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The direct joint probability method for estimating extreme sea levels Liu, Joan C. H. 2004

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T H E D I R E C T J O I N T P R O B A B I L I T Y M E T H O D F O R E S T I M A T I N G E X T R E M E S E A L E V E L S b y J O A N C. H. L I U B. A . Sc., The Un ivers i t y o f B r i t i sh Co lumb ia , 2002 A T H E S I S S U B M I T T E D I N A P A R T I A L C O M P L E T I O N O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F T H E M A S T E R S OF A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department o f C i v i l Engineer ing) W e accepted this thesis as con fo rming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A August 2004 ©Joan C. H . L i u , 2004 THE UNIVERSITY OF BRITISH COlUMRifl FACULTY OK GRADUATE STUDIES Library Authorization for sc ho,ar l y purposes m a y b e g r a n l e d 6 y l h e te ™ * ' " " ^ ^ Permission extensile c o p y i n 9 o f l h i s l h e s j ^ ^ ^ ^ ^ N a m e of A u t h o r (please print) D a t e ( d d / m m / y y y y ) Tit le o f Th6SiS m^ck SX C > Degree : 3 ~ Year : Departmentof _?£^lL_£ww^^ T h e Univers i ty o f B r i t i s r T c d u r ^ t e t/~ Q " Vancouve r , BC C a n a d a grad.ubc.ca/forms/?formlD=THS page 1 of 1 ABSTRACT The design o f coastal structures includes the key element o f est imat ing crest elevation. A crest height designed to protect against specif ied return periods avoids damages due to ove r f l ow ing and over topping. I n order to avoid over f l ow ing , the design sea levels should be at least at the design f lood level , also referred to as the extreme f l ood level , w h i c h is usual ly composed o f tides and storm surges. The extreme f lood level can be determined b y several approaches, such as the Annua l M a x i m a , Simple A d d i t i o n , Joint Probabi l i ty , and Revised Joint Probabi l i ty Methods. These methods have various l imi tat ions i n terms o f the required amount o f data, the representation o f contr ibut ing factors i n sea level f luctuat ions, the ab i l i ty to assess the j o i n t p robab i l i t y o f these factors, and the degree o f independence required o f the data. To m i n i m i z e over topping, i n addi t ion to consider ing tides and storm surges, the design sea levels should also include wave run-up. The design sea level , also referred to as the extreme sea level , includes the effects o f t ides, storm surges, and wave run-up. Wave run-up estimates are general ly based on the design f lood level and design wave c l imate, data for w h i c h are often dependent. Th is thesis develops the Di rect Joint Probabi l i ty M e t h o d for est imat ing extreme sea levels w h i c h s imultaneously considers tides, s torm surges, and wave run-up. Th is method has fewer l imi ta t ions than the prev ious ly ment ioned methods i n terms o f the assumption o f independent variables and the required amount o f data. Data for the C i t y o f R i chmond , Br i t i sh Co lumb ia , Canada, are used to demonstrate the Direct Joint Probab i l i t y M e t h o d , and results show that the method provides a reasonable estimate o f extreme sea levels, that is, the result ing estimates are w i t h i n the same range as other t rad i t ional ly appl ied methods. The results also indicate a large di f ference between design sea levels required for prevent ing ove r f l ow ing and those for prevent ing overtopping. The sea levels at R i c h m o n d are also increasing due to the i i c l imat ic and geologic effects. A hybr id o f the Di rect Joint Probabi l i t y and the Simple A d d i t i o n Methods is also appl ied in this thesis and is used to estimate extreme sea levels for regions facing long- term increases i n sea levels. The results o f the h y b r i d approach indicate that the cont r ibut ion to extreme sea level due to wave run-up increases w i t h long- term increases in sea levels. This can dramat ica l ly affect estimates o f extreme sea levels. 111 TABLE OF CONTENTS A B S T R A C T i i T A B L E O F C O N T E N T S i v L I S T OF T A B L E S v i i L I S T O F F I G U R E S v i i i L I S T OF A P P E N D I C E S x A C K N O W L E D G E M E N T S x i 1 I N T R O D U C T I O N 1 2 C O A S T A L S T R U C T U R E S A N D F L U C T U A T I O N S I N S E A L E V E L S 6 2.1 Impacts o f Sea Floods 6 2.2 Coastal Structures 8 2.2.1 Types o f Coastal Structures 8 2.2.1.1 Seawalls 9 2.2.1.2 Revetments 10 2.2.1.3 Dykes 11 2.2.2 Modes and Consequences o f Failures 11 2.2.2.1 Over topp ing and O v e r f l o w i n g 11 2.2.2.2 Instabi l i ty o f Outer Slope 13 2.2.2.3 Instabi l i ty o f Inner Slope 13 2.2.2.4 Scour 14 2.2.2.5 Geotechnical Failures 14 2.2.3 Des ign o f Coastal Structures 15 2.2.3.1 Crest E levat ion 17 2.2.3.2 Other Des ign Considerations 18 2.3 Fluctuat ions o f Sea Levels 19 2.3.1 Scient i f ic Classi f icat ion Scheme 21 2.3.2 Engineer ing Classi f icat ion Scheme 22 2.3.3 W i n d Waves 23 2.3.3.1 Characteristics o f W i n d Waves 24 2.3.3.2 Statistical Analys is o f Waves 24 2.3.3.2.1 Statistical Analys is o f Wave Height and Period 25 2.3.3.2.2 Wave Spectrum Analys is o f Wave He igh t and Per iod 26 2.3.3.3 Wave Predict ion : 27 2.3.4 Storm Surges 29 2.3.4.1 Generat ion o f S torm Surges 29 2.3.4.2 Characteristics o f Storm Surges 30 2.3.4.3 Est imat ion o f Storm Surges 30 2.3.4.3.1 Indirect Measurement o f Storm Surges 30 2.3.4.3.2 Numer i ca l Mode ls o f S torm Surges 31 2.3.5 As t ronomica l Tides 32 2.3.5.1 Generat ion o f Tides 33 2.3.5.2 Characteristics o f Tides 33 2.3.5.3 Measurements o f Tides and Establ ishment o f D a t u m 34 2.3.5.4 Predict ion o f As t ronomica l Tides 35 2.3.6 Tsunami 36 2.3.6.1 Generat ion o f Tsunami 37 2.3.6.2 Characteristics o f Tsunami 38 i v 2.3.7 E l N i n o 38 2.3.8 C l imato log ic and Geologic Effects 39 3 Q U A N T I T A T I V E M E T H O D S F O R E S T I M A T I N G E X T R E M E C O N D I T I O N S 50 3.1 Methods for Est imat ing F lood Levels 50 3.1.1 Probabi l i ty o f Exceedance and Return Period 51 3.1.2 A n n u a l M a x i m a M e t h o d 52 3.1.2.1 App l i ca t i on 53 3.1.2.2 Data Requirements 54 3.1.2.3 Assumpt ions 55 3.1.2.4 Advantages and Disadvantages 56 3.1.3 i?-Largest M a x i m a M e t h o d 56 3.1.3.1 Advantages and Disadvantages 58 3.1.4 S imple A d d i t i o n M e t h o d 58 3.1.4.1 Advantages and Disadvantages 60 3.1.5 Joint Probabi l i ty M e t h o d ( JPM) 60 3.1.5.1 App l i ca t i on 62 3.1.5.2 Data Requirements 64 3.1.5.3 Assumpt ions 65 3.1.5.4 Advantages and Disadvantages 66 3.1.6 Revised Joint Probabi l i ty M e t h o d ( R J P M ) 67 3.1.6.1 Advantages and Disadvantages 68 3.2 Est imat ion o f Ext reme Wave Condi t ions 69 3.2.1 L o n g - T e r m Dis t r ibu t ion o f Sea States 70 3.2.2 L o n g - T e r m D is t r ibu t ion o f Ind iv idua l Wave He igh t 70 3.3 Joint Probabi l i ty o f F lood Levels and Extreme Wave Condi t ions 71 3.4 Wave Run-up and Over topp ing Discharges 72 3.4.1 Wave Trans i t ion i n Shal low Water 73 3.4.2 Predic t ion o f Wave R u n - U p 74 3.4.2.1 Shore Protect ion Manua l ( S P M ) M e t h o d 75 3.4.2.2 V a n Der Meer-Janssen ( V D M J ) M e t h o d 77 3.4.3 Over topp ing Discharge 79 3.4.3.1 Est imat ion o f Over topp ing Discharges 79 3.4.3.1.1 Shore Protect ion Manua l ( S P M ) M e t h o d 80 3.4.3.1.2 V a n Der Meer ( V D M ) M e t h o d 81 3.4.3.1.3 Owen 's M e t h o d 82 3.4.3.1.4 Compar ison o f Methods for Predict ing Over topp ing Discharges 83 3.4.3.2 A l l o w a b l e Over topp ing Discharge 84 4 D I R E C T J O I N T P R O B A B I L I T Y M E T H O D ( D J P M ) 95 4.1 Descr ip t ion o f the D J P M 95 4.2 Data Requirements 98 4.3 Assumpt ions 99 4.4 Advantages and Disadvantages 99 5 S E A F L O O D P R O T E C T I O N I N R I C H M O N D , B.C 103 5.1 Surroundings 104 5.2 Settlements 104 5.3 F lood Concerns 105 5.3.1 R iver Floods 106 5.3.2 Sea Floods 106 5.3.3 Excessive A m o u n t o f Ra in 107 v 5.4 Current D y k e System 108 5.4.1 Legis la t ion 108 5.4.2 Descr ip t ion o f the D y k e System 109 5.4.3 D y k e Maintenance Program 110 5.5 Sea Leve l Var ia t ions 112 5.6 Data A v a i l a b i l i t y 114 6 M E T H O D O L O G Y 120 6.1 Input Parameters 120 6.2 Assessment o f the D J P M for Est imat ing Ext reme F lood Levels 123 6.2.1 App l i ca t ion o f the D J P M for Est imat ing Ext reme F lood Levels 123 6.2.2 App l i ca t i on o f the Annua l M a x i m a M e t h o d for Est imat ing Ext reme F lood Levels 124 6.2.3 App l i ca t i on o f the Simple A d d i t i o n M e t h o d for Est imat ing Ext reme F lood Levels 125 6.3 Est imat ing Ext reme Sea Levels fo r R i chmond 126 6.3.1 App l i ca t i on o f the D J P M for Est imat ing Extreme Sea Levels 126 6.3.1.1 App l i ca t i on o f D J P M for C o m b i n i n g Observed Sea Levels and Wave Run-up 127 6.3.1.2 App l i ca t ion o f D J P M for C o m b i n i n g Tides, S torm surges, and Wave run-up 129 6.3.2 App l i ca t i on o f the H y b r i d o f the D J P M and Simple A d d i t i o n M e t h o d for C o m b i n i n g Tides, Storm surges, Wave Run-up, and L o n g - t e r m Sea Leve l Rises 130 7 R E S U L T S A N D D I S C U S S I O N 137 7.1 Assessment o f the D i rec t Joint Probabi l i ty M e t h o d 137 7.1.1 Est imat ion o f Ext reme F lood Levels Us ing the D i rec t Joint Probabi l i ty M e t h o d 137 7.1.2 Est imat ion o f Ext reme F lood Levels Us ing the A n n u a l M a x i m a M e t h o d 138 7.1.3 Est imat ion o f Extreme F lood Levels Us ing the S imple A d d i t i o n M e t h o d 139 7.1.4 Assessment o f the Di rect Joint Probabi l i ty M e t h o d for Est imat ing Ext reme F lood Levels 139 7.2 Est imat ion o f Extreme Sea Levels using the D J P M and H y b r i d DJP-Simple A d d i t i o n M e t h o d 141 7.2.1 Est imat ions o f Extreme Sea Levels Us ing the Direct Joint Probabi l i ty M e t h o d 141 7.2.1.1 D J P M App l i ca t ion to Observed Sea Levels and Wave Run-up 141 7.2.1.2 D J P M App l i ca t ion to Tides, Storm Surges, and Wave Run-up 142 7.2.2 H y b r i d DJP-Simple A d d i t i o n M e t h o d App l i ca t i on to Tides, S torm Surges, Wave Run-up, and Long- te rm Sea Leve l Changes 143 7.2.3 Discussion o f the Est imat ions o f Extreme Sea Levels for R i chmond 143 7.3 Sources o f Error fo r Estimates o f Ext reme F lood and Sea Levels 145 8 C O N C L U S I O N S A N D R E C O M M E N D A T I O N S 161 R E F E R E N C E S •. 165 A P P E N D I C E S 170 Append ix A : Wave Run-up Curves 171 Append ix B: Over topp ing Parameters 178 Append ix C: L is t o f Notat ions.. . . . 180 v i LIST OF TABLES Table 2.1 Sea F lood His tor ica l Events 41 Table 2.2 Des ign Considerat ions and Cr i t i ca l Modes o f Fai lure o f Revetments f r o m Pi larczyk (2000) 43 Table 2.3 Characteristics o f Waves f r o m Thomson (1981) and Sorensen (1997) 44 Table 2.4 Saf f i r -S impson Hurr icane Damage Potential Scale from A b b o t t (1999) 44 Table 2.5 Storm Surge Mode ls Developed for Use in the Publ ic D o m a i n f r o m Re id (1990) 44 Table 3.1 P lo t t ing Posi t ion Formulas f r o m Wat t et al. (1989) , and Rao and H a m e d (2000) 85 Table 3.2 Va lue o f y f for Var ious Slope Surface Characteristics f r o m C E R C (1984) 85 Table 3.3 Reduct ion Factor Yf for Rough Slope f r o m Besley and A l l s o p (2000) 86 Table 3.4 Reduct ion Factor yf fo r Rough Slope f r o m Pi la rczyk (2000) 86 Table 3.5 Reduct ion Factor Yf for Rough Slope f r o m Headland et al. (2000) 86 Table 3.6 Empi r i ca l Coeff ic ients for Smooth Impermeable Simple Slop ing Coastal Structures f r o m Besley and A l l s o p (2000) 86 Table 3.7 Tolerable M e a n Discharge (m 3 / s per meter run) f r o m Besley and A l l s o p (2000) 87 Table 5.1 Data Ava i l ab i l i t y 116 Table 6.1 Descr ip t ion o f the Qua l i t y Codes from Mar ine Env i ronment Da ta Service Website 132 Table 7.1 Two-D imens iona l H is togram o f Tides and Storm Surges 149 Table 7.2 Screened Annua l M a x i m a Series o f Observed Sea Levels 150 Table 7.3 Summary o f Estimates o f Ext reme F lood Levels 151 Table 7.4 Summary o f Estimates o f Ext reme Sea levels 151 Table 7.5 Summary o f Estimates o f Ext reme Sea Levels under D i f fe rent Propagat ion Angles 152 Table 7.6 Summary o f the Estimates o f Ext reme Sea Levels Projected for D i f fe rent Years based on L o n g - T e r m Sea Leve l Changes 152 v i i LIST OF FIGURES Figure 2.1 Sea Leve l Record for M a r c h , 1964 45 Figure 2.2 Rubble M o u n d Seawall 46 Figure 2.3 Quarrystone Revetment 46 Figure 2.4 Typ ica l D y k e Section 47 Figure 2.5 F lowchar t o f Des ign Process for Revetments 47 Figure 2.6 Scient i f ic Classi f icat ion o f Factors A f fec t i ng Sea Leve l Fluctuat ions 48 Figure 2.7 Engineer ing Classi f icat ion o f Factors A f fec t i ng Sea Leve l Fluctuat ions 48 Figure 2.8 Deep-water Signi f icant Wave Height and Per iod M o n o g r a m ;.. 49 Figure 3.1 Example o f Two-D imens iona l H is togram for the J P M 88 Figure 3.2 Waves in Deep Water and Waves in Shal low Water 88 Figure 3.3 Wave R u n - U p 89 Figure 3.4 Wave R u n - U p on Smooth, Impermeable Slopes on a 1:10 B o t t o m w h e n d s / H 0 ' = 2.0 90 Figure 3.5 Wave R u n - U p Correct ion for Scale Factor 91 Figure 3.6 D e f i n i t i o n o f the Ang le o f the Wave At tack 92 Figure 3.7 Over topp ing Parameters a and Qo for Riprapped and 1:1.5 Structure Slope 93 Figure 3.8 Suggested L i m i t s for Over topp ing Discharges 94 Figure 4.1 Example o f Two-D imens iona l H is togram 100 Figure 4.2 Example o f Three-Dimensional H is togram . . . .100 Figure 4.3 F l o w D iag ram o f D J P M for C o m b i n i n g o f Tides, Surges, and W a v e Run-up 101 Figure 4.4 Example o f the Result o f the Di rect Joint Probabi l i ty M e t h o d fo r T w o Var iables 102 Figure 4.5 Example o f the Result o f the Di rect Joint Probabi l i ty M e t h o d fo r Three Var iables. 102 Figure 5.1 Hydrograph at Hope Gauge 117 Figure 5.2 Standard Cross Section o f the D y k e System 117 Figure 5.3 F lood Protect ion Features i n R i chmond 118 Figure 5.4 Sea D y k e o f R i c h m o n d a) L o o k i n g N o r t h b) L o o k i n g toward West 118 Figure 5.5 M a p o f Gauge Stations near R i c h m o n d 119 Figure 6.1 Long- te rm Trend o f Sea Levels at Point A tk inson 133 Figure 6.2 Construct ion D r a w i n g at West End o f B lunde l l Road in 1974 133 Figure 6.3 F l o w D i a g r a m o f the D J P M for Comb in ing Observed Sea Levels and Wave Run-up 134 Figure 6.4 F l o w D i a g r a m o f the D J P M for Comb in ing Tides, S to rm Surges, and Wave Run-up 135 Figure 6.5 F l o w D i a g r a m o f the H y b r i d o f the D J P M and Simple A d d i t i o n M e t h o d 136 Figure 7.1 Estimates o f Ext reme F lood Levels using the D J P M for Tides and Storm Surges... 153 Figure 7.2 Estimates o f Ext reme F lood Levels using the A n n u a l M a x i m a M e t h o d 154 Figure 7.3 Estimates o f Ext reme F lood Levels using the Simple A d d i t i o n M e t h o d for Tides and Storm Surge 155 Figure 7.4 Estimates o f Ext reme Sea Levels Us ing the D J P M for Observed Sea Levels and Wave Run-up 156 Figure 7.5 Estimates o f Ext reme Sea Levels U s i n g the D J P M for Tides, S torm Surges, and Wave Run-up 157 Figure 7.6 Estimates o f Ext reme Sea Levels using D J P M for Tides, S torm Surges, and Wave Run-up w i t h A n g l e Reduct ion Factor 158 Figure 7.7 Estimates o f Ext reme Sea Levels U s i n g the H y b r i d DJP-S imp le A d d i t i o n viii M e t h o d 159 gure 7.8 Estimates o f Ext reme F lood and Sea Levels using D i f fe ren t Methods 160 ix LIST OF APPENDICES A p p e n d i x A A 1. Wave Run-up Curve for djH0 = Oon a 1:10 B o t t o m Slope f r o m C E R C (1984) 171 A 2. Wave Run-up Curve for djH0 « 0.45 on a 1:10 B o t t o m Slope f r o m C E R C (1984) 172 A 3. Wave Run-up Curve for ds/H0 * 0.8 on a 1:10 B o t t o m Slope f r o m C E R C (1984) 173 A 4. Wave Run-up Curve for and djH0 > 3.0 on a 1:10 B o t t o m Slope f r o m C E R C (1984) . . 174 A 5. Wave Run-up Curve for ds/H0 = 3 on a Hor izonta l B o t t o m f r o m C E R C (1984) 175 A 6. Wave Run-up Curve o f ds/H0 = 5 on a Hor izon ta l B o t t o m f r o m C E R C (1984) 176 A 7. Wave Run-up Curve for djH'0 = 8 on a Hor izonta l B o t t o m from C E R C (1984) 177 A p p e n d i x B B 1. Over topp ing Parameters for a Smooth Ver t ica l W a l l on a 1:10 B o t t o m Slope f r o m C E R C (1984) 178 B 2. Over topp ing Parameters for a 1:10 Structure Slope on a 1:10 B o t t o m Slope from C E R C (1984) 179 x ACKNOWLEDGEMENTS I w o u l d l ike to express m y grati tude to those w h o made this master thesis possible. • N S E R C and U B C F lood Fund, for p rov id ing the f inancia l support to complete this degree. • Dr . Barbara Lence o f the Department o f C i v i l Engineer ing at U B C , supervisor, for p rov id ing valuable comments and constant rev iew ing o f m y thesis, and spending al l those weekends to w o r k w i t h me. • Dr . M i c h a e l Isaacson o f the Department o f C i v i l Engineer ing at U B C , supervisor, for p rov id ing his expertise i n coastal engineering, and p rov id ing s t imulat ing suggestions and encouragement. • Yaros lav Shumuk o f U M A Engineer ing, Dr . Sheng L i o f the Department o f C i v i l Engineer ing at U B C , and Scott Thoml inson o f M E D S , for their technical assistance w h i c h is v i ta l for the research. • D a v i d Brown lee and Ter ry Crowe o f the C i t y o f R i c h m o n d , for g i v ing me permission to use the R i c h m o n d data in m y thesis and g iv ing me an oppor tun i ty to w o r k on this interest ing subject. • F a m i l y members, Y i n - Y e n L i u , Grace L i u , Chr is t ina L i u , Thomas L i u , and Joseph L i u , for their supports and encouragement i n d i f f i cu l t t imes. • Fr iends, especial ly W a y n e L o and D a v i d Roche, for al l their help, support, interest, and valuable hints. • F ina l ly , K e v i n Chen, for his patient love w h i c h enables me to complete m y thesis. x i 1 INTRODUCTION W h i l e coastal areas prov ide the convenience o f t ransportat ion and abundance o f natural resources, ocean water m a y threaten the safety o f coastal residents because natural beaches m a y be too nar row and too rap id ly eroded to prov ide adequate protect ion from extreme waves and water levels. Numerous natural disasters related to sea f loods in the mar i t ime areas have been recorded w o r l d w i d e . The impacts o f sea f loods are usual ly s igni f icant as such events have re lat ive ly short wa rn ing periods. The devastating loss o f l ives and damage to proper ty resul t ing from sea f loods have led mar i t ime countries to b u i l d coastal protect ion structures inc lud ing seawalls, revetments, and dykes along the shoreline as the p r imary defence against sea f loods. One o f the key design considerations o f such systems is the crest elevat ion. Engineers and scientists have sought to better understand causes o f extreme sea condi t ions i n coastal areas and to better estimate these for sett ing appropriate standards for crest e levat ion o f coastal structures. T o date, m a n y lessons have been learned from European practices as countries such as the Netherlands are considered the leaders in coastal engineering. Th is thesis develops a mod i f i ca t ion o f the Joint Probabi l i ty M e t h o d (JPM) , the D i rec t Joint Probabi l i ty M e t h o d ( D J P M ) , to estimate extreme sea levels. The D J P M requires fewer assumptions and enables a broader range o f appl icat ions than exist ing methods for est imat ing f l ood and sea levels. A n example appl icat ion o f the method, based on the C i t y o f R i c h m o n d , B r i t i sh Co lumb ia (B.C. ) , locat ing i n western Canada, is also prov ided. In i t ia l l y , coastal protect ion systems were designed based on previous extreme events, such as the highest s torm surge observed. However , the g row ing importance o f these coastal systems, due to the increasing number o f communi t ies and faci l i t ies i n coastal areas, has led to improvements i n the design o f coastal structures based on more robust estimates o f extreme sea 1 condi t ions that have a g iven l i ke l ihood o f occurrence. Instead o f us ing histor ical events direct ly, extreme sea condi t ions are predicted i n order to select a crest elevat ion that is able to protect against potent ia l future events. The est imation o f extreme sea condi t ions involves three major steps: 1) ident i f icat ion o f the cr i t ica l factors o f extreme sea condi t ions, 2) invest igat ion o f various approaches for assessing and comb in ing these factors, and 3) estimations o f design sea levels inc lud ing extreme f lood and sea levels. Oceanographers have studied the characteristics o f sea level f luctuat ions and have ident i f ied components o f extreme condit ions. These components include extreme f l ood levels and extreme wave condi t ions. The components o f extreme f lood levels inc lude storm surges, astronomical t ides, and tsunami; however, extreme f lood level estimates exclude the inf luence o f waves on sea levels. I n addi t ion to f l ood levels, wave condi t ions can also cause var ious degrees o f damage to coastal structures, and thus extreme wave condi t ions are integrated i n estimates o f extreme sea levels. Nevertheless, controversy exists regarding the inc lus ion o f the wave cl imate because the damage caused b y waves m a y be smal l . W h i l e most design manuals suggest the use o f wave run-up to represent the impact o f waves on coastal structures, several recent studies focus on the ident i f icat ion o f a l lowable over topping discharges in assessing extreme sea levels. I n those studies, the ident i f ied al lowable over topping discharges are used as a basis for selecting design extreme sea levels. A f te r selecting the cr i t ical components o f extreme f lood and sea levels and assessing the available data, the extreme f lood and sea levels m a y be estimated w i t h various methods. Trad i t iona l ly , the A n n u a l M a x i m a M e t h o d is appl ied to estimate the extreme f lood levels. However , this method requires a data set col lected over a long per iod o f t ime, and cannot d ist inguish among the di f ferent components o f extreme f l ood levels. A l t h o u g h the i?