Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Effects of uncertainty in hydrologic model calibration on extreme event simulation Roche, Anthony David 2005

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2005-0307.pdf [ 27.56MB ]
Metadata
JSON: 831-1.0063331.json
JSON-LD: 831-1.0063331-ld.json
RDF/XML (Pretty): 831-1.0063331-rdf.xml
RDF/JSON: 831-1.0063331-rdf.json
Turtle: 831-1.0063331-turtle.txt
N-Triples: 831-1.0063331-rdf-ntriples.txt
Original Record: 831-1.0063331-source.json
Full Text
831-1.0063331-fulltext.txt
Citation
831-1.0063331.ris

Full Text

E F F E C T S O F U N C E R T A I N T Y FN H Y D R O L O G I C M O D E L C A L I B R A T I O N ON EXTREME EVENT  SIMULATION  by  ANTHONY DAVID ROCHE  B . A . S c , University o f Waterloo, 2000  A T H E S I S S U B M I T T E D TN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FORT H EDEGREE OF  M A S T E R OF APPLIED  SCIENCE  in  THE F A C U L T Y OF G R A D U A T E STUDIES  (CIVIL E N G I N E E R I N G -  HYDROTECHN1CAL)  THE UNIVERSITY OF BRITISH C O L U M B I A  A p r i l 2005  © A n t h o n y D a v i d Roche, 2005  Abstract C o m p u t e r m o d e l s representing the h y d r o l o g i c c y c l e as a s i m p l i f i e d system have b e c o m e a preferred t o o l for e s t i m a t i n g f l o o d s . H o w e v e r , scientific understanding o f the uncertainty inherent i n these m o d e l s has not kept pace w i t h their d e v e l o p m e n t and a p p l i c a t i o n . In m a n y cases it is i n c o r r e c t l y a s s u m e d that a l l uncertainty i n m o d e l structure, input data, and parameters is m i n i m i z e d or e l i m i n a t e d t h r o u g h c a l i b r a t i o n . T h e e n d result is a u b i q u i t o u s but u n k n o w a b l e degree o f m o d e l p r e d i c t i v e uncertainty that m a y or m a y not s i g n i f i c a n t l y affect the o u t c o m e o f any g i v e n a p p l i c a t i o n . E x t r a p o l a t i o n o f a m o d e l b e y o n d its c a l i b r a t i o n range (i.e., for extreme event s i m u l a t i o n ) i n v a r i a b l y results i n a substantial increase i n this uncertainty.  T h i s w o r k a i m s to p r o m o t e qualitative and quantitative understanding o f m o d e l p r e d i c t i v e uncertainty i n extreme event s i m u l a t i o n . It therefore b e g i n s w i t h a r e v i e w o f the m a n y sources c o n t r i b u t i n g to m o d e l p r e d i c t i v e uncertainty, an analysis o f their o r i g i n s and interdependencies, and a synthesis o f v a r i o u s methods for a n a l y z i n g uncertainty.  A s a pre-requisite step towards the larger g o a l o f r e d u c i n g o v e r a l l m o d e l p r e d i c t i v e uncertainty, this w o r k investigates the v a r i a b i l i t y i n estimates o f extreme f l o o d s (e.g., peak f l o w , t i m i n g , and v o l u m e ) i n t r o d u c e d b y subjective d e c i s i o n s m a d e d u r i n g c a l i b r a t i o n . M u l t i p l e automatic calibrations o f a c o n c e p t u a l h y d r o l o g i c m o d e l are c o n d u c t e d u s i n g different objective functions to evaluate c a l i b r a t i o n performance, r e s u l t i n g i n a c o l l e c t i o n o f n o n - i n f e r i o r parameter sets. E a c h parameter set is then u s e d to simulate an extreme event based o n h y d r o l o g i c data for the C o q u i t l a m L a k e watershed i n B r i t i s h C o l u m b i a , w h i c h is d e v e l o p e d for h y d r o p o w e r b y B C H y d r o . T h e c o m b i n e d output o f these extreme event s i m u l a t i o n s characterizes the relative v a r i a b i l i t y i n the hydrographs.  S i m u l a t i o n s are c o n d u c t e d u s i n g the U n i v e r s i t y o f B r i t i s h C o l u m b i a W a t e r s h e d M o d e l ( U B C W M ) , w h i c h is w i d e l y u s e d to describe and forecast watershed b e h a v i o u r i n m o u n t a i n o u s areas o f B r i t i s h C o l u m b i a . C a l i b r a t i o n s o f the U B C W M u t i l i z e the S h u f f l e d C o m p l e x E v o l u t i o n A l g o r i t h m ( S C E - U A ) , an effective and efficient o p t i m i z a t i o n - b a s e d automatic c a l i b r a t i o n routine. B e c a u s e automatic calibrations fail to capture the different k i n d s o f expert k n o w l e d g e  ii  inherent i n a m a n u a l c a l i b r a t i o n , extreme event hydrographs obtained u s i n g calibrated parameter sets are c o m p a r e d o n a relative rather than absolute basis.  R e s u l t s s h o w that automatic c a l i b r a t i o n m a y p r o v i d e a straightforward m e t h o d o f i d e n t i f y i n g potential areas w h e r e subject m o d e l s are over-parameterized w i t h respect to the c a l i b r a t i o n data. M o r e i m p o r t a n t l y , p r e l i m i n a r y results s h o w that the v a r i a b i l i t y is r e l a t i v e l y constrained amongst s i m u l a t i o n s based o n a P r o b a b l e M a x i m u m F l o o d ( P M F ) scenario, w i t h coefficient o f v a r i a t i o n for peak f l o w , event v o l u m e , and t i m e to peak o f 4 % , 1%, and 1% respectively. T h i s v a l u e is n e g l i g i b l e i n c o m p a r i s o n w i t h other uncertainties that d o m i n a t e extreme events l i k e the P M F . T h u s , the P M F - b a s e d s i m u l a t i o n s are r e l a t i v e l y i n s e n s i t i v e to the different measures o f c a l i b r a t i o n performance used. S i m i l a r trials u s i n g other m o d e l s w o u l d p e r m i t an estimate o f the extent to w h i c h one c o u l d expect to r e s o l v e divergent estimates through i m p l e m e n t i n g different but e q u a l l y v a l i d c a l i b r a t i o n s .  O b s e r v a t i o n s o f this w o r k are a p p l i c a b l e for the m a n a g e m e n t o f h y d r o p o w e r p r o d u c t i o n and f l o o d c o n t r o l for these watersheds.  These observations w i l l p r o v i d e insights into uncertainty i n  extreme event s i m u l a t i o n and m a y contribute to the i m p r o v e d management o f water, h y d r o p o w e r systems, and p u b l i c safety i n C a n a d a and around the w o r l d .  iii  Table of Contents ABSTRACT  ii  TABLE OF CONTENTS  iv  LIST OF TABLES  vi  LIST OF FIGURES  vii viii  ACKNOWLEDGEMENTS 1. 1.1 1.2 1.3 1.4 1.5  INTRODUCTION Guidance for the Reader Approaches for Estimating Presenting Floods for Decision-making Expressing Uncertainty Model Predictive Uncertainty in Context  2. 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6  HYDROLOGIC MODELLING Hydrologic Processes Scale in Hydrology Precipitation Runoff Evapotranspiration The Evolution of Hydrologic Modelling The Beginnings of Modelling The Philosophy of Model Development and Application Model Complexity in Context Model Classifications Statistical, Empirical, and Black Box Models Conceptual Models Physically-Based Models Other Classifications Model Calibration The Nature of Calibration Manual Calibration Automatic Calibration Methods for Automatic Calibration Evaluating Model Performance Hydrologic Model Validation  3. 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.4  UNCERTAINTY IN HYDROLOGIC MODELLING Classification of Uncertainty in Hydrologic Modelling Natural Variability Data Uncertainty Model Uncertainty Parameter Uncertainty Uncertainty and Calibration Techniques for Exploring Uncertainty Sensitivity Analysis Reliability Analysis Multi-Objective Analysis Generalized Sensitivity Analysis Equifinality Uncertainty Isolation Uncertainty and Extreme Event Simulation  floods  iv  1 2 3 6 8 10 13 14 14 15 18 24 26 26 28 30 31 32 33 36 39 41 42 45 47 51 60 67 70 72 76 79 88 94 100 103 106 109 113 119 120 123 126  4. 4.1  R E S E A R C H T O O L S AND METHODS OF ANALYSIS Tools and Case Studies  4.1.1 4.1.2 4.1.3 4.1.4 4.2  The University of British Columbia Watershed Model The SCE-UA Method Interfacing UBCWM and SCE-UA The Coquitlam Lake and Illecillewaet River Watersheds Experimental Design  4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 5. 5.1 5.2 5.3  5.4  5.4.1 5.4.2 5.5 5.6  141 142 146 147 149 152 152 156 170  Coquitlam Lake Watershed Calibrations with Expanded Parameter Space Illecillewaet River Watershed  170 173 179  Sensitivity to Initial Random Seed  183  Coquitlam Lake Watershed Illecillewaet River Watershed  183 186  Event Simulations Extreme Event Simulations  5.6.1 5.6.2  130 135 136 137 140  Selection of Objectives SCE-UA Calibrations Validation Event Simulations Simulations based on a PMF Scenario RESULTS AND DISCUSSION Preliminary Tests Synthetic Calibrations Calibrations against Observed Data  5.3.1 5.3.2 5.3.3  129 130  190 195  Simulations Based on a PMF Scenario Simulations Based on a PMP-Scale Event  195 197  6.  CONCLUSIONS  200  7.  FUTURE DIRECTIONS  208  8.  GLOSSARY OF ACRONYMS AND WATERSHED MODELS  210  9.  REFERENCES  213  APPENDIX A : SELECTION OF S C E - U A PARAMETER V A L U E S APPENDIX B : DETAILED CALIBRATION RESULTS  B. 1: Complete Parameter Sets B.2: Annual Statistics for All Calibrations B.3: Mean Monthly Statistics for All Calibrations B.4: Summary Results from Event-based Simulations  V  List of Tables T a b l e 2 - 1 : C o m m o n M e a s u r e s for Quantitative E v a l u a t i o n o f H y d r o l o g i c M o d e l Performance  64  T a b l e 4 - 1 : C o m m o n l y C a l i b r a t e d Parameters o f the U B C W M  134  T a b l e 4-2: E x p a n d e d Parameter R a n g e s for C o q u i t l a m L a k e W a t e r s h e d C a l i b r a t i o n s  145  T a b l e 4 - 3 : Index o f S C E - U A C a l i b r a t i o n s  146  T a b l e 4-4: Start and E n d Dates for M a j o r I n f l o w E v e n t s to C o q u i t l a m L a k e  148  T a b l e 5-1: R e s u l t s o f Synthetic C a l i b r a t i o n for C a m p b e l l R i v e r W a t e r s h e d  153  T a b l e 5-2: Parameter V a l u e s for S u c c e s s f u l Synthetic C a l i b r a t i o n s - C o q u i t l a m L a k e Watershed  168  T a b l e 5-3: Parameter V a l u e s for Successful Synthetic C a l i b r a t i o n s - I l l e c i l l e w a e t R i v e r Watershed  169  T a b l e 5-4: S u m m a r y Statistics for M u l t i - O b j e c t i v e C a l i b r a t i o n s o f C o q u i t l a m L a k e Watershed  171  T a b l e 5-5: Parameter V a l u e s for M u l t i - O b j e c t i v e C a l i b r a t i o n o f C o q u i t l a m L a k e W a t e r s h e d . . 174 T a b l e 5-6: S u m m a r y Statistics for M u l t i - O b j e c t i v e C a l i b r a t i o n s o f C o q u i t l a m L a k e W a t e r s h e d u s i n g E x p a n d e d Parameter Space  175  T a b l e 5-7: Parameter V a l u e s for M u l t i - O b j e c t i v e C a l i b r a t i o n s o f C o q u i t l a m L a k e W a t e r s h e d u s i n g E x p a n d e d Parameter Space  176  T a b l e 5-8: S u m m a r y Statistics for M u l t i - O b j e c t i v e C a l i b r a t i o n s o f I l l e c i l l e w a e t R i v e r Watershed  179  T a b l e 5-9: Parameter V a l u e s for M u l t i - O b j e c t i v e C a l i b r a t i o n s o f I l l e c i l l e w a e t R i v e r Watershed  182  T a b l e 5-10: S u m m a r y Statistics for C o q u i t l a m L a k e W a t e r s h e d S e e d S e n s i t i v i t y  184  T a b l e 5-11: Parameter V a l u e s for C o q u i t l a m L a k e W a t e r s h e d S e e d S e n s i t i v i t y  187  T a b l e 5-12: S u m m a r y Statistics for I l l e c i l l e w a e t R i v e r W a t e r s h e d S e e d S e n s i t i v i t y  188  T a b l e 5-13: Parameter V a l u e s for I l l e c i l l e w a e t R i v e r W a t e r s h e d S e e d S e n s i t i v i t y  191  T a b l e 5-14: S u m m a r y R e s u l t s for S t o r m 1 S i m u l a t i o n s  194  T a b l e 5-15: A v e r a g e R e s u l t s for A l l M u l t i - O b j e c t i v e E v e n t S i m u l a t i o n s  195  T a b l e 5-16: S u m m a r y R e s u l t s for P M F - b a s e d E x t r e m e E v e n t S i m u l a t i o n s  197  T a b l e 5-17: S u m m a r y R e s u l t s for P M P - b a s e d S t o r m S i m u l a t i o n s  199  vi  List of Figures F i g u r e 1-1: R i s k - U n c e r t a i n t y Interaction  9  F i g u r e 2 - 1 : F l o w c h a r t o f the S C E - U A A l g o r i t h m  57  F i g u r e 2-2: F l o w c h a r t o f the S C E - U A C C E Strategy  59  F i g u r e 3-1: A c c u r a c y and P r e c i s i o n as A n a l o g u e s for E r r o r and U n c e r t a i n t y  71  F i g u r e 3-2: M i n d M a p o f Uncertainties i n H y d r o l o g i c M o d e l l i n g  75  F i g u r e 3-3: M o d e l U n c e r t a i n t y i n P r a c t i c e  89  F i g u r e 3-4: N o r m a l i z e d Parameter Sets for a S i n g l e - O b j e c t i v e O p t i m i z a t i o n w h e r e O b j e c t i v e F u n c t i o n V a l u e s D i f f e r b y < 1%  105  F i g u r e 3-5: N o r m a l i z e d Pareto O p t i m a l P a r a m e t e r Sets for a M u l t i - O b j e c t i v e O p t i m i z a t i o n . . . 106 F i g u r e 3-6: H y d r o g r a p h R a n g e s A s s o c i a t e d w i t h a Pareto S o l u t i o n Set  117  F i g u r e 5-1: Parameter C o n v e r g e n c e for S y n t h e t i c C a l i b r a t i o n i n P r e l i m i n a r y T e s t i n g  155  F i g u r e 5-2: O b j e c t i v e F u n c t i o n E v o l u t i o n for C a l i b r a t i o n s w i t h 10, 15, and 20 c o m p l e x e s  156  F i g u r e 5-3: P a r a m e t e r C o n v e r g e n c e for S u c c e s s f u l Synthetic C a l i b r a t i o n s  158  F i g u r e 5-4: T y p i c a l A n n u a l H y d r o g r a p h s for S u c c e s s f u l Synthetic C a l i b r a t i o n s  159  F i g u r e 5-5: P a r a m e t e r C o n v e r g e n c e for U n s u c c e s s f u l Synthetic C a l i b r a t i o n s  160  F i g u r e 5-6: T y p i c a l A n n u a l H y d r o g r a p h s for U n s u c c e s s f u l Synthetic C a l i b r a t i o n s  161  F i g u r e 5-7: N o r m a l i z e d Parameter V a l u e s for Successful Synthetic C a l i b r a t i o n s  163  F i g u r e 5-8: N o r m a l i z e d Parameter V a l u e s for Synthetic C a l i b r a t i o n Seed S e n s i t i v i t y T r i a l s ... 164 F i g u r e 5-9: O b j e c t i v e F u n c t i o n E v o l u t i o n for Synthetic C a l i b r a t i o n S e e d S e n s i t i v i t y T r i a l s . . . . 166 F i g u r e 5-10: E x a m p l e A n n u a l H y d r o g r a p h s for Synthetic C a l i b r a t i o n S e e d S e n s i t i v i t y T r i a l s . 167 F i g u r e 5-11: T y p i c a l A n n u a l H y d r o g r a p h for M u l t i - O b j e c t i v e C a l i b r a t i o n o f C o q u i t l a m L a k e Watershed  172  F i g u r e 5-12: N o r m a l i z e d Parameter V a l u e s for M u l t i - O b j e c t i v e C a l i b r a t i o n o f C o q u i t l a m L a k e Watershed  172  F i g u r e 5-13: N o r m a l i z e d Parameter V a l u e s for M u l t i - O b j e c t i v e C a l i b r a t i o n s o f C o q u i t l a m L a k e W a t e r s h e d u s i n g E x p a n d e d Parameter S p a c e  177  F i g u r e 5-14: T y p i c a l A n n u a l H y d r o g r a p h for M u l t i - O b j e c t i v e C a l i b r a t i o n o f I l l e c i l l e w a e t R i v e r Watershed  180  F i g u r e 5-15: N o r m a l i z e d Parameter V a l u e s for M u l t i - O b j e c t i v e C a l i b r a t i o n s o f I l l e c i l l e w a e t R i v e r Watershed  181  F i g u r e 5-16: T y p i c a l A n n u a l H y d r o g r a p h for C o q u i t l a m L a k e W a t e r s h e d S e e d S e n s i t i v i t y  184  F i g u r e 5-17: O b j e c t i v e F u n c t i o n E v o l u t i o n for C o q u i t l a m L a k e W a t e r s h e d Seed S e n s i t i v i t y . . . 185 F i g u r e 5-18: N o r m a l i z e d Parameter V a l u e s for C o q u i t l a m L a k e W a t e r s h e d Seed S e n s i t i v i t y . . 186 F i g u r e 5-19: T y p i c a l A n n u a l H y d r o g r a p h for I l l e c i l l e w a e t R i v e r W a t e r s h e d Seed S e n s i t i v i t y . 189 F i g u r e 5-20: O b j e c t i v e F u n c t i o n E v o l u t i o n for I l l e c i l l e w a e t R i v e r W a t e r s h e d Seed Sensitivity  189  F i g u r e 5-21: N o r m a l i z e d Parameter V a l u e s for I l l e c i l l e w a e t R i v e r W a t e r s h e d Seed Sensitivity  190  F i g u r e 5-22: T y p i c a l H y d r o g r a p h C o m p a r i n g o f I n i t i a l W a t e r s h e d C o n d i t i o n s for S t o r m 1  192  F i g u r e 5-23: C o q u i t l a m L a k e W a t e r s h e d S t o r m 1 Pareto H y d r o g r a p h  193  F i g u r e 5-24: Pareto H y d r o g r a p h for P M F - b a s e d E x t r e m e E v e n t S i m u l a t i o n s  196  F i g u r e 5-25: P M P - b a s e d S t o r m Pareto H y d r o g r a p h  198  vii  Acknowledgements W h i l e this thesis is n o m i n a l l y the product o f a sole author, it c o u l d never have been c o m p l e t e d w i t h o u t the a d v i c e and support o f m a n y colleagues a n d friends.  F i r s t l y , I w o u l d l i k e to thank m y supervisor, D r . B a r b a r a L e n c e o f U B C C i v i l E n g i n e e r i n g . H e r input, guidance, and enthusiasm were i n d i s p e n s a b l e i n k e e p i n g m e o n course and focussed. S h e also r e v i e w e d this entire d o c u m e n t repeatedly throughout its e v o l u t i o n , and for that has captured m y a d m i r a t i o n as w e l l as m y gratitude.  S e c o n d l y , I w o u l d l i k e to thank m y i n f o r m a l thesis c o m m i t t e e at B C H y d r o - M u r r a y K r o e k e r , G r a h a m L a n g , and D r . D e s H a r t f o r d . T h e y h e l p e d to set m y course and are the "target a u d i e n c e " for w h o m this thesis is prepared. I w o u l d also l i k e to thank D r . M a r k u s W e i l e r o f U B C Forestry, w h o p r o v i d e d a n e x p e r i e n c e d h y d r o l o g i s t ' s r e v i e w a n d e x c e l l e n t insight as m y o f f i c i a l s e c o n d reader.  S e v e r a l p e o p l e assisted w i t h the a p p l i e d p o r t i o n o f this thesis, and i n d o i n g so m a d e m y j o b v e r y m u c h easier. I benefited greatly f r o m the assistance o f U B C P r o f e s s o r E m e r i t u s D r . M i c h a e l Q u i c k , p r i n c i p a l author o f the U n i v e r s i t y o f B r i t i s h C o l u m b i a W a t e r s h e d M o d e l , and E d m o n d Y u , w h o d i l i g e n t l y m a i n t a i n s and upgrades the U B C W M source code. D r . Z o r a n M i c o v i c , w h o puts the " e x p e r t " i n B C H y d r o ' s "expert c a l i b r a t i o n s " , p r o v i d e d a d v i c e o n c a l i b r a t i o n o f b o t h the U B C W M and h y d r o l o g i c m o d e l s i n general.  I salute D r . Q i n g y u n D u a n o f the U n i t e d States N a t i o n a l O c e a n i c and A t m o s p h e r i c A d m i n i s t r a t i o n and D r s . S o r o o s h S o r o o s h i a n and H o s h i n G u p t a o f the U n i v e r s i t y o f A r i z o n a , w h o d e v e l o p e d , support, and continue to advance the r e m a r k a b l e S C E - U A o p t i m i z a t i o n a l g o r i t h m u s e d e x t e n s i v e l y i n m y w o r k . D r . Steve B u r g e s o f the U n i v e r s i t y o f W a s h i n g t o n , D r . N i c k K o u w e n o f the U n i v e r s i t y o f W a t e r l o o , and m a n y other industry and research leaders also p r o v i d e d c o m m e n t a r y and g u i d a n c e as I w a s starting out  viii  O f course, I a m eternally grateful to those w h o s e assistance kept m e f i n a n c i a l l y solvent d u r i n g m y w o r k . P r i m a r y support for this endeavour w a s p r o v i d e d b y the N a t u r a l Sciences and Engineering Research C o u n c i l o f Canada ( N S E R C ) through a Post-Graduate Scholarship. A d d i t i o n a l f i n a n c i a l assistance w a s p r o v i d e d through s c h o l a r s h i p s and awards f r o m the U n i v e r s i t y o f B r i t i s h C o l u m b i a , U M A G r o u p L t d . , M r s . E a r l R . Peterson, the C a n a d i a n W a t e r R e s o u r c e s A s s o c i a t i o n , and a R e s e a r c h A s s i s t a n t s h i p w i t h m y supervisor, D r . B a r b a r a L e n c e o f C i v i l E n g i n e e r i n g at U B C . B C H y d r o also supported this project through their P r o f e s s i o n a l Partnership P r o g r a m , p r o v i d i n g a d v i c e , access to their f o r m i d a b l e l i b r a r y o f h y d r o l o g i c studies, a n d o f course the data that forms the basis o f the e x p e r i m e n t h e r e i n .  F i n a l l y , I w o u l d l i k e to thank m y colleagues at U B C , B C H y d r o , and K e r r W o o d L e i d a l , m y f a m i l y and friends, and m o s t e s p e c i a l l y m y w i f e D i a n e , w h o h a v e c o l l e c t i v e l y and e x h a u s t i v e l y put up w i t h m e t a l k i n g about this thesis for n e a r l y four years. I l o v e y o u a l l and a m pleased to say that I l o o k f o r w a r d to m o v i n g o n to the next b i g t h i n g .  ix  1. Introduction "Where there is no water, there is no life... we live by the grace of water." - National Geographic Special Edition, November 1993  T h e r e i s n o substance m o r e relevant o r m o r e necessary to the c o n t i n u a t i o n o f life o n E a r t h than water. It is a constant source o f sustenance, c o n v e n i e n c e , a n d c o n f l i c t , but rare extremes o f either presence o r absence c a n quite e a s i l y b e c o m e a matter o f life a n d death. A s such, there s h o u l d be little surprise that efforts to manage, m o v e , o r m i t i g a t e the benefits a n d hazards o f water h a v e b e e n c h r o n i c l e d for thousands o f years. Perhaps m o r e s u r p r i s i n g is the extent o f uncertainty that persists i n o u r u n d e r s t a n d i n g and p r e d i c t i o n o f the " w h e n " , " w h e r e " , and " h o w m u c h " o f water, and the l a c k o f tools a v a i l a b l e to a i d i n its characterization. F a i l u r e to appreciate the i m p l i c a t i o n s o f s u c h uncertainty c a n result i n spectacular and a w e s o m e consequences. T h i s thesis discusses the uncertainty i n u s i n g c o m p u t e r m o d e l s to p r e d i c t w a t e r s h e d response. It s p e c i f i c a l l y e x a m i n e s the influence o f uncertainty i n c a l i b r a t i o n objectives o n the p r e d i c t i o n o f f l o o d s . S e v e r a l n o n - i n f e r i o r parameter sets are u s e d to p r e d i c t r u n o f f f r o m an extreme event b a s e d o n a P M F scenario for the C o q u i t l a m R i v e r watershed above Coquitlam Lake.  F l o o d s are the m o s t o b v i o u s o f adverse consequences for a c i v i l i z a t i o n characterized b y l o w l a n d , r i p a r i a n , a n d coastal habitation. I n fact, floods are at once the m o s t c o m m o n a n d m o s t devastating o f natural disasters. O v e r t w o years ( 1 9 9 4 - 1 9 9 5 ) , f l o o d s constituted 5 0 % o f g l o b a l natural disasters, a n d w e r e r e s p o n s i b l e for 8,500 casualties ( R o s e n h a g e n a n d H a l p e r t , 1998). R e s u l t a n t s o c i a l a n d e c o n o m i c i m p a c t s c a n and do extend far b e y o n d the d i r e c t l y f l o o d e d areas ( G r e g o r y et a l . , 1996); for the p e r i o d above, total damages w e r e estimated at a p p r o x i m a t e l y U S $ 5 0 B ( R o s e n h a g e n a n d H a l p e r t , 1998). C l o s e r to h o m e , 2 0  t h  c e n t u r y C a n a d i a n floods h a v e  resulted i n several b i l l i o n d o l l a r s i n damages and at least 198 fatalities ( N a t u r a l R e s o u r c e s Canada, 2003).  P r o t r a c t e d p e r i o d s o f l o w f l o w (drought), d e c l i n i n g lake levels, a n d f a l l i n g water tables are another c o m m o n c o n c e r n for industry, government, the e n v i r o n m e n t , a n d the general p o p u l a t i o n .  1  An accurate assessment of low flows is critical to fish passage, aquatic habitat, irrigation, and water use planning (Pike and Scherer, 2003). Although the primary focus of this work lies in addressing uncertainty in hydrologic modelling of extreme floods, it would be remiss not to mention the possibility for cross-application of many concepts herein for low-flow prediction.  1.1  Guidance for the Reader  Defining the uncertainty surrounding flood management, or even flood magnitude estimation, is far beyond the scope of any single work. This thesis focuses on the technical uncertainties of hydrologic modelling, with an emphasis on the uncertainty incorporated through subjective model calibration. In particular, this work presents a synthesis of the current state of knowledge with regard to hydrologic model predictive uncertainty in the estimation of extreme floods. To illustrate the impact of the concepts discussed, an investigation of how model calibration affects the estimation of extreme flood events is undertaken. This introductory chapter takes an atypical form with the intent of placing later chapters in context. The chapter outlines some of the various ways that uncertainty can affect the selection of methods for flood estimation and the interpretation of their results. As this chapter is intended to provide the reader with an understanding of the larger context of uncertainty in which hydrologic models are applied, readers seeking a more focused discussion of applied hydrologic modelling may wish to proceed directly to Chapter 2 or 3. Chapter 2 provides a literature-based review of hydrologic modelling, beginning with a review of hydrologic processes and their potential contributions to uncertainty in modelling. The chapter explores a brief evolutionary history of hydrologic modelling, as well as the basics of model classification and various approaches for model calibration. This chapter is intended as a background for less experienced modellers or those seeking a basic review of uncertainty in hydrologic modelling. Readers comfortable with a comfortable understanding of the fundamentals of hydrologic modelling may wish to proceed directly to Chapter 3. Chapter 3 provides a more detailed literature-based discussion of the ways in which uncertainty in hydrologic modelling can be classified and explored. This chapter is intended for both beginning and experienced hydrologic modellers seeking to understand the various ways in  2  w h i c h uncertainty is i n t r o d u c e d into the m o d e l output. T h o s e f a m i l i a r w i t h the b o d y o f literature o n uncertainty i n h y d r o l o g i c m o d e l l i n g m a y w i s h to p r o c e e d d i r e c t l y to C h a p t e r 4.  C h a p t e r 4 describes an experiment that applies the U n i v e r s i t y o f B r i t i s h C o l u m b i a W a t e r s h e d M o d e l ( U B C W M ) to quantify the v a r i a b i l i t y i n t r o d u c e d i n extreme event s i m u l a t i o n through subjective assumptions made d u r i n g c a l i b r a t i o n . M u l t i p l e automatic calibrations o f a conceptual h y d r o l o g i c m o d e l are c o n d u c t e d u s i n g different measures o f performance to p r o v i d e a c o l l e c t i o n o f n o n - i n f e r i o r parameter sets. E a c h parameter set is then used to simulate an extreme event based o n h y d r o l o g i c data p r o v i d e d b y B C H y d r o . T h e c o m b i n e d output o f these extreme event s i m u l a t i o n s characterizes the relative v a r i a b i l i t y i n the hydrographs associated w i t h the use o f different c a l i b r a t i o n objectives.  C h a p t e r 5 discusses the results o f the e x p e r i m e n t o u t l i n e d i n C h a p t e r 4. C h a p t e r 6 p r o v i d e s c o n c l u s i o n s , and C h a p t e r 7 g i v e s r e c o m m e n d a t i o n s for future w o r k . These chapters w i l l l i k e l y be o f interest to a l l readers. C h a p t e r 8 p r o v i d e s a g l o s s a r y o f c o m m o n l y - u s e d abbreviations for ease o f reference.  1.2  Approaches for Estimating floods  T h e f l o o d s c a u s i n g greatest h a r m are those associated w i t h h i g h m a g n i t u d e and p o o r p r e d i c t a b i l i t y . E x t r e m e e x a m p l e s o f this c o n d i t i o n i n c l u d e debris f l o w s and floods i n steep terrain, w h i c h entrain r o c k fragments, earth, and air into the f l o o d f l o w ( H o r t o n , 1999). S i m i l a r l y , u n c o n t r o l l e d releases f r o m natural (e.g., m o r a i n e - d a m m e d ) or anthropogenic reservoirs t h r o u g h failure o f a containment structure are c o m m o n l y associated w i t h significant damages and fatalities. L a r g e , unpredictable f l o o d s c a n also result f r o m b l o c k a g e s o f the d o w n s t r e a m c h a n n e l (e.g., i c e , l o g o r debris j a m s ) , and backwaters f r o m other channels c o n c u r r e n t l y i n f l o o d c a n restrict f l o w and inundate significant upstream areas. A l l o f these cases are difficult ( i f not i m p o s s i b l e ) to predict effectively, and a l l are g e n e r a l l y b e y o n d the scope o f a watershed-scale h y d r o l o g i c m o d e l . It is appropriate to c o m m e n c e a d i s c u s s i o n o f h o w to select an approach for f l o o d e s t i m a t i o n w i t h a r e m i n d e r o f the i n s u f f i c i e n c y o f h y d r o l o g i c m o d e l s i n c h a r a c t e r i z i n g the c o m p l e t e f l o o d r i s k for certain situations. N o n e t h e l e s s , w i t h rare exceptions, e v e n these m o s t extreme cases are i n i t i a t e d b y a m o r e predictable, p u r e l y h y d r o l o g i c , h i g h - f l o w event.  3  A l m o s t every f l o o d scenario requires h e a v y r u n o f f f r o m upstream areas. Therefore, the greatest part o f s c i e n t i f i c effort i n f l o o d management has b e e n directed t o w a r d q u a n t i f y i n g the effects o f and r e l a t i o n s h i p b e t w e e n p r e c i p i t a t i o n and runoff. M a n y approaches have emerged for estimating the m a g n i t u d e o f a g i v e n h y p o t h e t i c a l f l o o d f l o w (e.g., a d e s i g n event). G e n e r i c a p p r o x i m a t i o n s s u c h as the r a t i o n a l m e t h o d or r e g i o n a l l y - d e r i v e d drainage area - discharge curves are g e n e r a l l y acceptable i n the absence o f detailed data or as c h e c k s o n m o r e detailed c a l c u l a t i o n s for s m a l l projects. Perhaps the s i m p l e s t m e t h o d for estimating f l o o d quantiles i n v o l v e s e x a m i n i n g h i s t o r i c a l events and i n f e r r i n g future events based o n those o b s e r v e d i n the past (e.g., f l o o d frequency analysis or p a l e o f l o o d h y d r o l o g y ) . M o r e c o m p l e x methods use c o m p u t e r a p p l i c a t i o n s to represent one or m o r e phases o f the h y d r o l o g i c c y c l e as a s i m p l i f i e d system. W h e r e v e r p o s s i b l e , m o r e than one m e t h o d o f e s t i m a t i n g the d e s i g n event s h o u l d be used ( N a t i o n a l R e s e a r c h C o u n c i l C a n a d a , 1989). R e g a r d l e s s o f the tool(s) i m p l e m e n t e d , it is c r u c i a l to r e c o g n i z e that e v e n the m o s t c o m p l e x tools are o n l y a p p r o x i m a t i o n s o f the natural system ( M c C u e n , 1973). E a c h approach is n a t u r a l l y subject to a u n i q u e set o f uncertainties.  