS)
Lake SUP MHG ERI ONT STC
Mean (cms) -10.126 (-.50%) -94.725 (-3.05%)
28.610 (3.86%)
55.033 (5.10%)
-3.729 (-1.37%)
Lag 1 autocorrelation -0.1641 0.1600 0.3595 0.3447 0.2084
Table IV-6. Variance-covariance matrix of the residuals
SUP MHG ERI ONT STC
SUP 7.2422e+006 3.2572e+006 5.4747e+005 -2.8236e+005 7.4626e+004
MHG 3.2572e+006 9.2817e+006 2.5794e+005 -1.5916e+005 1.1574e+005
ERI 5.4747e+005 2.5794e+005 5.4823e+006 3.7256e+006 3.3158e+005
ONT -2.8236e+005 -1.5916e+005 3.7256e+006 8.9381e+006 5.8482e+005
STC 7.4626e+004 1.1574e+005 3.3158e+005 5.8482e+005 2.1496e+005
Since significant temporal correlation is found in the residuals after fitting the multiple regression models, the error model was altered from white noise (with no serial correlation) to a correlated multivariate normal process as described in chapter 2, and applied in Section 3.6.
The optimized values of 5,..,1, iri are presented in Table 4-7, along with the temporal
correlation coefficients of the observed and simulated NBS series. It can be seen from the values that the generation procedure reproduces the temporal correlation of observed NBS series very well.
Table IV-7. Autocorrelation of the generated error term
Lake SUP MHG ERI ONT STC
ir 0.4720 -0.3570 0.4090 -0.7500 0.7930
Observed NBS autocorrelation 0.1463 0.1115 0.2842 0.2023 0.5070
Generated NBS autocorrelation 0.1438 0.1084 0.2807 0.1974 0.5009
26
Table IV.8 and IV.9 present the variance covariance matrix of observed and simulated NBS se-ries. It can be seen that the diagonal elements of the two tables are very close, while the non-diagonal elements are of the same order of magnitude.
The percent difference between the coefficients of the Variance-Covariance matrix of the gen-erated NBS time series to the coefficients of the Variance-Covariance matrix of the observed NBS series is presented in table IV-10. While there is no difference in the diagonal it is up to 38% elsewhere (covariance between Lake Erie and Lake Ontario NBS series). The covariance between Lake Superior and Lake Erie and the one between Lake Superior and Lake St-Clair are well reproduced (0% difference). The generation procedure presented in this chapter was therefore able to reproduce the variance of individual Lakes NBS but not always the cross-correlation.
Table IV-8. Variance-Covariance matrix of observed annual NBS series (×1E5 (m
3/s)
2)
SUP MHG ERI ONT STC
SUP 2.34 1.89 0.32 0.21 0.05
MHG 1.89 5.51 1.09 1.07 0.22
ERI 0.32 1.09 0.84 0.43 0.12
ONT 0.21 1.07 0.43 0.56 0.08
STC 0.05 0.22 0.12 0.08 0.04
Table IV-9. Variance-Covariance matrix of the generated NBS time series (×1E5 (m
3/s)
2)
SUP MHG ERI ONT STC
SUP 2.34 1.75 0.32 0.13 0.05
MHG 1.75 5.51 0.94 0.80 0.17
ERI 0.32 0.94 0.84 0.30 0.11
ONT 0.13 0.80 0.30 0.56 0.06
STC 0.05 0.17 0.11 0.06 0.04
27
IV.4
Table IV-10. relative difference in percent between the coefficients of the Variance-Covariance matrix of the generated NBS time series to the coefficients of the Variance-Covariance matrix of the observed NBS series
SUP MHG ERI ONT STC
SUP 0.00 7.41 0.00 38.10 0.00
MHG 7.41 0.00 13.76 25.23 22.73
ERI 0.00 13.76 0.00 30.23 8.33
ONT 38.10 25.23 30.23 0.00 25.00
STC 0.00 22.73 8.33 25.00 0.00
IV.5 Stochastic Sequences with Climate Change Emission Scenario A1B
and A2
Outputs from the third generation of the Canadian Circulation Model (CGCM3) were obtained from the Data Access Integration (DAI) website which is maintained by Environment Canada (http://loki.qc.ec.gc.ca/DAI/). CGCM3 predictor data for the SRES A2 and SRES A1B climate change scenario were obtained for the years of 1961-2000 and from 2001 – 2100. Scenarios A1B is used to represent a con-vergent world of rapid economic growth, a global population that reaches 9 billion in 2050 and then gradually declines, the quick spread of new and efficient technologies. It is considered as an optimistic scenario. Sce-nario A2 is used to represent a world of independently operating, self-reliant nations with a continuously in-creasing population, a regionally oriented economic development plus a slower and more fragmented tech-nological evolution. It is a pessimistic scenario. Using the two scenarios A1B and A2 will insure that the worst case and the best case possibilities are considered.
The predictors downloaded from the DAI website were averaged over the Lakes watersheds (assuming
the predictors values are constant on each grid-cell), and finally were standardized using the mean and the standard variation of the 1961-1990 period as in equation (9). This ensures the new predic-tors have the appropriate scaling to be used in the model developed in chapter V. The length of the generated sequences can only be as long as the length of the predictors. Therefore, instead of one continuous sequence of 50000 values, 500 sequences of 100 years (corresponding to years 2001-2100 were generated.
28
Chapter V. Annual to Monthly disaggregation of synthetic NBS series
The employed disaggregation model is the Stedinger model (Grygier and Stedinger, 1988; Grygier and Stedinger, 1990). In order to apply this model, data must be normally distributed. Data transformation must be performed in advance whenever any data does not meet this normality condition. The transformation and the applied Stedinger model are briefly described in the following two subsections.
V.1 Transformation
The applied disaggregation model works with normally distributed random variables. However, the Great Lakes NBS are commonly positively skewed. Therefore, the yearly and seasonal NBS must be transformed to meet the normality requirement. Two types of transformations, Gam-ma and Logarithm, (Salas et al., 2009) are applied for the monthly data of Great Lakes NBS while the annual NBS are statistically proven to be normally distributed. The detailed infor-mation is presented in Table VI.1. Skewness test (Salas et al., 1980) was used to check whether the transformed data is normally distributed or not. Almost all annual and monthly transformed data except the October data of Lake St. Clair accept the hypothesis of the skewness test that data is normally distributed
Table V-1. Transformation and Related Information (Ann=annual; 1, 2,…., 12=month; Gam=gamma transformation; Log=Logarithmic transformation; None=no transformation; A=normality hypothesis accepted; R=normality hypoth-esis rejected).
Applied Transf. Skewness Test
Applied Transf. Skewness Test
Seas Trans a Skew Result
Seas Trans a Skew Result
Lake
Su
pe
rio
r
Ann None 1.0 0.026 A
Lake
Mic
hig
an-H
uro
n
Ann None 1.0 -0.045 A
1 Gam 0.0 -0.034 A 1 Log 9500.6 0.031 A
2 Log 16840.3 0.140 A 2 Log 18480.0 -0.044 A
3 Gam 0.0 0.129 A 3 Log 4906.7 -0.071 A
4 Log 1022.8 -0.132 A 4 Log 5817.8 0.069 A
5 Gam 0.0 0.094 A 5 Log 8752.5 0.033 A
6 Log 1562.3 -0.096 A 6 Gam 0.0 0.055 A
7 Log 7850.7 0.093 A 7 Gam 0.0 0.008 A
8 Gam 0.0 -0.020 A 8 Gam 0.0 0.024 A
9 Log 5189.4 -0.043 A 9 Log 7227.6 -0.105 A
10 Log 8045.9 0.094 A 10 Log 20803.9 0.324 A
11 Gam 0.0 0.102 A 11 Log 9103.3 -0.001 A
12 None 1.0 0.062 A 12 Log 8421.5 -0.049 A
29
Lake
Eri
e
Ann None 1.0 0.126 A
Lake
On
tari
o
Ann None 1.0 0.230 A
1 Log 4060.3 0.335 A 1 Log 1275.6 0.066 A
2 Log 6850.6 0.071 A 2 Log 1632.8 -0.100 A
3 Log 1855.0 -0.081 A 3 Log 316.7 -0.115 A
4 None 1.0 -0.010 A 4 Log 38805.0 0.033 A
5 Log 593.5 -0.109 A 5 Log 0.0 -0.040 A
6 Gam 0.0 0.111 A 6 Log 295.6 -0.060 A
7 Log 1710.1 0.075 A 7 Log 1205.2 0.115 A
8 Log 1634.7 -0.094 A 8 Gam 0.0 -0.195 A
9 Log 1892.1 0.072 A 9 Log 1086.8 -0.160 A
10 Log 4717.8 0.116 A 10 Gam 0.0 -0.075 A
11 Log 4241.6 0.135 A 11 Gam 0.0 0.073 A
12 Log 3972.4 -0.073 A 12 Log 2958.2 0.094 A
Lake
St.
Cla
ir
Ann None 1.0 0.089 A
Lake
St.
Cla
ir
7 Log 140.0 -0.012 A
1 Log 1573.3 0.007 A 8 Log 200.0 0.046 A
2 Gam 0.0 0.024 A 9 Log 262.4 0.196 A
3 None 1.0 -0.042 A 10 Log 193.9 0.430 R
4 None 1.0 0.111 A 11 Log 126.9 -0.190 A
5 Log 553.0 -0.139 A 12 Log 1200.6 0.173 A
6 Log 492.9 -0.057 A
V.2 Application of the Disaggregation Method
For disaggregating annual NBS series to monthly, the Stedinger model (Grygier and Stedinger, 1988; Grygier and Stedinger, 1990) is a condensed contemporaneous model that is expressed as
,1,,, ΛDYCεBYAY (10)
where (1) A , C , and D are diagonal S × S parameter matrices (S=number of stations) and B
is a full S × S parameter matrix for season τ; (2) Y and ,Y are annual and monthly data vector
for year ν and season τ, respectively; (3) ,ε are the independent standard normal S × 1 column
noise vector; and (4) 1,, YWΛ are weighted seasonal flows, where the weights W (a
diagonal S × S matrix) depend on the type of transformations and the term ,Λ ensures that
additivity of the model is approximately preserved, i.e. the seasonal data summing to the annu-al data. Estimated parameters through SAMS2009 (Salas et al., 2009) are presented in Appendix A. For the first season (τ=1), 1C and 1D are null matrices (i.e. all elements for the matrices are
zero) and for the second season (τ=2), 2C is a null matrix. Further details are referred to
(Grygier and Stedinger, 1988; Grygier and Stedinger, 1990). Note that this Stedinger model does
30
not preserve the lag-1 correlation between the first season of a given year and the last season of the previous year. When using transformed data, the additivity constraint that seasonal flows sum to the annual flow is lost. However, the difference is small. One can correct the sea-sonal flows to meet this constraint. In this study, no corrections were made.
V.3 Disaggregation results
Detailed results are presented in detail in chapters VII and VIII where their statistical character-istics are compared to those of observations.
31
Chapter VI. Verification of the statistical properties of the dis-aggregated NBS time series (contemporary resid-ual supplies using global climate indices)
In the next sections, the statistical characteristics of the generated annual and monthly NBS se-quences are compared to those of historical NBS series. Only the results of Lake Superior and Lake Michigan-Huron with Georgian Bay are presented, as the quality of the fit is generally the same for all lakes. A number of key statistics were estimated to validate the model perfor-mance from the observed data and 500 generated sequences. These statistics are presented with boxplot. In a boxplot, a box displays the interquartile range (IQR) and whiskers extend to the maximum and minimum with horizontal lines at the 10th and 90th percentiles. The horizon-tal line inside the box depicts the median of the data. Also, the value of the statistic corre-sponding to the observed data is represented with a cross mark connected with a dotted line. Frequency plots were also used to compare the empirical CDF of the generated sequences to the empirical CDF of the observations. The frequency plots were generated using the Weibull Plotting Position formula applied to 500 samples of 100 year. The following graphs are present-ed for each of the five lakes (Superior, Michigan-Huron with Georgian Bay, Erie, Ontario and St-Clair)
A plot of the generated annual NBS series versus historical annual NBS series
A plot of the generated monthly NBS series versus historical monthly NBS series
A box-plot of the mean, standard deviation, skewness, lag-1 autocorrelation, minimum and maximum of the generated annual NBS, which are compared to the historical val-ues.