-Largest M a x i m a M e t h o d improves upon the Annua l M a x i m a Me thod , i t cannot treat the extreme f lood levels as a func t ion o f a number o f dist inct components. The Simple A d d i t i o n M e t h o d al lows 2 for the representation o f the di f ferent components o f extreme f l ood and sea levels, but it cannot associate the estimates w i t h an actual j o i n t probabi l i ty o f these components. T w o addit ional methods developed to estimate extreme f lood levels are the J P M and the Revised Joint Probabi l i ty M e t h o d (RJPM) . These two methods estimate the extreme f l ood levels as a funct ion o f var ious components w i t h dist inct probabi l i t ies o f occurrence. They are based on the assumptions that tides and storm surges are independent, and that the probab i l i t y density func t ion for s torm surges over a l im i ted t ime per iod represents the p robab i l i t y densi ty func t ion for the popula t ion o f al l possible storm surges. The f i rst assumption o f hav ing independent components l im i ts the applications o f these approaches, because the factors causing the f luctuat ions o f water levels are rarely independent o f each other. The D J P M is developed in this research to overcome this l im i ta t ion . The D J P M f inds the extreme f l ood and sea levels w i t h associated probabi l i t ies o f exceedance b y evaluat ing the func t ion o f the j o i n t probabi l i t ies o f the various components o f the extreme f lood and sea levels, respectively. The j o i n t probabi l i t ies o f the various components are obtained b y construct ing a mu l t i -d imens iona l h is togram for the discrete ranges o f the data for each component, where the dimensions o f the h is togram correspond to the number o f components considered. Th is method is appl ied in the case o f R i chmond , B. C , Canada. The C i t y o f R i chmond , B. C , Canada is a part o f the delta reg ion located along the Pacif ic Coast and faces potent ial threats o f sea f loods due to its l o w elevat ion relat ive to sea levels. The C i t y has per imeter dyke systems for bo th L u l u and Sea Islands. The current crest elevation o f dykes and revetments used in these systems has been constructed to meet design f lood level standards plus freeboard. A lack o f data available for pred ic t ing extreme condi t ions has lead to the use o f h istor ical extreme events to define the design f l ood level . Whether or not an increase in the crest elevat ion is necessary is a question that has been raised dur ing the recent strategic f l ood p lann ing for the Ci ty . T o respond to this question, a reassessment o f the extreme f lood and 3 sea levels is required. The est imat ion o f the extreme sea levels for R i c h m o n d requires the ident i f icat ion o f the cr i t ica l components cont r ibut ing to sea level f luctuat ions and an assessment o f the available data. The cr i t ical components include tides, s torm surges, wave run-up, and long- term sea level increases, and various types o f data related to these are available at nearby locations. The D J P M is an ideal approach for est imating extreme sea levels o f a complex si tuat ion such as the R i c h m o n d case, because i t is able to estimate the extreme sea levels composed o f t ides, s torm surges, and wave run-up. A l t h o u g h the A n n u a l M a x i m a M e t h o d and Simple A d d i t i o n M e t h o d can on ly estimate the extreme f l ood levels composed o f tides and storm surges, they are also appl ied herein i n order to assess the representativeness o f the D J P M . Resul t ing sea level f luctuat ions due to increases in long- term sea levels are also assessed using a hybr id o f the D J P M and the Simple A d d i t i o n Method . The objectives o f this thesis are to: 1. investigate the various causes o f variat ions in sea levels, 2. explore opt ions for est imating extreme sea condi t ions inc lud ing extreme f l ood levels and wave condi t ions, 3. develop and demonstrate the D J P M , a methodo logy for determin ing extreme sea levels, and 4. estimate the extreme sea levels for R ichmond , B.C., i n order to set the standard for this part icular dyke system. The f i rst object ive o f invest igat ing the f luctuat ions i n sea levels is addressed i n Chapter 2 w i t h a discussion o f the impacts o f sea f loods and the designs o f coastal structures. The second object ive o f iden t i f y ing and evaluat ing quanti tat ive methods for est imat ing extreme condit ions is addressed in Chapter 3. The D J P M is then developed in Chapter 4. Chapter 5 describes the case o f R i c h m o n d , B.C. and discusses the background, variat ions in sea levels, and data avai labi l i ty . Chapter 6 presents the methodo logy used to estimate extreme sea levels for R ichmond , and 4 Chapter 7 presents the resulting design crest elevation for the Richmond case, addressing the fourth and final objective. Chapter 8 provides a summary of the thesis findings and a discussion of the recommendations based on these findings. 5 2 COASTAL STRUCTURES AND FLUCTUATIONS IN SEA LEVELS The f irst part o f this chapter stresses the importance o f coastal structures b y rev iewing a l ist o f h istor ical catastrophic events caused b y sea f loods i n Section 2 . 1 . Section 2.2 presents the types o f structures used to prov ide a higher level o f protect ion i n coastal areas, focusing on the design processes and in part icular the design crest elevat ion o f the coastal structures. To better design the coastal structures, inc lud ing the determinat ion o f the crest elevations, the understanding o f ocean processes is necessary. Therefore, Section 2.3 investigates the factors that contr ibute to sea level f luctuat ions. 2.1 Impacts of Sea Floods Coastal areas of ten attract the gathering o f settlements and c iv i l izat ions due to the ease o f f i nd ing means o f t ransportat ion and sources o f food, but the dangers due to f lood ing increases w i t h p r o x i m i t y to oceans. The records o f events o f sea f loods and resul t ing damages are tangible evidence o f such dangers. The impacts o f sea f loods can be estimated for h istor ical events and a l ist o f such events is presented in Table 2 . 1 . A s shown i n Table 2 . 1 , sea f loods occurr ing at d i f ferent t imes and locations stem from di f ferent causes and result i n d i f ferent degrees o f damage due to differences in geological locat ion, oceanographic characteristics, and cl imate systems. M a r i t i m e countr ies, such as the Netherlands, Ind ia, Japan, Canada, and the Un i ted States o f Amer i can ( U S A ) , have experienced the cont inued threats o f sea f loods. A m o n g these countries, the Netherlands is the f i rst country to develop an ongoing log o f events and also one o f the f irst countries to implement dyke systems. The earliest recorded events are in 1228 i n Friesland and 1282 i n Zuyder Zee, bo th in the Netherlands. A s the elevat ion o f the Nether lands is lower than 6 the sea level , the country is more susceptible to sea f loods. A l t h o u g h the Netherlands already implemented dykes after 1228, s torm surges in 1282 st i l l managed to sweep the protected area. Learn ing gradual ly f r o m such histor ical events, the Netherlands has bu i l t and improved the dyke systems m i n i m i z i n g the potent ial impacts o f sea f loods. I n addi t ion, the histor ical events indicate that the c o m m o n sources o f major f loods are storms, tsunami, and a combinat ion o f h igh tides and w i n d . For example, storms are the cause o f sea f loods i n the events o f the 1954 f lood in Rhode Is land, U S A and the 1970 f l ood in Pakistan. Tsunami are among the sources that cause sea f loods resul t ing in the greatest level o f damages, and the cause o f several events, for example the 1896 f lood i n Saniku, Japan, the 1946 and 1960 f loods in B r i t i sh Co lumbia , Canada. A n example o f a sea f l ood or ig inat ing b y a combinat ion o f h igh t ide and w i n d is the 1953 f l ood in Nor thwest Europe. H is tor ica l events also suggest that sea levels are local ized dur ing extreme events. The 1964 event w h i c h occurred in Br i t i sh Co lumbia , Canada is o f part icular interest, and was caused b y an earthquake i n A laska. Figure 2.1 presents the sea level records f r o m four locations in B r i t i sh Co lumb ia , inc lud ing T o f m o , Port A lbe rn i , V ic to r ia , and Point A t k i n s o n dur ing this event. The tsunami generated an average peak wave height o f 3 m at T o f m o and an average peak wave ampl i tude o f 6 m at Port A lbe rn i , simultaneously. Other locations also experienced higher wave run-up. A l t h o u g h the distance between T o f m o and Port A l b e r n i is on ly about 125 k m , the severity o f sea f loods at these sites varies to a great extent. Therefore, estimates o f extreme sea levels should consider local condit ions. 7 2.2 Coastal Structures T o avoid potent ia l damages f r o m sea f loods, coastal structures are constructed, par t icu lar ly w h e n a natural beach cannot prov ide adequate protect ion. Coastal structures are posi t ioned relat ive to the shoreline according to their funct ions, and classif ied based on their relat ive pos i t ion to the shoreline as shore-parallel and shore-perpendicular. Focusing on shore-parallel structures, Section 2.2.1 introduces dif ferent types o f coastal structures. A l t h o u g h coastal structures serve to increase the level o f protect ion o f near shore areas from wave attacks, erosion, and sea f loods, these structures are not able to complete ly e l iminate potent ia l damage. W h e n a coastal structure is unable to accompl ish its funct ion, fa i lure occurs. Section 2.2.2 describes various fai lure modes, as understanding fai lure modes is essential to successful design o f the coastal structures. Section 2.2.3 describes the design o f the coastal structures. 2.2.1 Types of Coastal Structures Shore-perpendicular structures, such as T-gro ins, of fshore breakwaters, and art i f ic ia l headlands, are usual ly used to reduce sediment transport (Headland et al. , 2000). B y reducing sediment transport, the shore-perpendicular structures ind i rec t ly decrease the r isk o f f lood ing , because sediment transport to the near-shore area can increase the elevat ion o f the ocean beds, and thus increase sea levels. For example, the experience i n the Nether lands suggests a trend i n r is ing sea levels due to the increase o f sediments i n the ocean beds. Shore-paral lel structures, such as seawalls, revetments, and dykes, are used to prevent the cross-shore movement o f sand that occurs dur ing storms and the inundat ion o f near shore areas (Headland et al. , 2000). A l t h o u g h bo th structures prov ide protect ion to the coastal areas, this section focuses on the 8 shore-parallel structures, i n part icular sea wal ls , revetments, and dykes. Sea wal ls , revetments, and dykes possess certain simi lar i t ies and are usual ly classif ied b y their relat ive distance to the natural beach pro f i le . Revetments and dykes are located closer to the land, whereas seawalls are located somewhat seaward o f a typ ica l dyke or revetment locat ion (Headland et al. , 2000). 2.2.1.1 Seawalls Seawalls are massive structures bu i l t a long the coastl ine to protect near shore areas from wave impact , erosion damage, and f lood ing. They are the most w i d e l y used op t ion for coastal defence (Besley, 1999; U S A C E , 1995). Since seawalls are constructed closer to the sea than dykes and revetments, they are exposed to stronger waves on a near-cont inuous basis, such as s torm waves. Thus, they are designed to be re lat ive ly massive and have higher crest elevations in order to resist the fu l l force o f s torm waves. I n general, seawalls are composed o f an armour, base layer, f i l ter, toe protect ion, f i l ter layer, and backslope armour, as shown i n Figure 2.2. The armour is used to protect the seawall from over topping and to resist wave act ion w i thou t s igni f icant displacement o f armour mater ial (Besley, 1999). T w o c o m m o n materials for the base layer o f seawalls are rubble mound , i.e., rock, and mono l i th i c mater ia l , i.e., steel or concrete sheet p i le (Headland et al. , 2000). A f i l ter is a transi t ional layer o f gravel , smal l stone, or fabric placed between the under ly ing soi l and structure to prevent the m ig ra t ion o f f ine soi l particles through the voids, distr ibute the weight o f the armour uni ts, and permi t re l i e f o f pressures w i t h i n the soils ( C E R C , 1995). I n addi t ion, seawalls no rma l l y require extensive toe protect ion to prevent the occurrence o f scour. Acco rd ing to their funct ions, seawalls have various face shapes inc lud ing ver t ica l , near vert ical , s loping faces, stepped wal ls , and curved faces. 9 2.2.1.2 Revetments A revetment m a y be a coastal structure or a fac ing o f erosion resistant mater ial . This section refers to revetments as coastal structures, but they are c o m m o n l y used as a fac ing w h e n combined w i t h other coastal structures. The revetment system serves a number o f di f ferent funct ions, inc lud ing the prov is ion o f protect ion for exposed faces o f coastal f l ood embankments, reduct ion o f over topping discharges, l i m i t i n g o f damage to crests and rear faces, p rov is ion o f erosion protect ion at c l i f f bases and around new reclamations, and p rov is ion o f protect ion for w i n d w a r d faces o f reservoir dams and embankments (A l l sop and M c C o n n e l l , 2000). A l t h o u g h there are cases where i t is d i f f i cu l t to make a clear d is t inct ion between a seawal l and a revetment, a revetment is general ly located closer to the land than a seawall . Thus, a revetment is exposed to a lower level o f wave attack than a seawall , i t does not feature back slope armour. A revetment system has a number o f key components inc lud ing a cover or armour layer, f i l ter, toe protect ion, and crest protect ion (A l l sop and M c C o n n e l l , 2000). The armour layer provides basic protect ion against wave act ion, and the m a i n types o f mater ia l l i ke l y to be used are rock armour or r iprap (narrow or w i d e graded), stone or concrete b locks (loose or grouted), concrete b locks (loose, cable-t ied, sol id, cel lular in ter lock ing, or art iculated), concrete slabs (usual ly cast i n si tu), fabr ic - formed concrete mattresses, and asphaltic materials (open stone asphalt, b i tumen grout ing, or asphaltic concrete) ( C E R C , 1995; A l l sop and M c C o n n e l l , 2000). F igure 2.3 is an example o f a revetment w i t h rock armour. R i g i d armours, such as concrete slabs, are more massive but are unable to accommodate settlement o f the under ly ing mater ia l , whereas f lex ib le armours, such as r iprap, are l ighter and can tolerate vary ing amounts o f displacement ( C E R C , 1995). The f i l ter funct ions s im i la r l y as for seawalls. The toe protect ion for a revetment has a s imi lar funct ion as the toe protect ion for a seawall i n terms o f its prevent ion o f scour. 10 2.2.1.3 Dykes Simi lar to seawalls and revetments, dykes are barriers constructed to prevent f lood ing dur ing storms. They general ly feature a large and sometimes re la t ive ly impermeable core. The key components o f a dyke include an armour layer, toe protect ion, and f i l ter as shown in Figure 2.4. The armour layer serves to protect the dyke w i t h the armour of ten replaced b y revetments as a fac ing o f erosion resistant mater ial . The filter funct ions s im i la r l y to that for seawalls and revetments, and toe protect ion is essential w h e n scour becomes a p rob lem. Dykes can be appl ied w i t h other coastal protect ion structures, such as revetments. The dyke system in R ichmond , B.C., Canada is an example o f the dyke w i t h a revetment. 2.2.2 Modes and Consequences of Failures A l t h o u g h coastal structures have been constructed i n most mar i t ime countries, damages st i l l occur due to fai lures i n extreme condit ions as discussed i n Sect ion 2 . 1 . Var ious factors have caused coastal structures to be more vulnerable i n extreme condi t ions. Th is section investigates var ious fai lure modes, w h i c h include over topping and ove r f l ow ing , instabi l i ty o f outer slope, instabi l i ty o f inner slope, scour, and geotechnical fai lure. 2.2.2.1 Over topp ing and O v e r f l o w i n g Over topp ing and ove r f l ow ing are two modes o f fai lure occur r ing at the crest o f coastal structures. Over topp ing refers to when wave over topping or wave run-up exceeds an acceptable threshold (an acceptable threshold could be less than or equal to the crest elevat ion), whereas 11 o v e r f l o w i n g refers to when a f lood level exceeds the crest elevat ion. O v e r f l o w i n g can obv ious ly cause damage to coastal structures and coastal areas due to the invo lvement o f a greater amount o f water on a cont inuous basis, but there is controversy surrounding whether a single wave or wave run-up can cause the fai lure o f a coastal structure due to its instantaneous nature. A l l sop and M c C o n n e l l (2000) suggest that "over topp ing does not i tse l f consti tute structural fai lure, but heavy over topping m a y constitute a funct ional fai lure o f the defence, w h i c h m a y in turn endanger the stabi l i ty o f the crest or other elements o f the structure." P i la rczyk (2000) suggests that " i f a structure is overtopped, even b y m ino r splash, the stabi l i ty can be affected b y eroding the area above or behind the structure, remov ing soi l support ing the top o f the structure leading to the unrave l l ing o f the structure f r o m the top down , or increasing the vo lume o f water i n the soi l beneath the structure, contr ibut ing to drainage prob lems. " M o s t design guides permi t an acceptable threshold o f over topping discharge depending on the types o f usage o f areas behind the coastal structures, but they also require the determinat ion o f the potent ia l wave run-up to set the crest elevation. The design guidance regarding the crest e levat ion is further discussed in Section 2.2.3.1. I n the case where some over topping is a l lowed, the crest elevat ion is reduced. For a smal l amount o f over topping, a grass mat over c lay on the dyke surface is suff ic ient (Pi larczyk, 2000). Nevertheless, for heavy over topping, the crest should be increased un t i l an acceptable over topping discharge is achieved (A l l sop and M c C o n n e l l , 2000) . Figure 2.2 shows an example o f a wave w a l l incorporated i n the design o f a seawall , w h i c h may also be implemented i n the case o f heavy overtopping. 12 2.2.2.2 Ins tab i l i ty o f Outer Slope Instabi l i ty o f the outer slope is a type o f fa i lure occurr ing on the seaward slopes usual ly composed o f armour or revetments, when wave impact pressures f r o m wave attacks are greater than the strength o f the armour or revetments. Wave impact pressures are induced b y local ized breaking waves, and they general ly occur close to the water level (A l l sop and M c C o n n e l l , 2000). Th is type o f wave attack m a y lead to erosion and loss o f stabi l i ty o f the outer slope (Pi larczyk, 2000). Examples o f this type o f fa i lure inc lude local deformat ions and loss o f ind iv idua l cover layer elements (A l l sop and M c C o n n e l l , 2000). Damage from wave pressures can be m i n i m i z e d i f coastal structures possess suff ic ient size and thickness o f armour mater ia l , par t icu lar ly i n the region o f the structure exposed to the st i l l water level (A l l sop and M c C o n n e l l , 2000). 2.2.2.3 Ins tab i l i ty o f Inner Slope S imi la r ly , instabi l i ty o f the inner slope is also a type o f fa i lure i n armour units or revetments, w h e n up l i f t pressures f r o m wave attacks are greater than the strength o f the armour or revetments. Wave up l i f t pressures are created on the underside o f the armour layer or revetments dur ing wave d raw-down, w h e n the armour layer is l o w i n permeabi l i t y (A l l sop and M c C o n n e l l , 2000) . A n example o f instabi l i ty o f the inner slope is p ip ing . T o avo id instabi l i ty o f the inner slope, armour mater ial or revetments should be o f a suf f ic ient size and thickness (A l l sop and M c C o n n e l l , 2000). 