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Development of methods for regional flood estimates in the province of British Columbia, Canada Wang, Yuzhang 2000

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DEVELOPMENT OF METHODS FOR REGIONAL FLOOD ESTIMATES IN THE PROVINCE OF BRITISH COLUMBIA, CANADA by Yuzhang Wang B.Eng., Chengdu Geological College, China, 1983 M . S c , Chinese Academy of Sciences, China, 1990 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Forestry) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A 2000 © Yuzhang Wang, Vancouver, Canada, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be al lowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract Flood estimation in the province of British Columbia is often based on either single-site frequency analysis or graphical peak-flow regionalization procedures. These methods involve large uncertainties, especially at short-term record stations and ungauged sites, because of the vague selection of frequency distributions and delineation of homogeneous regions. To reduce these uncertainties and overcome the drawbacks of the current methods, an innovative regional frequency analysis was proposed in this study. L-moments were used for the three stages, namely delineating and testing homogeneous regions, identifying and fitting regional distributions, and developing regional functions for the transfer of information from gauged to ungauged watersheds. Based on the most recently available flood database, the province of British Columbia was divided into 19 homogeneous regions of which 14 are non-mixture regions and five are mixture regions. A mixture means in some years the annual floods are generated by one mechanism, while in other years they are generated by other physically different mechanisms. It was found that either the generalized logistic (GLOG) or the generalized extreme value (GEV) may be considered as the regional parent distribution for any of the non-mixture regions, whereas the non-parametric distribution can be used for the mixture regions. In the non-mixture regions, hierarchical approaches and regression models were developed for gauged and ungauged watersheds. For the hierarchical approaches, the first two parameters of the G L O G or GEV distribution were estimated from at-site data while the third parameter was from the region. For the regression models, the parameters of the G L O G or G E V distribution were regressed on the catchment size. In the mixture regions, a non-parametric method was combined with the regression method for the development of regional models. Monte Carlo simulation studies showed that the developed hierarchical approaches were substantially more accurate than the single-site methods, especially for long-term flood quantiles. In particular, it was shown that about three times more data were required for the single-site models to be as accurate as the developed hierarchical approaches. The proposed regression models were validated through split-sampling experiments. Statistical tests showed that the quantiles from the regression models were in good agreement with those from actual observations. ii CONTENTS A C K N O W L E D G M E N T S x i i A B S T R A C T ii C O N T E N T S iv L I S T O F T A B L E S v i . , L I S T O F F I G U R E S vii; N O T A T I O N S -ix A B B R E V I A T I O N S xii Chapter 1 I N T R O D U C T I O N . 1 1.1 Motivation 1 1.2 Objectives 5 1.3 Thesis Overview 7 Chapter 2 L I T E R A T U R E R E V I E W 9 2.1 Introduction 9 2.2 Single-Site Flood-Frequency Analysis 10 2.2.1 Distribution Selection 10 2.2.2 Commonly Used Distributions 12 2.2.3 Methods of Fitting Distributions 13 2.3 Regional Flood-Frequency Analysis 16 2.3.1 Delineation of Homogeneous Regions 16 2.3.2 Homogeneity Tests 19 2.3.3 Identification of the Regional Parent Distribution 21 2.4 Methods for Regional Flood-Frequency Analysis 22 2.4.1 Index-Flood Approach 23 2.4.2 Regional Regression Method 25 2.5 Mixed Processes in Flood-Data Series 27 2.6 Summary 29 Chapter 3 T H E O R E T I C A L B A C K G R O U N D A N D D E V E L O P M E N T 3 2 3.1 TheoryofL-Moments 3 2 3.2 Advantages of L-Moments 3 4 3.3 Homogeneity Testing 3 7 3.4 Identification of the Regional Parent Distribution 3 8 3.5 Hierarchical Approach to Regional Flood-Frequency Analysis 3 9 3 . 6 Non-Parametric Method 4 0 Chapter 4 * R E S E A R C H M E T H O D S 4 4 4.1 Study Area 4 4 4 . 2 Data Selection 4 5 4.2 .1 Test for Independence 4 7 4 . 2 . 2 Test for Trend 4 7 4 . 2 . 3 Test for Homogeneity in Time 4 7 4 . 2 . 4 Test for Randomness 4 8 4 . 3 Delineation of Homogeneous Regions 4 8 4.3 .1 Delineation of Homogeneous Regions Based on CDF 4 9 4 . 3 . 2 Delineation of Homogeneous Regions Based on Climatic and Physiographic Conditions 4 9 4 . 3 . 3 Testing the Significance of Mixed Processes in Flood-Data Series 5 0 4 . 3 . 4 Testing the Effect of Sampling Variability on the CDF Shape 5 1 4 . 3 . 5 Testing the Homogeneity of Delineated Hydrologic Regions 5 3 4 . 4 Identification of Regional Parent Distribution 5 7 4 . 5 Proposed Hierachical Frequency Models 5 9 4 . 6 Split-Sampling Experiment 6 0 4 . 7 Monte Carlo Simulation 6 2 Chapter 5 R E S U L T S A N D D I S C U S S I O N S 6 5 5.1 Data Screening 6 5 5.2 Classification of Homogeneous Regions 6 6 5.2.1 Homogeneous Regions Based on CDF 6 6 5 . 2 . 2 Homogeneous Regions Based on Climatic Conditions 6 7 5 . 2 . 3 Homogeneous Regions Based on Physiographic Conditions 6 9 5 . 2 . 4 Tests of Mixed Processes in Flood-Data Series 7 0 IV 5.2.5 Homogeneity Testing 72 5.3 Selection of Regional Parent Distribution 74 5.3.1 Regional Parent Distribution 74 5.3.2 Sensitivity Analysis of Skewness 75 5.4 Regional Models for Non-Mixture Regions 76 5.4.1 Models for the Generalized Logistic Distribution 77 5.4.2 Models for the Generalized Extreme Value Distribution 80 5.5 Regional Models for Mixture Regions 83 5.6 Residual Analysis for the Developed Regression Models 85 5.7 Extrapolation of the Developed Regression Models 86 5.7.1 Non-Mixture Regions 86 5.7.2 Mixture Regions 88 5.8 Validation of the Developed Regression Models 89 5.9 Comparison of the Proposed Hierarchical Regional Models and Single-Site Models 91 Chapter 6 S U M M A R Y A N D C O N C L U S I O N S 94 6.1 Summary of the Study 94 6.2 Major Conclusions 97 6.3 Contributions to Hydrological Sciences .;: 98 6.4 Recommendations for Future Studies 101 References 103 Appendix A Tables 116 Appendix B Figures 134 Appendix C Hydrometric Stations for This Study 183 Appendix D Examples of Computing Floods in Different Regions 199 List of Tables Table 2.1 Independent Variables Used in 10 Flood-Frequency Analyses 117 Table 3.1 L-Moment Relationships for Some Common Distributions 117 Table 5.1 Test for Independence of Annual Maximum Daily Flows 118 Table 5.2 Test for Trend of Annual Maximum Daily Flows 118 Table 5.3 Test for Randomness of Annual Maximum Daily Flows 119 Table 5.4 Test for Homogeneity of Annual Maximum Daily Flows 119 Table 5.5 Number of Stations in Each Region as Defined by CDF Shape 120 Table 5.6 Test for the Significance of Mixtures 120 Table 5.7 F-Test for Variance between Groups based on Record Length in Different Regions 121 -122 Table 5.8 A N O V A for Mixture Regions 123 Table 5.9 Test for the Effect of Sampling Variability 124 Table 5.10 Test of Homogeneity for Delineated Regions 124 Table 5.11 Summary of Homogenous Regions in BC 125 Table 5.12 Regional Weighted Average L-Skewness and L-Kurtosis for Non-Mixture Homogeneous Regions 126 Table 5.13 Best-Fit Distribution for Non-Mixture Homogeneous Regions 127 Table 5.14 Regional Models for G L O G Distribution 128 Table 5.15 Regional Models for GEV Distribution 128 Table 5.16 Regional Models for Mixture Regions 129 Table 5.17 Weighted Regional Average of L-Moments for Mixture Regions 130 Table 5.18 Analysis of Regression Slopes for Different Watershed Sizes 130 Table 5.19 Comparison between Actual and Predicted Floods for the Proposed Models 131 Table 5.20 Comparison of RMSE between Regional and Single-Site Methods 132-133 vi List of Figures Fig. 3.1 L-Moment Ratio Diagram 135 Fig. 3.2 Non-Parametric Density Construction by the Kernel Method 136 Fig. 4.1 Locations of the Hydrometric Stations in BC 137 Fig. 4.2 Typical Cumulative Distribution Functions with a Dog-Leg (08MA001) and without a Dog-Leg (08LE031) 13 g Fig. 4.3 Procedures for the Delineation of Homogeneous Regions \ 39 Fig. 5.1 Number of Stations and Average Sample Length vs. Drainage Area 140 Fig. 5.2 Annual Daily Maximum Flow by Month for Station 08MB005 140 Fig. 5.3 Annual Daily Maximum Flow vs. Rank for Station 08MB005 141 Fig. 5.4 Annual Daily Maximum Flow by Year for Station 08MB005 141 Fig. 5.5 Different Flood-Generating Mechanisms 142-143 Fig. 5.6 The First Classification of Homogeneous Regions Based on Shape of CDF 144 Fig. 5.7 Cumulative Distribution Function Plots of Station 08LB022 145 Fig. 5.8 Flood Occurrence during the Year by Region 146-148 Fig. 5.9 The Second Classification of Homogeneous Regions Adjusted for Similarity in the Timing of Annual Floods 149 Fig. 5.10 The Third Classification of Homogeneous Regions Adjusted for Similarity in Physiographic Conditions 150 Fig. 5.11 Classification of Homogeneous Regions Adjusted for Tests of Significance of Mixed Processes and Sampling Variability 151 Fig. 5.12 The Final Classification of Homogeneous Regions Adjusted for L-Skewness and L-Kurtosis 152 Fig. 5.13 Regional L-Moment Ratio Diagram 153 Fig. 5.14 Best-Fit Distribution for Each Homogeneous Region 154 Fig. 5.15 Sensitivity of the Flood Frequency Curve to L-Skewness (GEV) 155 Fig. 5.16 L-Coefficient of Variation vs. Drainage Area in BC 155 Fig. 5.17a Non-Parametric vs. Parametric Distribution for Station 08MG005 156-157 Fig. 5.17b Non-Parametric vs. Parametric Distribution for Station 08MG007 158-159 Fig. 5.17c Non-Parametric vs. Parametric Distribution for Station 08DB001 160-161 vii: Fig. 5.17d Non-Parametric vs. Parametric Distribution for Station 08EG011 162-163 Fig. 5.17e Non-Parametric vs. Parametric Distribution for Station 10CB001 164-165 Fig. 5.18 Two-Component Extreme Value Distribution in Feasible L-moment Space 166 Fig. 5.19 Parameters of GEV vs. Drainage Area for NMR2-1 167 Fig. 5.20 Sensitivity of the Mean to Sample Size 167 Fig. 5.21 Sensitivity of L-Coefficient of Variation to Sample Size 168 Fig. 5.22 Sensitivity of L-Skewness to Sample Size 168 Fig. 5.23a Location Parameter of G L O G vs. Drainage Area (NMR1-1) 169 Fig. 5.23b Scale Parameter of G L O G vs. Drainage Area (NMR1-1) 169 Fig. 5.24a Location Parameter of G L O G vs. Drainage Area (NMR1-2) 170 Fig. 5.24b Scale Parameter of G L O G vs. Drainage Area (NMR1 -2) 170 Fig. 5.25a Location Parameter of G L O G vs. Drainage Area (NMR8-1) 171 Fig. 5.25b Scale Parameter of G L O G vs. Drainage Area (NMR8-1) 171 Fig. 5.26a Location Parameter of G E V vs. Drainage Area (NMR8-2) 172 Fig. 5.26b Scale Parameter of GEV vs. Drainage Area (NMR8-2) 172 Fig. 5.27 Flood Quantiles vs. Drainage Area (MR3) 173-175 Fig. 5.27A Record Length vs. Drainage Area 176 Fig. 5.28 Comparison of Actual and Predicted Floods 177-179 Fig. 5.29a Accuracy of Estimating Flood Events for Various Return Periods for NMR2-1 180 Fig. 5.29b Accuracy of Estimating Flood Events for Various Return Periods for NMR8-2 180 Fig. 5.29c Accuracy of Estimating Flood Events for Various Return Periods for NMR8-3 181 Fig. 5.30a Influence of Sample Size on the Accuracy of the 100-yr Flood Event for NMR2-1 181 Fig. 5.30b Influence of Sample Size on the Accuracy of the 100-yr Flood Event for NMR8-2 182 Fig. 5.30c Influence of Sample Size on the Accuracy of the 100-yr Flood Event for NMR8-3 182 viii Notations A,: population mean in L-moment space /L,: population standard deviation in L-moment space Xr: r-th population L-moment statistic r 2 : population coefficient of variation in L-moment space r 3 : population skewness in L-moment space r 4 : population kurtosis in L-moment space j3r: probability weighted moment of order r Pr: unbiased estimator of /3r / ,: weighted average mean in L-moment space t2: weighted average standard deviation in L-moment space t3: weighted average skewness in L-moment space t4: weighted average kurtosis in L-moment space 12: sample L-coefiicient of variation in L-moment space r 3: sample L-skewness in L-moment space tA: sample L-kurtosis in L-moment space , tl'], : L-coefficient of variation, L-skewness, and L-kurtosis, at site x,, x 2 x n : a random sample of observations fir: r-th central moment about the origin jur: r-th absolute moment IX L : likelihood function S(%): homogeneity statistic cr2„A.s: variance of L-moments of the observed flood-data series a1 sim '• synthetic variance of L-moments QT : discharge of T-year return period Qm: mean annual flood (index flood) F(x): cumulative distribution function (CDF) /?,, B2: location parameters of the two components in a mixed distribution ax, a2: scale parameters of the two components in a mixed distribution a,, a2: relative weight of the two components in a mixed distribution Xhl: i-th smallest variable in a sample of size n H : homogeneity statistic by Hosking and Wallis (1993) vx, V2, Vj- between-site variability measures of L-moment statistics Vohs: value of an observed network of K, , V2 or yi Hv, c r „ : mean and standard deviation of the synthetic counterparts of vohs ZDIST : regional goodness-of-fit statistic r 4 ' J / s / : theoretical L-kurtosis of a given distribution G-U : standard deviation of f4 cr 2, a)2: time and space sampling variances f(x): probability density function K(x): kernel function h : smoothing factor from the sample Rm: hypothesis statistic for testing significance of mixed processes Go*' Q,OL ' 05os> 05oi : 50-year quantiles of populations L and S N: number of stations within a given region. Er: hypothesis statistic for testing sampling variability nl: sample length at site i h, k: two shape parameters of the Kappa distribution a, k: location, scale, and shape parameters of a distribution 12: coefficient of identification SEE: standard error of estimate r ( . ) : Gamma function E(%): error function actual : observed discharge of T-year return period predicted : estimated discharge of T-year return period xi: Abbreviations AP Antecedent Precipitation CFA Consolidated Frequency Analysis CDF Cumulative Distribution Function EV1 Extreme Value I GEV Generalized Extreme Value Distribution G L O G Generalized Logistic Distribution GPA Generalized Pareto Distribution L-Ck Coefficient of L-Kurtosis L~CS Coefficient of L-Skewness L~CV L-Coefficient of Variation LN3 Three-parameter Log-Normal Distribution LP3 Log-Pearson Type III Distribution MSE Mean Square Error M R Mixture Region N D Non-Parametric Distribution N M R Non-Mixture Region RMSE Root Mean Square Error TCEV Two-Component Extreme Value Distribution xii Acknowledgments Grateful thanks to Dr. Younes Ali la of the Faculty of Forestry, University of British Columbia for his outstanding supervision, constructive criticism, technical advice and encouragement throughout the study. I am also indebted to Dr. Douglas Golding and Dr. Peter Marshall of the Faculty of Forestry, University of British Columbia, and Dr. Wambuwa Kaluli of the Greater Vancouver Regional District for their support, assistance, discussions and comments during my stay in the University. They help me whenever I need help. I would like to acknowledge Dr. Denis Russell, Dr. Tony Kozak, Dr. Roy Sidle and Dr. Valerie LeMay for their teachings and personal help. These teachings and help led me to this stage easier. Acknowledgments are also extended to my colleagues Adam Stonewall, Ahmed Mtiraoui, Emmanuel Ladalla, David Luzi, Paula Calvert, Andrew Whitaker and Jaime Cathcart for their cooperation, assistance and discussions during my study. I would also like to thank the financial support for this research provided by Y . Alila through FRBC research grant HQ96330-RE, Graduate Scholarship and Graduate Fellowship from the University of British Columbia, and Scholarship from BC Hydro. Finally, I would like to express my appreciation to my family, especially my wife for her patience, encouragement and support. Without the moral support from my family there should not have been such a conclusion to this part in my life. x i i i Chapter 1 INTRODUCTION 1.1 Motivation Reliable estimates of floods are necessary to correctly size hydraulic structures, reduce the risk of their failures, and minimize downstream environmental damage. In British Columbia, flood estimates at gauged stations are often calculated using single-site frequency analysis of annual maximum flood series (Ministry of Forests, BC, 1995). Flood-data series are commonly assumed to follow a certain probability distribution and the parameters of the distribution are estimated using moments or maximum likelihood. Unfortunately, there are several sources of errors associated with this approach. The first source of error in single-site frequency analysis results from mis-identification of the underlying probability distribution. Different distributions may produce significantly different estimates for the same return period (Coulson, 1991). There is no firm theoretical basis for the exclusive use of one distribution over others when analyzing a relatively short annual flood series with the intention of predicting the magnitude of extreme events (Gumbel, 1958). In British Columbia, the distribution of extreme value type I (EV1 or Gumbel distribution) has often been chosen for estimating floods. Waylen and Woo (1983) used the EV1 as the parent distribution for fitting floods in the Fraser River basin. Loukas and 1 Quick (1995) also used EV1 as the parent distribution for comparing different flood estimation techniques for ungauged watersheds in coastal British Columbia. Other distributions, such as the three-parameter log-normal (LN3) distribution and the log-Pearson type III (LP3) distribution, have also been recommended in British Columbia (Reksten, 1987). However, it is questionable whether the correct distribution has been chosen without a mathematical test, particularly when floods are generated by a mixture of two populations. A mixture occurs when floods are generated by mixed processes. This means, within the same catchment, in some years the annual maximum floods are generated by one mechanism (for instance, rainfall), while in other years they are induced by a physically different mechanism (for instance, snowmelt or rain on snow). The second source of error in single-site frequency analysis results from sampling deficiencies, where the sample length is not long enough for a reliable statistical analysis. This is particularly true in the province of British Columbia, as most of the stations were established in the 1970's and the average record length of the available 960 hydrometric stations is less than 20 years. This may lead to unreliable flood estimates, especially for longer return periods. Under the Forest Practices Code of British Columbia (Ministry of Forests, BC, 1995), all new permanent and semi-permanent bridges and culverts that are an integral part of forest road infrastructures must be designed to accommodate a 100-year flood. It is clear that, for most watersheds in the province, the 100-year flood cannot be reliably estimated using single-site frequency analysis. One method that can be used to overcome these shortcomings is regional flood-frequency analysis. Regionalization can be considered equivalent to extension of the gauged network and can provide users with a better alternative for estimating floods. Regional flood-frequency analysis uses data from sites other than the site where flood estimates for specific return periods (quantiles) are needed. Because 2 more information is used than in the single-site flood-frequency approach, regional flood-frequency analysis has potential for greater accuracy in the final quantile estimates. However, information from other sites can be appropriately transferred only within a "homogeneous" region. Recent studies show that, in nearly all practical situations, regional flood-frequency estimators outperform single-site frequency estimators (Lettenmaier et al, 1987) Because of the advantage of regional flood-frequency analysis, flood estimates in British Columbia have also been based on a graphical peak-flow regionalization procedure. Reksten (1987) developed a peak-flow regionalization procedure of compiling, analyzing, and interpreting peak-flow data for estimating floods for ungauged watersheds in British Columbia. The province is separated into sub-regions, and the mean annual daily floods from the watersheds in these sub-regions are plotted against the basin area. The ratios of the floods for different return periods to the mean annual daily flood are computed for the gauged sites. Then, i f the drainage area of an ungauged site is known, the mean annual daily flood can be obtained from graphs and finally, the flood estimates of any return period can be calculated by multiplying the mean annual daily flood by a given growth factor. This procedure is also used to estimate instantaneous floods in the province. The reliability of any regional flood-frequency analysis is highly dependent on the degree of hydro logic homogeneity. Ingledow and Associates Ltd. (1969) delineated British Columbia into 29 hydrologic zones by assessing: (1) physiographic zones based on study of topographic maps, (2) climatic zones based primarily on maps of precipitation and temperature patterns, plus maps of climatic regions, and (3) patterns of mean annual, peak and low streamflows. Due mainly to the lack of suitable basin physiographic data and the inherent bias associated with flood statistics computed based on the conventional moments at the time, no attempt was made to check the boundaries of the 29 hydrologic zones by statistical or 3 mathematical techniques or to develop any regional equation for each zone. Reksten (1987) adopted the same set of hydrologic zones and reported that: " . . . until the boundary of these zones can be checked by statistical or mathematical techniques, it is unrealistic to assign to them the term homogeneous either in the physical or statistical sense. ... the delineation of boundaries for the statistically homogeneous zones will have to await the results of future studies which in turn will be dependent upon the availability of data." More recently, a group of hydrologists classified British Columbia into 41 hydrologic zones (Chapman et al., 1995). This hydrologic-zone map has the same problems as the hydrologic-zone derived by Ingledow and Associates Ltd. (1969). The authors recognized that defining the boundaries of the 41 zones involved a great deal of subjective assessment that combined the information in the data-based maps and the theoretical and practical hydrology knowledge of the group members. Moreover, it is questionable that these zones can be considered homogeneous, because no statistical or mathematical testing for homogeneity was done. Most of the distributions used in hydrology are for describing a single population, which implies that the maximum annual floods measured at the outlet of the same catchment are identically distributed. This assumption may not be valid. In British Columbia, mixed processes in flood-data series are produced by spatial variation of the flood-generating processes of both snowmelt and rainfall (Waylen and Woo, 1982; 1983). Melone (1985) concluded that floods do not result from the same flood producing mechanisms on all drainage basins in the coastal region of British Columbia. Floods may be induced by snowmelt in spring or summer, by rainfall in fall or winter, and by both snowmelt and rainfall during the year. 4 Furthermore, since Reksten's study in 1987, an extra 10 years of streamflow record has been collected. This amounts to some 12000 station-years of record and would result in a regional model with a better spatial and temporal coverage. In summary, the uncertainties associated with current practice, the availability of a longer hydrometric database, the need to account for mixed processes in flood data, and the advantages of regional flood-frequency analysis justify developing a new regional method that leads to more reliable flood estimates in British Columbia. 