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The effects of scale and storm severity on the linearity of watershed response revealed through the regional… Cathcart, Jaime Grant 2001

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THE EFFECTS OF S C A L E A N D STORM SEVERITY ON THE LINEARITY OF WATERSHED RESPONSE R E V E A L E D THROUGH THE REGIONAL L-M O M E N T ANALYSIS OF A N N U A L P E A K FLOWS by JAIME GRANT C A T H C A R T B.A.Sc. The University of British Columbia, 1987 M.A.Sc. The University of British Columbia, 1993 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES Institute of Resources and Environment Resource Management and Environmental Studies We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A September 2001 © Jaime Grant Cathcart, 2001 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 allowed without my written permission. W S T l T V T i e r Department of gETSeuftCeS. AMfr e.o»*HgAJT The University of British Columbia Vancouver, Canada Date DE-6 (2/88) A B S T R A C T Natural basins rarely demonstrate linear response patterns across a large range of spatial scales and storm severities, yet assumptions of linearity, both explicit and implicit, are common to most peak flow estimating techniques. The extent to which these assumptions are valid is one of the most intriguing and challenging questions facing hydrologists today. This thesis presents the results of an effort to shed some light on the darkness of our understanding of this issue. The study represents the first time that the relationship between storm severity and the linearity of peak flow response has been considered on a regional basis, in contrast with the more conventional approach of investigating linearity through the use of deterministic models of single watersheds. An effort is made to introduce some understanding of causative physical processes into the frequency analysis of extremes, thereby distinguishing the work from the common approach of applying mathematically rigorous but physically sterile curve-fitting exercises. The study focuses on the spatial scaling patterns of linear moment flood statistics, and plausible explanations are offered for observed regional scaling trends in terms of the various precipitation and runoff mechanisms that dominate at different scales and in different climates. The characteristics of these mechanisms are then linked back to the effects that variations in L-moment ratio statistics have on flood quantile estimates, and most importantly, the tail behaviour of flood frequency distributions. The research program mainly utilizes historical records of USGS peak flow values for Oregon State, but USGS equations for other states, as well as hourly streamflow and precipitation data from the Carnation Creek Watershed on the west coast of Vancouver Island, British Columbia, are also analyzed. The findings indicate that the primary flood statistics, namely the mean, L-coefficient of variation, L-skewness and L-kurtosis, generally demonstrate different degrees of spatial scaling linearity at small and large scales, with contrasting patterns for humid and arid regions. Furthermore, these variations in spatial scaling linearity are shown to translate into differences in the scaling patterns of return period quantiles, thereby demonstrating differences in storm linearity at different spatial scales. This in turn equates to differences in the tail behaviours of flood frequency distributions at different scales. The observed variations in linearity are attributed to the differing relative roles played by various moisture input and runoff generation mechanisms at different scales and under different climate regimes. To the author's knowledge, this work presents an original attempt to physically explain the tail behaviour of flood frequency curves. The results cast doubt on the validity of many commonly applied flood frequency analysis concepts, techniques and models, such as the delineation of geographically defined hydrologic regions, the statistical testing of hydrologic homogeneity, the mapping of flood statistics, the use of regional flood frequency distributions and the application of the index flood method across a broad range of spatial scales. Furthermore, the findings raise concerns about the common practice of applying rainfall-11 runoff concepts and models, such as the unit hydrograph approach, popular time of concentration equations and the Rational Method, without serious consideration of the significance of basin size or climate type. i i i T A B L E O F C O N T E N T S Page ABSTRACT ii LIST OF TABLES xiii LIST OF FIGURES xv NOMENCLATURE AND ACRONYMS xix ACKNOWLEDGEMENTS xxiii PREAMBLE xxiv Chapter 1.0 INTRODUCTION 1 1.1 HISTORICAL B A C K G R O U N D 1 1.2 OBJECTIVES 3 1.2.1 SCIENTIFIC OBJECTIVES 4 1.2.2 OPERATIONAL OBJECTIVES 4 1.3 THESIS OVERVIEW 5 2.0 REVIEW OF POPULAR STATISTICAL METHODS FOR PEAK FLOW ESTIMATION 9 2.1 INTRODUCTION 9 2.2 SINGLE SITE FREQUENCY ANALYSIS 10 2.2.1 DISTRIBUTION SELECTION 10 2.2.1.1 Theoretical Limitations of a Distribution 11 2.2.1.2 Shape of a Distribution 11 2.2.1.3 Flood Mixture 11 2.2.1.4 "Goodness-of-Fit" Tests 12 2.2.1.5 Common Practice 13 2.2.2 P A R A M E T E R ESTIMATION 14 2.2.2.1 Graphical Method 14 2.2.-2.2 Least Squares Method 15 2.2.2.3 Method of Moments 15 2.2.2.4 Method of Maximum Likelihood 16 2.2.2.5 Method of Linear Moments 16 2.3 REGIONAL FLOOD FREQUENCY ANALYSIS 21 2.3.1 DELINEATION OF HOMOGENEOUS REGIONS 22 2.3.2 JDENTIFICATION OF REGIONAL DISTRIBUTIONS 23 2.3.3 POPULAR REGIONAL FLOOD FREQUENCY MODELS 23 iv 2.3.3.1 The Index Flood Approach 23 2.3.3.1.1 General 23 2.3.3.1.2 The Forest Practices Code of BC (FPCBC) Model 24 2.3.3.1.3 The BC MOELP Model 25 2.3.3.1.4 Model Limitations 26 2.3.3.2 The Method of Direct Regression of Quantiles 28 2.3.3.3 The Method of Regression for Distribution Parameters 29 2.3.3.4 Russell's Bayesian Method 30 2.4 S U M M A R Y 31 3.0 DATA SELECTION AND SCREENING 32 3.1 INTRODUCTION 32 3.2 D A T A SELECTION 33 3.3 D A T A SCREENING 34 3.3.1 L E N G T H OF RECORD 34 3.3.2 FLOW REGULATION 36 3.3.3 BASIN A R E A 36 3.3.4 D A T A INDEPENDENCE, TREND, RANDOMNESS A N D HOMOGENEITY 38 3.3.5 HIGH A N D LOW OUTLIER FLOODS 40 3.3.6 TREATMENT OF ZERO FLOWS 41 3.3.7 REGIONS WITH INSUFFICIENT D A T A 42 3.3.8 C O M M O N PERIOD OF RECORD 42 3.3.8.1 Period Selection 43 3.3.8.2 Testing the Significance of the Climate Shift 45 3.3.9 DISCORDANT STATIONS 50 3.4 STATISTICS OF HISTORICAL A N N U A L P E A K FLOWS 52 3.5 CONCLUSION 52 4.0 REGION DELINEATION 53 4.1 INTRODUCTION 53 4.2 LITERATURE REVIEW 54 4.2.1 GEOGRAPHICAL CONVENIENCE 54 4.2.2 SUBJECTIVE PARTITIONING 55 4.2.3 OBJECTIVE PARTITIONING 56 4.2.4 CLUSTER ANALYSIS 56 v 4.2.5 OTHER M U L T I V A R I A T E TECHNIQUES 57 4.3 RESEARCH METHODS 57 4.3.1 G E N E R A L 57 4.3.2 MONTE C A R L O SIMULATION M O D E L 58 4.4 RESULTS AND DISCUSSION 58 4.4.1 BASIN PHYSIOGRAPHY A N D PRECIPITATION 58 4.3.2 P E A K FLOW CHARACTERISTICS 62 4.4.3 GEOGRAPHICAL PROXIMITY 66 4.4.4 WATERSHED DIVIDES A N D POLITICAL BOUNDARIES 66 4.4.5 N U M B E R OF STATIONS 66 4.4.6 DISCORDANCY M E A S U R E 69 4.4.7 VISUAL ASSESSMENT OF STATISTICAL SCALING PATTERNS 69 4.4.8 TESTING THE HYPOTHESIS OF HOMOGENEITY 69 4.4.8.1 General 69 4.4.8.2 Monte Carlo Simulation 72 4.4.8.3 Oregon Regression Plots 76 4.4.9 REGION DESCRIPTIONS 80 4.4.9.1 Western Regions 80 4.4.9.2 Transition Regions 82 4.4.9.3 Eastern Regions 83 4.5 CONCLUSIONS 84 5.0 INVESTIGATING THE SCALING OF MEAN ANNUAL PEAK FLOWS 86 5.1 INTRODUCTION 86 5.2 LITERATURE REVIEW 87 5.2.1 SCALING DEFINED 87 5.2.2 HISTORICAL B A C K G R O U N D 87 5.2.3 SCALING COEFFICIENT 88 5.2.4 SCALING EXPONENT A N D BASIN RESPONSE LINEARITY 89 5.2.5 REGION ARIDITY 91 5.2.5.1 Moisture Input Mechanisms 92 5.2.5.2 Runoff Mechanisms 92 5.3 RESEARCH METHODS 93 5.4 DISCUSSION 93 5.4.1 REGION ARIDITY 94 5.4.2 RAIN VERSUS SNOWMELT 97 VI 5.4.3 BASIN A R E A 97 5.4.3.1 Humid versus Arid Regions 99 5.4.3.2 Curvilinear Scaling Pattern 99 5.4.4 DIRECT PHYSICAL INTERPRETATIONS 101 5.5 CONCLUSIONS 102 6.0 OREGON: SCALING OF THE MEAN 103 6.1 INTRODUCTION 103 6.2 RESEARCH METHODS A N D B A C K G R O U N D 103 6.2.1 LINEAR REGRESSION TECHNIQUES 104 6.2.2 LINEAR A N D POWER EQUATIONS 104 6.2.3 BASIN A R E A AS THE O N L Y INDEPENDENT V A R I A B L E 105 6.2.4 SCALING COEFFICIENTS AND EXPONENTS 107 6.2.5 GROUPING D A T A ACCORDING TO BASIN S C A L E 107 6.2.6 TESTING B R E A K S IN TRENDLFNES 107 6.2.7 D A T A VARIABILTIY A N D RELIABILITY 108 6.3 RESULTS 109 6.3.1 STRENGTHS OF REGIONAL RELATIONS 109 6.3.1.1 Coefficient of Determination 109 6.3.1.2 Standard Error of Estimate 109 6.3.2 SCALING EXPONENTS 112 6.3.3 SCALING COEFFICIENTS 112 6.3.4 R A N G E OF M E A N P E A K FLOW ESTIMATES 112 6.4 DISCUSSION 114 6.4.1 STRENGTHS OF REGIONAL RELATIONS 114 6.4.1.1 General 114 6.4.1.2 Coefficient of Determination 116 6.4.1.3 Standard Error of Estimate 118 6.4.2 G E N E R A L REGIONAL SCALING PATTERNS 118 6.4.3 GROUPING REGIONS INTO M E G A REGIONS 118 6.4.4 ZONE A 121 6.4.5 S M A L L BASINS VERSUS L A R G E BASINS 121 6.4.5.1 Oregon Patterns 121 6.4.5.2 Comparison with Postulated Scaling Model 125 6.4.5.3 Testing the Validity of Simple Linear Extrapolation 125 6.4.5.4 Power or Linear Regression Models? 127 6.5 CONCLUSIONS 134 V l l 6.5.1 A L L AREAS 134 6.5.2 S M A L L BASINS VERSUS L A R G E BASINS 134 7.0 SCALING OF THE L-COEFFICIENT OF VARIATION 136 7.1 INTRODUCTION 136 7.2 LITERATURE REVIEW 137 7.2.1 SIMPLE SCALING 138 7.2.2 MULTI-SCALING 142 7.2.2.1 Empirical Studies 142 7.2.2.2 Derived Flood Frequency Studies 146 7.3 RESEARCH METHODS 147 7.4 RESULTS 148 7.4.1 REGIONAL S C A L N G TRENDS 148 7.4.2 P W M ANALYSIS 152 7.5 DISCUSSION 152 7.5.1 PHYSICAL EXPLANATIONS FOR S C A L N G TRENDS 152 7.5.1.1 Decreasing Trend 155 7.5.1.2 Increasing Trend 155 7.5.1.3 Constant Trend 157 7.5.2 HUMID VERSUS ARID CLIMATES 157 7.5.3 SIGNIFICANCE OF SNOWMELT 157 7.5.4 SIMPLE LINEAR EXTRAPOLATION 159 7.5.5 R E G O N A L L Y CONSTANT L - C V 159 7.5.6 L - C V VERSUS C V 159 7.5.7 P W M ANALYSIS 160 7.5.7.1 Value of the Scaling Exponents 160 7.5.7.2 Simple Scaling 160 7.5.7.3 Multi-Scaling 160 7.5.7.4 L-Cv Trends versus P W M Trends 163 7.6 CONCLUSIONS 163 8.0 SCALING OF THE L-SKEWNESS AND L-KURTOSIS 165 8.1 INTRODUCTION 165 8.2 LITERATURE REVIEW 166 8.2.1 SCALING OF THE L-SKEWNESS 166 8.2.2 SCALING OF THE L-KURTOSIS 169 8.2.3 CHOICE OF A FREQUENCY DISTRIBUTION 170 viii 8.3 R E S E A R C H M E T H O D S 170 8.3.1 G E N E R A L 170 8.3.2 Z - T E S T 171 8.4 R E S U L T S 171 8.4.1 L - S K E W N E S S 171 8.4.1.1 Regional Scaling Trends 173 8.4.1.2 Comparing the L-Cs to the L - C v 173 8.4.2 L - K U R T O S I S 176 8.4.2.1 Regional Scaling Trends 179 8.4.2.2 Comparing the L - C k to the L -Cs 179 8.4.3 Z - T E S T 179 8.5 D I S C U S S I O N 183 8.51 L - S K E W N E S S 183 8.5.1.1 Physical Explanations for Scaling Trends 183 8.5.1.1.1 General 183 8.5.1.1.2 Humid versus A r i d Climates 183 8.5.1.1.3 Storage Effects 185 8.5.1.2 Comparing the L-Cs to the L - C v 187 8.5.1.3 Skewness Separation 187 8.5.1.4 Mapping Skewness 190 8.5.1.5 Validity of the Gumbel (EV1) Distribution 190 8.5.1.6 Simple Linear Extrapolation 190 8.5.1.7 Regionally Constant L-Cs 190 8.5.2 L - K U R T O S I S 191 8.5.2.1 Physical Explanations for Scaling Trends 191 8.5.2.1.1 General 191 8.5.2.1.2 Humid versus A r i d Climates 191 8.5.2.2 Comparing the L - C k to the L -Cs 193 8.5.2.3 Simple Linear Extrapolation 193 8.5.2.4 Regionally Constant L-Cs 193 8.5.3 Z - T E S T 193 8.5.3.1 Distribution Selection 193 8.5.3.2 Implications of Distribution Selection 194 8.6 C O N C L U S I O N S 194 8.6.1 L - S K E W N E S S 194 8.6.2 L - K U R T O S I S 197 8.6.3 D I S T R I B U T I O N S E L E C T I O N 197 ix SCALING EXPONENT VERSUS RETURN PERIOD 199 9.1 INTRODUCTION 199 9.2 LITERATURE REVIEW 201 9.2.1 G E N E R A L 201 9.2.2 DECREASING SCALING EXPONENT 202 9.2.3 INCREASING SCALING EXPONENT 202 9.2.4 CONSTANT SCALING EXPONENT 203 9.2.5 S U M M A R Y 203 9.3 RESEARCH METHODS 204 9.3.1 USGS REGRESSION EQUATIONS 204 9.3.2 OREGON STUDY 204 9.4 RESULTS 205 9.4.1 USGS REGRESSION MODELS 205 9.4.2 OREGON STUDY 205 9.5 DISCUSSION 208 9.5.1 USGS REGRESSION MODELS 208 9.5.1.1 General 208 9.5.1.2 Arid versus Humid Climates 208 9.5.1.3 Small versus Large Basins 208 9.5.2 OREGON STUDY 209 9.5.2.1 General 209 9.5.2.2 L-Moment Ratios 209 9.5.2.3 Small versus Large Basins 210 9.6 CONCLUSIONS 210 10.0 A REGIONAL FREQUENCY MODEL FOR PEAK FLOW ESTIMATION IN OREGON 216 10.1 INTRODUCTION 216 10.2 LITERATURE REVIEW 217 10.2.1 AT-SITE M O D E L 217 10.2.2 INDEX FLOOD M O D E L 218 10.2.3 REGIONAL SHAPE ESTIMATION M O D E L 218 10.2.4 HIERARCHICAL REGIONAL M O D E L 218 10.2.5 FRACTIONAL MEMBERSHIP 218 10.3 RESEARCH METHODS 218 10.3.1 M O D E L DESCRIPTION A N D APPLICATION 218 x 10.3.2 M O N T E C A R L O T Y P E S I M U L A T I O N 219 10.4 R E S U L T S A N D D I S C U S S I O N 219 10.4.1 M O D E L A P P L I C A T I O N A T G A U G E D SITES 219 10.4.1.1 Station 14134000 - Salmon River near Government Camp 221 10.4.1.2 Station 14303200-Tucca Creek Near Blaine 223 10.4.2 M O D E L A P P L I C A T I O N A T U N G A U G E D SITES 226 10.4.3 C O M P A R I S O N O F S I N G L E - S I T E A N D R E G I O N A L M O D E L S 232 10.5 C O N C L U S I O N S 234 11.0 SCALING BEHAVIOUR AND THE RATIONAL METHOD 235 11.1 I N T R O D U C T I O N 235 11.2 L I T E R A T U R E R E V I E W 236 11.2.1 T H E R A T I O N A L M E T H O D 236 11.2.1.1 Runoff Coefficient 237 11.2.1.2 Time of Concentration 237 11.2.1.2.1 General 237 11.2.1.2.2 Different Measures 239 11.2.1.2.3 Scaling Relations 242 11.3 R E S E A R C H M E T H O D S 245 11.3.1 D A T A S E L E C T I O N 245 11.3.2 M E A S U R E D T I M E O F C O N C E N T R A T I O N 246 11.3.2.1 Minimum Time of Rise 246 11.3.2.2 Effects of Storm Intensity on the Tc 248 11.3.3 C O N S I D E R A T I O N O F L O G G I N G E F F E C T S 248 11.3.4 D E R I V E D T I M E OF C O N C E N T R A T I O N 248 11.3.4.1 Derived Tc 248 11.3.4.2 Precipitation Analyses 249 11.3.4.2.1 Extreme Rainfall 249 11.3.4.2.2 Mean Annual Precipitation 250 11.3.4.3 Peak Flow Analysis 250 11.3.4.4 Runoff Coefficients 251 11.4 R E S U L T S A N D D I S C U S S I O N 251 11.4.1 M E A S U R E D T I M E O F C O N C E N T R A T I O N 251 11.4.1.1 Development of a Tc Scaling Relation 251 11.4.1.2 Effects of Storm Intensity on the Tc 256 11.4.2 C O N S I D E R A T I O N O F L O G G I N G E F F E C T S 259 X I 11.4.3 DERIVED TIME OF CONCENTRATION 260 11.4.3.1 Extreme Rainfall 261 11.4.3.2 Mean Annual Precipitation 265 11.4.3.3 Peak Flow Analysis 266 11.4.3.4 Runoff Coefficients 269 11.4.3.5 Time of Concentration 276 11.4.4 TYPICAL TIME OF CONCENTRATION ESTIMATES 277 11.4.5 ESTIMATION OF THE Qioo 281 11.5 CONCLUSIONS 284 12.0 CONCLUSIONS AND RECOMMENDATIONS 286 12.1 G E N E R A L 286 12.2 MAJOR CONCLUSIONS A N D CONTRIBUTIONS 286 12.2.1 SCIENTIFIC FINDINGS 287 12.2.2 OPERATIONAL FINDINGS 292 12.3 IMPLICATIONS OF FINDINGS 294 12.3.1 SCALING LINEARITY 295 12.3.2 STORM LINEARITY 296 12.3.3 S U M M A R Y 297 12.4 RECOMMENDATIONS FOR FUTURE RESEARCH 297 13.0 REFERENCES 299 Appendices A PERIODS OF HISTORICAL RECORD FOR THE OREGON DATA 329 B SUMMARY OF SAMPLE STATISTICS FOR THE OREGON DATA 338 C REGRESSION STATISTICS 344 D SCALING OF THE L-STANDARD DEVIATION 359 E L-MOMENT RATIOS 368 F REGIONAL SCALING PATTERNS: MEAN OF ANNUAL PEAK FLOWS 385 G REGIONAL SCALING PATTERNS: L-STANDARD DEVIATION OF PEAK FLOWS 392 H REGIONAL SCALING PATTERNS: L-Cv OF PEAK FLOWS 399 I REGIONAL SCALING PATTERNS: L-Cs OF PEAK FLOWS 406 J REGIONAL SCALING PATTERNS: L-Ck OF PEAK FLOWS 413 K CARNATION CREEK DATA 420 xu L I S T O F T A B L E S Table Page 3.1 Summary of Statistical Values Used to Compare Regression Equations for Different Periods of Record 47 4.1 Regional Summary: Mean Annual Precipitation and Timing of Annual Peak Flows 65 4.2 Regression Statistics for the Mean and L-Standard Deviation of Annual Peak Flows 71 4.3 Summary of Regional Sample Sizes 72 5.1 Predominant Rainfall and Runoff Mechanisms in Humid and A r i d Regions 91 5.2 U S G S Regression Equations for Estimating Mean Annual Peak Flow 95 5.3 Summary of Factors Influencing the Value of the Exponent 102 6.1 Summary of Regional Regression Analyses for the Mean of Annual Peak Flows vs. Drainage Area (Q=c*Area b) 110 6.2 Summary of Mean Annual Peak Flow Estimates 114 6.3 Results of Testing Differences Between the Power Regression Curves for Small and Large Areas (Q=c*Area b) 126 6.4 Comparison of the Coefficient of Determination and Standard Error Values for Linear and Power Regression Curves 128 6.5 Summary of Regional Regression Analyses for the Mean of Annual Peak Flows vs. Drainage Area (Q=c+b*Area) 131 6.6 Results of Testing Differences Between the Linear Regression Curves for Small and Large Areas (Q=c+b*Area) 133 7.1 Summary of Regional Scaling Relationships for the L - C v (L-Cv = b*logA+c) 148 7.2 Regional Scaling Trends of the L - C v 149 7.3 Testing the Significance (at 5%) of the Slope of the L - C v Scaling Relationships 151 7.4 Regional Scaling Exponents for Different Probability Weighted Moments 152 8.1 Summary of Regional Scaling Relationships for the L-Cs (L-Cs = b*logA+c) 173 8.2 Regional Scaling Trends of the L-Cs 174 8.3 Comparison of Regional Scaling Trends of the L - C v and L-Cs 175 xi i i 8.4 Summary of Regional Scaling Relationships for the L - C k (L-Ck = b*logA+c) 176 8.5 Regional Scaling Trends of the L - C k 180 8.6 Comparison of Regional Scaling Trends of the L-Cs and L - C k 181 8.7 Distribution Selection: Z-test Results 182 8.8 Summary of Stations with Negative L -Cs Values 186 8.9 Comparison of Regional Scaling Trends of the L - C v and L-Cs (Al l Areas) 188 9.1 Changes in the Scaling Exponents with Increasing Return Period (from U S G S Regression Equations) 206 9.2 Number of U S G S Regions with Indicated Changes in the Exponent 205 9.3 Regional Summary of Exponent Change with Increasing Return Period 207 10.1 Distribution Parameter Estimators Used by Different Models 217 10.2 Statistics for Stations Used In Examples of Regional Regression Analysis for Gauged Sites 222 10.3 Statistics for Stations Used In Example of Regional Regression Analysis for A n Ungauged Site 229 11.1 Comparison of Hillslope and Channel Components of Tc Values 255 11.2 Summary of Tp and Peak Flow Values for Basin H and E 260 11.3 Station Summary of Annual Maximum Rainfall Statistics 262 11.4 Basin Summary of Annual Maximum Rainfall Statistics 262 11.5 Station Summary of Annual Precipitation Statistics 265 11.6 Basin Summary of Annual Precipitation Estimates 266 11.7 Summary of Carnation Creek Basin Characteristics and Peak Flow Data 267 11.8 B C M O E Runoff Coefficients (Coulson, 1991) 271 11.9 Comparison of Storm Precipitation Totals 272 11.10 Summary of Measured Total Storm Runoff Coefficients 273 11.11 Comparison of Storm Runoff Coefficients 274 11.12 Estimated Runoff Coefficients 276 11.13 Time of Concentration Estimates 278 11.14 Estimates of the Qioo by the Rational Method 282 12.1 Quantifying Basin Response Linearity 291 xiv L I S T O F F I G U R E S Figure Page 2.1 Clustering of Sample Conventional Moment Ratios Around the Population Value for an E V 1 Distribution (100 samples with n = 10, 25, 40 and 100) (Wallis, 1989) 19 2.2 Clustering of Sample L-Moment Ratios Around the Population Value for an E V 1 Distribution (100 samples with n = 10, 25, 40 and 100) (Wallis, 1989) 20 3.1 Flowchart of Data Screening Process 35 3.2 Hydrologic Regions of Oregon 37 3.3 Comparison of Regression Plots, with and without Areas > 3000 k m 2 39 3.4 Cumulative Departure of Annual Peak Instantaneous Discharge - Station 14193000 : Will iamina Creek near Will iamina 44 3.5 Cumulative Departure of Annual Peak Instantaneous Discharge - Station 14134000 : Salmon River near Government Camp 44 3.6 Moving 10 Year Average of Annual Peak Instantaneous Discharge - Station 14193000 : Will iamina Creek near Wil l iamina 46 3.7 Moving 10 Year Average of Annual Peak Instantaneous Discharge - Station 14134000 : Salmon River near Government Camp 46 3.8 Comparison of Regression Plots for Different Periods of Record - Region 6: L-Mean of Annual Peak Instantaneous Flows vs. Drainage Area 48 3.9 Comparison of Regression Plots for Different Periods of Record - Region 6: L-Standard Deviation of Annual Peak Instantaneous Flows vs. Drainage Area 48 3.10 Comparison of Regression Plots for Different Periods of Record - Region 12: L-Mean of Annual Peak Instantaneous Flows vs. Drainage Area 49 3.11 Comparison of Regression Plots for Different Periods of Record - Region 12: L-Standard Deviation of Annual Peak Instantaneous Flows vs. Drainage Area 49 3.12 Example of Discordant Stations In A Data Set 51 4.1 Physiographic Regions of Oregon 59 4.2 Relief Map of Oregon 60 4.3 Average Annual Precipitation of Oregon 61 4.4 Temperature Patterns Across Oregon 63 4.5 Influence of Largest Area Data Point in Zone A 68 4.6 Influence of Largest Area Data Point in Region 2, for Areas < 50 k m 2 68 4.7 Example of Outlier Stations Moved From One Region to Another 70 xv 4.8 Sampling Variability vs. Sample Size for the Mean 73 4.9 Sampling Variability vs. Sample Size for the L - C v 74 4.10 Sampling Variability vs. Sample Size for the L -Cs and L - C k 75 4.11 Examples of Residual Variability for Regression Plots of the Mean 77 4.12 Examples of Residual Variability for Regression Plots of the L - C v 78 4.13 Examples of Residual Variability for Regression Plots of the L-Cs 79 4.14 Comparison of Residual Scatter With and Without Scaling Consideration 81 5.1 Correlation Between Mean Annual Precipitation and U S G S Regression Equation Scaling Exponents 96 5.2 Correlation Between Mean Annual Precipitation and U S G S Regression Equation Coefficients 98 5.3 Correlation Between Mean Annual Precipitation and U S G S Regression Equation Peak Flow Values 98 5.4 Typical Scaling Patterns for Flood Quantiles in Humid and Ar id Regions (as suggested by Goodrich et al. (1997)) 100 6.1 Examples of Regional Scaling Plots for the Mean 111 6.2 Correlation Between Mean Annual Precipitation and Mean Annual Flood Scaling Exponents 113 6.3 Correlation Between Mean Annual Precipitation and Mean Annual Flood Scaling Coefficients 113 6.4 Correlation Between Mean Annual Precipitation and Estimated Peak Flow for Oregon 115 6.5 Regions 3: Mean of Annual Peak Instantaneous Flows vs. Drainage Area 117 6.6 Regions 12: Mean of Annual Peak Instantaneous Flows vs. Drainage Area 117 6.7 Regions 8: Mean of Annual Peak Instantaneous Flows vs. Drainage Area 119 6.8 Regions 9: Mean of Annual Peak Instantaneous Flows vs. Drainage Area 119 6.9 Comparison of Regional Regression Curves for the Mean of Annual Flood Series 120 6.10 Zone A : Mean of Annual Peak Instantaneous Flows vs. Drainage Area 122 6.11 Typical Scaling Patterns for the Mean Annual Flood in Oregon 123 6.12 Empirical Examples of Scaling Patterns for the Mean Annual Flood 123 6.13 Comparison Between Linear and Power Regression Curves for Region 1 129 6.14 Comparison Between Linear and Power Regression Curves for Region 8 130 6.15 Comparison of Linear Regional Regression Curves for Areas < 50 k m 2 132 6.16 Comparison of Linear Regional Regression Curves for Areas > 50 k m 2 132 xvi 7.1 Characteristics of Simple Scaling 139 7.2 Characteristics of Multi-Scaling 140 7.3 L - C v Scatter Plot for Flood Data used by Smith (1992) 144 7.4 L - C v Scatter Plot for A l l Stations in Oregon 144 7.5 Multi-Scaling Pattern Detected by Smith (from Gupta et al., 1994) 145 7.6 Examples of Regional Scaling Plots of the L - C v of Annual Flood Series 150 7.7 Example of the V-Shaped Scaling Pattern of the L - C v 153 7.8 Example of Decreasing L - C v for Both Small and Large Basins 153 7.9 Probability Weighted Moments: Variation of the Scaling Exponent for A l l Regions 154 7.10 Comparison of Average Regional Regression Curves for the L - C v 158 7.11 Example of Comparative Scaling Patterns for the L - C v and Cv of Peak Flows 161 7.12 Probability Weighted Moments: Examples of Simple Scaling 162 7.13 Probability Weighted Moments: Examples of Multi-Scaling 162 8.1 Examples of Regional Scaling Plots of the L-Cs of Annual Flood Series 172 8.2 Relationship Between Regional Average L - C v and L-Cs 177 8.3 Comparison of Regional Scaling Relationships for the L - C v and L-Cs 177 8.4 Examples of Regional Scaling Plots of the L - C k of Annual Flood Series 178 8.5 L-Moment Ratio Diagram 184 8.6 Comparison of Average Regional Regression Curves for the L -Cs 184 8.7 Comparison of L-Cs Scaling Relationships for Regions 4 and 5 189 8.8 Skewness Separation Effect (from Dawdy and Gupta, 1995) 189 8.9 Comparison of Average Regional Regression Curves for the L - C k 192 8.10 Effect of Distribution Selection on Flood Quantiles 195 9.1 Relationship Between Scaling Linearity and Storm Linearity 200 9.2 Example of Decreasing L - C v and Scaling Exponent 211 9.3 Example of Increasing L - C v and Scaling Exponent 212 9.4 Example of Increasing and Decreasing L - C v and Scaling Exponent 213 10.1 Regional Frequency Analysis Shape Model for Gauged Sites 220 10.2 Mean for Regional Regression Analysis Example: Station 14134000 224 10.3 L-St. Dev. for Regional Regression Analysis Example: Station 14134000 225 10.4 Statistics for Regional Regression Analysis Example: Station 14303200 227 xvi i 10.5 Regional Frequency Analysis Shape Model for Ungauged Sites 230 10.6 Regional Regression Analysis for Ungauged Site: Bow Creek 231 10.7 Influence of Sample Size on the Accuracy of 100 Year Flood Estimates 233 10.8 Potential Improvements to Flood Estimation Accuracy for Regional Models 233 11.1 Different Measures for the Time of Concentration 241 11.2 B C M O E T i m e o f Concentration Relationships (Coulson, 1991) 244 11.3 Carnation Creek Station Locations and M A P Isohyets 247 11.4 Measurement of Time of Rise 247 11.5 Example of Runoff Coefficient Determination 252 11.6 Comparison of Measured Time of Rise Value 254 11.7 Relationship Between Time of Rise and Peak Flow Intensity 257 11.8 Relationship Between Peak Flow and Rainfall Intensity 258 11.9 Intensity-Duration-Frequency Curves for Carnation Creek Basin B 263 11.10 Comparison of Mean Annual Intensity-Duration Curves 264 11.11 Comparison of 100-Year Intensity-Duration Curves 264 11.12 Comparison of Mean Annual Peak Flows for Oregon State (wettest regions) and Carnation Creek 268 11.13 Scaling Relationship for Estimates of the 100-Year Peak Flow 270 11.14 Runoff Coefficient vs. Peak Flow Intensity 275 11.15 Comparison of Different Time of Concentration Estimates 279 11.16 Basin H : Comparisons of Q100 Estimates Calculated with Different Tc 283 11.17 Basin B : Comparisons of Q100 Estimates Calculated with Different Tc 283 12.1 Typical Scaling Patterns for the Mean and their Causative Physical Mechanisms 288 12.2 V-Shaped Scaling Pattern for the L - C v with Causative Physical Mechanisms 290 xvi i i N O M E N C L A T U R E A N D A C R O N Y M S Population mean X Sample mean Population variance r, Population skewness Sample skewness Ji Population kurtosis 82 Sample kurtosis Pr Probability Moment of order r Pr Sample Probability Moment of order r K L-moment of order r K Sample L-moment of order r ?r L-moment ration of order r K Sample L-moment ratio of order r Vector containing the sample L-moment ratios for a site u Group average of u;. <*>, Regression coefficient of order I f*v Mean of N simulated values of V °\ Standard deviation of N simulated values of V t{,) Sample L - C v for site I .R Regional average L - C v , weighted proportionally for record length AO Sample L-Cs for site I ,R Regional average L-Cs , weighted proportionally for record length Ai) l4 Sample L - C k for site I tR M Regional average L - C k , weighted proportionally for record length r t h shifted Legendre polynomial DIST L - C k of a fitted distribution rR 4 Weighted average regional L - C k B4 Bias correction for T 4 Standard deviation of Rf Transformed coefficient of determination for exponential equations X Dummy variable ( Q T ) N Normalized T-year peak flow xix A Basin area a Regression coefficient A E S Atmospheric Environment Services b, m Scaling exponent B C British Columbia B C M O E L P B C Ministry of the Environment, Lands and Parks C Runoff coefficient C D F Cumulative distribution function C F A Consolidated frequency analysis Ck Coefficient of kurtosis Cs Coefficient of skewness c u Unbiased regression coefficient Cv Coefficient of variation d Duration of unit excess rainfall D i Discordancy measure E[.] Mathematical expectation F(x) Cumulative distribution function of x F P C B C Forest Practices Code of B C G E V Generalized Extreme Value distribution G L Generalized Logistic distribution G N Generalized Normal distribution G P Generalized Pareto distribution H Homogeneity statistic H-test Homogeneity test I Rainfall intensity I 2 Untransformed coefficient of determination for exponential equations IDF Intensity-duration-frequency k Regional scaling coefficient K Scaling factor K - S Kolmogorov-Smirnov L . Lag time L - C k Linear moment coefficient of kurtosis L-Cs Linear moment coefficient of skewness L - C v Linear moment coefficient of variation L-moment Linear moment xx Ls Total stream length L-stdev Linear moment standard deviation N Number of sites in a group (sample size) n Roughness coefficient n; record length for site i P3 Log Pearson Type UJ distribution P M F Probable maximum flood pdf probability density function P M P Probable maximum precipitation P W M Probability weighted moment Q Peak flow Q(A) Peak flow as a function of area Qmean Mean annual peak flow Q p Mean annual peak flow Q T Peak flow corresponding to return period T r Order of a series of values R Correlation coefficient R 2 Coefficient of determination R M Rational Method R M S E Root mean square error S Average slope of total stream length s 2 Sample variance SCS Soil Conservation Service S e Standard error of estimate S e u Untransformed standard error of estimate for exponential equations stdev Standard deviation s y p Standard error of prediction T Return period t Rainfall duration Tc Time of concentration Tr Time of rise U B C University of British Columbia U H Unit hydrograph U S United States U S G S United States Geological Survey xxi V Subscript denoting the transposition of a vector or matrix V Weighted measure of between site dispersion of sample L-moment ratios wsc Water Survey of Canada X Independent variable individual record X i cumulative departure y Dependent variable z Matrix of sums of squares and cross-products of uj and u Z goodness-of-fit statistic xxi i A C K N O W L E D G E M E N T S A great many people and organizations have contributed to this project and I am indebted to them all. In particular, I would like to thank: Dr. Younes A l i l a , for sharing his great technical knowledge and for pouring his heart and soul into this project. His enthusiasm and energy are limitless, and I am fortunate to have ridden on his wake. I cherish our friendship and wi l l remember fondly our many technical discussions and debates. Dr. Les Lavkulich, for helping me navigate the minefield of academia and for welcoming me into his interdisciplinary program in Resource Management and Environmental Studies. Dr. S.O. (Denis) Russell, for his guidance and for making my M . A . S c . experience so pleasant that I was encouraged to pursue a Ph.D. The principals and employees of Knight Piesold Ltd., for supporting my decision to return to academia, for accommodating my limited and sporadic work schedule, for providing me with free access to all their resources, for covering my tuition expenses, and for keeping a cheery outlook despite multiple extensions to my projected completion date. The Natural Sciences and Engineering Research Council (NSERC) of Canada, for providing a Post Graduate Scholarship for two years of study. The University of British Columbia, for providing a University Graduate Fellowship award for one year of study. M y family, for their undying support. M y parents, David and Diana, who taught me to reach up and take a chance in life; my wonderful wife, Mamie, whose unwavering love and support magnified the good times and diminished the tough times; and my precious daughter Sara, whose birth and growth during the course of this study brought me new perspective and great joy. xxin P R E A M B L E Over the past ten years, in my role as a practicing engineering hydrologist, I have been required to develop flood estimates for a large variety of conditions in various parts of the world. Every case is different, with the key variables being watershed size, climate aridity and basin physiography. Yet, in almost every situation, the recommended approaches are essentially the same, with little or no recognition of the differences amongst the physical processes that are responsible for the largest flows in each watershed. A primary example of this