13 2.2.2.4 Scour Scour of ten occurs at the toe o f coastal structures. Th is m a y cause instabi l i ty at the toe and eventual ly lead to fai lure throughout the entire structure (A l l sop and M c C o n n e l l , 2000; Pi larczyk, 2000). Factors that affect the severity o f toe scour inc lude wave break, wave run-up, backwash, wave ref lect ion, and the grain size d is t r ibut ion o f the beach or bo t tom materials (Pi larczyk, 2000). For example, the possib i l i ty o f scour increases w h e n a wave breaks near the toe o f a coastal structure. Wave breaking is a phenomenon part icular to shal low water, where the water depth is less than twice the height o f the wave. Thus, one approach for avo id ing scour is to locate the toe be low at least tw ice the design wave height. I n addi t ion, scour can be avoided b y construct ing toe protect ion using sheet pi les, w h i c h extend d o w n to be low the m a x i m u m predicted scour depth (A l l sop and M c C o n n e l l , 2000). A s ment ioned i n Section 2 .2 .1 , toe protect ion is a key component o f most coastal structures. 2.2.2.5 Geotechnical Fai lures Geotechnical fa i lure of ten occurs at the foundat ions o f coastal structures. Causes o f geotechnical fa i lure include seismic effects and the s lumping o f p o o r l y compacted sub-soil . This m a y lead to a s igni f icant deformat ion o f the armour layer and settlement o f the who le structure. Settlement is a c o m m o n prob lem for seawalls because o f their heavy weight . Construct ion m a y a l low for some settlement, but signif icant d i f ferent ia l settlement along a section o f the structure is l i ke l y to lead to local fa i lure and to decrease the overal l height o f the structure. Adequate compact ion before the construct ion and a f u l l seismic invest igat ion cou ld m i n i m i z e this type o f fai lure. 14 2.2.3 Design of Coastal Structures A complete design o f coastal structures involves four interconnected aspects: the funct ion o f the structure, the physical environment, the construct ion method, and the operat ion and maintenance (A l l sop and M c C o n n e l l , 2000; Pi larczyk, 2000). F igure 2.5 presents a f lowchar t o f the design process for revetments, but cou ld be ef fect ively appl ied w i t h m i n o r adjustments to most shore-parallel coastal structures. I n the process o f de f in ing the func t ion o f a structure, the objectives and cr i ter ia are established. Cr i ter ia are used to measure the objectives inc lud ing technical feasib i l i ty , economic feasibi l i ty, and social and po l i t i ca l feas ib i l i ty (Pi larczyk, 2000). A n example o f cr i ter ia constraining the technical feasib i l i ty is a suf f ic ient crest height to avo id over topping or ove r f l ow ing . The cr i ter ia for economic feasib i l i ty and social and po l i t i ca l feasib i l i ty are also used w h e n selecting alternative designs o f coastal structures. A s described in F igure 2.5, op t imiza t ion is of ten appl ied in the selection process. For instance, w h e n a structure is designed to protect against over topping, the cost o f construct ion increases due to the higher standard o f crest elevat ion. I n addi t ion, the construct ion method and operat ion and maintenance are impor tant considerations. The remainder o f this section focuses on the funct ion o f the structure and its physical environment i n the context o f technical feasibi l i ty . I n general, the m a i n object ive o f coastal structures is to p rov ide an increased level o f protect ion i n coastal areas wh i l e m i n i m i z i n g al l modes o f fa i lure w i thou t excessive maintenance. However , because o f the stochastic nature o f wave condi t ions and water levels, i t is impossible to design a coastal structure w i t h zero probabi l i ty o f fai lure. Thus, the design objectives should include the design l i fe o f the coastal structure and an acceptable r isk level dur ing the design l i fe. The design l i fe (ND) o f the coastal structure is typ ica l l y 50 to 100 years (A l l sop and M c C o n n e l l , 15 2000). The structures are designed to wi ths tand a design event (or events), w h i c h is (are) usual ly def ined b y a part icular combinat ion o f wave cond i t ion and water level . Each design event w i l l have a selected return per iod (TR) w h i c h indicates the average l i ke l ihood o f the event being exceeded in a g iven t ime per iod. The design return per iod should no rma l l y be s igni f icant ly longer than the design l i fe . The probab i l i t y o f exceedance o f the design event dur ing the l i fe o f the structure is the encounter probab i l i t y (pe) as described b y (A l l sop and M c C o n n e l l , 2000; Pi larczyk, 2000; Headland et al. , 2000): pe=\-{\-\ITR)N° 2.1 W h i l e the func t ion o f the structure sets the overv iew o f the design, "shore protect ion works cannot be designed w i thou t an adequate understanding o f local env i ronmenta l site cond i t i on " (Headland et al., 2000). Thus, an invest igat ion o f the physical env i ronment is conducted. The physical envi ronment related to the technical design o f coastal structures is described statist ical ly b y the f o l l o w i n g variables: topography, bathymetry, c l imate, water levels, wave cl imate, current system, coastal processes, and geotechnical features (Pi larczyk, 2000; Headland et al. , 2000). The topography and fronting bathymetry o f the area o f interest are essential to the funct ional and structural design o f coastal protect ion works , because the design o f a coastal protect ion scheme is of ten dictated b y water depth (Headland et al. , 2000). Structural requirements for resist ing the wave force m a y be reduced result when the fronting bathymetry is shal low as shal low near-shore bathymetry acts to break waves before they reach the structure (Headland et al. , 2000). Coastal processes, such as regional geomorphology, long- term shoreline changes, and sediment transport, should also be considered i n the long- term p lann ing o f coastal protect ion. Hydrau l ic condi t ions, such as water level and wave cl imate, are invest igated further i n Section 2.3. A l l variables serve as the input into the design o f components o f coastal structures. W h i l e the choice o f crest elevat ion is the focus o f this research and is discussed i n detai l i n Sect ion 2.2.3.1, other 16 design components are b r ie f l y rev iewed i n Section 2.2.3.2. Deta i led guidel ines can be found in the Shore Protect ion M a n u a l b y the Coastal Engineer ing Research Center ( C E R C ) o f U.S. A r m y Corp. o f Engineer ing ( U S A C E ) (1984). A l t h o u g h the coastal structures can be determined w i t h a higher level o f accuracy us ing the guidel ines for coastal protect ion compared w i t h experience-based designs in the past, P i larczyk st i l l cautions that " the solut ion be ing considered should st i l l be tested w i t h a scaled mode l , since no general ly accepted design rules exist for al l possible solut ions and circumstances" (Pi larczyk, 2000). 2.2.3.1 Crest E levat ion I n the past, the crest elevat ion was selected based on the highest observed storm surge level and an estimate o f wave characteristics f r o m histor ical records. The study o f the fai lure modes o f coastal structures led to a systematic approach for est imat ing crest elevation, where the crest elevat ion is selected to avo id over f low ing , over topping, and geotechnical fai lures. Components to be considered when selecting the crest height inc lude f l ood level , wave run-up, a l lowable over topping discharge, and anticipated structure settlement ( C E R C , 1995). The crest elevat ion should be at least at f lood level. The f lood level is a c o m m o n basis for the design o f coastal structures, because fai lure o f such structures typ ica l l y causes f loods i n l o w - l y i n g areas and ampl i f ies the impact o f waves. I n general, a f l ood level is p r i m a r i l y affected b y tides, be ing responsible for 97 % o f the impacts, and storm surges, being responsible for 3 % o f the impacts (Pugh and Vassie, 1980). B o t h factors are discussed i n Section 2.3, and the methods o f est imating extreme f lood levels are discussed in Section 3 .1 . To m i n i m i z e over topping, the extreme sea levels should inc lude bo th f lood levels and wave run-up. T w o issues arising from est imating extreme sea levels composed o f f l ood levels 17 and wave run-up are the 1) est imat ion o f the j o i n t p robab i l i t y o f water level and wave condit ions, and 2) est imat ion o f the damage caused b y over topping. The f i rst issue is addressed i n Section 3.3, and the second issue is discussed here. To prevent over topping, some suggest the use o f wave run-up to set the crest height as a conservative approach, under the assumption that on ly a s igni f icant wave run-up w i l l cause considerable damage to coastal structures ( C E R C , 1995). Assuming this is correct, two methods are c o m m o n l y used to determine conservative damage est imat ion: 1) using a smaller value o f wave run-up and 2) us ing a l lowable over topping discharges. A n example o f using a smaller value o f wave run-up can be found in Du tch practice for designing dykes. The crest elevat ion i n the D u t c h practice is def ined b y a combinat ion o f a static water level w i t h a 0 . 0 1 % probabi l i t y o f exceedance wh i l e wave run-up is der ived for wave condi t ions w i t h a 2 % probab i l i t y o f exceedance (Pi larczyk, 2000) . A larger probab i l i t y o f exceedance means the occurrence o f such an event is more frequent, and thus, the value o f the event is smaller. The extreme sea level or crest elevat ion value is then reduced due to the poss ib i l i ty o f an increased wave run-up occurr ing. The alternate method involves determining an a l lowable vo lume o f water that w o u l d result i n over topping or an a l lowable over topping discharge. W h i l e recent research in Europe has focussed on determin ing the al lowable over topping discharge, there is no consensus regarding the amount o f over topping discharge a l lowed. The est imat ion o f wave run-up and over topping is discussed in Sect ion 3.4. 2.2.3.2 Other Des ign Considerations Other design components o f coastal structures include slope, slope protect ion (i.e., armour uni t or revetment) , toe, and f i l ter selection. The slope o f a coastal structure inf luences the d is t r ibut ion o f wave forces, and thus also inf luences the choice o f materials suitable for slope 18 protect ion and the height o f the structure (Pi larczyk, 2000). Shal lower slopes lead to a distr ibuted reduced wave force and reduced wave run-up on the seaward slope. Thus, the design slope is usual ly 1:2 to 1:6, depending on the differences i n water levels and the wave condit ions (A l l sop and M c C o n n e l l , 2000). Slope protect ion provides stabi l i ty, and the design includes the selection o f materials and the determinat ion o f the thickness o f the materials. Each type o f cover layer for revetments has its o w n cr i t ical fai lure mode, wave loading, and strength, as shown in Table 2.2. The thickness or size o f the protect ion layer is determined b y a stabi l i ty cr i ter ion for wave attack, as discussed i n Sections 2.2.2.2 and 2.2.2.3. The design o f toe components includes the determinat ion o f elevat ion o f toe and types o f toe protect ion. A s ment ioned i n Section 2.2.2.4, w h e n scour is a concern for the design area, the toe should be constructed to a depth greater than the predicted scour, and should include toe protect ion (A l l sop and M c C o n n e l l , 2000). A f i l ter is a transit ional layer o f stones ranging w i t h a we igh t ranging f r o m 6 to 10% o f the armour we igh t i n order to distr ibute the we igh t o f the armour units ( C E R C , 1984). 2.3 Fluctuations of Sea Levels One o f the earliest studies descr ibing the factors in f luenc ing sea-level is b y N o m i t s u and Okamoto (1927). They suggest two pr inc ipa l classes o f cont r ibu t ing factors, the internal and external causes. The internal causes refer to the factors af fect ing changes i n the properties o f sea water, such as temperature, sal ini ty, and precip i tat ion. Examples o f external causes are atmospheric pressure, w i n d and the consequences o f the Cor io l is Ef fect upon the m o v i n g water mass. He lna (1944) cites r iver discharge as an addi t ional external cause. Galerk in (1960) proposes three classes o f factors: those related to the variat ions o f the phys ica l properties o f sea water, those related to the f luctuat ions i n the quant i ty o f water, and those af fect ing the uneven 19 dis t r ibut ion o f sea-level heights w i t h i n a basin. For example, r i ver discharge is one o f the factors causing changes i n the quant i ty o f water. M o r e recently, H i l m i et al. (2000) suggest that changes in sea levels are inf luenced by various factors, such as " t ida l mot ions, w i n d effects, atmospheric pressure, var ia t ion o f sea water density, r iver runof f , and (ocean) current system." A l t h o u g h an ocean current characterized by regular i ty as a cont inuous stream f l o w i n g along a definable path cou ld be t idal or non- t ida l , permanent or seasonal, hor izontal or ver t ica l , some indiv iduals argue that the ocean current system is associated w i t h water m o t i o n rather than the vert ical var iat ions i n sea levels (Thomson, 1981). I n general, the current system yie lds local f luctuat ions o f sea levels, except under the more extreme scenario o f E l N i n o . Therefore, E l N i n o is considered as a cr i t ical factor i n the est imat ion o f extreme sea levels and is discussed later i n this section. Recogniz ing the importance o f the changes o f physical propert ies and the quant i ty o f sea water, this section focuses on the generating mechanisms af fect ing the uneven d is t r ibut ion o f sea-level heights w i t h i n a basin, because these are of ten the causes o f extreme condit ions and thus are related to the est imat ion o f extreme sea levels. Sections 2.3.1 and 2.3.2 f i rst compare the differences in the c o m m o n terms used i n the scient i f ic and engineer ing classi f icat ion schemes concerning the generation mechanisms causing sea f luctuat ions. The generating mechanisms described in Sections 2.3.3 to 2.3.8 include w i n d waves, s torm surges, tsunami, t ides, E l N i n o , and c l imato log ic and geologic effects. A s waves approach the shores, the fo rward m o m e n t u m carries the waves up the coastal structures resul t ing i n wave run-up, w h i c h is described i n Sect ion 3.4. 20 2.3.1 Scientific Classification Scheme I n the scient i f ic context, sea levels are described as the super imposi t ion o f di f ferent types o f waves and seasonal and long- term f luctuat ions i n sea levels. Thus, the factors contr ibut ing to sea f luctuat ions can be classif ied into waves and non-waves. A s summar ized i n Figure 2.6, wave components inc lude w i n d waves, s torm surges, tsunami, and tides. A m o n g a l l wave components i n the scienti f ic context, the most signi f icant one af fect ing the degree o f var ia t ion i n sea level is the wave component generated by w i n d as wind- induced waves are accountable fo r approx imate ly 9 0 % o f the m a x i m u m wave heights achieved (Myers , 1969; O c h i , 1998). I n addi t ion, the non-wave components include E l N i n o , and c l imato log ic and geologic effects. Waves are def ined as an osci l la tory movement o f water o n or near the surface o f standing water. They ini t iate the osci l latory movement b y a disturbance appl ied to water, w i t h a restor ing mechanism fo rc ing the water toward equ i l i b r ium and f o r m i n g a d is tor t ion i n the opposite d i rect ion. The type o f disturbance and the type o f restor ing mechanism def ine the type o f waves, as summarized in Table 2.3. Table 2.3 presents the characteristics o f each mechanism: wave per iod (7 ) , wave length (Z) , wave height (H), and generating and restor ing forces. A wave per iod is the t ime that i t takes t w o successive crests to pass a f i xed po in t , and a wave length is the distance between t w o successive crests. A wave height is the di f ference i n surface elevat ion f r o m wave crest to t rough. Waves can be further classif ied in to deep-water waves, intermediate-water waves, and shal low-water waves, depending on the rat io o f water depth and wavelength. I f the total water depth (d) exceeds 2 5 % o f the wavelength, the waves are deep-water waves or short waves, such as w i n d waves. Deep-water or short waves are unaffected b y the bo t tom, so wave mot ions are ident ical to those i n in f in i te ly deep water. I f the water depth is between 2 5 % and 5% o f the wavelength, the waves are intermediate-water waves. Intermediate-water waves are part ly affected by the bo t tom topography. I f the water depth is less than 5 % o f the wave 21 length, the waves are considered to be shal low-water waves or long waves, such as tides tsunami. The i r characteristics are st rongly in f luenced b y var iat ions i n water depth and \ period. 2.3.2 Engineering Classification Scheme I n the engineering context, the effects on sea levels are composed o f waves and water level factors (see Figure 2.7). Here, waves have short wave periods o n the order o f seconds. Water levels affect the mean elevat ion o f the water w h e n averaged over a short per iod o f t ime, about one minute ( C E R C , 1984). Here, water levels have a l o w frequency o f occurrence, and are composed o f waves w i t h longer wave periods (i.e., minutes, hours, and days) and long- term sea level changes (i .e., seasonal and annual). The category o f "wa te r l eve l " includes s torm surges, tsunami, t ides, E l N i n o , and c l imato log ic and geologic effects. Water levels are c o m m o n l y measured by a s t i l l i ng -we l l type o f gauge, w h i c h is composed o f a f loat and an or i f ice. The f loat inside the st i l l w e l l moves upward or d o w n w a r d w i t h the water surface elevat ion, w i t h the movement recorded on a chart to prov ide a t ime series o f the observed water levels. The or i f ice at the bo t tom o f the w e l l serves to f i l te r out osci l lat ions hav ing a per iod smaller than one minute. A s discussed earl ier, f l ood levels are p r imar i l y composed o f t ides and storm surges, and do not include wave factors. Therefore f l o o d levels coincide w i t h the def in i t ion o f "water leve l , " and thus, the engineer ing classi f icat ion approach is more relevant for the design process. 22 2.3.3 Wind Waves A l t h o u g h waves m a y be generated b y various mechanisms, the most random f luctuat ion -o f the sea surface is general ly attr ibuted to w i n d waves. The generat ion o f w ind- induced waves is in i t ia ted b y the transfer o f energy from the w i n d to the water. Ripples on a ca lm water surface are the f irst s ign o f such energy transfer, and develop w h e n w i n d reaches a threshold veloci ty. Ripples or smal l w i n d waves are k n o w n as capi l lary waves, because the restor ing mechanism is the water tension. A s w i n d w i t h increasing strength b lows over an undisturbed distance for a per iod o f t ime, the waves g row higher and longer, and beg in to m o v e at greater speeds un t i l reaching a " f u l l y developed sea." A " f u l l y developed sea" is a cond i t ion i n w h i c h the rate o f energy input to the waves from the w i n d is balanced b y the rate o f energy dissipat ion due to wave breaking and surface water turbulence (Sorensen, 1997). W a v e break ing is a funct ion o f water depth fo r a g iven w i n d condi t ion, wave heights are smaller and wave periods are shorter i f generation takes place i n shal lower water ( C E R C , 1984). However , the cond i t ion o f " f u l l y developed sea" is not c o m m o n l y reached even in large storms. Th is m a x i m u m l i m i t depends on three factors: w i n d speed, w i n d durat ion, and fetch (Thomson, 1981 ; Sorensen, 1997). The w i n d durat ion is the length o f t ime the w i n d has been b l o w i n g at a part icular speed, and the fetch is an unobstructed distance over w h i c h the w i n d has been b l o w i n g i n the same di rect ion at the same speed. The w i n d fetch is important i n w i n d generation, because i t l im i ts the length o f t ime a group o f t ravel ing waves w i l l d i rect ly receive energy from a part icular w i n d system. A s the waves g row, the restor ing mechanism evolves f r o m water tension to grav i ty ; thus, the larger w i n d waves are cal led grav i ty waves. 