1.2 Objectives The primary objective of this study is to develop an improved regional method for estimating floods at both gauged and ungauged sites in British Columbia. This method will provide a new frontier in the design of hydraulic structures, including culverts and bridges under forest roads, that will reduce the risk of their failure and minimize the impact of environmental damage caused by flooding. The specific objectives of the thesis are: 1) to assess the regional variability of floods in British Columbia using the most recent scientific advances in stochastic hydrology, namely the linear moment statistics; 2) to investigate the extent of mixed populations in flood-data series at a site and in a regional context and their effect on flood estimates; 3) to delineate homogeneous hydrologic regions in British Columbia based on the extent of mixed processes in the flood-data series and similarities in climatic and physiographic characteristics; 5 4) to verify the homogeneity of the delineated regions using L-moment based statistical tests of homogeneity; 5) to identify the most suitable regional parent distribution within each delineated region using L-moment based regional goodness-of-fit testing; 6) to develop regional flood-frequency models for estimating flood quantiles at gauged sites within each homogeneous region; 7) to develop functional relationships for the transfer of flood information from gauged to ungauged sites within each homogeneous region; 8) to investigate ways of using the developed regional models, based on medium to large catchment data, for inferring small catchment flood estimates; 9) to evaluate the reliability of the proposed regional flood-frequency models using Monte Carlo simulation and split-sampling experiments. To the best of the author's knowledge, this is the first large-scale flood-frequency study ever conducted in British Columbia using the L-moments (Hosking, 1990). The L-moments, which are based on the linear combination of order statistics, have recently gained considerable momentum in the regional analysis of hydrologic extremes (Alila, 1994). The advantages of L-moments are that they can characterize a wider range of distributions than conventional moments, they are less sensitive to outliers in the data, and they suffer less from the effect of sampling variability (Hosking and Wallis, 1990). One of the major scientific contributions of this study is the development of an original innovative approach for delineating hydrologically-homogeneous regions. The new approach accounts for the fact that annual flood series in some areas are 6 identically distributed and in some other areas they are governed by mixed distributions. The new approach is successfully applied to the regionalization of flood data in the province of British Columbia where mixed flood processes have long been recognized. Within the context of the newly developed method for delineating homogeneous regions, a regional hierarchical flood-frequency approach is developed for estimating flood quantiles at gauged watersheds. This constitutes the second major scientific contribution of the study. In this hierarchical approach, different distribution parameters are estimated from different but nested sub-sets of data. It is demonstrated that this approach is not only theoretically and empirically valid, but also much more reliable than the single-site flood-frequency methods used in practice, particularly for longer return periods. While flood estimates are more often required at small-ungauged catchments, particularly in forestry applications, long-term flood data are often available only at larger watersheds. The third major scientific contribution of this study relates to investigating the extent to which small catchment flood estimates can be inferred from larger gauged watershed data: Based on this study, new physically based rules supported by empirical findings are established for extrapolating regional flood-frequency models to smaller catchments. 1.3 Thesis Overview Chapter 1, Introduction, presents the limitations of current methods used in flood-frequency analysis in the province of British Columbia and emphasizes the objectives and contributions of the study. 7 Chapter 2, Literature Review, reviews the literature with respect to single-site and regional frequency analysis, delineation of homogeneous regions and selection of regional parent distributions, and issues about mixtures in flood-data series. Unresolved problems or deficiencies and potential solutions by the proposed methods are highlighted. Chapter 3, Theoretical Background and Development, presents the theoretical background with emphasis on L-moments and their superiority to conventional moments, homogeneity tests, selection of regional parent distributions and the non-parametric approach. As well, Monte Carlo simulation is introduced. Chapter 4, Research Methods, describes the approach to data screening, the procedures for delineation of homogeneous regions, the methods for identifying regional parent distributions, and the techniques for evaluating the proposed models. Chapter 5, Results and Discussions, provides the research results of this study. Different regional models for estimating floods in the province of British Columbia are presented. Evaluations of the proposed models are conducted and technical issues discussed. Chapter 6, Summary and Conclusions, summarizes the study, provides major conclusions and lists recommendations for future research. 8 Chapter 2 LITERATURE REVIEW 2 . 1 Introduction Estimating floods is of fundamental importance in engineering hydrology. Flood estimates at gauged sites are often computed based on single-site flood-frequency analysis. This approach needs a sufficient flood-data series, otherwise, it may lead to large uncertainties or errors in flood estimates. It can not be applied at gauged sites with short flood-data series or at ungauged watersheds. Because of its advantages, regional flood-frequency analysis has been widely accepted for estimating floods. It involves three basic steps (Kite, 1977). The first is delineating homogeneous regions. The second is identifying a regional parent flood-frequency distribution for each delineated region. The third is developing regional models that can be applied at gauged and ungauged sites throughout each of the delineated homogeneous regions. Regional models are often parametrically developed. In such models, proper selection of a probability distribution is critical. However, the underlying parent distribution is not usually known. Flood-data series are commonly assumed to have been drawn from a single population distribution. Nevertheless, this basic 9 assumption is not always valid because floods may be generated by mixed processes. 2 . 2 Single-Site Flood-Frequency Analysis The single-site flood-frequency approach is based on analysis of data available at a given site. The basic assumption for single-site flood-frequency analysis is that the data are independently, identically and randomly distributed from a single population. This approach includes two steps: (1) selecting a probability distribution to describe the flood-data series, and (2) fitting the selected distribution to the observed data set. 2.2.1 Distribution Selection Many statistical distributions for flood-frequency analysis have been investigated in hydrology. Horton (1913) applied the normal distribution to describe annual flood series. Because annual flood series are often skewed and do not follow the normal distribution, other statistical distributions have to be considered. By using a log-normal distribution, Hazen (1914) found a better fit to annual flood-data series. Gumbel (1958) developed the theory of extreme values, which has been extensively used in hydrology. The Gumbel distribution, or EV1, is probably one of the most frequently used distribution in flood-frequency analysis (Bobee and Rasmussen, 1994). Jenkinson (1955) presented the generalized extreme value (GEV) distribution by combining the three asymptotic extreme-value distributions into a single form. The LN3 distribution was recommended by Chow (1954), with the argument that floods should be regarded as the result of a large number of independent factors. Houghton (1978) developed the five-parameter Wakeby distribution because it is more flexible when fitting the right hand tail (higher floods) and the left hand tail (lower floods) of observed flood-frequency curves. 10 Because of the lack of a theoretical basis for selecting an appropriate distribution for flood-frequency analysis, some countries recommend a distribution in an effort to standardize flood-frequency analysis procedures. In the USA and in Australia the LP3 was recommended (IEA, 1977; USWRC, 1982), whereas the G E V (NERC, 1975) and GLO (IH, 1999) were suggested in the U K . Gumbel (1958) stated that the goodness-of-fit of any distribution cannot be foreseen from theory. Conventionally, several distributions are fitted to the same observed data and the one that gives the "best" fit is selected. Graphical techniques in which data positions are plotted on probability paper can be used to select best-fit distributions (Dalrymple, 1960; Farmer and Fletcher, 1972) comparing the observed data with the theoretical form of the distribution. This procedure may produce a subjective conclusion and may limit the range of selected distributions. Some statistical goodness-of-fit tests such as Chi-Square, Kolmogorov Smirnov (Keeping, 1966) and Akaike Information Criterion (Akaike, 1974) have been used for selecting a distribution. However, serious reservations about the use of such tests have been raised (Matalas and Gilroy, 1968) because of the relative degree of subjectivity of the index of fit and the plotting position. It is also questionable to use these tests in regional flood-frequency analysis since they are single-site-based methods. To avoid having to select a frequency distribution and to deal with mixed processes in flood-data series, Adamowski (1985) introduced a non-parametric approach. The author stated that this approach does not require the assumption of any probability density functional form. He suggested the following approaches to estimate the probability density function: (1) kernel method, (2) orthogonal series methods, (3) penalized-likelihood methods, and (4) maximum-likelihood methods. 11 In the non-parametric kernel approach, which is detailed in section 3.6, selecting a smoothing factor (h) is important. The smoothing factor is similar to the class width in the histogram estimate of a density function. A large h value will lead to a unimodal shape for the density function, regardless of the multimodality of the data series, whereas a small h value will result in a distorted multimodal density shape regardless of the unimodality of the data series (Gingras and Adamowski, 1992). The determination of a smoothing factor is described in section 3.6. The selection of a kernel function is less critical than the choice of the smoothing factor. Adamowski (1985) used two different kernel functions: the rectangular kernel function and the Gauchy exponential kernel function. In other studies (Gingras and Adamowski, 1992; Sharma et al, 1997), the Gaussian kernel function was employed to describe the flood-data series. The non-parametric approach is one of very few approaches that account for mixtures in the flood-data series. Therefore, it was used for investigating flood characteristics in mixture regions in this study. 2.2.2 Commonly Used Distributions Many random variables in hydrology, such as maximum rainfall, maximum daily flow, and the lowest stream flow, may be described by one of several extreme-value distributions (Gumbel, 1958). As mentioned earlier, several studies have advocated the use of the EV1 distribution in British Columbia. The EV1 distribution has theoretical constant skewness and kurtosis values of 1.14 and 5.4, respectively. Depending on how these values compare to the observed skewness and kurtosis of the flood data, the EV1 distribution may or may not be appropriate for a particular region. In this study, the suitability of the EV1 distribution to the annual flood data in British Columbia will be evaluated. 12 The GEV distribution is also frequently used for estimating flood quantiles in both single-site and regional approaches (NERC, 1975). The GEV incorporates Gumbel's type I, II, and III extreme-value distributions. The cumulative distribution function of the G E V is given in section 4.4. Another important distribution, the LP3 distribution, was widely recommended for the description of floods in Australia (Pilgrim, 1987) and in the United States (USWRC, 1982). The U.S. Water Resources Council (USWRC) used the LP3 distribution in the Guidelines for Determining Flood Flow Frequency, known as Bulletin 17B. It has been widely applied since then. However, Bulletin 17B procedures were essentially finalized in the mid-1970s, so they did not benefit from subsequent advances in stochastic hydrology, namely the use of L-moments in regionalization techniques (Maidment, 1993). There is no doubt that diversity of climatic and physiographic characteristics increases as the area under study increases. The application of a standard distribution is questionable i f applied to a large area like the United States. The LN3 distribution is also commonly used for describing flood-data series. Details of the LP3 and LN3 distributions are described in section 4.4, because they were also investigated for fitting flood data in this study. 2.2.3 Methods of Fitting Distributions In single-site flood-frequency analysis, the parameters of any selected distribution are estimated from the observed data series, assuming that the flood series are independently, identically and randomly distributed. The most commonly used approaches for fitting distributions are the method of moments and the maximum likelihood method. 13 The method of moments is based on the relationship of the parameters of the distribution to its absolute moments about the origin. The r-th absolute moment is given by: jur=[y ma* (2.1) and the r-th central moment or moment about the origin is given by: /U\=fjx-ju]yf(x)dx (2.2) where r is the order of moment, //, is the population mean, and f(x) is the probability density function of variable x. Most of the two- or three-parameter distributions used in hydrology are described by the first two or three moments. In other words, the location, scale, and shape parameters are a function of the first-, second- and third-order moments, respectively. However, the accuracy of estimating higher order moments from the data series deteriorates rapidly as the order r increases and the sample size decreases. It has been shown that only the first order moment (the mean) estimated from the sample is an unbiased estimate of the population mean. The second and third order moments (the standard deviation and the coefficient of skewness, respectively) estimated from the sample are quite biased (Hazen, 1924; Wallis et al., 1974). Therefore, flood estimates from three-parameter distributions are less reliable because of the bias induced in estimating moments of higher orders. The maximum likelihood approach was first introduced by Fisher (1922). It is based on the idea that the joint probability distribution of the n observations in the data series is proportional to the product: 14 L = Y\f(x>,a,b,c,...) (2.3) where L is the likelihood function, f(x,,a,b,c,...) is the probability density function of x with a, b, c,..., being the parameters to be estimated. Maximum likelihood estimators have good statistical properties in large samples. For most distributions used in hydrologic practice, maximum likelihood estimators are asymptotically optimal (unbiased, minimum variance), but long sample records are generally needed for the asymptotic properties to prevail. Flood-data series are rarely long (typically 20-35 years) and for such sizes the maximum likelihood method may produce relatively poor estimators (Bobee and Rasmussen, 1994). For fitting a two-parameter distribution, both the method of moments and the maximum likelihood method provide nearly unbiased and almost equally efficient quantile estimates (Lowery and Nash, 1970; Samuelson, 1972; Watt and Nozdryn-Plotnicki, 1980). However, the method of moments was suggested for use because of its relative computational simplicity. When fitting a three-parameter distribution, there is no general agreement on which method is better. Fisher (1941) and Thorn (1958) showed that both approaches produced poor parameter estimates for highly skewed data. This is particularly important, because it was found, in this study, that the regional parent distributions in the province are either the generalized logistic distribution or the generalized extreme-value distribution. Both of these are three-parameter distributions. This implies that neither the method of moments nor the maximum likelihood method may provide appropriate parameter estimates. Greenwood et al. (1979) introduced probability-weighted moments. Landwehr et al. (1979) compared probability-weighted moments with the methods of moments and maximum likelihood estimation and showed that this technique can produce 15 unbiased parameter estimates when the data series are drawn from a random process. Similar findings were supported by many other studies (Lettenmaier and Potter, 1985; Lettenmaier et al., 1987; Wallis and Wood, 1985). Based on probability-weighted moments, Hosking (1990) introduced linear moment statistics (L-moments) which are more easily interpreted than probability-weighted moments (Details are discussed in chapter 3). Studies show that L-moments are virtually unbiased, less sensitive to outliers in flood-data series, and are better estimators of distribution parameters (Hosking, 1990; Royston, 1991). Therefore, L-moments were used in all stages in this study. 2.3 Regional Flood-Frequency Analysis Regional flood-frequency analysis is used for flood estimation at sites where little or no observations are available. It is based on the idea of transferring hydrologic information from some sites within a homogeneous region to ungauged sites, or to sites where the observed series of flood records are too short to permit a reliable estimation of floods. Studies show that even when a region is moderately heterogeneous, regional flood-frequency analysis can still yield more accurate quantile estimates than single-site frequency analysis (Lettenmaier and Potter, 1985; Lettenmaier et al., 1987; Hosking and Wallis, 1988; Potter and Lettenmaier, 1990) 2.3.1 Delineation of Homogeneous Regions A reliable regional model can be obtained by successfully delineating homogeneous regions (Bobee and Rasmussen, 1994; Burn et al., 1997; Greis and Wood, 1981; Hosking et al., 1985b; Lettenmaier et al, 1987). The correct grouping of different 16 catchments into homogeneous regions has become the critical issue in regional flood-frequency analysis. Delineating homogeneous regions is of particular importance because hydrologic information can be transferred accurately only within a region that is homogeneous. In general, the more homogeneous a region is, the more reliable the floods estimated by a regional frequency approach are. In the study by Dalrymple (1960), homogeneous regions were geographically defined through a "funnel" test by comparing the variability of the 10-year quantile estimates from each site in the region with that estimate expected i f sampling noise alone was responsible for the difference between sites. However, Wiltshire (1986) pointed out that this method is neither particularly powerful nor sensitive to a wide variety of extreme variables. Burn (1989; 1990) proposed the "region-of-influence" approach to determine the hydrological neighborhood or region of influence of a particular drainage basin. The site of interest is located at the center of a space with relevant flood statistics, catchment physiographic parameters, or a combination of both. The method involves choosing a distance threshold, and only sites whose distances to the target site do not exceed this threshold can be included in the region of influence. An advantage of this method is that each site can be weighted according to its proximity to the site of interest. However, the choice of the threshold is subjective (Bobee and Rasmussen, 1995). Cavadias (1990) developed a delineation method based on canonical correlation. This method is mathematically different from the region-of-influence approach, but it is still based on the same concept of determining the hydrological neighborhood or region of influence of a particular drainage basin. Groups of sites with similar flood response are visually identified in a space with two hydrologically homogeneous canonical variables and then subsequently interpreted in a space with 17 two physiographic canonical variables. A site of interest is then assigned to a particular region according to catchment characteristics. The problem with this method is that the grouping of sites is based on a subjective visual judgement and that there is no guarantee that a pattern can be found (Bobee and Rasmussen, 1995). In a comparative study of a number of regional flood-frequency analysis methods, GREHYS (1996a, b) showed that the neighborhood approach for the delineation of hydrologically homogeneous regions is superior to the fixed-region approach. However, one of the disadvantages of neighborhood approach is that, for each watershed, a group of network of homogeneous watersheds needs to be defined. This makes practice difficult. Also, the neighborhood approach uses physiographic data (slope of the main channel, percentage area of basin covered with forests, shape and orientation parameters of the watershed, etc.) and climate data (mean accumulated annual snow water equivalent, mean annual precipitation, etc.). When physiographic and climate parameters for gauged watersheds are not available (such as the case in the province of British Columbia), the neighborhood approach cannot be applied. Nathan and McMahon (1990) used cluster procedures to analyze homogeneous regions with emphasis on selecting and weighting the attribute variables describing the physiography and climate of the catchments. There is a great deal of subjectivity in determining the selection and weighting of variables, which limits the application of this approach. Nevertheless, grouping catchments of similar climatic and physiographic characteristics is necessary (but not sufficient) for a meaningful determination of homogeneous regions. For instance, the National Environment Research Council (NERC, 1975) divided the United Kingdom into 11 geographic homogeneous regions according to the contours from the largest catchments. Newfoundland, 18 Canada, was divided into two homogeneous regions according to the variability of climate factors (Panu et al., 1984). In other studies, flood-data statistics were used for grouping the catchments into homogeneous regions. Such parameters may include the coefficient of variation, the coefficient of skewness, and so on. In a study of regionalization of floods in Italy, Fiorentino et al. (1987) advanced a hierarchical approach to grouping sites into homogeneous regions. This method is based on the idea that the higher the order of a regional moment that is to be estimated, the greater the number of sites needed to produce an estimate with a given degree of reliability. Therefore, the coefficient of skewness (or parameters related to it) should be estimated from a large region, while the coefficient of variation (or parameters related to it) should be estimated from sub-regions. Using Monte Carlo simulation, they showed that the method produced significantly more accurate flood quantiles than the classic regional index approach and the single-site approach. Delineation of homogeneous regions was done in this study by grouping stations with similarities in the shape of the cumulative distribution function, classifying different climatic regimes, analyzing physiographic characteristics and investigating and testing mixtures in flood-data series. Homogeneous regions were further verified using homogeneity tests. 2.3.2 Homogeneity Tests The delineation of homogeneous regions is traditionally based on geographic, political, administrative or physiographic boundaries, and then assumed homogeneous in hydrologic response (Burn et al., 1997). Without verification by statistical tests, the assumption of homogeneity may not be justified when there is a large amount of spatial variability in the physiographic, climatic, and hydrologic 19 characteristics of the catchments in the region. The most common homogeneity measures are the dimensionless scale and shape parameters: the coefficient of variation ( C v ) and the coefficient of skewness ( C v ) (Bobee and Rasmussen, 1994). Assumptions of homogeneity in various orders of moments can be justified only by mathematical methods. Cunnane (1988) pointed out that determining a homogeneous region should be based on statistical tests of hypothesis. Pilon et al. (1991) suggested a variance ratio test for the homogeneity in a region, based on the concept of L-moments: S(%) = 2 _ 2 C* obs CT > v/„, <? obs (2.4) where <j20bs is the variance of L-moments of the observed flood-data series and a1 s,m is its synthetic counterpart based on a large number of network simulations. The authors indicated that cr2,,,,, represents the noise of the synthetic network due to sampling error for the particular L-moment, whereas cr 2^, represents a combination of both sampling error and real variation (signal) of the flood-data series. Therefore, S{%) represents the percentage of signal that is evident in the flood-data series, and the percentage of noise of the network is 100- S. However, this test is not convenient for delineating homogeneous regions because equation 2.4 provides only a qualitative description. Hosking and Wallis (1993) proposed a homogeneity test (H test) based on the notion that all stations have the same population L-moment statistics (i.e., L-coefficient of variation, L-skewness, and L-kurtosis) in a homogeneous region. Therefore, the weighted average L-moment statistics are considered as the best representative parameters of the network under study. The homogeneity of the 20 network of stations may be examined by quantifying the difference in the sampling variability of various L-moments between the observed network and it's synthetic counterpart. The statistical significance of the difference in the L-moment sampling variability of the two networks is assessed through Monte Carlo simulations using the H test (Hosking and Wallis, 1993). In this study, the H test was applied to all regions to validate regional homogeneity. Details of the test are given in sections 3.3 and 4.3. 2.3.3 Identification of the Regional Parent Distribution Another issue of regional flood-frequency analysis is the selection of the parent distribution for a particular homogeneous region. Regional flood-frequency analysis faces the same difficulty as the single-site approach does. That is, the true form of the underlying parent distribution is not known. As outlined earlier, no universal standard distribution exists. However, identification of the regional parent distribution is often based on standard practices or procedures. These procedures provide researchers and practitioners with convenient guidance, but their theoretical basis and the accuracy of the outcome are questionable. Although some commonly used distributions are recommended, such as the EV1, GEV, LN3, and LP3, it is arguable that a distribution can be considered suitable without appropriate mathematical tests. Identification of the regional parent distribution may be based on empirical goodness-of-fit tests. However, it has been shown that some regional estimators are not robust in terms of the selected distribution, which implies that mis-identifying the regional parent distribution may induce large errors in flood-frequency analysis (Lettenmaier et al, 1987) 21 L-moment diagrams (Schaefer, 1990; Pilon et ah, 1991; Vogel and Fennessey, 1993) are a popular tool for identifying the parent distribution for a particular region. In a homogeneous region, flood-data series are assumed to be from the same population. The shape indices (L-skewness and L-kurtosis) of the parent distribution can be represented by the regional averages based on the flood-data series from all sites in the region. Hosking and Wallis (1993) developed a statistical test based on L-moments for identifying the regional parent distribution. Because of its advantages, their method was applied in this study (section 3.4). 