A l t h o u g h any a p p r o a c h to estimating r u n o f f m a y be l o o s e l y c l a s s i f i e d as a " h y d r o l o g i c m o d e l " , the n e o - c l a s s i c a l d e f i n i t i o n o f h y d r o l o g i c m o d e l l i n g i n v o l v e s the use o f a c o m p u t e r m o d e l to predict f l o w s under g i v e n input c o n d i t i o n s . F o r this reason, a d i s t i n c t i o n is necessary b e t w e e n t w o different concepts o f a " m o d e l " . A fundamental h y d r o d y n a m i c m o d e l or concept based o n p h y s i c a l p r i n c i p l e s (e.g., k i n e m a t i c or d i f f u s i v e w a v e equation) is v e r i f i a b l e and repeatable i n appropriate c o n t r o l l e d experiments. A c o m p u t e r m o d e l effectively b u i l d s o n m o r e fundamental m o d e l s , a p p l y i n g t h e m to a c o m p l e x r e a l i t y w i t h m u l t i p l e assumptions and p o t e n t i a l c o d i n g errors ( S m i t h et a l . , 1994).  T h e r e are t w o established fundamental p h i l o s o p h i e s for d e f i n i n g a f l o o d magnitude. P r i o r to the 1970s, the m a j o r i t y o f floods c o m p u t e d for e n g i n e e r i n g d e s i g n and analysis w e r e defined p r o b a b i l i s t i c a l l y , i.e., an appropriate frequency for the d e s i g n event w a s selected or mandated, and the m a g n i t u d e defined as a co-requisite v a l u e ( B e r g a , 1998). T h e m o s t w e l l - k n o w n e x a m p l e o f this a p p r o a c h is f l o o d frequency analysis. K l e r n e s (2000b) argues s t r o n g l y against the h y d r o l o g i s t ' s l o n g s t a n d i n g dependence o n f l o o d frequency analysis to p r o v i d e frequency and m a g n i t u d e for an extreme event. M u c h - d i s c u s s e d p r o b l e m s w i t h f l o o d frequency analysis i n c l u d e the i m p a c t o f p l o t t i n g p o s i t i o n and the a p p l i c a b i l i t y o f the Independent and I d e n t i c a l l y  4  D i s t r i b u t e d R a n d o m V a r i a b l e ( E D R V ) concept. Perhaps m o s t i m p o r t a n t l y , a series o f N data cannot reasonably be expected to p r o v i d e r e l i a b l e i n f o r m a t i o n about p r o b a b i l i t i e s less than approximately  l/N.  Subsequent d e v e l o p m e n t o f m a t h e m a t i c a l and c o m p u t e r m o d e l s has contributed to the emergence o f a d e t e r m i n i s t i c approach for estimating potential f l o o d c o n d i t i o n s . A d e t e r m i n i s t i c analysis generates an estimate b y c o m b i n i n g a g i v e n set o f i n i t i a l c o n d i t i o n s and m o d e l l i n g the resultant h y d r o l o g i c response.  T h e t y p i c a l approach i n v o l v e s c o m b i n i n g observed extreme  h y d r o l o g i c a l factors u s i n g a h y d r o l o g i c m o d e l to o b t a i n a worst-case scenario. H o w e v e r , this a priori s p e c i f i c a t i o n o f input c o n d i t i o n s has engendered m u c h debate c o n c e r n i n g j u s t h o w extreme an event s h o u l d be c o n s i d e r e d .  S i n c e the 1950s, d a m and s p i l l w a y structures w i t h severe consequences o f failure have g e n e r a l l y adopted the " p r o b a b l e m a x i m u m f l o o d " ( P M F ) as a d e s i g n c r i t e r i a ( G r a h a m , 2 0 0 0 ) . T h e d e f i n i t i o n o f the P M F b e c a m e s o m e t h i n g o f a h o l y g r a i l for deterministic m o d e l l e r s , e s p e c i a l l y i n the r e a l m o f d a m safety. I n r a i n f a l l - d r i v e n watersheds, a P M F event w o u l d be d r i v e n b y the P r o b a b l e M a x i m u m P r e c i p i t a t i o n ( P M P ) , defined as the greatest depth o f p r e c i p i t a t i o n theoretically p o s s i b l e for a g i v e n l o c a t i o n , areal extent, and season ( H a n s e n et a l . , 1988).  Input  c o n d i t i o n s for watersheds that experience significant s n o w m e l t r u n o f f are somewhat m o r e c o n v o l u t e d . I n m a n y s n o w m e l t - d o m i n a t e d areas, the m a x i m u m f l o o d arises f r o m some c r i t i c a l c o m b i n a t i o n o f s n o w m e l t (as a f u n c t i o n o f a c c u m u l a t e d s n o w p a c k and temperature sequence) and h e a v y p r e c i p i t a t i o n ( p o s s i b l y o c c u r r i n g as r a i n - o n - s n o w ) .  T h e a p p l i c a t i o n o f the P M F as a d e s i g n event has r e c e n t l y b e g u n to be questioned.  Some  scientists b e l i e v e that the P r o b a b l e M a x i m u m F l o o d as defined above is too vague to a l l o w for its generic use as an a p p r o a c h for engineering d e s i g n a n d analysis. T h e r e is no current set o f standardized procedures for c a l c u l a t i n g the P M F , n o r is there a m e t h o d for q u a n t i f y i n g h o w p r o b a b l e the f l o o d r e a l l y i s . A d d i t i o n a l l y , there i s a dearth o f i n f o r m a t i o n c o n c e r n i n g b e h a v i o u r o f watersheds under f l o o d i n g o f P M F proportions. F a u l k n e r notes that h i s t o r i c floods o f r e c o r d for some rivers are i n i t i a t e d b y different processes than those p r e s u m e d to cause the P M F ( F a u l k n e r , 2 0 0 3 ) . S e v e r a l p r o m i n e n t researchers have noted that the pursuit o f q u a n t i f i c a t i o n ( " h o w to route") s h o u l d be superseded b y attempts to g a i n a better understanding o f the  5  processes involved in extreme event runoff ("what to route") (Burges, 2003; Cordova and Rodriguez-Iturbe, 1983). There is, therefore, significant uncertainty involved in selecting how any design flood should be defined. In fields such as structural engineering, government or other regulatory bodies provide guidance when calculating appropriate design conditions (e.g., load combinations); however, regulatory guidance on design flood selection is relatively limited. Since the 1970s, the move away from probabilistic flood estimation has resulted in a general shift awayfromprescriptive regulatory governance of flood selection (Berga, 1998). In the field of flood estimation, definitive and inflexible standards are of questionable value. In some cases, the design conditions suggested by prescriptive standards or guidelines do not include the critical case (Dumont and Dube, 2003). One possible alternative is presented in the guidelines published by the Canadian Dam Association in 1999 for use as best management practices (CDA, 1999). These guidelines link the selection of design floods for dams to the consequences of failure based on loss of life, environmental damage, and financial loss. Although the guidelines represent current best practice in the industry, they still lack quantitative guidance for selecting an appropriate flood condition when the consequences of failure are neither severe nor negligible. Recent thought challenges the common practice of using overly-conservative "worstcase" scenarios in such cases as inconsistent, unwarranted, and philosophically pessimistic (Lohani et al., 1997). However, in the absence of an alternative with a strong legal precedent, their use is likely to continue.  1.3  Presenting Floods for Decision-making  Most computer-based hydrologic models present a single, deterministic estimate for a flood situation with each application. However, such a simple representation of a flood is often not sufficient for effective decision-making, especially where significant safety, financial, or environmental factors are involved. The problems with simple representation are obvious: if the flood has been over-estimated, inefficient or unnecessary designs translate into higher costs; if the flood is underestimated, an unsafe design results. Between these two extremes lies a range of values that would be "acceptable" if design conditions were known with certainty.  6  In m o s t cases w e have at best a l i m i t e d understanding o f w h e r e o n the safety-vs.-efficiency c o n t i n u u m any g i v e n deterministic estimate lies. T h e presentation o f "representative", "average", or "best" results w i t h o u t a d e s c r i p t i o n o f their associated uncertainties gives an i l l u s i o n o f p r e c i s i o n and o b j e c t i v i t y , e s p e c i a l l y to those not f a m i l i a r w i t h the approaches or tools u s e d i n the estimate ( K e e n e y and W i n t e r f e l d t , 1989). T h e literature r e v i e w i n Chapters 2 and 3 is c o n c e r n e d w i t h i d e n t i f y i n g the v a r i o u s m e c h a n i s m s that render p r e c i s i o n and o b j e c t i v i t y unattainable i n h y d r o l o g i c m o d e l l i n g , at least at present.  T h o s e presenting f l o o d estimates for analysis must be c o n s c i o u s o f the g r o w i n g i n v o l v e m e n t o f the p u b l i c i n the d e c i s i o n - m a k i n g process. T h e portrayal o f f l o o d - m i t i g a t i o n projects as p r e v e n t i n g a l l or v i r t u a l l y a l l floods (i.e., b y d e s i g n i n g for the P M F o r other e x t r e m e l y rare events) has h a d a p o l a r i z i n g effect. T y p i c a l l y , p e o p l e are left either unaware o f the p o t e n t i a l for failure o r s k e p t i c a l o f the experts' analysis and d e s i g n ( L i n s l e y et a l . , 1992; S l o v i c , 1992). T h e d i s c u s s i o n o f l o w - p r o b a b i l i t y events i n the absence o f n u m e r i c a l data is p a r t i c u l a r l y difficult for the p u b l i c to interpret, as qualitative concepts l i k e " a s m a l l chance o f b e i n g e x c e e d e d " c a n have large ranges o f interpretation ( K e e n e y a n d W i n t e r f e l d t , 1989).  D i f f i c u l t i e s o f presenting the results o f a h y d r o l o g i c s i m u l a t i o n for d i s c u s s i o n or d e c i s i o n are m a g n i f i e d w h e n the i m p a c t o f that i n f o r m a t i o n is unclear. U n c e r t a i n t y and disagreement c a n arise i n translating f l o w s into water l e v e l s , failure p r o b a b i l i t i e s for h y d r a u l i c structures, or consequences for aquatic resources (e.g., C a i s s i e and E l - J a b i , 2 0 0 3 ) . T h i s is c o m p l i c a t e d w h e n different stakeholders have different v i e w p o i n t s and objectives ( G r e g o r y et a l . , 1996). M u t u a l l y agreeable objectives for i d e n t i f y i n g , m e a s u r i n g , and understanding i m p a c t s are a pre-requisite for any group-oriented analysis o f t e c h n i c a l issues ( F i e r i n g , 1976).  W h e r e non-experts must interpret uncertain f l o o d r i s k s , biases a n d beliefs c a n p l a y a significant role i n arguments and d e c i s i o n s . A v a r i e t y o f p e r s o n a l characteristics (e.g., personality, o p i n i o n s , values, e c o n o m i c or c u l t u r a l context) and situational factors (e.g., voluntariness o f exposure, f a m i l i a r i t y , control) have been n o t e d to i n f l u e n c e the r e l a t i o n b e t w e e n p e r c e i v e d r i s k , p e r c e i v e d benefit, and r i s k acceptance; few o f these factors are e x p l i c i t l y quantifiable ( G r e g o r y et a l . , 1996; S l o v i c , 1987; S l o v i c , 1992). Therefore, experts h a v e noted that the p u b l i c assigns t e c h n i c a l l y unsubstantiated perceptions o f r i s k to certain situations ( G r e g o r y et a l . , 1996).  7  E x p e r t s m u s t understand the i m p a c t s o f a n y biases that stakeholders m a y have, because this c a n p o t e n t i a l l y have as m u c h i m p a c t o n the q u a l i t y o f a d e c i s i o n as the t e c h n i c a l uncertainties.  F u r t h e r d i f f i c u l t i e s c a n arise i f non-expert stakeholders are r e q u i r e d to use i n t u i t i o n to interpret statistical i n f o r m a t i o n . E v e n e x p e r i e n c e d researchers m a y a v o i d s i m p l e p r o b l e m s (e.g., the g a m b l e r ' s fallacy) w h i l e f a l l i n g p r e y to the same biases under i n t u i t i v e j u d g m e n t o f m o r e intricate and less transparent p r o b l e m s ( T v e r s k y and K a h n e m a n , 1974).  1.4  Expressing Uncertainty  S i n c e uncertainty and r i s k are related at a fundamental l e v e l , a d e c i s i o n as to w h a t is acceptable s h o u l d address b o t h the degree o f r e s i d u a l r i s k that is " a c c e p t a b l e " to stakeholders and the uncertainty s u r r o u n d i n g the r i s k analysis. F i g u r e 1-1 illustrates w h y this is necessary; it s h o w s the r e l a t i o n s h i p b e t w e e n the r i s k and uncertainty, w h e r e r i s k is defined as the product o f frequency a n d consequence. N o t e that each c u r v e represents an e q u a l l y v a l i d but distinct interpretation o f r i s k . E a c h c u r v e is u n i q u e l y defined b y its l e v e l o f (un)certainty, expressed i n this case as c o n f i d e n c e l e v e l . T h e figure i m p l i e s that any arbitrary frequency w i l l be associated w i t h a range o f p o s s i b l e consequences, each w i t h a different co-requisite l e v e l o f uncertainty; l i k e w i s e , each consequence w i l l have a range o f p o s s i b l e frequencies. Therefore, it is i m p o s s i b l e to e x p l i c i t l y define an acceptable l e v e l o f r i s k w i t h o u t at least i m p l i c i t l y d e f i n i n g a c o r r e s p o n d i n g l e v e l o f acceptable uncertainty.  O n e o f the few l e s s - e x p l o r e d areas for p o l i c y d e v e l o p m e n t i n h y d r o l o g y and other risk-related fields is the d e t e r m i n a t i o n o f what constitutes "acceptable uncertainty". Precedents exist for the s e l e c t i o n o f a s i n g l e threshold v a l u e ; for e x a m p l e , transportation e n g i n e e r i n g r e g u l a r l y i m p l e m e n t s projects d e s i g n e d to meet safety standards for a s p e c i f i c percentile o f drivers. A m o r e direct e x a m p l e is g i v e n b y the U n i t e d K i n g d o m H e a l t h and Safety E x e c u t i v e ( H S E , 2001) t h r o u g h their a d o p t i o n o f 10" as an appropriate b o u n d a r y b e t w e e n tolerable and unacceptable 4  r i s k for situations i n w h i c h the r i s k is i m p o s e d or i n v o l u n t a r y .  A s s h o w n i n F i g u r e 1-1, different levels o f uncertainty require different l e v e l s o f p r o t e c t i o n and m a y therefore d r i v e different c h o i c e s a m o n g alternatives ( N R C , 2 0 0 0 a ) . Projects h a v i n g m a r g i n a l or indeterminate benefit-cost ratios c a n be the most sensitive to uncertainty, as the  8  log scale  90% confidence 5 0 % confidence 10% confidence log scale  Figure 1-1: Risk - Uncertainty Interaction (from p. 5-20, Lohani et at,  1997)  uncertainty m a y be sufficient to j u s t i f y or deter their a p p r o v a l ( N R C , 1995). F o r e x a m p l e , a s m a l l uncertainty i n the c o n d i t i o n s l e a d i n g to failure o f a levee c a n m a k e a large difference i n the o v e r a l l system r e l i a b i l i t y and therefore c a n i n f l u e n c e d e c i s i o n s p e r t a i n i n g to the project as a w h o l e ( N R C , 2 0 0 0 b ) . I n m a n y cases, it is reasonable to expect that the a v a i l a b l e i n f o r m a t i o n is insufficient to e v e n b e g i n to p r o p e r l y address cost-benefit studies. T h i s i s a n undesirable state o f affairs f r o m b o t h fiscal analysis and p u b l i c safety perspectives.  F o r the v a r i o u s reasons above, an e x p r e s s i o n o f the associated uncertainty s h o u l d be r e q u i r e d i n a l l r i s k analyses ( N R C , 1995). T o o often results are l i m i t e d to values c a l c u l a t e d w i t h the "best" estimates or b y a v e r a g i n g o v e r the f i n a l p r o b a b i l i t y d i s t r i b u t i o n . E v e n w h e r e uncertainties are a c k n o w l e d g e d and accounted for, answers p r o v i d e d b y scientists are often d i v o r c e d f r o m their uncertainty as they are passed " u p the tree" to stakeholders or d e c i s i o n - m a k e r s ( G r a y s o n et a l . , 1992b).  T h e m o s t t h o r o u g h a p p r o a c h to e x p r e s s i n g uncertainty i n v o l v e s presenting a f u l l d e s c r i p t i o n o f the u n c e r t a i n results to d e c i s i o n - m a k e r s (e.g., a c u m u l a t i v e d i s t r i b u t i o n s h o w i n g the c o n t i n u u m o f event m a g n i t u d e and p r o b a b i l i t y ) . A l t h o u g h this c a n sometimes be d i f f i c u l t to interpret, this  9  f u l l set o f results effectively a l l o w s the p o l i c y d e c i s i o n (i.e., w h a t constitutes acceptable r i s k ) to be separated from the t e c h n i c a l analysis. A l t e r n a t i v e l y , L o h a n i et a l . ( L o h a n i et a l . , 1997) argue for presenting m e a n values to describe m a g n i t u d e w h i l e i n c l u d i n g l o w p r o b a b i l i t y - h i g h consequence i n f o r m a t i o n w h e r e v e r the i m p a c t s warrant.  In cases w h e r e extensive quantitative i n f o r m a t i o n i s not a v a i l a b l e , the use o f successive a p p r o x i m a t i o n s c a n sometimes p r o v i d e acceptable b o u n d s for the quantity o f interest ( K e e n e y and W i n t e r f e l d t , 1989). E x a m p l e s i n c l u d e the l i m i t i n g frequency o f 10" i m p l i c i t i n the State o f 6  W a s h i n g t o n ' s d e f i n i t i o n o f a d e s i g n f l o o d for d a m s or the c o n c e p t u a l procedure for progressive refinement o f " U l t i m a t e L i m i t State" estimates ( F a u l k n e r , 2 0 0 3 ; H a r t f o r d et a l . , 2 0 0 1 ) .  T h e above d i s c u s s i o n s have illustrated s o m e o f the w a y s that uncertainty c a n manifest i n h y d r o l o g i c analysis t h r o u g h subjective c h o i c e s m a d e i n approach, event selection, acceptable risk, biases, a n d the i n c l u s i o n o f uncertainty i n an analysis. H o w e v e r , the process o f d e c i s i o n m a k i n g does not o c c u r i n a static e n v i r o n m e n t , and the potential for change is a m a j o r source o f uncertainty. C h a n g e s i n p o l i c y , k n o w l e d g e , t e c h n o l o g y , or potential consequences c a n a l l create a " m o v i n g target" effect w h e n attempting to address f l o o d s and their i m p a c t s ( F a u l k n e r , 2 0 0 3 ) . In m o s t cases, the factors are interdependent.  F o r e x a m p l e , Jarrett (1990) u s e d p a l e o h y d r o l o g i c  evidence to s h o w that large f l o o d s o b s e r v e d at h i g h e r elevations i n C o l o r a d o are l i k e l y due to debris f l o w s rather than intense h i g h - e l e v a t i o n p r e c i p i t a t i o n . T h i s change i n k n o w l e d g e affects potential consequences t h r o u g h a certain but indeterminate increase i n the expected frequency o f f l o w s o f s i m i l a r magnitude.  T h e r e s u l t i n g increase i n uncertainty m i g h t cause a p o l i c y shift i n  the r e g i o n , l e a d i n g to the i m p l e m e n t a t i o n o f a d d i t i o n a l structural or non-structural protective measures.  1.5  Model Predictive Uncertainty in Context  T h e broad-based nature o f the h y d r o l o g i c system leads G r a y s o n et a l . (1992b) to refer to the study o f h y d r o l o g y as " t r a n s s c i e n t i t l e " - i m p l y i n g the pursuit o f answers to questions asked o f science w h i c h cannot be answered b y science. T h e d i s c u s s i o n s o f the p r e c e d i n g sections d o not address the m a n y w a y s that uncertainty c a n manifest i t s e l f quantitatively w i t h i n the actual process o f m o d e l l i n g - the " s c i e n t i f i c " p o r t i o n o f a n y f l o o d study. These m o r e t e c h n i c a l uncertainties m u s t be c o m b i n e d w i t h v a l u e j u d g m e n t s , biases, and other n o n - t e c h n i c a l  10  uncertainties. T h e u l t i m a t e g o a l is to define the r o l e that uncertainty p l a y s , and identify h o w it interacts w i t h r e s u l t i n g r i s k s , options, and d e c i s i o n s for a n y g i v e n situation.  U n c e r t a i n t y is also a necessary c o n s i d e r a t i o n for h y d r o l o g i c m o d e l l i n g i t s e l f ( O ' C o n n e l l and T o d i n i , 1996). In a d d i t i o n to the caveats a p p l i c a b l e to c o m p u t e r m o d e l l i n g i n any f i e l d (e.g., garbage i n - garbage out), h y d r o l o g i c m o d e l s create a d d i t i o n a l challenges. M o s t i m p o r t a n t l y , the results generated b y c o m p u t e r s i m u l a t i o n s fuse together d i v e r s e types o f uncertainty, s u c h as those a r i s i n g f r o m natural processes and those related to m a t h e m a t i c a l representation.  A l e a t o r y uncertainty, or natural v a r i a b i l i t y , represents the v a r i a b i l i t y o f the p h y s i c a l w o r l d o n the a s s u m p t i o n that this v a r i a b i l i t y cannot be m i t i g a t e d . C o n v e r s e l y , the U . S . N a t i o n a l R e s e a r c h C o u n c i l (p. 4 1 , N R C , 2000a) relates the concept o f e p i s t e m i c o r k n o w l e d g e - b a s e d uncertainty to " a l a c k o f understanding o f events and processes, o r a l a c k o f data f r o m w h i c h to d r a w inferences". K n o w l e d g e - b a s e d uncertainty c a n be r e d u c e d under the correct c o n d i t i o n s . S i n c e m o s t h y d r o l o g i c m o d e l s d o not e x p l i c i t l y account for uncertainty i n any f o r m , an understanding o f the m a g n i t u d e and r e l a t i o n s h i p o f these t w o c o m p o n e n t s is c r u c i a l . U n d e r s t a n d i n g the uncertainty o f the a p p l i e d m o d e l s and processes is an o b v i o u s prerequisite for assessing the uncertainty i n a n y flood-related d e c i s i o n ( N R C , 1995).  W i t h i n the category o f e p i s t e m i c uncertainty, p o t e n t i a l sources for uncertainty i n c l u d e m o d e l structure, m a t h e m a t i c a l representation, m o d e l parameter v a l u e s , and input data errors ( L e i and S c h i l l i n g , 1996). These different types o f uncertainty i n h y d r o l o g i c m o d e l l i n g are often l u m p e d together i n an a p p l i e d h y d r o l o g i c a l context to create an aggregate " m o d e l p r e d i c t i v e uncertainty" ( N R C , 1995). A n y o f the types o f uncertainty l i s t e d above has the potential to influence results, and therefore a l l s h o u l d be addressed i n any uncertainty analysis ( V i c e n s et a l . , 1975).  It has often been assumed that m o d e l c a l i b r a t i o n c a n r e s o l v e and reduce a l l c o m p o n e n t s o f m o d e l p r e d i c t i v e uncertainty. H o w e v e r , c a l c u l a t i n g m o d e l p r e d i c t i v e uncertainty w i t h techniques s u c h as s e n s i t i v i t y analysis is i m p l i c i t l y dependent o n c a l i b r a t i o n ; it i s o f o n l y l i m i t e d v a l u e unless the m o d e l and data errors are k n o w n to be i n s i g n i f i c a n t ( L e i and S c h i l l i n g , 1996). Past analyses o f " m o d e l p r e d i c t i v e uncertainty" for h y d r o l o g i c m o d e l s h a v e s h o w n a w i d e range o f v a r i a t i o n i n m o d e l accuracy. T h e accuracies o f r a i n f a l l - r u n o f f s i m u l a t i o n s reported i n the literature are t y p i c a l l y i n f l u e n c e d b y factors s u c h as those presented b y M i c h a u d and S o r o o s h i a n (1994):  11  •  the i n c l u s i o n o r e x c l u s i o n o f m o d e l c a l i b r a t i o n ;  •  the i n c l u s i o n or e x c l u s i o n o f split-sample v a l i d a t i o n ;  •  the number, variety, and c l i m a t o l o g y o f storms e x a m i n e d ;  •  different understandings o f " g o o d " and " p o o r " s i m u l a t i o n s ;  •  b e n c h m a r k data r e l i a b i l i t y ;  •  the context o f results (e.g., real-time forecasting, s i n g l e h i s t o r i c results, or m u l t i p l e peak f l o w s f r o m a set o f h i s t o r i c a l storms);  •  r u n o f f d y n a m i c s (e.g., i n i t i a l c o n d i t i o n s , d o m i n a n t processes); and  •  m o d e l assumptions, parameter values, and spatial r e s o l u t i o n .  A t t e m p t i n g to address m a n y o f the m o s t c o m m o n p r o b l e m s i n the present era o f h i g h - p o w e r e d d i g i t a l computers has s o m e w h a t p r e d i c t a b l y l e d to the d e v e l o p m e n t o f some i m m e n s e l y c o m p l e x c o m p u t a t i o n a l m o d e l s o f h y d r o l o g y . H o w e v e r , theoretical rigour does not i n and o f i t s e l f l i m i t uncertainty, and c a n i m p l y a degree o f accuracy that m a y not exist ( G r a y s o n et a l . , 1992b). R a t h e r than i n c r e a s i n g m o d e l c o m p l e x i t y , progress i n r e d u c i n g m o d e l p r e d i c t i v e uncertainty w i l l l i k e l y depend o n the establishment o f a n e w p a r a d i g m that i n c l u d e s an acceptance o f uncertainty i n the results ( B e v e n , 2 0 0 2 ) . A best practice approach is needed to a l l o w professionals to m o v e o n to d e f i n i n g appropriate p r i n c i p l e s rather than arguing about h o w the impacts o f uncertainty s h o u l d be addressed ( F a u l k n e r , 2 0 0 3 ) .  12  2. Hydrologic Modelling "Models are like maps: never final, never complete until they grow as large and complex as the reality they represent." - James Gleick, from "Genius: The Life and Science of Richard Feynman "  T h i s chapter is intended as b a c k g r o u n d for less e x p e r i e n c e d m o d e l l e r s or for those s e e k i n g a b a s i c r e v i e w o f the fundamentals o f h y d r o l o g i c m o d e l l i n g . R e a d e r s s e e k i n g a m o r e a d v a n c e d and focussed d i s c u s s i o n o f uncertainty m a y w i s h to p r o c e e d d i r e c t l y to C h a p t e r 3.  A s o l i d understanding o f processes represented i n h y d r o l o g i c m o d e l l i n g is r e q u i r e d for any d i s c u s s i o n o f m o d e l p r e d i c t i v e uncertainty. Therefore, this chapter b e g i n s w i t h a s u m m a r y o f v a r i o u s processes important to m o d e l l i n g . A l t h o u g h the concepts are b a s i c , the n o v i c e m o d e l l e r is encouraged to c o n s i d e r the d i s c u s s i o n i n terms o f the potential uncertainty inherent i n m o d e l l i n g the m o r e c o m p l e x aspects o f the system. T h o s e f a m i l i a r w i t h the c o m p l e x i t y o f these processes and the related s i m p l i f i c a t i o n s and assumptions i m p l i c i t i n different h y d r o l o g i c m o d e l s w i l l l i k e l y w i s h to p r o c e e d d i r e c t l y to S e c t i o n 2.2. S e c t i o n 2.2 p r o v i d e s the reader w i t h insight into the a p p r o a c h and p h i l o s o p h y o f h y d r o l o g i c m o d e l d e v e l o p m e n t a n d a p p l i c a t i o n . I n S e c t i o n 2.3, a b r i e f o v e r v i e w o f the v a r i o u s k i n d s o f m o d e l s addresses their advantages and disadvantages. A detailed d i s c u s s i o n o f automatic and m a n u a l c a l i b r a t i o n f o l l o w s i n S e c t i o n 2.4, h i g h l i g h t i n g the n e e d for c o n s i d e r i n g uncertainty i n the c a l i b r a t i o n process. In particular, S e c t i o n 2.4 introduces the S h u f f l e d C o m p l e x E v o l u t i o n m e t h o d d e v e l o p e d at the U n i v e r s i t y o f A r i z o n a ( S C E - U A ) , w h i c h is u t i l i z e d i n the quantitative experiment o u t l i n e d i n C h a p t e r 4. T h i s chapter c o n c l u d e s w i t h a d i s c u s s i o n o f s o m e o f the factors l i m i t i n g progress i n h y d r o l o g i c m o d e l l i n g , w h i c h leads into the d i s c u s s i o n o f uncertainty i n C h a p t e r 3.  13  Hydrologic Processes  2.1  2.1.1  Scale in Hydrology  In nature, scales o f things are not arbitrary but tend to concentrate a r o u n d discrete states as a f u n c t i o n o f their m a t e r i a l substance and o f the balance b e t w e e n the interacting forces ( K l e m e s , 1983). S c i e n t i f i c progress has t y p i c a l l y b e e n s l o w e r i n d i s c i p l i n e s attempting to w o r k b e t w e e n d o m i n a n t scale levels than i n those w o r k i n g w i t h i n a s i n g l e scale (ibid.). H y d r o l o g y is a c l a s s i c e x a m p l e ; c o m p o n e n t processes c a n be active at m a n y different spatial and t e m p o r a l scales f r o m m i n u t e to g l o b a l . K l e m e s ( i b i d . ) generalizes the " c h a r a c t e r i s t i c " scale o f h y d r o l o g y as b e t w e e n 1 and 1000 k m i n space and 100 seconds to 100 years i n t i m e . H o w e v e r , s u c h generalizations 2  serve o n l y i n p h i l o s o p h i c a l d i s c u s s i o n s ; the p r a c t i c i n g h y d r o l o g i c m o d e l l e r must understand w h a t is o c c u r r i n g at a l l scales to a v o i d m a k i n g i r r e s p o n s i b l e s i m p l i f i c a t i o n s . A c c o r d i n g to S o n g a n d James (1992), h y d r o l o g i c processes c a n be c h a r a c t e r i z e d at five t y p i c a l scales, i n c l u d i n g the following:  •  laboratory scale - t y p i c a l l y less than 1 0 m , for d e s c r i b i n g detailed p h y s i c s o f watersurface interactions (e.g., i n f i l t r a t i o n ) or subsurface processes;  •  p l o t or h i l l s l o p e scale - t y p i c a l l y tens o f metres, for d e s c r i b i n g r u n o f f processes;  •  catchment scale - t y p i c a l l y hundreds to thousands o f metres, for c h a r a c t e r i z i n g the interaction o f v a r i o u s h i l l s l o p e s feeding into a single c h a n n e l ;  •  b a s i n or watershed scale - t y p i c a l l y tens to thousands o f k i l o m e t r e s , for c h a r a c t e r i z i n g the generation, storage and translation r o u t i n g o f a c h a n n e l n e t w o r k or r i v e r system; and  •  continental or g l o b a l scale - thousands o f k i l o m e t r e s and greater, for c h a r a c t e r i z i n g the atmospheric processes that d r i v e the h y d r o l o g i c c y c l e .  