23 2.3.3.1 Characteristics o f W i n d Waves The characteristics o f w i n d waves, w h i c h m a y be types o f capi l lary waves or grav i ty waves, can be found i n Table 2.3. The wave periods are in the range o f 0.1 to 30 sec, and their lengths are in the range o f 0.02 to 1000 m. In addi t ion, w i n d waves in the ocean can reach wave heights o f up to typ ica l l y near ly 3 m (Sorensen, 1997). W i n d waves are also a type o f deep-water wave, where the b o t t o m o f ocean does not affect the wave. The w i d e range o f periods, wavelengths, and heights contr ibute to the complex i t y o f oceans. 2.3.3.2 Statist ical Analys is o f Waves Analys is o f recorded or predicted data is used to describe and forecast w i n d waves or sea state. A sea state describes the force o f progressively h igher seas b y wave height, and is composed o f waves w i t h va ry ing height and per iod. W i n d is an essential element i n storm generation, thus wave statistics are also used to describe the severi ty o f s torm waves and extreme wave condi t ions. Wave data m a y be predicted or measured. Procedures for predic t ing waves are prov ided i n Section 2.3.3.3. Waves can be measured b y v isual observat ion, wave-r ider buoys, ship-borne accelerometers mounted in the hu l l , bo t tom-mounted pressure sensors, and more recently, satell ite laser alt imeters and over-the hor izon radar (Thomson, 1981). V isua l l y observed wave data are rout ine ly obtained from ships or f ish ing boats, but the data col lected are typ ica l ly not ve ry accurate. Instruments used to measure waves have higher accuracy in general, but they fa i l occasional ly dur ing major storms. Radar systems can also col lect direct ional in format ion . Waves are c o m m o n l y measured every three hours. Ana lys is o f wave records can be per formed in two ways: us ing a statistical approach or a wave spectrum analysis. The most 24 c o m m o n statistical indices, or parameters o f a sea state, are the s igni f icant wave height and peak wave per iod, and these m a y be der ived from either approach. The signi f icant wave height, denoted as Hs, is the average height o f the highest one-th i rd o f the observed or measured waves, and the peak wave per iod, denoted as Tp, is the wave per iod that corresponds to the m a x i m u m wave energy level i n the wave spectrum (Sorensen, 1997; Och i , 1998). 2.3.3.2.1 Statistical Analys is o f Wave Height and Per iod The statistical approach identi f ies the ind iv idua l waves i n the h is tory o f record and analyzes the statistical properties o f the height and periods o f the ind iv idua l waves. The ident i f icat ion o f waves from gauged wave record is c o m m o n l y conducted b y the zero up-crossing method, w h i c h is a standard method used to d iv ide a t ime series record o f waves into ind iv idua l components. The p r imary goal is to f i nd the appropriate d is t r ibut ion type for the probab i l i t y densi ty func t ion o f the wave height, p{H). Longuet -H igg ins (1952) state that the density func t ion o f waves is best def ined b y a Ray le igh Probabi l i t y D is t r ibu t ion (Sorensen, 1997; Och i , 1998). The Ray le igh Probabi l i ty D is t r ibu t ion can be wr i t t en as: where Hrms, the root mean square height, is a parameter o f the Ray le igh Probabi l i t y D is t r ibu t ion. The root mean square height is def ined as: rms 2.2 25 where Ht are the ind iv idua l wave heights i n a record conta in ing N waves (Sorensen, 1997; Och i , 1998). C o m b i n i n g the Ray le igh Probabi l i ty D is t r ibu t ion and the de f in i t ion o f the signi f icant wave height leads to the f o l l o w i n g representation o f s igni f icant wave height ( C E R C , 1984; Och i , 1998): Hs=\A\6Hrms*JlHrms 2.4 The height o f the most probable m a x i m u m wave in this part icular sea state, Hmax, can also be der ived b y the signi f icant wave height and the total number o f waves encountered, N, as described b y Equat ion 2.5 (Isaacson and Foschi , 2000). i 7 = / / . , - - - - ^ « 1 . 8 ~ 2 . 0 # . 2.5 N can be estimated b y the f o l l o w i n g equation: N = 3 6 0 0 — 2.6 77 where D is the durat ion i n hours, and T is the average wave per iod i n seconds. I f the most probable m a x i m u m wave height w i t h i n a storm is o f interest, D can represent the durat ion o f a storm in hours. For other cases, D represents the recording in terval i n hours. 2.3.3.2.2 Wave Spectrum Analys is o f Wave He igh t and Per iod A n alternate and more recent approach ut i l izes the wave spectrum o f the wave record based on a Fournier Analys is o f the record (Sorensen, 1997). Th is approach is di f ferent f r o m the 26 statistical approach, because the sea state is described in terms o f a spectrum o f wave energy. A spectrum o f wave energy is a p lo t o f the observed wave energy at each possible wave frequency against the wave energy, w h i c h is equal to the square o f wave ampl i tude. The total area under the spectral curve can be used to describe the degree o f sea severi ty and the total area is propor t ional to the total energy i n the wave f ie ld. The de f in i t ion o f s igni f icant wave height using the spectrum approach is ( C E R C , 1984; Och i , 1998): H, = 4 . 0 1 ^ 2.7 where mo is the area under the spectral density funct ion, and is def ined as ( C E R C , 1984; Och i , 1998): N 2 - o = Z ^ - 2.8 where a, is the ampl i tude o f wave /. Equat ion 2.7 can be used w i t h Equat ions 2.5 and 2.6 to estimate the most probably m a x i m u m wave height. 2.3.3.3 Wave Predic t ion There are two types o f wave predict ion: h indcast ing and forecasting. I n hindcast ing the predic t ion is based on past meteorological condi t ions, whereas i n forecast ing the predic t ion is based on predicted condit ions. A l t h o u g h the same procedures are used for bo th types o f wave predic t ion, the sources o f meteorological data are dif ferent. Several numer ica l models and parametr ic models for h indcast ing and forecasting exist each w i t h var ious computat ional and data requirements, and va l id i t y considerations. 27 The pred ic t ion o f w i n d waves can be accompl ished b y emp loy ing numer ica l models. These numer ica l models are developed to solve the equations described i n the Mi les-Ph i l l ips -Hasselmann Theory ( M P H Theory) . The M P H Theory developed b y M i l e s (1957) and Phi l l ips (1957), and mod i f i ed b y Hasselmann et al. (1973) describes the mechanism o f w i n d wave generation w i t h d i f ferent ia l equations. To solve these equations, the numer ica l methods require enormous computat ional t ime and meteorological data. S imp l i f i ca t ion o f the di f ferent ia l equations led to the development o f the s impl i f ied models, such as parametr ic models, to predict a smal l number o f parameters, such as wave height, wave per iod, and wave direct ion. A l t h o u g h the parametr ic models do not require a great deal o f computat ion t ime, they are va l id on ly under certain condi t ions, such as fe tch- l imi ted or dura t ion- l imi ted cases. The Sverdmp-Munk-Bretschneider ( S M B ) M e t h o d , developed b y Sverdrup and M u n k (1947) and mod i f i ed b y Bretschneider (1952), uses a set o f empir ica l monograms, such as that shown i n Figure 2.8, to prov ide a f irst order approx imat ion o f the signi f icant wave heights and periods based on the fetch length and durat ion o f w i n d . These empir ica l monograms are developed through f ie ld observations and analyses. For example, using Figure 2.8, the intersect ion o f the value o f a k n o w n fetch length located on the x-axis, and the value o f a k n o w n durat ion o f w i n d located on the y-axis is determined and used to approximate the signi f icant wave height and per iod. However , most monograms are restricted to deep water regions, because the S M B M e t h o d assumes that waves are generated under constant w i n d in deep or constant depth waters. Th is method remained the most w i d e l y used approach un t i l the J O N S W A P experiment (Sorensen, 1997). The JONSWAP-based curves inc lude results f r o m more recent tests and observations, and appear to be more accepted in the scienti f ic communi ty . However , neither results should be considered def in i t ive. A l t h o u g h the S M B curves have a long h is tory o f use, the JONSWAP-based curves have the advantage i n that the effects o f atmospheric stabi l i ty m a y be inc luded i n the calculat ion (V incent and Resio, 1990). 28 2.3.4 Storm Surges Storm surges are another wave component w h i c h cause f luc tuat ing sea levels and possess a close relat ionship w i t h w i n d . Section 2.3.4.1 discusses the generat ion o f s torm surges, Section 2.3.4.2 presents the characteristics o f storm surges, and Sect ion 2.3.4.3 investigates the numer ica l models o f s torm surges. 2.3.4.1 Generat ion o f Storm Surges Storm surges, the super-elevation o f the water surface, result f r o m the barometr ic pressure d i f ferent ia l and storm w i n d stress, w h i c h are the t w o condi t ions for a storm. Storm surges are a type o f in f ragrav i ty wave. W h e n storm surges approach the shore, the change in m o m e n t u m associated w i t h the breaking o f waves results i n a force that raises water levels at the shoreline. The extent to w h i c h water levels can rise depends on the in i t ia l water depth, the characteristics and behaviour o f the storm, the hydrography o f the basin, and the in i t ia l state o f the system ( C E R C , 1984). For example, since wave breaking is a phenomenon occurr ing when the water depth decreases, the effect o f s torm surges is considerable i n shal lower areas. Therefore, i n l o w - l y i n g areas, such as Ho l land , or the delta o f the Ganges R iver i n Bangladesh, there is extensive f lood ing caused b y storm surges, par t icu lar ly those that occur dur ing periods o f h igh astronomical tides (Thomson, 1981). 29 2.3.4.2 Characteristics o f S torm Surges A l t h o u g h the f requency o f occurrence o f surge events is less than that o f w ind- induced waves, the impact o f a storm surge is typ ica l l y greater than w ind- induced waves par t icu lar ly for l o w - l y i n g countries. A s indicated i n Table 2.3, for s torm surges the wave periods are i n the range o f minutes, and the wavelengths are in the range o f hundreds o f meters to hundreds o f k i lometres. The heights o f storm surges vary w i t h the pressure and w i n d speed. The Saff ir-S impson Hurr icane Damage Potential Scale relates variat ions i n pressure and w i n d speed to wave heights and potent ia l damage (see Table 2.4). A hurr icane is a s torm emanating i n the subtropics. I n general, the wave heights are less than 3 m i n deep water, but they can reach as h igh as 6 m wh i l e propagat ing into shal low areas. 2.3.4.3 Est imat ion o f Storm Surges Est imat ion o f the height o f a storm surge can be conducted i n several ways , using indirect measurement or numer ica l and empir ica l models. The empir ica l models are developed and va l id for regions near the N o r t h Sea only . The method o f indirect measurement requires bo th data o f observed sea levels and tides, whereas the numer ica l models require the use o f w i n d and pressure f ields. 2.3.4.3.1 Indirect Measurement o f S torm Surges Values o f observed sea levels, as ment ioned earlier, are the average o f the water levels after f i l te r ing out osci l lat ions w i t h a per iod smaller than a minute. A l t h o u g h observed sea levels 30 m a y be composed o f s torm surges, tsunami, and tides, tsunami usual ly destroy gauge stations and thus are rarely captured through measurement. Thus, the observed sea level general ly refers to tides and storm surges. Therefore, the t ime series o f s torm surges can be estimated b y the dif ference between observed water levels and predicted astronomical t ides, where the differences between water levels and astronomical tides are cal led residuals or residual water levels. G iven the t ime series o f residuals, design storm surges can be estimated f r o m statistical analyses, such as the Annua l M a x i m a Method . Unfor tunate ly , the instruments used to measure observed sea levels m a y not prov ide a rel iable measurement due to operat ional d i f f icu l t ies dur ing storms. Nonetheless, measured water levels are of ten used in combinat ion w i t h numer ica l models to estimate design s torm tides (Headland et al. , 2000). There is a p rob lem w i t h this approach when appl ied to data in semi-enclosed basins, such as bays and inlets, because semi-enclosed basins possess rap id and smal l sea-level changes, cal led seiches. Seiches are generated b y tides and local w inds . The effect o f seiches on sea levels is captured i n the measured observed sea levels. Thus, the di f ference between observed sea levels and tides i n such situations includes bo th storm surges and seiches. 2.3.4.3.2 Numer i ca l Mode ls o f S torm Surges Storm surges can also be estimated b y numer ica l models w h i c h simulate a hypothet ical design storm. Use o f numer ica l models of ten requires the knowledge o f the spatial and temporal d is t r ibut ion o f surface w i n d speed and di rect ion, the surface air pressure, the fo rward path, and the speed o f the design s torm (Sorensen, 1997). The models invo lve hydrodynamic cont inu i ty and m o t i o n equations, w h i c h describe the s torm surge generat ion process ( C E R C , 1984). I n general, the equations are approximated using a two-d imens iona l f in i te di f ference method. The 31 f in i te di f ference method f irst requires the construct ion o f a gr id system that covers the area to be modeled. E m p l o y i n g k n o w n input data and proper ly specif ied boundary condi t ions, the two hor izonta l f l o w components and the water surface elevat ion at successive t ime intervals are calculated at each g r id point . Several numer ica l models have been developed to analyze storm surges for di f ferent boundary condit ions, such as along the open coast, and in bays and estuaries. Some examples o f two-d imensional storm-surge models are l isted i n Table 2.5. Each model has its strengths and weaknesses, and the overal l c red ib i l i ty o f the mode l depends on the f o l l o w i n g elements: the consistency o f boundary constraints, the adequacy o f the resolut ion prov ided, and the ver i f i ca t ion o f the mode l b y the cal ibrat ion process (Reid, 1990). 2.3.5 Astronomical Tides Ast ronomica l tides are the alternate rise and fa l l o f the surface o f the sea due to the gravi tat ional attract ion between the sun, moon , and the earth. H i g h t ide is the highest point reached b y the sea, wh i l e l o w t ide is the lowest point that the water reaches as the sea surface fal ls. The t idal range is the dif ference between the height o f water at h igh and l o w tide i n any g iven place. As t ronomica l tides are important i n est imat ing the extreme sea level for the f o l l o w i n g reasons: 1) the range o f tides inf luences the mechanism o f water breaking at the near-shore area, and 2) the potent ia l impact o f a storm or tsunami m a y be very dependent on whether i t arrives at h igh or l o w t ide (Sorensen, 1997). A discussion o f the generation o f tides is prov ided i n Section 2 .3 .5 .1 , characteristics o f tides i n Section 2.3.5.2, measurement o f tides i n Section 2.3.5.3, and f ina l l y predic t ion o f tides i n Section 2.3.5.4. 32 2.3.5.1 Generat ion o f Tides As t ronomica l tides are waves generated b y the gravi tat ional attract ion o f the sun, m o o n , and other astronomical bodies act ing on the oceans. The strength o f the gravi tat ional force is inversely related to the square o f the distance between t w o objects. Thus, the variat ions i n distances between the sun and earth, and the m o o n and earth f r o m the rotat ion o f the earth-sun-m o o n system y ie ld the dif ference i n the strength o f the gravi tat ional force and the heights o f t idal waves. There are over 390 active t idal components based on the di f ferent combinat ions o f gravi tat ional forces between the sun and earth, and the m o o n and earth (Sorensen, 1997). For example, a lunar component is w h e n a po in t on the earth completes the f u l l ro tat ion to the same pos i t ion relat ive to the m o o n , where the t ime for the rotat ion or the per iod for a lunar cycle is 24 hours and 50 minutes. The various components combine i n d i f ferent ways at each coastal locat ion. The resultant t idal waves repeat themselves based on the di f ferent combinat ions. Thei r regular i ty makes t idal wave predic t ion stra ight- forward. 2.3.5.2 Characteristics o f Tides The characteristics o f tides are closely related to the active t ida l components, as each o f the 390 active t idal components has a per iod, phase angle, and ampl i tude. The variat ions in components y ie ld t ida l waves hav ing periods ranging f r o m 8 hours to 18.61 years. I n general, the periods are typ ica l l y 12 hours, or 24 hours and 50 minutes w h i c h correspond to the pr inc ipa l periods o f the solar and lunar cycles, respect ively (Thomson, 1981 ; Sorensen, 1997). The wavelengths can be thousands o f k i lometres. The wave heights are usual ly l o w as shown in Table 2.3. However , at some locations, the wave heights can be as h i g h as 6 m, because long 33 waves are s t rongly affected b y variat ions in sea depth. T ida l waves can also be categorized b y the number o f cycles o f the t idal wave per day. W h e n a t idal wave has one cycle or one crest and one t rough per day i t is cal led d iurnal , wh i l e a t idal wave hav ing t w o cycles per day is cal led semidiurnal . 2.3.5.3 Measurements o f Tides and Establ ishment o f D a t u m The purpose o f measuring t idal levels is to establish datum levels and to predict magnitudes o f tides. Tides are c o m m o n l y measured b y t ida l gauges (Thomson, 1981). I n Canada, the Canadian Hydrograph ic Service (CHS) is responsible for t ide data col lect ion. The C H S chooses the coastal locations for gauge stations, collects t ida l data, and produces the Canadian T ide and Current Table. A n ideal gauge station should experience the fu l l t idal range, should not be exposed to strong currents or waves, and should also be strategical ly located to aid mar ine navigat ion. Tides are measured b y t idal gauges. The types o f gauges inc lude the s t i l l ing w e l l , pressure gauge, and satellite al t imetry. These instruments are also used to measure observed sea levels. The s t i l l ing w e l l is the t radi t ional method o f measurement described i n Section 2.3.2. S t i l l ing we l ls are robust and re lat ive ly s imple to operate, but have some disadvantages, w h i c h include 1) h igh costs, 2) the need for a vert ical structure for m o u n t i n g the st i l l w e l l over deep water (i.e., adjacent to a c l i f f ) , and 3) the suscept ibi l i ty to erroneous readings (Pugh, 1987). The errors invo lved i n us ing s t i l l ing wel ls are due to swel l waves and par t ia l p lugg ing b y debris and l i v i ng organisms, l i m i t i n g accuracy to approx imate ly 0.02 m for water levels and t w o minutes for t ime (Thomson, 1981 ; Pugh, 1987). Thus, pressure gauges have been developed to measure t idal levels b y measuring the pressure at some f i xed po in t be low the sea surface and conver t ing 34 the pressure to water level . The advantages o f pressure gauges inc lude the ab i l i t y to be placed at of fshore locations, the lack o f suscept ibi l i ty o f records to be affected b y w i n d waves, the ease o f handl ing and the comparat ive ly good accuracy (Thomson, 1981). The m a i n disadvantages o f pressure gauges are 1) their suscept ibi l i ty to undesired dynamic pressure effects due to currents that f l o w past the or i f ice, and 2) their l imi tat ions due to the accuracy o f the pressure sensing transducer (Pugh, 1987). Satell i te a l t imetry can cover a large area ve ry rap id ly , but the results are d i f f i cu l t to interpret i n terms o f t ide levels (Pugh, 1987). A datum is used as the reference level for depths p lo t ted on naut ical charts and for descr ibing sea levels. Var ious types o f da tum exist, such as the chart ( C H A R ) datum and geodetic (GSC) datum. The C H A R datum is the reference fo r measur ing the height o f the tides and is the lowest water level dur ing normal tides. Consequently, the C H A R datum is subject to change i f the lowest water level is changed. The GSC datum is a f i xed datum chosen b y local authorit ies. 2.3.5.4 Predic t ion o f As t ronomica l Tides Due to the regular i ty i n the rotat ion o f the sun, m o o n , and earth system, the astronomical tides can be predicted w i t h a h igh accuracy. Predict ion o f astronomical tides f irst requires cont inuous observations for a year or more i n order to accurately estimate the t imes and heights o f h igh and l o w tides. The observed record o f sea level is then analyzed and decomposed into constituents b y harmonic analysis. The basis o f harmonic analysis is the assumption that the t idal variat ions can be represented b y a f in i te number o f harmonic terms or constituents. Each constituent or component has a specif ied ampl i tude, cyc l ic per iod, angular speed, and phase lag. 35 U p o n the determinat ion o f these parameters, the overal l equat ion o f the height o f tides can be expressed, as i n Equat ion 2.9 ( T a w n and Vassie, 1989): where At is the ampl i tude o f constituent i, Oi is the angular speed o f consti tuent i, Vi is the equ i l i b r ium t ida l phase at ^=0 for constituent i, gv is the phase lag o f constituents, and/7 and u, are the corresponding nodal corrections o f constituent i. The summat ion i n Equat ion 2.9 is over the number o f constituents, M, that can be resolved w i t h i n the length o f data col lected (Tawn, 1992; Ts imp l is and B lackman, 1997). The values o f Vi, oj , and f( are usual ly k n o w n , and At and g, are estimated using at least one year o f sea level data (Pugh and Vassie, 1980). Once the parameters have been estimated, hour ly t idal levels can be generated for the f u l l nodal cycle o f 18.61 years, because the t idal sequence is a determinist ic per iodic func t ion ( T a w n and Vassie, 1989; T a w n , 1992). 