2.4 Methods for Regional Flood-Frequency Analysis Two commonly used methods for regional flood-frequency analysis are the index-flood approach and the regional regression method. Studies in the 1980's showed that use of index-flood procedures should provide substantially improved flood quantile estimates (Wallis and Wood, 1985). On the other hand, regional regression models have long been used for predicting flood quantiles at ungauged sites due to their simplicity. Newton and Herrin (1982) concluded that this approach did as well or better than more complex rainfall-runoff modeling procedures in a nation-wide test in the USA. The British frequency analysis procedure represents an index-flood method (NERC, 1975). This approach involves using the G E V distribution to describe the flood-data series, whereas the original index-flood approach proposed by Dalrymple (1960) uses the EV1 as the parent distribution in a homogeneous region. 22 2.4.1 Index-Flood Approach In the original index-flood approach (Dalrymple, 1960), the procedure for estimating flood quantiles involves the following steps: 1) plot a frequency curve relating discharge to return period at each site based on all available data; 2) read off QT from each frequency curve; 3) estimate the mean annual flood Qm from the flood-data series at each site; 4) at each site, calculate the ratios of XT = for various T-year return periods and plot a dimensionless frequency curve relating Xr to T; 5) average all at-site dimensionless frequency curves to obtain what has been referred to as the "regional growth curve", assumed to be constant within the homogeneous region; 6) develop a relationship between the mean annual flood and catchment physiographic and climatic variables; 7) for an ungauged site, estimate the mean Qm from the developed relationship in 6); and 8) compute the required discharge of any return period using: Qr = Xr x fiL, (2-5) In the index-flood approach, the mean annual flood (index flood) can be predicted, in general, by: 23 Qm = f{P\,Pl-P,) (2.6) where px p2...pn are physiographic variables, climatic variables, or a combination of both. The index-flood approach is a regionalization technique with a long history in flood-frequency analysis (Dalrymple, 1960). It is popular among hydrologists and has been widely applied. As Maidment (1993) pointed out, the index-flood procedure is an accurate method when its assumptions are satisfied. The basic assumptions of the index-flood method are that the region under consideration is homogeneous in terms of the coefficient of variation (i.e., the C,, of the flood-data series is constant within the region) and that a relationship exists between the index-flood and selected physiographic and climatic variables. It was also assumed that the data at all sites in the homogeneous region followed an EV1 distribution. Potter and Lettenmaier (1990) showed that index-flood methods perform better than the method recommended in Bulletin 17B by the US Water Resources Council. In Canada, there are twelve studies for the various regions of the country that use the index-flood method (Watt et al, 1989). However, arguments about scale issues have received increasing attention in recent years. Gupta et al. (1994) and Gupta and Dawdy (1995) used simple-scaling and multi-scaling theories to analyze the characteristics of flood-data series in a region. They concluded that the index-flood approach is related to simple-scaling that indicates a constant coefficient of variation and a log-log linear relationship between flood quantiles and drainage areas. When flood-data series have multi-scaling properties, the assumption of the constancy of coefficient of variation is violated. This theory shows that the coefficient of variation for small catchments 24 increases, and for large catchments decreases as the drainage area increases. Also, the flood quantiles in multi-scaling do not show a log-log linear relationship with respect to drainage areas (Gupta et al, 1994; Gupta and Dawdy, 1995). It was found in this study that the assumption of constancy of the coefficient of variation for the index-flood method is violated (section 5.4). Therefore, the regression technique was adopted for the regionalization of floods in the province. 2.4.2 Regional Regression Method Regression is a classic method used to describe the relationship between a dependent variable and independent variables. Various regression models have been used in hydrology to predict the statistics of hydrologic variables (e.g., quantiles, means, and standard deviations) as a function of physiographic characteristics and other parameters in a region. Estimation of quantiles can be obtained using regression. Within a hydrologically homogeneous region, floods of various return periods can be assumed to depend upon the physiographic and climatic variables of individual watersheds. This requires selecting independent variables, such as watershed size, channel length, mean basin elevation, surface storage by lakes and swamps, drainage density, forest coverage, and precipitation, and selecting a regression model form (linear, power, logarithmic, exponential, polynomial, etc.). In most cases, experience and judgement have to be used to determine which independent variable(s) should be included in the regression model at the first stage. Choice of the final independent variables is often made on a statistical basis; that is, many variables are used in preliminary regressions and those that lack statistical significance are discarded. 25 Riggs (1973) reviewed the significant variables found in 10 published regional flood-frequency regressions (Table 2.1). Drainage area was the only variable used in the regression models for all 10 studies. Campbell and Sidle (1984) concluded that the size of watershed is the only variable significantly related to streamflows. It is evident that drainage area is the most important variable for explaining the variation in streamflows. Therefore, the most dominating variable, the size of watershed, was used for regression analysis in this study. Discussions on whether the drainage area on its own is enough to explain the variability of floods in British Columbia are given in section 5.6. Interpreting results from a regional regression analysis is not straightforward because the residuals contain both sampling variability and variation due to basin characteristics, and there is no measure of the relative amounts of each (Riggs, 1973). A small standard error of estimate indicates that there is little sampling variability in the flood-data series used. In practice, the assumption is that the residual is largely due to basin characteristics. This assumption is not often justified and more likely, the major part of the residual variation is due to sampling (Riggs, 1973). However, because of their simplicity, regional regression models are still the most commonly used method in flood-frequency analysis, even though some subjectivity is involved in selecting the independent variables and the underlying model form. The regression method was applied in this study to develop different regional models for non-mixture regions and for mixture regions, and was combined with other methods such as hierarchical approaches and non-parametric approaches. 26 2.5 Mixed Processes in Flood-Data Series If flood-data series are produced by more than one flood-generating mechanism, the population of the flood-data series is considered to be a mixed process or a mixed population. As summarized earlier, most of the commonly used distributions in hydrology assume that flood data are identically distributed (i.e., drawn from one single population). This may produce large errors in the estimates of floods because floods may be generated by mixed processes. The phenomenon of mixed processes has long been recognized in flood-data series. Potter (1958) demonstrated the evidence for two distinct populations by using the shape of the cumulative distribution function of floods. Many other researchers (e.g., Cox, 1966; Dickinson et al., 1992; Hirschboeck, 1987; Leytham, 1984; Singh, 1968) also have analyzed mixed processes and suggested solutions for dealing with mixed processes in flood-data series. Parametric frequency analysis of mixed processes can be performed by either unclassified or classified approaches. The unclassified approach is based on the idea that a set of flood data is a combined sample from two distinct distributions. The cumulative distribution function (CDF) in this case can be expressed as: F(x) = alF](x) + a2F2(x) (2.7) where o,+a 2 =l, 0<(a,, a2)<\, F^(x) and F2(x) are the cumulative distribution functions of the two distinct sub-populations, and a, and a2 are the relative weights of each component distribution. F,(x) and F2(x) are often replaced by two EV1 distributions or a combination of any other two distributions. Thus, the two-component mixed distribution may 27 become a two-component extreme-value distribution (TCEV). Many studies indicate that the TCEV approach is generally applicable where annual flood series are generated by two mechanisms (Rossi et al., 1984). Without classifying or splitting the data sample, the parameters of equation 2.7 can be estimated using the maximum likelihood method (Fiorentino et al., 1985) or the method of least squares (Singh, 1987). However, this approach may result in more uncertainties because the number of parameters to be estimated from the same data set is doubled for a mixture of two distributions. The classified approach to frequency analysis of floods with mixed processes was first introduced by Waylen and Woo (1982; 1983). To describe a mixed distribution of floods, they recommended: Assuming the mixture is formed by two EV1 distributions, equation 2.8 becomes: where /?,, /32, « , , and a2 are the location and scale parameters of population 1 and 2, respectively. This approach requires the classification of floods based on the generating mechanisms. For instance, i f the mixture is caused by seasonal differences an annual flood series may be formed for each season and the parameters of each component distribution are estimated accordingly. In many cases, however, mixed processes in floods may be caused by mechanisms other than those caused by seasonal differences. In these cases, the classification of floods requires F(x) = Fx(x)F2(x) (2.8) F(x) = exp[- exp(- a, (x - px)) - exp(- a2 {x - J32))] (2.9) 28 hydroclimatic data that are often not available. The classified approach to mixed distributions becomes, therefore, very impractical. Another way of analyzing mixed processes in flood-data series is by non-parametric methods. Because the non-parametric approach does not require assuming any functional form for density it is particularly effective in dealing with mixed observations generated by multiple mechanisms (Adamowski, 1985). To avoid the pitfalls of the parametric techniques of treating mixed distributions, the non-parametric method was adopted in this study. A detailed description of the non-parametric method is presented in section 3.6. 2.6 Summary To estimate floods at ungauged sites and to improve the accuracy of estimation of floods at gauged sites, regional flood-frequency approaches are used. In regional parametric flood-frequency analysis, the following three problems have to be solved: (1) identifying hydrologically homogeneous regions; (2) identifying the parent distribution of a given homogeneous region; and (3) developing regional estimation models for the identified homogeneous regions. Delineation of homogeneous regions has been researched in various studies and several delineation approaches have been developed, e.g., the "funnel" test by Dalrymple (1960), the hierarchical approach by Fiorentino et al. (1987), the region-of-influence approach by Burn (1989), canonical correlation by Cavadias (1990), and the L-moment regional H test by Hosking and Wallis (1993). However, none of these methods accounts for mixture in flood-data series, while it has been demonstrated that floods in the province, even though in the same watershed, may 29 be generated by different mechanisms. Therefore, the shape of CDF as an indicator of mixture is used in this study for delineating homogeneous regions. The determination of the regional parent distribution is usually based on either standardized practices or empirical goodness-of-fit tests. Standardization provides convenient guidance, but the theoretical basis is questionable. Studies show that empirical goodness-of-fit tests for selection of a regional distribution may not be robust and consequently may introduce large errors in the flood-frequency analysis. In British Columbia, there seems to be little agreement on the forms of distributions. Therefore, the EV1 distribution and others are often applied in flood-frequency analysis. In this study, the suitability of these distributions will be investigated using L-moment diagrams and L-moment based regional goodness-of-fit tests. The outcome of this investigation is important to practical application in flood-frequency analysis in British Columbia. The existence of mixed processes in flood-data series makes estimating floods complicated. Commonly used single-population distributions cannot be applied when floods are generated by mixed populations. There are various ways to deal with mixed processes but there is no general agreement on which method is better for solving mixed processes in flood-frequency analysis. Index-flood methods and regional regression methods are often used for estimating floods in regional flood-frequency analysis. Each of those approaches has advantages and disadvantages. If homogeneous regions are properly delineated based on similarities in physiographic and climatic characteristics and appropriately tested, and if the assumption of the index-flood approach is met, then the index-flood method might be a better choice as a regional flood-frequency analysis. On the other hand, regional regression methods are widely used in flood-frequency 30 analysis because of their simplicity, although some subjectivity is involved in the selection of independent variables and the underlying model form. In this study, regional models for flood-frequency analysis in British Columbia will be developed using regression methods combined with hierarchical and non-parametric approaches. The performance of the proposed regional models will be evaluated and compared to current practice using Monte Carlo simulations. 31 Chapter 3 THEORETICAL BACKGROUND AND DEVELOPMENT 3.1 Theory of L-Moments L-moments are different from conventional sample moments. Conventional moments are computed as sums of powers of the observations, whereas L-moments are obtained using linear combinations of the ordered sample values. Hosking (1990) defined the first four L-moments A , A 4 as follows: (3.1) 2:2 (3.2) -E(X 3 v • 3:3 2X2i + Z 1 : 3 ) (3.3) (3.4) where Xin is the / -th smallest variable in a sample of size n. 32 Thus, \ is the population mean and A^ is a measure of dispersion analogous to the standard deviation. Both A^ and the standard deviation are measures of the difference between two randomly selected values of X. However, the standard deviation assigns more weight to large differences, and hence to extreme sample values, than does A^ (Royston, 1991). Aj describes the difference between the upper- and lower-tail lengths of a sample of size three from the distribution of variables, with the median (X2J) as the central point. A4 provides an indication of the peakiness of the distribution. It is proportional to a weighted difference between the variable outer extremes and the central portion in samples of size four. When many samples are taken, A1 and A^ are considered as measures of the location and scale of the distribution, and the scale-free ratios are regarded as measures of skewness and kurtosis. For this reason, r 3 and r 4 are known as the coefficient of L-skewness (L -Cs) and L-kurtosis (L-Ck) (Hosking, 1990). Hosking also defined the ratio of A^ to A, as the L-coefiicient of variation r 3 = /£3/A2 (3.5) r 4 — / l 4 / A^ (3.6) (L-Cv). Hosking (1990) further showed that: 4 = A (3.7) A7=2ft-& (3.8) ^ = 6 A - 6 ^ + A (3.9) A4 = 20/?3-30/? 2+12#-/? 0 (3.10) 33 where the probability weighted moment of order r, f3r, was defined as 0 = [x{F)F'dF (3.11) and F stands for non-exceedance probability. Suppose that x, , x 2 x n is a random sample from the distribution of X and let xln < ... <xnn be the ordered sample values, then the following is shown as unbiased estimators of J3r (Landwehr et al, 1979): where i is the rank of a particular observation in the ordered sample. The estimation of L-moments of a random sample may be computed using equations 3.12. 3.2 Advantages of L-Moments A sample statistic is robust when its performance is not affected by changes in the population distribution. This means that the difference in performance between estimates of the same statistic from one data sample to another should be minimal even though the data contain outliers, as long as data are from the same distribution. Royston (1991) concluded that L-moments are generally less sensitive or fairly robust to outliers. Based on three data sets with sample sizes ranging from 162 to 2477, he showed that sub-samples with no outliers produced different values of conventional skewness and kurtosis compared to sub-samples containing outliers. ( / - l ) ( / - 2 ) - ( f - r ) 2) •••(/!->•)' ('•) (3.12) 3.7, 3.8, 3.9, and 3.10, after the values of ^ , ^ , ^ ... are computed using equation 34 However, L-skewness and L-kurtosis of the same sub-samples were found to be fairly consistent and stable around their population values. L-moments are almost unbiased, while conventional moments are both highly biased and highly variable, especially in small samples. This is because conventional moment estimators require squaring ( x 2 ) , cubing (x 3 ) , and quadrupling ( x 4 ) the observations, which gives more weight to observations far from the mean, especially when estimating the skewness and kurtosis. In flood-data series, occasional events may be several times smaller or larger than others. Such values may distort the information provided by other observations. Wallis (1989) demonstrated using simulation how L-moments are less biased than conventional moments. Another important advantage of L-moments is their efficiency and unboundedness. In this regard, the method of maximum likelihood has been shown to be asymptotically more efficient than the method of moments, especially when the observations indicate the need for a three-parameter distribution. Nevertheless, asymptotic theory does not always lead to the smallest variance in frequency analysis of hydrologic variables where the sample size is usually small (Hannan, 1987). However, it was shown that the method of L-moments is more efficient than the maximum likelihood technique for various sample sizes of up to 100 (Hosking and Wallis, 1987). The mathematical boundedness of skewness and kurtosis is a serious limitation of the method of moments (Kirby, 1974). Shape indices computed from observations are bounded, and can not reach or cover the full range of the population values. However, it was demonstrated that both L-skewness and L-kurtosis computed from sample sizes larger than four can take values covering the full range of the population values (Hosking, 1986). 35 Klemes (2000a, b) articulated some cautionary notes about the use of L-moments in hydrologic frequency analysis. He argued that high outliers in a flood-data series are important for extrapolating to long return period events in engineering design, as they define the upper leg of the flood frequency curve. Therefore, by using L-moments, which are less sensitive to these outliers, practitioners may be missing on the most important piece of information in the flood data series. However, Klemes (2000a, b) failed to point out that when the outliers are caused by sampling variability (for instance, a 100-year flood event in a 10-year sample) they should not be given an undue weight. In this case, it is more sensible to use a method that is less sensitive to outliers in the data, such as the L-moments. On the other hand, when the outliers are the result of differences in flood generating mechanisms (i.e., a mixture of two or more populations) the critical issue becomes in the distribution selection and not in the fitting technique. In this case, the appropriate use of a mixture distribution is the most sensible way of using the information provided by the outliers and the use of L-moments would still be more superior to conventional moments. In this study, statistical tests of hypothesis will be used to check the validity of the assumption that flood observations in the same data series are identically distributed. In regions where this assumption can not be validated, appropriate mixture distributions will be recommended. In summary, L-moments (e.g., L-coefficient of variation, L-skewness, and L-kurtosis) are less biased, more robust, and more efficient than conventional moments. These characteristics present significant advantages in hydrology for screening data to identify sites with questionable or unusual observations and for confirming i f a group of basins have sufficiently similar properties to yield the same distribution. They can also be used for parameter estimation of frequency distribution and regional quantile estimation. L-moments were used for all the 36 phases in developing the proposed regional frequency models (i.e., delineating and testing homogeneous regions, identifying and fitting regional distributions, and developing regional models) in this study. 3.3 Homogeneity Testing The H test proposed by Hosking and Wallis (1993) is based on the idea that all stations have the same population L-moments in a homogeneous region. The test statistic is given by: H=(K,h,-Mr) ( 3 1 3 ) where Vohs is the value of an observed network of K, , V2 or K 3 (defined in equations 4.4, 4.5, and 4.6) in a homogeneous region and nv and av are the mean and standard deviation of the synthetic counterparts of Vohs estimated from a large number of synthetically generated regions with the same characteristics as the real region (i.e., same number of stations, same record length at each station, and the same regional observed L-skewness and L-kurtosis values). The hypothesized homogeneous regions can be tested using a Monte Carlo simulation procedure, which is detailed in section 4.3. To avoid any commitment to a particular parent distribution for computing the synthetic counterparts, the four-parameter Kappa distribution is used in generating synthetic regions or network of sites (Hosking, 1988). The Kappa distribution includes the generalized logistic, generalized extreme-value and generalized Pareto (GPA) distributions as special cases. 37 3.4 Identification of the Regional Parent Distribution If it is determined that a region is homogeneous, flood-data series at all sites within this region could be assumed to be from the same parent distribution. The regional average L-skewness and L-kurtosis based on the flood-data series from all sites in the region would represent the L-moments of such a parent distribution. Furthermore, different distributions have different relationships between their respective population L-skewness and L-kurtosis values, as shown in the L-moment ratio diagram (Fig. 3.1) and in Table 3.1. The L-moment ratio diagram provides the possibility of distinguishing a specific distribution by its particular L-moment ratio. Distribution selection in L-moment analysis is performed by comparing L-moment ratios of L-skewness and L-kurtosis to the theoretical values. Averages of L -skewness and L-kurtosis within a homogeneous region can be plotted on an L-moment ratio diagram along with theoretical curves for various candidate distributions. If the point corresponding to the regional averages is located near the curve corresponding to a given distribution, this nearest distribution will be a reasonable choice for the parent distribution in this region. The selected distribution from the L-moment ratio diagram can be verified by a goodness-of-fit test (Hosking and Wallis, 1993). (3.14) where f4 is the regional average L-kurtosis of the observed network in the homogeneous region and r 4 D / 5 7 is the theoretical L-kurtosis, and a-h is the standard deviation of f4 obtained by repeated simulations of the homogeneous region with the DIST frequency distribution as a parent. 38 Based on Monte Carlo simulation performed by Hosking and Wallis (1993), the goodness-of-fit of a particular distribution should be considered acceptable at the 90% confidence level if |z| < 1.64. The Z-test was used in this study for choosing the best-fit parent distribution for a given homogeneous region. This test uses regional data as opposed to single-site information. It is therefore more reliable than single-site goodness-of-fit testing. 