C o m m o n a l i t i e s o f v e g e t a t i o n and l a n d use at each scale t y p i c a l l y have associated c o m m o n a l i t i e s o f u n d e r l y i n g h y d r o l o g i c a l m e c h a n i s m s or b e h a v i o u r s (e.g., a l p i n e v s . sub-alpine at the catchment scale; t r o p i c a l v s . temperate at the b a s i n scale) ( S i n g h , 1995b). V i e s s m a n and L e w i s  14  (1996) note that it is easiest to deal w i t h h y d r o l o g y at the watershed or r i v e r b a s i n scale due to the r e l a t i v e l y sharp boundaries o f the r u n o f f system. It is c o m m o n l y accepted, h o w e v e r , that no adjustment o f scale c a n place such w e l l - d e f i n e d boundaries o n the other components o f the water balance; the h y d r o l o g i c c y c l e is a c l o s e d system o n l y at the g l o b a l scale ( B e v e n , 2 0 0 0 ; K l e m e s , 1983; V i e s s m a n and L e w i s , 1996).  2.1.2 Precipitation P r e c i p i t a t i o n c a n take m a n y forms, but the t w o m o s t c o m m o n (i.e., r a i n and s n o w ) are o f greatest i m p o r t for h y d r o l o g i c m o d e l l i n g . T h e m a i n difference f r o m a h y d r o l o g i c a l perspective is the delayed r u n o f f response associated w i t h s n o w . T h i s difference is v e r y important f r o m the standpoint o f h y d r o l o g i c m o d e l l i n g , as r a i n - d o m i n a t e d basins p r o m o t e a focus o n the accurate capture o f i n d i v i d u a l events w h i l e s i m u l a t i o n o f s n o w m e l t - d o m i n a t e d basins is m o r e concerned w i t h seasonal p r e c i p i t a t i o n totals ( M i c o v i c , 2003a). O t h e r forms o f p r e c i p i t a t i o n (e.g., sleet, h a i l , graupel) are less c o m m o n ; their s i g n i f i c a n c e to h y d r o l o g i c m o d e l l i n g is therefore l i m i t e d .  There are three p r i m a r y categories o f p r e c i p i t a t i o n events: c o n v e c t i v e , orographic, and c y c l o n i c or frontal ( V i e s s m a n and L e w i s , 1996). C o n v e c t i v e p r e c i p i t a t i o n arises w h e n m o i s t air heated near the terrestrial interface rises and c o o l s , a n d t y p i c a l l y results i n short-term h i g h - i n t e n s i t y l o c a l p r e c i p i t a t i o n i n the area o f the updraft ( A M S , 2 0 0 0 ; H o r t o n , 1999).  Orographic  p r e c i p i t a t i o n results f r o m the l i f t i n g o f m o i s t air masses o v e r natural barriers s u c h as ridges or m o u n t a i n ranges, and is c o n t r o l l e d b y barrier slope, height o f barrier, and air m a s s stability ( Q u i c k , 1995; V i e s s m a n and L e w i s , 1996). F r o n t a l p r e c i p i t a t i o n results f r o m the interaction o f t w o air masses o f different density, almost i n v a r i a b l y segregated b y temperature ( A M S , 2000). T h e t e r m "frontal p r e c i p i t a t i o n " is m o s t significant i n its sense o f d i s t i n c t i o n f r o m c o n v e c t i v e and o r o g r a p h i c p r e c i p i t a t i o n ( V i e s s m a n and L e w i s , 1996).  T h e different m e c h a n i s m s o f p r e c i p i t a t i o n generation are distinct i n their b e h a v i o u r and s h o u l d be m o d e l l e d as s u c h w h e r e v e r p o s s i b l e , since factors s u c h as p r e c i p i t a t i o n and t i m i n g c a n often c o n t r o l the f l o o d response o f a b a s i n ( K o n r a d , 2 0 0 1 ) . F o r e x a m p l e , the v a r i o u s p r e c i p i t a t i o n events that l e d to the M i s s i s s i p p i R i v e r f l o o d o f 1993 w e r e not a m o n g the m o s t extreme events o f r e c o r d at a n y spatial scale. Stationary or s l o w - m o v i n g storm systems c a n also lead to f l o o d c o n d i t i o n s (e.g., S p r i n g C r e e k , C o l o r a d o i n O g d e n et a l . (2000) and K i c k a p o o C r e e k , T e x a s i n  15  S m i t h et a l . (2000) ). S u c c e s s i v e l y i n c r e a s i n g peaks o f r a i n f a l l f r o m a s t o r m m o v i n g i n the d o w n s t r e a m d i r e c t i o n represent the w o r s t case scenario f r o m a h y d r a u l i c standpoint. I n this case, subsequent w a v e s o f r u n o f f c a n propagate a n d overtake p r e c e d i n g w a v e s ( O g d e n et a l . , 2 0 0 0 ; Thapa and K h a n a l , 2001).  S i n c e v o l u m e , t i m i n g , a n d d i s t r i b u t i o n o f p r e c i p i t a t i o n are the m o s t significant factors i n d e t e r m i n i n g f l o o d m a g n i t u d e , it is n o surprise that techniques for the measurement o f p r e c i p i t a t i o n are w e l l d e v e l o p e d . T h e t w o m a i n sources for r a i n data are gauge n e t w o r k s a n d radar measurement.  R e c o r d i n g p r e c i p i t a t i o n gauges are the m o s t c o m m o n f o r m o f data used i n h y d r o l o g i c research, p r o v i d i n g c o n t i n u o u s p o i n t estimates o f p r e c i p i t a t i o n at a specific l o c a t i o n ( D u c h o n a n d E s s e n b e r g , 2 0 0 1 ) . T h e t w o most c o m m o n l y u s e d classes o f r e c o r d i n g gauges are t i p p i n g b u c k e t and w e i g h i n g gauges.  A t i p p i n g b u c k e t gauge i n v o l v e s a s m a l l , bi-stable b u c k e t h a v i n g t w o chambers, each w i t h a v o l u m e equivalent to a fraction o f m i l l i m e t e r o f r a i n . T h e presence o f water i n one side o f the bucket w i l l cause it to tip to that side, s p i l l i n g the f u l l c h a m b e r a n d a l i g n i n g the e m p t y c h a m b e r i n p o s i t i o n to c o l l e c t p r e c i p i t a t i o n . A datalogger records the n u m b e r o f tips w i t h i n a specified period.  A w e i g h i n g gauge records c o n t i n u o u s o r p e r i o d i c m e c h a n i c a l measurements o f the c u m u l a t i v e w e i g h t o f p r e c i p i t a t i o n b y u s i n g a r o l l p l o t o r electronic means, a n d therefore requires p e r i o d i c c a l i b r a t i o n i n the f i e l d ( D u c h o n a n d E s s e n b e r g , 2 0 0 1 ) . F o r a m o r e c o m p l e t e d i s c u s s i o n o f automatic gauges, the reader is referred to N y s t u e n et a l . (1996).  N o n - r e c o r d i n g r a i n gauges are m u c h m o r e straightforward but do not a l l o w the user to estimate r a i n f a l l rate o r intensity. R e g a r d l e s s , s u c h data h a v e p r o v e n useful to h y d r o l o g i c m o d e l l e r s i n the past (e.g., Faures et a l . , 1995). N o n - r e c o r d i n g r a i n gauges t y p i c a l l y consist o f an a m p l i f i e d c o l l e c t o r w h i c h i s m e a s u r e d m a n u a l l y against a f i x e d o r r e m o v a b l e scale. T h e i r s i m p l i c i t y has l e d to the d e v e l o p m e n t o f an extensive n e t w o r k o f amateur m e t e o r o l o g i c a l stations across the U n i t e d States ( N a t i o n a l W e a t h e r S e r v i c e , 2 0 0 3 ) .  16  T h e e x p a n s i o n o f D o p p l e r radar across N o r t h A m e r i c a p r o v i d e s an alternative f o r m o f p r e c i p i t a t i o n measurement.  W h i l e r e c o r d i n g r a i n gauges capture t e m p o r a l v a r i a b i l i t y for a g i v e n  l o c a t i o n , radar r a i n f a l l estimates c a n characterize spatial v a r i a b i l i t y for a g i v e n series o f t e m p o r a l snapshots. R a d a r c a n p r o v i d e t h r e e - d i m e n s i o n a l observations o v e r thousands o f square k i l o m e t r e s . T h e p h y s i c s i n v o l v e d i n radar measurement l i m i t its a c c u r a c y at greater distances, and u s u a l l y an expert must r e v i e w results p r i o r to use to a v o i d i n c o r p o r a t i o n o f a n o m a l i e s into the data set.  S o l i d p r e c i p i t a t i o n (i.e., s n o w ) is u s u a l l y m e a s u r e d w i t h a heated t i p p i n g bucket gauge or a w e i g h i n g gauge treated w i t h antifreeze to m e l t the s n o w o n contact. W h i l e m u c h less c o m m o n , radar c a n also be u s e d to estimate s n o w f a l l . C o l l i e r and L a r k e (1978) f i n d that radar measurements c a n h a v e as little as 1 3 % error w h e n c o m p a r e d to gauge-measured data.  More  generally, h o w e v e r , radar estimates o f s n o w f a l l h a v e not been v e r y successful ( X i a o et a l . , 1998). Studies i n v o l v i n g radar and ground-based s n o w measurements s h o u l d be interpreted w i t h care, as radar e s t i m a t i o n relationships c a n be dependent o n diverse factors s u c h as range, l o c a t i o n , temperature, s n o w f a l l type, and season ( i b i d . ; H u n t e r et a l . , 2 0 0 1 ) . In m o s t p r a c t i c a l cases, error is l i k e l y to be at least several times that reported b y C o l l i e r and L a r k e (1978), and c a n e a s i l y e x c e e d a factor o f t w o ( K r a j e w s k i , 2 0 0 5 ) .  S o n i c s n o w depth sensors and s n o w p i l l o w s are also c o m m o n l y used to r e c o r d variations i n s n o w depth and s n o w water equivalent o v e r t i m e . M a n u a l s n o w depth s a m p l i n g (i.e., s n o w courses) are r o u t i n e l y p e r f o r m e d for the purposes o f e s t i m a t i n g the s p r i n g freshet.  I n a less t r a d i t i o n a l  context, M a t t h e w s (1999) demonstrates h o w remote sensing c a n be a p p l i e d to determine the s n o w - c o v e r e d area for a b a s i n and thus p r o v i d e i n s i g h t into s n o w p a c k generation and d e p l e t i o n .  T h e r e are t w o general approaches for estimating s n o w m e l t : energy balance methods and i n d e x methods ( M a i d m e n t , 1993). T h e p h y s i c a l l y - b a s e d energy balance m e t h o d applies c o n t i n u i t y p r i n c i p l e s to the v a r i o u s energy fluxes o f a watershed ( V i e s s m a n and L e w i s , 1996). T h e m a i n d r a w b a c k o f the energy balance m e t h o d is that it requires significant amounts c o l l e c t e d f r o m w i t h i n the b a s i n (e.g., r a d i a t i o n , w i n d , v a p o u r pressure).  T h e v a r i o u s i n d e x - b a s e d methods for estimating s n o w m e l t use calibrated parameters and i n d e x variables to estimate s n o w m e l t o n a c a t c h m e n t - w i d e basis ( M a t t h e w s , 1999). Index m e t h o d s are  17  g e n e r a l l y less accurate but easier to a p p l y , r e q u i r i n g o n l y one o r t w o data series for their i n d e x variable(s). A i r temperature i s the m o s t c o m m o n l y u s e d v a r i a b l e due to its w i d e a v a i l a b i l i t y and strong c o r r e l a t i o n to s n o w m e l t processes. T h e W o r l d M e t e o r o l o g i c a l O r g a n i z a t i o n ' s 1986 c o m p a r i s o n o f s n o w m e l t m o d e l s c o n c l u d e s that the c h o i c e o f m o d e l u s u a l l y depends o n the intent o f the a p p l i c a t i o n and the nature o f the p r o b l e m and h y d r o l o g i c r e g i m e ( W M O , 1986).  2.1.3 Runoff A n u m b e r o f processes c o m p r i s e the transition o f p r e c i p i t a t i o n to streamflow. A c o m p l e t e t a x o n o m y o f the processes active i n a n y one catchment w o u l d s h o w that o v e r t i m e , the c o m p o s i t i o n o f o u t f l o w b y source changes d e p e n d i n g o n w h i c h processes are most active i n a v o l u m e t r i c sense. In m o s t areas, the in situ relative c o n t r i b u t i o n s cannot be measured d i r e c t l y ; h o w e v e r , quantitative estimates are often p o s s i b l e . I n cases w h e r e source c h e m i s t r y c a n be i d e n t i f i e d for each m e c h a n i s m , the u t i l i z a t i o n o f c h e m i c a l data and tracers a l l o w s researchers to e x p l o r e quantitative estimates for the p r o p o r t i o n a l o r i g i n o f r u n o f f waters (Hornberger and B o y e r , 1995).  F o r decades, studies have attempted to characterize the properties o f v a r i o u s r u n o f f responses. F o r e x a m p l e , D u n n e (1982) p r o v i d e s estimates o f t y p i c a l v e l o c i t i e s for o v e r l a n d , subsurface, and c h a n n e l f l o w s . T h i s s e c t i o n b r i e f l y describes the m e c h a n i c s o f and approaches for e s t i m a t i n g the translation o f p r e c i p i t a t i o n into runoff.  P r e c i p i t a t i o n m a y be intercepted b y vegetation or c o l l e c t i n m i n o r depressions i n the g r o u n d surface. These t w o intermediate processes are b o t h s i n k s for p r e c i p i t a t i o n , but neither contributes d i r e c t l y to runoff; m o s t o f the water detained b y these processes either infiltrates into the g r o u n d or evaporates after the r a i n stops ( P i k e and Scherer, 2 0 0 3 ) .  I n b o t h cases, e v e n the  m o s t detailed h y d r o l o g i c m o d e l s a p p r o x i m a t e the processes o n an areal basis, u s u a l l y u s i n g e m p i r i c a l parameters.  I n particular, seasonal variations i n i n t e r c e p t i o n are often not c o n s i d e r e d .  Infiltration refers to the process b y w h i c h water m o v e s f r o m the surface into the s o i l m a t r i x . Infiltration is constantly active i n a l l wetted permeable areas except those f r o m w h i c h active seepage is o c c u r r i n g . T h e rate o f i n f i l t r a t i o n is a function o f s o i l m o i s t u r e and c o n d i t i o n as w e l l as s o i l type. C o a r s e soils, w e l l - v e g e t a t e d l a n d , l o w s o i l m o i s t u r e , and a m a c r o - p o r o u s top layer (i.e., affected b y b u r r o w i n g insects and a n i m a l s ) a l l p r o m o t e h i g h i n f i l t r a t i o n rates (Fetter, 1994).  18  I n i t i a l s o i l m o i s t u r e at a g i v e n l o c a t i o n varies o v e r t i m e ; as a dependent c o n d i t i o n , the l i m i t i n g i n f i l t r a t i o n rate does l i k e w i s e . In situ s o i l m o i s t u r e c a n be assessed u s i n g a v a r i e t y o f apparatus (e.g., tensiometers or T D R ( T i m e - D o m a i n R e f l e c t o m e t r y ) probes), and i n f i l t r a t i o n rates for s o i l samples under different m o i s t u r e c o n d i t i o n s c a n be m e a s u r e d i n a laboratory. H o w e v e r , m a n y factors c a n i n f l u e n c e i n f i l t r a t i o n at a l o c a l l e v e l (e.g., l e a f litter a n d l o c a l topography). T h i s renders a d e t a i l e d characterization o f the i n f i l t r a t i o n process c o m p l i c a t e d e v e n for i d e a l c o n d i t i o n s ( V i e s s m a n and L e w i s , 1996).  Infiltrated water generally percolates v e r t i c a l l y d o w n w a r d t h r o u g h the unsaturated z o n e u n t i l m e e t i n g the water table. H o w e v e r , it is c o m m o n - e s p e c i a l l y w h e r e the catchment is d o m i n a t e d b y steep h i l l s l o p e s - to have h o r i z o n t a l f l o w above the n o m i n a l saturated z o n e , a process referred to as i n t e r f l o w . Preferential f l o w occurs t h r o u g h root v o i d s a n d other macropores i n the s o i l m a t r i x under saturated or unsaturated c o n d i t i o n s . H o w e v e r , i n t e r f l o w m o s t c o m m o n l y occurs o n steep h i l l s l o p e s w h e n p e r c o l a t i n g water encounters a z o n e o f l o w e r v e r t i c a l h y d r a u l i c c o n d u c t i v i t y (Fetter, 1994; R e f s g a a r d and S t o r m , 1995). T h e r e s u l t i n g p e r c h e d water table often triggers lateral subsurface f l o w , either t h r o u g h l o w e r - p e r m e a b i l i t y strata or s o i l macropores ( W e i l e r et a l . , 2 0 0 5 ) . T h u s , i n t e r f l o w c a n o c c u r under saturated or near-saturated c o n d i t i o n s e v e n t h o u g h the response occurs above the n o m i n a l water table ( i b i d . ) . T h i s is an important process for d e l i v e r i n g water to the v a l l e y b o t t o m at the h i l l s l o p e scale ( W e i l e r and M c D o n n e l l , 2004).  Interflow is d i f f i c u l t to quantify due to its transient nature. A n effective n u m e r i c a l d e s c r i p t i o n requires k n o w l e d g e o f subsurface strata topography, h y d r a u l i c c o n d u c t i v i t y , m a c r o p o r o s i t y , and c o n n e c t i v i t y , as w e l l as antecedent s o i l m o i s t u r e and groundwater c o n d i t i o n s . T h e scientific understanding o f i n t e r f l o w at the h i l l s l o p e scale has e v o l v e d c o n s i d e r a b l y o v e r the past few decades t h r o u g h the a p p l i c a t i o n o f n e w measurement techniques l i k e isotope and c h e m i c a l t r a c i n g ( W e i l e r et a l . , 2 0 0 5 ) . A l t h o u g h the p r e v a i l i n g understanding o f i n t e r f l o w has advanced, i n t e r f l o w is not w e l l represented i n m o s t h y d r o l o g i c m o d e l s . E v e n r e l a t i v e l y sophisticated h y d r o l o g i c m o d e l s l i k e M I K E S H E ( D H I S o f t w a r e ' s p o p u l a r v e r s i o n o f the Systeme H y d r o l o g i q u e E u r o p e e n ) calculate o n l y v e r t i c a l f l o w i n the unsaturated z o n e (Refsgaard and S t o r m , 1995). I n some cases (e.g., b u l k h y d r a u l i c c o n d u c t i v i t y ) , the s o i l properties g o v e r n i n g  19  i n t e r f l o w (e.g., b u l k h y d r a u l i c c o n d u c t i v i t y , i n c l u d i n g macropores) cannot be accurately assessed, since the act o f s a m p l i n g often changes the property i n question ( B e v e n , 2 0 0 2 ) .  Saturated-zone groundwater processes are c o m p a r a t i v e l y w e l l understood but i n m a n y cases just as d i f f i c u l t to quantify. G r o u n d w a t e r b a s e f l o w sustains surface water systems through d r y periods b y s l o w d e p l e t i o n o f subsurface storage. V o l u m e s and f l o w rates for groundwater are difficult to characterize because o f their strong dependence o n r e g i o n a l g e o l o g y . F u r t h e r c o m p l e x i t y arises f r o m the t h r e e - d i m e n s i o n a l nature o f subsurface f l o w . A l t h o u g h c o m p l e x , groundwater c a n be m o d e l l e d i n three d i m e n s i o n s (e.g., M I K E S H E i n R e f s g a a r d and S t o r m , 1995). H o w e v e r , m a n y m o d e l s assume f l o w i n the t h i r d d i m e n s i o n to be n e g l i g i b l e , a l l o w i n g for t w o - d i m e n s i o n a l analysis w i t h i n the saturated z o n e ( V i e s s m a n and L e w i s , 1996).  G r o u n d w a t e r measurements t y p i c a l l y i n v o l v e r e c o r d i n g the depth to w h i c h an unrestricted c o l u m n o f water w i l l rise at a n u m b e r o f i n d i v i d u a l locations. A series o f s u c h measurements can determine f l o w d i r e c t i o n but cannot f u l l y describe subsurface f l o w w i t h o u t a d d i t i o n a l information on hydraulic conductivity.  T y p i c a l l y , g r o u n d w a t e r f l o w c a l c u l a t i o n s are based o n equations d e v e l o p e d for a s i m p l e c o n t r o l v o l u m e s and b e n c h m a r k e d to the subject catchment u s i n g e m p i r i c a l parameters d e r i v e d f r o m f i e l d measurements.  R e g a r d l e s s o f the q u a l i t y o f the parameter estimation, equations d e r i v e d o n  the basis o f a h o m o g e n e o u s , i s o t r o p i c c o n t r o l v o l u m e are often i n a p p l i c a b l e at large scales due to heterogeneity a n d preferential f l o w p a t h w a y s ( B e v e n , 2 0 0 2 ) .  M o d e r n h y d r o l o g i c literature d i v i d e s f l o w o v e r the g r o u n d surface into t w o distinct m e c h a n i s m s . T h e d o m i n a n t m e c h a n i s m for a g i v e n r e g i o n depends o n c l i m a t e , topography, g r o u n d p e r m e a b i l i t y , and r a i n f a l l intensity.  H o r t o n or " i n f i l t r a t i o n e x c e s s " O v e r l a n d F l o w ( H O F ) , n a m e d for h y d r o l o g i c p i o n e e r R . E . H o r t o n , o c c u r s w h e n the p r e c i p i t a t i o n rate exceeds the i n f i l t r a t i o n rate. E x c e s s p r e c i p i t a t i o n runs o v e r the surface u n t i l it either infiltrates elsewhere or reaches an area o f surface a c c u m u l a t i o n (e.g., depression, l a k e , stream). H O F c a n o c c u r as a result o f e x t r e m e l y intense p r e c i p i t a t i o n or moderately-intense p r e c i p i t a t i o n f a l l i n g o n l o w - p e r m e a b i l i t y surface layers s u c h as e x p o s e d b e d r o c k , f r o z e n g r o u n d , asphalt, or e x t r e m e l y d r y s o i l . H O F is u s u a l l y the d o m i n a n t f o r m o f  20  o v e r l a n d f l o w observed i n a r i d e n v i r o n m e n t s due to the t y p i c a l c o m b i n a t i o n o f a dry, l o w p e r m e a b i l i t y surface layer and infrequent but intense p r e c i p i t a t i o n events.  Saturation O v e r l a n d F l o w ( S O F ) occurs w h e n a p r e c i p i t a t i o n event causes the water table to rise to the surface (e.g., at the base o f a h i l l s i d e ) , creating a seepage face ( D u n n e , 1982). T h e t e r m saturation o v e r l a n d f l o w refers to the c o m b i n a t i o n o f direct p r e c i p i t a t i o n o n saturated s o i l and f l o w e m e r g i n g f r o m an ephemeral seepage face. S O F is m o r e c o m m o n l y o b s e r v e d than H O F i n wet a n d temperate regions due to its dependence o n t o p o g r a p h y rather than intense p r e c i p i t a t i o n and l o w p e r m e a b i l i t y . S O F c a n be o b s e r v e d a l o n g creeks and stream b a n k s d u r i n g m o s t higherintensity p r e c i p i t a t i o n events. F l o o d i n g due to S O F is t y p i c a l l y associated w i t h p r o l o n g e d periods o f r a i n that substantially raise l o c a l water tables.  T h e v o l u m e o f surface f l o w is t y p i c a l l y c a l c u l a t e d b y subtracting estimated losses and i n f i l t r a t i o n f r o m estimates o f areal totals for r a i n and s n o w m e l t ( M i c h a u d and S o r o o s h i a n , 1994). T h e actual m e c h a n i c s o f surface f l o w w o u l d be n e a r l y i m p o s s i b l e to simulate i n d e t a i l , necessitating the use o f substantial a p p r o x i m a t i o n s . W h e r e detailed s i m u l a t i o n o f o v e r l a n d f l o w is r e q u i r e d , the m o s t w i d e l y - a p p l i e d a p p r o a c h is to use the St. V e n a n t equations to route f l o w as a t h i n sheet o f water m o v i n g o v e r a h o m o g e n o u s landscape o f constant or s m o o t h l y - v a r y i n g roughness. M o r e c o m p l e x c o m p u t a t i o n s are u s e d i n leading-edge c o m p u t e r software s u c h as D H I ' s M I K E S H E , w h i c h applies the Saint V e n a n t equations i n t w o h o r i z o n t a l d i m e n s i o n s (Refsgaard and S t o r m , 1995). These m e t h o d s i m p l i c i t l y assume that surface r u n o f f remains i n the f o r m o f characteristically t w o - d i m e n s i o n a l sheets rather than c o n s i d e r i n g w e l l d o c u m e n t e d but m o r e c o m p l e x r e a l - w o r l d b e h a v i o u r (e.g., r i l l i n g and b a c k w a t e r i n g a r o u n d u n e v e n m i c r o - t e r r a i n features).  These assumptions m a y u l t i m a t e l y p r o v i d e the correct answer for t i m i n g and v o l u m e  o f water r e a c h i n g the c h a n n e l , but bear little resemblance to the in situ process o n a h i l l s l o p e (Burges, 2 0 0 2 ) .  A l l r u n o f f e v e n t u a l l y m a k e s its w a y to the c h a n n e l n e t w o r k o f streams, r i v e r s , a n d lakes. T h e acceptable representation o f these surface water systems i n m o s t m o d e l s arises f r o m a f a i r l y t h o r o u g h understanding o f o p e n - c h a n n e l h y d r a u l i c s . H o w e v e r , m a n y r o u t i n g approaches s t i l l r e l y o n e m p i r i c a l and subjective methods o f a p p r o x i m a t i o n that are a p p l i c a b l e o n l y under specific c o n d i t i o n s (e.g., M a n n i n g ' s f o r m u l a ) . T h i s suggests that o u r a b i l i t y to m o d e l r i v e r  21  h y d r a u l i c s is a r g u a b l y " g o o d e n o u g h " (i.e., not unacceptable) rather than " g o o d " . Further, cases exist w h e r e the standard suite o f assumptions are i n v a l i d . F o r e x a m p l e , c h a n n e l losses c a n be important i n a r i d regions w h e r e H o r t o n i a n r u n o f f dominates, and ephemeral lakes c a n change size a p p r e c i a b l y d u r i n g r u n o f f events, m a k i n g m o d e l l i n g d i f f i c u l t ( M i c h a u d and S o r o o s h i a n , 1994; O g d e n et a l . , 2 0 0 0 ) .  S t r e a m f l o w measurement is c r i t i c a l to h y d r o l o g y , as it is the m o s t h e a v i l y used (and often the o n l y a v a i l a b l e ) i n d i c a t o r o f the h y d r o l o g i c a l b e h a v i o u r o f a watershed. Fortunately, s t r e a m f l o w measurement c a n be r e l a t i v e l y straightforward c o m p a r e d to measurement o f other h y d r o l o g i c a l processes (e.g., areally-distributed p r e c i p i t a t i o n a n d evapotranspiration).  Structures s u c h as  w e i r s , flumes, a n d culverts c a n be u s e d to measure f l o w d i r e c t l y through the w e l l - d o c u m e n t e d r e l a t i o n s h i p b e t w e e n their geometric properties a n d the l o c a t i o n o f the water surface at different f l o w rates. A c c u r a c i e s o f measurement v a r y w i t h the type o f structure.  A current meter, u s e d to measure f l u i d v e l o c i t y at a s i n g l e p o i n t , c a n also be u s e d to measure f l o w b y e m p l o y i n g a systematic pattern o f v e l o c i t y and depth measurements.  T h e product o f  each p a i r o f measurements is m u l t i p l i e d b y its c o r r e s p o n d i n g p o r t i o n o f the r i v e r cross-section. T h e results are then s u m m e d to attain an a p p r o x i m a t i o n o f the streamflow. B y repeating the above m e t h o d at a v a r i e t y o f different f l o w l e v e l s , a r e l a t i o n s h i p b e t w e e n ambient water surface e l e v a t i o n and discharge c a n be obtained. W i t h this stage-discharge relationship, measurements o f water stage (i.e., u s i n g a s t i l l i n g b a s i n or staff gauge) are e a s i l y converted into streamflow. T h i s m e t h o d is f a v o u r e d b y h y d r o m e t r i c agencies s u c h as the W a t e r S u r v e y o f C a n a d a and the U . S . Geological Survey (Moore, 2004).  G e o m o r p h o l o g i c changes c a n s i g n i f i c a n t l y affect the stage-discharge r e l a t i o n s h i p ; therefore, it must be updated o n a regular basis to preserve its accuracy. T h i s is the most c o m m o n m e t h o d for c a l c u l a t i n g s t r e a m f l o w ; m o s t g a u g i n g stations r e c o r d o n l y water l e v e l .  A l l o f the above methods o f m e a s u r i n g s t r e a m f l o w have c o n d i t i o n s under w h i c h they are either infeasible or p r o v i d e m e a n i n g l e s s results. In basins w h e r e c o l d e r temperatures are associated w i t h l o w streamflow, significant data c a n be lost or m a d e suspect due to i c e c o n d i t i o n s (e.g., E n v i r o n m e n t C a n a d a , 2 0 0 1 ) . G e n e r a l p r o b l e m s w i t h current m e t e r i n g at l o w f l o w s i n c l u d e s t r e a m f l o w v e l o c i t i e s at or b e l o w the meter's stall speed, insufficient depth for r e q u i r e d  22  submergence, a n d i n s u f f i c i e n t stream w i d t h to s a m p l e a sufficient n u m b e r o f v e r t i c a l sections ( P i k e a n d Scherer, 2 0 0 3 ) . T u r b u l e n t e n v i r o n m e n t s c a n also disrupt current m e t e r i n g , as v e l o c i t i e s m a y not be consistent e n o u g h at any g i v e n p o i n t to p r o v i d e a r e l i a b l e measurement. E v e n r e l a t i v e l y accurate measurement devices s u c h as w e i r s a n d f l u m e s are not e x e m p t f r o m p r o b l e m s ; their i n s t a l l a t i o n c a n affect the l o c a l f l o w r e g i m e , a n d c o n d i t i o n s must be a l l o w e d to s t a b i l i z e before results c a n be c o n s i d e r e d b r o a d l y a p p l i c a b l e ( M i c h a u d a n d S o r o o s h i a n , 1994).  L o w f l o w s , irregular cross-sections, and h i g h turbulence m a y p r e c l u d e the use o f s o m e o r a l l o f the above g a u g i n g techniques. H o w e v e r , i n m a n y s u c h cases, discharge c a n be m e a s u r e d u s i n g a c o n s e r v a t i v e tracer a n d the p r i n c i p l e o f mass balance. T h i s approach, c a l l e d salt d i l u t i o n g a u g i n g , i n v o l v e s i n j e c t i n g into the f l o w a s o l u t i o n c o n t a i n i n g a c h e m i c a l tracer o f k n o w n concentration. T h e i n j e c t i o n c a n be p e r f o r m e d as either a c o n t i n u o u s input o r a s l u g injection. T h e d i l u t i o n o f the tracer is then m e a s u r e d at a n appropriate distance d o w n s t r e a m . M o o r e (2004) cites c o m m o n table salt ( N a C l ) as the m o s t p o p u l a r tracer because it i s i n e x p e n s i v e , r e a d i l y a v a i l a b l e , e a s i l y measurable ( u s i n g e l e c t r i c a l c o n d u c t i v i t y ) , a n d n o n - t o x i c for the exposures c u r r e n t l y associated w i t h discharge measurements. B e c a u s e salt d i l u t i o n g a u g i n g relies o n attaining a c o m p l e t e lateral m i x o f the tracer s o l u t i o n , it is i d e a l l y suited for the irregular a n d h i g h l y turbulent e n v i r o n m e n t s t y p i c a l o f m o u n t a i n streams. F o r e n v i r o n m e n t a l and p r a c t i c a l reasons, salt d i l u t i o n g a u g i n g is less appropriate for f l o w s greater than about 15 m / s ( K i t e , 3  1993).  H o w e v e r , salt d i l u t i o n g a u g i n g also has l i m i t a t i o n s . M o s t i m p o r t a n t l y , the i m p l i c i t requirement for steady-state f l o w g e n e r a l l y l i m i t s salt d i l u t i o n g a u g i n g to discrete (as o p p o s e d to c o n t i n u o u s ) measurements o f discharge. T h e r e is also the p o t e n t i a l for e n v i r o n m e n t a l i m p a c t s (e.g., W o o d and D y k e s , 2 0 0 2 ) , as h i g h concentrations are t y p i c a l l y o b s e r v e d near the p o i n t o f injection. H o w e v e r , deleterious i m p a c t s are u n l i k e l y g i v e n the s h o r t - l i v e d a n d l o c a l i z e d nature o f these h i g h e r concentrations; concentrations d o w n s t r e a m o f the m i x i n g r e a c h are u s u a l l y far b e l o w c o m m o n l y - a c c e p t e d 4 8 - h o u r t o x i c i t y thresholds ( i b i d . ; M o o r e , 2 0 0 5 ) .  It is o f great s i g n i f i c a n c e to h y d r o l o g i c m o d e l l e r s that large floods often o v e r w h e l m f i x e d m e a s u r i n g structures, regardless o f the measurement approach. T h i s t y p i c a l l y m e a n s that data c o l l e c t i o n is interrupted o n the r i s i n g l i m b o f a f l o o d h y d r o g r a p h , a n d i s not r e s u m e d u n t i l repairs  23  c a n be effected (e.g., O g d e n et a l . , 2 0 0 0 ) . T h e result is that c r i t i c a l s t r e a m f l o w data - i.e., that o f the f l o o d peak and d u r a t i o n - is lost. I n s u c h cases, h y d r o l o g i s t s m u s t r e l y o n expert analysis o f q u a l i t a t i v e data s u c h as eye-witness accounts and h i g h - w a t e r m a r k s s u c h as debris scatter and tree scarring. A p p r o x i m a t e f o r m u l a e s u c h as the slope-area m e t h o d , an approach c o m m o n l y a p p l i e d i n s u c h cases, incorporate substantial uncertainty into peak f l o w estimates.  