situation is the almost universal acceptance and application of the index-flood method (Dalrymple, 1960). This approach, which assumes constancy of the shape and dispersion parameters of all common flood distributions, is liberally applied regardless of whether this assumption of constancy is physically plausible. Similarly, the standard unit hydrograph approach and the Rational Method, which both assume high degrees of linearity in watershed response, are commonly applied in both arid and humid watersheds, despite the different response patterns characteristic of the different climates. Other examples are the popular use of the Gumbel or Extreme Value Type I distribution for modelling observed flood data, and the selection of the Kirpich equation as the favoured means of estimating time of concentration values. The Gumbel distribution, which has a fixed shape parameter that dictates the thickness of the upper tail of the distribution, is commonly applied because of its simplicity, regardless of whether or not it reasonably represents the distribution of the observed flows. This practice is often defended with the reasoning that "It's the one everyone uses." This rationale is also frequently applied to the selection of a time of concentration (Tc) value, which is typically required for rainfall-runoff modelling. Hundreds of different Tc equations have been developed over the years, largely on the basis of data from small agricultural or experimental plots, but the equations are regularly applied without appreciation of their limitations, or recognition of the probable physical mechanisms that produced each measured response pattern. The "standard approach" line of reasoning has some merit, in that a level of comfort is offered to the practitioner who applies the standard of the day, but it has little technical value and it stifles the development and understanding of the science. If this approach was adopted in all disciplines, we would still be sending messages by carrier pigeon. Fortunately, however, mankind is rarely satisfied with the status quo. Every time I generated a new set of flood values, I found myself struggling with the lack of validation applied to the use of the various approaches and frustrated with my own ignorance about the different physical processes that contribute to flood flows. I wanted to know what conditions made each approach more or less applicable in each situation? Most modelling approaches assume that watersheds are linear systems, and therefore the degree of model applicability is largely dictated by the validity of this assumption. A large amount of information on watershed linearity is available in the literature, but it is xxiv largely qualitative in nature, as few researchers have attempted to quantify or delineate variations in linearity and link these patterns to the hydrologic processes that are associated with different climates and physiographic conditions. This lack of information provided the motivation to embark on the long journey that has culminated with the penning of this dissertation. xxv Chapter 1 INTRODUCTION 1.1 H I S T O R I C A L B A C K G R O U N D Hydrology is commonly known as the "science of water" (Viessman and Lewis, 1995). It is the field of study that focuses on the properties of water and its occurrence, distribution and movement on the earth. Like most natural sciences, it is based on the postulation that nature is rational. That is, "nature obeys well ordered rules which are understandable by human intelligence and conform to models that man constructs upon the basis of his intuition." (Davis, 1931) Man has been able to derive mathematical equations to explain natural phenomena, and to a remarkable degree, at the first order level, there has been accord between the expected and the observed. Central to the field of hydrology is the concept of a hydrologic cycle, the endless circulation of water between the earth and its atmosphere. Many ancient philosophers and medieval scholars speculated about the circulation of water, and offered many different explanations. The theory slowly evolved until about the beginning of the first millennium, when the Roman engineer Marcus Vitruvius combined the evaporation and rainfall ideas of the Greek philosophers Clazomenae and Theophrastus, with his concepts of infiltration and groundwater, to form the basis of the modern version of the cycle (Chow et al., 1988). These ideas were purely philosophical, and it was not until the Renaissance period, with the works of Leonardo da Vinci and his contemporaries, that there was a shift towards observational science. Scientists began recording observations of rain, snow and streamflow, and offered explanations for the relationships amongst the various components of the hydrologic cycle. During the 18 t h and 19 t h centuries, significant advances were made in hydraulic measurement and experimentation, resulting in the development of many hydraulic principles that are still in use today. For instance, the well known Bernoulli, Chezy, Darcy and Mannings equations were all developed during this time. In contrast, hydrologic concepts remained primitive and essentially undeveloped. Systematic observations of precipitation and streamflow, on a 1 reasonably large organized scale, did not generally begin until the middle to late 18l century (Linsley et al., 1982). Consequently, little quantitative work in hydrology was done until the early 1900's, when a reasonable database had developed and interest in flood control and irrigation provided the impetus for organized quantitative research. One topic that began to attract great attention at this time was the estimation of peak flows. Under the force of gravity, water flows downhill, and in the process, potential energy is converted into kinetic energy. In some instances, this energy has been successfully harnessed by man and used to his advantage, but more frequently, it has caused considerable damage. The extent of damage is directly associated with flow rate, which is a measure of the volume of water flowing per unit of time. Subsequently, if the magnitude of peak flows can be estimated or predicted, then steps can be taken to accommodate these flows and minimize the potential for damage. Engineers, through the nature of their practice, are typically faced with the task and responsibility of estimating or predicting the magnitude of expected flows. These estimates involve the concepts of risk and probability of occurrence, and are known as design flows. During the early part of the 20 t h century, design flow estimation techniques were largely limited to empirical methods such as the Rational Method, the Burkli-Ziegler equation and other simple rainfall-runoff models. In the 1930's, great advances were made in deterministic modelling, when Sherman (1932) introduced unit hydrograph theory. This approach is still popular today and unit hydrographs often serve as the fundamental units of sophisticated models. It is not clear in the literature when statistical approaches were first introduced in hydrology, although there is general agreement that the first applications were largely in the area of flood flow estimation (Haan, 1977). Considerable gains were made in the 1950's and 1960's, when procedures for plotting historical peak flow data and fitting parametric distributions to data sets were refined and largely automated with computer application. Surprisingly, despite incredible recent advances in computing power and an ever increasing historical peak flow database, there have been few significant improvements made to peak flow estimation techniques in the past 30 years. This situation can be attributed to a number of factors, including that established techniques are relatively simple and reasonably well understood; that when applied with typical published model parameter values they generally produce conservative (high) results; that they can produce reasonable results in the hands of experienced users; and that there is considerable room for estimation error given the randomness of flood occurrences. However, despite the relative successes of current models, there is still much room for improvement, and it is offered that big gains can only be achieved through a greater understanding and recognition of the complexity (re: non-linearity) of the physical processes that produce floods in natural systems. New sophisticated techniques designed to provide better statistical fits of modelled to observed data are continuously evolving, yet this effort is largely misdirected as the underlying assumptions of many modelling approaches do not have a strong basis in the physical world. This situation was succinctly summarized by Klemes (1971), when he stated that ... 2 " . . . the main emphasis in stochastic analysis of hydrologic processes, which basically is the domain of pure hydrology, has been on the fitting of various preconceived mathematical models to empirical data rather than arriving at a proper model from the physical nature of the process itself. The empirical data representing a hydrologic event are treated as a collection of abstract values that could pertain to anything or to nothing at all. Their hydrologic flavour, the physical substance that makes, for instance, a precipitation record an entity entirely distinct from, say, a record of stock fluctuations, is not reflected in the analysis. Thus, what we usually find is not, in fact, statistical or stochastic hydrology, but merely an illustration of statistical and probabilistic concepts by means of hydrologic data. Such an approach can hardly contribute to the hydrological knowledge." Klemes made this comment more than 25 years ago, but the situation today is little different. At a recent meeting of the American Geophysical Union in San Francisco ( A G U , 2000), the main thrust of many research presentations was the offering of new and sophisticated statistical procedures for improving the index flood approach, such as those by Sveinsson et al. (2000) and Niggli et al. (2000), but there was little recognition of the fact that the approach is largely invalid due to the scaling behaviour of flood statistics. A similar lack of understanding of the physical processes is also common to most deterministic models, as was noted by Yevjevich (1974). Deterministic models, which are purported to consider the physical mechanisms that interact to produce flows, typically oversimplify the complex interactions of climatic and hydrologic factors. The oversimplification of natural processes, such as the common assumption of a linear basin response, does not recognize the large number of factors that affect the behaviours of natural hydrologic systems, and consequently many models are unable to satisfactorily reproduce observed hydrologic patterns. 1.2 O B J E C T I V E S The primary objective of this thesis is to explore and gain a greater understanding of the linearity of peak flow response and the manner in which it is influenced by climatic and physiographic factors. This linearity refers to both scaling linearity, whereby hydrologic response differs according to spatial scale, and storm linearity, whereby hydrologic patterns are dictated by storm severity. The intent is to use statistical methods to reveal the extent of physio-climatic influences on flood patterns. The work represents the first application of a regional flood frequency model as an exploratory tool in this regard. It is believed that the combined use of L-moment statistics, a large regional data base covering a range of physio-climatic zones, and the adoption of scaling homogeneity as the definition of homogeneity, is unique, and that this particular combination of methods wil l reveal new insights into the non-linearity of hydrologic response. It is hoped that the results of the research presented in this thesis wil l contribute to a better understanding of hydrologic processes, and in turn provide a basis for assessing the validity of linearity assumptions common to many operational models as well as the evolving field of random cascade modelling theories. 3 Specific objectives directed towards this primary goal are outlined below and categorized as either scientific or operational. It is recognized that there is considerable overlap between these two categories, but the distinction is made to help the reader appreciate the scientific contributions of this thesis 1.2.1 S C I E N T I F I C O B J E C T I V E S (i) To investigate the extent to which hydrologically homogeneous regions can be delineated, whereby homogeneity is achieved if variations in flood statistics are solely explained by variations in spatial scale. (ii) To examine the relative effects of scale, climate and physiography on the mean annual peak flow, and to compare the results to the patterns suggested in the literature. (iii) To explore the effects of scale, climate and physiography on the L-coefficient of variation of peak flows, and to examine the results in the context of simple and multi-scaling theories. (iv) To analyze the influences of scale, climate and physiography on the L-skewness and L-kurtosis of peak flows, and to explain the observed behaviour in these terms. (v) To use flood statistics to reveal the degrees to which both scaling and storm linearity are influenced by changes in storm severity at different scales and under different climatic and physiographic conditions. (vi) To develop a framework for quantifying both relative and absolute degrees of basin response linearity. 1.2.2 O P E R A T I O N A L O B J E C T I V E S (i) To develop a regional model for estimating peak flows at gauged and ungauged sites, for an area that is similar to British Columbia in both climate and physiography. (ii) To determine the degree to which hydrologic information can be simply extrapolated from medium and large basins to small basins, and to what extent restrictions apply under different climatic and physiographic conditions. (iii) To assess the validity of using the index flood method for estimating peak flows. In particular, the investigation will focus on whether or not the assumed constancy of distributional parameters is valid, and if so, under what conditions. 4 (iv) To investigate the appropriateness of using the Rational Method to estimate peak flows for small, humid, mountainous basins, and to explore the link between basin response time (time of concentration) and the scaling linearity of peak flows. (v) To explore the conceptual legitimacy of using a single regionally representative flood distribution type, and to assess the implications of distribution selection on flood quantile values. 