3.5 Hierarchical Approach to Regional Flood-Frequency Analysis A hierarchical approach to regional flood-frequency analysis is based on the idea that different flood characteristics are assumed to be approximately constant over different spatial scales. Following Matalas and Gilroy (1968), Fiorentino et al. (1987) showed that if a regional parameter t90 was estimated as the mean of K site estimates Q}: (3.15) then, the mean square error (MSE) of the estimate can be described by: MSE(0o) = E \0o-0o ) K (3.16) 39 where E is the expected value, and a2 and co2 are the time and space sampling variances, respectively. The parameter Ke is the equivalent number of independent stations given by: Ke= (3.17) \ + {K-l)p0 and pg is closely related to the average inter-site correlation of annual maximum floods within the defined region. Equations 3.16 and 3.17 suggest that the accuracy of regional estimators is affected by: (1) time sampling variance, (2) space sampling variance, and (3) inter-station correlation. They imply that, by increasing the number of sites, one can obtain a reduction of time sampling variability that is partially counterbalanced by the space disturbance errors and is limited by inter-station correlation (Fiorentino et al, 1987). Theoretically, a regional model should involve a time-space trade-off that maximizes the benefits of combining data while minimizing the consequences of defining larger regions (Alila, 1994). This justifies the need for a hierarchical procedure for fitting the regional parent distributions. In the hierarchical approach, different distribution parameters are estimated from different, but nested, subsets of data. In other words, the L-skewness is estimated from regional information while the standard deviation and the mean may be estimated from sub-regional and at-site data, respectively. 3.6 Non-Parametric Method Non-parametric density estimation has been a part of the statistics literature since the 1960s and has been applied in stochastic hydrology since the 1980s (Schuster 40 and Yakowitz, 1986). The non-parametric method requires neither the assumption of a density function nor the estimation of parameters based on the mean, variance, and skewness. Instead, it requires the selection of a kernel function and a position smoothing factor. Let x,, x 2 , . . .x„ be a random sample of observations, the kernel estimate of probability density function /(*) at each fixed point x,. is given (Scott and Factor, 1981)by: nn 7:7 h where x is the flood variable, x, to x„ are the flood observations, n is the number of observations in the sample, K(x) is the assumed kernel function that is itself a probability density function, and h is a smoothing factor to be estimated from the sample. The principle of a kernel estimator, as described by equation 3.18, is that a probability density function (kernel) of a prescribed form (i.e., rectangular, Gaussian, etc.) is associated over a specified range (expressed by h) on either side of the observation. The non-parametric estimate of the probability density function is the sum of the kernels (Fig. 3.2). Adamowski (1985) conducted simulations using different types of kernels. The author concluded that the optimal and the most efficient kernel is the Epanechnikov kernel. However, this kernel is bounded, and so it is not desirable for use in flood-frequency analysis when an extrapolation of a density function is required. Therefore, with a very small loss of accuracy, a Gaussian kernel (i.e., a normal distribution with standard deviation equal to h) is used to present the density of 41 every point in flood-frequency analysis. The Gaussian kernel is given (Adamowski, 1985) by: K(x) = -£±= exp{ - {x - x,)2 / 2h2} (3.19) The non-parametric method is especially effective in dealing with historical observations with mixed processes in a region, where the floods are generated by multi-mechanisms or from more than one population. Assume that there are two sets of data in the observations, one is denoted by x2,...x and the other is denoted by x ,, x 2,...x , that happen to exceed the prescribed value Y during the m = [n - « 0) year historical record. The probability values of the two sets of data are estimated (Adamowski and Feluch, 1990; Adamowski, 1989) by: {x, < Y)i with probability pix < Y) = —— (3.20) {x, > Y}with probability p{x>Y) = ^  (3.21) and the unknown density function is estimated by a two-component mixed distribution: f(x) = fix < Y)pix <Y) + fix > Y)pix > Y) (3.22) where the density fix) is estimated non-parametrically by equation 3.18. The value of smoothing parameter h in equation 3.19 is determined from the cross-validation procedure that requires solving the following equation (Adamowski and Feluch, 1990): 42 - « b + Z 4 ( l - 2 c ^ ) ( A 2 : / - l ) - l ] = 0 (3.23). './=i;'<7 where c = l4ln l{n-\), Atj = (xf - x^j 14, and dtj = exp(A2y 14). In an attempt to avoid distributional assumptions when floods are generated by mixed processes, the non-parametric method was used in this thesis. As described above, the kernel function was assumed Gaussian, and the smoothing factor was estimated using the cross-validation procedure. 43 Chapter 4 RESEARCH METHODS 4.1 Study Area British Columbia is Canada's westernmost province, with an area of 950,000 km 2. It is essentially a mountainous region, except for the northeast coiner that includes a small portion of the Great Plains. In winter, the climate of British Columbia is dominated by the north-south orientation of the mountains and the Pacific Ocean. Moist eastward air masses from the Pacific Ocean are forced over these mountains resulting in relatively heavy rainfall on the west slopes and much less rainfall on the east slopes. Precipitation is mainly in the form of rainfall in the southwestern area and snow in other areas of the province. Frontal storm activities on the coast dominate annual precipitation in winter (November to March). Convective storm activities in the interior become important in the summer (May to August). The proportion of annual precipitation that falls in the form of snow as opposed to rain increases with latitude, elevation, and distance from the coast. Snowfall varies from less than 10% of winter precipitation at sea level in the south coastal region to almost 100% in the northern interior (Moore and McKendry, 1996). 44 In summer, relatively weak upper air masses in combination with the development of a persistent high pressure area above the Pacific Ocean results in fewer frontal systems moving through British Columbia. Consequently, considerably drier conditions result throughout most of the province. However, as the temperature increases in the summer, streamflows are generated by snowmelt (and occasionally rain on snowmelt) which dominate the annual runoff in almost all places in the province. The exception is the coastal mountains where most of the flow is still produced in winter. This is illustrated by the discharge histogram by month of occurrence in Fig. 5.8. 4.2 Data Selection Numerical analysis in this study was conducted on annual maximum daily discharges measured at unregulated streams located throughout the province. The decision to use daily discharge was based on the fact that the number of stations measuring daily flows is three times larger than the number of stations measuring instantaneous flows in the province. The stream gauge network in British Columbia is fairly new, as most stations were established in the 1970s. Stations are unevenly distributed with the majority concentrated in the southern areas of the province (Fig. 4.1). As a compromise between spatial and temporal coverage of data, all catchments with a minimum record length of 10 years were used. This threshold length of record may not lead to reliable at-site flood-frequency information but can still provide useful information in regional flood-frequency studies (Burn et al, 1997). A total of 511 annual flood series representing natural hydrologic regimes and with the minimum length of 10 years was obtained from the H Y D A T C D R O M of Water Survey of Canada, Environment Canada, 1994. As published by Environment Canada, the quality of the data is considered satisfactory. In regional flood-frequency analysis, it is assumed that data from 45 different sites in the same year are independent. If this assumption is violated, then the estimate of flood quantiles may be biased and inaccurate. However, the error is relatively minor because it is generally accepted that the errors in flood estimates are often due more to estimating the at-site mean than to estimating the coefficient of variation. It is mostly the latter flood statistic that is affected by inter-site dependence (Hosking and Wallis, 1988). Using Monte Carlo simulation to assess the effect of inter-site dependence on the regional probability weighted moment algorithm, Hosking and Wallis (1988) concluded that: • any bias in flood estimates is unchanged by the presence of inter-site dependence. • the accuracy of flood estimates decreases when inter-site dependence is present, but this is less important for practical applications than the bias in flood estimates due to heterogeneity (inequality of flood-frequency distributions in the region). • even when both heterogeneity and inter-site dependence are present and the form of the flood-frequency distribution is mis-specified, regional flood-frequency analysis is more accurate than at-site analysis. In view of the above spatial cross-correlation findings, it is presumed that the presence of inter-site dependence, i f there is any, does not pose serious problems to the accuracy and bias of flood-quantile estimates in this study. In flood-frequency analysis, it is also assumed that flood-data series are independent, random, homogeneous and without trend. These assumptions were verified using the Consolidated Frequency Analysis (CFA) package of Environment Canada (Pilon and Harvey, 1994). 46 4.2.1 Test for Independence Independence means that the probability of occurrence of an observation in the sample is not affected by any other observations in the same sample. A data series is temporally independent i f no observation in the data series has an influence on any observation following it. In a time series, independence can be measured by the significance of the correlation coefficient between N - l (N is the sample size) pairs of the / -th and the (/ +1) -th observations. If the correlation coefficient is not significantly different from zero, the independence of the flood-data series is assumed. The non-parametric Spearman rank order correlation coefficient test was used to examine the serial correlation in the data series. 4.2.2 Test for Trend If the successive observations of a time series are made during a period of gradually changing conditions, there will be a trend in the magnitude of the observations in the series when they are arranged in chronological order. Spearman's rank order correlation coefficient test was applied for the presence of trend and shifts and their significance in the flood-data series. 4.2.3 Test for Homogeneity in Time Homogeneity in time is usually assumed in flood-frequency analysis. This means that the flood statistics are invariant with respect to time. For instance, i f changes have occurred during the sampling period there should be differences between the means of the sub-samples before and after the changes. When two sub-samples of approximately the same size are selected for analysis the sums of the ranks of the two sub-samples should not differ significantly i f homogeneity in time exists. Assuming a normal distribution and that the two sub-samples have the same 47 variance, then the difference of the sub-sample means can be tested for significance using the t -distribution. These assumptions are usually not met in hydrology (Pilon and Harvey, 1994). Therefore, testing for homogeneity was accomplished using the Mann-Whitney split-sample test, which is a function of the sub-sample sizes and their sums of ranks. 4.2.4 Test for Randomness A simple random sample is one that is selected in such a way that any other sample could have resulted with equal likelihood. In hydrology, randomness is usually caused by the data series arising from natural causes. The non-parametric "runs above and below the median" test for general randomness was employed in this study. The test is based on the order or sequence in which the observations of each station are obtained. From the data of each station, the median was determined and the number of runs of observations above or bellow the median was counted. The median is used because the probability of exceeding the median is 0.5, regardless of the probability distribution from which the sample is drawn. Stations that failed any of the four tests above (at the 99% confidence level) were considered to be unreliable records and, therefore, were eliminated from further analysis. 4.3 Delineation of Homogeneous Regions Regional homogeneity is an important requirement in regional flood-frequency analysis. In this study, a great deal of attention was paid to delineating the homogeneous regions in the province. 48 4.3.1 Delineation of Homogeneous Regions Based on CDF Shape The extent of mixture in the maximum annual flood data of all gauged catchments was assessed through the shape of the empirical cumulative distribution function (CDF) relating the floods to their probability of occurrence or return period. If the CDF plots as a straight line on a probability paper, it was assumed that the annual floods are generated by a single population. On the other hand, a break in the slope in the CDF is an indication that the annual floods is generated by a mixed distribution. Potter (1958) was the first to use the shape of the CDF to identify mixture in the annual flood data. He presented evidence suggesting that the break in slope (or dog-leg) of the flood-frequency curve is the result of sampling from two different populations of floods. A l l CDFs were plotted using the Cunnane plotting position on normal probability papers as recommended by Pilon and Harvey (1994). As a next step, hydrologic zones based on geographic patterns to the shape of the CDF were delineated. Geographic patterns to the shape of the CDF have been used as evidence that these patterns are the results of real differences in catchment physiographic and climatic characteristics (Archer, 1981). 4.3.2 Delineation of Homogeneous Regions Based on Climatic and Physiographic Conditions The timing of the peak flow was employed in this study as a climatic index. Therefore, it can be used as a measure of similarity in catchment response (Reed, 1994). Basins with similarities in the timing and seasonality of flood may also have similarities in important physiographic and hydrologic response characteristics. Such basins can be considered as potential candidates for membership in the same region for regional frequency analysis. The spatial trend of the timing of floods was investigated across the province using the discharge histogram by month of occurrence. The province was delimited into zones of consistent timing of the 49 annual floods. These zones were subsequently superimposed on the regions already delineated using the shape of the CDF according to section 4.3.1. Physiographic conditions significantly affect hydrologic characteristics. The following five major regions of British Columbia (Holland, 1964) were used for further adjusting the boundaries of delineated hydrologic regions: 1) Coast Mountains, a northwest oriented strip parallel to the Pacific coast; 2) Interior Plateau, which includes the large central area and a part of the southern area; 3) Columbia Mountains and Southern Rocky Mountains, northwest oriented mountains in the southeast corner of the province; 4) Northern and Central Plateaus and Mountains, covering a vast area in the northern and central areas; and 5) Great Plains, a triangle-shaped area in the northeast of the province. 4.3.3 Testing the Significance of Mixed Processes in Flood-Data Series The difference in magnitude of the slopes of the lines forming a dog-leg CDF is a measure of the significance of mixed processes in flood data. Fig. 4.2 shows that a dog-leg is formed by two straight lines, which may be viewed as evidence of two populations generating floods. However, i f the dog-leg is not accentuated or the two straight lines coincide, the flood series at the station should still be considered to arise from one population. In this case, the criterion must be met that any two discharges with the same return periods from the two straight lines should be close enough. 50 To test the significance of mixed processes within each of the delineated mixture regions, a hypothesis test (the Rm statistic) was developed: R __LyI,flsL + a Qsos (4.1) where QIQS, QWL , QWS, and QXL are the quantiles as defined in Fig. 4.2 for the 10- and 50-year return periods, respectively, L and S represent the two different populations, and N is the number of stations within a given region. These quantiles were estimated from the two straight lines forming the dog-leg of the CDF. Theoretically, one single line results from the overlap of the two straight lines on which the two discharges QOS, QLAL, and QIL)S, g 5 0 / are chosen. Therefore, i f the RM statistic is significantly different from 1, the region was considered as a mixture region. The procedure for testing the significance of mixed processes in flood-data series consisted of the following steps: 1) determine the QQS, Q]0T, QSOS, and g50, from the CDF plot at each station; 2) compute the Rm statistic using equation 4.1 for the assumed mixture region; 3) test at the 99% confidence level if the Rm statistic is statistically different from 1 using Student's t -test; and 4) decide if the region is a mixture region. 4.3.4 Testing the Effect of Sampling Variability on the CDF Shape A dog-leg CDF may imply that floods are produced by two flood-generation mechanisms. However, the break in slope in the CDF could also be caused by 51 sampling variability (i.e., the shape of CDF may be affected by the record length (Potter, 1958)). In this study, an analysis of variance (ANOVA) was conducted to test the null hypothesis that the dog-leg is caused by a real difference in flood-generating mechanisms against the alternative hypothesis that the dog-leg is caused by sampling variability. This test was used with a significance level of one percent. An error statistic using the 50-year quantiles estimated from both lines of a CDF plot was developed: E _ (QsoL ~ QsOS ) (4.2) Q SOL As in equation 4.1, QI()S and g 5 0 /. are arbitrarily chosen. However, any other quantiles should result in the same conclusion. The concern here is not about the selection of a statistic itself, but about whether the statistic varies in terms of record length. A similar test was also used by Potter (1958) to investigate the effect of sample size on the shape of the CDF. The test procedure for equation 4.2 consisted of the following steps: 1) in each of the previously defined mixture regions, all stations were placed into three groups. The first group had stations with short record lengths, the second one had stations with long record lengths, while the third one had stations with intermediate record lengths. Grouping was done so that approximately the same number of stations was in each of the three sets; 2) at each station, the QXS and QKI were determined from the CDF plot and the error statistic ER was calculated using equation 4.2; 52 3) for each of the selected groups, the record-length weighted average of the error statistic Er was calculated; 4) an F-test was performed to check i f the variances of the error statistic Er were equal among groups; and 5) an A N O V A was performed to determine if there were significant differences in the record-length weighted average of the error statistic Er caused by grouping (i.e., by sampling variability). For a given region, i f the dog-leg is not caused by sampling variability, the difference between the Er values of the various groups should not be statistically significant. In such a case, this region should be considered to be a mixture region. 4.3.5 Testing the Homogeneity of Delineated Hydrologic Regions To perform the statistical homogeneity test H (section 3.3) for any of the delineated regions, the Kappa distribution was fitted using the weighted average mean (/,), standard deviation (t 2), skewness (f 3), and kurtosis (/ 4) in L-moment space. These statistics were estimated by: t. = EI,"/,"1 EN (4.3) where tr(l) is the value of tr at site i of the region, nj is the sample length at site i , and N is the number of sites within the region. The weighted average as opposed to simple average was used in equation 4.3 because the variance of estimated parameters of statistical models is inversely proportional to the sample length (Hosking and Wallis, 1991). 53 Samples drawn from this parent distribution were arranged to replicate the number of sites and the number of observations at each site. Through a Monte Carlo simulation, a large number (Nshll = 500) of synthetically generated networks (regions) were then replicated, and the following three measures of the between-site variability of sample L-moments were computed for each.network as follows: 1) based on the L-coefficient of variation (t2) only, the weighted standard deviation of t2 is calculated as V], 2) based on the L-coefficient of variation (t2) and L-skewness (f 3), the weighted average distance from the site to the network's mean on the t2 versus r3 space is then calculated as V2, / n, 3) based on the L-skewness (r 3 ) and L-kurtosis (r 4), the weighted average distance from the site to the network's mean on the t3 versus f4 space is calculated as V3, y ^.ll?-tf+<t?-!.)']" .... v3 = (4-6) 54 where t2, t3, and f4 are the network weighted average means of L-Cv, L- C , , and L-Ck, respectively; t(2], , and are the L-Cv, L-Cs, and L-Ck at site /, respectively. A FORTRAN computer program (Hosking, 1991) was employed for assessing homogeneity. The subsequent steps were followed: 1) For the observed network of sites the following values were computed: a) the mean /,, L-standard deviation l2, L-coefiicient of variation t2, L -skewness t3, and L-kurtosis tA at each station; b) the average regional weighted L-moment parameters 7,, t2, ti, and tA; c) the between-site L-moment variability measures yj, y2, and y using equations 4.4, 4.5, and 4.6. 2) The four-parameter Kappa distribution was fitted by the method of L-moments using the statistics computed in step (b) above. The CDF of the Kappa distribution is given by (Hosking, 1988): where ^ is the location parameter, a is the scale parameter, and h and k are the shape parameters. and (4.7) 55 3) For the synthetic network (or region): a) a synthetic network with the same number of sites and the same record length at each site was generated using equation 4.7; b) the mean /,, L-standard deviation l2, L-coefficient of variation t2, L-skewness t3, and L-kurtosis t4 were computed at each site; c) the between-site L-moment variability measures yt, y2, and K , were calculated; and d) steps (b) and (c) were conducted for 500 replicates of the synthetic network, and e) the mean nv and standard deviation av of each of the quantities yt, j / , and K 3 were determined based on the 500 replications. 4) Finally, the homogeneity statistic (H) was computed using equation 3.13. In light of the diversity of hydrologic conditions in British Columbia, and according to Monte Carlo simulation research by Hosking and Wallis (1993), a region under testing is defined homogeneous i f the H value is less than 1. The region is considered as possibly homogeneous if the H value is between 1 and 2. The region is declared heterogeneous when the //value is greater than or equal to 2. If a region is declared homogeneous, it is assumed that the between-site variability of L-skewness and L-kurtosis is not in excess of the variability caused by random sampling (i.e., spatial differences in these L-statistics result from the between-site variability of the record length within the region). 56 The above procedure of homogeneity testing was conducted for each of the delineated regions and is summarized in Fig. 4.3. The geographic boundaries of regions that did not pass the homogeneity test H were further adjusted until a satisfactory set of statistically homogeneous hydrologic regions was achieved. 4.4 Identification of Regional Parent Distribution In a non-mixture homogeneous region, flood-data series at all sites could be assumed to be drawn from the same regional parent flood-frequency distribution. If the between-site variability of L-skewness and L-kurtosis were mainly due to sampling variability, the regional weighted average L-skewness and L-kurtosis based on the flood-data series from all gauged sites in the region can be used to represent the L-moments of such a parent distribution. For each of these regions, the procedure used to identify the regional parent distribution involved the following: 1) compute the regional weighted-average values (with respect to record length) of L-skewness and L-kurtosis based on data from all sites within the region; 2) use the location of the computed regional L-skewness and L-kurtosis on the theoretical L-moment ratio diagrams to identify a possible distribution that may be assumed to represent the flood-data series within the region; and 3) use the goodness-of-fit Z test (equation 3.14) to further verify the selection of the regional parent distribution. The Z-test discriminates between five of the most frequently used distributions in the analysis of hydrologic extreme variables, namely the: 57 1. Generalized extreme value (GEV), with its cumulative distribution function given by: F(x) = exp^ - 1 - k(x-Q a for k * 0 (4.8a) x-C F(x) = exps - exp - for k = 0 (4.8b) V a J where ^ is the location parameter, a is the scale parameter, and k is the shape parameter. For k>0, the distribution has a finite upper bound at £+cc/k and corresponds to the EV3 distribution, for k <0, the distribution has a thicker right-hand tail and corresponds to the EV2 distribution. The EV1 distribution is obtained when k=0. 2. Pearson type I I I (P3), with its CDF expressed as: where T(.) is the gamma function, and ^ , a, and k are the location, scale, and shape parameters, respectively. 3. Generalized logistic distribution (GLOG), with its CDF given by: (4.9) (4.10) 58 in which ^, a, and k are the location, scale, and shape parameters, respectively. 4. Generalized log-normal distribution (LN3), with its CDF given by: F(x) = <t> 1 & V 1 (4.11) where O is the cumulative distribution function of the standard Normal distribution, is the location parameter, a is the scale parameter, and k is the shape parameter. 