A box-plot of the mean, standard deviation, skewness, lag-1 autocorrelation, minimum and maximum of the generated monthly NBS, which are compared to the historical val-ues
The empirical CDF of the annual observed and simulated NBS time series (the simulated NBS time series are split into several smaller time series of the same length as the ob-served NBS series, and an empirical CDF is plotted for each oxygen short series).
The empirical CDF of the monthly observed and simulated NBS series (same procedures described at the bullet above).
All these plots are intended to serve a qualitative assessment of the performance of the gen-eration procedure. The following arbitrary conventions were used to qualify the fit as perfect, excellent, good, poor or very poor based on of the above-described figures:
Boxplots: the fit is perfect if the point representing the observations falls exactly on the median line; the fit is excellent if the point representing observations is closer to the mean than to any quartile; the fit is good if the point representing the observations is in
32
the intrauterine range but is closer to one quartile than to the mean. The fit is poor if the point representing the observations is outside the interquartile range but within the 10% and 90%. The fit is very poor if it is below the 10% or above the 90% line.
Empirical CDF: if the empirical CDF observed NBS is exactly at the middle oxygen "cloud" of CDFs of simulated NBA, the fit will be considered perfect. If it lies within the upper and lower limits of the "cloud" and is generally closer to the center than to the upper and lower limits of the "cloud" then the fit is excellent. If it lies within the upper and lower limits of the "cloud" and is generally farther to the center than to the upper and lower limits of the "cloud" then the fit is good. If it sometimes happens to cross the up-per or lower limits of the "cloud" then the fit is poor. If it is generally outside the lower and upper limits of the “cloud” then the fit is very poor.
VI.1 Lakes NBS series cross-correlation
Figure VI-1. Cross-correlation of the annual observed (circle) and simulated (boxplot) data. Note that the numbers in x-axis indicates 1: Lake Superior, 2: Lakes Michigan-Huron, 3: Lake Erie, 4: Lake Ontario, 5: Lake St-Clair. The sequence for the cross-correlation along the x-axis is 1-2, 1-3, 1-4, 1-5, 2-3, 2-4, 2-5, 3-4, 3-5, and 4-5
Based on the performance criteria developed at the beginning of the chapter, the quality of the
reproduction of cross-correlation between simulated annual NBS can be qualified as follow:
Excellent: Superior and Ontario
33
Good: Superior and Erie/Erie and St-Clair
Poor and very poor: all the others
The generation methodology fails to reproduce spatial correlation in the system and this may
cause issues in the reproduction of extreme events at downstream lakes when the synthetic
NBS series will be routed through the hydrological model.
Figure VI-2. Cross-correlation of the monthly observed (circle) and simulated (boxplot) data. Note that the numbers in x-axis indicates 1: Lake Superior, 2: Lakes Michigan-Huron, 3: Lake Erie, 4: Lake Ontario, 5: Lake St-Clair. The sequence for the crosscorrelation along the x-axis is 1-2, 1-3, 1-4, 1-5, 2-3, 2-4, 2-5, 3-4, 3-5, and 4-5
Figure IV.2 shows that the quality of the reproduction of cross-correlation between simulated
monthly NBS is poor or very poor in the vast majority of cases.
VI.2 Verification of basic statistics in simulated annual and monthly NBS for Lake Superior
The purpose of this section is to check whether the mean, the standard deviation, the skew, the
lag 1 autocorrelation, the maximum and minimum of Lake Superior simulated NBs are repro-
duced at the annual and monthly scale using the quality measures defined at the beginning of
the chapter. The simulated and monthly NBS are presented on Figure VI-3.
34
Figure VI-3. Time series of Lake Superior for an example of the generated monthly (top) and annual (bottom) data
Figure VI-4. Basic statistics of the annual observed (+) and simulated data (boxplot) of Lake Superior
Figure VI-4 shows that at the annual level, the reproduction of the mean, standard deviation,
lag 1 autocorrelation and maximum of Lake Superior NBS series is excellent. The reproduction
of the maximum falls in the ‘good’ category while the reproduction of the skew is poor.
35
Figure VI-5. Basic statistics of the monthly observed (circle) and simulated (boxplot) data of Lake Superior
Figure VI-5 shows that at the monthly level, the reproduction of the mean, standard deviation,
lag 1 autocorrelation and skew of Lake Superior NBS series is excellent.
Figure VI-6. Empirical CDF of the annual observed (thick line) and simulated (thin grey lines) data of Lake Su-perior
36
Based on the quality criteria developed at the beginning of the chapter, figure VI-6 shows that
the reproduction of Lake Superior annual NBS CDF is good (the empirical CDF lies within the
upper and lower limits of the "cloud" but is quite often farther to the center than to the upper
and lower limits of the "cloud”)
Figure VI-7. Empirical CDF of the monthly observed (thick line) and simulated (thin grey lines) NBS of Lake Su-perior (m
3/s)
At the monthly scale, the reproduction of Lake Superior NBS CDF is varies between excellent
and good (figure VI-7).
VI.3 Verification of basic statistics in simulated annual and monthly NBS for Lake Michigan-Huron with Georgian Bay
The purpose of this section is to check whether the mean, the standard deviation, the skew, the
lag 1 autocorrelation, the maximum and minimum of Lake Michigan-Huron with Georgian Bay
simulated NBS are reproduced at the annual and monthly scale using the quality measures de-
fined at the beginning of the chapter. The simulated and monthly NBS are presented on Figure
VI-8.
37
Figure VI-8. Time series of Lakes Michigan-Huron for the observed monthly (top) and annual (bottom) data
Figure VI-3. Time series of Lakes Michigan-Huron for an example of the generated monthly (top) and annual (bottom) data
38
Figure VI-10. Basic statistics of the annual observed (circle) and simulated data (boxplot) of Lakes Michigan-Huron
Figure VI-8 shows that at the annual level, the reproduction of the mean, standard deviation,
skew, lag 1 autocorrelation and maximum of Lake Michigan-Huron with Georgian Bay NBS se-
ries is excellent. The reproduction of the minimum falls in the ‘good’ category.
Figure VI-11. Basic statistics of the monthly observed (circle) and simulated (boxplot) data of Lakes Michigan-Huron
39
Figure VI-5 shows that at the monthly level, the reproduction of the mean, standard deviation,
and skew of Lake Superior NBS series is excellent. Lag 1 autocorrelation is also very well repro-
duced for all months except January, where it falls in the ‘poor’ category. The CDF of the simu-
lated NBS are well reproduced (Excellent to good) at both the annual and monthly scale (Fig-
ures VI-12 and VI-13)
Figure VI-12. Empirical CDF of the annual observed (thick line) and simulated (thin grey lines) data of Lakes Michigan-Huron
40
Figure VI-13. Empirical CDF of the monthly observed (thick line) and simulated (thin grey lines) data of Lakes Michigan-Huron with Georgian Bay
VI.4 Overall performance of the generation procedure
Based on the results presented in sections VI.1 to VI.3, the generation procedure based on NL-
ARX and ENSO predictors’ successfully reproduces individual Lakes NBS mean, the standard de-
viation, the skew, the lag 1 autocorrelation, the maximum and minimum at both annual and
monthly scales. However, cross-correlations are not that well reproduced. Cross-correlation
plays an important role in the distribution of extreme events at downstream reaches when
flows are routed using a hydrological model. If the generated NBS series are used to simulate
the system, flow statistics for upstream lakes will be more reliable than flow statistics at down-
stream Lakes.
41
Chapter VII. Verification of the statistical properties of the dis-aggregated NBS time series (Contemporary Resid-ual Supplies using NCEP variables)
In the next sections, the statistical characteristics of the annual and monthly NBS sequences generated using NCEP variables are compared to those of historical NBS series. In this chapter, only the results of Lake Superior are presented. The quality assessment procedure is exactly the same as the one used in chapter VI.
VII.1 Lakes NBS series cross-correlation
At the annual scale, NBS cross correlation is well reproduced for Superior/Erie, Superi-
or/Ontario and Superior/St-Clair. The quality of the annual cross-correlation between Superior
and Michigan-Huron with Georgian Bay is at the limit between good and poor according to the
classification adopted at the beginning of chapter VI. However, comparing Figure VI-1 to Figure
VII-1, it is clear that cross-correlation is much better reproduced than with the NL-ARX models.
At the monthly time scale, the quality of the reproduction of cross-correlation varies between
good and poor (Figure VII-2), but are still better than monthly cross-correlation reproduced
with NL-ARX models (Figure VI-2).
42
Figure VII-1. Cross-correlation of the annual observed (circle) and simulated (boxplot) data. Note that the numbers in x-axis indicates 1: Lake Superior, 2: Lakes Michigan-Huron, 3: Lake Erie, 4: Lake Ontario, 5: Lake St-Clair. The sequence for the crosscorrelation along the x-axis is 1-2, 1-3, 1-4, 1-5, 2-3, 2-4, 2-5, 3-4, 3-5, and 4-5
Figure VII-2. Cross-correlation of the monthly observed (X) and simulated (boxplot) data. Note that the num-bers in x-axis indicates 1: Lake Superior, 2: Lakes Michigan-Huron, 3: Lake Erie, 4: Lake Ontario, 5: Lake St-Clair. The sequence for the crosscorrelation along the x-axis is 1-2, 1-3, 1-4, 1-5, 2-3, 2-4, 2-5, 3-4, 3-5, and 4-5
VII.2 Verification of basic statistics in simulated annual and monthly NBS for Lake Superior
The purpose of this section is to check whether the mean, the standard deviation, the skew, the
lag 1 autocorrelation, the maximum and minimum of Lake Superior NBS are reproduced at the
annual and monthly scale using the quality measures defined at the beginning of the chapter.
The simulated and monthly NBS are presented on Figure VI-8.
43
Figure VII-3. Time series of Lake Superior for the observed monthly (top) and annual (bottom) data
Figure VII-4. Basic statistics of the annual observed (circle) and simulated data (boxplot) of Lake Superior
Figure VI-4 shows that the mean, standard deviation, skew and lag one autocorrelation are very
well reproduced at the annual by the generation procedure (category=’Excellent’ according to
the classification adopted at the beginning of chapter VI). However, the maximum is overesti-
mated and the minimum is underestimated (category=’Poor’ according to the classification
44
adopted at the beginning of chapter VI). At the monthly scale, the mean, standard deviation,
skew and lag one autocorrelation are very well reproduced too (Figure VII-5, catego-
ry=’Excellent’).
Figure VII-5. Basic statistics of the monthly observed (X) and simulated (boxplot) data of Lake Superior
Figures VII-6 and VII-7 show that the CDF of the simulated NBS are well reproduced (Excellent to good) at both the annual and monthly scale.
45
Figure VII-6. Empirical CDF of the annual observed (thick line) and simulated (thin grey lines) data of Lake Su-perior
Figure VII-7 Empirical CDF of the monthly observed (thick line) and simulated (thin grey lines) data of Lake Su-perior
46
VII.3 Overall performance of the generation procedure
The NBS generation procedure presented in this chapter successfully reproduces individual
Lakes NBS mean, the standard deviation, the skew, the lag 1 autocorrelation, the maximum and
minimum at both annual and monthly scales. However, cross-correlations are not that well re-
produced but are better the ones obtained with NL-ARX. One source of concern is the fact that
the maximum of the Superior annual NBS is underestimated while the minimum is overesti-
mated. This may lead to underestimation of high flows and overestimation of low flows.
47
Chapter VIII. Statistics of Lakes Levels Statistics simulated with Contemporary Residual Supplies
The two synthetic NBS sequences described in chapters V and VI were routed using the coordi-nated Great Lakes Routing Model. The routing exercise was carried out by Dr Yin Fan and David Fay, from Environment Canada at Cornwall. For each of the five Lakes, the plots below will be used to assess if the simulated flows are representative of observed flows in the system.