S t r e a m f l o w entering a r e s e r v o i r c a n also be estimated w h e r e o u t f l o w s are c o n t r o l l e d or m o n i t o r e d and the v o l u m e o f the r e s e r v o i r i s k n o w n . I n f l o w s are estimated as the s u m o f total releases f r o m the r e s e r v o i r a n d a n y change i n storage. C h a n g e i n storage is t y p i c a l l y estimated b y m e a s u r i n g reservoir l e v e l a n d c o n v e r t i n g it to v o l u m e u s i n g a stage-storage c u r v e ; this often requires a n i m p l i c i t a s s u m p t i o n that the l a k e or reservoir is groundwater-neutral (i.e., neither a c c u m u l a t i n g f r o m n o r d i s c h a r g i n g to the subsurface). D u e to the p o t e n t i a l i m p a c t o f d y n a m i c effects l i k e w i n d set-up a n d w a v e s o n stage measurement, b a c k - c a l c u l a t e d estimates for r e s e r v o i r i n f l o w s s h o u l d be u s e d w i t h c a u t i o n .  2.1.4 Evapotranspiration E v a p o r a t i o n and transpiration together c a n account for as m u c h as 8 0 % o f h y d r o l o g i c a c t i v i t y i n a t y p i c a l b a s i n ( K l e m e s , 1986a). C a l c u l a t i o n s o f pure e v a p o r a t i o n are c o m m o n l y l i m i t e d i n a p p l i c a t i o n to d e t e r m i n i n g losses from lakes or reservoirs. A p p r o a c h e s i n c l u d e a p p l y i n g e m p i r i c a l adjustments to p a n e v a p o r a t i o n measurements, or u s i n g m o r e detailed data to e m p l o y water budget, energy budget, o r m a s s transfer techniques. I n s o m e cases, e v a p o r a t i o n has b e e n p r e - c a l c u l a t e d from data c o l l e c t e d at a c l i m a t e station ( E n v i r o n m e n t C a n a d a , 2 0 0 5 ) . H o w e v e r , f e w h y d r o l o g i c m o d e l s attempt to e x p l i c i t l y calculate evaporation.  T r a n s p i r a t i o n is essentially the e v a p o r a t i o n o f water taken up b y plants, shrubs, and trees. In a d d i t i o n to the p h y s i c a l factors g o v e r n i n g evaporation (e.g., exposure, heat fluxes), transpiration is k n o w n to v a r y w i d e l y w i t h plant species, density, and size (Fetter, 1994). A v a i l a b l e m o i s t u r e c a n also l i m i t transpiration w h e n s o i l m o i s t u r e drops b e l o w the p l a n t ' s w i l t i n g p o i n t ( V i e s s m a n and L e w i s , 1996). C h a n g e s i n season w i l l d i r e c t l y affect transpiration as plants r e s p o n d to e n v i r o n m e n t a l s t i m u l i (e.g., cf. d e c i d u o u s and b o r e a l forests). M e a s u r e m e n t o f transpiration alone is e x t r e m e l y d i f f i c u l t a n d must be undertaken i n c l o s e l y c o n t r o l l e d laboratory c o n d i t i o n s  24  w h i c h e l i m i n a t e external evaporation (e.g., u s i n g a potometer). T h e results o f s u c h experiments are n a t u r a l l y h i g h l y dependent o n a m b i e n t a n d constituent c o n d i t i o n s .  D u e to the d i f f i c u l t y o f o b t a i n i n g distinct estimates for each process independently, the term evapotranspiration ( E T ) is u s e d to represent the c o m b i n e d return o f water to the atmosphere through evaporation and transpiration ( P i k e and Scherer, 2 0 0 3 ) . E T i s the largest s i n k for p r e c i p i t a t i o n i n a l l but e x t r e m e l y h u m i d , c o o l c l i m a t e s (Fetter, 1994). R e f s g a a r d and S t o r m (1995) note that E T accounts for a p p r o x i m a t e l y 7 0 % o f annual p r e c i p i t a t i o n i n temperate zones. K l e m e s (1986a) points out the d i s c r e p a n c y b e t w e e n the large fraction o f h y d r o l o g i c a c t i v i t y i n a b a s i n a s c r i b e d to E T and its relative dearth o f treatment i n h y d r o l o g i c literature and practice.  E T p l a y s a m i n o r role d u r i n g short-term s t o r m events, since the air is t y p i c a l l y at or near its saturation p o i n t . F o l l o w i n g p r e c i p i t a t i o n events, d e p l e t i o n o f the s o i l m o i s t u r e t h r o u g h evapotranspiration creates a s o i l m o i s t u r e deficit. Infiltrated water must r e p l e n i s h this deficit before subsurface r u n o f f w i l l contribute to s t o r m f l o w .  T h e dependence o f transpiration o n s o i l m o i s t u r e is reflected i n the p a r a l l e l dependence o f E T o n s o i l m o i s t u r e . T o account for, T h o r n t h w a i t e (1944) c a l l s the upper l i m i t o f transpiration losses "potential E T " , d e f i n e d as "the water loss w h i c h w o u l d o c c u r i f at n o t i m e there is a d e f i c i e n c y o f water i n the s o i l for the use o f v e g e t a t i o n " . A s s o i l m o i s t u r e d e c l i n e s f r o m f i e l d c a p a c i t y to w i l t i n g p o i n t , there is s o m e uncertainty as to the rate at w h i c h evapotranspiration is affected (Fetter, 1994). T h e r e d u c e d rate o f E T r e s u l t i n g f r o m r e d u c e d s o i l m o i s t u r e is referred to as "actual evapotranspiration". A l t h o u g h s o i l water content c a n be m e a s u r e d w i t h s o m e p r e c i s i o n (at least as a p o i n t v a l u e ) , there is persistent uncertainty associated w i t h the c o n v e r s i o n b e t w e e n p o t e n t i a l and actual evapotranspiration.  P o i n t a p p r o x i m a t i o n s o f evapotranspiration for a particular plant a n d l o c a t i o n c a n be obtained u s i n g a l y s i m e t e r (Fetter, 1994), but s u c h estimates are t y p i c a l l y l i m i t e d due to in situ v a r i a b i l i t y o f species, size, and density. P a n e v a p o r a t i o n data have also been related to E T through e m p i r i c a l coefficients w i t h v a r y i n g degrees o f a c c u r a c y ( A l l e n et a l . , 1998).  M e t h o d s for c a l c u l a t i n g evapotranspiration i n h y d r o l o g i c m o d e l l i n g are g e n e r a l l y either e m p i r i c a l or based o n the mass balance a n d energy budget approaches noted above. T h e water  25  budget a p p r o a c h is o f little use i n h y d r o l o g i c m o d e l l i n g , since E T is c o m m o n l y r e q u i r e d to c l o s e a water balance w h e n estimating runoff, and energy-based approaches require types o f f i e l d data not c o m m o n l y a v a i l a b l e i n m e t e o r o l o g i c data sets (Fetter, 1994). T h e end result is that m a n y h y d r o l o g i c m o d e l s use p u r e l y e m p i r i c a l r e l a t i o n s h i p s that require a m i n i m u m amount o f data and w h i c h r e l y o n c a l i b r a t i o n to be l o c a l l y a p p l i c a b l e (e.g., Q u i c k , 1995).  M o r e a d v a n c e d methods s u c h as the P e n m a n - M o n t e i t h equation ( M o n t e i t h , 1965) d i r e c t l y calculate actual evapotranspiration u s i n g c o m p l e x energy balance inputs s u c h as net r a d i a t i o n a n d vegetative c a n o p y resistance to heat and v a p o u r transfer. H o w e v e r , the P e n m a n M o n t e i t h e q u a t i o n is s t i l l o n l y intended for use w i t h u n i f o r m expanses o f vegetation - a rare c o n d i t i o n i n the natural e n v i r o n m e n t ( A l l e n et a l . , 1998).  In general, e v e n the m o s t accurate m e t h o d s for e s t i m a t i n g E T h a v e substantial m a r g i n s o f error. F o r a deeper d i s c u s s i o n o f evapotranspiration m e t h o d s , the reader is referred to the w o r k o f Jensen et a l . (Jensen et a l . , 1990).  2.2  The Evolution of Hydrologic Modelling 2.2.1 The Beginnings of Modelling  M u s i n g s o n the nature o f h y d r o l o g y h a v e been traced b a c k as far as the p h i l o s o p h e r s o f A n c i e n t G r e e c e ; h o w e v e r , a scientific understanding o f the h y d r o l o g i c c y c l e d i d not b e g i n to emerge u n t i l the fifteenth century. E a r l y efforts to quantify h y d r o l o g i c v a r i a b l e s first began to appear i n the 17  th  century, w h i l e 1 8  th  century d e v e l o p m e n t s i n h y d r a u l i c theory and instrumentation l e d to  extensive e x p e r i m e n t a t i o n and e m p i r i c a l study throughout the 1 9  th  century ( V i e s s m a n and L e w i s ,  1996). A m o n g these, S i n g h and W o o l h i s e r (2002) i d e n t i f y the b e g i n n i n g s o f m a t h e m a t i c a l m o d e l l i n g o f h y d r o l o g y i n the rational m e t h o d o f M u l v a n y (1851) and the relationship b e t w e e n peak s t o r m r u n o f f and r a i n f a l l intensity d e v e l o p e d b y I m b e a u (1892).  T h e 1930s saw systematic f i e l d experiments c o n d u c t e d i n the U S M i d w e s t i n an attempt to understand the p h y s i c a l processes i n v o l v e d i n the r a i n f a l l - r u n o f f transition ( W o o l h i s e r , 1996). H y d r o l o g i c m o d e l l i n g b e g a n to emerge as a process for d e v e l o p i n g concepts, theories, and m o d e l s o f i n d i v i d u a l c o m p o n e n t s o f the h y d r o l o g i c c y c l e , s u c h as o v e r l a n d f l o w , c h a n n e l f l o w ,  26  infiltration, depression storage, evaporation, interception, subsurface f l o w , and base f l o w ( S i n g h and W o o l h i s e r , 2 0 0 2 ) . E a r l y h y d r o l o g y t y p i c a l l y i g n o r e d i n f o r m a t i o n a l uncertainty and adopted an either/or a p p r o a c h i n its f o r m u l a t i o n rather than c o m b i n i n g a l l a v a i l a b l e i n f o r m a t i o n ( V i c e n s et a l . , 1975). B u r g e s (2002) r e v i e w s the s i g n i f i c a n c e o f i n d i v i d u a l c o n t r i b u t i o n s b y H o r t o n , P e n m a n , D a r c y , and others to m o d e l s o f the v a r i o u s c o m p o n e n t processes. H o w e v e r , their c o l l e c t i v e progress w a s l i m i t e d b y the intensity o f the c o m p u t a t i o n s i n v o l v e d . F u r t h e r difficulties emerged as researchers p r o v i d e d strong e x p e r i m e n t a l e v i d e n c e o f the n o n - l i n e a r nature o f the r u n o f f process ( W o o l h i s e r , 1996).  T h e advent o f the d i g i t a l c o m p u t e r i n the 1960s w i t h its a b i l i t y to m a n a g e c a l c u l a t i o n s o f p r e v i o u s l y p r o h i b i t i v e c o m p l e x i t y l e d to an e x p l o s i o n o f interest and research i n h y d r o l o g y ( W o o l h i s e r , 1996). T h e first " r e a l " h y d r o l o g i c m o d e l s emerged (e.g., the Stanford W a t e r s h e d M o d e l , S W M ) , b e i n g c o n c e p t u a l i n nature w h i l e r e t a i n i n g a degree o f theoretical p h y s i c a l s i g n i f i c a n c e i n the c o n t r o l l i n g parameters.  T h r o u g h these tools, m o d e l l e r s w e r e first p r o v i d e d  w i t h the c a p a b i l i t y to c o m p r e h e n s i v e l y synthesize past events, predict future events, quantify extreme c o n d i t i o n s , evaluate anthropogenic i m p a c t s o n h y d r o l o g y , and thereby i m p r o v e the understanding o f h y d r o l o g y (p. 2 3 8 , F r e e z e and H a r l a n , 1969).  E v e n w h i l e the earliest c o n c e p t u a l m o d e l s w e r e b e i n g refined, s o m e researchers adopted a contrasting p a r a d i g m . T h e y p r o p o s e d that m o d e l s l i n k together e x i s t i n g but independent m a t h e m a t i c a l descriptions o f the v a r i o u s h y d r o l o g i c processes (e.g., F r e e z e and H a r l a n , 1969). T h e g o a l w a s the creation o f a c o m p r e h e n s i v e p h y s i c a l l y - b a s e d d i g i t a l h y d r o l o g i c m o d e l w h o s e output w o u l d c o m p l e t e l y describe the h y d r o l o g i c system. T h e d o m i n a n t r o l e o f spatial and sequential variations i n the m o d e l input and output is reflected i n F r e e z e and H a r l a n ' s v i s i o n o f s u c h m o d e l s as " t h r e e - d i m e n s i o n a l b o u n d a r y - v a l u e p r o b l e m s w i t h s p a t i a l l y and sequentially distributed inputs, s o l v e d b y n u m e r i c a l m e t h o d s " (p. 2 5 5 , i b i d ) .  A l t h o u g h researchers n o t e d early o n that data requirements for s u c h m o d e l s w o u l d be p r o h i b i t i v e e v e n for s m a l l h e a v i l y - i n s t r u m e n t e d research catchments, the i n t e r v e n i n g decades have seen w i d e a d o p t i o n o f F r e e z e and H a r l a n ' s framework. B e v e n presents one p o s s i b l e reason for the acceptance o f the f r a m e w o r k through his o b s e r v a t i o n that " d i f f i c u l t sciences [...] often aspire to  27  demonstrate progress and m a t u r i t y b y m o r e a d v a n c e d m a t h e m a t i c a l d e s c r i p t i o n s " (p. 2 0 3 , B e v e n , 2002).  M a n y different m o d e l s are n o w based o n variants o f either the Stanford W a t e r s h e d M o d e l ( " c o n c e p t u a l " m o d e l s ) or the F H 6 9 b l u e p r i n t ( " p h y s i c a l l y - b a s e d " m o d e l s ) . T h e v a r i o u s types o f m o d e l s are d i s c u s s e d i n m o r e detail i n S e c t i o n 2.3.3.  2.2.2 The Philosophy of Model Development and Application T h e a d o p t i o n o f h y d r o l o g i c m o d e l s into m a i n s t r e a m science and engineering has resulted i n the emergence o f t w o distinct goals w h e n d e v e l o p i n g (or a p p l y i n g ) a h y d r o l o g i c m o d e l . T h e first g o a l is research-oriented, attempting to increase o u r understanding o f h y d r o l o g y t h r o u g h e x p l o r a t i o n o f different assumptions and theories. T h e s e c o n d g o a l seeks effective p r e d i c t i o n o f real w o r l d b e h a v i o u r for a p p l i e d contexts s u c h as water resources management ( G r a y s o n et a l . , 1992b). E x p e r i m e n t a t i o n a d v a n c i n g the first g o a l cannot be c o n c l u d e d a priori to e x t e n d the understanding o f the s e c o n d , and v i c e v e r s a ( B e v e n , 1989).  N u m e r o u s m a t h e m a t i c a l m o d e l s o f the last century have b e e n d e v e l o p e d i n the s e c o n d sense, i.e., to address o n l y the h y d r o l o g i c a l v a r i a b l e o f interest at the t i m e ( F r a n c h i n i and P a c c i a n i , 1991). In contrast, several authors c o n t e n d that the p r e v a i l i n g "cost-effective", results-oriented a p p l i c a t i o n o f c o m p l e x n u m e r i c a l m o d e l s is o f l i m i t e d s i g n i f i c a n c e to true progress i n h y d r o l o g y (e.g., B e r g s t r o m et a l , 2 0 0 2 ; B r a b e n , 1985; K l e m e s , 1982, 2000a).  B e v e n (2000) contends that, i n the past, t e c h n o l o g i c a l rather than scientific progress has f u e l l e d the d e m a n d for h y d r o l o g i c m o d e l s . M a n y a c c l a i m e d " a d v a n c e s " i n h y d r o l o g i c m o d e l l i n g m o r e c o r r e c t l y reflect one o r m o r e o f ( B e c k , 1987; B e v e n , 2 0 0 0 ; B e v e n and F e y e n , 2 0 0 2 ) :  •  the t e c h n o l o g i c a l c a p a b i l i t y for enhanced data c o l l e c t i o n and management;  •  the w i d e s p r e a d i m p l e m e n t a t i o n o f m o r e c o m p l e x m o d e l s ;  •  the a p p l i c a t i o n o f m o r e , less, or different c a l i b r a t i o n ; or  •  the easy v i s u a l i z a t i o n o f results.  F u n d a m e n t a l m o d e l l i n g t e c h n o l o g y d e v e l o p e d decades ago is s t i l l i n use i n m a n y parts o f the w o r l d , i n part because n e w techniques have p r o v e n unable to i m p r o v e m o d e l a c c u r a c y or  28  efficacy w h e n the u n d e r l y i n g m o d e l structure or data are f a l l i b l e ( W o o l h i s e r , 1996; S i n g h and Woolhiser, 2002).  S i n c e the d e v e l o p m e n t o f c o m p u t e r m o d e l s o f h y d r o l o g y , p e r i p h e r a l p r o b l e m s l i k e m o d e l c a l i b r a t i o n have seemed to d o m i n a t e the focus o n process and perspective advocated b y H o r t o n i n the 1930s ( K l e m e s , 1986b). In general, h y d r o l o g i c m o d e l s require a s o u n d p h y s i c a l basis i f they are to be s c i e n t i f i c a l l y c r e d i b l e . Therefore, it is v i e w e d as a p o s i t i v e d e v e l o p m e n t that m o r e recent w o r k s h o w s a g r o w i n g return to process-oriented research and a focus o n f i e l d w o r k ( K l e m e s , 2000a).  S t r o n g p h y s i c a l r e a s o n i n g or e m p i r i c a l e v i d e n c e s h o u l d be u s e d to determine w h i c h representations and s i m p l i f i c a t i o n s are appropriate for a g i v e n natural system ( O ' C o n n e l l a n d T o d i n i , 1996). W o o l h i s e r (1996) notes that a g o o d m a t c h b e t w e e n d o m i n a n t in situ and m o d e l l e d processes is a pre-requisite for success; n e g l e c t i n g s u c h i n t u i t i v e precepts c a n lead to structural inadequacies and c a l i b r a t i o n errors ( G a n and B u r g e s , 1990b). F a u l k n e r et a l . (1998) and F r a n c h i n i and P a c c i a n i (1991) present s o m e s p e c i f i c e x a m p l e s o f inconsistencies o f methodology.  V e r y few m o d e l s and data sets are substantial e n o u g h to a l l o w m o d e l l e r s to e x p l a i n w h y a g i v e n h y d r o l o g i c m o d e l s i m u l a t i o n is c o n s i d e r a b l y different f r o m its c o r r e s p o n d i n g f i e l d measurements ( S m i t h et a l . , 1994). O f t e n , the l a c k o f a p a r a l l e l f i e l d p r o g r a m for m a n y m o d e l s interferes w i t h a d v a n c i n g an u n d e r s t a n d i n g o f the m o d e l structure a n d its influence o n results ( G r a y s o n et a l . , 1992b; M i c h a u d and S o r o o s h i a n , 1994; O ' C o n n e l l and T o d i n i , 1996; S o n g and James, 1992). I n the absence o f o b j e c t i v e l y v e r i f i a b l e truthing data, W o o l h i s e r (1996) b e l i e v e s a m o d e l l e r must constantly be a s k i n g questions s u c h as " D o e s this m a k e sense?", " W h a t is the uncertainty o f m y p r e d i c t i o n ? " , and " D o e s this l e v e l o f uncertainty render the analysis m e a n i n g l e s s ? " . Seibert and M c D o n n e l l (2002) argue that the m o d e l l e r s h o u l d also be c h e c k i n g the m o d e l for c o n s i s t e n c y and reasonableness against any "soft" (i.e., i m p r e c i s e or qualitative) data that m a y be a v a i l a b l e . L o a g u e and F r e e z e (1985) p o i n t out that the usefulness o f results often depends o n the m o d e l l e r ' s understanding o f the a p p l i e d m o d e l and its r e l a t i o n s h i p to the h y d r o l o g i c nuances o f the subject catchment.  29  P o o r o r uncertain m o d e l results s h o u l d i n s t i n c t i v e l y beget a return to the prototype rather than a d d i t i o n a l " f i n e - t u n i n g " o f a n extant m o d e l . K l e r n e s (2000a) contends that, i f the substance o f the m o d e l is h e l d sacrosanct, l i t t l e i n s i g h t c a n be gained into the prototype regardless o f the effort expended. U n f o r t u n a t e l y , p o o r m o d e l results are s e l d o m reported ( B e v e n , 2 0 0 0 ; G r a y s o n et a l . , 1992b). T h e m o d e l d e v e l o p e r ' s e m o t i o n a l investment often m a k e s it m o r e a p p e a l i n g to search for a n e w a p p l i c a t i o n for an unsatisfactory t o o l than to search for a n e w and better w a y o f d e a l i n g w i t h the p r o b l e m ( K l e m e s , 1983).  2.2.3 Model Complexity in Context U n t i l a p p r o x i m a t e l y a decade ago, l i m i t e d c o m p u t a t i o n a l c a p a b i l i t y w a s a substantive barrier to a d v a n c e d h y d r o l o g i c m o d e l l i n g . H o w e v e r , the o n g o i n g and accelerating g r o w t h o f p r o c e s s i n g p o w e r m a d e it i n e v i t a b l e that c o m p u t a t i o n a l a b i l i t y w o u l d s o o n outpace and surpass o u r a b i l i t y to m o d e l . W i t h an increased a b i l i t y to s o l v e n u m e r i c a l p r o b l e m s , studies o f the effects o f parameter v a r i a t i o n o n m o d e l results b e c a m e m o r e v i a b l e ( G u p t a et a l . , 1999). S i m u l t a n e o u s l y , the v a l u e o f i n v e s t i g a t i o n into s i m p l i f i c a t i o n s and a p p r o x i m a t e solutions (e.g., K u c z e r a , 1997) d i m i n i s h e d as c o m p u t a t i o n a l advances negated the m a i n advantage o f s u c h approaches.  The complexity o f  m o s t c o n t e m p o r a r y m o d e l s i s n o w constrained o n l y b y the degree o f c o m p l e x i t y appropriate to the m o d e l , subject, and research context.  T h e d e f i n i t i o n o f an " a p p r o p r i a t e " degree o f c o m p l e x i t y has b e e n and continues to be a major focus for d i s c u s s i o n and research. S o m e b e l i e v e that the r e q u i r e d l e v e l o f accuracy s h o u l d dictate the c h o i c e o f m o d e l i n a n y situation, since e v e n s i m p l e , easy-to-use m a t h e m a t i c a l m o d e l s c a n often e x p l a i n a large part o f s t r e a m f l o w variance ( B e v e n , 1989; D u n n e , 1982; G a r e n and B u r g e s , 1 9 8 1 ; J a k e m a n and H o r n b e r g e r , 1993). S m i t h et a l . (1994) p o i n t out that the theoretical r i g o r o f c o m p l e x m o d e l s is not an a priori guarantee o f accuracy. I n m a n y cases, s i m p l e r m o d e l s m a y g i v e answers o f the same q u a l i t y as their c o m p l e x counterparts, often at a l o w e r cost ( G a n et a l . , 1997; W o o l h i s e r , 1996). I n general, there i s a c o n t i n u u m o f trade-offs between the l i m i t e d process d e s c r i p t i o n o f the simplest m o d e l s and the uncertainty i n t r o d u c e d b y those o f h i g h e r c o m p l e x i t y ( H o r n b e r g e r et a l . , 1985).  T h e h y d r o l o g i s t s h o u l d keep i n m i n d that a l l m o d e l s are, b y d e f i n i t i o n , an abstraction o f r e a l i t y and are therefore to some extent incorrect ( W o o l h i s e r , 1996). I f a m o d e l ' s representation o f  30  r e a l i t y w e r e perfect and exhaustive, the m o d e l w o u l d not be a m o d e l o f the natural system but its d u p l i c a t e ( K l e m e s , 2000a). R e s u l t s for e v e n the best m o d e l s s h o u l d therefore be v i e w e d w i t h a degree o f s k e p t i c i s m . G r a y s o n et a l . (p. 855, 1994b) present a strong case that " m o d e l s s h o u l d not be a p p l i e d as substitutes for k n o w l e d g e " . Rather, the authors c o n t e n d that the proper use and interpretation o f m o d e l results c a n require m o r e k n o w l e d g e and h y d r o l o g i c i n s i g h t than w o u l d o t h e r w i s e be necessary.  2.3  Model Classifications  W h e n e x p l o r i n g an emergent f i e l d o f science, the i n i t i a l r e l a t i o n s h i p s d e v e l o p e d are generally s i m p l e and l e a d to o n l y l i m i t e d k n o w l e d g e and understanding.  S u c h r e l a t i o n s h i p s are l a b e l e d  " e m p i r i c a l " to d i s t i n g u i s h t h e m f r o m the " c a u s a l " relationships that characterize process d y n a m i c s ( K l e m e s , 1982). E m p i r i c a l relationships are often u s e d as c o n v e n i e n t s u m m a r i e s o f c o m p l e x c a u s a l chains. I n h y d r o l o g y the e m p i r i c a l approach has m a n y different labels, such as operational, p r e s c r i p t i v e , a n a l y t i c a l , and statistical, w h i c h a l l attempt to "understand" and " e x p l a i n " the b e h a v i o u r o f any system i n terms o f a r e l a t i v e l y few c o m p r e h e n s i b l e elements ( i b i d . ) . S c i e n t i f i c d i s c i p l i n e s u s u a l l y e v o l v e f r o m the c o n s t r u c t i o n o f e m p i r i c a l m o d e l s to the d e v e l o p m e n t o f c a u s a l m o d e l s after r e a c h i n g a f a i r l y a d v a n c e d stage o f d e v e l o p m e n t ; h y d r o l o g y is n o e x c e p t i o n . M o d e l s i n use today span a range o f c o m p l e x i t y f r o m p u r e l y e m p i r i c a l to h i g h l y detailed.  I n d i v i d u a l h y d r o l o g i c m o d e l s are m o s t c o m m o n l y c l a s s i f i e d a c c o r d i n g to their c o m p l e x i t y . T h r e e categories are t y p i c a l l y considered, n a m e l y e m p i r i c a l and b l a c k b o x m o d e l s , c o n c e p t u a l m o d e l s , and p h y s i c a l l y - b a s e d reductionist m o d e l s ( K u c z e r a and Parent, 1998). B o u n d a r i e s b e t w e e n the m o d e l types are not strict, and the roles o f m o d e l s i n different categories c a n be c o m p l e m e n t a r y rather than c o m p e t i t i v e ( O ' C o n n e l l and T o d i n i , 1996). G a n (1987) p r o v i d e s m o r e i n f o r m a t i o n o n the v a r i o u s classes o f m o d e l s .  A l t h o u g h m a n y studies o f the 1970's and early 80's c o m p a r e results o f m o d e l s selected f r o m w i t h i n a s i n g l e category, few c o m p a r e or contrast the results generated b y different types o f m o d e l s . T h e r e is e v e n s o m e debate about whether s u c h c o m p a r i s o n s are v a l i d g i v e n the d i s s i m i l a r i t i e s o f process and a p p l i c a t i o n across m o d e l types; S m i t h et a l . (p. 8 5 1 , 1994) refer to one c r o s s - c o m p a r i s o n study as " a c l a s s i c e x a m p l e o f apples versus oranges". T h e l a c k o f  31  c o m p a r i s o n studies m a k e s it d i f f i c u l t to assemble a robust synthesis o f e x p e r i m e n t a l c o n c l u s i o n s ( W o o l h i s e r , 1996). Further, the c h o i c e o f m o d e l for a g i v e n situation is frequently dictated b y a v a i l a b l e data or other factors not c o n s i d e r e d i n a c o m p a r i s o n study.  S i n g h (1995a) p r o v i d e s r e v i e w s o f m a n y m o d e l s o f v a r y i n g c o m p l e x i t y . A c o m p r e h e n s i v e and f a i r l y up-to-date l i s t i n g o f a l l watershed m o d e l s is enumerated b y S i n g h and W o o l h i s e r (2002).  2.3.1 Statistical, Empirical, and Black Box Models F r o m a s c i e n t i f i c perspective, statistical, e m p i r i c a l , a n d b l a c k - b o x m o d e l s represent the simplest class o f h y d r o l o g i c m o d e l . S i n g h and W o o l h i s e r (2002) trace the roots o f m a t h e m a t i c a l m o d e l l i n g i n h y d r o l o g y to the 1 9  th  century w o r k s o f M u l v a n y (1851) and I m b e a u (1892). M o r e  recent e x a m p l e s o f e m p i r i c a l m o d e l s i n h y d r o l o g y i n c l u d e statistical F l o o d F r e q u e n c y A n a l y s i s ( F F A ) , regression m o d e l s , and the transfer f u n c t i o n m o d e l s described b y J a k e m a n and H o r n b e r g e r (1993). M o d e l s i n this category c a n also i n c l u d e the s i m p l e c o r r e l a t i o n o f h y d r o l o g i c quantiles (e.g., f l o o d peak, characteristic u n i t h y d r o g r a p h , or l o w f l o w ) w i t h g e o l o g i c , g e o m o r p h i c , and c l i m a t o l o g i c variables ( D u n n e , 1982).  Statistical m o d e l s c o m m o n l y c o n s i d e r a s t r e a m f l o w t i m e series as a m a t h e m a t i c a l series h a v i n g general d e s c r i p t i v e properties s u c h as central tendency, variance, and autocorrelation. Parameters are u n l i k e l y to have p h y s i c a l s i g n i f i c a n c e . W h e n m o d e l s are v i e w e d as a p u r e l y m a t h e m a t i c a l construct, B e v e n (1989) notes that three to five parameters s h o u l d be sufficient to reproduce m o s t o f the i n f o r m a t i o n i n a h y d r o l o g i c a l r e c o r d . J a k e m a n and H o r n b e r g e r (1993) c o n c l u d e that the " p e r m i s s i b l e m o d e l c o m p l e x i t y " seems to be around s i x parameters.  G r i g g et a l . (1999) s t r o n g l y c a u t i o n against b l i n d reliance o n e m p i r i c a l l y - e s t i m a t e d floods (e.g., through F F A ) . K l e m e s (1982) presents a stronger v i e w , a r g u i n g that b y u t i l i z i n g the data to define an appropriate m o d e l , s u c h m o d e l s are s e l f - l i m i t e d and have n o j u s t i f i c a t i o n b e y o n d their u n d e r l y i n g data set. T h i s does not m e a n that s i m p l e m a t h e m a t i c a l m o d e l s are o f little use to the h y d r o l o g i s t ; o n the contrary, they c a n be appropriate or e v e n o p t i m a l i n c i r c u m s t a n c e s where the l i m i t a t i o n s o f the m e t h o d s are a c k n o w l e d g e d .  T h e generic term " b l a c k - b o x " m o d e l refers to a m o d e l i n w h i c h inputs are c o n v e r t e d into outputs u s i n g f o r m u l a s , c a l c u l a t i o n s , or p r e - p r o g r a m m e d relationships that the end user does not need to  32  see or understand to be able to use. A l t h o u g h a l l higher-order computer-based m o d e l s o f h y d r o l o g y c o u l d therefore be c o n s i d e r e d " b l a c k - b o x " m o d e l s , the m o r e c o m m o n interpretation refers to a m o d e l that - l i k e statistical and e m p i r i c a l m o d e l s - m a k e s no attempt to e m p l o y the k n o w n p h y s i c s o f the h y d r o l o g i c p h e n o m e n o n ( K u c z e r a and Parent, 1998). H o w e v e r , u n l i k e statistical or s i m p l e e m p i r i c a l m o d e l s , " b l a c k - b o x " m o d e l s c a n possess significant t e c h n i c a l c o m p l e x i t y . Substantial k n o w l e d g e is c o m m o n l y r e q u i r e d for set up, t r a i n i n g , and analysis o f process-oriented output characteristics l i k e p r e d i c t i v e uncertainty.  T h e concept o f a " b l a c k - b o x m o d e l " i n h y d r o l o g y i s perhaps m o s t c l e a r l y understood w h e n used to describe an a r t i f i c i a l n e u r a l n e t w o r k ( A N N ) . A N N s create a (potentially c o m p l e x ) set o f relationships between input and output data that is f u l l y dependent o n (and thus v a r i a b l e w i t h ) the set o f data u s e d to " t r a i n " the m o d e l . I n this w a y , an A N N m o d e l differs f r o m a l l other m o d e l types, w h i c h c o m b i n e a f i x e d set o f f o r m u l a s and algorithms w i t h v a r i a b l e parameters. A w o r k i n g understanding o f the c o m p l e x " b l a c k - b o x " m e c h a n i c s o f an A N N h y d r o l o g i c m o d e l w o u l d y i e l d little to no h y d r o l o g i c insight. S i n g h and W o o l h i s e r (2002) observe that e v e n the m o s t c o m p l e x A N N s do not m o d e l the internal processes o f a catchment.  