2.3.6 T s u n a m i Tsunami is a Japanese w o r d for harbour wave. A l t h o u g h the f requency o f occurrence o f tsunami is very smal l , tsunami have the potent ial for causing great destruct ion as discussed in Section 2.1 ( Impact o f Sea Floods). Therefore, tsunami are general ly not considered in the design o f coastal protect ion systems, unless the coastal systems are designated for cr i t ical faci l i t ies, such as bridges or power plants (Headland et al., 2000) . Sect ion 2.3.6.1 discusses the generation o f tsunami, and Section 2.3.6.2 presents the characteristics o f tsunami. M 2.9 36 2.3.6.1 Generat ion o f Tsunami Tsunami are generated b y underwater disturbances, such as faul t earthquakes, landslides, and volcanic eruptions. Tsunami generated b y landslides and volcanic eruptions on ly affect the areas near the source, whereas tsunami generated b y earthquakes can travel across an ocean basin (Camf ie ld , 1990). The most c o m m o n cause o f tsunami is earthquakes, but not al l earthquakes result i n tsunami. The condi t ions under w h i c h earthquakes generate tsunami include: 1) the earthquake is generated beneath the sea f loor and has a magni tude o f greater than 6.5 on the Richter scale, 2) the center o f the earthquake is not deeper than approx imate ly 100 k m be low the seabed, and 3) the earthquake causes an upward or d o w n w a r d displacement (Thomson, 1981 ; Sorensen, 1997). The th i rd cond i t ion is essential, because on l y an upward or downward displacement can cause d is tor t ion o f the ocean surface. The ma jo r i t y o f earthquakes in the Paci f ic Ocean cause a sideways s l ipp ing o f the sea f loor rather than an upward or downward displacement, so most earthquakes i n the Paci f ic Ocean do not result in tsunami. H a v i n g said this, 62 % o f al l tsunami occur i n the Paci f ic Ocean (Thomson, 1981). Waves generated b y tsunami are s imi lar to the waves generated b y d ropp ing a rock in a pond, i n w h i c h the waves m o v e ou tward f r o m the source region. I n general, wave heights generated b y tsunami decrease, but the number o f waves increase w i t h distance from the source region, because the energy level decreases as the distance f r o m the source region increases ( C E R C , 1984). The energy level is lost to the f r i c t ion caused b y " rubb ing against the b o t t o m " or natural obstacles, such as straits, passages, islands, and shoals. 37 2.3.6.2 Characteristics o f Tsunami The characteristics o f tsunami are discussed f r o m the po in t o f v i e w o f two scenarios: or ig inat ing i n condi t ions o f deep water, and propagat ion near the shore. W h e n a tsunami is generated in the ocean or deep water, i t consists o f a series o f waves that spread rap id ly away from their source where speeds can be over 900 k m / h (Thomson, 1980). The height o f a tsunami at sea is always smal l , rare ly exceeding 1 m, hav ing a wave length o f t yp ica l l y i n the range o f 100 to 400 k m (Thomson, 1980). Due to the characteristics o f long waves, the tsunami decelerates as i t approaches shal low areas, w h i l e the speed o f the tsunami is d i rect ional ly propor t ional to the depth o f water. Tsunami arrive as a series o f waves hav ing periods ranging from 10 to 60 minutes (Abbot t , 1999). Some waves m a y cause no th ing more than a series o f gentle rises and fal ls. The height o f tsunami can increase to as h igh as 10 m wh i l e approaching shal low areas due to shoal ing, refract ion, and resonance. The wave can be ampl i f ied dramat ical ly , par t icu lar ly w h e n the natural resonant frequency matches that o f the tsunami. Before a large tsunami is generated near the shore, the w i thd rawa l o f water from the beach is usual ly observed. Th is phenomenon is caused b y the large wave length. A s i t t yp ica l ly takes 10 to 50 m i n for a successive crest to pass a g iven po in t w i t h the g iven speed and wave length, of ten a t rough reaches the shore f irst, thus, creating the w i thd rawa l o f water. 2.3.7 El N i n o E l N i n o is an inter-annual event o f an ocean current car ry ing an i rregular f l o w o f unusual ly w a r m surface water. U p w e l l i n g along the Paci f ic Coast, where the co ld water f r o m the subsurface moves upward to the surface, occurs throughout the year serv ing to cool the ocean 38 water, cool the air, moderate the coastal temperatures and humid i t y , and b r i n g deep, nutr ient- r ich water upward (Thomson, 1984). However , dur ing an E l N i n o event, upwe l l i ng ceases near northern Peru and causes " a tongue o f the w a r m eastward- f lowing equatorial current to push southward about 6° S (Thomson, 1984). Because this occurs dur ing the Christmas season, this event is cal led E l N i n o meaning "Chr is t C h i l d . " A n E l N i n o event leads to a breakdown o f the trade w inds system over the equatorial Paci f ic Ocean and causes an abnormal ly h igh ra infa l l . The c l imat ic effects o f large-scale E l N i n o disturbances also cause f l ood ing (ma in ly due to excessive ra infa l l ) and drought condi t ions over a w ide area, sometimes extending as far as the Southern Paci f ic Ocean, Europe, A f r i c a , and As ia . The w a r m and nutr ient-poor waters cause great ecological damage. Such disturbances have taken place i n 1953, 1957-58, 1972-73, 1976, 1982-83, and 1992 (Thomson, 1984; N o r t h Amer i can Lake Management Society Websi te) . I t is bel ieved that E l N i n o events enhance the sea level var iab i l i t y for var ious durat ions depending on the locat ion relat ive to the equator (Subbot ina et al. , 1994). A l t h o u g h the impact o f E l N i n o on the sea levels is st i l l subject to ongoing research, i t is not general ly considered i n the design o f coastal protect ion systems. 2.3.8 C l i m a t o l o g i c a n d Geolog ic Ef fects W h i l e other components prov ide an explanat ion for short- term changes in sea levels, on a global level , the mean sea level is observed to be changing relat ive to land levels over a long per iod o f t ime. Sea level rise is important i n the design o f coastal structures, because i t increases the r isks o f ove r f l ow ing and over topping (Pi larczyk, 2000). Fur thermore, the sea level rise can also impact the shore through erosion w h i c h leads to an instabi l i ty o f coastlines. Thus, there is an increased need to inc lude the effect o f the long- term sea level rise i n the design o f coastal 39 structures, such as in the est imat ion o f extreme sea levels. T w o causes o f long- term sea level changes are c l imato log ic and geologic effects. The c l imato log ic effects are m a i n l y caused b y global w a r m i n g that results i n expansion o f sea water and me l t i ng o f glaciers. The trend o f sea-level rise is w e l l documented in the Un i ted States, w i t h research showing that the sea level rise dur ing the latest decades ranges f r o m 1 to 4 m m per year depending o n the locat ion (Bruun , 1989). For example, the East Coast o f the U S A has an average rate o f sea level rise o f 2 to 3.5 m m per year, whereas the G u l f Coast has an average rise o f 2 to 3 m m per year (Bruun , 1989). Other studies also indicate that the sea level rise on the Paci f ic Ocean dur ing the 2 0 t h Century is estimated to be at the rate o f 2 m m per year (Bruun , 1989; Church, 2002). Geological effects are due to possible tectonic up l i f t or subsidence o f the coast, where an up l i f t m o t i o n w i l l increase the relat ive sea level , and a subsidence m o t i o n w i l l decrease the sea level . The geological effects on sea level thus depend on the locat ion o f the coast and the possible coastal movements. 40 Table 2.1 Sea F lood His tor ica l Events Da te Place D e s c r i p t i o n D a m a g e 1228 Friesland, the Netherlands Sea f l ood i n Friesland 100,000 l ives lost 1282 Zuyder Zee, the Netherlands A great s torm broke th rough the natural dykes, and the N o r t h Sea flooded into the area i n one day. N / A Oct. 11, 1634 Northsea Coast, the Netherlands A storm destroyed the coastl ine o f N o r t h Friesland, and the flood accompanying this s torm swept away complete vi l lages. 15,000 l ives lost Oct. 31, 1876 Backergunge, Ind ia T ida l waves caused by a cyc lone flooded the r iver delta, and some areas were covered by 40 f t o f water. 100,000 l ives lost June 15, 1896 Sanr iku, Japan A n undersea earthquake about 100 mi les offshore t r iggered a t ida l wave 80 f t h igh (Abbot t , 1999). 27,000 l ives lost Sept. 8, 1900 Galveston, Texas, U S A A hurr icane w i t h a w i n d speed o f 120 m p h raised the water level a long the shore o f the G u l f o f M e x i c o 15 ft above the usual 2 ft o f the t ida l range ( A b b o t t 1999). 7,200 l ives lost June 23, 1946 Vancouver Is land, B r i t i sh Co lumb ia , Canada A n earthquake on the east coast o f Vancouver Is land caused the water level at F rank l in River i n A l b e r n i In let to rise 6.1 to 9.1 m above average. N / A Jan. 31 -Feb. 5, 1953 Nor thwest Europe A hurr icane and h igh tides f o l l o w e d by f loods led to dyke fa i lure and devastated N o r t h Sea coastal areas. The storm surge rose 3 to 3.5 m above the normal h igh water; the waves topped over the dykes (Pi larczyk, 2000). 1,850 people l ives lost and over $250 m i l l i o n o f damages Aug., 1954 Providence, Rhode Is land, U S A Hurr icane Carol raised the water level to 16 ft above normal level . Property damages o f $41 m i l l i o n 1960 T o f i n o (Vancouver Is land), B r i t i sh Co lumbia , Canada A n earthquake o f magni tude o f 9.5 i n Chi le caused a 1.2-m wave at To f ino . Wave run-up was higher i n many areas. N / A Oct. 31, 1960 Chit tagong, Pakistan A cyclone and storm surge caused m u c h damage. Thousands o f l ives and houses were lost Feb. 16, 1962 H a m b u r g and western coastal areas, Germany H i g h t ide and strong w i n d condi t ions led to the fai lure o f the levee system. 315 l ives lost Mar. 27, 1964 Br i t i sh Co lumb ia , Canada A n earthquake o f magni tude 8.5 i n A laska caused a 2 .4-m wave at Property damages o f $8.4 m i l l i o n 41 Tof ino . Wave run-up was higher i n many areas. Wave ampl i tude i n Port A l b e r n i was over 6 m (Thomson, 1981). Aug.,, 1969 Miss iss ipp i , U S A Hurr icane Cami l le caused the storm surge that reached a m a x i m u m o f 6.9 m above mean sea level (Sorensen, 1997; Abbot t , 1999). 256 l ives lost; property damage o f $1 b i l l i o n Nov. 13, 1970 East Pakistan A cyc lone-dr iven t ida l wave f r o m the Bay o f Bengal led to f lood ing . 200,000 k i l l ed ; over 100,000 miss ing Sept., 1972 At lan t ic Seaboard Remnants o f Hurr icane Agnes caused f lash f lood ing . 134 l ives lost; property damage o f $3 m i l l i o n Feb., 1976 Bay o f F u n d y , N S and N B , Canada Hurr icane force w inds accompanied the Groundhog Day storm. Tens o f m i l l i ons o f damages Feb. 27-March 5, 1983 Cal i fo rn ia , U S A Pacif ic storms caused severe f l ood ing along the Ca l i fo rn ia coast. 13 l ives lost; property damage estimated at $200 m i l l i o n Sept. 17-21, 1989 Caribbean Sea and the U S A East Coast Hurr icane H u g o brought a storm surge o f 5.2 m in height and caused f lash f lood ing (Abbot t , 1999). 11 l ives lost; property damage about $7 b i l l ions o f dollars April 30, 1991 Bangladesh, Ind ia 145 m p h cyclone struck coastal areas and offshore islands causing 6-meter waves. 150,000 k i l l ed ; property damage about $1 b i l l i o n Nov. 5 - 7, 1992 The Phi l ippines Trop ica l s torm The lma caused 3-m f loodwaters on Leytes, Samar, and Negros Islands. A b o u t 3,400 k i l l e d ; 50,000 homeless Jul. 17, 1998 Papua N e w Guinea Three tsunami, possibly spurred by an undersea landslide f o l l o w i n g an earthquake, destroyed entire vi l lages i n the northwest prov ince o f Sepik. A t least 2,000 found or presumed dead September, 1999 M a r i t i m e Provinces, Canada Trop ica l s torm Harvey and Hurr icane Gert f looded O x f o r d , N o v a Scotia. Property damage o f 12 m i l l i o n . Jan., 2000 M a r i t i m e Provinces, Canada A n intense win ter storm occurred i n the Mar i t imes dur ing a run o f very h igh tides. The storm surge caused coastal f l ood ing i n the G u l f o f St. Lawrence af fect ing P E I , N B , and N S . $20 m i l l i o n o f damages March, 2000 Prince Edward Island and the G u l f o f St. Lawrence coast o f N e w Brunswick , Canada The remains o f a t ropical s torm h i t the G u l f o f the St. Lawrence w i t h w i n d gusts o f 120km/h causing ocean waves o f 7 to 11 m and massive damage to coastal infrastructure. N / A 42 Table 2.2 Design Considerations and Critical Modes of Failure of Revetments from Pilarczyk (2000) Type of cover layer Critical failure mode Determinant wave loading Strength Sand/gravel Initiation of motion Transportation of material Profile formation Velocity field in waves Weight, friction Dynamic "statlibty" Clay/grass Erosion Deformation Maximum velocity Wave impact Cohesion Grass roots Quality of clay Riprap Initiation of motion Deformation Maximum velocity Seepage Weight Friction Permeability of sub-layer/core Gabions and sand, or cement ' mattresses, including geotextiles Initiation of motion Deformation Rocking Abrasion/corrosion of wires U . V . light Maximum velocity Wave impact Climate Vandalism Weight Blocking Wires Large unit Permeability including sub-layer Placed blocks, including block mats Lifting Bending Deformation Sliding Overpressure Wave impact Thickness, friction, interlocking Permeability, including sub-layer/ geotextile Cabling/pins Asphalt Erosion Deformation Lifting Maximum velocity Wave impact Overpressure Mechanical strength Weight 43 Table 2.3 Characteristics o f Waves f r o m Thomson (1981) and Sorensen (1997) Type Period Wavelengths Wave Heights Generating Mechanisms Restoring Mechanisms Capillary Waves (ripples, wavelets) Less than 0.1 s Less than 2 cm Less than 10 cm Wind, pressure fluctuations Surface tension " CJL . — Gravity waves (chop, sea, swell) 0.5 s to 30 s 10cm to 1000 m Over 10 cm and less than 3 m Wind Gravity ' r—_ £ — Infragravity waves On the order of minutes Hundreds of meters to hundreds of kilometers Typically less than 3 m but can exceed 6 m during significant storms Storm system (winds and atmospheric pressure gradients) Gravity Tsunami 5 min to more than 1 h Hundreds of kilometers Less than 1 m at sea, but can increase to as high as 10 m when approaching shallow areas Volcanic activity, landslides, and tectonic uplifting Gravity Tides Mainly 12 '/2 h and 25 h Thousands of kilometers Low height in deep ocean, but up to 6 m in some locations Gravitational attraction of sun and moon Gravity and Coriolis force Table 2.4 Saf f i r -S impson Hurr icane Damage Potential Scale f r o m A b b o t t (1999) Barometric Pressure Wind Speed (mph) Storm Surges (m) Damages Category 1 Over 28.94 i n 7 4 - 9 5 1 . 2 2 - 1 . 5 2 M i n i m a l 2 td. Category 2 2 8 . 5 0 - 2 8 . 9 3 9 6 - 1 1 0 1 . 8 3 - 2 . 4 4 Moderate Category 3 27.91 - 2 8 . 4 9 1 1 1 - 1 3 0 2 . 7 4 - 3 . 6 6 Extensive 2 tL Category 4 2 7 . 1 7 - 2 7 . 9 0 131 - 155 3 . 9 7 - 5 . 4 7 Ext reme 2 tL . Category 5 Less than 27.17 Over 155 Over 5.47 Catastrophic Table 2.5 Storm Surge Mode ls Developed for Use i n the Publ ic D o m a i n f r o m Re id (1990) Code Name Originator Agency User SPLASH Jelesnianski Nat iona l Weather Service SLOSH Jelesenianski and Chen Nat iona l Weather Service SURGE II Reid et al. Coastal Engineer ing Research Center SSURGE Wanstrath Waterways Exper iment Stat ion CESSM Tetra Tech Federal Emergency Management Agency WIFM But ler Waterways Exper iment Stat ion 44 27 28 M A R C H 29 J ' i i J I I I I I I i 1 I 1 1 I t Figure 2.1 Sea Leve l Record for M a r c h , 1964 F r o m Thomson (1981) 45 Wave wall leach Oceon Csp stone 92 to683-Kg if the Mit»mg;6w«h surface is n u » M | W shall!» required »e place the ocean 0,6-m- '* side to« ot Ei h5-m ' Eievdtion- vanes occording to beoch surface. 0.1-m — J 1.5. 0 , 3 - i M Core materia! 92-kg to chips m i n . 2 5 % > 2 0 - k g Note Where wells exist modify section by omittmq rock on lonrjside Figure 2.2 Rubble M o u n d Seawall F r o m C E R C (1984) Figure 2.3 Quarrystone Revetment F r o m C E R C (1984) 46 ARMOR STONE 2 LAYERS W 3 0 0 0 IE Ds - 2.5 FT 1 ,... BASELINE OF CONSTRUCTION \ ,?, ; / „ . 8" CRUSHED STONE ROADWAY Figure 2.4 Typ ica l D y k e Section F r o m Headland et al. (2000) G a t h e r i n f o r m a t i o n requ i red fo r d e s i g n Assessment of revetment material options Genera-tion of alterna-tive de-signs Design of initial cross-sect ion Design Detailed design Construction aspects Revetment function "111 1 1 Revetment performance - - - - - j . - I f data Hydraulic conditions j-* a v2i i -_ — -j. a b ) e Geotechnical conditions 1 Geometry j A s s e s s wave conditions & water levels JT. Constraints Derivation of water levels Prediction of waves at coastal j locations/on inland waters i Slope Crest elevation [ w Cover layer thickness i 1 ! Overtopping & scour i - *-( Material-specific design methods i Optimization & selection of final option Filter l l ' - • I Toe protection i>l Crest protection I l l I —w Preparation of slope, tolerances, specifications Inspection, maintenance & repair Figure 2.5 F lowchar t o f Design Process for Revetments F r o m A l l sop and M c C o n n e l l (2000) 47 Sea Fluctuations I Figure 2.6 Scient i f ic Classi f icat ion o f Factors A f f e c t i n g Sea Leve l Fluctuat ions Storm Surges Tsunami Sea Fluctuations Tides E l N i n o C l imato log ic / Geologic Effects Figure 2.7 Engineer ing Classi f icat ion o f Factors A f f e c t i n g Sea Leve l Fluctuat ions 48 49 3 QUANTITATIVE METHODS FOR ESTIMATING EXTREME CONDITIONS Estimates o f extreme condit ions used i n the design o f coastal structures require quant i f icat ion o f the extreme f l ood levels and wave condi t ions associated w i t h a design probab i l i t y and determinat ion o f wave run-up and over topp ing condi t ions. Th is Chapter discusses the quanti tat ive methods for est imating extreme condi t ions. Sections 3.1 and 3.2 discuss the methods for est imating extreme sea f loods and extreme wave condit ions, respectively. Section 3.3 describes the j o i n t p robab i l i t y o f f l ood levels and extreme wave condi t ions that occur i n nature, w h i c h is essential for est imat ing extreme sea levels that include wave effects. F ina l ly , Section 3.4 provides the methods for est imat ing wave run-up and over topping condit ions. 3.1 Methods for Estimating Flood Levels Methods for est imating f lood levels range from statistical to determinist ic approaches. Statist ical analysis is the convent ional method for determining an extreme f l ood level ; however, due to the assumptions used i n the statistical methods, such an approach is usual ly l im i ted b y the avai labi l i ty o f long- term data. The statistical methods include the A n n u a l M a x i m a Me thod and A n n u a l /^-largest M a x i m a Me thod , w h i c h are described i n Sections 3.1.2 and 3.1.3, respectively. These statistical approaches require long- term data, and a c o m m o n p rob lem encountered b y coastal engineers is the lack o f such data, par t icu lar ly for sea levels and waves. In addi t ion, the statistical approaches consider f l ood levels as a single independent var iable, but f l ood levels are 50 actual ly in f luenced b y a range o f factors, such as tides and storm surges. The determinist ic approaches incorporate s imp l i f ied representations o f sea condi t ions that are composed o f on ly the cr i t ical components that contr ibute to sea level f luctuat ions. These approaches require data col lected for a re la t ive ly short per iod o f t ime, but the complexi t ies o f f l ood levels may not be f u l l y characterized. The Simple A d d i t i o n M e t h o d introduced i n Section 3.1.4 is a determinist ic approach that does not consider the probab i l i t y associated w i t h each component. The Joint Probabi l i ty M e t h o d (JPM) introduced i n Section 3.1.5 incorporates the components that affect sea level f luctuat ions, and the probabi l i ty associated w i t h the var ious components. The most successful appl ications o f the J P M have been for t ida l ly dominant sites. The Revised Joint Probabi l i ty M e t h o d (RJPM) presented i n Section 3.1.6 was developed to overcome the restr ict ions o f the J P M . Before presenting these methods, the concept o f a probab i l i t y o f exceedance and return per iod associated w i t h an extreme cond i t ion is in t roduced i n Section 3.1.1. 