1. A plot of the generated NBS sequences, superimposed with the historical mean and the historical 95% confidence interval (i.e. the mean plus 1.96 times the standard deviation). Results are considered satisfactory if the simulated flows mostly fall within the 95% confidence intervals of the historical flows
2. CDFs of 500 100-year sub-series extracted from the generated NBS vs. a CDF of his-torical NBS series. Results are acceptable if the empirical CDF of the historical flows lie within the limits of ‘cloud’ of CDFs generated with the simulated values
3. A frequency plot of the sorted sorted average annual values and generated NBS se-ries versus 1/(1-p), where p is the exceedance probability. The frequency plot is cal-culated using Weibull Plotting Position formula. Results are satisfactory if the two plots agree on the lower values (i.e. the probability of low levels are similar for ob-served and simulated flows)
4. A frequency plot of the sorted sorted average annual values and generated NBS se-ries versus 1/(p), where p is the exceedance probability. The frequency plot is calcu-lated using Weibull Plotting Position formula. Results are satisfactory if the two plots agree on the higher values (i.e. the probability of high levels events are similar for observed and simulated flows)
The plots are presented below for Lake Superior only as similar patterns were obtained for all lakes.
48
VIII.1 Lake Superior levels statistics(contemporary supplies using ENSO)
Figure VIII-1 presents the simulated mean annual Lake Superior levels superimposed with the
mean historical levels and 95% historical confidence intervals. The simulated values are cen-
tered on the historical mean and most of the simulated values lie within the 95% confidence
interval.
Figure VIII-1. Superior, frequency plot (Weibull Probability Plot) of annual series
49
Figure VIII-2. Superior, Simulated and mean historical levels
Figure VIII-2 shows that the CDF of annual mean lake levels is well reproduced since the empiri-
cal CDF of observations lies in the middle of the ‘cloud’ of CDF of similar-length samples drawn
from the simulated NBS series.
Figure VIII-3. Superior, frequency plot (Weibull Probability Plot) of annual series (low tail)
50
Figure VIII-4. Superior, frequency plot (Weibull Probability Plot) of annual series (high tail)
Figures VIII-3 and VIII-4 show a good fit at the upper and lower tails of the frequency plots be-
tween historical and simulated values. Therefore the generation procedure was able to capture
the frequency of low and high waters levels for the historical period.
VIII.2 Lake Superior levels statistics(contemporary supplies using NCEP
variables)
Figure VIII-5 shows that the simulated mean annual lake levels are centered on the historical
mean and that most of the simulated values lie within the 95% confidence intervals.
51
Figure VIII-5. Superior, frequency plot (Weibull Probability Plot) of annual series
The empirical CDF or mean annual levels lies in the middle of the ‘cloud’ of CDF of similar-
length samples drawn from the simulated NBS series (Figure VIII-6), but it is very close to the
lower limit for higher lake levels, and very close to the higher limits for lower Lake levels. There-
fore simulated flows tend to overestimate high lake levels and underestimate low lake levels.
52
Figure VIII-6. Superior, Simulated and mean historical levels
Figure VIII-6. Superior, frequency plot (Weibull Probability Plot) of annual series (low tail)
Figure VIII-6 shows that
Figure VIII-7. Superior, frequency plot (Weibull Probability Plot) of annual series (high tail)
53
Figures VIII-6 and VIII-7 show a reasonable at the upper and lower tails of the frequency plots
between historical and simulated values. Therefore the generation procedure was able to cap-
ture the frequency of low and high waters levels for the historical period.
54
Chapter IX. Lake Levels Statistics (Sequences Accounting for Climate Change)
500 sequences of 100 years of synthetic NBS were generated for each of the A1B and A2 scenarios. These sequences were routed using the coordinated Great Lakes Routing Model. The climate change induced variations of both annual NBS and lake levels are presented here for periods 2001-2025, 2026-2050, 2051-2075 and 2076-2100. The 1900-2008 and 1976-2000 his-torical average NBS and lake levels are plotted for comparison purpose.
IX.1 Lake Superior
IX.1.1 Net Basin Supply
Figure IX-1. Superior NBS variations under climate change (upper panel: scenario A1B; lower panel: scenario A2)
Under scenario A1B, Both the 1900-2008 and 1976-2000 average annual NBS are above the
higher quartile of the simulated NBS. Under this scenario, Inflows to Lake Superior will system-
atically be lower than what have been observed, with very small variations from period to peri-
od. Under scenario A2, the median lake level will start decreasing around 2050. The reduction
in NBS will reach up to 50% in 2076-2100.
55
IX.1.2. Lake Levels
Figure IIX-2. Superior levels variations under climate change (upper panel: scenario A1B; lower panel: scenar-io A2)
Lake levels follow the same trends as the Net Basin supply discussed in section VI.1.1. Levels
will be systematically lower than historical levels, with an important decrease at the end of the
century under scenario A2 (Figure IX-2).
IX.2 Lake Michigan Huron with Georgian Bay
IX.2.1 Net Basin Supply
Under scenario A1B, lake Michigan-Huron NBS will remain close to the historical means until
2050 then decrease slightly (Figure IX-3). Both the 1900-2008 and the 1976-2000 mean annual
NBS supplies stay in the interquartile range of the simulated NBS. Under scenario A1B, the me-
dian NBS will be slightly above historical values for the first 25 years, and then stay below the
historical means for the remaining 75 years. The trend is not uniform as the median NBS for
2051-2075 is below both the median NBS for 2026-2050 and the one of 2076-2100.
56
Figure IIX-3. Michigan-Huron NBS variations under climate change (upper panel: scenario A1B; lower panel: scenario A2)
IX.2.2 Lake Levels
The graph on Figure IX-4 shows that under both scenarios Lake Michigan-Huron with Georgian
Bay mean level will be below the historical means and will keep decreasing under both climate
change scenarios. The decrease will be more marked under scenario A2.
IX.3 Lake Erie
IX.3.1. Net Basin Supply
Figure IX-5 shows that under scenario A1, Lake Erie NBS will be stable and close to historical
mean. However under scenario A2, there will be a significant increasing trend in Lake Erie’s NBS
between 2000 and 2100.
Figure IX-4. Erie NBS variations under climate change (upper panel: scenario A1B; lower panel: scenario A2)
57
IV.3.2 Lake Levels
Despite the stable or increasing projected NBS, Lake Erie’s levels projections are below histori-
cal levels because of the influence of upstream Lakes (Superior and Michigan-Huron). Levels will
be decreasing under both scenarios and the decrease will be more pronounced under scenario
A2.
Figure IX-5. Erie levels variations under climate change (upper panel: scenario A1B; lower panel: scenario A2)
IX.4 Lake St-Clair
IX.4.1. Net Basin Supply
Projected NBS supplies for lake St-Clair are close to or above historical levels for all
four periods and both scenarios (Figure IX-7). Under both scenarios, the supply is the
highest in the 2001-2025 period, decrease dusing the 2026-2050 and 2052-2075
58
periods and then increase again during the last period without reaching the
magnitude of the first period..None of the two scenarios seems to be dryer or wetter
than the other.
Figure IIX-6. St-Clair NBS variations under climate change (upper panel: scenario A1B; lower panel: scenario A2)
IX.4.2 Lake Levels
Despite its increasing projected NBS, Lake St-Clair’s levels projections are below historical levels
because of the influence of upstream Lakes (Superior and Michigan-Huron). Levels will be de-
creasing under both scenarios and the decrease will be more pronounced under scenario A2.
59
Figure IIX-7. St-Clair levels variations under climate change (upper panel: scenario A1B; lower panel: scenario A2)
60
IX.5 Conclusion
The two synthetic NBS sequences accounting for climate change were compared to historical
values then used to simulate the levels of the four upstream lakes (Superior, Michigan-Huron
with Georgian Bay, Erie and St-Clair). The two larger and upper lakes displayed significant de-
creases in both NBS and levels. The decreases were more pronounced under scenario A2. The
two last lakes displayed stable or increasing NBS, but their levels were driven down by Lake Su-
perior and Lake St-Clair. Overall, both scenarios predict significant level decrease throughout
the great lakes system. These finding should however be factored by the large uncertainty in
the predictors (GCM outputs), the relationships between predictors and NBS, and finally the
statistical characteristics of the generated synthetic NBS. The main conclusion is that any new
management plan for the great lakes system should be able to cope with lower inflows and
lower levels for the next century.
61
References
Amadou, A., Seidou, O., Ouarda, T.B.M.J., 2009. Nonlinear ARX models of the Interconnexxion between the Great Lakes NBS components and Global Climate Indices. report submitted to the International Joint Commission. 185p.
DAI CGCM3 Predictors (2008): Sets of Predictor Variables Derived From CGCM3 T47 and NCEP/NCAR Reanalysis, version 1.1, April 2008, Montreal, QC, Canada, 16 pp
Fagherazzi, L., Guay, R. and Spark, Salas, J.D., Sveisson, O., (2005). Stochastic modeling and simulation of the Great Lakes-St-Lawrence river system. Hydr0-Quebec Production.
Grygier, J.C., Stedinger, J.R., 1988. Condensed Disaggregation Procedures and Conservation Corrections for Stochastic Hydrology. Water Resources Research, 24(10): 1574-1584.
Grygier, J.C., Stedinger, J.R., 1990. SPIGOT, A Synthetic Streamflow Generation Software Package, School of Civil and Env. Engr., Cornell University, Ithaca, NY.
Lee, T., Ouarda, T.B.M.J., 2010. Multivariate Stochastic Simulation of Climate Indices with Empirical Model Decomposition R-1188, INRS-ETE, Quebec.
Ljung, L. (2007). "System Identification Toolbox 7 User's Guide," The MathWorks
Mantua, N.J., Hare, S.R., Zhang, Y., Wallace, J.M., Francis, R.C., 1997. A Pacific interdecadal climate oscillation with impacts on salmon production. Bulletin of the American Meteorological Society, 78(6): 1069-1079.
Rodionov, S., Assel, R.A., 2003. Winter severity in the Great Lakes region: a tale of two oscillations. Clim Res, 24(1): 19-31.
Salas, J.D., Delleur, J.W., Yevjevich, V., Lane, W.L., 1980. Applied Modeling of Hydrologic Time Series. Water Resources Publications, Littleton, Colorado, 484 pp.
Salas, J.D., Sveinsson, O., Lee, T.S., Lane, W., Frevert, D., 2009. Developments on Stochastic Analysis, Modeling, and Simulation (SAMS 2009), World Environmental and Water Resources Congress 2009: Great Rivers. ASCE, Kansas City, MO, pp. 4840-4849.
Torrence, C., and Compo, G.P. (1998). A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc. 79:61-78
Tsonis, A.A., Elsner, J.B., Sun, D.-Z., 2007. The Role of El Niño—Southern Oscillation in Regulating its Background State, Nonlinear Dynamics in Geosciences. Springer New York, pp. 537-555.