H o w e v e r , the a b i l i t y o f A N N s to r e c u r s i v e l y learn f r o m data c a n save substantial t i m e i n m o d e l d e v e l o p m e n t , e s p e c i a l l y w h e r e traditional parameter e s t i m a t i o n techniques are not c o n v e n i e n t ( S i n g h and W o o l h i s e r , 2 0 0 2 ) . F o r m o r e i n f o r m a t i o n o n A N N s , the reader is directed to a t w o part, detailed d i s c u s s i o n o f the role o f A N N s i n h y d r o l o g y authored b y the A S C E (2000a, 2000b).  2.3.2 Conceptual Models C o n c e p t u a l h y d r o l o g i c m o d e l s c o m p r i s e an intermediate l e v e l o f c o m p l e x i t y . T h e y generally represent the h y d r o l o g i c c y c l e as several interconnected subsystems, each o f w h i c h simulates a c o m p o n e n t process through e m p i r i c a l l y or h e u r i s t i c a l l y - d e t e r m i n e d but p h y s i c a l l y - p l a u s i b l e functions ( D u a n et a l . , 1992). C o n c e p t u a l m o d e l s capture b r o a d features o f catchment response but are c o m p u t a t i o n a l l y and i n f o r m a t i o n a l l y straightforward, attempting to balance structural s i m p l i c i t y against the p h y s i c s o f the p r o b l e m (p. 2 1 7 , F r a n c h i n i and P a c c i a n i , 1991). T h e intended spatial and t e m p o r a l scales o f a p p l i c a t i o n often have a p r o f o u n d influence o n m o d e l structure d e v e l o p m e n t ( S i n g h , 1995b).  33  B e v e n (1989) r e m i n d s h y d r o l o g i s t s that a c o n c e p t u a l m o d e l presents an a p p r o x i m a t i o n o f the real w o r l d and therefore m u s t introduce significant p o t e n t i a l for error a n d uncertainty. I n particular, H o r n b e r g e r et a l . (1985) note that s o m e parts o f a c o n c e p t u a l m o d e l m a y have a stronger basis i n scientific or p h y s i c a l theory than others. B o u n d a r y c o n d i t i o n s u s e d to define the b e h a v i o u r s o f the v a r i o u s c o m p o n e n t s u b - m o d e l s m a y also be neglected o r changed, d i v o r c i n g the m o d e l structure from its b a s i c c o n s t i t u t i v e assumptions ( F r a n c h i n i a n d P a c c i a n i , 1991).  T h e r e are different degrees o f p h y s i c a l basis e v e n w i t h i n the category o f c o n c e p t u a l m o d e l s . U n s u r p r i s i n g l y , m o r e c o m p l e x m o d e l s t y p i c a l l y require greater effort i n c a l i b r a t i o n a n d a p p l i c a t i o n than s i m p l e r m o d e l s . In general, the l e v e l o f c a l i b r a t i o n d i f f i c u l t y is d i r e c t l y related to the n u m b e r o f parameters a n d c o m p l e x i t y o f m o d e l structure. F r a n c h i n i a n d P a c c i a n i (1991) a p p l y several c o n c e p t u a l m o d e l s o f v a r y i n g c o m p l e x i t y to a f o u r - m o n t h s i m u l a t i o n , r e p o r t i n g that a l l but one o f the m o d e l s generate acceptable s i m u l a t i o n s o f the r e c o r d e d discharges. S i g n i f i c a n t l y , the m o r e c o m p l e x m o d e l s require m u c h m o r e effort i n c a l i b r a t i o n than the most abstract m o d e l , a l t h o u g h the authors report that it is g e n e r a l l y useless to attempt to identify l i n k a g e s b e t w e e n the in situ a n d m o d e l l e d r u n o f f processes i n the m o s t abstract case (ibid.).  C e r t a i n structural features m a y l i m i t the a b i l i t y o f a c o n c e p t u a l m o d e l to represent the h y d r o l o g i c response o f a catchment. F o r e x a m p l e , the t y p i c a l " c o n c e p t u a l " representation o f the subsurface i n v o l v e s p a r t i t i o n i n g it into t w o o r m o r e discrete z o n e s w i t h f i x e d storage capacities and i n f i l t r a t i o n thresholds. T h i s t y p i c a l l y results i n a crude representation o f processes s u c h as i n f i l t r a t i o n , p e r c o l a t i o n , and evapotranspiration ( G a n a n d B u r g e s , 1990a). K l e m e s (p. 102, 1982) c r i t i c i z e s s u c h s i m p l i c i t y for "[taking] shortcuts to f i l l the v o i d b e t w e e n the data a n d the goals w i t h l o g i c a l l y p l a u s i b l e assumptions that are s o m e t i m e s correct but often w r o n g and, m o r e often than not, i n d i v i d u a l l y untestable". E x p l i c i t v a l i d a t i o n o f the m o d e l structure is u s u a l l y not feasible due to the abstract nature o f the m o d e l processes ( G a n a n d B u r g e s , 1990a).  T h e h y d r o l o g i c characteristics o f a catchment m a y also p r e c l u d e g o o d s i m u l a t i o n b y a c o n c e p t u a l m o d e l , as certain c o n d i t i o n s are m o r e e a s i l y r e a l i z e d i n a c o n c e p t u a l framework than others. G a n a n d B u r g e s (1990b) observe p o o r c a l i b r a t i o n results from the a p p l i c a t i o n o f a c o n c e p t u a l m o d e l to a catchment h a v i n g m u l t i p l e h y d r o l o g i c a l l y - d i s t i n c t subcatchments w i t h a  34  c o m m o n outflow.  M o s t c o n c e p t u a l m o d e l s have been d e v e l o p e d for a p p l i c a t i o n to temperate or  wet climates, m a k i n g s i m u l a t i o n o f d r y catchments (i.e., those i n w h i c h less than 2 0 % o f r a i n f a l l b e c o m e s runoff) m o r e d i f f i c u l t due to the greater s i g n i f i c a n c e o f i n f i l t r a t i o n and evapotranspiration ( G a n et a l . , 1997).  T h e abstraction o f p h y s i c a l r e a l i t y i n a conceptual m o d e l l i m i t s the t e m p o r a l r e s o l u t i o n that c a n be r e a l i z e d . T h e s e n s i t i v i t y o f a m o d e l to the c h o s e n timestep is greatly increased as the timestep approaches and exceeds the b a s i n residence t i m e ( A r n a u d et a l . , 2 0 0 2 ) . G a n and B u r g e s (1990b) f i n d c l o s e agreement b e t w e e n observed and s i m u l a t e d m e a n f l o w rates at the d a i l y scale, but the c o m p a r i s o n deteriorates for lesser timesteps. M i c o v i c (2003a) b e l i e v e s that satisfactory results for a watershed-scale c o n c e p t u a l m o d e l w i l l p r o v e unattainable for any t i m e step less than one hour.  T h e abstract, " g r e y - b o x " nature o f conceptual m o d e l s i s d i s t i n g u i s h e d b y the need to calibrate one or m o r e parameters u s i n g o b s e r v e d data ( K u c z e r a a n d Parent, 1998). L i k e e m p i r i c a l m o d e l s , c o n c e p t u a l m o d e l s are best suited to applications w h e r e c o n d i t i o n s d u r i n g the c a l i b r a t i o n p e r i o d are h y d r o l o g i c a l l y s i m i l a r to those o f the s i m u l a t i o n p e r i o d (p. 1 8 6 S , K l e m e s , 1986a. W h i l e i d e n t i f y i n g a u n i q u e a n d o b j e c t i v e l y v e r i f i a b l e set o f "true" parameter values is u s u a l l y i m p o s s i b l e , G a n a n d B i f t u (1996) argue that a c o n c e p t u a l l y - r e a l i s t i c parameter set is a prerequisite for g o o d p e r f o r m a n c e i n forecasting or p r e d i c t i o n . H o w e v e r , earlier w o r k b y G a n ( G a n , 1987; G a n and B u r g e s , 1990a) s h o w s that e v e n a c o n c e p t u a l l y - r e a l i s t i c parameter set cannot guarantee g o o d performance.  B e c a u s e the parameters o f c o n c e p t u a l m o d e l s are d e f i n e d b y the data, the r e s u l t i n g structure i s not general i n its a p p l i c a b i l i t y ( B e v e n , 2 0 0 2 ) . L i k e e m p i r i c a l m o d e l s , there is n o basis for a s s u m i n g that a c o n c e p t u a l m o d e l c a n effectively extrapolate b e y o n d its c a l i b r a t i o n experience ( G a n and B u r g e s , 1990a). R e d u c i n g dependence o n c a l i b r a t i o n b y decreasing the n u m b e r o f parameters i n a c o n c e p t u a l m o d e l w o u l d have the undesirable effect o f t r a n s f o r m i n g the "greyb o x " (i.e., conceptual) representation o f the watershed into a p u r e l y b l a c k - b o x d e s c r i p t i o n (Seibert, 2 0 0 0 ) . A d d i t i o n a l d i s c u s s i o n c o n c e r n i n g the e x t r a p o l a t i o n o f calibrated m o d e l s is p r o v i d e d i n S e c t i o n 3.4.  35  2.3.3 Physically-Based Models In m a n y fields i n c l u d i n g h y d r a u l i c s , the m o s t accurate f o r m o f m o d e l l i n g i n v o l v e s a " p h y s i c a l m o d e l " - an exact, scaled representation o f the natural system d e s i g n e d and constructed under c o n t r o l l e d c o n d i t i o n s for i n v e s t i g a t i v e purposes. H o w e v e r , the large areal extent o f m o s t watersheds c o m b i n e s w i t h s m a l l - s c a l e h y d r o l o g i c a c t i v i t y to p r e c l u d e the use o f p h y s i c a l m o d e l s for a l l but the most basic h y d r o l o g i c experiments. T h e m o s t c o m p l e x h y d r o l o g i c m o d e l s are instead based o n h i g h l y detailed m a t h e m a t i c a l representations o f the v a r i o u s in situ processes. S u c h m o d e l s are referred to as p h y s i c a l l y - b a s e d . In contrast to p h y s i c a l m o d e l s , p h y s i c a l l y based m a t h e m a t i c a l m o d e l s a p p l y p a r t i a l differential equations to represent the v a r i o u s processes w i t h i n the b o u n d a r y c o n d i t i o n s o f the b a s i n (Freeze and H a r l a n , 1969). Therefore, w h i l e not as exact as a w e l l - d e v e l o p e d p h y s i c a l m o d e l , a p h y s i c a l l y - b a s e d h y d r o l o g i c m o d e l is n o m i n a l l y capable o f s i m u l a t i n g i n detail the i n t e r n a l processes and variables o f a catchment.  T h e r e seems to be little argument that m o d e l s o f laboratory-scale systems are p h y s i c a l l y - b a s e d , i n c o m p a r i s o n w i t h m o d e l s operating at the catchment or b a s i n scales ( W o o l h i s e r , 1996). S o m e argue that p h y s i c a l l y - b a s e d m o d e l s are, i n essence, an attempt to "scale u p " the processes o f the laboratory scale to larger contexts (e.g., K u c z e r a and Parent, 1998). T h e s c a l i n g and t r a n s p o s i t i o n o f processes b e t w e e n d o m i n a n t scales c a n l e a d to p o t e n t i a l inconsistencies w i t h their d e f i n i n g theoretical assumptions. T h e strengths and l i m i t a t i o n s o f process a n d data s c a l i n g are d i s c u s s e d further i n S e c t i o n 3.1.  A s a result o f their c o m p l e x i t y and d e t a i l , p h y s i c a l l y - b a s e d m o d e l s have greater data requirements than their s i m p l i f i e d counterparts.  A s a m i n i m u m , physically-based models  g e n e r a l l y require the f o l l o w i n g four k i n d s o f input data (Freeze and H a r l a n , 1969):  •  m o d e l d e f i n i t i o n i n f o r m a t i o n (e.g., g r i d size, t i m e step, and t o p o g r a p h i c data);  •  m e t e o r o l o g i c a l input data (e.g., the time-variant f l u x o f water at the s o i l surface);  •  f l o w parameter estimates (e.g., M a n n i n g ' s n, h y d r a u l i c c o n d u c t i v i t y ) ; and  •  m a t h e m a t i c a l basis a n d structure (i.e., the equations and functions that define the model).  36  Contemporary models can also incorporate diverse data from digital elevation models, remote sensing imagery, chemical tracer experiments, and groundwater level monitoring. The intensive data dependence of physically-based models implies that their performance is strongly dependent on the completeness, accuracy, and representativeness of the input data. Physically-based models are arguably more "realistic" than models which must be calibrated to historical data in a curve-fitting exercise (Beven, 2001). Physically-based models offer the possibility of hydrologic relevance under conditions beyond those of the process-recorded history, and may identify efficient shortcuts which can then be used to improve empirical or conceptual models (Klemes, 1982). In particular, research has shown that physically-based models offer improved performance over calibrated models for situations where physical and hydrologic descriptions of the watershed are available but a long-term gauging record is not (Michaud and Sorooshian, 1994). Physically-based models are also able to generate distributed predictions and thereby evaluate changes in the constituent conditions of a watershed (Beven, 2002). However, internal calculations for points within the catchment are not necessarily representative of physical reality and are easily divorced from their uncertainty (Grayson et al., 1992b). Nonetheless, the distributed predictions of these models have proved popular in diverse fields where distributed hydrologic results act as inputs to other physical, chemical, biological, environmental, or ecological models (Singh, 1995b). Two classes of criticism are commonly directed at physically-based models: firstly, strictly speaking, physically-based models almost never have an absolute physical basis (i.e., in the sense that it is highly unusual that all parameters can be determined a priori from research or field measurements); secondly, physically-based models are more likely to be mis-used than simpler models (Woolhiser, 1996). Grayson et al. (1994a) conclude that their inability to effectively represent the large- and small-scale spatial variability of rainfall is likely to eliminate any theoretical advantages that models of this type might otherwise possess. Others disagree, cautioning against "indicting" physically-based models on the basis of a single set of poor or non-representative results (Smith et al., 1994; Woolhiser, 1996).  37  In m a n y cases, c o n c e r n about the p r e d i c t i v e c a p a b i l i t i e s o f p h y s i c a l l y - b a s e d m o d e l s appears to c o m e i n part f r o m d i f f i c u l t i e s i n a p p l i c a t i o n e x p e r i e n c e d b y those w h o do not understand the m o d e l ( W o o l h i s e r , 1996). O ' C o n n e l l and T o d i n i (1996) propose that rather than a b a n d o n i n g p h y s i c a l l y - b a s e d m o d e l l i n g , one s h o u l d r e - a l i g n expectations o f w h a t c a n be a c h i e v e d , and o n what t i m e scale.  In the m o s t precise sense, a " p h y s i c a l b a s i s " i m p l i e s that a m o d e l does not require c a l i b r a t i o n . T h i s is r a r e l y the case i n practice, r e s u l t i n g i n an i n a b i l i t y to c o m p l e t e l y a b a n d o n e m p i r i c a l , c a l i b r a t e d parameters ( M a d s e n , 2 0 0 3 ) . F o r e x a m p l e , the p r o m i n e n t p h y s i c a l l y - b a s e d m o d e l M I K E S H E uses calibrated coefficients for c a l c u l a t i n g o v e r l a n d and c h a n n e l f l o w , w h i l e f l o w t h r o u g h m a c r o p o r e s i n the unsaturated z o n e is m o d e l l e d b y an " e m p i r i c a l bypass f u n c t i o n " (Refsgaard a n d S t o r m , 1995). F o r this reason, an u n c a l i b r a t e d p h y s i c a l l y - b a s e d m o d e l c a n be at a distinct disadvantage w h e n c o m p a r e d d i r e c t l y w i t h calibrated m o d e l s ( L o a g u e and F r e e z e , 1985).  C a l i b r a t i o n o f p h y s i c a l l y - b a s e d m o d e l s is d i f f i c u l t due to frequent m a t h e m a t i c a l overparameterization, w h i c h i n turn results from a l a c k o f d e f i n i t i v e a priori parameter v a l u e estimation. W o o l h i s e r (1996) argues that the need for c a l i b r a t i o n often justifies a r e d u c t i o n i n m o d e l d i m e n s i o n a l i t y . In s o m e cases, " e x c e s s " parameters c a n be r e p l a c e d b y s i m p l i f y i n g assumptions. F o r e x a m p l e , the M I K E S H E m o d e l c a n be u s e d w i t h a lesser c o m p l e m e n t o f parameters i f the data are insufficient to support its f u l l d i m e n s i o n a l i t y , a feature s p e c i f i c a l l y i n t e n d e d to reduce the r i s k o f overparameterization (Refsgaard and S t o r m , 1995). T h e i m p a c t o f reductions i n m o d e l c o m p l e x i t y or parameter d i m e n s i o n a l i t y o n the degree o f " p h y s i c a l b a s i s " has not b e e n w i d e l y d i s c u s s e d i n the literature.  W o o l h i s e r (1996) identifies a d i c h o t o m y b e t w e e n the g r o w i n g acceptance o f p h y s i c a l l y - b a s e d m o d e l s i n the e n g i n e e r i n g c o m m u n i t y and the s k e p t i c i s m o f m a n y i n the research c o m m u n i t y . S o m e h a v e gone so far as to suggest that s i m p l e r m o d e l s m a y be m o r e appropriate i n certain situations ( i b i d . ) . R e g a r d l e s s o f their p r a c t i c a l u t i l i t y i n a forecasting or p r e d i c t i v e context, it is g e n e r a l l y agreed that p h y s i c a l l y - b a s e d m o d e l s have great potential for e x p l o r i n g the details o f h y d r o l o g i c interactions and i n v e s t i g a t i n g the fundamentals o f r u n o f f processes ( D u n n e , 1982; G a n and B u r g e s , 1990a).  38  2.3.4 Other Classifications M o d e l s c a n be c l a s s i f i e d b y factors other than c o m p l e x i t y o f m o d e l structure. S i n g h (1995b) presents a f a i r l y c o m p r e h e n s i v e o v e r v i e w o f alternative c l a s s i f i c a t i o n systems. M o r e p r o m i n e n t d i s t i n c t i o n s i n c l u d e the t o p o g r a p h i c a l representation o f the watershed ( l u m p e d o r distributed), the d u r a t i o n o f the m o d e l (single event or c o n t i n u o u s s i m u l a t i o n ) , and the m a t h e m a t i c a l approach (stochastic or d e t e r m i n i s t i c ) . C l a s s i f i c a t i o n a c c o r d i n g to these factors is not n e c e s s a r i l y strict.  A m o n g s t the above, the c l a s s i f i c a t i o n m o s t significant to the c o n s i d e r a t i o n o f uncertainty i s a r g u a b l y the d i s t i n c t i o n b e t w e e n l u m p e d and distributed m o d e l s . I f the spatial scale o f m o d e l c o m p u t a t i o n i s large i n c o m p a r i s o n w i t h the scale o f in situ spatial v a r i a t i o n , m o d e l parameters b e c o m e p h y s i c a l l y unmeasurable and assume "effective v a l u e s " ( W o o l h i s e r , 1996). S u c h m o d e l s are referred to as " l u m p e d " . M o s t l u m p e d m o d e l s use a s i n g l e set o f representative values for each a p p l i c a t i o n (i.e., each watershed or catchment). C o n c e p t u a l m o d e l s are almost a l w a y s l u m p e d to s o m e degree, whereas distributed m o d e l s h a v e been d e v e l o p e d i n response to the detailed representation r e q u i r e d b y p h y s i c a l l y - b a s e d m o d e l s ( H o r n b e r g e r and B o y e r , 1995). T y p i c a l l y , the distributed m o d e l breaks d o w n a catchment into s m a l l e r response units or g r i d cells. A d i s c u s s i o n o f the r e l a t i o n s h i p b e t w e e n v a r i a b i l i t y , scale, and distributed m o d e l l i n g is i n c l u d e d i n S e c t i o n 3.1.1.  H y d r o l o g i c m o d e l s p r o v i d e either c o n t i n u o u s or single-event s i m u l a t i o n . B o t h c o n t i n u o u s and event-based m o d e l s step t h r o u g h their a p p l i c a t i o n p e r i o d u s i n g discrete timesteps. H o w e v e r , c o n t i n u o u s m o d e l s simulate h y d r o l o g i c c o n d i t i o n s for a p r o l o n g e d p e r i o d , whereas event-based m o d e l s are l i m i t e d to c o n s i d e r a t i o n o f a single r u n o f f event. E v e n t - b a s e d m o d e l s t y p i c a l l y do not simulate E T , s o i l m o i s t u r e d e p r e c i a t i o n or catchment recession.  E i t h e r type o f m o d e l c a n be u s e d for a g i v e n a p p l i c a t i o n , but one type m a y be better suited than the other i n certain situations. F o r e x a m p l e , i m p l e m e n t i n g the event-based m o d e l K I N E R O S R a l l o w s Faures et a l . (1995) to c o m p i l e a statistical representation o f peak estimates from m u l t i p l e s i m u l a t i o n s o f s t o r m r u n o f f w i t h o u t h a v i n g to extract the i n f o r m a t i o n from a c o n t i n u o u s t i m e series. In a different study, C o o p e r et a l . (1997) find that objective functions a p p l i e d to the entire data set y i e l d better a p p r o x i m a t i o n s o f the synthetic parameter set than objective functions that  39  o n l y e x a m i n e peak o r l o w f l o w quantiles. T h e results o f C o o p e r et a l . (ibid.) i m p l y that a c o n t i n u o u s m o d e l is m o r e appropriate for their study.  E v e n t - b a s e d m o d e l s require a "snapshot" o f watershed c o n d i t i o n s at the b e g i n n i n g o f each s i m u l a t i o n . M e a s u r e m e n t s o f these c o n d i t i o n s are s e l d o m a v a i l a b l e and therefore must be assigned a priori o r c a l i b r a t e d as a d d i t i o n a l parameters.  O n e c o u l d argue that event-based  m o d e l s exchange the uncertainty i n m o d e l l i n g E T a n d s o i l m o i s t u r e for uncertainty i n i n i t i a l c o n d i t i o n s . F o r cases w h e r e i n i t i a l c o n d i t i o n s for a n event-based s i m u l a t i o n are obtained f r o m a p r e l i m i n a r y m o d e l , the user s h o u l d be aware that the s i m u l a t i o n is s t i l l subject to uncertainty i n antecedent c o n d i t i o n s ; i t i s m e r e l y h i d d e n as a c o m p u t e d v a r i a b l e ( G r a y s o n et a l . , 1992a). M i c h a u d and S o r o o s h i a n (1994) present a n e x a m p l e o f a h y d r o l o g i c m o d e l ( K T N E R O S ) a p p l i e d w i t h i n i t i a l c o n d i t i o n s excerpted f r o m the output o f another m o d e l . A l t h o u g h c o n t i n u o u s m o d e l s also require i n i t i a l c o n d i t i o n s , a n accurate estimate o f the i n i t i a l c o n d i t i o n s i s less important i f data series i s s u f f i c i e n t l y l o n g . I n s u c h cases, a " s p i n - u p " (or " w a r m - u p " ) p e r i o d c a n b e used. T h e i d e a i s that, o v e r t i m e and regardless o f i n i t i a l c o n d i t i o n s , m o d e l l e d watershed c o n d i t i o n s w i l l g r a d u a l l y a p p r o a c h those in situ. Therefore, p r o v i d e d a sufficiently l o n g data series i s a v a i l a b l e , actual starting c o n d i t i o n s are irrelevant ( H o u g h t o n - C a r r , 1999). Output f r o m the spin-up p e r i o d i s not u s e d to evaluate m o d e l performance.  T h e proper d u r a t i o n o f the " s p i n - u p " p e r i o d s h o u l d b e d e t e r m i n e d f r o m the t i m e r e q u i r e d for m o d e l l e d watershed c o n d i t i o n s to converge w i t h in situ c o n d i t i o n s . I n practice, spin-up times range f r o m several w e e k s to several years ( B i n g e m a n , 2 0 0 1 ; M a d s e n , 2 0 0 0 ; T h y e r et a l . , 1999; V r u g t et a l . , 2 0 0 3 ) .  F i n a l l y , a l t h o u g h this thesis i s w r i t t e n l a r g e l y i n the context o f deterministic h y d r o l o g i c m o d e l s , the reader s h o u l d b e aware o f the range o f stochasticity that c a n b e accounted for i n a h y d r o l o g i c m o d e l . M o s t h y d r o l o g i c m o d e l s currently i n use are f u l l y deterministic; f r o m the perspective o f the m o d e l , a l l parameters and data are k n o w n w i t h certainty. H o w e v e r , it is p o s s i b l e for a m o d e l to incorporate p r o b a b i l i s t i c representation o f b o t h parameters and data. In m o s t cases, stochastic m o d e l s use repeated M o n t e C a r l o s i m u l a t i o n s to establish a p r o b a b i l i t y - m a g n i t u d e output w h i c h supplants the s i m p l e m a g n i t u d e estimate p r o v i d e d b y deterministic m o d e l s .  40  P r o b a b i l i t y distributions for stochastic m o d e l parameters are u s u a l l y either specified a priori b a s e d o n expert k n o w l e d g e or p u b l i s h e d data, or estimated f r o m a field s a m p l i n g p r o g r a m .  The  m o d e l is r u n repeatedly u s i n g r a n d o m samples f r o m the d i s t r i b u t i o n o f each parameter.  Stochastic m o d e l s , s u c h as the one espoused b y K u c h m e n t a n d G e l f a n (2002), are t y p i c a l l y u t i l i z e d for extreme event p r e d i c t i o n . E x t r e m e events are m o s t often characterized b y significant uncertainty regarding antecedent c o n d i t i o n s and flood-generating processes. A l t h o u g h stochastic m o d e l s are designed to p a r t i a l l y quantify m o d e l p r e d i c t i v e uncertainty, the user s h o u l d be aware that results w i l l t y p i c a l l y be l i m i t e d b y the least accurate constituent parameters, c o m p o n e n t s , or probabilities.  2.4  Model Calibration  E v e r y h y d r o l o g i c m o d e l l e r faces t w o fundamental challenges:  the first is to select an appropriate  m o d e l for the study site; the s e c o n d is to determine parameter values s u c h that the m o d e l c l o s e l y simulates the in situ b e h a v i o u r ( S o r o o s h i a n and G u p t a , 1995). P o o r parameter s p e c i f i c a t i o n c a n lead to p o o r s i m u l a t i o n regardless o f m o d e l s o p h i s t i c a t i o n (e.g., M i c h a u d and S o r o o s h i a n , 1994). Therefore, careful set-up a n d i n i t i a l i z a t i o n is necessary to m a x i m i z e the r e l i a b i l i t y o f the m o d e l ( G u p t a e t a l . , 1999).  M o d e l i d e n t i f i c a t i o n (e.g., system and b o u n d a r y d e f i n i t i o n , data specification, and d e l i n e a t i o n o f relationships between variables) cannot be treated as entirely distinct from m o d e l c a l i b r a t i o n (e.g., selection o f e v a l u a t i o n criteria, i d e n t i f i c a t i o n o f an i n f o r m a t i v e data set, and parameter o p t i m i z a t i o n ) . A s the focus o f this w o r k i n v o l v e s neither d e s i g n i n g n o r altering a m o d e l , the m o d e l i d e n t i f i c a t i o n phase is l a r g e l y set b y the selection o f an extant h y d r o l o g i c m o d e l . H o w e v e r , even w h e n a p p l y i n g a k n o w n m o d e l , the c a l i b r a t i o n phase s h o u l d be pursued w i t h due c o n s i d e r a t i o n o f the context o f the structure and properties o f the i n d i v i d u a l m o d e l ( S o r o o s h i a n and G u p t a , 1985).  T h r e e stages s h o u l d be e x p l o r e d i n m o d e l a p p l i c a t i o n : c a l i b r a t i o n , v a l i d a t i o n , and an assessment o f r e l i a b i l i t y ( S i n g h , 1995b). T h e b u l k o f this C h a p t e r deals w i t h m o d e l c a l i b r a t i o n . V a l i d a t i o n is discussed further i n S e c t i o n 2.4.6, w h i l e r e l i a b i l i t y ( i n terms o f the uncertainty i n m o d e l results) is the subject o f C h a p t e r 3.  41  In general, a p p l y i n g a m o d e l requires e s t i m a t i n g values for t w o types o f parameters: " p h y s i c a l " parameters w h i c h c a n be measured (e.g., watershed area) and " p r o c e s s " parameters w h i c h represent watershed properties not d i r e c t l y measurable (e.g., t i m e l a g constants for runoff) ( S o r o o s h i a n and G u p t a , 1995). C a l i b r a t i o n i n v o l v e s b a c k - c a l c u l a t i o n o f the " p r o c e s s " parameters t h r o u g h trial-and-error c o m p a r i s o n s o f p r e d i c t i o n s to r u n o f f records ( D u n n e , 1982). S o r o o s h i a n and G u p t a (1995) observe that a priori j u d g m e n t and experience are frequently u s e d to i d e n t i f y a range or bounds for each " p r o c e s s " parameter d u r i n g the p r e l i m i n a r y stages o f c a l i b r a t i o n . I n general, the use o f expert j u d g m e n t i n s u c h matters s h o u l d c o m p l e m e n t rather than replace other evidence ( K e e n e y a n d W i n t e r f e l d t , 1989).  T h e i n t e n t i o n o f c a l i b r a t i o n s h o u l d be clear before c o m m e n c i n g - i.e., i s the g o a l s i m p l y to a c h i e v e the best p o s s i b l e fit for the c a l i b r a t i o n set, or does it i n v o l v e the i d e n t i f i c a t i o n o f a u n i q u e a n d realistic parameter set that c l o s e l y represents the p h y s i c a l system ( S o r o o s h i a n and G u p t a , 1983). S u c h general considerations are d i s c u s s e d i n S e c t i o n 2 . 4 . 1 , w h i l e Sections 2.4.2 and 2.4.3 address m a n u a l and automatic c a l i b r a t i o n , respectively.  2.4.1 The Nature of Calibration U n t i l recently, c a l i b r a t i o n has arguably b e e n the m o s t i n t e n s i v e l y investigated aspect o f h y d r o l o g i c m o d e l l i n g ( B e v e n , 2 0 0 0 ) . T h r o u g h m a n y studies, threads o f consistency have emerged i d e n t i f y i n g v a r i o u s pre-requisites for a successful c a l i b r a t i o n .  In the current era o f r e d u c e d f u n d i n g for h y d r o m e t r i c and m e t e o r o l o g i c data n e t w o r k s , it is fitting that p r i m a r y c o n s i d e r a t i o n be afforded to the data. A l t h o u g h e v e n a s i n g l e year o f c a l i b r a t i o n data c a n assist i n parameter s p e c i f i c a t i o n , c a l i b r a t i o n is m u c h m o r e sensitive to data i n f o r m a t i v e n e s s than to length o f r e c o r d ( G a n et a l . , 1997; G u p t a et a l . , 1998; S o r o o s h i a n et a l . , 1983). R e f s g a a r d (1994) c o n c l u d e s that three to five years o f s t r e a m f l o w data constitute an adequate basis for c a l i b r a t i o n o f a c o n t i n u o u s m o d e l . Y a p o et a l . (1996) f i n d that i n f o r m a t i v e n e s s o f the data set is o p t i m a l u s i n g an 8-year span o f s t r e a m f l o w data. W h i l e m u c h less c o m m o n , s o m e guidance o n the appropriate s i z e o f a c a l i b r a t i o n data set for event-based m o d e l s is also a v a i l a b l e : S o r o o s h i a n and G u p t a (1995) r e c o m m e n d that the n u m b e r o f data used i n c a l i b r a t i o n e x c e e d 20 times the n u m b e r o f parameters b e i n g estimated. N a t u r a l l y , a m o d e l s i m u l t a n e o u s l y calibrated against m u l t i p l e data series (e.g., m u l t i p l e streamflow measurement  42  l o c a t i o n s , or s t r e a m f l o w and groundwater together) w i l l t y p i c a l l y require a shorter p e r i o d o f data than the same m o d e l c a l i b r a t e d against catchment o u t f l o w alone.  