3.1.1 Probability of Exceedance and Return Period The m a i n object ive o f the extreme analysis o f f l ood levels or wave condi t ions is to determine the future probab i l i t y o f the occurrence o f an event. The future probab i l i t y o f occurrence o f an event can be characterized b y the probab i l i t y o f exceedance (p), w h i c h is the probab i l i t y o f a f l ood level being exceeded in any one t ime per iod, or the return per iod (TR), w h i c h is the average number o f t ime periods between occurrence o f events equal to or greater than a g iven f lood level . TR is then the reciprocal o f p. 51 To obta in p and TR, a l low J[x) to be the probab i l i t y densi ty func t ion for the variable o f interest (x). Then F(x) is the cumulat ive d is t r ibut ion func t ion for this variable. The probabi l i ty that x w i l l not exceed an extreme event (z) i n any one t ime per iod as def ined by: z F(z)= \f(x)dx 3.1 -co \ The probab i l i t y o f z be ing exceeded i n any given year, or the probab i l i t y o f exceedance (p(z)) is described as: CO p(z) = 1 - F(z) = \f(x)dx 3.2 z I n addi t ion, the return per iod o f z, w h i c h is denoted as TR(Z), is the reciprocal o f p(z) (Pugh and Vassie, 1980, Wat t et al. , 1989): T ( Z ) = _ L = 3.3 p(z) l - F ( z ) In this thesis, the variable o f interest, x, cou ld represent either sea level or wave height. The value z therefore represents the extreme f lood level or extreme wave height. The magni tude o f the extreme f lood or extreme wave condi t ion, z, corresponds to a part icular exceedance probab i l i t y (p(z)) and return per iod (TR(z)). 3.1.2 Annual Maxima Method The A n n u a l M a x i m a M e t h o d is a frequency analysis o f the histor ical f l ood levels. This section presents the appl icat ion, data requirements, assumptions, and advantages and disadvantages o f this method. I t is important to keep i n m i n d the poss ib i l i ty that the basic f lood 52 data be ing processed m a y contain substantial errors resul t ing f r o m human fai l ings and the d i f f i cu l t y o f measur ing h igh f loods par t icu lar ly dur ing extreme condi t ions (Wat t et al. , 1989). 3.1.2.1 App l i ca t i on The appl icat ion o f the Annua l M a x i m a M e t h o d involves the f o l l o w i n g steps: selecting a sample in the f o r m o f an available data series, f i t t ing a theoret ical p robab i l i t y d is t r ibut ion to the sample, and determin ing the extreme f lood levels based on the theoretical p robab i l i t y d is t r ibut ion. I n the analysis o f h istor ical f l ood levels, the m a x i m u m value o f hou r l y sea levels for each year is f i rst ident i f ied and extracted from the or ig ina l set o f sea level data to f o r m the m a x i m u m annual series. The m a x i m u m annual series is then screened for miss ing data and outl iers. M iss ing data series are categorized into broken records and incomplete records. A broken record is general ly due to f inancia l or manpower constraints, and is not related to the occurrence o f an event, thus " the di f ferent record segments should no rma l l y be combined and treated as cont inuous record" (Wat t et al., 1989). A n incomplete record m a y result from extreme events, and may indicate an inaccurate recording o f an extreme event. Therefore, miss ing data should be estimated i n a manner consistent w i t h the data col lect ion agency (Wat t et al. , 1989). The annual m a x i m a series should also be screened for outl iers, w h i c h refer to observations i n a data set w h i c h depart s ign i f icant ly f r o m the m a i n trend o f the data. A n out l ier cou ld be the result o f an error i n measurement, i n w h i c h case i t w i l l distort the interpretat ion o f the data. Env i ronment Canada recommends that "no special measures be adopted to ident i fy and treat outl iers unless one or more very unusual event o f a special nature are k n o w n to have occur red" (Wat t et al., 1989). 53 The screened annual m a x i m a series is ranked in descending order according to observed sea levels. The probab i l i t y o f exceedance is estimated for each data po in t b y app ly ing p lo t t ing pos i t ion fo rmula . Examples o f the p lo t t ing pos i t ion fo rmu la are shown in Table 3 . 1 , and the most c o m m o n l y used fo rmu la are the W e i b u l l and Cunnane. A s indicated in Table 3 .1 , the empir ica l p robab i l i t y o f exceedance or p lo t t ing posi t ion, pm, is a func t ion o f the number o f data points (N) and the rank o f the data value (mn). Each data value represents an extreme water level w i t h an empir ica l p robab i l i t y o f exceedance w h i c h is then used to f i t a d is t r ibut ion type. T w o approaches for de f in ing the theoretical d is t r ibut ion are used for annual m a x i m a series. The t radi t ional approach plots data points on di f ferent types o f d is t r ibut ion paper, where the return per iod is p lo t ted on the x-axis and the water level is p lot ted on the y-axis. A l inear t rend is determined among the data points and the d is t r ibut ion that is most appropriate is selected based on the best-f i t l inear funct ion. Theoret ical d istr ibut ions can also be determined using the parameters o f the d is t r ibut ion types, and examples o f the d is t r ibut ion types c o m m o n l y used are the L o g - N o r m a l ( L N ) , and Extreme Value ( E V ) Dis t r ibut ions inc lud ing the General ized Extreme Va lue ( G E V ) , Ext reme Value Type I or Gumbe l ( E V I ) , and Ext reme Va lue Type I I I or W e i b u l l ( E V I I I ) Dis t r ibut ions. 3.1.2.2 Data Requirements The A n n u a l M a x i m a Me thod requires a long record o f observed sea levels. W h i l e there is no standard for the m i n i m u m length o f data set for a reasonable est imat ion, two sources o f l i terature prov ide some guidance on this matter. Pugh and Vassie (1980) suggest that " the number o f years o f data required is not expl ic i t , but estimates based on fewer than 25 years o f data are un l i ke l y to be re l iable." Viessman, and Lewis (1996) also suggest that the required 54 length o f the data set depends on the projected quanti le, where the "ex t rapo la t ion using any o f the graphical aids is not recommended beyond t w o t imes the per iod o f reco rd " (Viessman, and Lewis , 1996). Therefore, g iven the general practice o f coastal engineering that requires the est imat ion o f extreme f lood levels hav ing a m i n i m u m return per iod o f 200 years, using the latter rule, the data col lect ion required for the return per iod should be at least 100 years durat ion. 3.1.2.3 Assumpt ions The appl icat ion o f the Annua l M a x i m a M e t h o d requires several assumptions regarding the statistical characteristics o f the data, the length o f the data set, and the theoretical d is t r ibut ion. The statistical characteristics include randomness, independence, non-homogenei ty and stat ionari ty (Wat t et al. , 1989). For randomness to ho ld the f luctuat ions o f the variable must arise f r o m natural causes. Independence refers to the requirement that there should be no relat ionship between two consecutive events. For example, the dependence between successive hour ly sea levels is strong due to tides, wh i l e the dependence between annual m a x i m u m values is general ly weak. Non-homogene i ty impl ies that al l the elements o f the data series originate f r o m di f ferent populat ions or are caused b y di f ferent storms. Stat ionari ty requires that the mean and variance o f the data are invar iant w i t h respect to t ime. There are statistical tests for the latter three cr i ter ia (e.g., the M a n n - W h i t n e y test can be used to test stat ionari ty and homogenei ty) . The use o f a large set o f data is also required because i t provides a fair size o f sample to represent the actual cond i t ion (Wat t et al. , 1989), and i t is necessary to j u s t i f y the asymptot ic arguments w h e n the G E V Dis t r ibu t ion is appl ied to the annual m a x i m a series (Tawn , 1992). I n addi t ion, the Annua l M a x i m a M e t h o d requires the assumption that al l data share the same theoretical d is t r ibut ion type (i.e., they are ident ica l ly distr ibuted). Fai lures to meet these 55 assumptions do not necessary i m p l y the complete inva l id i t y o f the method, but the users should be cautious w h e n extrapolat ing or interpolat ing f r o m the theoretical d is t r ibut ion. 3.1.2.4 Advantages and Disadvantages The A n n u a l M a x i m a M e t h o d is the most c o m m o n l y used method o f est imat ing extreme f lood levels. However , there are several disadvantages o f u t i l i z ing this method. The method does not specify the physical properties o f the f l ood levels, w h i c h are compr ised o f tides and storm surges. The largest meteorological surge i n a part icular year is un l i ke l y to be considered unless i t happens to coincide w i t h h igh t ide (Pugh and Vassie, 1980). However , s torm surges are c o m m o n l y the govern ing factor i n an extreme condi t ion, so a d is t r ibut ion o f s torm surges imbues a more mean ingfu l insight into the est imat ion o f actual extreme condi t ions. Therefore, the A n n u a l M a x i m a M e t h o d , w h i c h is incapable o f representing sea levels as a func t ion o f various factors, cannot por t ray the impact o f storm surges. Another disadvantage o f this method is the requirement for a long- term data col lect ion, though on ly one value f r o m each year is u t i l ized for the annual m a x i m a series. The data extract ion process m a y exclude some o f the extreme events, since on l y one event i n a year is used. For example, one year m a y have lower values o f water level i n general, but the m a x i m u m value from this year is s t i l l considered an extreme. 3.1.3 if-Largest Maxima Method The Annua l M a x i m a M e t h o d is improved b y the i?-Largest M a x i m a M e t h o d , w h i c h is the same as the A n n u a l M a x i m a M e t h o d except that the i?-Largest M a x i m a M e t h o d extracts R number o f m a x i m u m values from each year instead o f on l y one value f r o m each year. This 56 method was f irst appl ied to estimate extreme sea levels b y Pi razzol i (1983) i n a study o f sea levels i n Venice. The sea level i n Venice is steadily r is ing, so i t is impor tant to examine R extreme values per year to ident i fy the trend (Ts impl is and B lackman, 1997). The i?-largest M a x i m a M e t h o d shares the same assumptions as the A n n u a l M a x i m a M e t h o d but wh i le the assumptions regarding the statistical cr i ter ia for the data set, the long- term data set, and the d is t r ibut ion type m a y be met, the assumptions regarding independence and non-homogenei ty cr i ter ia m a y not. Because the same storm can cont inue for hours and y ie ld a number o f extreme sea levels for a part icular year, extreme sea levels from the same storm are c lear ly dependent or homogeneous. Therefore, there should be a screening process to ensure that the i?-largest values for each year are from di f ferent storms. A n example o f such a screening process involves selecting a s torm length, r, i n hours, such that the i?-largest values f r o m the same year are chosen to be at least r hours apart (Ts impl is and B lackman, 1997). Because more than one m a x i m u m value is selected from one year, this method requires on ly jus t over ten years o f data for a rel iable est imat ion ( T a w n and Vassie, 1989). S imi lar to the Annua l M a x i m a Me thod , the i?-largest values from each year are f i t to a suitable extreme value d is t r ibut ion, and the theoretical d is t r ibut ion is used to estimate the extreme f lood levels. The suggested d is t r ibut ion type for this method is the G E V Dis t r ibu t ion. Assuming that each ind iv idua l extreme value belongs to the same G E V Dis t r ibu t ion , the density func t ion o f the i?-largest values (Z/, Z2,...,ZR) for each year is shown to be (Smi th , 1986; Ts impl is and B lackman, 1997): f(ZL,Z2T...,ZR) = g-R exp 1 — it + — I Kk I l og l-k Zj -/u 3.4 where Z/ R denotes the R extreme values for the year, fx is the mean o f Z/, Z2,...,ZR, C is the standard deviat ion o f Zj, Z2,...,ZR, and k is the parameter o f the d is t r ibut ion. Equat ion 3.4 is 57 va l id under the f o l l o w i n g condit ions: 1) Zi>Zi>.>ZR, and 2) l - k (Z > 0 . I f Z,j, Z2J,..., V s J Z(R.i)>n, ZRn, are the largest values for each year, n, Equat ion 3.4 approximates the j o i n t density o f the i?-larger values for each year through / / „ , gn and kn, where ju„, c„ and kn are the mean, standard deviat ion, and d is t r ibut ion parameter o f the ^- largest values for n years. F ina l ly , the extreme sea levels can be interpolated or extrapolated w i t h a selected probab i l i t y o f exceedance and return per iod b y emp loy ing the theoretical d ist r ibut ion. 3.1.3.1 Advantages and Disadvantages B y extending the use o f the data set, this method renders considerably more precise estimates o f the extreme levels than the Annua l M a x i m a M e t h o d , because more than one h igh water value is taken f r o m one year. Furthermore, the use o f R number o f m a x i m u m values f r o m each year a l lows for a connect ion to regional c l imate var iab i l i ty , w h i c h i n effect requires a shorter data set to ensure the re l iab i l i t y o f this method (Ts imp l is and B lackman, 1997). However , the method does not overcome the fundamental problems observed for applications o f the A n n u a l M a x i m a Me thod , that is, the lack o f account ing for sea-level data that characterize the various components o f sea level f luctuat ions. 3.1.4 Simple Addition Method The Simple A d d i t i o n M e t h o d estimates extreme sea levels b y s imp ly adding extreme values o f a l l factors w h i c h are considered signi f icant to sea level variat ions. The value for each factor is estimated i n one o f two ways: either i t is the probable m a x i m u m value o f the data i n the 58 history o f record or i t is the quanti le obtained using the A n n u a l M a x i m a M e t h o d or the i?-Largest Me thod . Ker , Pr iestman & Associates Engineer ing L t d . ( K P A ) describes another procedure for selecting appropriate values for tides and storm surges. The two data sets, used fo r the purpose o f compar ison, are the m o n t h l y m a x i m a o f the observed water levels (i.e., t ides plus storm surges) and residuals (i.e., s torm surges) occurr ing at the t ime o f the m o n t h l y peak water level. Sea levels are estimated i n di f ferent combinat ions i n this w o r k , such as m o n t h l y m a x i m a or median t idal waves plus residuals. I n addi t ion, the probab i l i t y o f extreme values is also extracted b y t w o methods: selecting the highest annual values and selecting the highest values in the entire data set ( K P A , 1992). A l t h o u g h the study indicates that the results o f extreme sea levels using di f ferent data sets and probabi l i t ies are simi lar, i t shows the degree o f var iety i n choosing data sets and methods. Due to the conservative nature o f the S imple A d d i t i o n M e t h o d , adjustments m a y be made. K P A (1994) demonstrates such adjustments i n w h i c h the f ina l est imat ion o f extreme sea level is determined b y subtract ing 0.44 m f r o m the sum o f the extreme water level components. A n example o f the second approach to estimate values for the cr i t ica l factors is presented in SeaConsult Mar ine Research L td . (1985). Here, the extreme sea level is calculated b y adding the 200-year storm surge and m a x i m u m tides (Hodgins, 1985). I n general, the factors invo lved i n the est imat ion inc lude tides, storm surges, and al lowance for errors and other factors. For example, the study conducted b y K P A (1990) estimates the extreme sea level b y adding the h igh tides, m a x i m u m storm surges, addi t ional al lowance for concurrent local w i n d setup, numer ica l errors i n data, w i n d chop, E l N i n o effects, and freeboard. Freeboard is considered in order to account for factors such as wave run-up, seiches, rise in sea level , and settlements. The value o f freeboard is general ly regulated b y local authorit ies. I n this study, the values for each i tem are estimated as the probable m a x i m u m values based on histor ical data. 59 3.1.4.1 Advantages and Disadvantages The Simple A d d i t i o n M e t h o d offers the advantages o f be ing s imple and invo l v ing m i n i m a l computat ion t ime. However , this method is also very conservat ive, because al l factors are assumed to occur at the same t ime. I n fact, the probab i l i t y o f the m a x i m u m effects f r o m al l factors occurr ing at the same t ime is very low. There is also the d i f f i c u l t y i n determin ing a reasonable value for each factor. There is a great var iety i n the methods o f est imating the value and the probab i l i t y associated w i t h any est imation is subjective, because no actual j o i n t p robab i l i t y is calculated. 3.1.5 Joint Probability Method (JPM) The importance o f obta in ing accurate estimates o f extreme sea levels enhanced the need for improvements i n the precis ion o f the result ing estimates and i n the ab i l i t y to use shorter data series ( T a w n and Vassie, 1990). The J P M was developed b y Pugh and Vassie (1980) to f u l f i l l these two objectives. The major dif ference between this method and the aforementioned methods is that i t recognizes the various components causing sea level variat ions that are considered i n the engineering context as discussed i n Section 2.3. The three ma in components o f sea level variat ions considered i n the J P M are the mean sea level (Z0(t)), t idal level (X(t)), and meteoro log ica l ly induced level (Y(t)). The mean sea level refers to the long- term variat ions caused b y c l imato log ic and geologic effects and can be ident i f ied b y standard regression o f the long- term record o f the observed sea level . However , this te rm is usual ly assigned a constant value or a value o f zero, because i t general ly is small 60 compared w i t h t idal and meteoro log ica l ly induced levels (Ts impl is and B lackman, 1997). The t idal levels are general ly predicted based on measured t ide data as described i n Section 2.3.5.4. The meteoro log ica l ly induced sea level is i n general the effect o f s torm surges and in some cases is due to a combinat ion o f s torm surges. Thus, the sea level , C(t), is def ined as fo l lows (Pugh and Vassie, 1980): at) = Z0(t) + X(t) + Y(t) 3.5 W i t h the avai lab i l i ty o f observed sea level data (<f(0) a n d tide data (X(t)), and an assumption o f ZQ(t) = 0 , the storm surge can be expressed as: Y(t) = at)~X(t) 3.6 I f the probab i l i t y density func t ion o f s torm surges is fiiy), the probab i l i t y densi ty func t ion o f tides can be represented b y fi{£-y). The J P M assumes that tides and storm surges are independent, thus the probab i l i t y density funct ion o f the total sea level can be expressed as (Pugh and Vassie, 1980): co /(£)= \fA£-y)fs(y)dy 3.7 -co The cumulat ive densi ty funct ion, F( C, ) , for a g iven extreme sea level , z , is: F(z)=)f(C)dC 3.8 -co Given the cumulat ive density funct ion o f the sea level , any probab i l i t y o f exceedance and return per iod can be obtained b y app ly ing Equations 3.2 and 3.3. The f o l l o w i n g sections describe the detai led appl icat ion o f the J P M , the data requirements and assumptions for the method, and its advantages and disadvantages. 61 3.1.5.1 App l i ca t i on Throughout this descr ipt ion o f the J P M , i t is assumed that hou r l y data are available for the factors that contr ibute to sea level f luctuat ions, al though the J P M m a y be appl ied for data col lected at any short t ime interval . I n the case o f hour l y data, the J P M is used to estimate the sample probab i l i t y density funct ions for each cont r ibut ing factor i n Equat ion 3.7, to estimate the hour ly j o i n t p robab i l i t y o f exceedance based on these, and to use the hou r l y j o i n t probabi l i ty o f exceedance to estimate the annual j o i n t p robab i l i t y o f exceedance. The corresponding return per iod m a y also be determined based on Equat ion 3.3. Because the J P M can be appl ied for data col lected for short t ime intervals, it can be used in cases w i t h short data records, on the order o f years, p rov id ing that there are suff ic ient and accurate data records w i t h i n those years. For example, assuming avai lab i l i ty o f hour ly tides and observed sea levels, the process o f est imat ing the hour l y j o i n t probabi l i ty o f exceedance begins w i t h comp i l i ng the t ime series o f hour l y tides and storm surges. The t ime series o f hour ly tides are i n general p rov ided b y local hydrographic services, and the hour ly s torm surges m a y be determined b y indirect measurements (i.e., subtract ing the tides f r o m the observed sea levels) or numer ica l models. Then, the t ime series o f tides (X(ij) and storm surges (Y(t)) are used to construct histograms for the tides and storm surges. W i t h reference to Figure 3 . 