Part D
Predictability of Climate Indices with Time Series Models
Dr. Taesam Lee and Taha Ouarda Institut National de la Recherche Scientifique, centre Eau et Environnement
Dr. Ousmane Seidou University of Ottawa
Revised December 2011
i
Predictability of climate indices with time series models Taesam Lee1,2, Taha B.M.J. Ouarda2, and Ousmane Seidou3 1Gyeongsang National University 2INRS-ETE 3University of Ottawa
Presented to: International Joint Commission
Date: March 2011 Revised: December 2011
ii
Predictability of climate indices
using time series models
Taesam Lee1,2
, Taha B.M.J. Ouarda2, and Ousmane Seidou
3
1 Dept. of Civil Engr., Gyeongsang National University,
900 Gajwa-dong, Jinju-si, Gyeongsangnam-do, 660-701, South Korea
2 Chaire du Canada en Estimation des Variables Hydrométéorologiques
INRS-ETE 490, de la Couronne Québec (Québec) G1K 9A9, CANADA
3Dept. of Civil Engr., University of Ottawa
161 Louis Pasteur Office A113, Ottawa, ON, K1N 6N5, CANADA
Presented to:
International Joint Commission
iii
Contents
Acknowledgements ............................................................................................................ viii
Executive Summary............................................................................................................ viii
1. Introduction .................................................................................................................. 1
2. Mathematical description of applied models ................................................................ 2
2.1. ARMA ................................................................................................................... 2
2.1.1. Model Description ............................................................................................ 2
2.1.2. Parameter estimation and model selection ........................................................ 3
2.1.3. Predicting ARMA processes ............................................................................. 4
2.2. GARCH ................................................................................................................. 5
2.2.1. Definitions and representations of GARCH( qp ~,~ ) ........................................... 5
2.2.2. Prediction method of GARCH( qp ~,~ ) ............................................................... 6
2.3. DLM ...................................................................................................................... 8
2.3.1. State Space models and DLM ........................................................................... 8
2.3.2. Kalman filter for parameter estimation and prediction ................................... 11
2.4. EMD and NSOR.................................................................................................. 13
3. Data Description ......................................................................................................... 15
4. Predicting Monthly ENSO.......................................................................................... 16
4.1. Preliminary analysis and application methodology for monthly ENSO index ... 16
4.2. Results ................................................................................................................. 17
5. Predicting Monthly PDO ............................................................................................ 19
5.1. Preliminary analysis and application methodology for monthly PDO index...... 19
iv
5.2. Results ................................................................................................................. 20
6. Summary and Conclusions ......................................................................................... 21
Notation: .............................................................................................................................. 22
References ........................................................................................................................... 24
v
Lists of Tables
Table 1. Basic statistics of the annual ENSO index.
Table 2. AIC of the ARMA(p,q) model for the monthly ENSO index.
Table 3. RMSE values corresponding to the monthly ENSO index for the considered 20 years.
Table 4. Correlation between observed versus predicted monthly ENSO index values from
different models for the considered 20 years.
Table 5. Basic statistics of the annual PDO index.
Table 6. AIC of the ARMA(p,q) model for the monthly PDO index.
Table 7. Comparison of RMSE values of the monthly PDO index for the considered 20 years.
Table 8. Correlation between observed versus predicted monthly ENSO index values from
different models for the considered 20 years of the PDO index.
Table 9. Comparison of the RMSE values of the monthly PDO index for ARMA(5,0), EMD-
NSOR, and ARMA(5,0)-GARCH(1,1) models for the considered 20 years.
Lists of Figures
Figure 1. Annual (top panel) and monthly (bottom panel) ENSO time series.
Figure 2. Autocorrelation function (ACF) of the monthly ENSO index.
Figure 3. Seasonal variations of time series and statistics for the monthly ENSO index. (a)
Spaghetti plot of time series for each year and (b)-(d) monthly statistics.
Figure 4. Spectral density of the monthly ENSO index. Note that g(f) presents the smoothed
sample spectral density at frequency f (see Salas et al. 1980).
Figure 5. Scatter plots of the monthly ENSO index.
vi
Figure 6. RMSE values corresponding to the monthly ENSO index for the considered 20 years
(1990-2009) for ARMA(1,0), ARMA(4,0), ARMA(7,3), and ARMA(8,5) models. Note that the
x-axis represents the lag time (h).
Figure 7. Same as Figure 6 but for the models ARMA(4,0), Tr(1)-AR(4), ARMA(4,0)-
GARCH(1,1), and EMD-NSOR.
Figure 8. Correlation corresponding to the monthly ENSO index for the considered 20 years
(1990-2009) for ARMA(1,0), ARMA(4,0), ARMA(7,3), and ARMA(8,5) models.
Figure 9. Same as Figure 8 but for the models ARMA(4,0), Tr(1)-AR(4), ARMA(4,0)-
GARCH(1,1), and EMD-NSOR.
Figure 10. Predicting the monthly ENSO index using ARMA(4,0) model for lag time h=1,…,6
months and for the considered 20 years (1990-2009). Note that the red-cross line represents the
observed values and the black solid line represents the mean prediction while the gray regions
show 95 percent upper and lower limits of the mean.
Figure 11. Same figure as Figure 10 but using ARMA(8,5) model.
Figure 12. Same figure as Figure 10 but using Tr(1)-AR(4) model.
Figure 13. Same figure as Figure 10 but using ARMA(4,0)-GARCH(1,1) model.
Figure 14. Annual (top panel) and monthly (bottom panel) PDO time series for 1900-2009.
Figure 15. Autocorrelation function (ACF) of the monthly PDO index.
Figure 16. Seasonal variations of time series and statistics for the monthly PDO index. (a)
Spaghetti plots of time series for each year and (b)-(d) monthly statistics.
Figure 17. Scatter plots of the monthly PDO index.
Figure 18. RMSE values corresponding to the monthly ENSO index for the considered 20 years
(1990-2009).
vii
Figure 19. Correlation corresponding to the monthly ENSO index for the considered 20 years
(1990-2009).
Figure 20. Predicting the monthly PDO index using ARMA(9,7) model for lag time h=1,…,6
months and for the considered 20 years (1990-2009). Note that the red-cross line represents the
observed values and the black solid line represents the mean prediction while the gray regions
show 95 percent upper and lower limits of the mean.
Figure 21. Same figure as Figure 20 but using ARMA(28,0) model.
viii
Acknowledgements
The authors are grateful to the International Joint Commission for the management of the Great
Lakes for its financial support. The authors wish also to acknowledge the help of Reza Modarres
and Deepti Joshi, Ph.D. students at INRS-ETE.
Executive Summary
The objective of the current study was to predict climate indices that would affect Great Lakes
system Net Basin storage in the future. Preliminary studies have shown that in comparison to all
other available indices, the monthly El Niño-Southern Oscillation (ENSO) index and Pacific
Decadal Oscillation (PDO) index show the most significant teleconnection with net basin supply
(NBS) in the Great Lakes system. A number of time series models have been tested including a
traditional Autoregressive Moving Average (ARMA), Dynamic Linear Model (DLM),
Generalized Autoregressive Conditional Heteroscedasticity (GARCH) and the non-stationary
oscillation resampling (NSOR) technique. These models were used to predict the monthly ENSO
and PDO indices in order to eventually predict the NBS. The results showed that the DLM and
GARCH models performed well in predicting the monthly ENSO index. For the monthly PDO
index, predicted values from the traditional ARMA model presented a good agreement with the
observed values for lags lesser than a certain time period. Finally, the forecasted climate indices
were employed to predict the NBS of the Great Lakes system.
1
1. Introduction
The climate system is teleconnected to hydrology. A measure of different parts of the climate
system can be provided by indices such as the El Niño-Southern Oscillation (ENSO) index
(2007), and the Pacific Decadal Oscillation (PDO) index (Mantua et al., 1997). Such relations
are termed as ‘teleconnections’ (Wallace and Gutzler, 1981; Rodionov and Assel, 2000;
Alexander et al., 2002; Chiew and McMahon, 2002; Rodionov and Assel, 2003; Rogers and
Coleman, 2003; Hanson et al., 2004). For example, Rodionov and Assel (2003) found that a
substantial difference of the large-scale atmospheric circulation associated with PDO and ENSO
leads to an abnormally mild winter in the Great Lakes region.
As a result, these climate indices can be used to improve predictions of hydro-
meteorological variables (Changnon, 2004; Thomas, 2007; Kalra and Ahmad, 2009; Devineni
and Sankarasubramanian, 2010; Immerzeel and Bierkens, 2010; Westra and Sharma, 2010). A
number of methods have been developed to predict climate indices (Chen et al., 2004; Cheng et
al., 2010). These are mainly based on Global Climate Models (GCM) (Schneider et al., 1999;
Wu and Kirtman, 2003; Kirtman and Min, 2009). However, GCM based prediction is an
expensive undertaking. Prediction and retrospective data required to run the models are not
always available beyond the atmospheric scientific community. In the current study, therefore,
we predicted climate indices based on time series models which are easy to implement.
For this study, autoregressive moving average (ARMA) (Brockwell and Davis, 2003), a
traditional time-series model, was the main model used to predict climate indices. The
Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model (Engle, 1982) and
the Dynamic linear model (DLM) (West and Harrison, 1997; Petris et al., 2009) were also tested
2
along with the nonstationary oscillation resampling (NSOR) technique developed by Lee and
Ouarda(2010a). Nonlinear time series models (Fan and Yao, 2003; Ahn and Kim, 2005) could
have been tested, but due to the absence of nonlinear serial dependence in the considered climate
indices, these models were omitted.
From preliminary studies and literature review, it was found that the NBS components of
the Great Lakes system can be better predicted by incorporating teleconnections with predicted
climate indices, particularly ENSO and PDO. Thus, the primary objective of the current study
was to predict these two monthly climate indices using time series models so that the simulated
indices could be incorporated into the prediction of NBS components for the Great Lakes system.
A mathematical description of the applied time series models is outlined in section 2,
followed by an explanation of the employed climate indices in section 3. The functioning and
strengths of the predicted climate indices, ENDO and PDO, are shown in sections 4 and 5,
respectively. Summary and conclusions are presented in section 6.
2. Mathematical description of applied models
2.1. ARMA
2.1.1. Model Description
Xt is called an ARMA(p, q) process if Xt is stationary and if for every t
3
qtqttptptt ZZZXXX ...... 1111 (1)
where Zt is white noise with zero mean (i.e. 0Z ) and variance 2
Z (Salas et al., 1980;
Brockwell and Davis, 2003). Xt is said to be an ARMA(p, q) process with mean X if XtX is
an ARMA(p, q) process. Eq.(1) can also be simply expressed as
tt ZBXB )()( (2)
where p
pBBB ...1)( 1 , q
q BBB ...1)( 1 , and B is backward shift operator.
Xt in Eq.(2) can be further expressed as the
0
)()(
)(
j
jtjttt ZZBZB
BX
(3)
where )(B = )(
)(
B
B
.
2.1.2. Parameter estimation and model selection
A number of methods to estimate the parameters of the ARMA process (described by Eq.(1))
have been developed such as the Yule-Walker estimation (Yule, 1927; Walker, 1932), Burg’s
algorithm based on the forward and backward prediction errors (Burg, 1978), the innovations
algorithm (Brockwell and Davis, 1988), the Hannan-Rissanen algorithm (Hannan and Rissanen,
1982), and maximum likelihood estimation (MLE) (Brockwell and Davis, 2003).
The Yule-Walker estimation is derived by multiplying each side of Eq.(1) by Xt-j , j=0,1,…,
p+q and taking expected values on both sides. The obtained relations are termed as Yule-Walker
equations. These equations can be solved for θi (i=1,2,…q) and φj (j=1,2,…,p) by using the
estimates of the lagged second moments
4
In MLE, supposing Xt is a Gaussian time series, the likelihood of )',...,( 1 nn XXX (given
by Eq.(4) below) is maximized to estimate the parameters, where n is the number of records.
)'2/1exp()det()2()(12/12
nnnn
nL XCXCψ (4)
when ψ is the parameter set of the model. In Eq.(4),
)'( nnn E XXC , ],,[ 2
Zθφψ , )',...,( 1 pφ , and )',...,( 1 qθ
Note that the right side of Eq.(4) can be described as a function of φ , θ , and 2
Z . In the current
study, MLE was used to estimate the parameters of the ARMA model.
The Akaike Information Criterion (AIC), proposed by Akaike (1974), is used to compare
models having a different number of parameters. According to the criterion, the model with
lowest AIC value is considered to perform the best. The AIC criterion is given by
))(log(22AIC ψLnpar (5)
where npar is the number of parameters. Hurvich and Tsai (1989) introduced the bias corrected
version of AIC (denoted as AICC) which is defined as
)1/()1(2AICAICC parparpar nnnn (6)
2.1.3. Predicting ARMA processes
To predict Xn+h , for lag h(>0), a linear combination of the available data values [ Xn , Xn-1 ,…, X1]
that minimizes mean squared error must be found. The h-step ahead prediction Xn+h is given by
][...][][...][)(ˆ1111 qhnqhnphnphnn ZZXXhX (7)
For each j (j=1,..,p) and for the quantities with coefficients j , the corresponding value for
jhnX is substituted, if it is known. Otherwise, for each k = 1…h-1, khnX must first be
5
evaluated, and then substituted into the expression for jhnX . Simarly, for each j (j=1,..,q) and
for the quantities with coefficients j , jhnZ is substituted with its corresponding value. If
jhnZ is unknown, a value of 0 can be used. A complete description the ARMA process can be
referred to in Brockwell and Davis (Brockwell and Davis, 2003).