W h i l e h y d r o l o g i c m o d e l c a l i b r a t i o n g e n e r a l l y i m p l i e s c o m p a r i s o n o f o b s e r v e d and s i m u l a t e d catchment o u t f l o w , the use o f a d d i t i o n a l i n f o r m a t i o n c a n increase c o n f i d e n c e i n the representativeness o f the m o d e l . C a l i b r a t i o n c a n also i n c l u d e c o m p a r i s o n s to o b s e r v e d t i m e series o f depth to the water table, s n o w depth or s n o w water equivalent, s o i l m o i s t u r e , stable natural isotopes (e.g., o x y g e n - 1 8 ) , a l k a l i n i t y and p H , or n i t r o g e n l o a d i n g s ( B e r g s t r o m et a l . , 2 0 0 2 ) . C a l i b r a t i o n against different data types m a y a l l o w insights into different aspects o f the m o d e l . F o r e x a m p l e , the a p p l i c a t i o n o f c o n s e r v a t i v e c h e m i c a l tracer data p r o v i d e s an o p p o r t u n i t y to i m p r o v e descriptions o f water p a t h w a y s and residence times ( i b i d . ) .  A t the outset, a m o d e l l e r must decide w h a t aspects o f watershed b e h a v i o u r w i l l f o r m the basis for c a l i b r a t i o n , and h o w those b e h a v i o u r s w i l l be m e a s u r e d ( G u p t a et a l . , 1998). Further, the properties o f the m o d e l ( i n c l u d i n g m o d e l error) need to be r e c o g n i z e d a n d c o n s i d e r e d w h e n d e s i g n i n g a c a l i b r a t i o n strategy ( Y a p o et a l . , 1996).  M o s t i m p o r t a n t l y , a l l processes that are  h y d r o l o g i c a l l y relevant i n the catchment must be activated d u r i n g c a l i b r a t i o n . U n a c t i v a t e d process parameters cannot be c o n s i d e r e d calibrated and are l i k e l y to cause p o o r m o d e l p e r f o r m a n c e i f activated d u r i n g later s i m u l a t i o n s or i n other scenarios ( G a n and B u r g e s , 1990b; L e i a n d S c h i l l i n g , 1996). S o r o o s h i a n and G u p t a (1985) u t i l i z e data sets c h o s e n s p e c i f i c a l l y to ensure a c t i v a t i o n o f a l l aspects o f the m o d e l .  D a t a sequences c o n t a i n i n g a h i g h degree o f h y d r o l o g i c v a r i a b i l i t y are m o r e l i k e l y to activate the v a r i o u s o p e r a t i o n a l m o d e s o f the m o d e l and thus result i n m o r e r e l i a b l e parameter estimates ( S o r o o s h i a n and G u p t a , 1995; Y a p o et a l . , 1996). W e t years (or catchments) tend to be better suited for c a l i b r a t i o n against catchment o u t f l o w than d r y years (or catchments).  T h i s is due to  their m o r e v a r i a b l e f l o w r e g i m e and the d o m i n a n c e o f o u t f l o w - as o p p o s e d to E T - as a d r i v i n g f l u x ( G a n , 1987; G a n et a l . , 1997; T h y e r et a l . , 1999). Q u a l i t y o f response is also less consistent for d r y antecedent c o n d i t i o n s ( G a n and B u r g e s , 1990b). Y a p o et a l . (1996) r e c o m m e n d u s i n g a data set c o n s i s t i n g o f the wettest p e r i o d o n r e c o r d . T h i s r e c o m m e n d a t i o n is supported b y the experience o f the U S N a t i o n a l W e a t h e r S e r v i c e ( H o g u e et a l , 2 0 0 0 ) . G a n (1987) attributes the acceptable p e r f o r m a n c e o f h i s m o d e l under extreme input to g o o d representation o f the processes  43  important to extreme event runoff within the calibration data. He generally notes that including major flood events in the calibration data series can improve the adequacy of calibration for extreme conditions (ibid.). One may be inclined to conclude that a model should be calibrated on conditions similar to those expected during its application. However, Klemes (1986b) points out that this is not logical if the goal of calibration is to ascertain an appropriate model structure rather than merely provide the best fit for the observed data. If the goal is an appropriate model structure, Klemes advises calibrating with hydrologically dissimilar data before validating on conditions similar to those of the target simulation. Models relying solely on calibration data are often overparameterized. In such cases, the process of establishing appropriate interaction between the model and the available data can overshadow the search for a good representation of the system (Kuczera, 1997). In a large number of cases, varying parameter values obtained from multiple calibrations of the same model generate similar goodness-of-fit results. Seibert (2000) find that the aggregate parameter range defined by multiple "successful" calibrations encompasses approximately one-third of the feasible parameter space. Non-identifiability (i.e., the inability to identify a unique and optimal parameter set) can manifest itself in two ways. Firstly, distinct output functions may yield similar results for quantitative evaluation criteria. Secondly, distinct parameter sets may generate identical output functions. The latter is generally of more concern to hydrologic modellers (Sorooshian and Gupta, 1985). Sorooshian and Gupta (1983) and Yapo et al. (1996) present several reasons why a model parameter may be poorly identifiable in the second sense: •  parameter interdependence: parameters interact strongly with other parameters, allowing for compensating variations in the feasible parameter space;  •  parameter nonstationarity: parameter location and precision are correlated to varying watershed or data characteristics not accounted for during calibration;  •  data noninformativeness: the data do not contain the hydrologic conditions required to properly identify the parameters; 44  •  c r i t e r i o n inadequacy: the e v a l u a t i o n c r i t e r i a cannot p r o p e r l y extract the i n f o r m a t i o n c o n t a i n e d i n the data;  •  m a t h e m a t i c a l d i f f i c u l t y : the c a l i b r a t i o n is constrained b y l o c a l features o f the response surface; and  •  i n s e n s i t i v i t y : variations i n the values o f the parameters do not s i g n i f i c a n t l y affect the m o d e l output or the c a l i b r a t i o n c r i t e r i o n .  T e s t i n g for g l o b a l i d e n t i f i a b i l i t y is i m p r a c t i c a l i n m o s t cases w h e r e it is not i m p o s s i b l e ( S o r o o s h i a n and G u p t a , 1985). Therefore, i n m o s t situations, i d e n t i f y i n g stable and realistic parameter v a l u e s is as important as f i n d i n g values w h i c h p r o d u c e a g o o d fit to the o b s e r v e d data ( M i c o v i c , 1998). L a t e r chapters o f this w o r k focus o n the need to understand and quantify the potential i m p a c t o f alternate solutions o n m o d e l s i m u l a t i o n s .  2.4.2 Manual Calibration T h e m o s t c o m m o n approach to m o d e l c a l i b r a t i o n is m a n u a l (sometimes c a l l e d "expert") c a l i b r a t i o n ( H o g u e et a l . , 2000).  T y p i c a l l y , an expert user f o l l o w s a s e m i - i n t u i t i v e trial-and-  error process, adjusting parameter values to i m p r o v e the " c l o s e n e s s " o f fit between observed data and s i m u l a t e d output ( B o y l e et a l . , 2 0 0 0 ) . A l t h o u g h v i s u a l i n s p e c t i o n o f the observed and s i m u l a t e d hydrographs is the t y p i c a l standard measure o f performance, statistical c o m p a r i s o n s and other factors are c o m m o n l y used i n c o n j u n c t i o n w i t h qualitative analysis o f fit ( H o g u e et a l . , 2 0 0 0 ) . E m p h a s i s i s u s u a l l y p l a c e d o n m a t c h i n g peaks, f l o o d v o l u m e s , recession slopes, and base flow (ibid.).  T h e e v a l u a t i o n process c o n v e n t i o n a l l y i n v o l v e s a q u a l i t a t i v e synthesis o f v i s u a l i n s p e c t i o n a n d n u m e r i c a l criteria, and often proceeds t h r o u g h sequential c a l i b r a t i o n o f parameter groups or objectives. T h e intent o f sequential c a l i b r a t i o n i s to s i m p l i f y the process b y c o n s i d e r i n g fewer variables s i m u l t a n e o u s l y . T h i s is a c h i e v e d b y first i s o l a t i n g and o p t i m i z i n g those parameters or objectives h a v i n g the greatest influence o v e r the results w h i l e n e g l e c t i n g secondary objectives and less-sensitive parameters. H o w e v e r , the p o t e n t i a l for parameter interaction i m p l i e s that  45  sequential c a l i b r a t i o n - unless h i g h l y iterative - c a n l o c k the parameter values i n a trap set up b y c o m p e n s a t i n g errors ( B e r g s t r o m et a l . , 2 0 0 2 ) .  T h e process o f m a n u a l c a l i b r a t i o n is h i g h l y l a b o u r - i n t e n s i v e . It c a l l s for a considerable degree o f t r a i n i n g , experience, a n d effort, and requires a substantial c o m m i t m e n t o f t i m e and expertise for e v e n the most expert user ( B o y l e et a l . , 2 0 0 0 ; G a n , 1987; H o g u e et a l . , 2 0 0 0 ) . A c h i e v i n g acceptable results requires that the m o d e l l e r have a w o r k i n g understanding o f the m o d e l , the data, and the watershed system ( B o y l e et a l . , 2 0 0 0 ) .  M a n y organizations h a v e d e v e l o p e d systematic procedures to be f o l l o w e d b y users i n an attempt to standardize the process. T h e U S N a t i o n a l W e a t h e r S e r v i c e i n p a r t i c u l a r has d e v e l o p e d a m a n u a l procedure that p r o v i d e s excellent m o d e l c a l i b r a t i o n s but is k n o w l e d g e - i n t e n s i v e and therefore not e a s i l y transferred b e t w e e n p e o p l e or m o d e l s ( B o y l e et a l . , 2 0 0 0 ) . W h i l e the c a l i b r a t i o n procedure f o l l o w e d b y a user m a y be standardized, the i n d i v i d u a l ' s adjustments w i l l be i n f l u e n c e d o n p e r s o n a l s k i l l s and experience. H o u g h t o n - C a r r (1999) finds that v a r i o u s experts offer d i f f e r i n g perspectives o n what constitutes " g o o d " or " b a d " performance, and w i l l t y p i c a l l y evaluate p e r f o r m a n c e differently.  M a n u a l c a l i b r a t i o n relies o n the conceptual r e l a t i o n s h i p b e t w e e n m o d e l parameters and watershed characteristics to support the h y d r o l o g i s t ' s e x p e c t a t i o n that adjustment o f a g i v e n parameter v a l u e i n a p a r t i c u l a r d i r e c t i o n w i l l have a predictable effect o n the m o d e l output ( G u p t a et a l . , 1999). S o r o o s h i a n and G u p t a (1995) state that the m a i n weakness o f m a n u a l c a l i b r a t i o n i s i n not k n o w i n g w h e n to quit, since e v e n experts m a y not a l w a y s be able to assess the p o t e n t i a l for further i m p r o v e m e n t . T h e s e m i - i n t u i t i v e nature o f w o r k i n g through the c o m p l e x process o f m a n u a l c a l i b r a t i o n c a n l e a d to c o n s i d e r a b l e frustration ( L a n , 2 0 0 1 ) .  O n e o f the greatest advantages o f m a n u a l c a l i b r a t i o n i s the a b i l i t y to incorporate e x p l i c i t a l l o w a n c e for p o t e n t i a l data errors ( B o y l e et a l . , 2 0 0 0 ) . W h e r e s i m u l a t i o n q u a l i t y is p o o r due to erroneous or non-representative data, the user c a n choose to ignore the erroneous data i n favour o f those b e l i e v e d to be m o r e correct.  46  2.4.3 Automatic Calibration A u t o m a t i c c a l i b r a t i o n uses n u m e r i c a l c o m p a r i s o n s b e t w e e n o b s e r v e d and s i m u l a t e d data to o p t i m i z e parameter values w i t h o u t user intervention ( M a d s e n , 2 0 0 3 ) . A n automatic m o d e l c a l i b r a t i o n is t y p i c a l l y o b j e c t i v e i n nature and r e l a t i v e l y easy to i m p l e m e n t , but focuses o n n u m e r i c a l o p t i m i z a t i o n a n d m a y thus result i n a h y d r o l o g i c a l l y unrealistic parameter set ( B o y l e et a l . , 2 0 0 0 ; H o g u e et a l . , 2 0 0 0 ) . H i s t o r i c a l l y , this has l e d the h y d r o l o g i c m o d e l l i n g c o m m u n i t y to be h i g h l y s k e p t i c a l o f automatic c a l i b r a t i o n i n an a p p l i e d context.  N o n e t h e l e s s , a substantial b o d y o f o p i n i o n suggests that m o d e r n state-of-the-art automatic c a l i b r a t i o n methods c a n be c o n s i d e r e d a v i a b l e alternative to expert c a l i b r a t i o n for some purposes ( G a n and B i f t u , 1996; G u p t a et a l . , 1999). O n e s u c h situation is where m u l t i p l e objectives a d d significant c o m p l e x i t y to the c a l i b r a t i o n process ( M a d s e n , 2 0 0 3 ) . A n o t h e r e x a m p l e is the w o r k o f Seibert and M c D o n n e l l (2002), w h o use m u l t i - o b j e c t i v e c a l i b r a t i o n to demonstrate that automatic m e t h o d s do not necessarily p r e c l u d e the i n c o r p o r a t i o n o f qualitative or i m p r e c i s e data. A u t o m a t i c c a l i b r a t i o n c a n also be used as an adjunct to m a n u a l c a l i b r a t i o n ; automatic o p t i m i z a t i o n a l g o r i t h m s are c o m m o n l y used for f i n e - t u n i n g an expert hydrologist's m a n u a l c a l i b r a t i o n ( D u a n et a l . , 1994b).  T h e e v o l u t i o n o f automatic c a l i b r a t i o n has been m o t i v a t e d b y the need for s i m p l i c i t y , speed, objectivity, confidence, and less reliance o n hard-to-find expert calibrators ( H o g u e et a l . , 2 0 0 0 ; S o r o o s h i a n and G u p t a , 1995). M o s t research regarding automatic c a l i b r a t i o n has been c a r r i e d out for l u m p e d c o n c e p t u a l m o d e l s ( M a d s e n , 2003). R e c e n t l y - d e v e l o p e d m u l t i - o b j e c t i v e automatic c a l i b r a t i o n tools m a y increase the a p p l i c a t i o n o f automatic c a l i b r a t i o n to m o r e c o m p l e x distributed catchment m o d e l s as they are adopted b y distributed m o d e l l e r s s e e k i n g c o n s i s t e n c y w i t h o b s e r v e d data throughout the catchment.  T h e i m p l e m e n t a t i o n o f an automatic c a l i b r a t i o n is defined b y the s e l e c t i o n o f its v a r i o u s c o m p o n e n t s : objective function, search a l g o r i t h m , c a l i b r a t i o n data, and search t e r m i n a t i o n criteria ( H o g u e et a l . , 2 0 0 0 ) . T y p i c a l l y , the automatic p o r t i o n o f a n y c a l i b r a t i o n routine has the f o l l o w i n g steps i n c o m m o n ( G u p t a et a l . , 1999):  47  •  identify c o m p o n e n t s ;  •  o b t a i n an i n i t i a l estimate o f the v a l u e s (or ranges) for the parameters;  •  execute the m o d e l ;  •  measure performance; a n d  •  a p p l y a search a l g o r i t h m to f i n d parameter values that i m p r o v e the v a l u e o f the objective function.  T h e q u e s t i o n o f " w h e n to q u i t " is addressed for automatic c a l i b r a t i o n t h r o u g h e x p l i c i t s p e c i f i c a t i o n o f t e r m i n a t i o n criteria. A p p r o p r i a t e t e r m i n a t i o n c r i t e r i a are necessary to a c h i e v e the desired balance b e t w e e n effectiveness a n d efficiency. T e r m i n a t i o n c r i t e r i a t y p i c a l l y i n c l u d e parameter c o n v e r g e n c e ( m e a n i n g that u n i q u e v a l u e s have b e e n i d e n t i f i e d for each parameter), f u n c t i o n c o n v e r g e n c e ( m e a n i n g the o b j e c t i v e f u n c t i o n cannot be s i g n i f i c a n t l y i m p r o v e d ) , o r a m a x i m u m n u m b e r o f iterations (to a v o i d endless loops) ( H o g u e et a l . , 2 0 0 0 ) .  A l t h o u g h parameter c o n v e r g e n c e i s u s u a l l y the best i n d i c a t o r o f a m a t h e m a t i c a l l y - o p t i m a l parameter set, n o n e o f the above c r i t e r i a c a n c o n c l u s i v e l y ascertain that the g l o b a l o p t i m u m has b e e n reached, s i n c e p o o r l y - c h o s e n c r i t e r i a c a n halt the search procedure before it locates the best a v a i l a b l e parameter set ( S i n g h a n d W o o l h i s e r , 2 0 0 2 ; S o r o o s h i a n et a l . , 1983).  T h e strength o f an automatic p r o c e d u r e depends o n h o w w e l l its d e s i g n reflects the v a r i o u s factors important to o b t a i n i n g a successful c a l i b r a t i o n i n each specific case. T h e i d e n t i f i c a t i o n o f these factors has b e e n the subject o f s i g n i f i c a n t research ( B o y l e et a l . , 2 0 0 0 ) . M u c h early research focused o n i m p r o v i n g search m e t h o d s that c o n v e r g e d to one o f the m u l t i p l e points o r clusters c o m p r i s i n g the l o c a l o p t i m a o f the response surface ( K u c z e r a , 1997). A l t h o u g h m a n y a d v a n c e d search techniques c a n n o w a v o i d b e i n g trapped b y l o c a l o p t i m a , the p r o b l e m is far f r o m s o l v e d . W h i l e n e w procedures m a y be able to p r o d u c e r e l i a b l e estimates o f g l o b a l o p t i m a for c o m p l e x p r o b l e m s (e.g., D u a n et a l . , 1994b), the focus o f these procedures is s t i l l d i r e c t e d at o p t i m i z a t i o n rather than h y d r o l o g i c m o d e l c a l i b r a t i o n .  D a t a uncertainty, m o d e l structure uncertainty, a n d dependence o n the c a l i b r a t i o n procedure (e.g., objective f u n c t i o n selection) i m p l y that the best p o s s i b l e representation o f the system w i l l not n e c e s s a r i l y c o i n c i d e w i t h the g l o b a l o p t i m u m o f the objective f u n c t i o n ( V r u g t et a l . , 2 0 0 3 ) . G a n  48  (1987) observes objective function values a p p r o a c h i n g their theoretical l i m i t for some c a l i b r a t i o n s , despite "gross s i m u l a t i o n errors" o b v i o u s i n the c o r r e s p o n d i n g hydrographs. H o u g h t o n - C a r r (1999) emphasizes that o p t i m i z a t i o n a l g o r i t h m s l a c k a n e x p e r t ' s k n o w l e d g e o f m o d e l structure and cannot j u d g e w h e n a parameter set is unrealistic.  A u t o m a t i c c a l i b r a t i o n methods are t y p i c a l l y c o m p a r e d o n t w o scales: a c c u r a c y o f s o l u t i o n (i.e., effectiveness) and e f f i c i e n c y o f convergence. E f f i c i e n c y o f convergence for an automatic c a l i b r a t i o n m e t h o d refers to h o w q u i c k l y the final s o l u t i o n is i d e n t i f i e d , i m p l i c i t l y representing the amount o f c o m p u t a t i o n a l effort r e q u i r e d (e.g., r e q u i r e d n u m b e r o f f u n c t i o n evaluations) ( T h y e r et a l . , 1999). Robustness o f s o l u t i o n s (i.e., the v a r i a t i o n i n solutions across m u l t i p l e trials) is another potential c r i t e r i o n for e v a l u a t i o n ( C o o p e r et a l . , 1997).  T h e emergence o f automatic c a l i b r a t i o n techniques has reduced the degree o f user t r a i n i n g r e q u i r e d to o b t a i n r e a l i s t i c - l o o k i n g results f r o m a h y d r o l o g i c m o d e l . H o w e v e r , the s i m p l i f i e d , " h a n d s - o f f ' m o d e l c a l i b r a t i o n process a n d the " o b j e c t i v i t y " inherent i n automatic c a l i b r a t i o n also increase the potential for error, as there i s a net loss o f expert guidance i n i n t u i t i v e l y assessing the f e a s i b i l i t y o f parameter values and sets. T h e k n o w l e d g e c o n t r i b u t i o n o f the expert user i n m a n u a l c a l i b r a t i o n a v o i d s m a n y o f the p i t f a l l s a w a i t i n g the incautious automatic calibrator. C o n s i d e r the case o f o b v i o u s data error: m e t h o d s o f automatic c a l i b r a t i o n i m p l i c i t l y assume that a l l data are e q u a l l y correct. A n automatic procedure m a y distort the c a l i b r a t i o n to compensate for data error that an expert w o u l d q u i c k l y and e a s i l y disregard.  T h e concept o f an " o b j e c t i v e " automatic c a l i b r a t i o n must also be v i e w e d w i t h c a u t i o n , as results h a v e been s h o w n to depend o n subjective d e c i s i o n s s u c h as the c r i t e r i a to be o p t i m i z e d and the p e r i o d o f c a l i b r a t i o n data ( B e r g s t r o m et a l . , 2 0 0 2 ) . R e s u l t s o f c o m p a r i s o n s for v a r i o u s automatic c a l i b r a t i o n a l g o r i t h m s m a y also be affected b y c h a n g i n g the d i m e n s i o n a l i t y o f the parameter space (Seibert, 2 0 0 0 ) . G a n et a l . (p. 9 7 , 1997) present a study o f differences between automatic c a l i b r a t i o n s a p p l i e d w i t h a v a r i e t y o f m o d e l s , o p t i m i z a t i o n a l g o r i t h m s , data, and objective functions.  M o r e specific l i m i t a t i o n s o f automatic m o d e l c a l i b r a t i o n have b e e n discussed b y m a n y authors. S o m e o f the m o r e w i d e l y a c k n o w l e d g e d challenges i n c l u d e the f o l l o w i n g :  49  •  the i n a b i l i t y o f any single performance measure to p r o p e r l y address a l l the important characteristics o f the system ( G a n , 1987; M a d s e n , 2 0 0 0 ) ;  •  interdependence o f parameter v a l u e s , r e s u l t i n g i n uncertain and v e r y s l o w i m p r o v e m e n t s ( G a n , 1987; M o o r e and C l a r k e , 1981);  •  p r o b l e m s w i t h m o d e l or parameter i d e n t i f i a b i l i t y , s u c h that appreciable changes i n parameter v a l u e cause little or no change i n objective f u n c t i o n ( M o o r e and C l a r k e , 1981; S o r o o s h i a n and G u p t a , 1983);  •  the i m p o s s i b i l i t y o f e x h a u s t i v e l y e x p l o r i n g the feasible r e g i o n due to d i s c o n t i n u i t i e s , non-differentiable regions, or other c o m p l e x i t i e s o f the response surface ( G a n , 1987; M o o r e and C l a r k e , 1981);  •  the potential for m o d e l d i v e r g e n c e g i v e n p o o r i n i t i a l parameter values ( G a n , 1987); and  •  treatment o f each c o m b i n a t i o n o f parameter v a l u e s as " t h e o r e t i c a l l y p o s s i b l e " w h e n s o m e m a y be i n c o m p a t i b l e w i t h science o r p h y s i c s ( K l e m e s , 2000a).  O b v i o u s l y , automatic c a l i b r a t i o n is not a facile s o l u t i o n to m o d e l c a l i b r a t i o n , and cannot entirely replace m a n u a l i n v o l v e m e n t i n the process. U s e r i n f o r m a t i o n is r e q u i r e d to c u s t o m i z e the approach and evaluate the process ( M a d s e n et a l . , 2 0 0 2 ) . M u l t i p l e studies have s h o w n that g l o b a l o p t i m i z a t i o n performance i m p r o v e s as parameter ranges are r e d u c e d or constrained (e.g., F r a n c h i n i et a l . , 1998; G a n and B u r g e s , 1990a; K u c z e r a , 1997; Seibert and M c D o n n e l l , 2 0 0 2 ) . T h u s , l i m i t i n g the parameter space to be searched is one e x a m p l e o f a k e y area for expert user input.  T h e above-noted potential i n a d e q u a c y o f a single measure o f performance is not a l w a y s i n s u r m o u n t a b l e , since automatic c a l i b r a t i o n is not l i m i t e d to the single-stage g l o b a l o p t i m i z a t i o n o f general objectives. O f t e n , the c a l i b r a t i o n p r o b l e m is r e d u c e d to sub-problems, each o f w h i c h m a y be s o l v e d m a n u a l l y or b y u s i n g different o p t i m i z a t i o n techniques ( G u p t a et a l . , 1999).  50  H o w e v e r , one must be careful to a v o i d the p o s s i b i l i t y o f an o p t i m i z a t i o n a l g o r i t h m negating expert k n o w l e d g e i n c o r p o r a t e d earlier i n the process.  M o d e l - s p e c i f i c k n o w l e d g e - b a s e d expert systems are one e x a m p l e o f a h y b r i d approach.  A  k n o w l e d g e - b a s e d expert system uses o p t i m i z a t i o n to automate v a r i o u s stages o f a m a n u a l c a l i b r a t i o n ( M a d s e n et a l . , 2 0 0 2 ) . These processes are d e s i g n e d to y i e l d results at least c o m p a r a b l e to the expert process w h i l e i m p r o v i n g efficiency. B o y l e et a l . (2000) propose u s i n g expert k n o w l e d g e to select feasible parameter ranges and values for d o m i n a n t parameters, then i m p l e m e n t i n g a n automatic search to m i n i m i z e parameter uncertainty w i t h a focus o n the potential for parameter interaction. H o g u e et a l . (2000) propose a M u l t i s t e p A u t o m a t i c C a l i b r a t i o n S c h e m e ( M A C S ) that sequentially applies automatic c a l i b r a t i o n to different subsets o f m o d e l parameters. S u b j e c t i v e and statistical evaluations o f M A C S s h o w a significant i m p r o v e m e n t i n e f f i c i e n c y o v e r the m a n u a l approach, w i t h c o n s i s t e n t l y c o m p a r a b l e results. H o w e v e r , the authors c a u t i o n that techniques s u c h as M A C S are not yet ready for o p e r a t i o n a l use (ibid.).  It is important to d i s t i n g u i s h between successful testing o f automatic c a l i b r a t i o n techniques and a readiness for " r e a l - w o r l d " a p p l i c a t i o n ; the first does not n e c e s s a r i l y i m p l y the second.  Much  research and testing is p u r s u e d o n " s t e r i l i z e d " data to a l l o w better assessment o f the effectiveness and e f f i c i e n c y o f the c a l i b r a t i o n procedure. H o w e v e r , results f r o m s u c h tests are significant o n l y i n a research context. B e f o r e b e i n g a p p r o v e d for p r a c t i c a l a p p l i c a t i o n , automatic techniques s h o u l d be e x t e n s i v e l y tested w i t h the type, quantity, and q u a l i t y o f data encountered i n the f i e l d ( H o g u e et a l . , 2 0 0 0 ) .  2.4.4 Methods for Automatic Calibration M e t h o d s for automatic c a l i b r a t i o n encompass a range o f c o m p l e x i t y . T h i s section explores some o f the m a n y techniques a v a i l a b l e to the h y d r o l o g i c m o d e l l e r , b e g i n n i n g w i t h some s i m p l e techniques a n d p r o g r e s s i n g to those o f higher c o m p l e x i t y .  T h e simplest o f automatic c a l i b r a t i o n methods is M o n t e C a r l o S i m u l a t i o n ( M C S ) . " M o n t e C a r l o " refers to the general process o f r a n d o m s a m p l i n g f r o m a pre-determined statistical p o p u l a t i o n w i t h the g o a l o f generating an approximate s o l u t i o n to a quantitative p r o b l e m . F o r a M o n t e C a r l o procedure, the a c c u r a c y o f the a p p r o x i m a t i o n is related to the n u m b e r o f trials. I n  51  the h y d r o l o g i c m o d e l l i n g context, M o n t e C a r l o S i m u l a t i o n t y p i c a l l y i n v o l v e s repeated trials o f the candidate m o d e l u s i n g parameter values r a n d o m l y s a m p l e d f r o m a priori parameter distributions. In most cases w h e r e M C S is used as a c a l i b r a t i o n t o o l , the p r o b a b i l i t y d i s t r i b u t i o n is assumed to be u n i f o r m for each parameter to reflect the ignorance o f the m o d e l l e r w i t h regard to the " m o s t l i k e l y " parameter values.  T h e M C S technique is straightforward, preserves n o n l i n e a r interactions w i t h i n the c a l c u l a t i o n s , and is not dependent o n l i m i t i n g assumptions about an appropriate d i s t r i b u t i o n o f parameter values ( B i n l e y et a l . , 1991). T h e process has n o m e m o r y and each trial is independent. Therefore, a M o n t e C a r l o a p p r o a c h is c o n c e r n e d w i t h neither i m p r o v i n g objectives n o r u p d a t i n g the parameter distributions.  T h e chances that p u r e l y r a n d o m s a m p l i n g w i l l q u i c k l y result i n a near o p t i m a l s o l u t i o n are n e g l i g i b l e . E v e n i f f o u n d , there is little chance that the o p t i m a l s o l u t i o n c o u l d be r e c o g n i z e d as a n y t h i n g m o r e than the best m e m b e r o f a finite set; a r a n d o m search w i t h o u t d i r e c t i o n requires enumeration o f a statistically large p o r t i o n o f the parameter space before n e a r - o p t i m a l i t y c a n be d e c l a r e d w i t h c o n f i d e n c e ( L a n , 2 0 0 1 ) . Therefore, one c a n c o n c l u d e that w h i l e M C S is a g o o d t o o l for e x p l o r i n g the parameter space or i d e n t i f y i n g a g o o d starting p o i n t for m o r e detailed procedures, it is not i t s e l f a strong t o o l for automatic c a l i b r a t i o n . H o w e v e r , M o n t e C a r l o theory p l a y s a r o l e i n m o s t a d v a n c e d o p t i m i z a t i o n algorithms.  M o s t c o m m o n l y - u s e d m e t h o d s for automatic c a l i b r a t i o n i n v o l v e a directed search i n the f o r m o f o p t i m i z a t i o n . T h e e x p l i c i t g o a l o f o p t i m i z a t i o n m e t h o d s is the systematic adjustment o f parameter values i n search o f a set that m i n i m i z e s (or alternatively, m a x i m i z e s ) the v a l u e o f an objective function or set o f functions ( C o o p e r et a l . , 1997). A n a l y t i c a l l y - b a s e d solutions to o p t i m i z a t i o n p r o b l e m s require response surface c o n d i t i o n s s u c h as continuity, c o n v e x i t y , and t w i c e differentiability, w h i c h (at least i n c o m b i n a t i o n ) are u n c o m m o n i n the context o f a p p l i e d h y d r o l o g i c m o d e l l i n g ( i b i d . ) . T h i s precludes the use o f a n a l y t i c a l or d e r i v e d solutions, l e a v i n g n u m e r i c a l s o l u t i o n as the o n l y v i a b l e approach.  N u m e r i c a l l y - b a s e d o p t i m i z a t i o n c a n i n v o l v e l o c a l - s e a r c h or g l o b a l search strategies. L o c a l search strategies m o v e i n the d i r e c t i o n o f i m p r o v i n g solutions (e.g., d o w n w a r d gradient) u n t i l no further i m p r o v e m e n t is p o s s i b l e . E x a m p l e s o f l o c a l search methods i n c l u d e the Pattern S e a r c h  52  M e t h o d , the R o s e n b r o c k M e t h o d , S e q u e n t i a l Q u a d r a t i c P r o g r a m m i n g ( S Q P ) , and the N e l d e r M e a d A l g o r i t h m , also c a l l e d the S i m p l e x M e t h o d ( F r a n c h i n i et a l . , 1998; H o g u e et a l . , 2 0 0 0 ) .  