5. Generalized Pareto (GPA), with its CDF: F(x) = l-Qxp{k-l\og[l-k(x-C)/a]} k * 0 (4.12a) F(x) = \-exp{-(x-C)/a} k = 0 (4.12b) where a, and k are the location, scale, and shape parameters of the distribution, respectively. For a mixture region, both the TCEV and the non-parametric methods were investigated. However, it was found that the TCEV is not appropriate for the flood data in the province (section 5.4). Therefore, the non-parametric method was used for analyzing the mixed processes in flood-data series. 4.5 Proposed Hierarchical Frequency Models In section 5.2 it is shown that each delineated region can be considered homogeneous in the L-skewness and L-kurtosis. Therefore, all catchments within 59 the same region can be represented by the same values of L - Cv and L-Ck. These findings justified the need for a hierarchical flood-frequency model within each delineated region where different parameters of the regional parent distribution were estimated based on different but nested subsets of data. To fit the identified regional frequency distribution at any catchment, a hierarchical approach composed of two levels was proposed. At the first level, the L-skewness and L-kurtosis of the annual flood series were computed based on regional information by equation 4.3. At the second level, the mean and L-standard deviation of the annual flood series were estimated from at-site data. The computed regional L-skewness and L-kurtosis and the at-site mean and L-standard deviation were then used to fit the identified regional distribution by the method of L-moments. Monte Carlo simulations described in section 5.9 will demonstrate how the proposed hierarchical frequency models result in more accurate flood quantiles than the single-site frequency analysis. The proposed hierarchical approach is only suitable for the non-mixture regions where conventional parametric frequency distributions can be justified. In the mixture region, where the nonparametric frequency distribution has been adopted, a multiple regression approach relating flood quantiles to the physiographic characteristics of the catchment is advocated. 4.6 Split-Sampling Experiment Regression models in this study were developed to estimate floods at ungauged catchments. Regression models relate the flood quantile of a particular return period to the physiographic (and climatic) parameters of the catchment. A split-sampling experiment in space was conducted to validate these regression models. In each 60 region, the station with the longest record of observations was selected for this analysis. This experiment consisted of the following steps: 1) compute for each long-term station the observed floods, {QT))HSENED> f ° r different return periods using the non-parametric method. The non-parametric method was applied for the observed quantiles in order to avoid any commitment to a particular flood-frequency distribution that might bias the results; 2) use the proposed regional models to calculate the predicted floods, (@r)predicted' ^ o r t n e s a m e r e t urn periods as in step 1, assuming no streamflow record available; and 3) compare the differences between the observed and the predicted flood values using the error equation: £(%) = 100 "-Y) "(fir) ' 1 ' observed \ 1 * predicted ( a ) (4.13) \ V ^ ' 'observed J The flood quantile, {QT \hseneJ, is estimated from the at-site long-term flood record, which justifies its use as a reference in the split sampling validation experiment. Ali la (2000) used a similar approach validating a set of rainfall frequency equations for Canada. 6 1 4.7 Monte Carlo Simulation Monte Carlo simulation is a data-generation technique in which synthetic data are generated from a given probability distribution. Monte Carlo simulation has been widely used in statistical hydrology to study the probabilistic behavior of hydrological processes. In general, Monte Carlo simulation is used to obtain a model of output given a probability distribution for the input. The transformation from the input to the output is made by a mathematical model that represents the behavior of the physical system under study. In flood-frequency analysis, two basic concerns should be considered with respect to Monte Carlo simulation. One is model uncertainty (selection of a parent distribution). For a particular situation, the problem is how to select a model among several alternatives. Because hydrologic processes are inherently complex, i f an inappropriate parent distribution is selected, the output of the simulation will definitely mislead a decision maker. Usually, the parent distribution of a flood-data series is unknown. Fortunately, Monte Carlo simulations in this study were based on distributions that had been statistically identified using L-moment theory and, therefore, conclusions can be obtained with more confidence. Another concern in Monte Carlo simulation is parameter uncertainty. This uncertainty arises because the model parameters can be estimated only from historical samples, which are usually small and contain errors. In general, the variance of estimated parameters is a decreasing function of the sample size. This, in turn, implies that the variance of the estimate will decrease as the sample size increases. As discussed earlier, because of their unbiased and robust properties, L-moments were used for determining the distribution parameters of the model. 62 Monte Carlo simulation was applied in this study with the purpose of evaluating the performance of the developed hierarchical models in comparison to single-site flood-frequency analysis. It consisted of the following steps: 1) fit the identified regional distribution to the regional L - C , , and given values of the mean and L - Cv (observed typical values); 2) use the fitted distribution to estimate the Q2, Q5, Q]0, Q20, Q50, Qm, Q200 , Q500, and Qmo. These are the true population flood values for some arbitrarily selected return periods; 3) use the above fitted distribution to generate a large number of samples of the same size «,; 4) fit the regional distribution by the method of L-moments to each synthetic sample using a single-site approach. In this case the sample estimates of the mean, L - Cv and L - C v should be used in fitting the distribution; 5) use the fitted distribution in 4) to calculate the single-site estimates of the Q2, Qs ' Q\0 ' Q-20 ' QsO ' QlOO ' ^200 ' S5OO' 2l000 > 6) repeat steps 4) and 5) for various sample sizes n, =10, 20, 30, 40, 50, 60, 70, 80, 90, and 100. This will allow us to assess the effect of sample size on the reliability of floods using single-site frequency approach; 7) fit the regional distribution by the method of L-moments to each synthetic sample using the developed hierarchical approach. In this case, use the sample estimates of the mean and L-Cv, but the regional value of L-Csof step (1) above; 63 8) use the fitted distribution in 7) to calculate the regional estimates of the Q2, Q5, (2l0> 220> £?50> 0100 > £?200 > £?500 > a I l a ' Si 000 '•> a n < ^ 9) repeat steps 7) and 8) for various sample sizes n,=10, 20, 30, 40, 50, 60, 70, 80, 90, and 100. This will allow us to assess the effect of sample size on the reliability of floods using regional frequency analysis. To compare the performances between single-site and regional approaches, the root mean square error (RMSE) was used in this study. RMSE reflects the dispersion of estimates from the population values and comprises both the bias and precision as defined as: where Bias refers to the difference between the expected value of an estimate and the population value, while Variance refers to the spread of the estimates. (4.14) 64 Chapter 5 RESULTS AND DISCUSSIONS 5.1 Data Screening There were 476 out of 511 stations that passed the tests of randomness, trends, independence, and homogeneity. Station names, numbers, drainage areas, and record lengths are listed in Appendix C. The average record length of the selected stations was 24 years (the longest record was 90 years, and the shortest was 10 years). As shown in Fig. 5.1, most of the stations have drainage areas over 100 km1. Approximately 25% of the stations had drainage areas smaller than 100 km2. As an example, results of tests of randomness, trend, independence, and homogeneity are given in detail in Tables 5.1 to 5.4 and Fig. 5.2 to 5.4 for station 08MB005. This station had a continuous record of annual maximum daily flows from 1971 to 1994. Test results indicated that the data was random and homogeneous, and did not display significant dependence and trend. Fig. 5.2 indicates that the maximum flows for this station have been in June, July, and August. For a qualitative assessment of the data regarding trends or jumps, the annual daily maximum flow was plotted over rank in Fig. 5.3 and over year in Fig. 5.4. Visual assessment of these plots confirmed that there was no significant evidence of trends or jumps in the data. Shifts (jumps) and trends in flood-data series can be produced gradually or instantaneously. For example, a large clear-cut in a basin may quickly result in a 65 shift, whereas a gradual forest insect infestation or climate change may produce trends in the flood-data series. In regional frequency analysis, trends in flood data caused by climate variability can be treated in different ways. For instance, one could use data from only the hydrometric stations with a common record length during a particular period where the climate could be considered stable. However, doing so could be at the expense of reducing the spatial and temporal data coverage of the study region. In British Columbia, the bulk of the streamflow data have been collected during the last 40 years and therefore no attempt was used to reduce the hydrometric data from the various stations to a common recording time period. Church (1997) used also this same period (1961 to 1990) as the global period for his analysis in a recent regionalization study. 5.2 Classification of Homogeneous Regions 5.2.1 Homogeneous Regions Based on CDF Mixtures in different regions in the province may be caused by totally different hydrologic processes. On the coast, for instance, mixtures may be caused by different types of storms or the same type of storm with different antecedent moisture conditions (e.g., rain on very wet antecedent conditions versus rain on relatively dry antecedent conditions). In the interior, mixtures may be caused by snowmelt versus rainfall or rain on snowmelt. In other areas, mixtures could be caused by different types of snowmelt mechanisms (e.g., melt caused by long-wave as opposed to short-wave radiation). Fig. 5.5 shows how the antecedent precipitation (AP) can be used as an index to classify flood events according to the generating mechanism. However, analysis of AP cannot be done for all stations in British Columbia because of the lack of precipitation data. Therefore, the shape of the CDF at each station in the province was checked to determine i f there were 66 mixed processes in the flood-data series. The spatial trends of the CDF were used to divide the entire province into mixture and non-mixture regions. According to the shape of the CDF, the province was divided into three contiguous non-mixture regions (NMR) and three contiguous mixture regions (MR) as shown in Fig. 5.6. For each of the NMRs, the majority of CDFs were straight lines, and for each of the MRs, the majority of CDFs had a dog-leg. The number of stations in each region as defined by the CDF shape is given in Table 5.5. Checking for dog-legs in flood-data series involves some subjectivity. For example, one may argue that the CDF plot of station 08LB022 in Fig. 5.7 suggests a mixture in the flood data because the shape of the plot is not completely straight. However, the flood-data series from this watershed was considered identically distributed. Such subjectivity in the delineation of homogeneous regions is not too critical, because: 1) the CDF was only one of a group of criteria that were used to define homogeneous regions, and 2) the shapes of CDFs were further verified by testing the significance of mixed processes in flood-data series and testing whether the dog-leg was caused by a real mixture or the result of sampling variability. Details are given in section 5.2.4. 5.2.2 Homogeneous Regions Based on Climatic Conditions As described in section 4.3.2, the timing of flood occurrence was used for classifying homogeneous regions. Checking the histograms of flood peaks by month of occurrence revealed that all identified regions except MR3 had floods that occurred in the same period or season of the year. Fig. 5.8 displays the contrast in 67 the timing of floods between the delineated regions. For instance, further classification of MR3 was necessary and the original MR3 was split into five new regions (MR3 through MR7) as shown in Fig. 5.9. Most floods in N M R 2 occurred in May and June, while those in NMR3 occurred in June and July. This one-month flood delay between NMR2 and NMR3 may be due to slower temperature increase in the spring or early summer in NMR3. As shown in Fig. 5.9, NMR3 is located in the northwestern corner of the province where snowmelt floods are generated relatively late. Because the Coast Mountains are an effective barrier to moist air flow to the east, in the interior areas the climate is warm and dry in summer and colder and less arid in winter, and precipitation is mainly in the form of snow. Therefore, when temperature gradually increases, summer streamflows are generated by snowmelt. Histograms of floods by month of occurrence in MR1 , 2, 3, 4, and 5 illustrate this flood-generating mechanism (Fig. 5.8). This phenomenon dominates the annual runoff in almost all interior areas in the province. It is evident that flood occurrences in MR6 and 7 differ from those of other regions (Fig. 5.8). In MR6, there are both summer floods and winter floods. They may be explained as rainfall-induced floods and snowmelt-induced floods. MR6 covers the Coast Mountain areas where precipitation is both rainfall (in lower elevation areas) and snow (in higher elevation areas) in winter. The portion of the precipitation that is rainfall becomes streamflow soon and the portion that is snow accumulates on the ground as a source for streamflow in the coming summer when temperature increases. It is interesting to note that MR7 is the only region in the province where almost all floods occur in winter (October to February). Unlike MR6, where both rainfall and snowmelt generate floods, winter rainfall dominates the flood process in MR7. Very 68 few stations, for example, stations 08NB074 and 08HD001 in the Central Highlands of Vancouver Island, occasionally have snowmelt floods in the summer because of some higher elevations in the watersheds. Each of the three NMRs and seven MRs can be considered as homogeneous in both CDF shape and timing of flood at this stage of the study. In other words, both the shapes of CDFs and the timing of floods are similar within each of the 10 regions mapped in Fig. 5.9. 5.2.3 Homogeneous Regions Based on Physiographic Conditions The physiographic features of British Columbia can be described simply as mountainous systems of which northwest oriented mountains (for example, Central Mountains of Vancouver Island, Coast Mountains, and Rocky Mountains) play an important role in the hydrologic regime, as they are perpendicular to the eastward movement of the frontal systems. Because of steep slopes and considerable elevation range in the coastal regions, orographic effects are prominent. Both the Strait of Georgia and the Interior Plateau are rain shadow areas (Waylen and Woo, 1983). However, delineation based on CDF and the timing of flood occurrence did not reflect the rain shadow area of the Strait of Georgia. Hence it was necessary to consider the Strait of Georgia as a separate region (MR8). This resulted in the exposed coast of BC being a separate region, MR9, as indicated in Fig. 5.10. The flood magnitude on the windward and leeward sides of the non-exposed coastal mountains is different. Watersheds on the windward side have higher peak flows. Floods are caused by large rainfall amounts in winter and by significant snowmelt in summer. The magnitude of peak flows on the leeward side is low because of the rain shadow effect, and peak flows are caused mostly by snowmelt in summer, with the time of flood occurrences similar to summer floods on the windward slopes. 69 Therefore, differences in flood-generating mechanisms (rainfall vs. snowmelt) and differences in the magnitude of floods between the windward side and the leeward side of the mountains have to be taken into account when delineating the province into homogeneous regions. This resulted in the old MR6 of Fig. 5.9 being split into two regions, MR6 and 7, as shown in Fig. 5.10. The Rocky Mountains are an important physiographic system in the province. Also, the Great Plains, a triangle-shaped area in the northeast of the province, is another unique physiographic unit. Therefore, the boundaries of these two regions which were originally delineated on the basis of the analysis of CDF and timing of flood occurrence were modified to reflect their distinct physiographic characteristics as indicated in Fig. 5.10. Based on the above analysis, British Columbia was divided into three non-mixture regions and nine mixture regions. The boundaries of these regions are shown in Fig. 5.10. 5.2.4 Tests of Mixed Processes in Flood-Data Series (1) Significance of mixed processes For each of the mixture regions of Fig. 5.10, a test of significance of mixed processes in flood-data series (section 4.3.3) was applied. Two of the previously assumed mixture regions did not have significantly mixed processes at the 99% confidence level as shown in Table 5.6. Therefore, these two regions, previously delineated MR1 (southeast British Columbia) and MR9 (along the exposed coastal areas at lower elevations), were considered as non-mixture regions. 70 (2) Effect of sampling variability on CDF shape A dog-leg on a CDF plot could be caused by either mixed processes or sampling variability. To distinguish their differences, the statistic Er described in section 4.3.4 was employed. The F-test results in Table 5.7 indicate that the differences in the variances of the statistic Er among groups were not significantly different within each of the seven remaining mixture regions listed in Table 5.6. Therefore, A N O V A was applied to test i f sampling variability had any effect on the statistic Er within each of the mixture regions and the results are given in Tables 5.8 and 5.9. Tables 5.8 and 5.9 suggest that three mixture regions (i.e., the Rocky Mountains, Northern Interior, and Arrow Lake), based on previous procedures, fail the test. Dog-legs in these three regions do not appear to be caused by mixed processes, but rather by sampling variability. This indicates that the plot of CDF is expected to be a straight line without a dog-leg, if enough observations were available for all stations in the region. The three regions that failed the test at this stage were re-classified as non-mixture regions. The remaining mixture regions are MR1 the North-eastern Great Plains, MR2 the Coast Mountains (1), MR3 the Coast Mountains (2), and MR4 the Georgia Depression, respectively (Fig. 5.11). Among the five coastal regions, NMR5 is the only one in which the mixture was found to be nonsignificant according to the procedure of delineation and several statistical tests of significance. It can be argued that it makes more sense to consider NMR5 as being a mixture region like the other four coastal regions. Among all the regions in the province, NMR5 is the region with the fewest number of stations (eight stations only). This might have contributed to the nonsignificance of mixture in NMR5. However, when comparing this region with other four coastal regions, it was found that NMR5 is exposed to the coast and has the lowest elevation. Because 71 this region is below the usual snow-line, the annual maximum daily flood is dominated by rainfall only in winter. This physical reason for NMR5 being different from the other coastal regions is shown by comparing the monthly discharge histograms among the coastal regions (Fig. 5.8). 5.2.5 Homogeneity Testing Based on the analyses in this chapter, it was hypothesized that catchments that meet the following criteria are candidates for membership in a homogeneous region: 1) similar CDF shapes (dog-leg vs. straight-line); 2) similar climate indexed by the timing of floods; 3) consistent physiographic conditions; 4) similar levels of significance of mixed processes in flood-data series; and 5) similar effect of sampling variability on CDF shape. Within a homogeneous region, the population L-moment ratios from all sites should be the same. However, sample L-moment ratios may differ due to sampling variability. In addition, the spatial and temporal variability of L-moments are supposed to reflect the spatial and temporal variability of catchment characteristics that govern flood mechanisms. Therefore, the degree of variability of L-moments indicates the significance of the homogeneity of catchment characteristics within each region. Testing of homogeneity was done through a comparison of the between-site variability in sample L- moments within each region with what would be expected in an equivalent homogeneous region. 72 The H test (section 4.3.5) was applied to the 12 previously delineated regions (Fig. 5.11). In this study, the purpose of delineating homogeneous regions was to select a regional distribution so that a regional model can be developed. Therefore, only y} (equation 4.6), the test based on L-skewness and L-kurtosis, was used in the analysis because distribution selection is normally performed by comparing the L-moment ratios of L-skewness and L-kurtosis to theoretical values (section 3.4). Results of the H test displayed in Table 5.10 show that seven of the 12 regions may be considered homogeneous in both L — C v and L — Ck. The remaining five regions (NMR1, NMR2, NMR7, NMR8, and MR2) are heterogeneous and need to be further delineated into sub-regions to achieve homogeneity. Based on the boundaries of the watersheds, further classification of these five regions into homogeneous sub-regions was done by regrouping stations and modifying the boundaries. The final homogeneous regions are shown in Fig. 5.12. Table 5.11 shows the results of the H test for the final 19 regions. It can be seen that all 19 regions are homogeneous in both L-skewness and L-kurtosis with a minimum H value of 0.00 and a maximum H value of 1.91. In summary, 14 non-mixture homogeneous regions and five mixture homogeneous regions in British Columbia were delineated based on: 1) shape of cumulative distribution function of the annual floods; 2) timing of the annual flood events; 3) physiographic characteristics; 4) test for significance of mixed processes; 5) test for effect of sampling variability on CDF shape; and 73 6) test for homogeneity in L-skewness and L-kurtosis. This is the first time that the province has been delineated into homogeneous regions with a comprehensive procedure that combined consideration of physiography, climate, flood mixture, and statistical testing. This study demonstrated the successful application of an innovative procedure for the delineation of homogeneous regions in the province. The same procedure may also be used for other studies such as analysis of instantaneous flows and low flows. 5.3 Selection of Regional Parent Distribution 5.3.1 Regional Parent Distribution The regional weighted average L-skewness and L-kurtosis were computed for each of the 14 non-mixture regions and are given in Table 5.12. For each region, the regional ratio of r3 and /4were superimposed on a theoretical L-moment ratio diagram as shown in Fig. 5.13. The ratios were found to be closely distributed around the line of either G L O G or GEV. These results were confirmed by the Z-test as described in section 3.4. The Z-test discriminates between five of the most commonly used single population distributions in hydrology. Hence, this test can only be applied in the delineated non-mixture regions. Of the five candidate distributions, the G L O G and G E V distributions gave the best fit to 11 and 3 regions, respectively (Table 5.13 and Fig. 5.14). In other words, the G L O G and GEV distributions were most suitable for describing the flood-data series in British Columbia. These results are consistent with recent L-moment based regional flood-frequency studies. A goodness-of-fit test based on L-moments as suggested by Hosking and Wallis (1991) showed that both winter and summer low-flow data were best fitted by a G L O G and G E V distributions. Vogel and Wilson (1996) reviewed various studies of flood-frequency analysis using L-moments. They 74 concluded that "even though these studies involve flow samples throughout the world (Australia, New Zealand, Canada, the United States and Bangladesh), all studies recommend the use of G E V distribution" based on L-moment ratio diagrams. 5.3.2 Sensitivity Analysis of Skewness As was mentioned in chapter 1, several previous publications on flood-frequency analysis advocated the use of the Gumbel distribution in British Columbia. The argument behind the use of this two-parameter distribution is its simplicity and particularly because it does not require an estimate of the skewness. The skewness is not reliable when estimated from short at-site records using the conventional method of moments. This thesis advocates a regional approach in which the skewness is estimated not from a site but regionally. Regional estimates are based on longer record lengths and are, therefore, more reliable than at-site estimates. In addition, this study used the method of L-moments that has been proven much more reliable than the conventional method of moments. In the delineated homogeneous regions, L-skewness and L-kurtosis are constants in each region, but vary from one region to another. A sensitivity analysis was conducted to reflect the effect of a change in the L-skewness from one region to another on the flood-frequency curves and consequently on flood estimates. Fig. 5.15 shows the results of the analysis for the GEV distribution with L-skewness values of 0.10, 0.14, 0.18, 0.22, 0.26, and 0.30, respectively. These values of L -skewness are typical for homogeneous regions in the province. The Gumbel distribution is only a special case of the GEV distribution with an L-skewness value of 0.1699. It is clear that the flood estimates are sensitive to the L-skewness values 75 especially for a return period longer than 20 years. The errors of flood estimates using the Gumbel distribution with a fixed L-skewness, as opposed to using the different constant regional L-skewness values computed from different regions, might be significant. Based on the results of this study, the G L O G and GEV distributions should be used to estimate flood quantiles in the 14 non-mixture hydrologic regions of British Columbia. 