2.2. GARCH
Engle (1982) introduced the Autoregressive Conditional Heteroscedasticity (ARCH) model
to generalize the assumption of a constant one-period forecast variance. A natural generalization
of the ARCH process, known as GARCH, to allow for past conditional variances in the
conditional variance equation was proposed by Bollerslev (1986). The fundamental concept of
the GARCH model is that the current variance is dependent on past values of the series. Thus,
the conditional variance is expressed as a linear function of the squared past values of the series
(Engle and Kroner, 1995). The GARCH models have been widely used in Econometrics and
other fields (Bollerslev et al., 1992; Engle, 2001; Engle, 2002; Bosley et al., 2008). A few
applications in hydrometeorological fields have been shown in Elek and Márkus (2004), Ahn
and Kim (2005), and Wang et al. (2005). A brief description of the GARCH models and their
prediction procedure is presented in the following subsections.
2.2.1. Definitions and representations of GARCH( qp ~,~ )
A process Zt is called a GARCH( qp ~,~ ) process if it satisfies the following properties:
(i) 0),|( tuZZE ut (8)
(ii) 222 )()(),|( ttutt BZBtuZZVar (9)
6
where, the parameters of the GARCH process ( ,
q
i
i
i BB
~
1
)( and
p
j
j
j BB
~
1
)( ) exist.
The likelihood of the GARCH process is given by
n
t t
tt
ZL
12
22/12 exp)2()(
ψ (10)
whereψ is the parameter set of the GARCH process. These parameters are estimated by MLE
(Francq and Zakoian, 2010) based on the likelihood given by Eq. (10).
2.2.2. Prediction method of GARCH( qp ~,~ )
Equations(8) and (9) can be conveniently rewritten as the following (Andersen et al., 2003;
Francq and Zakoian, 2010)
ε
ε
ε
ε
ε
Z
Z
Z
... ...
... ...
... ...
... ...
...
-β ... -β βα β α
ω
ε
ε
ε
Z
Z
Z
t
t
pt-
t-
t-
t-r
t-
t-prr
pt-
t-
t
t-r
t-
t
0
0
0
0
01000
00100
0000
0010
0001
0
0
0
0
0
~
2
1
2
2
2
2
1~111
1~
1
2
1
2
1
2
(11)
where 2222 1 ttttt )σ(ησZε , )1,0(~ Nt and )~,~max( qpr .
The matrix form of Eq. (11) can be simplified to
tmtt )( 11
2
11
2
eeΓΞeΞ (12)
7
where ie is a vector with all components equal to zero except the ith
component, which is equal
to 1. The parameter matrix is Γ and 2
tΞ is the vector representing the column vector on the left
hand side of equation (11).
Recursively, the h-step ahead GARCH( qp ~,~ ) process is expressed as
21
0
111
2 ))(( t
hh
i
ihtr
i
ht ΞΓeeeΓΞ
(13)
The h-step ahead predictor for the conditional variance from the GARCH( qp ~,~ ) process is
1
0
2
,
1~
0
2
,
22 )|()|(r
j
jthj
p
i
ithihthttht ZIEIZE (14)
where It contains information about 2
it and 2
jtZ presented in the last two summation terms of
Eq.(14), and
1
1
1 )...(' eΓΓ1e h
h
11, ' ir
h
hi eΓe for i=0,…, p~ -1
1~for '
1~0for )('
11
111
,,…, r-pi=
-p,…, i=
i
h
iri
h
hieΓe
eeΓe (15)
where 1 is an identity matrix.
As an example, the predictor of the popular GARCH(1,1) process is illustrated as
2
1
1
11
1
0
2
11111
2 )()()()|( t
hh
i
t
ihi
tht ZIE
(16)
8
2.3. DLM
2.3.1. State Space models and DLM
State space models consider time series data as output from a dynamic system perturbed by
random disturbances (Künsch, 2001; Migon et al., 2005). A DLM is one of the important classes
of state space models (West and Harrison, 1997; Petris et al., 2009). A DLM is specified for a
vector tX by a normal distribution of the m-dimensional state vector ( tΛ ), where tX has s
variables ( 1s ) . At time t=0,
),( 000
CmΛ N (17)
together with a pair of equations for each time 1t ,
tttt VΛFX ),0(~ V
tt N CV (18)
tttt WΛGΛ 1 ),0(~ W
tt N CW (19)
where tF and tG are known ms and mm matrices; tV and tW are mutually independent
error sequences with Gaussian (normal) distribution; 0m and
0C are initial condition of the
mean and covariance of the state vector tΛ ; and V
tC and W
tC present the time dependent
covariance matrix. Note that Eq.(18) is the observation equation for the model defining the
sampling distribution of tX conditional on the quantity tΛ , while Eq.(19) is the evolution state
system equation, defining the time evolution of the state vector.
If tF and tG are constant for all t, then the model is referred to as a time series DLM
(TSDLM) and if the covariance matrices V
tC and W
tC are constant for all time t, then it is referred
9
as a constant DLM (CDLM). In the current study, we use the constant time series DLM
(TCDLM) such that FF t , GG t , VV
t CC and WW
t CC .
The ARMA model in Eq.(1) is represented by a TCDLM model as
ttX F (20)
ttt Z 1G (21)
where
] 0 0 1 [ F (22)
' ... 1 11 r (23)
and
0 ... 0 0
1 ... 0 0
0 ...
0 ... 1 0
0 ... 0 1
1
2
1
r
r
G (24)
and }1,max{ qpr , 0j for j>p and 0j for j>q.
Furthermore, the kth
order polynomial trend model (Godolphin and Harrison, 1975;
Abraham and Ledolter, 1983), denoted as Trend(k+1), for a univariate time series is described by
the DLM , as
] 0 0 1 [ F (25)
1 0 ... 0 0 0
1 1 ... 0 0
...
0 ... 1 1 0
0 ... 0 1 1
G (26)
10
and
),...,( 22
1 kWW
W diag C and VC = 2
V (27)
The random walk plus noise model, or local level model, (Petris et al., 2009) is a special
case of the polynomial trend model (Trend(1)) defined by
ttt VX ),0(~ 2
Vt NV (28)
ttt W 1 ),0(~ 2
Wt NW (29)
where s=m=1 and F=G=1. Also, the linear trend model, Trend(2) is presented by Equations. (25),
(26), and (27) as
] 0 1 [F (30)
1 0
1 1 G (31)
and VC = 2
V and ),( 22
21 WW
W diag C .
The ARMA model and the polynomial trend model can be combined through TCDLM
representation and is denoted as Trend(k+1)-ARMA(p,q). For example, the combination of the
Trend (2)-ARMA(2,0) model is given as
]0 1 0 1 [F (32)
0 0 0
0 0 0
0 0 1 0
0 0 1 1
2
1
G (33)
and VC = 2
V and )0,,,( 222
21 ZWW
W diag C .
11
2.3.2. Kalman filter for parameter estimation and prediction
In the description of the DLM models given in section 2.3.1, it was noted that all the variables
follow a normal distribution. Therefore, they can be completely determined by the first and
second moments (i.e. mean and variance). The Kalman filter (Kalman, 1960) gives us the
solution for the intricate problem of parameter estimation and prediction for the DLM. The
Kalman filter (Snyder, 1985) is an algorithm for efficiently doing exact inference in a linear
dynamic system. Three propositions for Kalman filtering, smoothing, and prediction are
described in the following text. Kalman filtering and smoothing are employed during parameter
estimation, while Kalman prediction is used during model simulation.
Proposition 1 (Kalman filtering) Consider the DLM given by Eqs. (17)-(19). Let
),(| 11:11
tttt N CmxΛ (34)
where 1:1 tx represents the observed data of the series X for the time period starting from 1 to t-1.
Then,
(i) The one-step-ahead predictive distribution of tΛ given that 1:1 tx is normal with
parameters
11:1 )|( ttttt E mGxΛa (35)
W
ttttttt Var CGCGxΛR
')|( 11:1 (36)
(ii) The one-step-ahead predictive distribution of tX given that 1:1 tx is normal with
parameters
ttttt E aFxXf )|( 1:1 (37)
V
ttttttt Var CFRFxXQ ')|( 1:1 (38)
12
(iii) The filtering distribution of tΛ given that t:1x is normal with parameters
)(')|( 1
:1 ttttttttt E fXQFRaxΛm (39)
ttttttttt Var RFQFRRxΛC1
:1 ')|( (40)
In time series analysis, one is often confronted with a case where one wants to reconstruct
the behavior of the system (i.e. backward estimation of all the observed states). This process,
known as smoothing recursion, can be stated in terms of means and variances in the following
manner.
Suppose that n observations x1,…, xn ; denoted by n:1x are available , then
Proposition 2 (Kalman smoothing)
If ),(~| 11:11
S
ttnt N CsxΛ , then
),(~| :1
S
ttnt N CsxΛ (41)
where
)(')|( 11
1
11:1
ttttttntt E asRGCmxΛs (42)
ttt
S
ttttttnt
S
t Var CGRCRRGCCxΛC 1
1
111
1
11:1 )(')|( (43)
Similar to the cases of filtering and smoothing described in Propositions 1 and 2, due to
the normality assumption, the distribution of the predicted values can be explicitly described for
lag h≥1 as follows
Proposition 3 (Kalman prediction)
(i) The distribution of htΛ given that t:1x is normal with parameters
)1()|()( :1 hEh thtthtt aGxΛa (44)
W
hthtthtthtt hVarh CGRGxΛR ')1()|()( :1 (45)
13
where tt ma )0( and tt CR )0(
(ii) The distribution of tX given that 1:1 tx is normal with parameters
)()|()( :1 hEh thtthtt aFxXf (46)
V
thtthtthtt hVarh CFRFxXQ ')()|()( :1 (47)
Note that in TCDLM, propositions 1-3 can be significantly simplified for cases when FF t ,
GG t , VV
t CC and WW
t CC for all t.
MLE is applied to estimate DLM parameters, which maximizes likelihood defined as
n
t
ttttt
n
t
tL1
1
1
)()'(2
1log
2
1)( fXQfXQψ (48)
where,ψ represents the parameter set in Eqs. (18) and (19). Note that ft is the Kalman filtering
expression of the one-step-ahead predictive distribution of tX given that 1:1 tx is normal, as shown
in Eq. (37). The optimization problem in Eq. (48) is solved through the Limited memory
Broyden– Fletcher–Goldfarb–Shanno method for Bound-constrained optimization (L-BFGS-B)
(Petris et al., 2009). This is the only method accepting restrictions in parameter spaces.
Furthermore, the Bayesian parameter estimation procedure for the DLMs has been established
assuming prior distributions of the parameters (West and Harrison, 1997; Petris et al., 2009).
2.4. EMD and NSOR
Lee and Ouarda (2010b) proposed a stochastic simulation model to adequately reproduce the
smoothly varying nonstationary oscillation (NSO) processes embedded in the observed data. The
proposed model employed a decomposition technique (Huang et al., 1998; Huang and Wu, 2008),
called Empirical Mode Decomposition (EMD). In addition to that, the nonparametric time series
14
models, k-nearest neighbor resampling (Lall and Sharma, 1996) and block bootstrapping, were
employed. This is called NSO resampling (NSOR). The overall procedure for EMD-NSOR
prediction is given below:
(1) Decompose the concerned time series (Xt) into a finite number of intrinsic mode
functions (IMFs).
(2) Select significant IMF components using a significance test (Wu and Huang, 2004) and
subjective criteria (Lee and Ouarda, 2010a).
(3) Fit stochastic time series models according to the nature of the components determined
in step (2). In the current study, significant IMF components are modelled using NSOR
(discussed later) and the residuals are modelled using first order autoregressive (AR(1))
model.
(4) Predict the IMF components using the fitted models (NSOR and AR(1)).
(5) Sum up the forecasted IMFs from each model.