T h e Pattern S e a r c h M e t h o d ( H o o k e and Jeeves, 1961) e x p l o r e s the response surface around the current p o i n t and then forces e x p l o r a t i o n i n the d i r e c t i o n o f greatest i m p r o v e m e n t ( F r a n c h i n i et a l . , 1998). T h e Pattern S e a r c h M e t h o d deals o n l y w i t h the v a l u e o f the objective function and therefore does not require c o n t i n u i t y or differentiability. T e r m i n a t i o n occurs w h e n either the response surface shows n e g l i g i b l e i m p r o v e m e n t i n any d i r e c t i o n , o r w h e n the calculated step s i z e o f e a c h i m p r o v e m e n t step reaches a specified l i m i t ( i b i d . ) .  T h e R o s e n b r o c k M e t h o d ( R o s e n b r o c k , 1960) approximates a gradient search w i t h o u t r e q u i r i n g the e v a l u a t i o n o f derivatives. It s y s t e m a t i c a l l y explores i m p r o v e m e n t s i n the objective f u n c t i o n v a l u e a l o n g a set o f o r t h o g o n a l vectors i n parameter space, r o t a t i n g the ordinate system i n the d i r e c t i o n o f the c o m p u t e d gradient once a l l vectors have b e e n e x p l o r e d . H o r n b e r g e r et a l . (1985) a p p l y the R o s e n b r o c k m e t h o d to the h y d r o l o g i c m o d e l T O P M O D E L , but f i n d that parameter e s t i m a t i o n is unreliable.  T h e S Q P a l g o r i t h m applies a c o n s t r a i n e d - o p t i m i z a t i o n v e r s i o n o f N e w t o n ' s m e t h o d and is e x t r e m e l y efficient at c o n v e r g i n g to a l o c a l o p t i m u m . F r a n c h i n i et a l . (1998) demonstrate the p o t e n t i a l for u s i n g S Q P as a " f i n e - t u n i n g " step i n c a l i b r a t i n g c o n c e p t u a l r a i n f a l l - r u n o f f m o d e l s .  O n e o f the most robust a n d p o p u l a r o f l o c a l - s e a r c h techniques is the d o w n h i l l s i m p l e x a l g o r i t h m d e s i g n e d b y N e l d e r and M e a d (1965), often referred to as the S i m p l e x M e t h o d . T h e N e l d e r M e a d d o w n h i l l s i m p l e x a l g o r i t h m s h o u l d not be confused w i t h the S i m p l e x M e t h o d o f D a n t z i g (1951) used i n linear p r o g r a m m i n g .  T h e N e l d e r - M e a d a l g o r i t h m b e g i n s w i t h a s i m p l e x o f n+1 points i n n - d i m e n s i o n a l parameter space a n d rank-orders t h e m best to w o r s t a c c o r d i n g to their objective function value. T h e w o r s t p o i n t is then reflected t h r o u g h the c e n t r o i d o f a l l other p o i n t s , r e p l a c i n g its parent i f the r e f l e c t i o n results i n an i m p r o v e d objective f u n c t i o n value. I f the reflected p o i n t is better than the best m e m b e r o f the s i m p l e x , the s i m p l e x is e x p a n d e d i n the d i r e c t i o n o f the reflected point. A s i m i l a r c o n t r a c t i o n step is p u r s u e d i f the reflected p o i n t is w o r s e than the w o r s t m e m b e r o f the s i m p l e x .  53  T e r m i n a t i o n t y p i c a l l y occurs w h e n the standard d e v i a t i o n o f objective f u n c t i o n values across the s i m p l e x drops b e l o w a s p e c i f i e d v a l u e .  D u a n et a l . (1992) f i n d that the N e l d e r - M e a d S i m p l e x M e t h o d outperforms other l o c a l search m e t h o d s because it adjusts to the response surface and c a n a v o i d b e i n g trapped b y the l o c a l o p t i m a w i t h i n a r e g i o n o f attraction. H o w e v e r , there are t w o n o t e w o r t h y l i m i t a t i o n s w i t h regard to the S i m p l e x M e t h o d and its treatment o f parameters. F i r s t l y , its effectiveness is substantially d i m i n i s h e d w h e n d e a l i n g w i t h m o r e than five to seven c o n c u r r e n t l y o p t i m i z e d parameters ( B u r g e s , 2 0 0 3 ) . S e c o n d l y , because the o r i g i n a l N e l d e r - M e a d A l g o r i t h m i s unconstrained, b o u n d e d parameters must be transformed before they c a n be o p t i m i z e d ( G a n , 1987).  L o c a l - s e a r c h m e t h o d s c a n be a p p l i e d as either single-step processes or successive o p t i m i z a t i o n s . T h e r e are t w o p o s s i b l e approaches for a p p l y i n g multi-step l o c a l search methods, n a m e l y repeated trials o f a s i n g l e a l g o r i t h m (e.g., D u a n et a l . , 1992) or c o n s e c u t i v e a p p l i c a t i o n o f different search strategies (e.g., F r a n c h i n i et a l . , 1998). W h i l e s u c h m u l t i - s t e p l o c a l searches are m o r e robust than a single trial, any i m p r o v e m e n t o v e r the s i n g l e - t r i a l case is m o s t l i k e l y due to the reduced d o m i n a n c e o f the v a r i o u s p r o b l e m s affecting each i n d i v i d u a l t r i a l .  L o c a l search m e t h o d s are frequently c h a l l e n g e d and often defeated b y n u m e r o u s systematic p r o b l e m s . L o c a l o p t i m a c a n trap a search a l g o r i t h m before it reaches the g l o b a l s o l u t i o n , necessitating user i n t e r v e n t i o n before the search c a n p r o c e e d ( S o r o o s h i a n and G u p t a , 1995; T h y e r et a l . , 1999). P o o r objective function s e n s i t i v i t y to c h a n g i n g parameter values and significant parameter interaction c a n also pose d i f f i c u l t i e s , e s p e c i a l l y as the search approaches the o p t i m u m . O t h e r c o m m o n p r o b l e m s i n c l u d e non-uniqueness o f o p t i m a l solutions and constitutive assumptions that are i n c o m p a t i b l e w i t h the h y d r o l o g i c data ( C o o p e r et a l . , 1997). T h e s e d i f f i c u l t i e s h a v e l e d researchers to c o n c l u d e that techniques d e s i g n e d to locate the o p t i m u m p o i n t o f a l o c a l i z e d r e g i o n are m a t h e m a t i c a l l y insufficient for l o c a t i n g g l o b a l o p t i m a i n c o m p l e x situations s u c h as h y d r o l o g i c m o d e l l i n g (e.g., D u a n et a l . , 1992; M c C u e n , 1973).  M a n y o p t i m i z a t i o n techniques have been d e s i g n e d to a v o i d the m o s t c o m m o n p r o b l e m s encountered b y l o c a l m e t h o d s ( C o o p e r et a l . , 1997; M a d s e n , 2 0 0 3 ) . T h e s e g l o b a l (sometimes c a l l e d parallel) search strategies i n c l u d e , a m o n g others, A d a p t i v e R a n d o m Search, G e n e t i c A l g o r i t h m s , S i m u l a t e d A n n e a l i n g , and the S h u f f l e d C o m p l e x E v o l u t i o n a l g o r i t h m S C E - U A  54  ( H o g u e et a l . , 2 0 0 0 ; M a d s e n et a l . , 2 0 0 2 ) . G l o b a l o p t i m i z a t i o n m e t h o d s ( G O M s ) are c o m m o n l y v i e w e d as h a v i n g a l o c a l phase ( t y p i c a l l y r e p l i c a t e d from w e l l - s t u d i e d p r e - e x i s t i n g l o c a l search techniques) and a g l o b a l phase that a v o i d s entrapment at l o c a l o p t i m a . Therefore, the m o s t significant differences b e t w e e n alternative G O M s are f o u n d i n their e x p l o r a t i o n o f the p r o b l e m space ( C o o p e r et a l . , 1997).  A d a p t i v e R a n d o m S e a r c h ( M a s r i et a l . , 1980) is a r g u a b l y the simplest e x a m p l e o f a G O M . A d a p t i v e R a n d o m S e a r c h chooses a f i x e d n u m b e r o f r a n d o m samples from each o f several nested sub-regions o f the parameter space. T h e nested sub-regions are t y p i c a l l y separated i n size b y an order o f m a g n i t u d e but are a l l centred o n the same focal. A f t e r each subset has b e e n e x a m i n e d , the s a m p l e d parameter set y i e l d i n g the best objective f u n c t i o n v a l u e b e c o m e s the f o c a l for the n e x t iteration. T h e search continues u n t i l the best p o i n t is repeatedly i d e n t i f i e d at the smallest s u b - r e g i o n o f the search. D u a n et a l . (1992) report that the effectiveness o f A R S is greatly i m p r o v e d w h e n c o u p l e d w i t h a l o c a l S i m p l e x search.  S i m u l a t e d A n n e a l i n g ( K i r k p a t r i c k et a l . , 1983) is based o n observations o f the energym i n i m i z i n g processes undergone b y c o o l i n g metals. T h e process w a l k s t h r o u g h the parameter space, e x p l o r i n g n e w parameter c o m b i n a t i o n s b y e v a l u a t i n g their objective f u n c t i o n v a l u e . A n y p o i n t that i m p r o v e s the objective f u n c t i o n b e c o m e s the starting p o i n t for the next e x p l o r a t i o n . B e i n g a d e r i v a t i v e o f the M e t r o p o l i s process, S i m u l a t e d A n n e a l i n g also accepts i n f e r i o r p o i n t s w i t h a s p e c i f i e d p r o b a b i l i t y . T h e a l g o r i t h m must be set up to a l l o w a large n u m b e r o f " u p h i l l " steps, but not so large as to m a k e the process c o m p l e t e l y r a n d o m ( T h y e r et a l . , 1999).  G e n e t i c A l g o r i t h m s c o m p r i s e a s p e c i a l case o f G O M s that shares the fundamental c o m p o n e n t s o f g l o b a l o p t i m i z a t i o n but is d i s t i n g u i s h e d b y its search technique. A l t h o u g h research into G e n e t i c A l g o r i t h m s b e g a n i n the 1970s, the w o r k o f G o l d b e r g (1989) w a s the catalyst for their w i d e r a d o p t i o n i n d i v e r s e fields o f o p t i m i z a t i o n .  L i k e other m e t h o d s , G e n e t i c A l g o r i t h m s use quantitative measures o f performance as objective functions. H o w e v e r , G e n e t i c A l g o r i t h m s search the parameter space b y e m u l a t i n g genetics (i.e., t h r o u g h the p r o b a b i l i s t i c s e l e c t i o n a n d r e c o m b i n a t i o n o f extant parameter values) rather than u s i n g M o n t e C a r l o or gradient-based search techniques to generate s u c c e s s i v e i m p r o v e m e n t s (Seibert, 2 0 0 0 ) . G e n e t i c s e l e c t i o n a l l o w s i n d i v i d u a l s w h i c h are better suited to their e n v i r o n m e n t  55  (i.e., parameter sets w h i c h p r o d u c e better m o d e l s i m u l a t i o n s ) to have a h i g h e r p r o b a b i l i t y o f r e p r o d u c i n g ( F r a n c h i n i et a l . , 1998).  V a r i o u s " r u l e s " are specified to g o v e r n the creation o f n e w parameter sets. C o m m o n e x a m p l e s o f different rules i n c l u d e d u p l i c a t i o n o f a parent (reproduction), r e - c o m b i n a t i o n o f elements  from  m u l t i p l e parents into a n e w i n d i v i d u a l (crossover), and selection o f r a n d o m values from the feasible range or a sub-range b o u n d e d b y the parent values (mutation) ( L a n , 2 0 0 1 ; Seibert, 2 0 0 0 ) . R u l e s are t y p i c a l l y a p p l i e d o n a p r o b a b i l i s t i c basis. I m p l e m e n t i n g an appropriate balance o f rules and p r o b a b i l i t i e s is c r u c i a l for the success o f the a l g o r i t h m (Seibert, 2 0 0 0 ) .  A s for m a n y G O M s , parameter values c a n either be u s e d i n their o r i g i n a l f o r m (e.g., Seibert, 2 0 0 0 ) or m o d i f i e d into a format m o r e amenable to the processes used i n genetic a l g o r i t h m s ; G o l d b e r g (1989) uses b i n a r y representations o f parameter values. In either case, discrete rather than c o n t i n u o u s parameters are required. N a t u r a l l y , therefore, the G A ' s o p t i m i z a t i o n reflects an integer rather than c o n t i n u o u s - v a r i a b l e s o l u t i o n . W i t h o u t careful c o n s i d e r a t i o n o f the d i s c r e t i z a t i o n process, any G O M that produces discrete solutions m a y y i e l d near- or e v e n subo p t i m a l solutions w i t h a r t i f i c i a l l y precise parameter values.  R e s u l t s o f studies a p p l y i n g G A s to h y d r o l o g i c m o d e l s are m i x e d (e.g., F r a n c h i n i and G a l e a t i , 1997; L a n , 2 0 0 1 ; Seibert, 2 0 0 0 ) . W a n g et a l . (1995) r e c o m m e n d c o u p l i n g a G e n e t i c A l g o r i t h m w i t h a secondary l o c a l - s e a r c h a l g o r i t h m to i m p r o v e results. F o r a c o m p l e t e d i s c u s s i o n o f the d e v e l o p m e n t o f genetic a l g o r i t h m s , the reader is referred to the s e m i n a l w o r k b y G o l d b e r g (1989).  O n e p a r t i c u l a r l y w e l l - d o c u m e n t e d and successful g l o b a l o p t i m i z a t i o n technique for automatic c a l i b r a t i o n o f h y d r o l o g i c m o d e l s is the S h u f f l e d C o m p l e x E v o l u t i o n a l g o r i t h m ( S C E - U A ) , d e v e l o p e d b y D u a n et a l . (1994a) at the U n i v e r s i t y o f A r i z o n a . T h e S C E - U A m e t h o d applies a v e r s i o n o f the N e l d e r - M e a d a l g o r i t h m u s i n g concepts from natural b i o l o g i c a l e v o l u t i o n to address the i n e f f i c i e n c i e s o f the N e l d e r - M e a d process ( D u a n et a l . , 1992). T h e S C E - U A m e t h o d is d e s i g n e d s p e c i f i c a l l y for the c a l i b r a t i o n o f h y d r o l o g i c m o d e l s and is based o n a synthesis o f ideas i n c l u d i n g a c o m b i n a t i o n o f deterministic and p r o b a b i l i s t i c approaches, c o m p e t i t i v e e v o l u t i o n , c o m p l e x shuffling, and systematic e v o l u t i o n o f a p o p u l a t i o n s p a n n i n g the parameter space ( D u a n et a l . , 1994b).  56  T h e w e l l - k n o w n N e l d e r - M e a d a l g o r i t h m is the basis o f the l o c a l search c o m p o n e n t for the S C E U A m e t h o d . H o w e v e r , it is the g l o b a l f r a m e w o r k that m a k e s the S C E - U A m e t h o d p o w e r f u l a n d unique. D u a n et a l . (1992) describe the process i n detail; their flowchart o f the g l o b a l search f r a m e w o r k is i n c l u d e d herein as F i g u r e 2 - 1 . T h e e x c h a n g e o f i n f o r m a t i o n between points a c h i e v e d b y s h u f f l i n g i n d i v i d u a l s b e t w e e n c o m p l e x e s - is w h a t a l l o w s the search to a v o i d b e i n g trapped b y l o c a l o p t i m a . T h e entire p o p u l a t i o n w i l l converge t o w a r d the n e i g h b o r h o o d o f the g l o b a l o p t i m u m , p r o v i d e d the i n i t i a l p o p u l a t i o n size is s u f f i c i e n t l y large (ibid.).  Input:  n = dimension, p = number of complexes m = number of points in each complex Compute: sample size s = p x m  Sample s points at random in Q. Compute the function value at each point.  I  Sort the s points in order of increasing function value. Store them in D.  Partition D into p complexes of m points i.e., D = (A , k = 1, ... ,p) k  I I  Evolve each complex A , k = 1, ... , p  CCE algorithm (see Figure 2-2)  Replace A , k = 1,... , p into D  No  Convergence criteria satisfied? Yes  Figure 2-1: Flowchart of the SCE-UA Algorithm (from p. 1027, Duan et al., 1992)  57  T h e separation o f p o i n t s into independent c o m p l e x e s a l l o w s each c o m p l e x to e x p l o r e the response surface i n different directions. T h e e v o l u t i o n o f each c o m p l e x is d r i v e n b y the " c o m p e t i t i v e c o m p l e x e v o l u t i o n " ( C C E ) strategy, i n w h i c h the m a j o r i t y o f n e w o f f s p r i n g are generated b y the N e l d e r - M e a d procedure ( N e l d e r and M e a d , 1965). D u a n et a l . (1992) p r o v i d e a detailed e x p l a n a t i o n o f h o w each c o m p l e x is e v o l v e d ; F i g u r e 2-2 is a flowchart o f the C C E strategy excerpted f r o m their w o r k . T h e v a r i o u s process parameters o f the a l g o r i t h m c o n t r o l v a r i o u s aspects o f the o p t i m i z a t i o n . T h e s e parameters c a n i n f l u e n c e the e f f i c i e n c y and effectiveness o f performance, and the user must ensure that a l l S C E - U A parameters are assigned appropriate values to m a x i m i z e the p r o b a b i l i t y that the a l g o r i t h m w i l l succeed. D u a n et a l . (1994b) describe the v a r i o u s parameters i n detail and e x p l o r e the s e n s i t i v i t y o f the a l g o r i t h m to parameter values. T h e S C E - U A m e t h o d m a y not a l w a y s be able to r e a c h g l o b a l convergence, but u s u a l l y produces at least n e a r - o p t i m a l results ( G a n and B i f t u , 1996). I n the decade since it w a s first d e v e l o p e d , the S C E - U A m e t h o d has b e e n studied e x t e n s i v e l y w i t h a n emphasis o n establishing its performance relative to other established approaches for o p t i m i z a t i o n . T h e consensus i n the literature i s that the S C E - U A m e t h o d p r o v i d e s solutions that are equal to or better than a l l other procedures for automatic c a l i b r a t i o n o f h y d r o l o g i c m o d e l s ( B u r g e s , 2 0 0 3 ; G a n and B i f t u , 1996; S i n g h and W o o l h i s e r , 2 0 0 2 ; S o r o o s h i a n and G u p t a , 1995). It has b e e n c o m p a r e d f a v o u r a b l y to G e n e t i c A l g o r i t h m s ( C o o p e r et a l . , 1997; K u c z e r a , 1997), semi-automated methods ( G u p t a et a l . , 1999), Pattern S e a r c h and S e q u e n t i a l Quadratic P r o g r a m m i n g ( F r a n c h i n i et a l . , 1998), S i m u l a t e d A n n e a l i n g ( C o o p e r et a l . , 1997; T h y e r et a l . , 1999), and the M u l t i - S t a r t S i m p l e x M e t h o d ( D u a n et a l . , 1992; G a n a n d B i f t u , 1996; S o r o o s h i a n et a l . , 1993). A l t h o u g h the a l g o r i t h m t y p i c a l l y requires tens o f thousands o f executions o f the h y d r o l o g i c m o d e l , m o s t studies also c o n c l u d e that the S C E - U A m e t h o d is the m o s t efficient automatic c a l i b r a t i o n t o o l a v a i l a b l e .  D e s p i t e the successes o f the S C E - U A m e t h o d i n the r e a l m o f automatic c a l i b r a t i o n , T h y e r et a l . (p. 7 7 3 , 1999) c a u t i o n that " S C E - U A s h o u l d not be v i e w e d as a panacea".  The S C E - U A method  is s t i l l subject to general challenges l i k e p o o r m o d e l f o r m u l a t i o n , unrepresentative or error-laden data, non-uniqueness o f o p t i m a l solutions, and s e n s i t i v i t y to objective function.  58  C^Jnfro  Given dimension n, complex A, and number of points m in A, select q, a, (3, where 2 <=q <=m, a>=l,/3>= 1. Set i = 1.  Assign a triangular probability distribution to A: 2(w + l - / ) . , Pi = — -,i = \,—,m m(m +1)  Select q points from A according to Pi. Store them in B and their relative positions in A in L. Set j = 1.  Sort B and L in order of increasing function value. Compute the centroid of Ui, ... , u_i and let u be the worst point in B. q  q  Compute r = 2g - u (reflection step). q  . » r within 0 ?  _  Generate a point z at random in H. Set r = z.  JyYes Compute f . r  Generate a point z at random in H. Compute  Replace B into A according to L and sort A in order of increasing function value.  =i+ i  r<  Yes  Return to SCE  Figure 2-2: Flowchart of the SCE-UA CCE Strategy (from p. 1028,Duan et al, 1992)  59  G a n and B i f t u (1996) suggest that these issues i m p l y a c r i t i c a l need to evaluate the g l o b a l o p t i m i z a t i o n features o f the S C E - U A m e t h o d w h e n it is a p p l i e d to a r e a l - w o r l d catchment, since e v e n the b a s i c concept o f a g l o b a l o p t i m u m m a y not m a k e sense i n an a p p l i e d context.  It is relevant to e m p h a s i z e a g a i n the general d i s t i n c t i o n b e t w e e n m e t h o d s w h i c h attempt to identify the g l o b a l o p t i m a o f an o b j e c t i v e f u n c t i o n and methods a i m e d at attaining the best p o s s i b l e representation o f a natural system. E v e n w h e r e a u n i q u e g l o b a l o p t i m u m c a n be f o u n d for a s p e c i f i c objective function, b o t h the g l o b a l o p t i m u m and the s i m u l a t i o n q u a l i t y s t i l l d e p e n d o n the c h o i c e o f objective f u n c t i o n as w e l l as the v a r i o u s m o d e l and data uncertainties. V r u g t et al. (2003) p o i n t out that, w h i l e the S C E - U A m e t h o d i s able to i d e n t i f y the g l o b a l m i n i m u m for a p r o b l e m , it has h a d little success i n d r i v i n g f o r w a r d the quest for a u n i q u e "best" parameter set.  2.4.5 Evaluating Model Performance M o d e l performance is evaluated d u r i n g c a l i b r a t i o n to p r o v i d e d i r e c t i o n to the search for a better representation o f the natural system. P e r f o r m a n c e is then re-evaluated d u r i n g v a l i d a t i o n to support the hypothesis that the m o d e l is c o r r e c t l y representing the prototype. E s t a b l i s h i n g " g o o d " m o d e l performance u s u a l l y i n v o l v e s attaining an " a c c e p t a b l e " goodness o f fit b e t w e e n m o d e l output and h i s t o r i c a l r e c o r d u n d e r the a s s u m p t i o n that c o n d i t i o n s o f a p p l i c a t i o n w i l l be s i m i l a r to those e x p e r i e n c e d d u r i n g c a l i b r a t i o n and v a l i d a t i o n ( K l e m e s , 1986b). S u c h evaluations o f m o d e l "correctness" a p p l y s o l e l y to the m o d e l output and cannot be extended to i m p l y " h y d r o l o g i c a l soundness" o f the m o d e l structure (ibid.).  M a d s e n (2000) proposes u s i n g four b a s i c factors to evaluate a h y d r o l o g i c m o d e l : water balance (average catchment r u n o f f v o l u m e ) , h y d r o g r a p h shape, t i m i n g / rate / v o l u m e o f peak f l o w s , and l o w f l o w agreement. M o d e l c a l i b r a t i o n g u i d e l i n e s g i v e n b y B u r g e s (2002) focus o n the following:  •  m a i n t a i n i n g annual, seasonal, w e e k l y , and d a i l y water balance;  •  m a t c h i n g h y d r o g r a p h shape, peak v a l u e s , and peak t i m i n g ;  •  e n s u r i n g p r e d i c t e d E T <potential E T for the r e g i o n ;  •  v e r i f y i n g s i m u l a t e d water storage fluctuations w i t h p r e c i p i t a t i o n patterns;  •  c o n f i r m i n g parameter values are consistent w i t h catchment properties;  60  •  m a t c h i n g surface f l o w to base f l o w ratio to s o i l and g e o l o g i c a l c o n d i t i o n s ;  •  k e e p i n g c o m p a r i s o n s consistent w i t h the a c c u r a c y and errors o f recorded data.  O b j e c t i v e s t y p i c a l l y have different units o f measurement and often cannot be aggregated into a s i n g l e measure. S u c h objectives are c a l l e d n o n - c o m m e n s u r a t e .  C h o o s i n g a s o l u t i o n based o n  m u l t i p l e objectives u s u a l l y requires trade-offs b e t w e e n n o n - c o m m e n s u r a t e objectives ( R e v e l l e et a l . , 1997).  In h y d r o l o g i c m o d e l c a l i b r a t i o n , n o c r i t e r i a or set o f c r i t e r i a for m e a s u r i n g m o d e l performance c a n be d e c l a r e d i n c o n t r o v e r t i b l y superior to a l l other techniques ( K l e m e s , 1986b; S o r o o s h i a n et a l . , 1983). I n a study assessing the performance o f several m o d e l s u s i n g m u l t i p l e categories o f criteria, H o u g h t o n - C a r r (1999) finds that n o m o d e l performs w e l l i n a l l categories, and n o s i n g l e category adequately describes m o d e l performance. T h e c r i t e r i a c h o s e n s h o u l d therefore be related to the purpose to w h i c h the calibrated m o d e l w i l l be a p p l i e d ( K l e m e s , 1986b). M a n u a l c a l i b r a t i o n s a n d some r e l a t i v e l y recent automatic c a l i b r a t i o n a l g o r i t h m s (e.g., B a s t i d a s et a l . , 2 0 0 1 ) c o n s i d e r m u l t i p l e c r i t e r i a i n a n attempt to reduce the r e l i a n c e o n any single measure o f performance.  A p p r o a c h e s to e v a l u a t i o n c a n be quantitative, qualitative, o r b o t h ( H o u g h t o n - C a r r , 1999). Often, a c o m b i n a t i o n o f statistical, g r a p h i c a l , and i n t u i t i v e measures are u s e d to evaluate a g i v e n m o d e l . F o r e x a m p l e , G a n (1987) u t i l i z e s statistical and g r a p h i c a l c o m p a r i s o n s , w h i l e also c o n s i d e r i n g p h y s i c a l p l a u s i b i l i t y o f parameter values g i v e n the catchment characteristics and c a l i b r a t i o n data. H o u g h t o n - C a r r (1999) finds that quantitative and qualitative approaches p r o v i d e different i n f o r m a t i o n : w h i l e qualitative c r i t e r i a are a m b i g u o u s , quantitative c r i t e r i a demonstrate a r e l a t i o n s h i p b e t w e e n f l o w r e g i m e and performance.  R e l i a n c e o n any one a p p r o a c h to the e x c l u s i o n o f others is i n a d v i s a b l e . F o r e x a m p l e , subjective g r a p h i c a l analysis m a y not c l e a r l y r e v e a l consistent biases. Perhaps m o r e i m p o r t a n t l y , quantitative c o m p a r i s o n s o f p a i r e d values from t w o t i m e c a n e a s i l y result i n unjustified and m i s l e a d i n g l e v e l s o f error i f m o d e l or data t i m i n g is w r o n g or uncertain. I f t i m i n g is s l i g h t l y off, large errors are created for r i s i n g and f a l l i n g l i m b s . S t a t i s t i c a l c o m p a r i s o n o f t w o s u c h t i m e series c o u l d result i n a h i g h l e v e l o f error not supported b y g r a p h i c a l analysis. A s an alternative,  61  B u r g e s (2002) proposes c a l i b r a t i o n based o n storm v o l u m e s and peak f l o w s as a m u l t i - o b j e c t i v e approach. B e r g s t r o m et a l . (2002) m a k e a l l o w a n c e for uncertainty i n m o d e l l i n g n i t r o g e n l e v e l s b y c o n s i d e r i n g the best m a t c h o f observed and s i m u l a t e d n i t r o g e n concentrations w i t h i n ± 3 days o f the n o m i n a l date.  A l t h o u g h g r a p h i c a l c o m p a r i s o n s are u s u a l l y c o n c e r n e d w i t h evaluations o f s i m u l a t e d and o b s e r v e d hydrographs, other types o f b o t h quantitative and qualitative data c a n be expressed g r a p h i c a l l y . Different g r a p h i c a l techniques c a n illustrate the properties and patterns o f a data set (e.g., b o x p l o t s , scattergrams, transformations).  H o u g h t o n - C a r r (1999) m a k e s g o o d use o f  g r a p h i c a l c o m p a r i s o n b y p l o t t i n g t w o different performance measures o n the x - and y-axes, w h i l e H o g u e et a l . (2000) i m p r o v e the representation o f recessions w h i l e m a i n t a i n i n g perspective for h i g h e r f l o w s b y a p p l y i n g a partial l o g transformation.  G r a p h i c a l analysis c a n also help to c l e a r l y and c o n c i s e l y o r g a n i z e related data. P l o t t i n g the t i m e series o f r a i n f a l l , catchment o u t f l o w , and r e s i d u a l d a i l y f l o w v o l u m e error o n a single graph is an effective w a y o f assessing performance and i s p a r t i c u l a r l y useful i n h i g h l i g h t i n g systematic errors i n the m o d e l ( B u r g e s , 2 0 0 2 ; G a n , 1987).  W h i l e m a n u a l calibrations use b o t h quantitative and qualitative c r i t e r i a to evaluate m o d e l performance, automatic c a l i b r a t i o n a l g o r i t h m s r e l y e x c l u s i v e l y o n quantitative c a l c u l a t i o n s . Q u a n t i t a t i v e measures c a n be as s i m p l e as direct c o m p a r i s o n o f f l o w v o l u m e , peak f l o w , or t i m e to peak ( L o a g u e and F r e e z e , 1985). M o r e c o m p l e x quantitative measures s u c h as m a x i m u m error, R o o t M e a n Square E r r o r ( R M S E ) , C o e f f i c i e n t o f D e t e r m i n a t i o n ( R ) , efficiency, and 2  coefficient o f r e s i d u a l m a s s are t y p i c a l l y based o n the relationship b e t w e e n each quantile o f the o b s e r v e d t i m e series and its s i m u l a t e d counterpart.  W h i l e u s u a l l y a p p l i e d to the entire t i m e  series, c o n s i d e r a t i o n o f a subset o f results c a n p r o v i d e i n s i g h t into the performance o f a specific m o d e l component.  T h e standard approach to quantitative e v a l u a t i o n i n v o l v e s d e f i n i n g some measure o f the length o f a v e c t o r c o m p o s e d o f the t i m e series o f r e s i d u a l errors ( G u p t a et a l . , 1998). T h e process o f c a l i b r a t i o n i s then v i e w e d as an iterative attempt to f i n d values o f the m o d e l parameters that m i n i m i z e the " l e n g t h " o f the vector. T h e process is c o m p l i c a t e d b y the l a c k o f an u n a m b i g u o u s l y correct w a y to define the length o f the error v e c t o r ( G u p t a et a l . , 1998; L a n ,  62  2 0 0 1 ) . Further, a n y statistical measure e x p l i c i t l y c o m p a r i n g the residuals o f t w o t i m e series a r g u a b l y places m o r e w e i g h t o n t i m i n g than o n m a t c h i n g f l o w patterns ( B u r g e s , 2 0 0 2 ) . T o d i n i (1988) points out that statistical techniques based o n analysis o f residuals neglect the p h y s i c a l characteristics o f the m o d e l , a v o i d i n g rather than t a k i n g advantage o f p r i o r expert k n o w l e d g e i n t r i n s i c to the m o d e l structure.  P o p u l a r measures for quantitative e v a l u a t i o n i n c l u d e coefficients o f linear c o r r e l a t i o n (e.g., the coefficient o f d e t e r m i n a t i o n , R ) , coefficients o f e f f i c i e n c y (e.g., N a s h - S u t c l i f f e efficiency, E ! ) , 2  and various equations related to regression and c u r v e fitting theory (e.g., s i m p l e least squares, S L S ) ( L a n , 2 0 0 1 ) . A w i d e v a r i e t y o f statistics are a v a i l a b l e , a l t h o u g h an i n d i v i d u a l user w i l l generally have a set o f preferred measures. M a n y users o f the U B C W M prefer a m o d i f i e d v e r s i o n o f N a s h - S u t c l i f f e e f f i c i e n c y that corrects for o v e r a l l v o l u m e error (e.g., L a n , 2 0 0 1 ; M i c o v i c , 1998). T h e U S N a t i o n a l W e a t h e r S e r v i c e uses a w i d e v a r i e t y o f measures, i n c l u d i n g D a i l y R o o t M e a n S q u a r e d E r r o r , T o t a l M e a n M o n t h l y V o l u m e S q u a r e d E r r o r , M e a n and M a x i m u m A b s o l u t e E r r o r , N a s h - S u t c l i f f e efficiency, B i a s (mean d a i l y error), P e a k D i f f e r e n c e , F i r s t L a g A u t o c o r r e l a t i o n , and N u m b e r o f S i g n C h a n g e s (i.e., the n u m b e r o f times the sequence o f residuals changes sign) ( G u p t a et a l . , 1998).  