1.3 T H E S I S O V E R V I E W As stated, the primary purpose of this thesis is to explore and gain a greater understanding of the linearity of watershed response with respect to peak flows, and the manner in which it is influenced by climate and physiography. The study follows the unique approach of detecting linearity through the patterns and variations of flood statistics that are commonly utilized in applying regional flood frequency analysis techniques. The reader will note throughout the text that inferences are made on the basis of scatter plot trends that in many instances are not statistically significant. When viewing these plots, the reader is urged to put aside concerns about statistical significance and instead focus on the possible hydrological and operational significances of the patterns. This task may prove difficult for those readers with strong analytical schooling, but it does have considerable merit, as conveyed by Klemes (1971). " . . . Short of large samples of data, the statistician washes his hands of responsibility in evaluating alternatives, considering most differences insignificant in the light of the actual information content of the data. But the problem is that a statistically insignificant difference in a parameter, say the coefficient of skewness, may lead to a difference of the order of millions of dollars in reservoir cost, which no engineer can treat as an insignificant one. Not being able to base his decision for a particular alternative on statistical evidence, the engineer has no choice but to select one which is most plausible from the hydrological (physical) viewpoint, even if the hydrologic information is only of a qualitative nature. Here we come to the point where the physical justifiability of a particular concept may play a significant role." Another point that should be noted by the reader is that flood data for British Columbia (BC) would have been utilized for this research if possible, as the author is most interested in understanding the behaviour of floods in B C . However, there is not sufficient small basin historical flood record in B C to permit meaningful analysis over a full range of scales, so data for the state of Oregon, which has similar climate and physiography to B C , were selected. It is recognized that there are a number of differences between the two areas, such as their geologic histories and associated soil types and depths, but it is believed that the information derived from the Oregon data will offer valuable insights into the behaviour of floods in BC. The text of this thesis is set out in the manner outlined below. The text does not follow the conventional format of presenting a separate chapter titled "Literature Review," but rather is 5 written essentially in journal paper format, with a small separate literature review covering each different topic, as required. It is believed that this approach better serves the reader by highlighting pertinent information. Chapter 1, Introduction, begins with a brief historical overview of the development of the hydrological sciences, with particular emphasis on the evolution of peak flow estimation techniques, and concluding with a view towards the primary weakness of current modelling efforts. The motivating factors for undertaking this project are then discussed, and the primary objectives of the thesis are listed. This is followed by an overview discussion that includes a brief description of each chapter. Chapter 2, Review of Popular Statistical Methods, presents a general review of statistical peak flow estimating techniques, plus an assessment of the statistical modelling approaches currently popular in British Columbia. It should be noted that some of the topics discussed in this chapter are revisited later in the thesis, but while Chapter 2 presents a general overview, later discussions provide greater detail on topics particularly relevant to their respective chapters. Readers who are particularly familiar with regional frequency analysis techniques may wish to skip most of Chapter 2 and concentrate on the final sections that describe the regional models particular to B C . Chapter 3, Data Selection and Screening, describes the reasoning behind the selection of USGS Oregon State flow records as the primary data set used for this research work. In addition, the extensive data screening process used in this study is described in detail. Chapter 4, Region Delineation, outlines the process used to delineate Oregon State into 12 hydrologically homogeneous regions. This process involved the consideration of numerous physical factors, such as climate, physiography and station density, in addition to the constraint that the regions contain a sufficient number of stations, which cover a broad enough range in drainage sizes, so as to facilitate an investigation into scaling patterns. The delineated regions are then assessed through a comparison with the results of a Monte Carlo type model to determine the extent to which homogeneity is achieved. The chapter concludes with a summary of the key characteristics of each region. Chapter 5, Investigating the Scaling of Mean Annual Peak Flows, explores and summarizes the scaling patterns of the mean that are offered in the literature. A discussion of general scaling patterns is presented and the results of specific studies are reviewed, with particular attention given to the patterns evidenced by a large set of regression equations that were developed by the USGS for a number of states in the US. Chapter 6, Oregon: Scaling of the Mean, presents regional scaling relations for the mean of annual peak flows for Oregon data, and discusses the factors that are likely responsible for the observed scaling patterns. Scaling relations for all basin sizes are described initially, and then small and large basins are considered separately. Furthermore, the validity of extrapolating scaling plots for large basins to small basins is assessed statistically and practically. 6 Chapter 7, Scaling of the L-Coefficient of Variation, explores the scaling behaviour of the L-Cv of annual peak flows in Oregon and relates the observed patterns to climatic and physiographic influences and flood flow response mechanisms. In addition, the findings are compared to scaling patterns noted in the literature, with particular reference made to the triangular shaped multi-scaling pattern proposed by Smith (1992) and Gupta et al. (1994). As well, the validity of assuming regional constancy in the L-Cv, which is required by the index flood approach for flood estimation, is investigated. Finally, the question of whether inferences can be made about the scaling behaviour of the L-Cv for small catchments, on the basis of the scaling behaviour displayed by flows from medium and large catchments, is addressed. Chapter 8, Scaling of the L-Skewness and L-Kurtosis, presents regional scaling relations for the L-Cs and L-Ck, and discusses the factors that appear to influence these relations. In particular, an effort is made to assess how these higher order L-moment statistics are affected by scale and/or the climatic and physiographic conditions attributed to a particular region. As well, the validity of assuming regional constancy in the L-Cs and/or the L-Ck is investigated, and the question of whether inferences can be made about these statistics for small catchments on the basis of their values for medium and large catchments, is addressed. In addition, the use of these statistics to select a distribution type, through the application of a formalized statistical test, is discussed, along with an assessment of how distribution selection can affect flood estimates. Chapter 9, Scaling Exponent Versus Return Period, considers the relationship between storm severity and scaling linearity and ties the observed behaviour to the climatic and physiographic conditions characteristic of particular regions. The investigation initially considers the patterns revealed by a large set of USGS peak flow regression equations that were developed for much of the US, and then focuses on the analysis of Oregon State data. Chapter 10, A Regional Frequency Model for Peak Flow Estimation in Oregon, presents the details of a regional frequency model for estimating peak flows in Oregon. This model represents the culmination and combination of all of the information contained in Chapters 3 through 10. It can be applied to estimate peak flows at both gauged and ungauged sites, and examples for each situation are provided. In addition, the details of a Monte Carlo type simulation study are presented to demonstrate the potential advantages of using a regional model, rather than a single-site approach, at gauged locations. Chapter 11, Linearity Implications on the Time of Concentration and the Rational Method, demonstrates the connection between the scaling pattern of peak flows and the corresponding scaling behaviour of time of concentration (Tc) values. This chapter examines the flood response patterns demonstrated by a series of five nested basins that comprise a very humid forested watershed, and presents an overview of the Rational Method (RM) and its applicability for estimating peak flows in steep, humid catchments. Measures of Tc for five basins are determined and the implications of using these measures to estimate peak flows with the R M are discussed. These results are then compared with 7 Tc and peak flow values estimated with some of the more popular Tc equations and the R M . Chapter 12, Conclusions and Recommendations, summarizes the major findings of this research and offers recommendations for the application of these findings. In addition, the limitations of this research are discussed and possible directions for further related research studies are suggested. Finally, the key implications of the findings are discussed in the broad context of hydrologic research and applications. Chapter 13, References, presents a list of the publications that were used as references for the work presented in this thesis. 8 Chapter 2 REVIEW OF POPULAR STATISTICAL METHODS FOR PEAK FLOW ESTIMATION 2.1 I N T R O D U C T I O N Statistical methods are applied in hydrology in an effort to account for the randomness of climatic variables and streamflow values. In general, hydrologic models involve stochastic components to either generate random input variables (i.e. rainfall) or analyze random output variables (i.e. flows). The generation of random input variables typically involves a random number generator, such as a Monte Carlo type simulator, whereas the analysis of random output variables typically involves the use of frequency analysis techniques. For the sake of brevity, and because input variable generation is very model specific, this discussion is limited to flood frequency analysis techniques. Flood frequency analysis is a procedure for estimating how often a specific flood event will occur, or how large a flood wil l be for a particular probability of exceedence or recurrence interval. This flood, with its associated probability of exceedence, is known as a quantile. The procedure typically involves the estimation of distributional parameters and the extrapolation of cumulative distribution functions to generate extreme flood values. The procedure is performed on a single site or regional basis, depending largely on the availability of data. In general terms, single site analysis is performed when a particular basin has sufficient historic peak flow data to adequately represent the flood population and permit the generation of realistic flood estimates. Regional analysis, on the other hand, is typically performed when there are little or no historic flow data for a particular site, but data are available for other local basins with similar hydrologic characteristics. Regardless of which approach is used, frequency analysis assumes that the flood observations are random, independent and identically distributed. These assumptions are largely untested in most 9 instances, even when traditional nonparametric statistical tests (Pilon and Harvey, 1994) are applied, because of the large sampling variability associated with the small sample sizes typical of available flood records. 2.2 S I N G L E S ITE F R E Q U E N C Y A N A L Y S I S Single site flood frequency analysis involves the fitting of a mathematical frequency distribution to a set of historic flow data. This process consists of two key steps: (1) the selection of an appropriate distribution type and (2) the estimation of the distribution's parameters on the basis of the available data. 2.2.1 D I S T R I B U T I O N S E L E C T I O N There is no definitive means of selecting the most appropriate distribution for a sample of flood events. Since the inception of flood frequency analysis, there has been great debate on the topic of distribution selection. Numerous theoretical curves have been fitted to flood data, dating back to the work of Horton (1913), who first used the normal distribution to describe the behaviour of floods. It was soon recognized, however, that annual flood series are often skewed, which led to use of the lognormal distribution (Hazen, 1914a). The development and popularity of many other skewed distributions then followed, with the most commonly applied distributions now being the Gumbel, the Generalized Extreme Value, the Log Pearson Type 111 and the 3 Parameter Lognormal (Bobee et al., 1993; Kite, 1978; Pilon and Harvey, 1994; Watt et al, 1989). The proponents of each distribution have been able to show some degree of confirmation for their particular distribution by comparing theoretical results and measured values. However, there is no theoretical basis for justifying the use of one specific distribution for modelling flood data (Bobee et al, 1993), and long-term flood records show no justification for the adoption of a single type of distribution (Benson, 1962a). Large studies on distribution selection have been completed in some countries, most notably in the US (USWRC, 1967) and Britain (NERC, 1975; Institute of Hydrology, 1999), resulting in the general adoption of specific distributions. In Canada, however, such a study has never been completed, and the selection of a distribution has not been given governmental direction. Rather, distribution selection is left to the individual, although the EV1 distribution is somewhat of a practical standard in this country. Its popularity is a result of its simplicity and ease of application, as well as the considerable exposure and demonstration it receives through the Rainfall Frequency Atlas for Canada (Hogg and Carr, 1985). This atlas adopted the EV1 distribution as the standard for extreme rainfall analysis in Canada. The suitability of a candidate distribution should be evaluated based on the distribution's ability to reproduce features of the data that are of particular importance in modelling. This statement seems simple and straightforward, but in practice, it is very difficult to assess 10 relative and absolute performance of different distributions. Many factors should be considered in distribution selection, including those discussed in the following sections. 2.2.1.1 Theoret ical L imi ta t ions of a Dist r ibut ion Some distributions are bound at their upper or lower tails, and these bounds may restrict a distribution's ability to model peak flows. Flows are necessarily bound at zero, so there is some theoretical argument to discount distributions that are unbound at the lower tail (i.e. the normal distribution). In practice, however, if interest is focussed on the upper tail of a distribution, the shape of the lower tail is not particularly relevant. The shape of the upper tail is of much greater significance. Some distributions have an upper bound, which can be interpreted as representing a basin's physical limitations for generating flood flows. However, distributions that are unbound in the direction of extreme events are favoured for modelling, since "the requirement that a distribution have a physically realistic upper bound may compromise the accuracy of quantile estimates at the return periods that are of real interest"(Hosking and Wallis, 1997). 2.2.1.2 Shape of a Dis t r ibut ion The basic shape of a distribution, which is largely reflected in the coefficient of skewness value, can be used to help determine how applicable a distribution may be in a particular situation. Most flood populations are heavily skewed in a positive direction. It follows, therefore, that they can best be modelled with theoretical distributions that have the ability to assume a similarly skewed form. This requirement disqualifies any symmetrical distributions, such as the Normal distribution. It also limits the applicability of distributions that have fixed skewness values, such as the E V Type I distribution. This distribution has a constant skewness coefficient of 1.14 (L-Cv = 0.1699), so unless a data sample has a skewness value of similar magnitude, there is little justification to adopt the E V Type I distribution rather than a more flexible distribution. 2.2.1.3 F lood M ix tu re Most commonly used probability distributions are shaped with a single peak supported by smooth rising and falling limbs. In some instances, peak flows at a site may be the result of two or more mechanisms (i.e. snowmelt and rainfall), or may be due to two or more distinct climate patterns (i.e. E l Nino and La Nina) and a two-peak distribution may be required to accurately model this phenomenon, known as flood mixture (Potter, 1987; Diehl and Potter, 1987). Certain specialized distributions, such as the non-parametric distribution (Adamowski, 1985) and the two-component extreme value distribution (Rossi et al, 1984), are particularly well suited for this application. Another approach is to separate the flood events into two categories based on time of occurrence, or years of occurrence, and fit a distribution to each series. The product of the two fitted cumulative distribution functions then serves as an estimate of the annual flood distribution (Waylen and Woo, 1982, 1983; Jarrett and Costa, 1982). From a practical standpoint, it remains to be shown whether the 11 recognition and consideration of flood mixture results in any significant benefit to flood quantile estimation (Diehl and Potter, 1987). Preliminary single-site studies (Mtiraoui, 1999) indicate that ignoring flood mixture typically results in the overestimation of quantiles for large return periods (i.e. > 50 years). However, in some instances, ignoring mixture may result in the underestimation of extreme events (Waylen and Woo, 1982). Little is known about the conditions that lead to these two different results, the magnitude of these effects, or the significance of flood mixture in regional models. These and other flood mixture concerns are being investigated with research that is currently in progress (Mtiraoui, 1999). 