1 , for N data points, n intervals for tides (X) and m intervals for storm surges (Y), the relat ive frequencies, f, i n the h igh l igh ted cells represent the probabi l i t ies o f occurrence o f tides and storm surges for the g iven intervals. For example, flXm) is the relat ive f requency o f tides in the interval o f Xm.i to Xm, and, i f the interval size is X + X suf f ic ient ly smal l , is an estimate o f the probab i l i t y o f occurrence fo r X = —si 2-. The series o f probabi l i t ies o f occurrence for each variable m a y be used to construct its sample probabi l i ty 62 density funct ion. A p p l y i n g Equat ion 3.7, the product o f the sample p robab i l i t y density funct ions o f tides and storm surges yields the sample j o i n t p robab i l i t y densi ty func t ion o f tides and storm surges. The sample j o i n t p robab i l i t y density func t ion o f tides and storm surges are used in Equat ion 3.8 to determine the sample j o i n t cumulat ive density func t ion and the sample j o i n t p robab i l i t y o f exceedance based on Equat ion 3.2. Deno t ing the sample j o i n t p robab i l i t y o f exceedance asPN(Z), the j o i n t hour ly probabi l i ty o f exceedance, p\(z), is determined based on: pN(z) = \-[l-p,(z)f •' 3.9 where z is the sea level composed o f tides and storm surges. For large values o f N, Equat ion 3.9 can be approximated as: , / > „ ( z ) * l - e x p [ - 3 . 1 0 I f the values o f sea levels exhibi t persistence (i.e., dependence between values i n adjacent t ime periods) the hou r l y p robab i l i t y o f exceedance m a y be adjusted. The J P M has been adapted based on the assumption o f one-dependence, w h i c h impl ies that the probab i l i t y density funct ion o f each sea level depends on ly on the value o f the previous sea level (Pugh and Vassie, 1980). Here, the hou r l y j o i n t p robab i l i t y o f exceedance in a set o f N samples can be determined b y so lv ing the f o l l o w i n g equation (Pugh and Vassie, 1980): p w ( z ) = l - e x p NPi(z) v A( Z) 3.11 where pi(z) is the probab i l i t y o f exceeding z i n t w o successive t ime periods and can be determined b y construct ing the histograms based on two-hour intervals. Pugh and Vassie (1980) indicate that " the adjustment for correlat ion (i.e., 1 - ^ 2 ^ ) is so smal l compared w i t h the uncertainties from other causes that, i n v iew o f the accuracy required, i n practice i t is not 63 requ i red. " The ul t imate goal o f the J P M is to determine the annual j o i n t p robab i l i t y o f exceedance and return per iod i n years. Thus, the relat ionship between the annual j o i n t p robab i l i t y o f exceedance and the hour l y j o i n t p robab i l i t y o f exceedance needs to be established. I f py(z) is def ined as the annual j o i n t probabi l i ty o f exceedance, py(z) can be described as a funct ion ofpi(z) (Pugh and Vassie, 1980): pY(z) = l -exp[ -8766p i (z)a] 3.12 where a = 1 for independent hour ly sea levels, and a = 1 - ^ 2 ^ for dependent hour l y sea AO) levels. The argument o f the exponential described i n Equat ion 3.12 is very smal l i n the case o f extreme values due to the l o w probabi l i ty o f exceedance; thus, Equat ion 3.12 can thus be mod i f i ed to (Pugh and Vassie, 1980): pY(z) = %166px{z)a 3.13 The annual j o i n t p robab i l i t y o f exceedance m a y be used to determine the corresponding return per iod b y app ly ing Equat ion 3.3. 3.1.5.2 Data Requirements The J P M requires two types o f data, the hour l y t ide and hou r l y observed sea level at the same locat ion. A s suggested b y Pugh and Vassie (1980), t w o years o f data w i l l p rov ide a rel iable est imat ion. Because short- term data are used i n this method, the J P M requires " a better observational accuracy and greater di l igence i n data processing ( than the A n n u a l M a x i m a M e t h o d ) " (Pugh and Vassie, 1980). I n part icular, the process o f obta in ing the t ime series o f s torm surge levels requires careful considerat ion because t i m i n g errors m a y occur w h e n di f ferent 64 t ime scales are used for the t ime series o f tides and observed sea levels. T i m i n g errors can distort the probab i l i t y d is t r ibut ion o f storm surges, w h i c h is a govern ing factor for extreme f lood levels. Thus, the t i m i n g should be corrected to w i t h i n one or t w o minutes (Pugh, 1989). I t is also impor tant to choose the data hav ing no bias regarding a part icular season, because the ampl i tude and frequency o f storm surges vary seasonally. Nevertheless, w h e n accurate data are not available, T a w n and Vassie (1990) suggest that the J P M is appl icable i f at least ten years o f data are available. 3.1.5.3 Assumpt ions The appl icat ion o f the J P M is va l id under a few assumptions. W h i l e hour l y sea levels must be independent, the independent assumption for sea levels is weak fo r t w o possible reasons, due to the t ida l and storm surge inf luences. T ida l inf luences are the usual cause o f dependent sea levels at t ide-dominant sites, whereas storm surges, w h i c h can last fo r hours, are the c o m m o n cause o f dependent sea levels at s torm-dominant sites. Therefore, the assumption must be relaxed so that the probab i l i t y density func t ion o f each sample depends o n l y on the value o f the previous sample (i.e., the assumption o f one-dependence). I n addi t ion, tides and storm surges are assumed to be independent for Equat ion 3.7 to be va l id , and adherence to this assumption is site specif ic. The assumption o f the independent relat ionship between tides and storm surges is usual ly true except i n shal low water where there is s igni f icant t ide and storm surge interact ion ( T a w n and Vassie, 1989). Tides and storm surges interact i n shal low water areas in that the largest storm surges are restricted f r o m occurr ing at h igh t ide ( T a w n , 1992). I n order to satisfy this assumption, Pugh and Vassie (1980) suggest f ind ing the s torm surge distr ibut ions for each interval o f t ides. This increases the computat ion t ime, and is not appl icable i n the cases w i t h 65 more than t w o dependent variables. Another assumption is that the density func t ion o f storm surges over a g iven per iod should represent the probab i l i t y densi ty func t ion for the populat ion o f al l s torm surges. Th is assumption is the key to the stabi l i ty o f the J P M and must ho ld i f the t idal and storm surge probabi l i t ies are to be combined to compute the probabi l i t ies o f total sea levels as described b y Equat ion 3.7 (Pugh and Vassie, 1980). U s i n g data i n Aberdeen and N e w l y n , Br i ta in , Pugh and Vassie (1980) conclude that "the surge act iv i ty is a stat ionary process and that a few years o f data gives a representative sample o f the surge popu la t ion . " 3.1.5.4 Advantages and Disadvantages The advantages o f this approach include: 1) the physical factors af fect ing sea levels are ident i f ied and incorporated, 2) stable values are obtained from a re la t ive ly short per iod o f data, 3) the associated probabi l i t ies are not based on extrapolat ion, 4) less restr ict ive physical assumptions are required, and 5) separation o f the determinist ic t ida l signal from the sea level should, i n pr inc ip le , p rov ide a better w a y o f est imating the stochastic s torm surge signal (Pugh, 1987; T a w n , 1992; Ts impl is and B lackman, 1997). The f i rst advantage is signif icant, because the extreme values caused b y h igh storm surges w i t h l o w tides can be ident i f ied b y separating the observed sea levels into components o f t ide and storm surge levels. The J P M provides for a more expl ic i t ident i f icat ion o f the actual extreme condit ions. The second advantage is pract ical for coastal engineers, because short- term data col lect ion is a rout ine p rob lem for coastal engineering projects. However , one o f the p r imary disadvantages is that this method cannot be appl ied to the est imat ion o f extreme sea levels composed o f any number o f dependent variables due to the u t i l i za t ion o f Equat ion 3.7 for obta in ing the j o i n t p robab i l i t y densi ty funct ion o f factors cont r ibut ing to extreme sea levels. Thus far, the J P M has on l y been appl ied to the 66 est imat ion o f extreme sea levels composed o f tides and storm surges. I n addi t ion, the assumption o f one-dependence i n sea levels i n the appl icat ion cou ld lead to signi f icant bias i n estimates o f extreme sea level. Th is is a weak assumption i n storm-dominant areas, where storms cou ld last for several hours. The assumption that the empir ica l s torm surge d is t r ibut ion represents the true populat ion m a y not ho ld except for t ide-dominant sites. For surge-dominant sites, this assumption is va l id on ly i f the number o f samples is suf f ic ient ly large to represent the true d is t r ibut ion. Another disadvantage is that the J P M w i l l not produce a probab i l i t y o f sea levels greater than the sum o f the largest observed storm surge combined w i t h the highest recorded astronomical t ide, w h e n i n fact there is a poss ib i l i ty o f a higher sea level occurr ing ( T a w n and Vassie, 1989). 3.1.6 Revised Joint Probability Method (RJPM) The Revised Joint Probabi l i ty M e t h o d ( R J P M ) der ived b y T a w n and Vassie (1989) attempts to account for some deficiencies o f the J P M caused b y the assumptions requi r ing one-dependence i n sea levels and representative distr ibut ions o f storm surges w i t h i n a given per iod. The R J P M is applicable for bo th t ide-dominant and surge-dominant sites, requ i r ing on ly one year o f the t ide and observed sea level data. The under ly ing concept, assumptions, and procedures o f the R J P M are s imi lar to the JMP. The R J P M introduces the extremal index, S, to relax the assumption o f one-dependence o f sea levels and replace a i n Equat ion 3.13: pN(z) = N8n(z)px(z) 3.14 where PN(Z) is the j o i n t p robab i l i t y funct ion, and N is the number o f observations i n one year. The te rm, NSn(z), can be thought o f as the effect ive number o f independent observations i n N observations. The extremal index ranges f r o m zero to one and can be explained i n the inverse 67 f o r m , d„~ (z), w h i c h is the mean over topping t ime o f sea level z for each independent excursion. I t is determined for di f ferent values o f z using the or ig inal t ime series o f hou r l y sea levels. For example, i f a storm lasts for on ly one hour i n a storm-dominant site, the extremal index w o u l d be 1 - EJS^L a s assumed in the J P M . The R J P M also introduces a mod i f i ca t ion to the determinat ion o f the probab i l i t y density funct ion for s torm surges to account for insuf f ic ient data o f s torm surges for representing the true populat ion. Instead o f f i nd ing the probab i l i t y density func t ion for s torm surges using the h is togram, m a x i m u m values o f storm surges are extracted f r o m each independent event to generate a probab i l i t y densi ty funct ion, w h i c h is represented b y any E V Dis t r ibu t ion . Th is method can be coupled w i t h a procedure o f spatial transfer, i f the site o f interest has less than one year o f data available. The extreme in fo rmat ion is transferred f r o m a site w i t h an extensive sea level record to a neighbour ing site w h i c h has on ly a l im i ted data series ( T a w n and Vassie, 1990). The distr ibut ions o f tides and storm surges f r o m the site o f interest, w h i c h has fewer data, can be represented as a func t ion o f the d is t r ibut ion o f tides and storm surges o f the reference site where the data are adjusted w i t h a t ime lag between the t w o locations and correlat ion coeff ic ients. The extreme sea levels can be estimated w i t h less than a year o f data using this approach. 3.1.6.1 Advantages and Disadvantages A l t h o u g h the R J P M requires the computat ion o f the extremal index, dn(z), i t provides a more stat ist ical ly sound approach, and provides a higher level o f conf idence. The in t roduct ion o f Sn(z) is not on ly capable o f addressing the dependence o f sea levels but also guards against the effect o f a smal l number o f observations on the est imat ion o f p robab i l i t y o f exceedance and 68 return per iod. I n addi t ion, the use o f the extreme value func t ion for s torm surges can better represent the true popula t ion o f storm surges. W h e n the records o f data co l lect ion are re lat ively short, the R J P M m a y prov ide an improved est imat ion o f the extreme sea levels. However , the R J P M have the same def ic iency as the J P M in terms o f be ing incapable o f f i nd ing the j o i n t p robab i l i t y o f extreme sea levels composed o f a number o f dependent variables. 3.2 Estimation of Extreme Wave Conditions Due to the stochastic nature o f waves, extreme wave condi t ions inc lud ing both wave heights and periods are ideal ly obtained f r o m long- term data o f 15 to 20 years (Herb ich, 1990). There is no established procedure for the selection o f the design wave per iod. The general practice is to use a scatter d iagram o f the wave heights and associated periods based on the actual wave data, each po in t o f w h i c h has a j o i n t p robab i l i t y o f occurrence, and to select a wave height and per iod combinat ion based on structural design object ives. Th is section focuses on determin ing the design wave heights direct ly. The t w o approaches for selecting a design wave height take into considerat ion the long- term d is t r ibut ion o f sea states and the long- term d is t r ibut ion o f ind iv idua l waves. Isaacson and Foschi (2000) determine the dif ference i n the estimations o f extreme wave heights using di f ferent approaches. They show that the m a x i m u m wave height i n a sea state w i t h a specif ied return per iod m a y be s ign i f icant ly di f ferent than the m a x i m u m ind iv idua l wave height. They demonstrate that the return per iod o f an ind iv idua l wave height m a y be lower than the return per iod o f a sea state (Isaacson and Foschi , 2000). M o s t guidel ines suggest the use o f the long- term d is t r ibut ion o f sea states to estimate design wave condi t ions, but one should keep in m i n d that the m a x i m u m wave heights der ived b y the long- term d is t r ibut ion o f sea states may occur more of ten than expected. Th is section discusses 69 both approaches: the long- term d is t r ibut ion o f sea states and the long- te rm dis t r ibut ion o f ind iv idua l waves. 3.2.1 Long-Term Distribution of Sea States The parameters o f sea states, inc lud ing s igni f icant wave height, can be der ived f r o m measured or predicted wave data, and measured or predicted sea state values are then used to estimate extreme wave heights. W h e n long- term data for sea states are available, the recommended procedure is s imi lar to the Annua l M a x i m a M e t h o d . The procedure involves 1) extract ing the m a x i m u m value o f signi f icant wave heights or ind iv idua l wave heights for each year, 2) f i t t ing the annual m a x i m a series to a theoretical d is t r ibut ion type, and 3) interpolat ing and extrapolat ing the design sea state or signi f icant wave height w i t h a selected probabi l i ty o f exceedance or return per iod. The c o m m o n l y used d is t r ibut ion types for extreme wave condi t ions are the Gumbe l , Frechet, W e i b u l l , and L o g - N o r m a l Dis t r ibut ions (Isaacson and Foschi , 2000). A l t h o u g h no single d is t r ibut ion funct ion is preferred for extreme wave statistics, the Gumbel D is t r ibu t ion and L o g - N o r m a l D is t r ibu t ion are most w i d e l y used (Goda, 1990). G iven the estimated s igni f icant wave height o f a specif ied return per iod, emp loy ing Equat ion 2.5 al lows the est imat ion o f m a x i m u m ind iv idua l wave height w i t h i n the design sea state. 3.2.2 Long-Term Distribution of Individual Wave Height The use o f the long- term dist r ibut ion o f ind iv idua l wave heights to estimate the extreme wave cond i t ion is developed b y Battjes (1970). Assumpt ions i nvo lved i n this approach include 1) the sea state must remain unchanged between t w o successive observat ion hours, and 2) the 70 i nd iv idua l wave height can be described b y the Ray le igh D is t r i bu t ion (Goda, 1990). The cumulat ive probab i l i t y funct ion o f ind iv idua l wave heights (H) is described as a funct ion o f s igni f icant wave heights and periods (Goda, 1990; Isaacson'and Foschi , 2000) : where p(H\Hs) is the condi t ional probabi l i ty func t ion o f ind iv idua l wave heights under a g iven sea state, and p(Hs,T) denotes the j o i n t p robab i l i t y o f the signi f icant wave height and wave per iod. The probab i l i t y o f exceedance and return per iod can be obtained b y app ly ing Equations 3.2 and 3.3. 3.3 Joint Probability of Flood Levels and Extreme Wave Conditions B o t h f lood levels and wave condi t ions are components o f extreme sea levels. Sections 3.1 and 3.2 discuss methods for est imating f lood levels and extreme wave condi t ions, respectively, but the j o i n t p robab i l i t y o f f l ood levels and extreme wave condi t ions is important when est imat ing extreme sea levels. However , the d i f f icu l t ies o f incorporat ing wave statistics i n a j o i n t p robab i l i t y approach are as fo l lows: 1) there is an interact ion between wave probab i l i t y and water levels, due to coastal refract ion changing as sea levels change, and 2) condi t ions w h i c h produce m a x i m u m storm surges are also l i ke l y to produce h igh waves f r o m a part icular d i rect ion (Pugh and Vassie, 1980). Coastal refract ion creates a dependence between waves and water levels and the degree o f dependence between waves and water levels varies from site to site. H igher degrees o f dependence between storms surges and waves exist because o f the relat ionship between m a x i m u m storm surges and waves. However , i n order to apply the J P M for mul t i -var iables f r o m observational data, the variables must be either complete ly independent or 3.15 71 complete ly dependent (Hawkes et al. , 2002). Thus, the J P M is general ly s imp l i f ied i n order to estimate extreme sea levels. The s impl i f i ca t ion approaches invo lve : 1) calculat ing the design wave height w i t h an assumed water level , 2) calculat ing the design water level w i t h an assumed wave height, or 3) calculat ing extreme wave heights and water levels and app ly ing the Simple A d d i t i o n M e t h o d (Besley, 1999). These three approaches have di f ferent applications. For example, one o f the applications o f the f irst method is i n situations where wave heights are l im i ted b y the depth o f the water, because wave heights pose more randomness under this condi t ion. Hawkes and Hague (1994), and S i m m (1996) apply the J P M b y der iv ing the t ime series o f waves and water levels and analyzing correlat ion and j o i n t probabi l i ty . Hawkes et al. (2002) apply a M o n t e Car lo S imula t ion approach to f i nd the j o i n t p robab i l i t y o f three variables. The three variables used i n this study are water level , s igni f icant wave height, and mean wave per iod, where the d is t r ibut ion for each variable is used to simulate a synthetic set o f data. The synthetic set o f data is used to determine the j o i n t probabi l i ty , and the dependence issue is dealt w i t h b y a statistical mode l for dependence. The pr inc ipa l advantage o f app ly ing M o n t e Carlo S imula t ion w i t h the J P M is that the higher extreme sea levels, w h i c h are not observed or measured w i t h i n the per iod o f data col lect ion, m a y be repl icated b y the M o n t e Carlo S imulat ion. However , these approaches ignore di f ferent components o f extreme f l ood levels i n order to compromise the inc lus ion o f extreme wave condi t ions i n the est imat ion o f extreme sea levels. 3.4 Wave Run-up and Overtopping Discharges Wave run-up and over topping discharges are t w o impor tant factors in f luenc ing the crest height o f a coastal structure, because they determine the amount o f freeboard required to prevent or m i n i m i z e over topping. A s waves propagate into shal low water, some waves start to break. 