S o m e o f the m o r e c o m m o n functions used i n e v a l u a t i n g h y d r o l o g i c m o d e l performance are presented i n T a b l e 2 . 1 , w i t h descriptions, n u m e r i c a l f o r m u l a t i o n s , and e x a m p l e s o f studies i n w h i c h each has been a p p l i e d . T h e s e c o m m o n functions generally each require a set o f assumptions r e g a r d i n g the statistical d i s t r i b u t i o n o f the output data errors, w h i l e errors i n input data are t y p i c a l l y i g n o r e d ( G u p t a et a l . , 1998). T h e coefficient o f d e t e r m i n a t i o n ( R or D ! ) is the square o f the coefficient o f linear c o r r e l a t i o n , a 2  p o p u l a r measure o f the strength o f relationship b e t w e e n t w o sets o f data. F o r independent data w i t h no c o r r e l a t i o n whatsoever, R and R are z e r o ; for h i g h l y - c o r r e l a t e d data sets, R and R 2  2  approach unity. T h e coefficient o f d e t e r m i n a t i o n is defined as the quotient o f e x p l a i n e d v a r i a t i o n and total v a r i a t i o n . L a n (2001) and Seibert and M c D o n n e l l (2002) note the difference b e t w e e n the case o f perfect c o r r e l a t i o n ( R = 1) and that o f perfect s i m u l a t i o n (i.e., zero error, w h e r e 2  q im = qobs)s  T h i s difference suggests that R alone is not a g o o d measure o f data s i m i l i t u d e for an 2  automatic c a l i b r a t i o n against a single data series. T h e l o w c o r r e l a t i o n (0.14) b e t w e e n the  63  Table 2-1: Common Measures for Quantitative Evaluation of Hydrologic Model Performance  Symbol / Acronym  Full Name  Description  R (D!)  Coefficient of Determination  Proportion of variance in q (x) that can be explained by the modelled q (y)  2  Numerical Formulation obs  r 2  _  [«5jcy-(Zx)(Zy)] 2  HMLE  Heteroscedastic Maximum Likelihood Estimator  Sum of weighted squared errors (w e ). Assumes errors are Gaussian and heteroscedastic, with zero mean and covariance matrix V = o I  2  2  (1)  (Lan, 2001) (Micovic, 1998) (Franchini and Pacciani, 1991)  (2)  (Gan, 1987) (Sorooshian et al., 1983)  (3)  (Gan, 1987) (Hornberger et al., 1985) (Houghton-Carr, 1999)  (4)  (Cooper et al., 1997) (Lan, 2001) (Micovic, 1998) (Gupta etal, 1999) (Houghton-Carr, 1999)  (5)  (Cooper et al., 1997) (Gupta etal., 1999) (Hogue et al., 2000)  (6)  (Cooper et al., 1997) (Gan, 1987) (Hornberger et al., 1985) (Houghton-Carr, 1999) (Sorooshian et al., 1983)  2  [«Ex -(Zx) l«Zy -(£v) J  sim  Example Applications  2  -  t  t  HMLE  = ^ ± w , e ,  2  ^ n  f\w,  %  |  2  LAD  E!  Least Absolute Difference; Absolute Least Value  Nash - Sutcliffe Efficiency  Sum of the absolute values of the errors (i.e., q - q ) for each pair of q and q obs  obs  = Y\ °bs 1=1  -qj\  a  n 2J E\-\  \Ssim ~ Qobs ]  , = 1  n  0  (qobs -  SLS or SSE  n  LAD  sira  A variant of the Coefficient of Determination that measures the relative magnitude of "noise" ( q i m - q b s ) to "information" s  RMSE; DRMS  sim  (=1  qobs)  (Daily) Root Mean Square Error  The square root of the mean of squared errors (i.e., q - q ). It is sometimes referred to as DRMS when calculated for a daily time series.  Simple Least Squares or Sum of Squared Errors  A simple summation of the squared errors (i.e., q - q ) for each point in time series t  obs  obs  sim  sim  RMSE=tf (q -q Y j  SLS  =  ohs  sim  YiQobs ;=1  ~<lsiJ  2  coefficient o f d e t e r m i n a t i o n ( D ! ) a n d r e s i d u a l v o l u m e error ( d V ) o b s e r v e d b y M i c o v i c ( M i c o v i c , 1998) supports this c o n c l u s i o n .  T h e N a s h - S u t c l i f f e e f f i c i e n c y statistic ( E ! ) measures the p r o p o r t i o n o f v a r i a b i l i t y i n the o b s e r v e d f l o w series that is e x p l a i n e d b y the h y d r o l o g i c m o d e l ( L o a g u e a n d F r e e z e , 1985). N a s h - S u t c l i f f e e f f i c i e n c y measures the r e l a t i v e m a g n i t u d e o f n o i s e ( q j - q b ) to i n f o r m a t i o n (q b - q ean)- E ! is s  m  0  s  0  S  m  better suited to e v a l u a t i o n o f h y d r o l o g i c s i m u l a t i o n s than the c o e f f i c i e n t o f d e t e r m i n a t i o n because it evaluates the m a g n i t u d e a n d shape o f difference b e t w e e n o b s e r v e d and s i m u l a t e d hydrographs.  H o w e v e r , the N a s h - S u t c l i f f e E ! i s b i a s e d for data sets w i t h large total v a r i a n c e for w h i c h the r e s i d u a l v a r i a n c e ( i n the numerator) is d o m i n a t e d b y the o b s e r v e d v a r i a n c e ( i n the d e n o m i n a t o r ) ( L a n , 2 0 0 1 ) . L a r g e events are w e i g h t e d m o r e h e a v i l y i n the sense that r e s i d u a l error for a large event w i l l h a v e m u c h m o r e effect o n the statistic than the r e s i d u a l error for a s m a l l event, e v e n i f the t w o are the same as percentages o f event m a g n i t u d e . T h e p o o r r e p r o d u c t i o n o f l o w f l o w p e r i o d s i n Seibert (2000) is attributed to the b i a s o f the N a s h - S u t c l i f f e e f f i c i e n c y statistic t o w a r d m a t c h i n g h i g h - f l o w events at the expense o f l o w - f l o w events. L o a g u e a n d F r e e z e (1985) observe that, i n general, N a s h - S u t c l i f f e efficiencies are better w h e n c a l c u l a t e d for v o l u m e a n d p e a k f l o w than for peak f l o w t i m i n g . R e c e n t w o r k has s h o w n that i m p r o v e m e n t is p o s s i b l e i n s o m e cases b y c a l c u l a t i n g the N a s h - S u t c l i f f e E f f i c i e n c y for the natural l o g a r i t h m s o f the d i s c h a r g e t i m e series (i.e., E ! = f [ l n ( Q ) ] ) ( W e i l e r , 2 0 0 5 ) .  In a G e n e t i c A l g o r i t h m c a l i b r a t i o n o f the U B C W M , L a n ( 2 0 0 1 ) demonstrates that, across the 2 0 t h generation p o p u l a t i o n , i n c r e a s i n g E ! is not n e c e s s a r i l y associated w i t h decreasing r e l a t i v e v o l u m e error dV/V.  T o e m p h a s i z e c o n s e r v a t i o n o f mass a n d c l o s i n g o f the water balance, the  U B C W M U s e r ' s M a n u a l ( Q u i c k , 1994) presents the m o d i f i e d N a s h - S u t c l i f f e statistic E O P T ! , d e f i n e d as:  E O P T ! = E ! -1 JJqobs - Sq im | / Sq bs S  0  (7)  T h e E O P T ! statistic is used for e v a l u a t i n g the p e r f o r m a n c e o f the U B C W M .  T h e t w o quantitative c r i t e r i a m o s t w i d e l y u s e d for m o d e l e v a l u a t i o n i n the literature are the S i m p l e Least Squares E r r o r ( S L S ) and the R o o t M e a n Square E r r o r ( R M S E ) statistics. T h e S L S  65  statistic is the s u m o f the squares o f the i n d i v i d u a l r e s i d u a l errors (i.e., S L S = 2(q bs - q im) )• A s 2  0  S  for other statistics, data error c a n l e a d to p r o b l e m s w h e n a p p l y i n g S L S since equal w e i g h t is g i v e n to b o t h v a l i d and erroneous quantiles i n the t i m e series. H o w e v e r , for S L S , any d i s t o r t i o n is exacerbated t h r o u g h the squaring o f large r e s i d u a l errors ( L a n , 2 0 0 1 ) . Least absolute difference functions are better suited to erroneous data because the errors are not squared before summation.  T h e R M S E statistic is defined as the square root o f the s u m o f the i n d i v i d u a l r e s i d u a l errors, effectively c o m p u t i n g the standard d e v i a t i o n o f the m o d e l p r e d i c t i o n error ( G u p t a et a l . , 1999). W h e n r e s i d u a l errors are c a l c u l a t e d at a d a i l y timestep, R M S E is s o m e t i m e s l a b e l e d D a i l y R o o t M e a n Square E r r o r ( D R M S ) .  T h e R M S E statistic assumes the presence o f G a u s s i a n ,  independent error w i t h h o m o g e n e o u s v a r i a n c e ; the r e s u l t i n g i m p l i c i t bias t o w a r d m a t c h i n g the highest events o f r e c o r d renders the R M S E statistic p o o r l y a p p l i c a b l e to data c o n s i s t i n g m o s t l y o f l o w f l o w s w i t h a few large events ( G a n and B i f t u , 1996; Y a p o et a l . , 1996). T h e tendency o f the R M S E objective f u n c t i o n to p r o v i d e g o o d peak estimates w h i l e a l l o w i n g strong bias i n other parts o f the h y d r o g r a p h is r e a d i l y apparent at intermediate steps o f the c a l i b r a t i o n process ( H o g u e et a l . , 2 0 0 0 ) . B y a s s u m i n g h o m o g e n e o u s v a r i a n c e i n error, S L S and R M S E i m p l y acceptance o f the argument p r o p o s e d b y A r n a u d et a l . (2002): that since the m a g n i t u d e o f error is not i n c r e a s i n g w i t h size, absolute error is constant and therefore relative error i n the data s h o u l d be expected to decrease for larger events.  I f the a c c u r a c y o f l o w - f l o w s i m u l a t i o n is a p r i o r i t y , the Heteroscedastic M a x i m u m L i k e l i h o o d E s t i m a t o r ( H M L E ) , d i s c u s s e d i n detail b y S o r o o s h i a n et a l . (1983), is a better c h o i c e than the R M S E o r S L S statistics. Y a p o et a l . (1996) i m p r o v e l o w - f l o w s i m u l a t i o n b y u s i n g the H M L E as an o b j e c t i v e f u n c t i o n instead o f R M S E , a l t h o u g h performance o n h i g h e r f l o w s deteriorates.  The  H M L E assumes output errors are G a u s s i a n w i t h zero m e a n and uncorrelated, heterogeneous variance. T h e heteroscedastic basis o f the H M L E a l l o w s for the existence o f m a g n i t u d e dependent errors i n s t r e a m f l o w measurement ( S o r o o s h i a n et a l . , 1 9 8 3 ; Y a p o et a l . , 1996). B o t h the R M S E and the H M L E require the independent and i d e n t i c a l l y distributed d e s i g n a t i o n for r e s i d u a l errors; i n neither case is the a s s u m p t i o n o f error independence supported ( Y a p o et a l . ,  66  1996). Independent c a l i b r a t i o n u s i n g the R M S E and H M L E c r i t e r i a c a n result i n drasticallydifferent values for m a n y o f the parameters ( S o r o o s h i a n et a l . , 1993).  In the l i m i t i n g case o f h o m o g e n e o u s v a r i a n c e , the H M L E statistic reduces to the S L S .  In a study  c o m p a r i n g S L S and H M L E , S o r o o s h i a n et a l . (1983) attribute the better c a l i b r a t i o n - p e r i o d statistics o b t a i n e d w i t h the S L S c a l i b r a t i o n to its strong c u r v e - f i t t i n g a b i l i t i e s ; the H M L E statistics are f o u n d to be superior for the v a l i d a t i o n p e r i o d . O n e i m p l i c a t i o n o f these findings is that automatic c a l i b r a t i o n s u s i n g the S L S statistic m a y sacrifice p h y s i c a l r e a l i s m i n f a v o u r o f fitting the c a l i b r a t i o n data set. M e a s u r e s w i t h a p o o r p h y s i c a l basis s h o u l d be a v o i d e d i f p o s s i b l e , as they c a n e a s i l y lead to o v e r f i t t i n g ( K l e m e s , 1982).  2.4.6  Hydrologic Model Validation  M a n y m o d e l s c a n be calibrated to a p p r o x i m a t e a g i v e n data set w i t h o u t c o r r e c t l y representing the  in situ processes. Therefore, a m o d e l m u s t be tested o n a distinct and independent data set to c o n f i r m that the uncertainty i n a p p l i e d m o d e l p r e d i c t i o n s is acceptable ( K l e m e s , 1986b; O ' C o n n e l l a n d T o d i n i , 1996). T h i s process is m o s t c o m m o n l y referred to as " m o d e l v a l i d a t i o n " . E l s e w h e r e i n the literature, the process is referred to as " c o n f i r m a t i o n " o r " m o d e l performance e v a l u a t i o n " (e.g., G u p t a et a l . , 1999).  M o d e l v a l i d a t i o n is also sometimes referred to as " v e r i f i c a t i o n " ( G u p t a et a l . , 1999). A l t h o u g h the terms " v a l i d a t i o n " and " v e r i f i c a t i o n " are often u s e d interchangeably, their respective m e a n i n g s are distinct. A " v a l i d a t e d " m o d e l must be i n t e r n a l l y consistent and have n o k n o w n or detectable flaws. In contrast, a " v e r i f i e d " m o d e l is one w h o s e truth has been demonstrated (Oreskes et a l . , 1994). It m a y b e c o n v e n i e n t to t h i n k o f " v e r i f i c a t i o n " as the l i m i t state reached as the q u a l i t y o f the v a l i d a t i o n process approaches p e r f e c t i o n and the size and scope tend to infinity.  H o r n b e r g e r and B o y e r (1995) c o n t e n d that e s t a b l i s h i n g truth is not and s h o u l d not be the g o a l o f v a l i d a t i o n , since e v e n a perfect m a t c h b e t w e e n o b s e r v e d and p r e d i c t e d data does not v e r i f y a m o d e l . O r e s k e s (p. 6 4 2 , 1994) argues that v e r i f i c a t i o n is o n l y p o s s i b l e for " c l o s e d systems i n w h i c h a l l the c o m p o n e n t s o f the system are established i n d e p e n d e n t l y and k n o w n to be correct". H y d r o l o g i c m o d e l s cannot constitute c l o s e d systems due to factors s u c h as i n c o m p l e t e  67  k n o w l e d g e , c o n t i n u u m theory, and s c a l i n g o f n o n - a d d i t i v e properties (ibid.).  Therefore,  h y d r o l o g i c m o d e l s cannot be t r u l y v e r i f i e d .  T y p i c a l criteria e x a m i n e d d u r i n g h y d r o l o g i c m o d e l v a l i d a t i o n i n c l u d e m a t h e m a t i c a l rigour, absence o f bias, and closeness o f fit. O b v i o u s l y , success w i t h these criteria does not necessarily i m p l y scientific o b j e c t i v i t y or h y d r o l o g i c a l insight ( K l e m e s , 2000a). F o r L e i and S c h i l l i n g (p. 8 1 , 1996), a successful v a l i d a t i o n means o n l y that "the m o d e l is not rejected for this v e r y task i n this v e r y situation". In this sense, a successful v a l i d a t i o n s h o u l d be considered as supporting evidence for, rather than p r o o f of, an acceptable representation o f reality ( B e r g s t r o m et a l , 2 0 0 2 ) . W h i l e the p r o b a b i l i t y o f a correct m o d e l representation i m p r o v e s w i t h i n c r e a s i n g d i v e r s i t y o f v a l i d a t i o n data, there is little basis for extrapolating c o n c l u s i o n s b e y o n d the t e m p o r a l , spatial, and m a g n i t u d e l i m i t s o f the v a l i d a t i o n data set ( B e v e n , 1989; K l e m e s , 1986b; O r e s k e s et a l . , 1994). T h u s , the data u s e d for v a l i d a t i o n s h o u l d i d e a l l y be h y d r o l o g i c a l l y s i m i l a r to c o n d i t i o n s expected i n the a p p l i e d s i m u l a t i o n s .  A l t h o u g h v a l i d a t i o n cannot c o n c l u s i v e l y v e r i f y a g i v e n m o d e l , p o o r m o d e l performance, as m e a s u r e d against observed data i n the v a l i d a t i o n phase, is an o b v i o u s indicator o f error. H o w e v e r , the results o f v a l i d a t i o n are c o m m o n l y insufficient to c o n c l u s i v e l y support or refute a m o d e l , e s p e c i a l l y w h e r e v a l i d a t i o n data are l i m i t e d ( G o o d r i c h and W o o l h i s e r , 1994; O r e s k e s et a l . , 1994). F o r m o d e l s that purport to simulate internal catchment responses, c o m p a r i s o n o f o u t f l o w hydrographs is an insufficient test o f v a l i d i t y ( B e v e n , 1989). B e r g s t r o m et a l . (2002) r e c o m m e n d that measurements other than discharge be c o n s i d e r e d to further validate the m o d e l . W h e r e v a l i d a t i o n is i n c o n c l u s i v e , the c o r r e s p o n d i n g l a c k o f confidence s h o u l d be reflected i n results.  T h e q u a l i t y o f s i m u l a t i o n m a y appear to deteriorate f r o m c a l i b r a t i o n to v a l i d a t i o n g i v e n the c u r v e - f i t t i n g nature o f the c a l i b r a t i o n process and the a s s u m e d independence o f the v a l i d a t i o n data set. G i v e n a successful c a l i b r a t i o n , the change i n p e r f o r m a n c e f r o m c a l i b r a t i o n to v a l i d a t i o n s h o u l d be modest to n e g l i g i b l e (e.g., G a n and B i f t u , 1996; G a n et a l . , 1997; M a d s e n , 2 0 0 3 ) . G a n and B i f t u (1996) f i n d that m o d e l p e r f o r m a n c e deteriorates m o r e b e t w e e n c a l i b r a t i o n and v a l i d a t i o n w h e n "stronger" automatic c a l i b r a t i o n approaches s u c h as the S C E - U A m e t h o d , are used. T h i s m a y be an e x a m p l e o f " o v e r f i t t i n g " a data set w i t h a p o w e r f u l o p t i m i z a t i o n t o o l .  68  There is often a temptation to return to the c a l i b r a t i o n phase or otherwise alter parameter values to i m p r o v e the fit o f the m o d e l to the v a l i d a t i o n data. H o w e v e r , s u c h c o n t a m i n a t i o n o f the v a l i d a t i o n process a u t o m a t i c a l l y precludes its success (Oreskes et a l . , 1994). B e v e n (1989) notes a further l i m i t a t i o n for event-based m o d e l s , since i f b o u n d a r y or i n i t i a l c o n d i t i o n s must be calibrated for a v a l i d a t i o n event, it cannot be c o n s i d e r e d a true v a l i d a t i o n . I f a c o n t i n u o u s m o d e l is b e i n g used i n an independent v a l i d a t i o n test, a l l o w a n c e must be m a d e for a " s p i n - u p " p e r i o d to a v o i d any influence o f i n i t i a l c o n d i t i o n s o n the v a l i d a t i o n p e r i o d .  L o a g u e and F r e e z e (1985) assert that, i n the case o f a successful c a l i b r a t i o n and i n the absence o f o v e r f i t t i n g o f parameters or v a l i d a t i o n b e y o n d the c a l i b r a t i o n range, errors i n c a l i b r a t i o n a n d v a l i d a t i o n data s h o u l d be statistically s i m i l a r . T h e m o s t c o m m o n approach to h y d r o l o g i c m o d e l v a l i d a t i o n applies this p r i n c i p l e to the s o - c a l l e d " s p l i t - s a m p l e " test. T h e split-sample test i n v o l v e s the subjective d i v i s i o n o f a single r e c o r d into t w o parts w i t h one used for c a l i b r a t i o n and the other reserved for v a l i d a t i o n . I f the r e c o r d l e n g t h is not sufficient to be split 50/50, the r e c o r d s h o u l d be split t w i c e , e.g., 7 0 / 3 0 and 3 0 / 7 0 , and v a l i d a t i o n must pass o n b o t h sets for the m o d e l to pass this test ( K l e m e s , 1986b). M e r z a n d B l o s c h l (2004) present a case study i n w h i c h they use K l e m e s ' (1986b) concept o f s p l i t - s a m p l e v a l i d a t i o n to g o o d effect.  A split-sample test is not necessarily r e l i a b l e i n a l l cases (e.g., G u p t a et a l . , 1999), e s p e c i a l l y g i v e n the t y p i c a l l a c k o f independence b e t w e e n the c a l i b r a t i o n and v a l i d a t i o n data series. I f the split-sample test i s suspect, a m o r e in-depth analysis o f m o d e l residuals is l i k e l y required. S u c h situations are not addressed herein. H o w e v e r , it is w o r t h n o t i n g that qualitative analysis must also p l a y a r o l e i n m o d e l v a l i d a t i o n , as parameter v a l u e s must be feasible, realistic, and able to l e n d a degree o f certainty to s i m u l a t i o n s and forecasts ( S o r o o s h i a n et a l . , 1983).  69  3. Uncertainty in Hydrologic Modelling "It is far better to foresee foresee  at  even without certainty  than not to  all." - Henri  Poincare  T h i s chapter is intended for b o t h b e g i n n i n g and e x p e r i e n c e d h y d r o l o g i c m o d e l l e r s s e e k i n g to understand the v a r i o u s w a y s i n w h i c h uncertainty is i n t r o d u c e d into the m o d e l output.  Those  f a m i l i a r w i t h the b o d y o f literature o n uncertainty i n h y d r o l o g i c m o d e l l i n g m a y w i s h to p r o c e e d d i r e c t l y to C h a p t e r 4. D e s p i t e significant progress, h y d r o l o g i c m o d e l s are s t i l l far f r o m a c h i e v i n g desired l e v e l s o f a c c u r a c y and certainty. O f t e n , the best that c a n be a c h i e v e d under i d e a l c o n d i t i o n s is to predict b e h a v i o u r w i t h a certain degree o f " c o n f i d e n c e " - the c o m p l e m e n t o f w h i c h is a degree o f r e s i d u a l uncertainty. A n y output f r o m a h y d r o l o g i c m o d e l s h o u l d therefore be a n a l y z e d w i t h regard to its significant i n the context o f the associated m o d e l p r e d i c t i v e uncertainty ( B i n l e y et a l . , 1991). H o w e v e r , uncertainty i t s e l f is often indeterminate or u n i d e n t i f i a b l e ; it c a n be quantitative, qualitative, or u n k n o w n i n character ( L e i and S c h i l l i n g , 1996). D i f f e r e n t aspects o f m o d e l b e h a v i o u r m a y h a v e different degrees o f uncertainty, and i d e n t i f y i n g the sources o f uncertainties i n m o d e l results c a n be e x t r e m e l y d i f f i c u l t ( G a r e n and B u r g e s , 1981). T h i s chapter p r o v i d e s an o v e r v i e w o f uncertainty i n the context o f h y d r o l o g i c m o d e l l i n g .  A d i s t i n c t i o n must be m a d e b e t w e e n the concepts o f error (or accuracy) and uncertainty. A l t h o u g h sometimes u s e d interchangeably, the t w o concepts are distinct i n nature. " E r r o r " is the difference b e t w e e n a c o m p u t e d or measured v a l u e and its true or t h e o r e t i c a l l y correct counterpart. Therefore, the t e r m " e r r o r " is p r o p e r l y u s e d w h e n the o b s e r v e d response, process, or o u t c o m e has been o b s e r v e d to be incorrect or i n v a l i d . " U n c e r t a i n t y " , o n the other h a n d , is the c o n d i t i o n that the c o m p u t e d or measured output may differ f r o m the b a s e l i n e response i n m a g n i t u d e , process, o r p r o b a b i l i t y . It is u s u a l l y expressed i n a relative or p r o b a b i l i s t i c sense, as o p p o s e d to error, w h i c h i s an absolute quantity.  T h e concept o f uncertainty is analogous i n some w a y s to " p r e c i s i o n " i n the w e l l - k n o w n e x a m p l e o f a c c u r a c y i n m a r k s m a n s h i p . A s illustrated i n F i g u r e 3-1, a result c a n be accurate but not  70  precise; s i m i l a r l y , it c a n be p r e c i s e but not accurate. B y e x t e n s i o n , m o d e l results c a n be accurate but uncertain, or certain but erroneous. Ideally, b o t h certainty and a c c u r a c y w i l l be present; m o r e c o m m o n l y , neither c a n be established c o n c l u s i v e l y .  (c) accurate but not precise  (d) accurate and precise  Figure 3-1: Accuracy and Precision as Analogues for Error and Uncertainty  A p p l i e d m o d e l l e r s s o m e t i m e s tend to focus o n the m o r e e a s i l y - i d e n t i f i a b l e issue o f m o d e l a c c u r a c y at the expense o f g i v i n g due c o n s i d e r a t i o n to uncertainty. Indeed, i f m o d e l l i n g experts struggle to understand the uncertainties i m p l i c i t i n their m o d e l s , it w o u l d be nai've to assume that p r a c t i c i n g h y d r o l o g i s t s w i l l g i v e s u f f i c i e n t l y c o m p r e h e n s i v e c o n s i d e r a t i o n to uncertainty i n their results ( G r a y s o n et a l . , 1994b).  71  L a r g e o r difficult-to-resolve uncertainties i n results c a n cause users to exert pressure o n the m o d e l d e v e l o p e r to " i m p r o v e " the m o d e l ( K l e m e s , 1982). I f " i m p r o v e m e n t s " (e.g., m o r e data, better understanding) are not e a s i l y a c h i e v a b l e , s o m e m o d e l l e r s resort to attempting to extract m o r e i n f o r m a t i o n f r o m the data t h r o u g h m a t h e m a t i c a l m a n i p u l a t i o n ( i b i d . ) . M o d e l developers and p r o m o t e r s s h o u l d ensure that there is an appropriate l e v e l o f awareness and d i s c u s s i o n o f m o d e l l i n g uncertainties amongst the c o m m u n i t y o f users ( G r a y s o n et a l . , 1994b).  A t h o r o u g h understanding o f the sources o f uncertainty a n d a consistent t a x o n o m y is necessary to facilitate awareness and d i s c u s s i o n i n the m o d e l l i n g c o m m u n i t y . T h e s e t o p i c s are e x p l o r e d i n S e c t i o n 3.1. Substantial uncertainty w i l l persist i n the outputs o f e v e n a w e l l - c a l i b r a t e d h y d r o l o g i c m o d e l ( B i n l e y et a l . , 1991). A d i s c u s s i o n o f the i n t e r a c t i o n b e t w e e n the c a l i b r a t i o n process a n d m o d e l p r e d i c t i v e uncertainty is c o n t a i n e d i n S e c t i o n 3.2.  A l t h o u g h a l l aspects o f uncertainty cannot be m e a s u r e d o b j e c t i v e l y , a n u m b e r o f approaches exist for e x p l o r i n g the i m p a c t o f uncertainty o n a m o d e l o r d e c i s i o n . A n o v e r v i e w o f v a r i o u s m e t h o d s for a n a l y z i n g uncertainty is p r o v i d e d i n S e c t i o n 3.3.  F i n a l l y , the p r e v i o u s l y - d i s c u s s e d caveats o f m o d e l e x t r a p o l a t i o n i m p l y that uncertainty m u s t increase c o n s p i c u o u s l y w h e n s i m u l a t i n g extreme events. S p e c i f i c considerations o f uncertainty for extreme event s i m u l a t i o n are d i s c u s s e d i n S e c t i o n 3.4.  3.1  Classification of Uncertainty in Hydrologic Modelling  O v e r t i m e , m a n y c l a s s i f i c a t i o n schemes for uncertainty have b e e n p r o p o s e d . L o h a n i et a l . (1997) p r o v i d e perhaps the most fundamental, p r o p o s i n g that t w o b a s i c types o f uncertainty exist: what is not k n o w n at a l l , a n d errors i n what is k n o w n . A l t h o u g h h i g h l y significant, s u c h a perspective is p h i l o s o p h i c a l i n nature and does not address the m o r e p r a c t i c a l q u e s t i o n o f sources o f uncertainty.  B e c k (1987) proposes c l a s s i f y i n g uncertainty a c c o r d i n g to the processes t h r o u g h w h i c h it is i n t r o d u c e d into the p r o b l e m f o r m u l a t i o n , i.e., t h r o u g h p r i o r assumptions a n d k n o w l e d g e , m o d e l  72  i d e n t i f i c a t i o n , a n d p r e d i c t i o n . T h i s a p p r o a c h is m u c h better suited to understanding the sources o f uncertainty i n h y d r o l o g i c m o d e l l i n g , but is s t i l l s o m e w h a t general i n nature.  V i c e n s et a l . (1975) c o n s i d e r uncertainty as b e l o n g i n g to one o f t w o b a s i c categories: natural a n d i n f o r m a t i o n a l . T h e U S N a t i o n a l R e s e a r c h C o u n c i l (p. 4 1 , N R C , 2 0 0 0 b ) has adopted the same perspective, d e f i n i n g natural v a r i a b i l i t y a n d k n o w l e d g e uncertainty as f o l l o w s :  "Natural variability - sometimes called aleatory uncertainty deals with inherent variability in the physical world; [...] In the water resources context, uncertainties related to natural variability include things such as stream flow, assumed to be a random process in time, or soil properties, assumed to be random in space. Natural variability is also sometimes referred to as external, objective, random, or stochastic uncertainty. " "Knowledge uncertainty - sometimes called epistemic uncertainty deals with a lack of understanding of events and processes, or with a lack of data from which to draw inferences; by assumption, such lack of knowledge is reducible with further information. The word epistemic is derived from the Greek "to know. " Knowledge uncertainty is also sometimes referred to as functional, internal, or subjective uncertainty." T h e N R C (2000a) observes that these t w o uncertainties affect c a l c u l a t i o n s o f r i s k differently, a n d cautions that the t w o s h o u l d be c l e a r l y d i s t i n g u i s h e d i n practice. N o n e t h e l e s s , the d i s t i n c t i o n b e t w e e n the t w o is s o m e w h a t arbitrary a n d h y p o t h e t i c a l i n nature. P e r c e p t i o n a n d context t y p i c a l l y g o v e r n the d i s t i n c t i o n , since different assumptions m a y cause natural uncertainties to b e c o m e k n o w l e d g e uncertainties a n d v i c e v e r s a ( N R C , 2 0 0 0 b ) .  V i c e n s et a l . (1975) d i v i d e i n f o r m a t i o n a l ( k n o w l e d g e ) uncertainty into t w o b a s i c components, c o r r e s p o n d i n g to uncertainty i n m o d e l structure a n d i n parameter values. L a t e r studies a d d a t h i r d c o m p o n e n t c o r r e s p o n d i n g to uncertainty a r i s i n g from uncertainty o r error i n o b s e r v e d data (e.g., G a r e n a n d B u r g e s , 1 9 8 1 ; L o a g u e a n d F r e e z e , 1985). E v e r y h y d r o l o g i c study is subject to v a r y i n g degrees o f a l l three components. T h e s e three c o m p o n e n t s are c o l l e c t i v e l y l a b e l l e d m o d e l p r e d i c t i v e uncertainty.  73  M e l c h i n g et a l . (p. 2 2 7 5 , 1990) s u m m a r i z e the r e s u l t i n g four b a s i c classifications o f uncertainty as f o l l o w s :  •  natural v a r i a b i l i t y , w h i c h refers to "the r a n d o m t e m p o r a l and areal fluctuations inherent i n natural processes";  •  data uncertainty, w h i c h i n c l u d e s measurement i n a c c u r a c y and errors, the adequacy o f the data to represent  in situ c o n d i t i o n s , and any data h a n d l i n g , t r a n s m i s s i o n , or  transcription errors;  •  m o d e l parameter uncertainty, w h i c h reflects " v a r i a b i l i t y i n the d e t e r m i n a t i o n o f the proper parameter values to use i n m o d e l l i n g a g i v e n causative event"; and  •  m o d e l structure uncertainty, w h i c h characterizes "the a b i l i t y o f the m o d e l to accurately reflect the watershed's true p h y s i c a l r u n o f f process".  C r i t i c a l analysis o f this t a x o n o m y encourages e x p l o r a t i o n o f relationships b e t w e e n s p e c i f i c sources o f uncertainty i n h y d r o l o g i c m o d e l l i n g . T h e m i n d m a p s h o w n i n F i g u r e 3-2 is the result o f one s u c h analysis. T h e figure s h o w s the central g o a l o f i m p r o v i n g e s t i m a t i o n o f extreme events, surrounded b y v a r i o u s issues that i n h i b i t progress. M a j o r issues connect to the g o a l as trunks, w i t h sub-issues s p e c i f i e d as branches. T h e major areas o f focus are analogous to the types o f uncertainty d i s c u s s e d above, w i t h event uncertainty c o r r e s p o n d i n g to natural v a r i a