A brief summary of the NSOR for the selected IMF component(s) is given below:
(1) A block length, LB, is randomly generated from a discrete distribution (e.g., Geometric
or Poisson). Poisson distribution was chosen for the current study as it was already
applied in a previous study (Lee and Ouarda, 2010a). More information on the
selection of this discrete distribution in block bootstrapping can be found in Lee
(2008). The related parameter is selected using variance inflation factor (VIF) (Wilks,
1997; Lee and Ouarda, 2010c).
(2) The weighted distances between the current and observed values and the change rates
of the current and observed values are estimated for each observed value. The
15
variances in the change rate and the original sequences are employed as weights. Here,
the change rate is defined as the difference between the current and immediately
previous value of an IMF component.
(3) The time indices of k-smallest distances among the observed record length, where k is
the tuning parameter, are estimated by Nk as a heuristic approach (Lall and
Sharma, 1996).
(4) One k time index from step (3) is selected, with the weighted probability ( jw , j=1,…k)
given by
k
j
j
j
jw
1'
'/1
/1
(5) The LB change rate values in the subsequent time level of the selected index are taken
and then combined with the previous state level to form the real domain values.
3. Data Description
For the current study, ENSO and PDO were selected as they are known to be teleconnected with
the hydro-climatological variables of the Great Lakes system (Amadou et al., 2010). Therefore,
these indices have been employed to forecast the NBS of the Great Lakes system. A brief
description of each of these climate indices is given in the following paragraphs.
ENSO is a climatic pattern occurring across the tropical Pacific Ocean, causing climate
variability on 3-7 year periods (Alexander et al., 2002). Among various ENSO indices
(Trenberth, 1997), the multivariate ENSO index developed by Wolter and Timlin (1993) was
16
employed in the current study. The dataset, ranging from 1950-2009, was downloaded from
http://www.esrl.noaa.gov/psd/people/klaus.wolter/MEI/.
The PDO index represents the principal component of the sea-surface temperature
anomalies in the North Pacific Ocean, polewards of 20oN. The most commonly used PDO index,
developed by Mantua and Zhang and their colleagues (Mantua et al., 1997; Zhang et al., 1997),
was employed in the current study with the dataset ranging from 1900-2009. It was downloaded
from http://jisao.washington.edu/pdo/PDO.latest.
4. Predicting Monthly ENSO
4.1. Preliminary analysis and application methodology for monthly
ENSO index
The monthly and annual time series of the employed ENSO index is presented in Figure 1. The
monthly time series presents strong persistence, shown in Figure 2, while the annual time series
shows weak serial dependence (only 0.285 for lag-1 autocorrelation function (ACF) in Table 1)
during the period 1950-2009. As shown in Figure 3, the monthly statistics of the ENSO index
does not show seasonal variation. The spectral density of the monthly ENSO index, as shown in
Figure 4, presents no seasonality. The scatter plots in Figure 5 reveal the linear relations for
different lags of the monthly ENSO index. From Figure 5, it is evident for each case that there is
a high density of low magnitude values in the scatter plot, with points becoming more sparse for
higher magnitude values. This suggests the existence of heteroscedasticity (i.e. differing
variance). Therefore, the GARCH model was also applied to this index. Furthermore, DLM,
EMD-NSOR, and different orders of ARMA(p,q) models have been tested.
17
Results from the following models are presented:
(1) ARMA(1,0)
(2) ARMA(4,0)
(3) ARMA(7,3)
(4) ARMA(8,5)
(5) DLM: Trend (1)-ARMA(4,0), simply denoted as Tr(1)-AR(4)
(6) ARMA(4,0) – GARCH(1,1)
The selection of the order of the ARMA models was based on the AIC given by Eq. (5).
The AIC values of ARMA(p,q) for p=0,…,10 and q=0,…,10 are presented in Table 2. Even if
ARMA(8,5) presents the smallest AIC, lower order models with AIC values slightly higher than
that of ARMA(8,5) were selected. ARMA(4,0) is an example of this. (Note that ARMA(4,0) has
the second smallest AIC). In the DLM and GARCH models, the ARMA model was selected as a
base model. A lower order ARMA model was preferred due to parsimony. Therefore, ARMA
(4,0) was selected as a base model for the DLM and GARCH models. Higher orders of the
ARMA models were also tested, but these simulations did not produce better results than
ARMA(4,0) (data not shown).
4.2. Results
To validate model performance, the first 40 years of the monthly ENSO index records (1950-
1989) were used to fit the models. Monthly forecasting was done for the 20 years of ENSO index
records between 1990-2009. Depending on the selected model, different numbers of predictors
were used to make forecasts for each month. For example, for ARMA(4,0) model, four
preceding months were used as predictors. Hence, to predict for January 1990, four months from
18
September-December 1989 were used. For further details, refer to section 2 (the mathematical
description of applied models).
The correlation and root mean square error (RMSE) between forecasted and observed
values were used to assess model performance. The RMSE results are tabulated in Table 3 and
plotted in Figure 6 and Figure 7 whereas the correlation results are tabulated in Table 4 and
plotted in Figure 8 and Figure 9. Note that lower RMSE and higher correlation characterize a
better model.
Model output displayed in Figure 6 shows ARMA(7,3) and ARMA(8,5) have slightly
lower RMSE values than ARMA(4,0) for each monthly lag, This difference can be considered to
be negligible. The RMSE values of ARMA(4,0) are, however, much lower than ARMA(1,0) for
each monthly lag. In Figure 7, Tr(1)-AR(4) and ARMA(4,0)-GARCH(1,1) models showed better
performance in comparison to ARMA(4,0). Very little difference was observed between the
RMSE values of the Tr(1)-AR(4) and ARMA(4,0)-GARCH(1,1) models. The EMD-NSOR
model had the poorest performance. The reason for this can be attributed to the fact that the
EMD-NSOR model was invented to characterize long-term oscillation patterns (Lee and Ouarda,
2010a), whereas the current work is a short-term prediction application.
Analysis of correlation between observed and forecasted values indicates slightly different
behavior in terms of model performance compared to RMSE results. This is illustrated in Table
4, Figure 8 and Figure 9. It is evident from Figure 8 that there is a negligible difference between
the performance of ARMA(4,0) and higher order ARMA models (i.e. ARMA(7,3), ARMA(8,5))
until h=8; after this higher order models show slightly better performance. Model performance
for correlation values in Figure 9 is different from RMSE performance in Figure 7. From Figure
9, it can be said that Tr(1)-AR(4) performs better than ARMA(4,0) for shorter lags (h=2-7
19
months); for longer lags, (h=8-12 months) the performance of ARMA(4,0) becomes better. The
ARMA(4,0)-GARCH(1,1) model consistently performs better than the other models over all
monthly lags. This is because the monthly ENSO index shows heteroscedasticity for all lags
(evident in Figure 5), which is reproduced well by GARCH models (Engle, 2002).
The prediction results for lag h=1-6 months are presented for ARMA(4,0), ARMA(7,3),
Tr(1)-AR(4), and ARMA(4,0)-GARCH(1,1) in Figures 10, 11, 12, and 13, respectively. As the
lag time (h) increases, the 95 percent upper and lower Confidence limits get wider. For years
1997 and 1998, prediction results reduces with lag for each tested model.
5. Predicting Monthly PDO
5.1. Preliminary analysis and application methodology for monthly
PDO index
The monthly and annual time series of the employed PDO index is presented in Figure 14. The
monthly time series presents strong persistence as shown in Figure 15 while the annual time
series shows significant serial dependence (0.5245 of lag-1 ACF in Table 5) during the period
1900-2009. As shown in Figure 16, the monthly statistics of the PDO index does not show much
seasonality. The scatter plots of monthly PDO index as shown in Figure 17 indicates a linear
relationship for all lags (h=1,…,12). Unlike the ENSO index, heteroscedasticity is not observed
for the PDO index. The analysis tested two DLM models and several different order ARMA(p,q)
models.
In particular, the results of the following models are presented:
(1) ARMA(1,0)
20
(2) ARMA(5,0)
(3) ARMA(9,7)
(4) ARMA(28,0)
(5) DLM: Trend(1) and ARMA(1,0), simply denoted as Tr(1)-AR(1)
(6) DLM: Trend(2) and ARMA(2,0), simply denoted as Tr(2)-AR(2)
The selection of the order of ARMA models was based on the AIC given in Eq. (5). The
AIC values corresponding to different orders of ARMA(p,q) are presented in Table 6. The table
shows that ARMA(9,7) is the best choice because the corresponding AIC value is minimized.
Similar findings for the same data set has been reported in a study by Nairn-Birch et al. (2009),
which focused on the simulation of the PDO index. A relatively lower-order model, ARMA(5,0),
and higher-order model, ARMA(28,0), with no moving average term (i.e. q=0), were also tested.
It was found that for the ARMA(p,q) models without the moving average term (i.e. q=0),
ARMA(28,0) corresponded to the lowest AIC value (data not shown). Both DLM models were
also tested with the results noted below.
5.2. Results
To validate model performance, the first 90 years of monthly PDO index records (1900-1989)
were used to fit the models. Monthly forecasting was done for each lag h=1,2,…12. for 20 years
of PDO index records (1990-2009).
Root mean square error (RMSE) and correlation between forecasted and observed values
were recorded and are presented in Tables 7 and 8, respectively. These results are also
graphically illustrated in Figure 18 and 19.
21
Figure 18 shows that the higher order ARMA models (i.e. ARMA(9,7) and ARMA(28,0))
perform better than the lower order ARMA models (i.e. ARMA(1,0) and ARMA(5,0)). The
RMSE value of ARMA(28,0) was found to be much lower than that of ARMA(9,7) for each lag.
Both DLM models performed poorly for all lags in comparison to the selected ARMA models.
Higher order ARMA models combined with the trend component of the DLM did not improve
model performance (data not shown).
The correlation between forecasted and observed values shown in Table 8 and Figure 19
did not coincide with the RMSE results. Although the ARMA(28,0) model performs the best for
shorter lags (h<8), the ARMA(9,7) model shows the least amount of correlation between
forecasted and observed values for all lags. For longer lags, (h>8), the lower-order ARMA
models (ARMA(1,0) and ARMA(5,0) ) show the best performance.
The prediction results for lag times of 1 to 6 months are presented for ARMA(9,7) and
ARMA(28,0) in Figure 20 and Figure 21, respectively. As lag time (h) increases, the 95 percent
upper and lower confidence limits get wider. Because of this, simulations for all models
corresponding to a lag time of h>6 do not have much statistical significance.
The EMD-NSOR model was also tested. Although isolated cases showed EMD-NSOR was
an improvement (data not shown), overall model performance was not better than lower-order
ARMA models (see Table 9). In addition to this, the ARMA-GARCH model was not found to
produce better results than the ARMA model (Table 9).
6. Summary and Conclusions
Climate indices can provide a good representation of the current climate system and therefore
can be good predictors of hydro-meteorological variables such as the NBS components of the
22
Great Lakes system. In the current study, two monthly climate indices (ENSO and PDO) were
predicted for a lag time of 12 months using different time series models ranging from the
traditional ARMA model to the DLM, GARCH, and EMD-NSOR models.
For the ENSO index, the results lead to the conclusion that the ARMA(4,0)-GARCH(1,1)
model is superior to the other tested models. Although the DLM model Tr(1)-AR(4) produced
the lowest RMSE values, correlation performance measurement revealed that the Tr(1)-AR(4)
did not perform well for longer lags (i.e. h>8). The GARCH process was the most successful at
predicting the ENSO index because of its ability to handle heteroscedasticity.
For the PDO index, simulation results showed the ARMA models to be superior to the
other tested models. For all tested models, predicted values for lags longer than 6 months,
showed wide confidence limits. This implied that the simulation results were not statistically
meaningful for lags longer than 6 months. The long-term oscillation model, EMD-NSOR, was
not capable of accurately predicting climate indices in the short-term.
The results from this study show that incorporating predicted ENSO and PDO climate
indices into the calculation of NBS components of the Great Lakes system can improve NBS
estimates.