5.4 Regional Models for Non-Mixture Regions The attempt to develop regional models for flood-frequency analysis was made by comparing the two commonly used approaches: the index-flood and regional regression. The index-flood approach is related to simple scaling and implies a constant coefficient of variation within the homogeneous region (Gupta and Dawdy, 1995). Flood-data series in British Columbia may have multi-scaling properties. In L-moment space, most of the 19 delineated regions are highly heterogeneous in L — CV. Column 5 of Table 5.11 indicates that the average value of H based on the 19 delineated region is 6.85, which is larger than 2 (limit for which a region may be considered homogeneous). It was also found that the L — CV apparently varied with the drainage area (Fig. 5.16). The plot shows the trend that the L ^ C v f o r small catchments increases, and for large catchments decreases. Consequently, the assumption of constancy of the coefficient of variation for the index-flood approach was violated. Therefore, the regional regression method was adopted in this study. 76 5.4.1 Models for the Generalized Logistic Distribution The generalized logistic distribution is defined by the location (^), scale (a), and shape (k) parameters. These three parameters were computed using L-moments: where A,, , and T 3 are as defined previously. The shape parameter k is related only to L-skewness (equation 5.1). However, it was shown that all delineated regions are homogeneous in L-skewness and L-kurtosis (section 5.2.5). In other words, L-skewness and L-kurtosis were constants or, practically, the between-site variability of L-skewness and L-kurtosis was not conspicuously different within any of the delineated homogeneous regions. Therefore, the shape parameter k can be represented by the regional weighted value, averaging the k from each station weighted by the record length of observations. 5.4.1.1 Estimation of Flood Quantiles at Gauged Sites Given a gauged station, the shape parameter k of G L O G remains constant (Table 5.14), while the other two parameters can be calculated (equations 5.2 and 5.3) using the lower order L-moments \ and A7 estimated from at-site data. Once the k = - r . (5.1) a = sinA:;r (5.2) (5.3) 77 three parameters of the G L O G distribution are estimated, flood quantiles for any return period(T = ) can be computed by: 1 - F(x) x = £+a[l-((\-F(x))/Fix))"} Ik (5.4) The above method is known as the hierarchical approach. Other studies have shown that the hierarchical approach is much better than the regional index approach and single-site method (Fiorentino et al, 1987; Alila, 1999). In the hierarchical approach, different distribution parameters are estimated from different, but nested, subsets of data. At gauged stations, it is more appropriate that the first two parameters of the G L O G distribution are estimated from at-site data while the third parameter is kept constant for the region. In other words, the L-kurtosis and L-skewness are estimated from regional information while the mean and standard deviation are estimated from at-site data. On the one hand, the use of this type of model is often supported by the fact that more data are needed for estimating higher order moments than for estimating lower order moments. This is one way of overcoming the associated uncertainties of making statistical inferences from short sample data. On the other hand, by using at-site data to estimate lower order moments as opposed to regression models as developed in the next sub-section, the likelihood of ignoring potential site-specific effects of watershed factors other than the drainage area on the distribution parameters is minimized. This is because the regression equations assume the drainage area as the sole factor affecting the distribution parameters. Although it was demonstrated (section 5.5) that regression model residuals are not in excess of what could be expected from sampling noise, other physiographic and climatic factors may affect floods. 78 5.4.1.2 Estimation of Flood Quantiles at Ungauged Sites Because a regional parent distribution was selected using the L-moment ratio diagram and verified by the Z test, it is assumed that the parent distribution can be applied to a region and that the parameters of the distribution are dependent on physiographic characteristics (the drainage area in this study). As shown previously, the G L O G distribution was selected for describing flood-frequency characteristics for 11 non-mixture homogeneous regions in the province (Fig. 5.14). Therefore, the parameters of G L O G were considered as dependent variables related to watershed size through regression. These parameters were then proposed for estimating floods for any return period at an ungauged watershed. Relationships between drainage area and the location parameter or scale parameter a of G L O G were created using five mathematical models, namely, linear, power, logarithmic, exponential, and polynomial. The final regression equation for each region was determined by choosing the model with the best fit (Table 5.14). Therefore, to apply the model in the 11 G L O G distribution regions, the flood estimation procedures at ungauged sites were: a) decide which of the 11 G L O G regions the ungauged watershed belongs to; b) decide on the regional L - Cs and L-Ck for the region using Table 5.12; c) compute the drainage area of the watershed in km2 from available maps or digital elevation models; d) find the constant shape parameter k and calculate parameters of a and £ of the G L O G distribution for the region using the regression models in Table 5.14; and 79 e) use the G L O G distribution (equation 5.4) as an estimator to compute floods, x, in m1 Is for a given probability or return period. An example of calculating a given return period flood is given in Appendix D. 5.4.2 Models for the Generalized Extreme Value Distribution The GEV distribution was shown to be the parent distribution for three of the non-mixture homogeneous regions in the province (section 5.3.1). Details on the GEV distribution were provided in section 4.4. The parameters of G E V can be computed using equations 5.5, 5.6, and 5.7. According to Hosking et al. (1985a), the following approximation has an accuracy of better than 9 x 10"4 (for - 05 < r 3 < 0.5) because no explicit solution for the shape parameter k is available: The shape parameter A: is a function of L-skewness. As discussed earlier, all delineated regions in the province are homogeneous in L-skewness and L-kurtosis (section 5.2.5). Therefore, for a given non-mixture region where the G E V is the parent distribution, parameter k was represented by the regional average k weighted by the record length of all stations in the region. The other two parameters of the G E V were determined using L-moments: k « 7.8590c+2.9554c2 (5.5) where 2 log 2 c = 3+T3 log3 \k (5.6) a = 80 a\ •[l-r(l + A:)] (5.7) k where r(.) denotes the gamma function, and A, and Aj are as defined previously. 5.4.2.1 Estimation of Flood Quantiles at Gauged Sites Following the same procedures described in section 5.4.1, estimations of flood quantiles at gauged stations were achieved using a hierarchical approach. Using equations 5.6 and 5.7, the location parameter £ and the scale parameter a of the GEV distribution were estimated from at-site data, while the shape parameter k was chosen from Table 5.15 based on regional analysis. Then, using equation 5.8, 5.4.2.2 Estimation of Flood Quantiles at Ungauged Sites Again, the relationships between drainage area and location parameter ^ or scale parameter a were developed using five mathematical models, namely, linear, logarithmic, power, exponential, and polynomial. Final regression equations for each region were determined by choosing the model with the best fit (Table 5.15). To apply the model in the three G E V distribution regions, the flood estimation procedures were: a) decide which of the three GEV regions the ungauged watershed belongs to; b) decide on the regional L - CK for the region using Table 5.12; flood quantiles for any return period (j -\-F{x) •) were computed by: (5.8) 81 c) compute the drainage area of the watershed in km2 from available maps or digital elevation models; d) find the constant shape parameter k and calculate parameters of a and C, of the G E V distribution for the region using the regression models in Table 5.15; and e) use the G E V distribution (equation 5.8) as an estimator to compute floods, x, in m2 /s for a given probability or return period. An example of calculating a given return period flood is given in Appendix D. The regression equations were developed for ungauged sites in all 14 non-mixture regions. These equations were based on the drainage areas alone. Firstly, physiographic and climatic parameters other than drainage area were not available in the original data set. Secondly, many other physiographic parameters, such as main channel length and channel slope, are usually a function of drainage area (Gray, 1961). It might not be appropriate to include all these variables in the same regression model due to multicolinearity. More importantly, it was found that the drainage area alone is sufficient to explain the majority of the variability in this study (section 5.6). The coefficient of identification (I2) was used to measure the extent of correlation among the untransformed variables because the proposed regression models were based on log-log space. By checking the coefficient of identification of the regression models, it was found that the parameters of the parent distribution of each individual non-mixture region were significantly related to the drainage area (Tables 5.14 and 5.15). The average of the coefficient of identification I2 is 0.87 for estimating the scale parameter a and 0.86 for estimating the location parameter <£\ NMR5 had the smallest coefficient of identification (72=0.63). This may result from the relatively 82 small number of stations that were used for building the regression, because NMR5 has the least number of stations (n=9). The standard error of estimate (SEE) is a measure of the variation between regression estimates and the data points used in deriving the regression equations. In this thesis, the standard error of estimate was presented as a percentage value, relative to the mean of the observed values. Values of SEE for the proposed regression models were given in Table 5.14 and 5.15. 5.5 Regional Models for Mixture Regions As discussed in section 2.5, because most parametric methods assume that observations are from the same population, the non-parametric method was used for analyzing mixed processes in flood-data series for regions in which floods are from more than one population. When comparing the fit of the non-parametric and some of the parametric distributions to the observed data, the superiority of the non-parametric approach is clearly shown, especially for higher return periods (Fig. 5.17). For example, the 100-year flood quantile for station 08MG007 was less than 200 m31s using the non-parametric approach, however, it approximately ranged from 230 m3 Is to 300 w 3 Is using different parametric distributions. For each station in the five mixture regions, the C F A package of Environment Canada (Pilon and Harvey, 1994) was used in the non-parametric flood-frequency analysis as described in section 3.6. The floods for return periods of 2, 5, 10, 20, 50, 100, and 200 years were computed for each station. These return periods were selected because they are most commonly used in practice. Different flood quantiles were then regressed on the drainage area using the five mathematical models described earlier and the best fit regression model 83 was chosen (Table 5.16). The coefficient of identification (I2) and the standard error of estimate (SEE) in percentage were given in Table 5.16. Therefore, for the five mixture regions in the province, the flood estimation procedures were as follows: a) decide which of the mixture regions is appropriate for a given watershed; b) compute the drainage area of the watershed in km2 from existing maps or digital elevation models; and c) use the corresponding regression equation for the relevant region (Table 5.16) to calculate the floods with return periods of 2, 5, 10, 20, 50, 100, and 200 years. An example of calculating a given return period flood is given in Appendix D. The TCEV distribution was suggested for mixture regions in some studies (Fiorentino et al., 1985; Rossi et al., 1984). However, the regional L-moments (Table 5.17) indicated that the T C E V was not appropriate for this study because only certain combinations of L-skewness and L-kurtosis agree with the requirement of a T C E V distribution (Gabriele and Arnell, 1991). Fig. 5.18 shows that most of the regional L-skewness and L-kurtosis were out of the TCEV feasible space. The non-parametric approach is suitable for the mixture regions because it is among the very few approaches that accommodate mixture in flood-data series. Also, the non-parametric method was selected because of its practicality. Splitting a flood time series into sub-samples corresponding to the component distributions making up the mixture is not required when using the nonparametric method. Classification of flood events of a time series based on flood-generating mechanisms requires 84 climate and meteorological data that are often not available in practice. Notwithstanding these advantages, the nonparametric method has its own limitation. The nonparametric method may not be reliable in extrapolating to longer return-periods and, therefore, this method must be employed cautiously. 5.6 Residual Analysis for the Developed Regression Models In this study, only the drainage area was used for developing the regression models. It is unfortunate that other parameters, such as mean basin elevation, length of main channel, slope, and percentage of area covered by lakes and swamps, were not available. Many may argue that the drainage area is insufficient for the regression equations (Benson, 1962). NMR2-1 was chosen to illustrate this discussion because it had the longest average record length of 34.9 years. Suppose a regression model is correctly chosen. If the dependent variable could be perfectly explained by the selected independent variable(s) and i f the at-site record lengths were sufficiently long, all data plots would fall on the regression line. However, this is never the case. For instance, the residuals of the developed regression model of Fig 5.19 are relatively high and might be caused by parameters other than the drainage area that were not included in the model and/or by sampling variability. To investigate the causes of these residuals, a Monte Carlo simulation based on the GEV distribution was conducted. Figs. 5.20, 5.21, and 5.22 display the results of the simulations for the sample size versus the mean, L-coefficient of variation, and L-skewness, respectively. Sample size significantly affected the mean, L-coefficient of variation, and L-skewness. In other words, it affected the three parameters of the identified regional parent distributions. When sample size increased, the plots from the simulations converge towards the population values of the mean, L-coefficient of variation, and L-skewness of 3242.97 m31 s, 0.4919, and 0.3791, respectively. For sample sizes less than 40 years, which is generally the 85 case in British Columbia, there is about 40%, 45%, and 80% sampling variability around the mean, L-coefficient of variation, and L-skewness, respectively. Comparing the values of residuals in Fig. 5.19 with the variation of the mean, L -coefficient of variation, and L-skewness in Figs. 5.20, 5.21, and 5.22, the Monte Carlo simulation demonstrated that the residuals of the developed regression models could be explained mostly by sampling variability as opposed to signal caused by physiographic parameters other than drainage area. These results are consistent with the findings of Benson (1952) who demonstrated that even the estimate of mean annual floods could vary significantly from the true population value because of sampling variability. He estimated that 12 years of record were needed to define the mean annual flood within 25% with 95% confidence limits. Therefore, it is concluded that, in the absence of other physiographic and climatic variables, drainage area is adequate for developing the regression models in the province. 5.7 Extrapolation of the Developed Regression Models It is worth noting that most of the flood data used in this study are for medium to large catchments. Of the 476 stations, 75% are in drainage areas larger than 100 km2, and only 5% of them are in drainage areas smaller than 10 km2. Consequently, the question arises as to whether it is appropriate to extrapolate the proposed regression models to smaller catchments. The t -test was used to investigate the extrapolation capabilities of the proposed regression models. 5.7.1 Non-Mixture Regions NMR1-1 and NMR1-2 in the southern Rocky and Columbia Mountains, and NMR8-1 and NMR8-2 in the Interior Plateau were used for the analysis because 86 they have relatively larger numbers of small gauged watersheds and they are typical of the various climatic and physiographic regimes in the province of British Colombia. The location and scale parameters of the regional parent distribution (GLOG or GEV) were plotted against the size of the gauged watersheds. Parameters were split into two groups according to the size of drainage area. Regression functions were built for drainage areas smaller than 100 km1 and larger than 100 km1 (Fig. 5.23 to 5.26). The t -test showed that the difference between the two slopes of the regressions was not statistically significant at the 99% confidence level, or that there was no slope break on the regression, for NMR1-1 and NMR1-2 in the southern Rocky and Columbia Mountains (Table 5.18). The similarity in the slope of the regression lines may be the result of a similar flood-generating mechanism in both small and large watersheds. In the southern Rocky and Columbia Mountains, precipitation is mainly in the form of snow and snowmelt dominates the response of both small and large watersheds. This supports the conclusion that the developed regression models can be extrapolated to smaller watersheds in this area. However, the t -test comparing the slopes of the regression equations showed significant difference for NMR8-1 and NMR8-2 in the Interior Plateau (Table 5.18), indicating that the regression models should not be extrapolated to smaller catchments. The change of the regression slope in this area may be explained by the fact that summer convective storms affect the flood response of smaller watersheds to a greater degree than larger watersheds. As mentioned in section 4.1, the summer convective rainfall is important because of its high intensity, short duration, covering only small areas in space. Therefore, in small watersheds, floods may be caused by convective cells because of the relatively drier climate in the Interior Plateau, while in large watersheds floods are still caused by snowmelt. Thus, the 87 difference in the flood-generating mechanism imposes significantly different slopes of the proposed regression models for small and large catchments. 5.7.2 Mixture Regions Investigating the extrapolation capabilities of the regression models in mixture regions was also conducted using the t -test. Flood quantiles at the various gauged watersheds within MR3 were separated into two groups based on watershed size and regressed on drainage area (Fig. 5.27). MR3 was chosen because it had the largest number of gauged watersheds smaller than 100 km1. The two slopes of the regressions for areas under 100 km1 and above 100 km1 were not statistically different (Table 5.18). Therefore, it was concluded that the developed regression models can be extrapolated to smaller watersheds in the Coast Mountains. This conclusion is consistent with the climatic conditions. Frontal systems dominate flood generation over the entire area of Coast Mountains. Unlike the convective storms in the Interior Plateau, the long-duration and moderate-intensity frontal storms cover large areas of thousands of square kilometers. The influence of frontal storms on floods is similar for both large and small watersheds, resulting in the same slope for the two regression functions (Table 5.18 and Fig. 5.27). From the residuals on the log-log plots between the drainage area and the parameters of the regional distribution of the flood quantiles, it can be seen that small catchments are associated with larger residuals in general (Figs. 5.23 to 5.27). These residuals may be due to the selected regression model being inappropriate and/or the drainage area not being enough to explain the spatial variability of floods. However, it has been illustrated (section 5.6) that the watershed size is sufficient for developing the regional models in British Columbia. It is interesting to note as shown in Fig. 5.27A that larger catchments have generally much longer flood-data records than smaller watersheds. Most likely, the residuals in small 88 catchments could be due to insufficient streamflow records. Therefore, the estimates of flood quantiles should be more reliable using extrapolation from the developed regression models than using the models developed based on small watersheds alone if there are only few records available. In summary, the regression models developed in this study could comfortably be extrapolated to small watersheds where precipitation is mainly in the form of snow and where precipitation is produced by frontal systems. On the other hand, the developed regression models should not be extrapolated to smaller watersheds where the flood-generating mechanisms over small and large watersheds are expected to be different. This is particularly the case in the drier belt of the province where rainfall in the form of high-intensity, short-duration convective cells is thought to dominate the flood response of small catchments. For these areas, other methods should be investigated for estimating floods for small-ungauged watersheds. Where a sufficient database of gauged watersheds is available, separate regression models need to be developed for small and large watersheds. The definition of small versus large watersheds, defined by the location of the break of slope in the regression model, needs to be investigated in future studies. 5.8 Validation of the Developed Regression Models A split-sampling experiment was applied to evaluate the reliability of the proposed regression models for the delineated homogeneous regions. In each region, the station with the longest record length was selected for computing the observed floods. The non-parametric method was used for computing the observed quantiles in order to avoid any commitment to a particular parent distribution for frequency analysis. Assuming that no observations were available at the selected stations, the predicted floods were computed using the regional regression models for the homogeneous regions. 89 Quantiles of 5, 10, 20, 50, and 100 years computed from the observed data and from the proposed models were compared (Table 5.19 and Fig. 5.28). Analysis of variance showed that floods from the proposed models were statistically the same as those from the observed network at the 95% confidence level for all 19 regions. The maximum relative error across all return periods ranged from -40.4% to 25.9%. However, the relative error averaged across the 19 regions did not exceed 14% for all return periods. Although the points of the plots in Fig. 5.28 do not fall exactly on the 45 degree line, the floods of 5, 10, 20, 50, and 100 year return periods from the proposed models are in good agreement with those from the observations. The relative errors for MR2-2, ranging from -6.8% to -40.4%, were much greater than for other regions. This may be due to unreliable floods from the observed network because the selected station (08EE008) for MR2-2 had the shortest record length (34 years) compared to other regions. Table 5.19 shows that the models may either overestimate or underestimate the observed floods. This may be explained by sampling variability. It is important to note the general trend of increasing relative errors with increasing return periods (Table 5.19). Computed floods for short return periods are more reliable than those of longer return periods when record length is short. Monte Carlo simulations in section 5.6 support this conclusion (i.e., the reliability of the mean, L-coefficient of variation, and L-skewness of flood observations depended on sample size). Although the results of the split-sampling experiment showed the proposed models were reliable, it is worth noting that the application of these models at ungauged sites should not be conducted in isolation. Practitioners should always check if there are physical reasons for which the particular ungauged watershed may be different from neighbouring gauged watersheds. In such a case, the proposed models alone may not be dependable. 