2.2.1.4 "Goodness-o f -F i t " Tests Numerous goodness-of-fit tests have been developed, and although none generates strongly conclusive results, they provide a measure of the coherence of distributions with actual data. A l l approaches involve some subjectivity and are generally limited to the range of observed data. One of the simplest tests is a graphical technique described by Dalrymple (1960). This test involves the combination of various plotting positions and probability papers, and the observed data are plotted and compared to the theoretical form of the distribution. Different probability papers and plotting positions are associated with different distributions, and a visual assessment is made to determine how well a data set follows a particular distribution. Most goodness-of-fit tests are directly computational and do not include a graphical component. The most common tests, which are often offered in statistical software packages, such as BestFit (Palisade, 1994), are the chi-square, Anderson-Darling and Kolmogorov-Smirnov tests. A l l three tests involve the development of a test statistic that measures discrepancies between sample points and the concurrent probability density function (pdf) of a particular distribution. The statistics are compared to standardized statistics, which vary according to levels of significance, and the results of these comparisons indicate whether the hypothesized pdf "fits" or not. Generally, the Anderson-Darling test is the most powerful test (Stephens, 1986), followed by the Kolmogorov-Smirnov test, and then the chi-square test. Caution should be exercised when using these tests as they are all subject to various limitations. For instance, the Kolmogorov test considers each data point separately rather than collectively, so the test may reject the fit of the hypothesized distribution if one data point is a poor fit, regardless of how well the remaining data follow the distribution. The chi-square test, in turn, is hampered by the fact that it involves the separation of data into classes and the test results are sensitive to the number of classes selected. A major criticism of these tests is that in many cases the test statistics of a number of distributions have values close to one another. This situation is a result of the data fitting process, which forces each distribution to have so much in common that differences between them are masked, making it difficult to determine which is the best fitting distribution 12 (Cunnane, 1985). Furthermore, each test measures different aspects of the goodness of fit, so the results of the tests are not always in agreement (Onoz and Bayazit, 1995). Another type of goodness-of-fit test is the moment ratio test. The moment ratio test compares the pairings of sample estimates of moment ratios (the Cv (coefficient of variation), Cs (coefficient of skewness) and Ck (coefficient of kurtosis)), with their theoretical population counterparts, for a range of assumed distributions. The distribution with the closest match between theoretical and sample values is selected as the most likely parent distribution. This test is often criticized because of its susceptibility to sampling error, which can be significant when estimating higher order moments with small data samples (Wallis et al., 1974; Haan, 1977), and the distribution selection is sensitive to small changes in the moments. In addition, conventional sample moments are biased downwards, and the use of moment-ratio diagrams, without allowance for this bias, results in erroneous inferences (Cunnane, 1985). Much of this criticism has been quelled by the introduction of L-moment statistics (Section 2.2.2.5), in place of conventional moment statistics, and L-moment ratio tests have found extensive use in recent flood frequency research (Chowdhury et al., 1991; Pearson, 1991; Hosking and Wallis, 1993; Stedinger et al., 1993; Vogel and Fennessey, 1993; Karim and Chowdhury, 1995; and Vogel and Wilson, 1996) and are particularly useful for screening-out inappropriate distributions (Ben-Zvi and Azmon, 1997). However, one problem that has not been alleviated with the introduction of L-moments is that sample points often plot very close to the theoretical values of many distributions, making it difficult to determine the best fitting distribution (Onoz and Bayazit, 1995). Furthermore, Klemes (2000b) contends that the geometry of the weighting functions employed by L-moments steers regional L-moments towards those of the G E V distribution. However, there is no direct mathematical proof of this contention, and the results of a number of recent empirical studies suggest otherwise (Institute of Hydrology, 1999; Wang, 2000). 2.2.1.5 Common Practice The selection of a distribution type is difficult and poses a major challenge to all practitioners of the flood frequency approach. In many instances, tradition and convenience often play a significant role in the selection of a flood distribution. Typically, the directives for applying the various methods lack clarity, and faced with the large number of distributions proposed in the literature, practitioners wil l frequently use familiar methods, regardless of known deficiencies. Another common practice is to fit flood data to a variety of distributions and then to select the distribution providing the best visual fit and/or the most conservative (highest) flow estimates. This process has been greatly facilitated with the development of flood frequency software packages such as the Consolidated Frequency Analysis (CFA) program, which was produced by Environment Canada (Pilon and Harvey, 1994). A serious problem that has arisen with the proliferation of user-friendly software packages is that the mathematics of frequency analysis is largely hidden. Many users generate flood estimates without 13 understanding the mathematics and limitations of the approach and the various distributions involved. As cautioned by Reich and Renard (1981), "We may overlook the value judgments made by experienced hydrologists who recognize causative physical processes that produce flood peaks in our fascination with new statistical and computational tools." This is of great concern because considerable error in flood quantile estimation can result from the misspecification of the parent distribution, particularly for extreme events. 2.2.2 P A R A M E T E R E S T I M A T I O N Once a distribution has been selected for frequency analysis it must be fitted to the data set. Distributions are defined by a set of parameters, and numerous techniques are available for parameter estimation. These techniques are designed to minimize differences between estimated parameters and population values, which is achieved by minimizing biases and maximizing the efficiency of the estimates. Bias is a measure of the tendency to give estimates that are consistently higher or lower than the true value, while efficiency is a measure of the dispersion of estimates about their own mean. Other desirable properties of parameter estimators are robustness and unboundedness. A statistic is said to be robust when its value is unaffected by outliers in a sample, while boundedness refers to the situation where a shape index computed from a sample is restricted and cannot achieve the population value. Various methods of estimation ate available and the five most common approaches are described in the following sections, in ascending order of efficiency (Kite, 1978) from least to most efficient. In practice, the first two methods do not find much application because they produce less satisfactory results than the remaining methods. The Method of Moments is the most popular approach for fitting data samples to probability distributions, but it can produce erroneous estimates if the sample sizes are small. The Method of Maximum Likelihood is more theoretically correct, as it produces more efficient parameter estimates (i.e. less error). However, it is more mathematically complex and difficult to apply, and for some probability distributions, there are no analytical solutions. The Method of L-Moments is the newest technique and it is only starting to gain recognition in general practice, although it has found considerable use in academic circles. It should enjoy continued increased use as it is not subject to the same magnitude of errors inherent in the Method of Moments, and mathematical solutions are available for the parameters of many common distributions. 2.2.2.1 Graphical Method The graphical estimation method involves the visual fit of a theoretical distribution to a sample data set by plotting the sample data points and simply drawing the "best fit" curve. A number of points on the curve are then selected according to number of parameters to be estimated. These points are then used to develop a series of equations, which are simultaneously solved to generate estimates of the theoretical function parameters. This 14 technique is facilitated with special distribution plotting papers that produce straight line plots when the data sample fits the particular distribution (Kite, 1978). 2.2.2.2 Least Squares Method The least squares estimation method is similar to the graphical method except that it uses a mathematical approach, rather than visual approach, to fit a theoretical function to an empirical distribution (i.e. the sample data points). The numerical algorithm involved minimizes the sum of the squares of the deviations of observed points from the theoretical function. In other words, it minimizes the distances between the observed data points and the estimated population distribution curve. The minimization procedure results in a set of equations that are simultaneously solved to generate parameter values for the theoretical function. 2.2.2.3 Method of Moments The method of moments, also known as the method of conventional moments, or the method of product moments, is an approach that computes the moments of a selected distribution based on a sample data set. Population parameters are estimated by (i) equating sample moments to corresponding population moments; (ii) defining population moments in terms of population parameters; and (3) solving for the unknowns. A numerical moment is similar in concept to a physical moment in that it is essentially equivalent to a force being applied at a distance from a point. For example, Chow et al. (1988) describe the first order moment as follows: " i f the data values are each assigned a hypothetical "mass" equal to their relative frequency of occurrence (1/n) and it is imagined that this system of masses is rotated about the origin x = 0, then the first moment of each observation X J about the origin is the product of its moment arm X J and its mass 1/n, and the sum of these moments over all the data is the sample mean." I st 1 " Moment: u v = E[x] = ^  xf(x)dx => x=— ^ jxi (2.9) The first moment is equated to the mean and is a measure of central tendency; the second moment is equated to the variance and is a measure of dispersion; the third is equated to the coefficient of skewness and is a measure of symmetry; while the fourth is equated to the coefficient of kurtosis and is a measure of peakedness. 2 n d : a2=E 2 X! (x<~ *y ( x - u , ) 2 J = £ ( * - u - ) f(x)dx => s 2 = ^ l (2.10) n-\ 3 r d : y E\(x-U r i rr 3 «Efo-*)3 - f {x-v)f{x)dx g , = (2.11) 15 4 " : J2 = l = \ [ (x-n) f(x)dx ^ g 2 = ( 4 (2.12) Most of the popular distributions used in hydrology have location and scale parameters that are functions of the first two moments, respectively. Both the third and fourth moments are associated with shape parameters, but only the third moment finds common application. The accuracy of estimating moments from data samples deteriorates with decreasing sample size and increasing moment order. The only unbiased estimator is the mean, while all higher order moments are significantly biased, which can lead to significant errors in flood quantile estimates (Alila, 1994). 2.2.2.4 Method of Maximum Likelihood The method of maximum likelihood is based on the idea that the best value of a parameter of a probability distribution should be that value which maximizes the likelihood or joint probability of occurrence of the observed sample. This approach is more mathematically complex than the method of moments and it involves the development of a likelihood function, which is given as: ;=1 The likelihood function is the product of the assumed probability density functions for each sample. The partial derivative of the natural logarithm of the likelihood function, with respect to each population parameter, is determined, and then set to zero, in order to maximize the likelihood function. This process results in a series of n equations with n unknowns, which are then solved to produce estimates of the population parameters. The method of maximum likelihood is generally preferred over the method of moments, for although it also produces biased parameter estimates, it is more efficient for large sample sizes and has more desirable statistical properties (Haan, 1977). However, the method of moments is still commonly employed because in some instances, maximum likelihood estimators cannot be reduced to simple formulas and complex numerical methods are required for their solution (Stedinger et al., 1993). 2.2.2.5 Method of Linear Moments In an effort to overcome some of the deficiencies of the methods of moments and maximum likelihood, Greenwood et al. (1979) developed the method of probability weighted moments, which are defined by the equation: 16 \ R J f „ \ (2.14) Probability weighted moments have been used for estimating the parameters of probability distributions by Landwehr ET AL. (1979a, b), Hosking ET AL. (1985b), and others. However, they have not been readily adopted by the hydrologic community because they are difficult to interpret and are not easy to equate to the conventional moment statistics of mean, standard deviation, skewness and kurtosis. Hosking (1990) found that the information conveyed by these conventional statistics was carried in certain linear combinations of the probability moments, which are known as linear moment statistics or L-moments. Hosking (1990) defined the r t h L-moment of a random variable x with the equation: K = E[XP;_,{F{X\ (2.15) where, P*_x(.) is the r t h shifted Legendre polynomial and F(x) is the cumulative distribution function of x. Higher order statistics are described by L-moment ratios, which are given by the equation: x = - T. = (2.16) In terms of measures of distributional shape, \ \ is the mean of a distribution, which is a measure of location; A.2 is a measure of scale or dispersion (analogous to the standard deviation and known as the L-standard deviation); T3 is a measure of symmetry (analogous to the coefficient of skewness and known as the L-skewness or L - C s ) ; and X4 is a measure of peakedness (analogous to the coefficient of kurtosis and known as the L-kurtosis or L-Ck). Another statistical parameter that is often employed in hydrologic studies is the coefficient of variation. Its L-moment equivalent is the L-coefficient of variation or L-Cv . Estimating distributional parameters with L-moments is similar to the approach that uses conventional moments, except that it involves generating functions with linear moments rather than the more common product moments. That is, the estimators of the parameters of a population are determined by defining the population parameters in terms of the population L-moments, equating the sample L-moments to the corresponding population moments, then solving for the population parameters. For any distribution, L-moments can be calculated in terms of probability weighted moments, as demonstrated below for sample L-moments (l r): /,=P 0 (2.17) /2=2P,-|30 (2.18) / 3 = 6 p 2 - 6 p , + p 0 (2.19) 17 /4 = 20(33 -30p2 +12P.