Notation:
t : time index
Xt : time dependent variable
Xt : vector of multivariate time dependent variables
Zt : time independent white noise variable whose square is time dependent in the
representation of the GARCH model
23
p, q : model order of the ARMA model
, : parameters of the ARMA model
n , npar : number of observations and parameters, respectively
h : prediction lag time
)(ˆ hX n : h-step ahead prediction, Xn+h
L(.) : likelihood
B : backward shift operator
,2 : mean and variance
C : covariance matrix
ψ : parameter set of a model
, :parameters of the GARCH model
tΛ :m-dimensional state vector
tt WV , :mutually independent error sequences with normal distribution
tt GF , : parameter and evolution matrices in the DLM model
24
References
Abraham, B. and Ledolter, J., 1983. Statistical methods for forecasting, 445 pp.
Ahn, J.H. and Kim, H.S., 2005. Nonlinear modeling of El Nino/southern oscillation index.
Journal of Hydrologic Engineering, 10(1): 8-15.
Akaike, H., 1974. A new look at the statistical model identification. IEEE Transactions on
Automatic Control, 19(6): 716–723.
Alexander, M.A. et al., 2002. The atmospheric bridge: The influence of ENSO teleconnections
on air-sea interaction over the global oceans. Journal of Climate, 15(16): 2205-2231.
Amadou, A., Seidou, O. and Ouarda, T.B.M.J., 2010. Nonlinear ARX models of the
interconnexion between the Great Lakes NBS components and global climate indices. in
preparation.
Andersen, T.G., Bollerslev, T., Diebold, F.X. and Labys, P., 2003. Modeling and forecasting
realized volatility. Econometrica, 71(2): 579-625.
Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of
Econometrics, 31(3): 307-327.
Bollerslev, T., Chou, R.Y. and Kroner, K.F., 1992. ARCH modeling in finance. A review of the
theory and empirical evidence. Journal of Econometrics, 52(1-2): 5-59.
Bosley, T.M. et al., 2008. The clinical spectrum of homozygous HOXA1 mutations. American
Journal of Medical Genetics Part A, 146A(10): 1235-1240.
Brockwell, P.J. and Davis, R., 1988. Simple consistent estimation of the coeffcients of a linear
filter. Stochastic Process Application, 28: 47-59.
Brockwell, P.J. and Davis, R.A., 2003. Introduction to Time Series and Forecasting. Springer,
456 pp.
Burg, J., 1978. A new analysis technique for time series data. Modern Spectrum Analysis,
NATO Advanced Study Institute of Signal Processing with emphasis on Underwater
Acoustics. John Wiley & Sons Inc, New York. , 334 pp.
Changnon, S.A., 2004. Temporal behavior of levels of the Great Lakes and climate variability.
Journal of Great Lakes Research, 30(1): 184-200.
Chen, D., Cane, M.A., Kaplan, A., Zebiak, S.E. and Huang, D.J., 2004. Predictability of El Nino
over the past 148 years. Nature, 428(6984): 733-736.
Cheng, Y.J., Tang, Y.M., Zhou, X.B., Jackson, P. and Chen, D.K., 2010. Further analysis of
singular vector and ENSO predictability in the Lamont model-Part I: singular vector and
the control factors. Climate Dynamics, 35(5): 807-826.
Chiew, F.H.S. and McMahon, T.A., 2002. Global ENSO-streamflow teleconnection, streamflow
forecasting and interannual variability. Hydrological Sciences Journal-Journal Des
Sciences Hydrologiques, 47(3): 505-522.
Devineni, N. and Sankarasubramanian, A., 2010. Improving the Prediction of Winter
Precipitation and Temperature over the Continental United States: Role of the ENSO
State in Developing Multimodel Combinations. Monthly Weather Review, 138(6): 2447-
2468.
25
Elek, P. and Márkus, L., 2004. A long range dependent model with nonlinear innovations for
simulating daily river flows. Natural Hazards and Earth System Science, 4(2): 277-283.
Engle, R., 2001. GARCH 101: The use of ARCH/GARCH models in applied econometrics.
Journal of Economic Perspectives, 15(4): 157-168.
Engle, R., 2002. New frontiers for arch models. Journal of Applied Econometrics, 17(5): 425-
446.
Engle, R.F., 1982. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance
of United Kingdom Inflation. Econometrica, 50(4): 987-1007.
Engle, R.F. and Kroner, K.F., 1995. Multivariate Simultaneous Generalized Arch. Econometric
Theory, 11(1): 122-150.
Fan, J. and Yao, Q., 2003. Nonlinear Time Series - Nonparametric and Parametric Methods.
Springer, New York, 552 pp.
Francq, C. and Zakoian, J.-M., 2010. GARCH Models: Structure, Statistical Inference and
Financial Applications, Chippenham, United Kingdom, 504 pp.
Godolphin, E. and Harrison, P., 1975. Equivalence theorems for polynomial projecting
predictors. . Journal of the Royal Statistical Society Series B-Stat Methodology, B 35:
205 – 215.
Hannan, E.J. and Rissanen, J., 1982. Recursive estimation of mixed autoregressive-moving
average order. Biometrika 69: 81-94.
Hanson, R.T., Newhouse, M.W. and Dettinger, M.D., 2004. A methodology to asess relations
between climatic variability and variations in hydrologic time series in the southwestern
United States. Journal of Hydrology, 287(1-4): 252-269.
Huang, N.E. et al., 1998. The empirical mode decomposition and the Hilbert spectrum for
nonlinear and non-stationary time series analysis. Proceedings of the Royal Society of
London Series a-Mathematical Physical and Engineering Sciences, 454(1971): 903-995.
Huang, N.E. and Wu, Z.H., 2008. A review on Hilbert-Huang transform: method and its
applications to geophysical studies. Reviews of Geophysics, 46(2).
Hurvich, C.M. and Tsai, C.L., 1989. Regression and time series model selection in small samples.
Biometrika, 76: 297-307.
Immerzeel, W.W. and Bierkens, M.F.P., 2010. Seasonal prediction of monsoon rainfall in three
Asian river basins: the importance of snow cover on the Tibetan Plateau. International
Journal of Climatology, 30(12): 1835-1842.
Künsch, H.R., 2001. State space and hidden Markov models In Complex Stochastic Systems
Chapman and Hall/CRC, Boca Raton, FL.
Kalman, R.E., 1960. A New Approach to Linear Filtering and Prediction Problems. Transactions
of the ASME-Journal of Basic Engineering, 82(Series D): 35-45.
Kalra, A. and Ahmad, S., 2009. Using oceanic-atmospheric oscillations for long lead time
streamflow forecasting. Water Resources Research, 45(3).
Kirtman, B.P. and Min, D., 2009. Multimodel ensemble ENSO prediction with CCSM and CFS.
Monthly Weather Review, 137(9): 2908-2930.
Lall, U. and Sharma, A., 1996. A nearest neighbor bootstrap for resampling hydrologic time
series. Water Resources Research, 32(3): 679-693.
Lee, T. and Ouarda, T.B.M.J., 2010a. Long-term prediction of precipitation and hydrologic
extremes with nonstationary oscillation processes. Journal of Geophysical Research-
Atmospheres, 115(D13107): doi:10.1029/2009JD012801.
26
Lee, T. and Ouarda, T.B.M.J., 2010b. Prediction of climate nonstationary oscillation processes
with empirical mode decomposition. Journal of Geophysical Research-Atmospheres, In
Publication: doi:10.1029/2010JD015142.
Lee, T. and Ouarda, T.B.M.J., 2010c. Stochastic simulation of nonstationary oscillation hydro-
climatic processes using empirical mode decomposition. In Review.
Lee, T.S., 2008. Stochastic simulation of hydrologic data based on nonparametric approaches, Ph.
D. Dissertation, Colorado State University, Fort Collins, CO., USA, 346 pp.
Mantua, N.J., Hare, S.R., Zhang, Y., Wallace, J.M. and Francis, R.C., 1997. A Pacific
interdecadal climate oscillation with impacts on salmon production. Bulletin of the
American Meteorological Society, 78(6): 1069-1079.
Migon, H., Gamerman, D., Lopez, H. and Ferreira, M., 2005. Bayesian dynamic models. In: D.
Day and C. Rao (Editors), Handbook of Statistics. Elsevier, New York, pp. 553-588.
Nairn-Birch, N. et al., 2009. Simulation and estimation of probabilities of phases of the Pacific
Decadal Oscillation. Environmetrics, 22(1): 79–85.
Parzen, E., 1967. Time Series Anlaysis Papers. Holden-Day, San Francisco.
Petris, G., Petrone, S. and Campagnoli, P., 2009. Dynamic Linear Models with R. Springer, New
York, 252 pp.
Rodionov, S. and Assel, R., 2000. Atmospheric teleconnection patterns and severity of winters in
the laurentian Great Lakes basin. Atmosphere-Ocean, 38(4): 601-635.
Rodionov, S. and Assel, R.A., 2003. Winter severity in the Great Lakes region: a tale of two
oscillations. Climate Research, 24(1): 19-31.
Rogers, J.C. and Coleman, J.S.M., 2003. Interactions between the Atlantic Multidecadal
Oscillation, El Nino/La Nina, and the PNA in winter Mississippi valley stream flow.
Geophysical Research Letters, 30(10).
Salas, J.D., Delleur, J.W., Yevjevich, V. and Lane, W.L., 1980. Applied Modeling of Hydrologic
Time Series. Water Resources Publications, Littleton, Colorado, 484 pp.
Schneider, E.K. et al., 1999. Ocean data assimilation, initialization, and predictions of ENSO
with a coupled GCM. Monthly Weather Review, 127(6 II): 1187-1207.
Snyder, R.D., 1985. Recursive Estimation of Dynamic Linear Models. Journal of the Royal
Statistical Society. Series B (Methodological), 47(2): 272-276.
Thomas, B.E., 2007. Climatic fluctuations and forecasting of streamflow in the lower Colorado
River Basin. Journal of the American Water Resources Association, 43(6): 1550-1569.
Trenberth, K.E., 1997. The definition of El Nino. Bulletin of the American Meteorological
Society, 78(12): 2771-2777.
Tsonis, A.A., Elsner, J.B. and Sun, D.-Z., 2007. The Role of El Niño—Southern Oscillation in
Regulating its Background State, Nonlinear Dynamics in Geosciences. Springer New
York, pp. 537-555.
Walker, G.T., 1932. On periodicity in series of related terms. Proceedings of the Royal Society A,
131 518-532.
Wallace, J.M. and Gutzler, D.S., 1981. Teleconnections in the geopotential height field during
the Northern Hemisphere winter. Monthly Weather Review, 109(4): 784-812.
Wang, W., Van Gelder, P.H.A.J.M., Vrijling, J.K. and Ma, J., 2005. Testing and modelling
autoregressive conditional heteroskedasticity of streamflow processes. Nonlinear
Processes in Geophysics, 12(1): 55-66.
27
West, M. and Harrison, J., 1997. Bayesian Forecasting and Dynamic Models. Springer, New
York, 700 pp.
Westra, S. and Sharma, A., 2010. An Upper Limit to Seasonal Rainfall Predictability? Journal of
Climate, 23(12): 3332-3351.
Wilks, D.S., 1997. Resampling Hypothesis Tests for Autocorrelated Fields. Journal of Climate,
10(1): 65-82.
Wolter, K. and Timlin, M.S., 1993. Monitoring ENSO in COADS with a seasonally adjusted
principal component index, Proc. of the 17th Climate Diagnostics Workshop.
NOAA/NMC/CAC, NSSL, Oklahoma Clim. Survey, CIMMS and the School of Meteor.,
Univ. of Oklahoma, Norman, OK, pp. 52-57.
Wu, R. and Kirtman, B.P., 2003. On the impacts of the Indian summer monsoon on ENSO in a
coupled GCM. Quarterly Journal of the Royal Meteorological Society, 129(595 PART B):
3439-3468.
Wu, Z.H. and Huang, N.E., 2004. A study of the characteristics of white noise using the
empirical mode decomposition method. Proceedings of the Royal Society of London
Series a-Mathematical Physical and Engineering Sciences, 460(2046): 1597-1611.