90 5.9 Comparison of the Proposed Hierarchical Regional Models and Single-Site Models Following the procedures in section 4.7, a Monte Carlo simulation was conducted to assess the accuracy of the proposed hierarchical regional models compared to the single-site frequency analysis commonly used in practice. NMR2-1 was chosen for the simulation because this region had the longest average record length (34.9 years). The G E V distribution was fitted using the regional L-skewness of 0.096 and the lower order L-moments A, of 460 m 3 / s and A^ of 54.2 m3 I s. The values of A, and A^ were assumed arbitrarily, but they were typical weighted averages for stations in the region. Then, the population values of flood estimates were computed using the fitted G E V distribution for 2, 5, 10, 20, 50, 100, 200, 500, and 1000 year return periods, respectively. These were assumed to be the true population values of the floods. The Monte Carlo simulation was conducted by synthetically generating a large number of samples (10,000) of size «,=10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 using the prescribed GEV parent distribution. Each sample was used to fit the GEV distribution by L-moments. In the single-site model, the GEV was fit using the actual statistics computed from each sample. However, in the proposed hierarchical regional model, the mean and L-Cv were estimated from the generated sample while the L - CS was held constant at the regional value of 0.096. The regional L-skewness varied widely among the 19 delineated regions. To assess the effect of the magnitude of the regional L-skewness on Monte Carlo simulation results, another two regions in southern British Columbia (NMR8-2 and NMR8-3) were considered. NMR8-2 is governed by the GEV distribution while NMR8-3 is described by the G L O G distribution. Following the same procedures as shown 91 above for NMR2-1, the simulations were conducted, using \ of 46.1 m31 s, of 9.78 m31 s and the regional L-skewness of 0.16 for NMR8-2, and using A, of 316 m3I s, of 32.3 m31 s and the regional L-skewness of 0.256 for NMR8-3. The simulation results for different return periods are summarized in Tables 5.20a, 5.20b, and 5.20c, with a sample size of 24 (approximately the average record length of observations in British Columbia). The RMSE values show that both regional and single-site models are in good agreement for return periods less than 20 years. For example, the R M S E values are 43.87 m31 s and 49.50 m3I s for 20-year flood quantile using the regional and single-site methods, respectively. However, these values become 58.20 m31s and 96.84 m3 Is for 100-year flood quantile using the regional and single-site methods, respectively (Table 5.20a). Therefore, the regional model substantially outperformed the single-site model for longer return periods. The same results are also displayed in Figs. 5.29a, 5.29b, and 5.29c, which show the variation of the R M S E with the return period for both single-site and regional models. As shown in Fig. 5.29, the flood estimate for the 100-year return period using the regional model is about 1.65 times more accurate than using the single-site model. Figs. 5.30a, 5.30b, and 5.30c display the variation of the R M S E with the sample size for the 100-year return-period flood. These figures clearly show the superiority of the proposed hierarchical regional flood-frequency approach. About three times more data are required for the single-site model to be as accurate as the proposed hierarchical regional model. The overall results of the Monte Carlo simulations substantiate that the method of L-moments provides virtually unbiased estimators of the flood quantiles regardless 92 of sample size. This was true for both models and confirmed the findings of Hosking et al. (1985a) and Alila (1994). 93 Chapter 6 SUMMARY AND CONCLUSIONS 6.1 Summary of the Study In British Columbia, flood estimates are usually based on single-site frequency analysis and graphical peak-flow regionalization procedures. As discussed, there are several drawbacks to the current practice, especially for short-term record stations and ungauged locations. Therefore, improved regional models were proposed in this study. L-moments, which are more efficient, less biased and less sensitive to outliers in flood-data series than the conventional moments, were used to delineate homogeneous regions, identify regional parent distributions, estimate the distribution's parameters, and develop regional models to provide more reliable flood estimates. Based on the similarities in the climatic, physiographic and hydrologic regimes, and the analysis of mixed processes and sampling variability, the province of British Columbia was delineated into regions that were homogeneous in L-skewness and L-kurtosis. The homogeneity of the delineated regions was validated by the use of L-moment based statistical tests. The province was divided into 14 non-mixture homogeneous regions and five mixture homogeneous regions. Each region had a constant L-skewness and L-kurtosis that defined a specific regional parent distribution. 94 By applying theoretical L-moment diagrams and L-moment based regional goodness-of-fit tests, it was concluded that the generalized logistic (GLOG) and generalized extreme-value (GEV) distributions were valid regional parent distributions in 11 and three non-mixture homogeneous regions, respectively. As mentioned in chapter 1, the Gumbel distribution is usually used for flood-frequency analysis in British Columbia. The Gumbel is a particular case of the G E V family of distributions. However, sensitivity analysis of L-skewness revealed that the errors of quantile estimates using the Gumbel distribution with a fixed L-skewness (0.1699) were substantial compared to those using the G E V distribution with different constants of regional L-skewness. Mixed processes have long been recognized in flood-data series in the province. Mixtures were analyzed by examining the shape of the cumulative distribution function and checking the histograms of flood peaks by month of occurrence. The existence of mixed processes was further confirmed by testing the significance of mixed processes and the effect of sampling variability. It was concluded that five homogeneous mixture regions, where floods are generated by mixed processes, exist in the province. Of the five mixture regions, four are located along the northwest-orientated Coast Mountains, while the other is located in the Great Plains in the northeast of the province. New regional flood-frequency models were developed for the 19 identified homogeneous regions. In a non-mixture region, the selected parent distribution (either G L O G or GEV) can be applied. At gauged stations, flood quantiles can be computed using a hierarchical approach. The first two L-moments were determined from at-site observations while the constant L-skewness is from the region. At ungauged sites, the parameters of the G L O G or GEV distribution were regressed on watershed size. Given the drainage area, the estimation of quantiles can be obtained by the procedures described in section 5.4. 95 For mixture regions, a non-parametric method was combined with the regression method to develop regional models based on the drainage area. The estimation of quantiles can be obtained by the procedures described in section 5.5. Because most of the watersheds in this study have medium to large drainage areas, extrapolation to small watersheds was investigated based on whether there was a slope break in the regression of flood quantile on watershed size. Results showed that the extrapolation can be used in areas where the precipitation is mainly in the form of snow or where frontal systems dominate the floods. However, in areas where convective cells play an important role in the flood response of small watersheds, the proposed regression models can not be extrapolated. Analysis of the regression residuals using Monte Carlo simulation revealed that the drainage area on its own is sufficient for developing regression models in both non-mixture regions and mixture regions in the province. Most of the residuals associated with developed regression models may be explained by sampling variability. To validate the proposed regression models in the delineated homogeneous regions, split-sampling experiments were used. Results showed that flood quantiles from the proposed models were in good agreement with those from observations. Analysis of variance showed that the two sets of flood quantiles were not statistically different at the 95% confidence level for all 19 regions. Finally, another Monte Carlo simulation was conducted to assess the accuracy of the proposed hierarchical regional models compared to single-site frequency analysis. Results showed that the hierarchical models were more reliable than their single-site counterparts. 9 6 6.2 Major Conclusions The following conclusions may be drawn from this study: 1. The province was delineated into regions that are homogeneous in flood characteristics using an innovative procedure that combines the study of mixed processes in flood-data series, sampling variability, and climatic and physiographic regimes. The homogeneity of the delineated regions was verified by the use of L-moment based statistical tests. The newly developed procedure is of particular significance for application in future regional flood, low-flow, and rainfall frequency studies in and outside British Columbia. 2. The province was delineated into two categories of hydrologically-homogeneous regions: non-mixture regions where floods can safely be assumed to be generated by one single population, and mixture regions where floods are generated by combination of more than one population. 3. Within the non-mixture regions, annual floods are best described by either the generalized logistic (GLOG) or generalized extreme-value (GEV) parent distributions. These distributions were identified using regional goodness-of-fit tests. These tests are superior to classical goodness-of-fit tests because they use regional as opposed to single-site data (Cong et al., 1993). The recommended distributions differ from the current distributions often used for flood-frequency analysis in British Columbia. The new findings are important to professional practice in flood-frequency analysis in the province as mis-identification of regional parent distributions may result in significant errors in flood quantiles. 4. Within the mixture regions, the nonparametric frequency distribution gave a much superior fit to the annual flood series than any of the commonly used 97 single population distributions. The non-parametric method is a more appropriate distribution for mixture regions. 5. Innovative regional flood-frequency models were developed in different regions in British Columbia. In non-mixture regions, hierarchical frequency models and regression equations were developed for estimating floods at gauged and ungauged watersheds, respectively. In mixture regions, a non-parametric approach was combined with the regression method for the development of regional equations for estimating floods at gauged or ungauged watersheds. 6. Monte Carlo simulations and split-sampling experiments showed that the proposed regional hierarchical frequency models and regression equations outperformed the current methods used in the province. 6.3 Contributions to Hydrological Sciences As a result of this study, the following original contributions were made: 1. Regional hydrologic homogeneity should account for the fact that floods in some areas may be generated by a single population while in some other areas by more than one population. Delineation of homogeneous regions in regional flood-frequency analysis can be achieved through the simultaneous investigation of flood-generation mechanisms, climatic and physiographic characteristics, and statistical testing of homogeneity. 2. There appears to be a relationship between the shape of flood-frequency curves (CDF) and the physiographic and climatic zones of the province. The spatial trends of the CDF shape are a testimony that the dog-leg is the result of a 98 mixture in the flood data as opposed to sampling variability and can not be ignored in practice. 3. The L-skewness and L-kurtosis of annual flood series may be considered approximately constant over hydrologic regions that are homogeneous in climatic and physiographic characteristics. The regional L-skewness and L-kurtosis values vary significantly from one hydrologic region to another across the province which reflects the contrasting climate and physiographic conditions of these regions. Consequently, a standard flood-frequency distribution with constant L-skewness and L-kurtosis, such as the EV1 distribution, applicable to all the regions cannot be justified. 4. It was demonstrated that the GEV and G L O G distributions gave the best fit to flood data in the 14 non-mixture regions of the province. Both are emerging as the most suitable frequency distributions for flood data in many parts of the world (Vogel and Wilson, 1996). 5. The regional L-moment ratios of observed flood data in the five-mixture regions of the province fell outside the feasible TCEV L-skewness versus L-kurtosis space. This demonstrated that the TCEV might not be always appropriate for fitting flood data generated by mixed processes. This contradicts the hydrologic literature which advocates the TCEV for fitting mixtures in flood data (Rossi et al, 1984; Fiorentino et al, 1987; Cannarozzo et al, 1995). 6. The L-coefficient of variation may not be constant over any of the delineated 19 regions. The L-coefficient of variation appears to vary not only with climate and physiography but also with the size of the watershed. As a result, the index-flood approach to regional flood-frequency analysis may not be justified anywhere in the province. This is inconsistent with recent research published by 99 Gupta et al. (1994) and Gupta and Dawdy (1995) who contend that the assumptions of the index-flood approach is justified in snow dominated regions. 7. The hierarchical approach to regional flood-frequency analysis where different distribution parameters are estimated from different but nested sub-sets of data is much more reliable than single-site flood-frequency analysis. Approximately three times more data were required for the single-site flood-frequency method to be as accurate as the regional hierarchical approach. 8. With the proper delineation of homogeneous regions, the drainage area alone may be sufficient for explaining the spatial variability of floods. The unexplained variance of the developed regression models was, to a large extent, accounted for by sampling variability. This is in agreement with the simple and multi-scaling theories of Gupta et al. (1994) and Gupta and Dawdy (1995) who advocate the delineation of homogeneous regions on the basis of scale being the sole parameter required to identify the flood-frequency distribution of any watershed within a homogeneous region. 9. When long-term streamflow data are available only for medium to large watersheds, such data should be used cautiously for inferring small catchment flood estimates. Regional regression models based on data for medium to large watersheds can be extrapolated only to regions where precipitation is either mainly snow or frontal rainfall systems covering wide areas (humid and sub-humid conditions). In these cases, maximum annual discharges from small and large watersheds result from the same flood-generating mechanism. 10. Regional regression models based on data from medium to large watersheds may not be extrapolated to small catchments in arid or semi-arid regions. In these regions, floods from small catchments are generated by convective rainfall 100 cells while those from large watersheds may be generated mainly by snowmelt. It is hypothesized that this change in flood generating mechanism causes a break in the slope of the regression models. 6.4 Recommendations for Future Studies 1. It was shown that the two-component extreme-value distribution could not parametrically describe the flood-data series in mixture regions in the province. It would be valuable to investigate and develop other combined parametric distributions able to describe floods in British Columbia. To do so it is suggested that flood-data series be separated based on flood-generating mechanisms. 2. Even though it was demonstrated that drainage area might be sufficient for building the regression models in the study, other variables dealing with watershed characteristics, climatic conditions, and forest impacts should be worth investigating to improve regression equations. 3. In this study, small watersheds have been arbitrarily defined as having a drainage area smaller than 100 km1. A more objective definition of small, medium, and large watersheds may be achieved by investigating regional flood-characteristics in areas with a larger database of small gauged watersheds than that in the province of British Columbia. 4. In this study, the province was delineated into 19 hydrologic regions homogeneous in the L-skewness and L-kurtosis. A l l of these regions were found to be excessively heterogeneous in the L-coefficient of variation. 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E., 1986, Regional flood-frequency analysis, I: Homogeneity statistics, Hydrological Sciences Journal, 31(3), 321-333. 115 Appendix A Tables 116 Table 2.1 Independent Variables Used in 10 Flood-Frequency Analyses, Reviewed by Riggs (1973) Variable 1 2 3 4 5 6 7 8 9 10 Drainage Area X X X X x x x x X X Main-Channel Slope X X X x x Percentage of Basin Covered by Lakes and Swamps X x x x X Mean Annual Precipitation X x X X Mean Annual Runoff x T-year 24-hour Rainfall X X Average Degree below Freezing in January X Orographic Factor X Elevation X X Number of Thunderstorm Days X x Main-Channel Length x Ratio of Runoff to Precipitation X Mean Annual Snowfall x Average Number of Wet Days per Year x Shape Factor x Geographical Factor x X Table 3.1 L-Moment Relationships for Some Common Distributions (Hosking, 1990) Distribution h Uniform 0 0 Normal 0 0 Exponential 0.3333 1.1667 Gumbel 0.1699 0.1504 G E V -k (1+5A;2)/6 k : Parameter of G E V 117 Table 5.1 Test for Independence of Annual Maximum Daily Flows (08MB005-CHILCOTIN RIVER BELOW BIG CREEK) - - - SPEARMAN TEST FOR INDEPENDENCE 08MB005 CHILCOTIN RIVER BELOW BIG CREEK ANNUAL MAXIMUM DAILY FLOW SERIES 1971 TO 1994 DRAINAGE AREA = 19300.00 SPEARMAN RANK ORDER SERIAL CORRELATION COEFF = -.090 D.F.= 21 CORRESPONDS TO STUDENTS T = -.414 CRITICAL T VALUE AT 5% LEVEL = 1.721 NOT SIGNIFICANT - 1% - = 2.518 NOT SIGNIFICANT I n t e r p r e t a t i o n : The n u l l h y p o t h e s i s i s Chat the c o r r e l a t i o n i s zer o . At the 5% l e v e l of s i g n i f i c a n c e , the c o r r e l a t i o n i s not s i g n i f i c a n t l y d i f f e r e n t from z e r o . That i s , the d a t a do not d i s p l a y s i g n i f i c a n t s e r i a l dependence. Table 5.2 Test for Trend of Annual Maximum Daily Flows (08MB005-CHILCOTIN RIVER BELOW BIG CREEK) SPEARMAN TEST FOR TREND 08MB005 CHILCOTIN RIVER BELOW BIG CREEK ANNUAL MAXIMUM DAILY FLOW SERIES 1971 TO 1994 DRAINAGE AREA = 19300.00 SPEARMAN RANK ORDER CORRELATION COEFF = -.037 D.F.= 22 CORRESPONDS TO STUDENTS T = -.171 CRITICAL T VALUE AT 5% LEVEL =-2.074 NOT SIGNIFICANT 1% - =-2.819 NOT SIGNIFICANT I n t e r p r e t a t i o n : The n u l l h y p o t h e s i s i s t h a t the s e r i a l ( l a g - o n e ) c o r r e l a t i o n i s z e r o . At the 5% l e v e l o f s i g n i f i c a n c e , the c o r r e l a t i o n i s not s i g n i f i c a n t l y d i f f e r e n t from z e r o . That i s , the d a t a do not d i s p l a y s i g n i f i c a n t t r e n d . 118 Table 5.3 Test for Randomness of Annual Maximum Daily Flows (08MB005-CHILCOTIN RIVER BELOW BIG CREEK) RUN TEST FOR GENERAL RANDOMNESS 08MB005 CHILCOTIN RIVER BELOW BIG CREEK ANNUAL MAXIMUM DAILY FLOW SERIES 1971 TO 1994 DRAINAGE AREA = 19300.00 THE NUMBER OF RUNS ABOVE AND BELOW THE MEDIAN (RUNAB) = 14 THE NUMBER OF OBSERVATIONS ABOVE THE MEDIAN(Nl) = 12 THE NUMBER OF OBSERVATIONS BELOW THE MEDIAN(N2) = 12 Range at 5% l e v e l of s i g n i f i c a n c e : 8. t o 18. NOT SIGNIFICANT I n t e r p r e t a t i o n : The n u l l h y p o t h e s i s i s t h a t the data a r e random. At the 5% l e v e l o f s i g n i f i c a n c e , the n u l l h y p o t h e s i s cannot be r e j e c t e d . That i s , the sample i s s i g n i f i c a n t l y random. Table 5.4 Test for Homogeneity of Annual Maximum Daily Flows (08MB005-CHILCOTIN RIVER BELOW BIG CREEK) MANN-WHITNEY SPLIT SAMPLE TEST FOR HOMOGENEITY 08MB005 CHILCOTIN RIVER BELOW BIG CREEK ANNUAL MAXIMUM FLOW SERIES 1971 TO 1994 DRAINAGE AREA= 19300.00 SPLIT BY TIME SPAN, SUBSAMPLE 1 SAMPLE SIZE= 11 SUBSAMPLE 2 SAMPLE SIZE= 13 MANN-WHITNEY U = 69.0 CRITICAL U VALUE AT 5% SIGNIFICANT LEVEL = 42.0 NOT SIGNIFICANT 1% - - = 31.0 NOT SIGNIFICANT I n t e r p r e t a t i o n : The n u l l h y p o t h e s i s i s t h a t t h e r e i s no l o c a t i o n d i f f e r e n c e between the two samples. At the 5% l e v e l o f s i g n i f i c a n c e , t h e r e i s no s i g n i f i c a n t l o c a t i o n d i f f e r e n c e between the two samples. That i s , they appear t o be from the same p o p u l a t i o n . 119 Table 5.5 Number of Stations in Each Region as Defined by CDF Shape Straight Line CDF Dog-Leg CDF Region Number of Stations % Number of Stations % NMR 1 65 94.2 4 5.8 N M R 2 67 87.0 10 13.0 NMR 3 21 91.3 2 8.7 MR 1 3 30.0 7 70.0 MR 2 2 09.5 21 90.5 MR 3 22 08.0 252 92.0 Total 180 37.8 296 62.2 Table 5.6 Test for the Significance of Mixtures Previous Region Mean STDV Number of Stations Test Re-Classified Name of Regions MR 1 0.926 0.079 10 Fail NMR 4 Creston MR 2 0.800 0.122 23 Pass MR 1 Rocky Mountains MR 3 0.691 0.133 20 Pass MR 2 Northeastern Plains MR 4 0.791 0.117 38 Pass MR 3 Northern Interior MR 5 0.731 0.195 89 Pass MR 4 Arrow Lake MR 6 0.771 0.150 28 Pass MR 5 Coast Mountains (1) MR 7 0.727 0.197 52 Pass MR 6 Coast Mountains (2) MR 8 0.845 0.103 38 Pass MR 7 Georgia Depression MR 9 0.907 0.096 9 Fail NMR 5 Exposed Coast Mean: Regional Mean ol STDV: Standard Deviatk F Rm Value an 120 Table 5.7 F-Test for Variance between Groups based on Record Length in Different Regions MR 1 Rocky Mountains Group 1 Group 2 Group 1 Group 3 Group 2 Group 3 Mean 0.238 0.362 0.238 0.077 0.362 0.077 Variance 0.024 0.015 0.024 0.008 0.015 0.008 Observations 8 8 8 7 8 7 df 7 7 7 6 7 6 F 1.5 3 3.08 1.95 p Critical 3.79 4.21 4.21 MR 2 Northeastern Plains Group 1 Group 2 Group 1 Group 3 Group 2 Group 3 Mean 0.341 0.354 0.341 0.340 0.354 0.340 Variance 0.025 0.025 0.025 0.008 0.025 0.008 Observations 7 7 7 6 7 6 df 6 6 6 5 6 5 F 1.02 3.26 3.22 F Critical 4.28 4.95 4.95 MR 3 Nort iern Interior Group 1 Group 2 Group 1 Group 3 Group 2 Group 3 Mean 0.337 0.266 0.337 0.169 0.266 0.169 Variance 0.021 0.020 0.021 0.006 0.020 0.006 Observations 14 12 14 12 12 12 df 13 11 13 11 11 11 F 1.0 6 3.29 3.11 F Critical 2.78 2.78 2.82 MR 4 Arrow Lake Group 1 Group 2 Group 1 Group 3 Group 2 Group 3 Mean 0.408 0.347 0.408 0.221 0.347 0.221 Variance 0.053 0.049 0.053 0.026 0.049 0.026 Observations 28 32 28 29 32 29 df 27 31 27 28 31 28 F 1.0 8 2.03 1.8 8 F Critical 2.87 1.89 1.87 121 Table 5.7 (cont.) F-Test for Variances between Groups based on Record Length in Different Regions MR 5 Coast Mountains (1) Group 1 Group 2 Group 1 Group 3 Group 2 Group 3 Mean 0.286 0.271 0.286 0.284 0.271 0.284 Variance 0.043 0.033 0.043 0.020 0.033 0.020 Observations 10 10 10 8 10 8 df 9 9 9 7 9 7 F 1.29 1.9 3 1.63 F * Critical 3.18 3.68 3.68 MR 6 Coast Mountains (2) Group 1 Group 2 Group 1 Group 3 Group 2 Group 3 Mean 0.262 0.268 0.262 0.202 0.268 0.202 Variance 0.021 0.019 0.021 0.020 0.019 0.020 Observations 18 18 18 16 18 16 df 17 17 17 15 17 15 F 1.0 B 1.0 3 0.9 6 p Critical 2.27 2.36 2.36 MR 7 Georgia Depression Group 1 Group 2 Group 1 Group 3 Group 2 Group 3 Mean 0.321 0.348 0.321 0.326 0.348 0.326 Variance 0.034 0.036 0.034 0.040 0.036 0.040 Observations 12 12 12 14 12 14 df 11 11 11 13 11 13 F 1.43 2.31 1.5 B F 1 Critical 2.82 2.64 2.64 122 Table 5.8 A N O V A for Mixture Regions Region Source of Variation S S df MS F F Critical MR 1 Between Groups 0.194 2 0.097 5.819 3.806 Within Groups 0.217 20 0.017 Total 0.411 22 MR 2 Between Groups 0.001 2 0.000 0.019 3.682 Within Groups 0.321 17 0.021 Total 0.322 19 MR 3 Between Groups 0.165 2 0.082 4.994 3.295 Within Groups 0.528 35 0.017 Total 0.693 37 MR 4 Between Groups 0.521 2 0.261 6.122 3.103 Within Groups 3.660 86 0.043 Total 4.181 88 MR 5 Between Groups 0.003 2 0.002 0.054 3.120 Within Groups 2.403 25 0.032 Total 2.406 27 MR 6 Between Groups 0.024 2 0.012 0.617 3.328 Within Groups 0.575 49 0.020 Total 0.599 51 MR 7 Between Groups 0.143 2 0.072 0.463 3.295 Within Groups 5.403 35 0.154 Total 5.546 37 123 Table 5.9 Test for the Effect of Sampling Variability Region Group 1 Group 2 Group 3 A N O V A Test New Region Mean STDV Mean STDV Mean STDV F F Critical MR 1 0.238 0.154 0.362 0.123 0.077 0.088 5.819 3.806 fail NMR6 MR 2 0.341 0.158 0.354 0.157 0.340 0.088 0.019 3.682 pass MR1 MR 3 0.337 0.145 0.266 0.141 0.169 0.080 4.994 3.295 fail NMR7 MR 4 0.408 0.230 0.347 0.221 0.221 0.161 6.122 3.103 fail NMR8 MR 5 0.286 0.200 0.271 0.178 0.284 0.143 0.054 3.120 pass MR2 MR 6 0.458 0.226 0.299 0.230 0.267 0.157 0.617 3.328 pass MR3 MR 7 0.262 0.143 0.268 0.128 0.202 0.145 0.463 3.295 pass MR4 Mean: Mean of the Error Statistic Er STDV: Standard Deviation Table 5.10 Test of Homogeneity for Delineated Regions Region Name Number of Record (H-value) Stations Length (yr) L -Cs /L -Ck NMR1 Castlegar 69 30.8 2.65 NMR2 Interior Plateau 77 28.5 2.25 NMR3 North-Western Plateau 23 19.2 1.91 NMR4 Creston 10 28.0 1.61 NMR5 Exposed Coast 9 31.0 0.82 NMR6 Rocky Mountains 23 28.4 1.40 NMR7 Northern Interior 38 21.4 2.48 NMR8 Arrow Lake 89 24.2 2.68 MR1 North-Eastern Plains 20 21.1 0.00 MR2 Coast Mountains(l) 52 26.6 2.92 MR3 Coast Mountains(2) 28 28.0 0.93 MR4 Georgia Depression 38 23.2 1.90 124 Table 5.11 Summary of Homogeneous Regions in BC Name Region Number of Stations Record Length (yr) (H-value) L-Cv L-Cs/L-Ck Castlegar NMR1-1 38 31.0 9.38 -0.54 NMR1-2 31 30.7 7.95 1.79 Interior Plateau NMR2-1 23 34.9 12.55 1.04 NMR2-2 25 23.7 9.95 0.15 NMR2-3 29 26.8 10.45 1.45 North-Western Plateau NMR3 23 19.2 3.79 1.91 Creston NMR4 10 28.0 0.95 1.61 Exposed Coast NMR5 9 31.0 1.68 0.82 Rocky Mountains NMR6 23 28.4 8.48 1.40 Northern Interior NMR7-1 22 21.9 8.51 1.69 NMR7-2 16 20.8 1.39 1.25 Arrow Lake NMR8-1 33 20.8 13.22 1.86 NMR8-2 31 27.7 10.58 1.50 NMR8-3 25 24.2 