-P 0 (2.20) Correspondingly: mean (2.21) (2.22) L-stdev l2 L-Cv (2.23) L-Ck L-Cs (2.24) (2.25) Since sample estimators of L-moments are linear combinations of ranked sample observations, they provide an advantage over the standard method of product moments, which involves the squaring and cubing of observations. The squaring and cubing of values can introduce large biases and reduce the robustness of statistics when sample sizes are small, because too much weight is given to very large or very small values. The method of L-moments has been shown to generate less biased and more robust parameter estimators than the method of moments (Wallis, 1989; Royston, 1991). In fact, L-moments estimators are nearly unbiased for all sample sizes and all distributions (Vogel and Fennessey, 1993), while product moments show remarkable bias for sample sizes less than one hundred (Fischer, 1929; Wallis et al., 1974). This is particularly significant for regional flood frequency analysis because sample L-moment statistics will converge on the population value, while the bias in conventional moment statistics almost guarantees that they will not converge on the population value. This characteristic of the different moment theories is demonstrated by comparing Figures 2.1 and 2.2, which were borrowed from Wallis (1989). These figures present the results of a Monte Carlo type simulation, which generated a number of sample data sets from a single parent distribution. The L-moment statistics clearly cluster around the population values, while the conventional moment statistics demonstrate significant bias. Other advantages of L-moment statistics are that they are virtually unbound (Hosking, 1986), and that they are generally more efficient than maximum likelihood (Hosking et al., 1985b; Hosking and Wallis, 1987) and conventional moment statistics. Despite the apparent advantages of L-moment statistics, not all researchers are convinced of their merit. In particular, Klemes (2000b) vigorously attacks their use. He is concerned that L-moments artificially impose a structure upon a data set and de-emphasize the importance of observed extremes, and by doing so lead to the underestimation of extreme design events. However, he fails to mention that standard practice requires that extreme outliers receive special treatment in conventional moment analysis for the specific purpose of de-18 •11.0.1 so- EV L N = 10 •*•• 9 ;o • S 4 . 0 -S t • « 7 .0 -o * 3 . 0 - • —*+ 5 5 . 0 -2 . 0 - 3 0 -1 . 0 - • ' i " • : r . I . .• I • • I , I — i 1 .0 -- 2 . 0 - 1 . 0 0 . 0 1 . 0 2 . 0 3 . 0 Skewness - 2 . 0 v T . O o ! o 1 .0 2*0 3 . 0 Skewness 1 1 . O i 9 . 0 7 . 0 -5 5 . 0 -3 . 0 -1 . 0 EV I, N,= 40 1 1 . 0 9 . 0 : « 7 . 0 -© 5 5 .0 H 3 . 0 1.0 EV I, M = 1 0 0 —" T T -2.0 -1.0 0.0 1.0 2.0 3.0 -2.0 -1.0 0.6 1.0 2.0 3.0 Skewness * Skewness Figure 2.1 - Clustering of Sample Conventional Moment Ratios Around the Population Value for an EV1 Distribution (100 samples with n = 10,25,40 and 100) (Wallis, 1989) 19 d:6-m • in o 0.4^ _J 0.2-J 0.0--0.2-EVL N - 1 0 -0J2 0.0 0.2 0J4 " 0.6 0.8 L-Skewne3S 0.8-i 0.6-CI • © 0.4-•'. i —i 0.2 o:oj -0.2^  EV I. N = 25 • IV V i " ! ' . ! ! ' " . ! " " ! " " ! " " ! " " ) -012 0.0 0.2 0.4 0.6 0.8 L-Skewnest O.81 0.6 0:44 0.2 0.0 -0.2 EV I. N= 40 -04 o-o 04 0.4 0.6 0.8 C-Skewnes* 0.8 0.6 Sb .4 .•3.' 0:2 j 0.0^  - 0 . 2 4 ^ T T r EV I N ? 100 'I MIIIMMI.IIIIIIIHII.nl I I HI, ll|IHI| -0.2 0.0 0.2 0.4 0.6 0.8 L-Skewness Figure 2.2 - Clustering of Sample L-moment Ratios Around the Population Value for an EV1 Distribution (100 samples with n = 10,25,40 and 100) (Wallis, 1989) 20 emphasizing their importance. Klemes views L-moments to be part of the "scientific smoke screen(s)" (Klemes, 2000a) used by proponents of frequency analysis to hide their ignorance, and argues that rather than advancing the practice of flood estimation, the use of L-moments is counterproductive in that it adds to the "aura of rigor and science" (Klemes, 2000b) surrounding a practice that involves considerable assumption and approximation. While many of the points presented in Klemes' two papers are valid and deserve consideration, it should be recognized that his criticism of L-moments is based largely on his objection to the practice of frequency analysis in general, and his belief that most practitioners of this approach are not aware of its short-comings. If one accepts frequency analysis as an approach for estimating flood quantiles, then L-moments provide undeniable advantages over conventional moments. This is particularly true when considering regional trends in higher order moment statistics, as is done in this thesis. The use of L-moments permits the delineation of regional trends that otherwise might be obscured by biases and sampling variability. 2.3 R E G I O N A L F L O O D F R E Q U E N C Y A N A L Y S I S Regional flood frequency analysis employs the same basic principles and techniques as single site flood frequency analysis, but it involves the derivation of flood data on a regional, rather than a single site, basis. Sufficient information to allow for the realistic estimation of extreme flood events is seldom available at a site, and as a result, a variety of techniques have been developed for using climatic and hydrologic data from nearby and similar locations to compensate for short or non-existent site records. In theory, i f a set of N data sites can be identified in a hydrologic region, each having a similar flood frequency distribution and n years of record, then collectively, there is a possibility that there wil l be the equivalent of Nn data values for generating quantile estimates at any location within the region. In practice, however, this is never the case, because all flood records within a region are not statistically independent (Hosking and Wallis, 1988), and because the frequency distributions at each site are never identical (Hosking and Wallis, 1997). Nonetheless, it is generally advocated that a regional approach be favoured over a single site approach, even when long historic flow records are available for specific sites (Cunnane, 1985; Potter, 1987; Potter and Lettenmaier, 1990; Hosking and Wallis, 1997), because a regional approach effectively increases sample size, which in turn minimizes the effects of sampling variability (Stedinger, 1983) and results in more accurate quantile estimates. The exception to this rule is the situation where the site and regional statistics are substantially different. In this instance, the pooling of at-site and regional data may be counterproductive (Kuczera, 1983), as the noted difference may be due to significant atypical hydrometeorological differences between the region and the particular site. In general, the best approach is to use a regional model but also consider specific site records and geoclimatic information, i f available (Russell, 1982; Wall et al, 1987). 21 2.3.1 DELINEATION OF HOMOGENEOUS REGIONS A large number of different flood frequency models have been developed (Watt et al, 1989), but all of them have a common element, which is the delineation of hydrologically homogeneous regions. A hydrologically homogeneous region is generally defined as an area that exhibits a consistent pattern of hydrologic response. In terms of peak flow analysis, it typically refers to a collection of streamflow sites, each of which is assumed to have data drawn from the same frequency distribution. As hydrologic response patterns are the result of interactions between climate and physiography, regions are often delineated on the basis of associated variables. For example, many studies have utilized regional regression models and the signs and magnitudes of residuals to create regions (Wandle, 1977). In other instances, regions are simply defined by the location of watershed divides, political boundaries or land resource allocations. From a theoretical standpoint, these considerations are not always valid, but they often impose practical limits which the hydrologist must respect. Regional boundaries are also defined in terms of the similarity of flood characteristics or statistics, and this has been the focus of much research work. Dalrymple (1960) describes a homogeneity test which compares the variations of calculated ratios of the 10 year flood to the mean annual flood, against the variations that could be attributed to chance alone. Other examples are provided by DeCoursey (1973), who uses discriminant analysis and canonical correlation techniques, and the U S I A C W D (1983), who recommend the use of a constant regional coefficient of skewness. In addition, Tasker (1982) and Wiltshire (1986b) advocate the use of cluster analysis techniques to create hydrologic regions, while, more recently, Hosking and Wallis (1997) have suggested a number of nonparametric statistics for the identification of data sites that comprise a homogeneous collection. Hosking and Wallis use tests that compare the between-site variability of statistics based on L-moments with what would be expected of a homogeneous region. The concept of a region implies geographical closeness, and traditionally, homogeneous regions have been defined as sets of neighbouring streamflow stations. Acreman and Wiltshire (1987), Burn (1990) and Hosking and Wallis (1997) challenge the notion that geographical closeness is necessarily an indicator of similarity of the frequency distributions. They argue that homogeneous data sets should be used, rather than regions, with the data sets delineated by measuring the variables that influence the frequency distribution at each site and then grouping together those sites that are adjacent in some suitably defined space of site characteristics. These characteristics could be latitude and longitude, but they could also be altitude, mean annual precipitation, or any other appropriate variable. This logic carries considerable merit, particularly if data stations are widely dispersed and regions are geographically large. However, from a practical point of view, geographical regions are easier to define and interpret, and i f the regions are sufficiently small, they wil l necessarily tend to have similar characteristics, such as elevation and precipitation. For exploring the scaling behaviour of flood statistics, the most appropriate definition of hydrologic homogeneity is that provided by Gupta et al. (1994). They define a region as 22 homogeneous i f it is geographically contiguous and the differences amongst the probability distributions of peak flows within the region depend solely on differences in basin scale. 2.3.2 IDENTIFICATION OF REGIONAL DISTRIBUTIONS The identification of a regional distribution involves many of the same considerations as the selection of a distribution for a single site, as outlined in Section 2.2.1. The primary difference is that in regional modelling the flood characteristics of a number of sites are considered cumulatively, as opposed to on an individual basis. Currently, the only practical regional discriminatory test available is the moment ratio test, as described in Section 2.2.1.4. 2.3.3 POPULAR REGIONAL FLOOD FREQUENCY MODELS In many countries, most notably the US and Britain, national organizations have recommended specific regional flood estimation techniques. For instance, Bulletin 17B of the U.S. Interagency Advisory Committee on Water Data (1983) recommends the use of a regionally based skew coefficient for fitting flood data to the Log Pearson Type III distribution. In Canada, there is no specifically recommended procedure, and the choice of techniques for estimating peak flows is largely left to the individual. Many provincial agencies, such as the B C Ministry of Environment, Lands and Parks (BC M O E L P ) , Water Management Division, provide some regional flood summary statistics and general estimation guidelines, which help facilitate the process. In B C , a variety of regional flood frequency approaches are typically utilized (Loukas and Quick, 1995b), with the most common being the Index Flood Approach; the Method of Direct Regression of Quantiles; the Method of Regression for Distribution Parameters and Russell's Bayesian Method. 2.3.3.1 The Index Flood Approach 2.3.3.1.1 General The Index Flood approach was developed by Dalrymple (1960) and is arguably the most widely used flood regionalization procedure, worldwide. This approach is comprised of two key components that are developed for each hydrologically homogeneous region. The first step is the generation of dimensionless frequency or growth curves, which represent the ratios of floods at various frequencies to a specified index flood (typically the mean annual flood), for each station within a region. A regional growth curve is then derived as the median peak flow ratio at each return period. The second step is to generate a regression relationship between the characteristics of the drainage areas and the index floods for each location. Flood magnitudes can be estimated at any location in the region by combining estimates of the index flood with the regional frequency curve. The Index Flood approach is simple to develop and apply, but it has the following general limitations, as noted by Benson (1962b): 23 1. Due to sampling variability, the index flood for stations with short periods of record may not be representative of long-term values. This situation may result in significant differences in peak flow ratios for stations within a region. 2. The standard homogeneity test used to define homogeneous regions is based on the similarity of the 10-year peak flow to the mean annual flow. However, similarity at the 10-year ratio does not always correspond to similarity at higher return period ratios, and can lead to erroneous conclusions. 3. The method combines frequency curves for all catchment sizes and ignores the effect that catchment size has on peak flow ratios. In general, peak flow ratios tend to vary inversely with catchment size. The larger the catchment, the flatter the frequency curve and the lower the peak flow ratios (Dawdy, 1961). Use of a regional growth curve developed on the basis of growth curves for basins varying substantially in size wil l likely result in the underestimation of peak flows for small basins and the overestimation of peak flows for large basins. This effect is particularly strong when estimating floods with large return periods. Many variations of this approach have been developed (Kite, 1978), with modifications typically being applied to the definition of a homogeneous region, to the development of regional frequency curves through the use of different distributions, and/or the development of different techniques for determining the index flood. In B C , two different versions of the Index Flood approach have been advocated in recent years: the Forest Practices Code of B C Model and the B C M O E L P Model. 2.3.3.1.2 The Forest Practices Code of BC (FPCBC) Model The Forest Practices Code of British Columbia: Community Watershed Guidebook (BC Government, 1996) recommends the use of an index flood model that is based on the work of Church (updated in Church (1997)) and is comprised of a standard scaling equation and regionally based isolines of equation parameters. The return period flow of interest is expressed by the equation: <2_ = kA* (2.26) where, Qmean = mean annual peak flow k = regionally based value A = drainage area x = 0.68 = provincial scaling exponent This approach involves the calculation of Q m e a n for all gauged sites within a region using standard single-site frequency analysis techniques. A common scaling exponent of 0.68 is then assumed for all regions in the province, as suggested by Church (1997), who concluded that scale effects are independent of the primary controls of runoff, k values for each gauged 24 site are then determined and isolines of k values are plotted, with k values found to range from 7.0 in extremely wet areas to 0.1 in very dry areas. Q m e a n values for any site within a region can then be estimated by knowing the drainage area and interpolating for k from the plot of k isolines. Estimates of the Qioo are then obtained by multiplying the Q m e an by an assumed growth factor of 3.0. 