Yule, G.U., 1927. On a method of investigating periodicities in disturbed series, with special
reference to Wolfer's sunspot numbers. Philosophical Transactions of the Royal Society
London Series A, 226 267-298.
Zhang, Y., Wallace, J.M. and Battisti, D.S., 1997. ENSO-like interdecadal variability: 1900-93.
Journal of Climate, 10(5): 1004-1020.
28
Table 1. Basic statistics of the annual ENSO index.
Mean St.Dev Skew Lag1
ACF Max Min
0.0464 0.7677 -0.064 0.2854 1.6541 -1.6155
Table 2. AIC of the ARMA(p,q) model for the monthly ENSO index.
AR\MA 0 1 2 3 4 5 6 7 8 9 10
0 2029 1223.1 795.5 551.6 386.4 302.8 256.5 222.7 180.7 175.7 163.9
1 254.8 169.1 167.6 146.2 145.2 146.1 144.4 143.8 144.5 146.4 145.5
2 155.7 145.4 139.1 136.8 136.0 137.9 139.2 141.2 142.9 144.8 145.1
3 154.2 155.5 142.6 137.0 137.7 151.7 151.6 146.4 146.4 144.4 145.5
4 135.2 137.2 136.9 136.4 138.4 140.5 141.1 145.0 146.2 145.8 144.4
5 137.2 138.9 137.8 138.4 140.8 142.8 143.8 143.8 144.6 147.9 148.5
6 137.4 137.5 141.2 140.3 143.5 144.4 144.4 145.4 147.6 145.2 148.6
7 137.9 140.7 141.1 135.9 137.5 141.1 148.8 149.6 143.2 141.7 143.4
8 139.0 141.8 143.1 137.6 135.5 134.0 145.0 148.1 142.9 145.7 145.3
9 140.8 142.8 139.0 141.1 142.1 143.3 143.1 143.2 144.6 146.5 149.2
10 142.7 143.1 147.0 145.3 137.9 149.2 136.5 138.7 139.5 142.7 141.8
29
Table 3. RMSE values corresponding to the monthly ENSO index for the considered 20 years.
Lag ARMA(1,0) ARMA(4,0) ARMA(7,3) ARMA(8,5) Tr(1)-
AR(4)
ARMA(4,0)-
GARCH(1,1)
EMD-NSOR
1 0.30 0.27 0.27 0.27 0.26 0.26 0.40
2 0.49 0.45 0.46 0.46 0.43 0.44 0.53
3 0.64 0.59 0.60 0.60 0.56 0.56 0.65
4 0.76 0.71 0.72 0.72 0.67 0.67 0.76
5 0.84 0.79 0.80 0.80 0.74 0.75 0.87
6 0.91 0.84 0.85 0.85 0.79 0.80 0.96
7 0.95 0.88 0.89 0.89 0.83 0.85 1.04
8 0.99 0.91 0.91 0.91 0.86 0.88 1.11
9 1.02 0.93 0.93 0.93 0.89 0.90 1.17
10 1.04 0.94 0.95 0.95 0.91 0.92 1.23
11 1.05 0.95 0.96 0.96 0.93 0.94 1.28
12 1.06 0.95 0.96 0.96 0.95 0.95 1.33
Note that lag-h represents the prediction lag time (see h in Eq.(7))
Table 4. Correlation between observed versus predicted monthly ENSO index values from
different models for the considered 20 years.
Lag ARMA(1,0) ARMA(4,0) ARMA(7,3) ARMA(8,5) Tr(1)-
AR(4)
ARMA(4,0)-
GARCH(1,1)
EMD-
NSOR
1 0.94 0.96 0.96 0.95 0.95 0.96 0.81
2 0.84 0.87 0.87 0.87 0.87 0.88 0.68
3 0.72 0.77 0.77 0.77 0.78 0.79 0.54
4 0.59 0.64 0.64 0.64 0.66 0.68 0.37
5 0.48 0.54 0.54 0.54 0.56 0.59 0.21
6 0.38 0.44 0.45 0.45 0.47 0.50 0.06
7 0.30 0.36 0.37 0.37 0.37 0.41 -0.10
8 0.22 0.30 0.31 0.31 0.28 0.34 -0.23
9 0.15 0.23 0.25 0.25 0.19 0.26 -0.33
10 0.09 0.17 0.21 0.20 0.10 0.20 -0.42
11 0.03 0.12 0.17 0.16 0.01 0.14 -0.45
12 -0.01 0.08 0.14 0.14 -0.07 0.10 -0.47
30
Table 5. Basic statistics of the annual PDO index.
Mean St. Dev. Skew Lag1 ACF Max Min
0.0388 0.7687 0.0674 0.5245 -1.948 1.9942
Table 6. AIC of the ARMA(p,q) model for the monthly PDO index.
AR\MA 0 1 2 3 4 5 6 7 8 9 10
0 3202.8 2736.6 2553.7 2469.0 2440.0 2406.5 2381.8 2354.6 2351.4 2353.3 2343.9
1 2381.7 2320.0 2317.8 2316.9 2318.2 2320.0 2320.7 2321.6 2353.5 2319.9 2317.9
2 2333.2 2316.2 2314.0 2316.0 2319.6 2321.6 2320.0 2321.1 2319.9 2319.7 2318.0
3 2326.3 2315.8 2317.0 2317.6 2309.6 2308.8 2310.8 2311.7 2320.0 2310.1 2308.8
4 2322.6 2317.2 2319.5 2307.1 2308.0 2308.6 2322.5 2325.6 2321.8 2310.1 2311.5
5 2318.7 2320.7 2321.5 2308.5 2317.8 2312.3 2313.2 2311.4 2309.9 2310.3 2312.2
6 2320.7 2322.7 2318.2 2326.5 2320.1 2307.1 2309.3 2311.5 2313.3 2310.4 2312.3
7 2322.4 2324.3 2317.2 2325.3 2316.3 2310.2 2311.2 2313.5 2313.4 2309.4 2314.9
8 2324.3 2320.1 2312.7 2312.3 2315.2 2312.9 2319.1 2301.7 2298.9 2315.2 2315.1
9 2324.1 2320.8 2320.9 2310.0 2317.7 2307.1 2320.1 2298.6 2311.3 2313.1 2313.2
10 2316.2 2316.9 2318.7 2310.1 2308.9 2312.3 2311.2 2300.6 2313.1 2314.3 2315.6
31
Table 7. Comparison of RMSE values of the monthly PDO index for the considered 20 years.
Lag ARMA(1,0) ARMA(5,0) ARMA(9,7) ARMA(28,0) Tr(1)-
AR(1)
Tr(2)-
AR(2)
1 0.545 0.583 0.569 0.552 0.585 0.569
2 0.754 0.774 0.748 0.731 0.793 0.804
3 0.869 0.883 0.844 0.824 0.923 0.943
4 0.935 0.946 0.900 0.872 1.005 1.026
5 0.976 0.982 0.933 0.902 1.052 1.074
6 0.997 0.996 0.954 0.921 1.071 1.096
7 1.004 0.989 0.960 0.925 1.069 1.101
8 1.008 0.981 0.968 0.934 1.064 1.102
9 1.012 0.979 0.978 0.944 1.058 1.101
10 1.016 0.983 0.989 0.955 1.065 1.110
11 1.020 0.991 1.004 0.971 1.075 1.120
12 1.022 1.002 1.013 0.982 1.088 1.132
Table 8. Correlation between observed versus predicted monthly ENSO index values from
different models for the considered 20 years of the PDO index.
Lag ARMA(1,0) ARMA(5,0) ARMA(9,7) ARMA(28,0) Tr(1)-
AR(1)
Tr(2)-
AR(2)
1 0.866 0.827 0.831 0.832 0.841 0.835
2 0.709 0.654 0.659 0.672 0.681 0.670
3 0.559 0.507 0.508 0.538 0.536 0.529
4 0.438 0.404 0.390 0.445 0.418 0.422
5 0.338 0.332 0.301 0.374 0.326 0.342
6 0.264 0.285 0.227 0.321 0.263 0.289
7 0.249 0.279 0.195 0.300 0.241 0.269
8 0.260 0.285 0.170 0.285 0.224 0.257
9 0.269 0.284 0.149 0.268 0.199 0.243
10 0.272 0.270 0.132 0.250 0.163 0.222
11 0.254 0.237 0.094 0.214 0.120 0.187
12 0.231 0.191 0.055 0.180 0.065 0.147
32
Table 9. Comparison of the RMSE values of the monthly PDO index for ARMA(5,0), EMD-
NSOR, and ARMA(5,0)-GARCH(1,1) models for the considered 20 years.
Lag ARMA(5,0) EMD-
NSOR
ARMA(5,0)
GARCH(1,1)
1 0.583 0.807 0.602
2 0.774 0.994 0.809
3 0.883 1.156 0.929
4 0.946 1.257 0.990
5 0.982 1.327 1.017
6 0.996 1.352 1.027
7 0.989 1.337 1.024
8 0.981 1.324 1.020
9 0.979 1.327 1.018
10 0.983 1.338 1.021
11 0.991 1.350 1.029
12 1.002 1.360 1.045
33
Prediction of Monthly ENSO
34
Figure 1. Annual (top panel) and monthly (bottom panel) ENSO time series.
35
Figure 2. Autocorrelation function (ACF) of the monthly ENSO index.
36
Figure 3. Seasonal variations of time series and statistics for the monthly ENSO index. (a)
Spaghetti plot of time series for each year and (b)-(d) monthly statistics.
37
Figure 4. Spectral density of the monthly ENSO index. Note that g(f) presents the smoothed
sample spectral density at frequency f (see Salas et al. 1980) and the smoothing function of
Parzen (1967).
38
Figure 5. Scatter plots of the monthly ENSO index.
39
Figure 6. RMSE values corresponding to the monthly ENSO index for the considered 20 years
(1990-2009) for ARMA(1,0), ARMA(4,0), ARMA(7,3), and ARMA(8,5) models. Note that the
x-axis represents the lag time (h).
40
Figure 7. Same as Figure 6 but for the models ARMA(4,0), Tr(1)-AR(4), ARMA(4,0)-
GARCH(1,1), and EMD-NSOR.
41
Figure 8. Correlation corresponding to the monthly ENSO index for the considered 20 years
(1990-2009) for ARMA(1,0), ARMA(4,0), ARMA(7,3), and ARMA(8,5) models.
42
Figure 9. Same as Figure 8 but for the models ARMA(4,0), Tr(1)-AR(4), ARMA(4,0)-
GARCH(1,1), and EMD-NSOR.
43
Figure 10. Predicting the monthly ENSO index using ARMA(4,0) model for lag time h=1,…,6
months and for the considered 20 years (1990-2009). Note that the red-cross line represents the
observed values and the black solid line represents the mean prediction while the gray regions
show 95 percent upper and lower limits of the mean.
44
Figure 11. Same figure as Figure 10 but using ARMA(8,5) model.
45
Figure 12. Same figure as Figure 10 but using Tr(1)-AR(4) model.
46
Figure 13. Same figure as Figure 10 but using ARMA(4,0)-GARCH(1,1) model.
47
Prediction of Monthly PDO
48
Figure 14. Annual (top panel) and monthly (bottom panel) PDO time series for 1900-2009.
49
Figure 15. Autocorrelation function (ACF) of the monthly PDO index.
50
Figure 16. Seasonal variations of time series and statistics for the monthly PDO index. (a)
Spaghetti plots of time series for each year and (b)-(d) monthly statistics.
51
Figure 17. Scatter plots of the monthly PDO index.
52
Figure 18. RMSE values corresponding to the monthly ENSO index for the considered 20 years
(1990-2009).
53
Figure 19. Correlation corresponding to the monthly ENSO index for the considered 20 years
(1990-2009).
54
Figure 20. Predicting the monthly PDO index using ARMA(9,7) model for lag time h=1,…,6
months and for the considered 20 years (1990-2009). Note that the red-cross line represents the
observed values and the black solid line represents the mean prediction while the gray regions
show 95 percent upper and lower limits of the mean.
55
Figure 21. Same figure as Figure 20 but using ARMA(28,0) model.