5.91 -0.53 North-Eastern Plains MR1 20 21.1 5.51 0.00 Coast Mountains MR2-1 23 29.4 6.55 1.91 MR2-2 29 24.3 1.91 1.23 MR2-3 28 28.0 6.93 0.93 Georgia Depression MR3 38 23.2 4.53 1.90 125 Table 5.12 Regional Weighted Average L-Skewness and L-Kurtosis for Non-Mixture Homogeneous Regions Name Region Number of Observations Regional Weighted Average L-Cs Regional Weighted Average L-Ck Castlegar NMR1-1 1029 0.122 0.166 NMR1-2 782 0.052 0.157 Interior Plateau NMR2-1 715 0.096 0.144 NMR2-2 557 0.127 0.186 NMR2-3 749 0.084 0.170 North-western Plateau NMR3 685 0.121 0.199 Creston NMR4 279 0.075 0.157 Exposed Coast NMR5 263 0.151 0.190 Rocky Mountains NMR6 539 0.173 0.176 Northern Interior NMR7-1 NMR7-2 483 0.164 0.198 332 0.140 0.189 Arrow Lake NMR8-1 688 0.216 0.213 NMR8-2 744 0.160 0.163 NMR8-3 555 0.256 0.206 126 Table 5.13 Best-Fit Distribution for Non-Mixture Homogeneous Regions Name Region Average Record Length (yr) Number of Stations Best-Fit Distribution Z-value Castlegar NMR1-1 31.0 38 G L O G 0.77 NMR1-2 30.7 31 G L O G 0.71 Interior Plateau NMR2-1 34.9 23 G E V -1.40 NMR2-2 23.7 25 G L O G -0.87 NMR2-3 26.8 29 G L O G -0.03 North-Western Plateau NMR3 19.2 23 G E V -1.34 Creston NMR4 28.0 10 G L O G 0.74 Exposed Coast NMR5 31.0 9 G L O G -0.56 Rocky Mountains NMR6 28.4 23 G L O G 0.55 Northern Interior NMR7-1 21.9 22 G L O G -1.18 NMR7-2 20.8 16 G L O G -0.65 Arrow Lake NMR8-1 20.8 33 G L O G -0.53 NMR8-2 27.7 31 G E V -1.25 NMR8-3 24.2 25 G L O G 0.11 127 Table 5.14 Regional Models for G L O G Distribution Region Regional Models for Parameters of G L O G Constant k a = a A" C = cAJ a b I2 SEE(%) c d I2 SEE(%) NMR1-1 -0.1263 0.0221 0.9304 0.92 15.43 0.1116 0.9655 0.93 20.07 NMR1-2 -0.0554 0.0402 0.8704 0.91 18.32 0.1707 0.9456 0.93 21.26 NMR2-2 -0.1439 0.1195 0.6484 0.70 30.56 0.2191 0.8076 0.70 36.57 NMR2-3 -0.0972 0.0358 0.8527 0.89 11.39 0.1026 0.9751 0.90 17.62 NMR4 -0.0794 0.0194 1.0661 0.91 25.60 0.0473 1.2213 0.87 17.98 NMR5 -0.1529 0.5047 0.8676 0.70 22.74 1.4899 0.9248 0.63 16.55 NMR6 -0.1773 0.0039 1.2016 0.93 9.87 0.0103 1.3513 0.91 13.25 NMR7-1 -0.1765 0.0067 1.0061 0.96 16.31 0.0158 1.1715 0.98 18.40 NMR7-2 -0.1543 0.0150 0.9759 0.89 14.91 0.0370 1.0992 0.90 19.20 NMR8-1 -0.2468 0.0045 1.0381 0.89 10.82 0.0061 1.2129 0.90 10.24 NMR8-3 -0.2691 0.0070 1.2078 0.91 24.42 0.0288 1.3034 0.91 30.43 A : Drainage Area in km 2 I2: Coefficient of Identification SEE: Standard Error of Estimate Table 5.15 Regional Models for G E V Distribution Region Regional Models for Parameters of G L O G Constant k a = a A" £ = cAd a b I2 SEE(%) c d I2 SEE(%) NMR2-1 0.1549 0.0688 0.8802 0.94 18.12 0.1525 0.9741 0.91 26.07 NMR3 0.0666 0.0677 0.8575 0.75 32.80 0.2538 0.8532 0.78 27.75 NMR8-2 -0.0030 0.0153 1.0210 0.88 13.59 0.0155 1.1373 0.81 19.14 A : Drainage Area in km 2 I2: Coefficient of Identification SEE: Standard Error of Estimate 128 Table 5.16 Regional Models for Mixture Regions Model for MR1 QT = cA d Quantile a a Go Go G o Goo G200 Coefficient c 0.1140 0.2739 0.5729 0.8843 0.9691 1.0164 1.0501 d 0.9356 0.8857 0.8242 0.8057 0.804 0.8025 0.8019 I2 0.93 0.95 0.91 0.88 0.87 0.88 0.88 SEE(%) 8.19 17.13 9.77 20.48 29.10 15.57 22.04 Model for MR2-1 QT = cAd Quantile a G Go Go Go Goo Goo Coefficient c 0.2478 0.4541 0.6243 0.9894 1.1082 1.1168 1.1376 d 0.9399 0.8869 0.8631 0.8120 0.8086 0.8154 0.8172 I2 0.89 0.87 0.80 0.82 0.85 0.86 0.86 SEE(%) 16.35 27.12 26.80 19.54 20.43 14.14 23.60 Model for MR2-2 QT = cAd Quantile G a Go Go G o Goo Goo Coefficient c 0.1298 0.1922 0.2543 0.3097 0.3337 0.3492 0.3625 d 1.0097 0.9991 0.9873 0.9815 0.9947 0.9934 0.9920 I2 0.88 0.85 0.85 0.81 0.81 0.81 0.83 SEE(%) 20.18 30.38 26.45 18.55 24.26 33.63 31.15 Model for MR2-3 QT = cAd Quantile a G Go Go G o Goo Goo Coefficient c 0.7431 1.2568 1.7951 2.2265 2.4425 2.4851 2.5703 d 0.8558 0.8251 0.8034 0.7978 0.8013 0.8055 0.8047 I2 0.91 0.90 0.88 0.88 0.87 0.85 0.89 SEE(%) 26.81 17.20 24.39 31.79 28.74 16.23 22.54 Model for MR3 QT = cAd Quantile G G Go Go G o Goo G200 Coefficient c 0.7627 1.2389 1.4463 1.7070 1.9732 2.0506 2.1088 d 0.9263 0.9135 0.9224 0.9120 0.9023 0.9043 0.9065 I2 0.90 0.91 0.90 0.87 0.88 0.84 0.86 SEE(%) 14.15 15.62 25.25 19.06 11.18 23.04 27.38 A : Drainage Area in km 2 I2: Coefficient of Identification SEE: Standard Error of Estimate 129 Table 5.17 Weighted Regional Average of L-Moments for Mixture Regions Name Region Number of Stations Record Length (yr) Weighted Regional Average of L-moment L-Cs L-Ck North-eastern Plains MR1 20 21.1 0.318 0.215 Coast Mountains MR2-1 23 29.4 0.135 0.230 MR2-2 29 24.3 0.216 0.214 MR2-3 28 28.0 0.298 0.215 Georgia Depression MR3 38 23.2 0.088 0.137 Table 5.18 Analysis of Regression Slopes for Different Watershed Sizes Region Number of Watersheds Estimate Regression Slope Difference between v4<100 ^>100 b1(^<l00) b2(^>100) b1 and b2 NMR1-1 8 28 Location C, 1.3392 0.9383 Not Significant Scale a 1.2188 0.8926 Not Significant NMR1-2 16 13 Location £ 1.0940 0.8712 Not Significant Scale a 1.0533 0.8148 Not Significant NMR8-1 16 18 Location C, 1.9138 1.1729 Significant Scale a 1.6203 0.9935 Significant NMR8-2 20 11 Location ^ 1.6099 1.1196 Significant Scale a 1.1852 0.9969 Not Significant a 0.8607 0.6772 Not Significant a 0.8340 0.6677 Not Significant G o 0.8341 0.6619 Not Significant MR3 23 14 Q20 0.8148 0.6795 Not Significant O o 0.7931 0.6821 Not Significant 000 0.7964 0.6791 Not Significant Q7.00 0.7998 0.6778 Not Significant A: Drainage Area in km 2 130 "a d3 CO C O CM CN cn 00 i n oo cn i n co cd co CO oo in CO CO i n CM CD 00 o CO cn o CN CD in "5 CN in in co in oo in CN m d m oo o C N CN CO CO 00 o CD CD CN cn in oo CO cn m oo CO o CO i n o co CN cn CN m cn oo CN CN CD cn CO "50" CD > s I o o CN CO 00 cn CO CD CD o CN 00 o CN CN "5 o o CD Cd "D 0 •5 o CD CO c o cn CD cd C >.| CD c o '•+-* CD -4—» CO cd o o LU —> 0 0 CN CN Cd o 00 X 0 0 i n Cd o o LU CO o o o < 0 0 CO Cd o o o o LU z oo C O I 0 0 Cd o o CD o o Cd CO CO CD o I oo CN Cd CO oo o o LU LU 0 0 C N I CN Cd in i n o o o 0 0 CO I C N Cd 131 Table 5.20a Comparison of R M S E between Regional and Single-Site Methods (NMR2-1) Return Period (yr) Regional Single-Site Bias (w3 Is) RMSE (w 3 Is) Bias(m 3 Is) RMSE (w 3 /s) 2 0.14 21.31 0.40 23.58 5 0.26 29.87 -1.26 29.90 10 0.36 37.02 -1.58 37.44 20 0.47 43.87 -0.58 49.50 50 0.62 52.31 3.32 73.11 100 0.75 58.20 8.58 96.84 200 0.87 63.68 16.12 125.85 500 1.04 70.32 29.97 173.13 1000 1.17 74.92 43.70 216.60 Table 5.20b Comparison of R M S E between Regional and Single-Site Methods (NMR8-2) Return Period (yr) Regional Single-Site Bias(m 3 /s) RMSE(m 3 /s) Bias(m 3 /s) RMSE(w 3 /s) 2 0.03 3.77 0.10 4.08 5 0.07 6.02 -0.25 5.78 10 0.09 7.85 -0.39 7.79 20 0.11 9.69 -0.27 10.92 50 0.14 12.12 0.51 17.20 100 0.16 13.94 1.74 23.85 200 0.18 15.75 3.70 32.49 500 0.21 18.12 7.70 47.70 1000 0.24 19.89 12.05 62.88 132 Table 5.20c Comparison of R M S E between Regional and Single-Site Methods (NMR8-3) Return Period (yr) Regional Single-Site Bias (m31 s) R M S E (m3 Is) Bias (m31 s) R M S E (m3 Is) 2 0.15 12.50 0.47 11.83 5 0.39 25.18 -1.36 19.46 10 0.57 36.31 -2.51 30.76 20 0.77 49.16 -2.90 49.54 50 1.10 69.67 -0.82 92.00 100 1.39 88.58 4.59 145.19 200 1.74 111.09 15.85 227.66 500 2.31 147.66 46.33 411.33 1000 2.84 181.61 89.08 645.00 133 Appendix B Figures 134 L-Skewness L-moment ratios of some common distributions (or the values of usual interest E •^exponential, G « Gumbel, L-logistic U » uniform, GPA-generalized Pareto, GEV-generalized extreme value, GLO» generalized logistic, G N O -generalized normal, GAM - gamma, while vertical crosses are the lower bound for afl other distributions. Fig. 3.1 L-Moment Ratio Diagram (Wallis, 1989 and Hosking, 1990) O is the average L-moment ratio of observations. 135 0.0 10.0 20.0 30.0 X VALUES 40.0 so.o Fig. 3.2 Non-Parametric Density Construction by the Kernel Method (Gingras and Adamowski, 1992) 136 137 Qm Q OL t — (1 ft********* " / Qm Reourrenoe Interval la Years Recurrence Interval In Years Fig. 4.2 Typical Cumulative Distribution Functions with a Dog-Leg (08MA001, top) and without a Dog-Leg (08LE031, bottom) 138 139 22 15 i Less than 10 57 • # 51 19 126 49 m 25 • Number of Stations BAverage Sample Length (years) 98 1 1 29 37 27 J ] 35 1 31 100-499 500-999 1000-4999 5000-9999 10000- Larger than 49999 50000 Drainage Area (sq. km) Fig. 5.1 Number of Stations and Average Sample Length vs. Drainage Area Jan Feb | Mar Apr May Jun I Jul Aug Sep Oct Nov Dec Fig. 5.2 Annual Daily Maximum Flow by Month for Station 08MB005-CHILCOTIN RIVER BELOW BIG CREEK 140 Fig. 5.3 Annual Daily Maximum Flow versus Rank for Station 08MB005-CHILCOTIN RIVER BELOW BIG CREEK •LUI .1 iminmiiimmim E o I i n t ldeo idas ufoo Fig. 5.4 Annual Daily Maximum Flow by Year for Station 08MB005-CHILCOTIN RIVER BELOW BIG CREEK 141 oo O Snowmelt generated floods with AP<30 mm • Rainfall generated floods with 30<AP<130 mm A Rainfall generated floods with AP>130 mm 50 100 150 One Week Antecedent Precipitation (mm) 200 250 Fig. 5.5 Different Flood-Generating Mechanisms (Station 08EG011-ZYMAGOTITZ RIVER NEAR TERRACE) o ° o O Snow and rain on snow floods with AP<150 mm • Rainfall floods with AP>150 mm 50 100 150 One Week Antecedent Precipitation (mm) 200 250 Fig. 5.5 (cont.) Different Flood-Generating Mechanisms (Station 08FE003-KEMANO RIVER ABOVE POWERHOUSE TAILRACE) 142 2500 2000 in E "D O O 1500 ID i 1000 < 500 O ofib o o o Snow and rain on snow generated floods with AP<115 mm • Rainfall generated floods with AP>115 mm 50 100 150 One Week Antecedent Precipitation (mm) 200 250 Fig. 5.5 (cont.) Different Flood-Generating Mechanisms (Station 08EF005-ZYMOETZ RIVER ABOVE O.K. CREEK) 143 144 * * l 8 t 0 T . 0 1 145 <£>0 OH t C N O O O C D T j - C M O saouaj jnooo jo jsqiun[\| saouBjjnooo jo jaquiriN CD C o cn oo <£> C O m • u - C M O o o t o i r c M o saouaj jnooo jo jaquinN saouaj jnooo J° JaquinN 146 If) or co in co CM T -ssouaj jnooo jo jaqiurtN CD c o O CO CO ^ C M saouaj jnooo jo jaquinN 5 T f C M O C O C O - t f C M O saouaunooo jo jaquinN C O ^ C M O O O C O ^ - C M O saouaj jnooo jo jaquinN 147 ID lO T J - CO CN T -saouajjnooo io j a q a i n f g c g a) i c » .2 £ cn c a) z LY. £ >^ ^ _Q w (TJ .2 «> -{3 >" t o a> = - C 00 CD O c a) CO CO Z3 13 OJ o co c &_ O O O T3 O O CD W (0 <D O C O o o §* CO o i n a3 .5> E a) O O f - l D l O T f c O C M T - o saouaj jnooo jo jeqiunN 148 135° 130° 125° 120° 115° Fig. 5.9 The Second Classification of Homogeneous Regions Adjusted for Similarity in the Timing of Annual Floods 149 150 151 135° 130* 1251 120° 115° Fig. 5.12 The Final Classification of Homogeneous Regions Adjusted for L- Skewness and L-Kurtosis 1 5 2 Fig. 5.13 Regional L-Moment Ratio Diagram 153 135° 130° 125° 120° 115° 19 G E V I 16 G L Q G 18 ND 55° G L O G ND 1 G L O C ND 14 G L O G 13 G E \ 1 Exposed Coast 2 Georgia Depression 3 Coast Mountains (1) 4 Coast Mountains (2) 5 Coast Mountains (3) 6 Arrow Lake (1) 7 Arrow Lake (2) 8 Arrow Lake (3) 9 Castlegar (1) 10 Castlegar (2) 11 Creston 12 Rocky Mountains 13 Interior Plateau (1) 14 Interior Plateau (2) 15 Interior Plateau (3) 16 Northern Interior (1) 17 Northern Interior (2) 18 Northeastern Plains 19 Northwestern Plateau 50° G L O G 5 L 0 ( 8 , G L O G > GLOG: Generalized Logistic Distribution GEV: Generalized Extreme Value Distribution ND: Non-Parametrc Distribution ND \ ND 1 0 . G L O C G L O G G L O G Fig. 5.14 Best-Fit Distribution for Each Homogeneous Region 1 5 4 60 (A I 50 o u. E D E x 5 30 40 co Q D C C < 20 10 From top to bottom, L-skewness equal to 0.30, 0.26, 0.22, 0.18, 0.14 and 0.10, respectively. 10 100 1000 Return Period (yr) Fig. 5.15 Sensitivity of the Flood Frequency Curve to L-Skewness (GEV) 0.8 0.7 c 0.6 o 'S 0.5 > I 0-4 'o 8 0.3 o 0.2 0.1 0 • •• • •• • • • • 10 100 1000 10000 Drainage Area (sq. km) 100000 1000000 Fig. 5.16 L-Coefficient of Variation vs. Drainage Area in B.C. 155 r 1.25 2 5 10 20 60 100 600 Recurrence Interval In Years 1.25 2 6 Itoourrenoe Entwtl In Y««rs 10 20 SO 100 Fig. 5.17a Non-Parametric (ND) vs. Parametric Distribution for Station 08MG005-LILLOOET RIVER NEAR PEMBERTON (ND-top, G E V -bottom) 156 1.25 2 6 Recurrence Interval La Years No I I I I I I I I I I I 1.003 1.06 126 2 6 10 20 60 100 500 Recurrence InUrral i n Yean Fig. 5.17a (cont.) Non-Parametric vs. Parametric Distribution for Station 08MG005- LILLOOET RIVER NEAR PEMBERTON (LN3-top, LP3-bottom) 157 1.003 1.05 1J36 2 6 10 20 50 100 500 Recurrence Interval In Years CO o o I I I 1 1 1 1 I ' I ' 1.003 1.06 126 2 6 10 20 60 100 600 Reourrenoe Interval In Years Fig. 5.17b Non-Parametric (ND) vs. Parametric Distribution for Station 08MG007-SOO RIVER NEAR PEMBERTON (ND-top, G E V -bottom) 158 o o o I « « 1 1 • 1 ' 1 • 1 " 1.003 1.05 1.25 2 6 10 20 50 100 500 Recurrence Interval In Years CO o , , 1 1 1 1 1 1 1 1 1 O 1^ 1 1 1 1 1 1^ 1 I I I 1.003 1.06 1J25 2 6 10 20 50 100 500 Recurrence Interval In Years Fig. 5.17b (cont.) Non-Parametric vs. Parametric Distribution for Station 08MG007- s o o R I V E R N E A R P E M B E R T O N (LN3-top, LP3-bottom) 159 1003 1.06 1J26 2 6 10 20 60 100 600 Reourrenoe Interval In Years < * 1.003 1.06 1.26 2 6 10 20 60 100 600 Beourrenoe Interval In Years Fig. 5.17c Non-Parametric (ND) vs. Parametric Distribution for Station 08DB001-NASS RIVER ABOVE SHUMAL CREEK (ND-top, G E V -bottom) 160 1.003 1.06 1J26 2 6 10 20 50 100 500 Recurrence Interval In Years Recurrence Interval In Years Fig. 5.17c (cont.) Non-Parametric vs. Parametric Distribution for Station 08DB001 - NASS RIVER ABOVE SHUMAL CREEK (LN3-top, LP3-bottom) 161 CO O O I L J i- _ _ 1 1 1 1 1 1 1.003 1.05 1.26 2 5 10 20 50 100 500 Recurrence Interval Ln Yeara co o O I 1 ' • • • 1.003 1.06 1J36 2 6 10 20 60 100 500 Recurrence Interval In Years Fig. 5.17d Non-Parametric (ND) vs. Parametric Distribution for Station 08EG011 -ZYMAGOTITZ RIVER NEAR TERRACE (ND-top, G E V -bottom) 162 1.003 1.05 1J36 2 5 10 20 50 100 500 Recurrence Interval In Years H * * 1.003 1.05 1J35 2 5 10 20 GO 100 500 Recurrence Interval In Years Fig. 5.17d (cont.) Non-Parametric vs. Parametric Distribution for Station 08EG01 1 - Z Y M A G O T I T Z R I V E R N E A R T E R R A C E (LN3-top, LP3-bottom) 163 -ft* - f — / 1.06 125 2 6 10 20 60 100 600 R«ourreaoe Interval la Years * * * 1.003 1X16 126 2 6 10 20 60 100 600 Reourrenoe Interval in Years Fig. 5.17e Non-Parametric (ND) vs. Parametric Distribution for Station 10CB001-SIKANNI RIVER NEAR FORT NELSON (ND-top, G E V -bottom) 164 1J25 2 6 JO 20 60 100 500 Recurrence Interval In Years £ s • < • * ^ « 1.003 1.25 2 S 10 20 60 100 SOO Reourrvnoe Interval In Yean Fig. 5.17e (cont.) Non-Parametric vs. Parametric Distribution for Station 10CB001- SIKANNI RIVER NEAR FORT NELSON (LN3-top, LP3-bottom) 165 0 50 0 4 0 8 I 0-30 i + 0-20 O K ) 0 20 0-30 0-4O 0-SO 0-60 0-70 L - S k a w n a s s Fig. 5.18 Two-Component Extreme Value Distribution in Feasible L-moment Space (Gabriele and Arnell, 1991) 0 and X are parameters of T C E V ; + presents the weighted regional mean of L-Cs and L-Ck for the five mixture regions. 1 6 6 10000 1000 3 E to co 100 10 0.1 7000 6000 5000 c I 3000 2000 1000 0 A Scale parameter o Location parameter 10 100 1000 Drainage Area (sq. km) 10000 100000 Fig.5.19 Parameters of G E V vs Drainage Area for NMR2-1 • • • • • • , . • • • • . *• . * *t *"T • ^ - • • * • m ' m * "'. * \ v. v . \ • 10 20 30 40 50 60 70 80 90 100 Sample Size (yr) Fig. 5.20 Sensitivity of the Mean to Sample Size 167 ft^frfr . .V . ft. . • „ • • • ft • ^ ^ 0 10 20 30 40 50 60 70 80 90 100 Sample Size (yr) Fig. 5.21 Sensitivity of L-Coefficient of Variation to Sample Size • ft ft • • •• • • ft • • • , ft • - • • • • ft«W • . - —,----m--mr • ft ft • • • • • • • - • -• / • ft • • > • • • ft •• • •• ' ft ft • • • ft • •• 10 20 30 40 50 60 70 80 90 100 Sample Size (yr) Fig. 5.22 Sensitivity of L-Skewness to Sample Size 168 100000 I 10000 o o _ l o z. 1000 o l_ 3 a> E CO CL c o ra o o _ i 100 10 0.1 10 • Location Parameter for Drainage Area i Smaller than 100 sq.km O Location Parameter for Drainage Area Larger than or Equal to 100 sq.km 100 1000 Drainage Area (sq.km) 10000 100000 Fig. 5.23a Location Parameter of G L O G vs. Drainage Area (NMR1-1) o o _ i o s— O w_ 0) E « CO CL » 8 CO 1000 100 10 0.1 0.01 Scale Parameter for Drainage Area Smaller than 100 sq.km i O Scale Parameter for Drainage Area i Larger than or Equal to 100 sq.km 10 100 1000 Drainage Area (sq.km) 10000 100000 Fig. 5.23b Scale Parameter of G L O G vs. Drainage Area (NMR1-1) 169 o • Location Parameter for Drainage Area • Smaller than 1 0 0 sq.km O Location Parameter for Drainage Area ! Larger than or Equal to 1 0 0 sq.km j 10 100 Drainage Area (sq.km) 1000 10000 Fig. 5.24a Location Parameter of G L O G vs. Drainage Area (NMR1-2) • Scale Parameter for Drainage Area Smaller than 1 0 0 sq.km O Scale Parameter for Drainage Area Larger than or Equal to 1 0 0 sq.km 10 100 Drainage Area (sq.km) 1000 10000 Fig. 5.24b Scale Parameter of G L O G vs. Drainage Area (NMR1 -2) 170 10000 E 1000 o o d 100 o 3 aS E co co Q_ c g o o 10 0.1 0.01 o O • Location Parameter for Drainage Area Smaller than 100 sq.km O Location Parameter for Drainage Area Larger than or Equal to 100 sq.km 10 100 1000 Drainage Area (sq.km) 10000 100000 Fig. 5.25a Location Parameter of G L O G vs. Drainage Area (NMR8-1) 1000 o o _l o o 0) a> E 03 k_ 03 CL CO o CO 100 10 0.1 0.01 1 • Scale Parameter for Drainage Area Smaller than 100 sq.km O Scale Parameter for Drainage Area Larger than or Equal to 100 sq.km 10 100 1000 10000 Drainage Area (sq.km) Fig. 5.25b Scale Parameter of G L O G vs. Drainage Area (NMR8-1) 100000 171 1000 • Location Parameter for Drainage Area | Smaller than 100 sq.km j O Location Parameter for Drainage Area j Larger than or Equal to 100 sq.km | 10 100 Drainage Area (sq.km) 1000 10000 Fig. 5.26a Location Parameter of G E V vs. Drainage Area (NMR8-2) 1000 1 7 2 1000 100 in E o, O 10 • 2-yr Flood for Drainage Area Smaller than 100 sq.km i 02-yr Flood for Drainage Area Larger j than or Equal to 100 sq.km 10000 10 100 Drainage Area (sq. km) 1000 Fig. 5.27 Flood Quantiles vs. Drainage Area (MR3) 10000 1000 in J . 100 o 10 • 5-yr Flood for Drainage Area Smaller than 100 sq.km 05-yr Flood for Drainage Area Larger than or Equal to 100 sq.km 10 100 Drainage Area (sq. km) 1000 10000 Fig. 5.27(cont.) Flood Quantiles vs. Drainage Area (MR3) 173 10000 1 10 100 1000 10000 Drainage Area (sq. km) Fig. 5.27(cont.) Flood Quantiles vs. Drainage Area (MR3) 174 10000 1000 100 10 10000 • 50-yr Flood for Drainage Area Smaller than 100 sq.km O 50-yr Flood for Drainage Area Larger than or Equal to 100 sq.km 10 100 Drainage Area (sq. km) 1000 10000 Fig. 5.27(cont.) Flood Quantiles vs. Drainage Area (MR3) CO E o 1000 100 10 • 100-yr Flood for Drainage Area | Smaller than 100 sq.km ! 0100-yr Flood for Drainage Area j Larger than or Equal to 100 sq.km i 10 100 Drainage Area (sq. km) 1000 10000 Fig. 5.27(cont.) Flood Quantiles versus Drainage Area (MR3) 175 10000 1000 m E a. 100 10 • 200-yr Flood for Drainage Area Smaller than 100 sq.km O 200-yr Flood for Drainage Area Larger than or Equal to 100 sq.km 10 100 Drainage Area (sq. km) 1000 10000 Fig. 5.27(cont.) Flood Quantiles vs. Drainage Area (MR3) 10 100 1000 Drainage Area (sq. km) 10000 100000 Fig. 5.27A Record Length vs. Drainage Area 176 10000 T=5 Years 10 100 1000 Actual Flood (cms) T=10 Years 10000 1 10 100 1000 10000 Actual Flood (cms) Fig. 5.28 Comparison of Actual and Predicted Floods 177 T=20 Years Actual Flood (cms) T=50 Years 10000 • — 1000 100 10 10 100 1000 Actual Flood (cms) 10000 Fig. 5.28(cont.) Comparison of Actual and Predicted Floods 178 T=100 Years 1 10 100 1000 10000 100000 Actual Flood (cms) Fig. 5.28(cont.) Comparison of Actual and Predicted Floods 179 0.00 • i i : — I : i i — : : : : : I i i—:— I 1 10 100 1000 Return Period (yr) Fig. 5.29a Accuracy of Estimating Flood Events for Various Return Periods for NMR2-1 (Sample Size = 24) 70.00 0.00 I i i : : : : : • I j • : : : : : : j ; i : 1 10 100 1000 Return Period (yr) Fig. 5.29b Accuracy of Estimating Flood Events for Various Return Periods for NMR8-2 (Sample Size = 24) 180 700.00 Return Period (yr) Fig. 5.29c Accuracy of Estimating Flood Events for Various Return Periods for NMR8-3 (Sample Size = 24) -Single Site Model •Regional Model 0 10 20 30 40 50 60 70 80 90 100 110 Sample Size (yr) Fig. 5.30a Influence of Saple Size on the Accuracy of the 100-yr Flood Event for NMR2-1 181 Fig. 5.30b Influence of Saple Size on the Accuracy of the 100-yr Flood Event for NMR8-2 4 0 . 0 0 3 5 . 0 0 30 .00 r 2 5 . 0 0 O " J 2 0 . 0 0 or 15.00 10 .00 5 .00 0 .00 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 Sample Size (yr) Fig. 5.30c Influence of Saple Size on the Accuracy of the 100-yr Flood Event for NMR8-3 182 Appendix C Hydrometric Stations for this Study 183 co E re is co o co Q > O z o O CL| Q> CO < 0) o c V 3 o u O T3 O O i t ' ico E |3 <l re S CO u. c re CO Ol _ .E 2 re < IO CO c o re CO CA ja O CO 1 Z C M T -00 CO C M C D m C M C O C O C M T — T — C M C O T - O) T - (O N T f (f l n T — C O CD •f ^ oo N - T- C M o , i n C M a a O o co in -tf o o m co iygicocoococov-g ~ co t n r r-o O o m O oo T— O T— CM O CM CM o CO O O CD CD CO co CO O CD T— T— CN CO o CO CN co o CM ' - ^ ? ?C 1 C M ' C M 0 ? £ ^ 0 > ° 2 2 N ^ r o S S m s ^ N ^ ! : 3 S N ® ^ ^ W N < o s c \ i c o c \ i c \ i T - N U ) i o i n c o o c » c o re ICO o o lO CO i n i n o CM CO o CM CO N- oo o o co o o T— T— T— CO •<* •<r •<* 5 m o o o o o o o o o o o O O o o < < < CO CD CD CD CD O CD CD CD CD CD CD z z z z z z z z z z z z z z z co co oo CO co co co co co CO co co oo co oo o o o o o o o o o o o o o o o m co N co i - in CO N N N O O o o o o o o CD CD CD CD X X z z z z z z z N I - O T - C M ( \ | ( N O ! 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L L L L O O O O O O o o c o c o c o c o o o c o o o c o c o c o O O O O O O O O O O O CO r - C M LO CO o o C O o o O o o o o o o o O o o o o o o Q Q U J CD CD CD CD CD CD CD CD CD o o o o 0 0 o o o o o o o o o o o o o o o o o o o o o 5 8 x m 2 co 0 0 o sc LU LU or o or U J t X to < 2 U J > O oo < or U J > or 2 < 3 a w or x to 2 < 3 o co or 55 z sc U J U J or o or yj Qj 3 o Q < o or z o sc LU LU oo or o < >-o or \-z o h- LU £ o CO CO z yj < § 9 or z o < sc co or U J > _ or ° 5 § >-LU U J or z < or U J > or U J to I CO C M C M C O C O or o CO z o o z CM U J or o co « CD 5 z m LU ^ • i - c o n c o c o m c o f ' - N T - I O T - C D C O C D ' - C O r -CO CM T— C M C M C M T - ' t - T - T - T - T - C N C O T - T l - T -i— CNCNi— T - C O T - L O C N • « - N - t N T - C O T - C O C o i 2 ( J > L n O L O CM CO CN J - CO CO m C 0 T - 0 0 C M C 0 C 0 N ' - < - N - C O T - T - r - T - C N ' - t - ' t T t C O C N C M LO LO O O < < o o o ^— N " LO LO CO CO CO CO o o o o o < < < < < O CD CD CD CD o o 0 0 c o 0 0 c o o o o o o T— C O c o CM o o V™" o o o o o < < < m X X X X o o 0 0 0 0 0 0 o o o o 197 0> E ro '1 CO u CO Q > O z o co O 2 °-t « O = !* o "» o c = 3 >- -i fl) 2 E o-l 2 co S xi o LL C co o> .E £ « < (0 O ci 1 « 8 c o CO DC < LU * z LU OT LU LU >-< X I-3 o Si  Si > or LU _j m < z or LU > a: co z § o cr: m O 1 < z < z I-< or UJ LU CO 0_ < o LL. O LU z g CO or UJ > Q LU or LU > or *r z < co o =! § 5 O LU 3 X o > or LU LU < or LU > or LU 5 3 Id i i 3 I co < or or UJ LU > o o or LU Qj £ si LU or LU LU or > CJ or LU c5 E z E o < m u o o or LU _ i CL < LU - i h-i l o £ CO co LU CO LU LU z > or or < x or Si or Si ui co < or LU > or x CO O LU LU m co or or LU LU > > or or i £ * b o CO i-° 5 LU 3 Z CO or or z sr LU LU LU LU or or o o Q Q < LU or z o D O z H Z X or Si z or LU > or co o or 6 o or t L U 3 CO CD LU § or Si z LU £ LU or or o o z <r O 2 co 2 ? X I-3 O LU X LU LU or o >-Q> CO or co o o < LU u 3 z ^ 5 | UJ" § S 8 >-LU _J CO CO O LU CO LU O CO CO or < LU > o CO < or LU > or 2 LU LU or I-co CO o CM I-< or * LU LU % 2 2 5 or o 2 ° 5 o (M S Tf in CM ^ CM ( O t O N N C O ^ O O N N O O n N i s s i z o T T f c o n r N ^ i f i i n N n r - i N T f n o i c D i n T f S T - T f n n n T f T f r - T T c o x r T f T f C M c o T f L O C M C M T T — T - T - T - C O LO T - CM T - T CM T T - T - CM T - 1 -C O C M T - C O C M T - C M T - T f T - r - CM CM CM T -T - CM T— T - T f CM CM CM CM CO LO CM T f C O T f r — T f C O r — r— r ^ T - c o T - T - c o c o c M C M c o T - c o c o c n c n r ^ ^ ^ I c o c o L O c o c M T - T - CD T— CO LO C O CD ^ 00 CD Tf , _ T f 00 C M C O C D O C D C O C D C M C M C M o CO 0 ) 0 ( 0 C O C D 00 C M C O C O 00 C D L O co co CO LO T f CO CM C O C O T - CM L O CD l O l O 00 CO CO CO T - T— T— O 00 O T f T f T - C O C M T - T -CM T f o i o r - o o L O T - L o o o c M O T f i o c o o T -T - C M T — C O C M C M T f C O T — C M C M T — T - T - T - C M C O T f L O C M T f o C M C M co r--o o o ~ LO T— T— LO T— CM CO T f T CO CO O CD CO I T f LO CO CD x:— L O CO N O T - T - O O O O C D O T — C M C M r ^ - C D O O O C M T r i O c M 0 0 0 0 0 0 0 0 0 0 0 0 0 ° O O O O O O T - T - T - T — T - T - O CQLtlCDCQ[rjCQQQQU.LLLLU.OlIIIIIIIIIII< | X l x x x x x x x x x x x 5 5 5 5 5 5 5 5 5 5 5 5 5 X c o o o c o c o o o o o o o o o c a o o o o o o o o o o o o c o o o o o o o o o o o o o c o c o o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O O O O 198 Appendix D Examples of Computing Floods in Different Regions 199 Examples of computing floods in different homogeneous regions in BC Given a watershed with a drainage area of 150 km 2. The 100-yr floods in different regions can be obtained as following: A. In a non-mixture region with G L O G distribution, for example NMR1-1 Constant shape parameter (from Table 5.14) k = -0.1263 Scale parameter (from Table 5.14) a = aAh = 0.0 2 2 1 * 1 5 0 0 9 3 0 4 = 2.339 cms Location parameter (from Table 5.14) = cAd = 0.1116* 150° 9 6 5 5 =14.083 cms Using equation 5.4, Qm can be computed -0.1263" 2.339 0 0 0 " -0.1263 1-0.99 ' 0.99 + 14.083 Qm - 28.652cms B. In a non-mixture region with GEV distribution, for example NMR2-1 Constant shape parameter (from Table 5.15) k = 0.1549 Scale parameter (from Table 5.15) a = aAh = 0.0688 * 150° 8 8 0 2 = 5.662 cms Location parameter (from Table 5.15) £ - cAd = 0.1525 * 150 0 9 7 4 1 =20.091 cms Using equation 5.8, Qm can be computed: e » = ^ [ 1 - ( - | n 0 ' 9 9 ) , " " ] + 2 0 0 9 1 Qm =38.719cms C. In a mixture region, for example MR1 Using Table 5.16, Qm can be computed: Q]Q0 = 1.0 1 64 * 1 50° 8 0 2 5 = 56.673cms 200 

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