2.3.3.1.3 The BC MOELP Model The B C M O E L P favours another adaptation of the index flood approach (Reksten, 1987), and used it to develop models for eight regions throughout B C during the period of 1987 to 1992. The model development involved splitting the province into several regions, deriving regional relationships between drainage area and mean annual peak daily flow, deriving regional flood frequency growth curves and estimating regional mean (assumed constant) ratios of instantaneous to daily flows. For an example, see Chapman et al. (1991). These regional models are currently under revision according to a modified procedure, as outlined in the British Columbia Streamflow Inventory report (Coulson and Obedkoff, 1998), and updated models are currently available for three regions (Obedkoff, 1998; 1999; 2000b). This report presents a comprehensive summary of various analyses of historical precipitation and streamflow records for B C , as well as a model for estimating flows at ungauged sites. The model is based on an index flood approach that assumes a scaling exponent of 0.785 for all areas of British Columbia, for all return period flows ( Q T ' S ) . The value of the exponent was determined by plotting and comparing peak flows against drainage area for 41 hydrologic zones across the province. This model differs from the one developed by Church (1997) and adopted by the F P C B C , in that rather than involving isoline plots of k values for the mean annual peak daily flow, it utilizes isoline plots of the instantaneous Qio and Qioo values, which are both normalized for a 100 km 2 watershed. The Qio and Qioo values were obtained from historical records through standard frequency analysis procedures. These procedures involved the use of log-Pearson Type III, three parameter lognormal or Pearson Type III distributions, which were assigned on a regional basis. The resulting Qio and Qioo values were than normalized with the equation: n „ riooV'785 Q r { N ) = Q r X { — ) (2-27) where, Q T ( N ) = normalized T year (10 or 100 year) peak instantaneous flow (m3/s) Q T = corresponding T year peak instantaneous flow (m3/s) for area A A = drainage area (km2) This equation is based on the simple scaling relation of equation 2.26. Values of Qio(N) and Qioo(N), for ungauged watersheds, can be obtained from the isoline plots, and adjusted for different sized watersheds, through manipulation of equation 2.27. The 10-year and 100 year peak flow estimates can then be converted to another recurrence interval by using growth curves determined for each gauged station within a region. 25 This approach, as with the F P C B C model, is simple and easy to use, and like many models, it can generate useful information, provided the user is aware of its restrictions. 2.3.3.1.4 Model Limitations The two flood index modelling approaches developed for B C have a number of limitations, including the following: Both Models 1. The selection of a province wide scaling exponent is a simplifying assumption that ignores regional differences. In both models, the exponent is assumed to be constant for the entire province and to not change with drainage area or location. Church (1997) used standard covariance analysis techniques to conclude that a common scale factor is appropriate, but qualifies this with the statement that "the data permit one to suppose that there are common scale effects , even though the results do exhibit some variability (which might be related to the samples, or might be real)." He further supports this approach with the observation that "there is a substantial chance that the scale exponent in individual regions is biased by the particular set of available stations." This bias may be introduced by the purpose of the gauging program, such as investigating the potential for hydroelectric power, or by having several gauges on one river or river system. Coulson and Obedkoff (1998) conclude that the return interval peak instantaneous flow scaling factor is constant by "plotting and comparing peak flow against drainage area for each hydrologic zone." The authors offer no detailed description of the techniques employed, so it is difficult to determine the validity of the constancy assumption. However, as the historical peak instantaneous flow data set is considerably smaller than the peak daily data set, and as estimates of the Qjo and Gjoo are subject to considerably greater error than estimates of the mean, greater error is likely associated with the assumption in the B C M O E L P model than in the F P C B C model. Regardless of which model is used and what qualifying statements are offered, however, it remains that regional differences in the scaling factor are not recognized. 2. The paucity of peak flow stations in B C makes it difficult to develop reasonable contours or interpolate between contours with any accuracy. The gauging network is poorly distributed, both regionally and in terms of drainage basin areas. This is recognized by Church (1997), who states that the "resolution is consistent with provincial scale regional analysis, with strategic water resources planning requirements, and with global modelling of environmental change", but "is not compatible with needs to make reliable estimates of flows in individual ungauged basins." This qualification is not noted in either the Forest Practices Code or the BC Streamflow Inventory publications. This limitation is most significant for the B C Streamflow Inventory plots, as they were developed with a smaller database and the estimation of flood quantiles involves errors associated with parameter estimation and distribution selection. 26 3. No objective measure of the strength or effectiveness of the models is provided with either model. A regional data fitting procedure is not utilized in either case, so a statistical measure of "fit" cannot be employed. Rather, data for each station is fitted to a provincial equation on a station by station basis. While this procedure incorporates site specific information, it is more susceptible to sampling variability than a regional smoothing approach, which effectively increases the data sample size. Without a statistical measure of "fit", it is not possible to determine whether variations in the computed k or Q T values are due to legitimate differences in hydrologic signals, or are simply manifestations of poor models. 4. The historical flood database in BC is largely limited to medium and large basins (i.e. areas > 50 km 2), so both models are limited in application to basins of this size. Little data are available to indicate how well the regional relations can be extrapolated to small basins. 5. The ratios of instantaneous to daily peak flows (I/D) are not constant within a region, as assumed by the F P C B C model, but vary with basin size and antecedent moisture conditions (Loukas and Quick, 1995a), and possibly other variables. Some stations exhibit decreasing I/D ratios with increasing daily discharge, while other exhibit increasing ratios. For the B C M O E L P approach, large scatters of I/D values are noted within each region, making estimates difficult and of doubtful accuracy. The Forest Practices Code of BC Model 6. The use of a constant growth factor of 3.0 to convert the index flood to the associated Qioo is a gross simplification of the index flood approach. It assumes a constant growth curve for all streams in the province, which likely results in significant estimation error. The BC MOELP Model 7. The homogeneous regions used in the BC M O E L P analysis appear to have been delineated largely in terms of climate and physiography, but no explicit definition of hydrologic homogeneity is provided, so it is not possible to assess the appropriateness of the delineated regions. 8. The development of dimensionless regional growth curves, as part of the B C M O E L P model, assumes that the regional coefficient of variation (Cv) is constant and does not vary with drainage area or any other factor. Many researchers have challenged this notion. In the early sixties, Dawdy (1961) and Benson (1962a) argued that this assumption of constancy is not valid because empirical evidence suggests that the Cv tends to decrease with increasing drainage area. More recently, Smith (1992), Gupta and Dawdy (1994), and Bloschl and Sivapalan (1997) have all demonstrated that the Cv often follows a multiple scaling pattern, whereby the Cv changes systematically with changing basin scale. 27 9. Standard frequency analysis techniques were used as part of the B C M O E L P approach to generate return period peak flow estimates for each gauged site. The frequency distribution types used for the frequency analyses were selected largely on the basis of results from the Kolmogorov-Smirnov (K-S) goodness-of-fit test. As discussed in Section 2.2.1.4, this test can produce unreliable results. In addition, it is not clear whether a single distribution type was applied in each region, or whether different distributions were applied for each site within a region, depending on the K-S test results. General practice is to use a single distribution type for each homogeneous region, as consistency in distribution type is considered a condition of homogeneity. This is the approach being utilized for the revised B C M O E L P models. 10. A l l distributional parameter estimates in the B C M O E L P model were developed with product moment or maximum likelihood estimators, which may have contributed significant errors to the regional growth curves. Successful application of these models requires considerable judgement and interpretation, particularly when estimating peak design flows for small ungauged watersheds. This situation is explicitly recognized in a report titled "Guide to Peak Flow Estimation for Ungauged Watersheds in the Vancouver Island Region (Nanaimo)" (Chapman Geoscience Ltd., 1999). This report lists a number of limitations to the index flood approach and advocates the use of a variety of different estimation techniques, depending on the quality and availability of data. In addition, in recognition of the paucity of peak flow data in B C , no regional curves or equations are developed in the report. Rather, regional data summaries and plots are provided, and the user is required to use his own "judgement, experience and understanding of hydrology, and hydrologic/climatic principles" to derive curves or equations. The index flood approach, as it has been applied in B C , is appropriate for generating first approximations of return period peak flows, for medium and large basins. However, it is subject to the many limitations discussed, and as such, is fundamentally inadequate for estimating peak design flows, and particularly so for small, ungauged watersheds. The available database is insufficient to reflect hydrologic patterns at a sufficiently small scale, and the assumption of a province wide scaling factor ignores regional differences. 2.3.3.2 The Method of Direct Regression of Quantiles The method of direct regression of quantiles is well established, particularly in the US where it has been employed by the US Geological Survey (USGS) to develop peak flow estimation equations for every state (Jennings et al, 1994). This approach uses multivariate regression analysis techniques to relate the discharge of a T-year flood to a basin's physiographic and climatic variables. For each basin within a hydrologically homogeneous region, a traditional single-site flood frequency analysis is performed. The resulting quantile estimates, for each return period of interest, are then related to watershed variables. Drainage area, main channel slope, and annual precipitation are the three most commonly used watershed and 28 climatic characteristics in the US. Drainage area is by far the most significant indicator variable, and in six states, it is the only variable required (Jennings et al., 1994). Ordinary-least-squares regression techniques were initially favoured for the mathematics of this approach, but these have since been replaced by weighted-least-squares regression (Stedinger and Tasker, 1985; Tasker and Stedinger, 1986) and generalized-least-squares regression (Stedinger and Tasker, 1985, 1986) techniques. These two newer techniques are considered superior because they account for errors introduced by using data of unequal record lengths. Multivariate regional regression analysis was first introduced in the US by Langbein (1947), but did not gain much support until it was advocated by Benson (1962b). He urged the USGS to use this regression approach, as opposed to the index flood method, because it incorporates more information about the physiography of basins and is able to better reflect the basin to basin variation of peak flow patterns. The approach has found considerable application in Canada, with Watt et al. (1989) listing ten studies, but it has never reached the organized federal scale of the US. The biggest limitations of this approach are that it requires a large amount of physiographic and climatic data for development, the regression equations cannot be readily used for basins whose characteristics are out of the range of those basins used for developing the equations, and it does not permit users to incorporate judgement into the flood estimation procedure. Experience has shown that practitioners wil l often use readily available regression equations without knowing much about the data used to derive them. 2.3.3.3 The Method of Regression for Distribution Parameters The method of direct regression for distribution parameters is similar to the approach used for the direct regression of quantiles, except that regression techniques are used to relate basin characteristics to flood sample statistics, as opposed to quantiles. Knowing the required characteristics of a basin, the regression equations permit the estimation of appropriate sample statistics, which can then be used to generate a flood frequency distribution for the particular site. This approach is subject to many of the same limitations as the direct regression of quantiles, but is sometimes preferred, particularly by experienced hydrologists, because the user can have more influence on the development of a flood frequency curve. This is particularly true when the selection of distribution type is left to the user and/or when regression relations are provided as plots, as well as equations. One of the major criticisms of this approach is that the regression relations for higher order moment statistics are not always statistically significant (Watt et al, 1989), but this is of less concern with the use of L-moment statistics. This approach has found little application in Canada (Watt et al, 1989), but is currently the subject of a large research study at the University of British Columbia (Alila, 1999a). The first phase of this project is now largely complete, as presented by Wang (2000). 29 2.3.3.4 Russell's Bayesian Method The Bayesian method proposed by Russell (1982) generates flood quantile estimates from a compound probability distribution, which is generated as a weighted combination of a series of twenty-five individual distributions. The model assumes that either the Extreme Value Type I or Lognormal distributions are applicable for all flood data, and the twenty-five component distributions are each determined by a mean, a standard deviation and a weight or marginal probability. Model inputs of mean and standard deviation are estimated from regional values, and specified as low, probable and high estimates. The low estimates are those that have a 90% chance of being exceeded, the probable estimates are the "best" estimates, while the high estimates are those that have a 90% chance of not being exceeded. The user is required to make a subjective selection of the mean and standard deviation values corresponding to each of these probabilities. The model then computes the values, which fall halfway between the low and probable, and probable and high, values. Marginal probabilities of 0.169, 0.206, 0.250, 0.206 and 0.169 are then assigned to each of the mean and standard deviation values, from low to high, respectively. These values correspond to the areas under the normal curve between 0 and 20%, 20% and 40%, 40% and 60%, 60% and 80%, and 80% and 100%, respectively. The five mean and five standard deviation values, each with an assigned marginal probability, represent twenty-five combinations that are used to develop twenty-five frequency distributions. The probability of any particular combination of mean and standard deviation is determined by multiplying the probabilities assigned to them, and then normalizing to make the sum of the probabilities, of all 25 distributions, equal to one. The magnitude of a flood associated with any return period is simply computed as the weighted sum of the flood estimates from all component distributions, with results presented as low, probable and high values. The low and high values represent the 90% confidence limits. Flood estimates can be revised if site specific information is available. This information may take the form of recorded floods, the largest flood in a number of years, or a flow that has not been exceeded in a number of years. The model considers this information by revising the weights of the component distributions according to Bayes' theorem. Bayes' theorem is an established statistical procedure that utilizes prior, posterior and conditional probabilities to revise estimates based on new information (De Neufville, 1990). The principal strength of Russell's Bayesian approach is that it allows the user to incorporate information into the flood estimation process that otherwise would not be directly and formally considered. In addition, it provides a simple platform for incorporating engineering judgement into the flood estimation process. Through the selection of low, probable and high flood parameters, a user is able directly reflect his confidence in the estimation of the flood parameters. A narrow bracket reflects a high level of confidence, while a wide bracket reflects a high level of uncertainty. This level of uncertainty is conveyed in the results by the difference between the high and low flood estimates. Another strength of this model is that it does not make any assumptions about the scaling properties of the coefficient of variation of extreme historical flows. The regional methodology used to estimate mean and standard deviation values is independent of the model and does not influence the model's operation. 30 Russell's model is available in digital form and is easy to apply. However, despite finding some local use (Cathcart and Russell, 1995), it is not well known, particularly outside British Columbia. Its lack of use is more a reflection of its limited exposure and distribution, than of technical deficiency. However, the model does have some limitations, which could be addressed with relatively minor revisions. The most significant limitation is that the model only utilizes the two parameter Extreme Value Type I and lognormal distributions, which are not always the most appropriate distributions for all flood data. This restriction can lead to significant errors, as discussed in Section 2.2.1. The model is also hampered by its use of product moment statistics to estimate distributional parameters. The problems and concerns associated with this assumption are discussed in Sections 2.2.2.3 and 2.2.2.5. 2.4 S U M M A R Y Statistical methods are well established in hydrology and are useful tools for developing flood estimates. They are the most commonly applied flood estimation techniques, and despite the recent musings and proclamations of Klemes (2000a, 2000b), they wil l continue to find widespread application. The mathematical components of statistical methods continue to evolve with ever increasing sophistication, and while there is merit to this pursuit, greater gains are to be found in developing a better understanding of the link between the physical components of watersheds and the statistics commonly used to describe flood behaviour. The common phrase "garbage in, garbage out" is very apt for describing the results of many statistical flood models. While the statistical techniques may be "state of the art," the results are flawed if the fundamental hydrological components of the model are flawed. Various regional flood frequency models have been developed for application in B C , but all of them have fundamental deficiencies that cannot be overcome by improving the applied mathematical techniques. One example is the erroneous assumption of moment ratio scaling invariance that is inherent in the index flood approach. If any of the common moment ratio flood statistics, better known as the coefficients of variation, skewness and kurtosis, vary with scale, then the index flood approach is invalid. Another example is the notion that scale effects are independent of the primary controls of runoff, as assumed in the models developed by the B C M O E and the B C Forest Practices Code. If this assumption is incorrect, and scaling patterns are influenced by variations in climate, physiography and hydrologic mechanisms, then the current models are not valid, as their scaling equation exponents must vary from region to region. Therefore it was decided that the research would not focus on improving mathematical techniques, which appears to be the current favoured area of hydrologic research according to the bulk of current journal publications and the focus of presentations at the most recent Amercian Geophysical Union Conference ( A G U , 2000). Rather, it is believed that a greater contribution to the science of peak flow analysis and our ability to generate realistic and appropriate design flows can be made by improving our understanding of how peak flow statistics are affected by watershed characteristics such as climate and scale. 31 Chapter 3 DATA SELECTION AND SCREENING This chapter discusses the factors behind the selection of data from Oregon State for a study that is largely motivated by a desire to improve the estimation of peak flows in British Columbia. In addition, each of nine steps that were taken as part of the data screening process are discussed in detail as part of an effort to produce a body of work that is as transparent as possible. It is understood that the screening process used for this research is not perfect and may be subject to some criticism, but nothing has been hidden and readers are free to draw their own conclusions. The reader should note that some steps in the data screening process were done in conjunction with the region delineation procedure described in Chapter 4, and that an iterative process was required. As a result, many references are made to numerical regions (i.e. Regions 1 to 12) that are not fully described until Chapter 4, and the reader is referred to Chapter 4 if details are required. In particular, procedures described in Sections 3.3.7, 3.3.8 and 3.3.9 all required the screening of data on a regional basis, so preliminary groupings of the data were initially employed and then the screening processes were repeated once the regional groupings were refined. The primary objective of the work described in this chapter was to screen the selected historical peak flow data set to remove any data points that might unduly influence the results of this study. For example, an anomalously large value in small sample set would tend to skew flood statistics to the extent that they could not be considered representative of the population from which the samples were drawn. As a result, the large value is considered to have an "undue influence" on the statistics and would be removed from the sample set. 3.1 I N T R O D U C T I O N 32 3.2 D A T A S E L E C T I O N The historical peak flow database for British Columbia is largely limited to basins with areas greater than 50 km 2 . In fact, of the 342 automated streamflow stations currently or previously operated by Environment Canada, only 43 stations, or 13% of the total, represent basins smaller than 50 km 2 . Consequently, the results of any regional peak flow studies conducted with B C data are necessarily restricted to medium and large basins. However, in many instances, peak flow estimates are required for small basins. Common practice is to apply a simple linear extrapolation of the scaling relations for large areas to small areas, and assume one frequency distribution pattern for all basin sizes. This practice is routinely done despite the generally accepted idea that small basins have different hydrologic response patterns than large basins. Small basins are typically located at higher elevations and subsequently experience greater precipitation and have steeper topography and different vegetation and geology. In addition, small basin flows are more directly influenced by the spatial and temporal variation of precipitation. How are differences in basin climate and physiography manifested in flood statistics? Are the statistical patterns for large and small basins significantly different, and if so, how? One of the key objectives of this thesis is to try to answer these questions, with particular focus on BC ' s watersheds. Therefore, it was necessary to acquire a historical instantaneous peak flow database of adequate duration and with a reasonably even distribution of basin sizes, from a region with a similar climate and physiography to B C . Such a database was acquired from the US Geological Survey for the state of Oregon. Historical instantaneous peak flow records are available for 810 stations scattered throughout the state, for durations ranging from 1 year to 137 years. 293 of these stations, or 36% of the total, represent basins with areas less than 50 km 2 . It is recognized that the differences between conditions in B C and those in Oregon, such as geologic histories and associated soil types and depths, may limit the applicability of Oregon results to B C . However, with this understanding, it is believed that the information derived from the Oregon data will offer valuable insights into the behaviour of floods in B C . A l l discussions in this thesis refer to basin areas greater than 50 km 2 as medium and large, and basin areas less than 50 km 2 as small. This delineation is somewhat arbitrary, as is seems that everyone has a slightly different idea of what constitutes a small basin. A popular definition for a small basin is a basin that experiences reasonably constant rainfall, in space and time, and has hillslope runoff, as opposed to channel runoff, as the dominating runoff process. Of course, these conditions vary with physiography and climate, so the limiting size will also vary. Ponce (1989) presents a good discussion of this topic, and quotes upper limits ranging from 0.65 to 12.5 km 2 . The B C M O E L P defines a small basin as one with an area of less than 25 km 2 , while Church (1997) suggests that below about 100 km 2 , scale effects do not systematically affect runoff. In other studies, Pilgrim (1989) and Cordery and Pilgrim (1993) have demonstrated some consistency in basin response for basins up to 250 km 2 , while Smith (1992) identified a scaling break at about 50 km 2 . For this study, 50 km 2 was selected as the small basin upper limit. In agreement with Smith's findings, preliminary investigations with Oregon data indicated a possible break in statistical scaling relations 33 around this basin size. As well, from a practical standpoint, the 50 km 2 size resulted in a reasonable split of the available data set. 3.3 D A T A S C R E E N I N G Peak flows are difficult to measure and, as a result, historical flood records always contain some error. Measurement errors typically result from gauge misreadings or instrument and gauge malfunctions, while the most common calibration errors result from undetected rating curve shifts and the extrapolation of stage-discharge relations. In addition, the largest floods often exceed the measuring range of gauges, so peak flows must be estimated by utilizing the slope-area method and post-flood field measurements, which are inherently inaccurate. However, as the data were supplied by the USGS, who adhere to strict practices designed to minimize measurement and calibration errors, the flow values were considered to be of adequate quality. The USGS data were not independently checked for errors, but were screened to assess their suitability for analysis. This screening process involved a number of considerations and the application of various criteria. Some criteria were applied prior to any grouping of the data, while others were applied both during and after the delineation of hydrologic regions. This screening process identified 523 stations as being inappropriate for analysis, resulting in a final data set of 287 stations, 113 (39%) with basin areas of less than 50 km 2 and 174 (61%) with areas of greater than 50 km". Figure 3.1 outlines the steps of the screening process and indicates the number of stations eliminated at each step. The reader should note that over half of the available data set was eliminated simply because of insufficient record length or substantial regulation of flows. The details of each step are presented in the following discussions. If should be noted that flood mixture was not considered as part of the data screening process due to the difficulties associated with classifying hydrological data on the basis of moisture input mechanism (rain, rain-on-snow, snowmelt) and the fact that mixture is inherent within each mechanism (particularly rain in arid regions) and therefore not eliminated by classification anyway. Furthermore, it is not clear whether the elimination of mixture effects is desirable in itself, as mixture is associated with non-linearity in the peak flow patterns. 3.3.1 L E N G T H O F R E C O R D A l l stations with less than 10 years of record were eliminated from the data set. This 10 year period was selected somewhat arbitrarily, but it has some precedent (Burn et al, 1997) and it served as a reasonable balance between maximizing the number of stations for analysis and minimizing the sampling error associated with small sample sizes. Burn et al. (1997) argue that while a minimum 10 year record length may not lead to reliable at-site flood frequency information, it does provide useful information for regional flood frequency studies. Application of the minimum record length criterion reduced the data set to 602 stations, with 212 stations servicing areas of less than 50 km 2 . 34 O CO C O (su on ,o -*-» c S3 CO T3 f« O o I Dis oo H H z 93 o CC 00 fN s S o o V A r -i—1 ON a o o •c ^ +2 «« £ fi « •< ^ .c w co 03 ro oo ro 35 3.3.2 FLOW REGULATION A l l stations that were identified as having significantly regulated flows were removed from the data set. Flow regulation results in the reduction of peak flows and is generally associated with the presence of natural or manmade lakes or reservoirs within a watershed, or with the diversion of water for consumptive use. The term "significantly" is used to differentiate between stations whose peak flows are affected more or less than 10%. Given the errors associated with peak flow measurement, a regulating effect of less than 10% is not considered significant, and these stations were retained in the data set. This 10% value corresponds to a peak flow classification system that was presented by Lystrom (1970) and adopted by the USGS, but no information is available on how the magnitude of the regulating effect is determined. The screening of regulated stations was a lengthy and difficult process because the USGS does not appear to publish an up-to-date listing that identifies flow records as regulated or unregulated (Miller, 1998). Rather, the information had to gleamed from a variety of sources including the USGS website, and publications by Lystrom (1970) and Campbell (1984), along with a variety of USGS Water-Data Reports (example: Alexander et al, 1985a). In addition, stations whose peak flows showed little variation (i.e. coefficient of variation less than 0.2) were reviewed and removed from the data set i f mapping indicated the presence of large lakes. It is not known whether all regulated stations were identified and removed from the data set, but it is believed that the screening process was adequately successful, and that any remaining regulating effects are minimal. This process eliminated 210 stations, reducing the data set to 392 stations. 3.3.3 BASIN AREA Consideration was given to limiting the size of the drainage basins accepted into the study. Very large basins may integrate runoff from different regions and therefore may not reveal regionally distinctive results. Exactly what size constitutes a very large basin is not clearly defined, and it varies from region to region depending on the size of a region and its topography and climate. Church (1997), in his regional study of British Columbia stream flows, limited his basins to areas of less than 10,000 km 2 , but indicated that 1,000 km 2 would have been more