72 A f t e r waves break, the m o m e n t u m keeps the water m o v i n g up the face o f a beach or shore structure. Wave run-up, Rw, is def ined as the m a x i m u m vert ical height above the st i l l water level. W h e n the wave run-up exceeds the crest height o f a coastal structure, over topping occurs. Th is section discusses over topping as a vo lume, and as an a l lowable over topping discharge. Wave run-up and over topping discharges depend on a large number o f variables, such as geometr ic variables and wave condit ions. The transi t ion o f waves enter ing shal low water is discussed i n Sect ion 3 .4 .1 , t w o approaches for predic t ing wave run-up are described i n Section 3.4.2, and approaches for predic t ing the over topping discharges are discussed i n Section 3.4.3. 3.4.1 Wave Transition in Shallow Water The transi t ion f r o m deep to shal low water changes the wave characteristics. A s waves propagate f r o m deep water to shal low water, wave heights in i t i a l l y decrease sl ight ly, and then the wave heights rap id ly increase un t i l break-point is reached. U n l i k e the wave height, the wavelengths shorten as the depth decreases, and the decrease proceeds at a rate propor t ional to the decrease in the wave speed. The increase o f wave heights and decrease o f wavelengths yields an increase in the steepness o f the waves, that is, an increase i n the ratio o f the wave heights to the wave lengths. Conversely, the rat io o f water depth to wavelengths decreases and reaches a lower l i m i t for shal low-water waves. Once waves fa l l into the category o f shal low-water waves, the waves are affected b y the sea bot tom. I n addi t ion, an asymmetr ical wave pro f i le starts to develop i n shal low water. The asymmetry develops around the hor izontal axis as the wave crest steepens and the wave t rough f lattens. A compar ison o f waves i n deep and shal low water is shown in Figure 3.2. Figure 3.2 indicates that the posi t ive and negative sides o f the wave pro f i le i n deep water are simi lar, whereas the peaks o f the wave pro f i le are much 73 sharper than the troughs in shal low water (Och i , 1998). The asymmetr ica l waves are unstable and eventual ly break, w h e n the wave steepness reaches about 1/7 (Abbot t , 1999). A wave breaks for t w o basic reasons: 1) the wave becomes so steep that i t can no longer support its o w n weight and the crest collapses, or 2) the speed o f the water that moves w i t h the crest o f the wave exceeds the speed o f the wave, a l l ow ing the water i n the crest to overtake the wave f o r m , and causing i t to fa l l and break (Thomson, 1980). A f te r breaking, waves have smaller ampli tudes, and they m a y cont inue to break i f the condi t ion stated above is reached or they m a y not break. The breaking or non-breaking waves f ina l l y advance onto the shore and run up the coastal structures. The m a x i m u m distance above the mean sea level that the wave runs up a coastal structure is def ined as the wave run-up. The vo lume o f wave run-up over topping the coastal structures is the over topping discharge. 3.4.2 P r e d i c t i o n o f W a v e R u n - U p Wave run-up is the m a x i m u m vert ical height above the st i l l water level to w h i c h the water from an incident wave w i l l rise on the beach or structure, denoted as Rw i n Figure 3.3 (Sorensen, 1997; C E R C , 1984). Wave run-up is a func t ion o f var ious factors, such as wave characteristics, geometr ic variables, and f l u id properties. The wave characteristics include the wave heights, per iod, length, celer i ty, and energy. The geometr ic variables include water depth, slope o f the beach, slope o f the structure, shape o f the structure, slope relat ive roughness, and angle o f wave approach. The f l u id properties include mass densi ty and dynamic viscosity. However , the predic t ion o f wave run-up is made possible b y s i m p l i f y i n g the parameters, and by ignor ing some variables. The f o l l o w i n g two sections introduce t w o methods to predict wave run-up: the Shore Protect ion Manua l (SPM) M e t h o d and the V a n der Meer and Janssen ( V D M J ) 74 Method . The t w o approaches use empir ica l relat ionships der ived f r o m exper imental results, and the di f ferent experiments per formed dist inguish the t w o methods. For instance, the S P M M e t h o d uses deep-water wave characteristics to predict the wave run-up, whereas the V D M J M e t h o d requires the shal low-water wave characteristics. Futhermore, the S P M M e t h o d predicts the wave run-up i n general, whereas the V D M J M e t h o d estimates the wave run-up associated w i t h the 50-year wave. The app l iab i l i t y o f the approaches m a i n l y depends on the data avai labi l i ty . The Federal Emergency Management A g e n c y ( F E M A ) suggests that the S P M M e t h o d provides the mean wave run-up instead o f the extreme value o f wave run-up when coastal structures are compr ised o f composite slopes (Headland et al . , 2000) . 3.4.2.1 Shore Protection Manual (SPM) Method The Shore Protect ion Manua l ( S P M ) M e t h o d for est imat ing wave run-up was developed b y Stoa (1979), w h o re-analyzed the results f r o m several exper imental investigat ions b y Brunn , et al. (1984). The S P M M e t h o d recognizes the f o l l o w i n g variables as parameters o f wave run-up: water depth at structure toe (ds), bo t tom slope i n f ront o f a structure (6), incident deep-water wave height (H0), incident deep-water wave per iod (7 ) , structure shape and roughness, and scale effects, as shown i n Figure 3 .3(CERC, 1984). The first four parameters are inc luded i n the empir ica l relat ionship: Equat ion 3.16 defines the-relat ive wave run-up or the rat io o f wave run-up on smooth and R smoothjab 3.16 impermeable slopes i n small-scaled physica l models (R, 'smoothjab ) to incident deep water wave height (H0). The relat ive wave run-up, RsmoolhJah/H0, is a func t ion o f the bo t tom slope (6), 75 relat ive depth or ratio o f water depth to incident deep water wave height (ds./H0), and relat ive steepness (HQ/gT2). The empir ica l relat ionship can be i l lustrated b y a set o f wave run-up curves. Figure 3.4 contains a set o f such curves for wave run-up on smooth and impermeable slopes on a 1:10 bo t tom slope w h e n ds/H0 « 2 . 0 . S imi lar graphs have been developed for djH'0=0, ds/Ho*0A5, ds/H'0 *>0.S, and djH0> 3.0 on 1:10 bo t tom slope, and djHQ = 3 , ds/H0 = 5 , and ds/H0 = 8 on a hor izontal bo t tom, and these curves are inc luded i n Append ix A . The other parameters that affect the wave run-up are structure roughness and scale effect, and the pred ic t ion o f the actual wave run-up is adjusted for these by : R.. 3.17 where yy is the reduct ion factor for slope roughness and porosi ty , and ys is the adjustment factor, for scale effects. The concept o f a reduct ion factor for slope roughness is int roduced b y Battjes, et al. (1984) , w h o recognize the effect o f roughness and poros i ty on wave run-up and suggest the use o f a roughness and porosi ty correct ion factor to a l low for wave run-up calculations for a range o f structure slope characteristics. Th is roughness reduct ion factor, y/, is the ratio o f wave run-up or relat ive wave run-up on rough permeable or other non-smooth slope to the wave run-up on a smooth impermeable slope. That is: _ trough _ ^-rough/Hp / D D / ZJ' ^smooth ^smooth / / 7 0 The values o f jf for var ious types o f surfaces are l isted i n Table 3.2. S imi la r ly , the correct ion factor for scale effects, ys, represents the ratio o f wave run-up der ived for scaled physical models or other non-smooth slopes to the wave run-up under the actual physica l sett ing. The value o f ys is a func t ion o f wave heights and slope and is summarized i n the curves shown i n Figure 3.5. 76 I n conclus ion, the predic t ion o f wave run-up begins w i t h iden t i f y ing al l parameters, inc lud ing water depth at structure toe (ds), bo t tom slope i n f ront o f a structure (6>), incident deep-water wave height (Ho), incident deep-water wave per iod (T), and surface materials. Then, two ratios are calculated, the wave steepness ratio (HQ/gT2) and relat ive depth rat io (ds/H0). The relat ive depth rat io is used to ident i fy an appropriate set o f wave run-up curves. Acco rd ing to the wave run-up curves, the relat ive wave run-up is obtained w i t h the slope and the wave steepness rat io. The actual wave run-up is estimated according to Equat ion 3.18, w h i c h adjusts wave run-up for a range o f roughness and porosi ty, and scale effects. 3.4.2.2 Van Der Meer-Janssen (VDMJ) Method I n European pract ice, wave run-up is of ten indicated b y Ru,2%, representing the wave run-up level above the st i l l water level w h i c h exceeds 2 % o f the i n c o m i n g waves, or the 50-year waves. A c c o r d i n g to a series o f tests under various factors, V a n der M e e r and Janssen (1994) expressed the average value o f the wave run-up for a smooth straight slope as: H. 1-5 3.19 where Hs is the signi f icant wave height near the toe o f the structure, yn is the reduct ion factor for a shal low foreshore, y / i s the reduct ion factor for slope roughness, yp is the reduct ion factor for an obl ique wave attack, and £ e q is the equivalent breaker parameter for a slope w i t h a berm. The reduct ion factor for a shal low foreshore (%) is inc luded w h e n the relat ive depth ratio is between 1 and 4, i n w h i c h waves start to break. Thus, y/, can be estimated as: 77 r, = 1 - 0 . 0 3 4 - 3.20 The reduct ion factor for slope roughness is l isted i n Table 3.3, Table 3.4, and Table 3.5. They l ist alternative slope roughness reduct ion factors for a range o f armour and construct ion materials. The angle o f the wave attack can also inf luence the wave run-up. V a n der Meer and de W a l l (1993) describe wave run-up as a func t ion o f ob l ique ly i n c o m i n g waves and direct ional spreading. Th is invest igat ion suggests that yp for the 2 % wave run-up is estimated by: 7 ^ = 1 - 0 . 0 0 2 2 / ? 3.21 where /? is the angle o f the propagat ion d i rect ion w i t h respect to the normal al ignment axis o f coastal structures. Equat ion 3.21 requires /? to be measured i n degrees, and the def in i t ion o f /? is shown i n Figure 3.6. Since the V D M J M e t h o d uses the shal low-water waves i n the pred ic t ion o f wave run-up, wave break ing is a phenomenon necessary to be taken into considerat ion dur ing the calculat ion o f wave run-up. The equivalent breaker parameter for a slope w i t h a b e r m can be estimated b y (Pi larczyk, 2000) : ^eq yb^op 3.22 where jb represents the reduct ion factor fo r a berm, and £ o p represents the breaker parameter. I n addi t ion, the breaker parameter is obtained b y (Pi larczyk, 2000; Besley and A l l sop , 2000): £ = , t a n 6 ? 3.23 °" p7rHsl(gT2p) where 9 denotes the bo t tom slope i n front o f a structure, and Tp denotes the peak per iod o f the wave spectrum. The reduct ion factor, can be estimated b y (Headland et al . , 2000) : 78 L, 'berm w 1-0.5 3.24 where Bw denotes berm width, Lberm denotes berm length, and dh is the wave depth over the 3.4.3 O v e r t o p p i n g D ischarge When wave run-up exceeds the crest height of coastal structures, overtopping occurs. Although overtopping is considered one of the failure modes of coastal structures, there is extensive debate over the allowable level of overtopping discharge that minimizes the structural damage and reduces costs by allowing for lowering crest elevation. This section presents methods for estimating overtopping discharge in Section 3.4.3.1, and discusses the amount of the allowable overtopping discharge in Section 3.4.3.2 3.4.3.1 Estimation of Overtopping Discharges Since overtopping occurs due to a large wave run-up, the overtopping discharge is a function of parameters similar to those affecting the wave run-up. Overtopping discharges are commonly described as the peak or the mean overtopping discharge during a design event. Three methods for predicting mean overtopping discharge are the Shoreline Protection Manual (SPM), Van Der Meer (VDM), and Owen's Method. While the SPM Method is commonly used in North America, the VDM Method and Owen's Method are used in Europe. berm. Equation 3.24 only applies to cases where the product o£yhYfYp ^ 3.0. 79 3.4.3.1.1 Shore Protect ion Manua l ( S P M ) M e t h o d The S P M M e t h o d , recommended b y the U.S. A r m y Corps, o f Engineers (1984), describes the vo lume rate o f wave over topping as a func t ion o f the structure height, water depth at the structure toe, structure slope, and structure shape. The empir ica l relat ionship was f irst der ived b y Savi l le and Ca ldwe l l (1955), w h o investigated over topp ing rates and wave run-up heights on small-scaled laboratory models. Savi l le (1958) used a larger scaled mode l to tests the results obtained i n earlier research. The latter research indicates that the over topping rate per uni t length o f structure can be expressed b y Equat ion 3.25 or Equat ion 3.26, where both h-d equations are va l i d under the condi t ion o f 0 < H - ds ~< R and 0 < - ~< 1.0 ( C E R C , 1984): R Q = (gQlH0 )V2e 0 2 Hl a h h - > f h - d < R Q = (gQlK ) , / 2 e L 0.1085 , [ R+h-d, -—log. -R-h+d, 3.25 3.26 where Q is the over topping rate hav ing the units o f vo lume per uni t t ime per uni t structure length, H0 is the equivalent deep-water wave height, h is the height o f the structure crest above the bo t tom, ds is the depth at the structure toe, Rw is the wave run-up on the structure that w o u l d occur i f the structure were h igh enough to prevent over topping, and a* and Qu* are empi r ica l ly determined coeff ic ients that depend on incident wave characteristics and structure geometry. Values o f a* and Q0* are funct ions o f wave steepness (H0/gT2) and relat ive depth (ds/H0), and the approx imat ion o f bo th values for various slope and structure types are summarized as empir ica l relat ionships i n graphical format. Figure 3.7 provides the empir ica l relat ionship for structures hav ing r ip-rap and 1:1.5 slope. Values o f a* and Qn* are estimated b y f irst ident i fy ing whether waves are breaking or non-breaking. The breaking o f waves can be determined using 80 Equat ion 3.23, where waves are breaking w h e n £ , p < 2, and waves are not breaking when £ o p > 2. Empi r i ca l curves for other types o f structures are p rov ided in Append ix 2. 3.4.3.1.2 V a n Der Meer ( V D M ) M e t h o d V a n der Meer and de W a l l (1993) present the most recent approach for est imating the method o f determin ing the average over topping discharge for break ing waves and non-breaking waves separately. Thus, the method requires the ident i f icat ion o f the status o f breaking for the design condi t ions, and the wave breaking can be determined b y the breaker parameter as described i n Equat ion 3.23. Under the cond i t ion o f breaking waves (£op < 2 ) , the average over topp ing discharge is expressed as a func t ion o f a dimensionless parameter o f over topping discharge (Qb) (Pi larczyk, 2000): The dimensionless over topping discharge parameter is estimated using a dimensionless crest height (Rb). The relat ionship between the two parameters is as described (Pi larczyk, 2000; Besley and A l l sop , 2000) : I n addi t ion, Rb is estimated b y the f o l l o w i n g relat ionship, w h i c h is va l i d under the condi t ion o f 0.3 < Rb <2 (Pi larczyk, 2000; Besley and A l l sop , 2000): over topping discharge, Q, w h i c h has the units o f m 3 / s per m or L/s per m. They suggest a 3.27 Qb = 0 . 0 6 e x p ( - 5 . 2 / ? 4 ) 3.28 81 RC ]2TOHs l{gT2p) i Hs tand ybyhyfyp Rb=^-^ ' 7 " 3.29 where Rc is the crest height relat ive to st i l l water level. B y obta in ing the value o f Rb and emp loy ing Equat ions 3.27 and 3.28, Q can be estimated. S imi la r ly , for non-breaking waves (^ o p < 2), the dimensionless over topping discharge (Q„) is g iven b y (Pi larczyk, 2000): The relat ionship o f the dimensionless over topping discharge and dimensionless crest height is: C 2 „=0 .2exp( -2 .6 i?„ ) 3.31 where Rn is the dimensionless crest height. The predic t ion o f over topp ing discharges can be achieved b y f i rst obta in ing the value Rn. The dimensionless crest height is estimated as: R 1 Rn=— 3.32 H , rbrhyfyP U p o n the determinat ion o f R„, Q can be estimated b y emp loy ing Equat ions 3.30 and 3 .31. 3.4.3.1.3 Owen 's M e t h o d Owen ' s M e t h o d (Owen , 1980) predicts over topping discharge, Q, and involves the est imat ion o f t w o dimensionless parameters: the dimensionless over topp ing discharge parameter (Q*) and the dimensionless crest height parameter (R*). The over topp ing discharge (Q) is estimated: 82 Q'=QI(gTmHs) 3.33 The dimensionless parameters, Q* and R*, are funct ions o f the roughness coeff ic ient (/y) and coeff ic ients for al l slope angles (A l l sop and M c C o n n e l l , 2000; Besley and A l l s o p ; 2000) : Q' = Aexpi-BR* / rf) 3.34 R* =RJTm{gHsr 3.35 where A and B are empir ica l coeff icients that depend on the pro f i le o f coastal structures. Owen der ived values o f A and B for s imple coastal structures w i t h slopes o f 1:1 to 1:5. They are summarized i n Table 3.6. 3.4.3.1.4 Compar ison o f Methods for Predict ing Over topp ing Discharges The compar ison o f the V D M M e t h o d and Owen ' s M e t h o d shows the f o l l o w i n g advantages and disadvantages for each method. The equation used i n Owen ' s M e t h o d is s imple to apply, but requires di f ferent empir ica l coeff ic ients for di f ferent w a l l prof i les. On the other hand, the V D M M e t h o d is more compl icated to use, but does not require di f ferent coeff icients for di f ferent types o f coastal structures. However , the reduced number o f empi r ica l coeff icients i n the V D M M e t h o d can also reduce the sensi t iv i ty o f the response to di f ferent condit ions. Besley and A l l sop (2002) suggest that Owen 's M e t h o d is s t i l l the most appropriate means o f est imating over topping discharges for smooth, s imple s loping and bermed coastal structures for the w ider range o f condi t ions around the U.K. coastl ine. 83 3.4.3.2 A l l o w a b l e O v e r t o p p i n g D ischarge There is no general recommendat ion for acceptable levels for over topp ing for coastal structures, because al lowable over topping discharges depend on locat ion, use o f the structure, and type o f structure. However , some guidelines current ly exist. A simple guidel ine is found in standard D u t c h practice, for w h i c h a safe value o f over topping discharge is approx imate ly 0.002 m 3 / s for a grassed crest (Pi larczyk, 2000). T o be more conservat ive, recent Du tch practice indicates that this value can be increased to 0.005 m 3 / s or even to 0.001 m 3 / s for a " g o o d " qual i ty grass mat on a c lay sub-layer (Pi larczyk, 2000). Th is guidel ine is not general for other types o f coastal structures. O w e n (1980) suggests a number o f l i m i t i n g discharges for more general purposes, as shown in Figure 3.8. The al lowable over topping discharges are suggested for funct ional and structural safety. Another example o f guidel ines for a l lowable over topping discharge is g iven b y S i m m (1991), as shown in Table 3.7. Table 3.7 presents di f ferent l imi ts for embankments ( w i t h back slopes) and revetments (w i thout back slopes), but this guidel ine is for structural safety only . 84 Table 3.1 P lo t t ing Posi t ion Formulas f r o m Wat t et al. (1989), and Rao and H a m e d (2000) Method Equation Comments California Pm =™„IN Used in part o f Canada. Hazen m - 0 . 5 P"= N N o t i n general use. Weibull P = m" F m N + l Recommended i n Bu l l e t i n 17B (U.S. Interagency A d v i s o r y Commit tee on Water Data 1982). Chegodayev mn - 0 . 3 P m ~ N + 0.4 N o t i n general use. Cunnane mn- 0.4 P m ~ TV+ 0.2 Used by the Water Resources Branch o f Env i ronment Canada (P i lon et al. 1985) Gringorton m - 0.44 P m ~ Nm + 0 . 1 2 Blom m„ - 0 . 3 7 5 P m ~ N + 0.25 Adamowski mn - 0.24 P m ~ TV+ 0.5 Table 3.2 Va lue o f Yf fo r Var ious Slope Surface Characteristics f r o m C E R C (1984) SloDe Surface Characteristic Placement Yf £_ _ — —~ ' Smooth, impermeable 1.0 • J • — Concrete blocks Fi t ted 0.90 Basalt blocks Fi t ted 0 . 8 5 - 0 . 9 0 Gobi blocks Fi t ted 0 . 8 5 - 0 . 9 0 Grass 0 . 8 5 - 0 . 9 0 One layer of quarry stone (impermeable foundation) R a n d o m 0.80 Ouarrvstone Fi t ted 0 . 7 5 - 0 . 8 0 Round auarrystone Random 0 . 6 0 - 0 . 6 5 A el . " ~ ; " Three lovers of auarrvstone (impermeable foundation) R a n d o m 0 . 6 0 - 0 . 6 5 t i i «- 1 * K ' Ouarrvstone R a n d o m 0 . 5 0 - 0 . 5 5 Concrete armour unit R a n d o m 0 . 4 5 - 0 . 5 0 85 Table 3.3 Reduct ion Factor j y f o r Rough Slope f r o m Besley and A l l s o p (2000) Armour type Y f J * — • Smooth, impermeable 1.0 Rough concrete 0.85 Pitched stone in mrtar 0 . 7 5 - 0 . 8 Rock armour, 2 layer 0 . 5 - 0 . 6 Hollow cube armour units, one layer 0.5 Dolos armor units 0.4 Stabit armor units 0 . 3 5 - 0 . 4 Tetravods. two layers 0.3 Table 3.4 Reduct ion Factor y / fo r Rough Slope f r o m Pi larczyk (2000) Tvpe of slope Y f J " A— • Smooth, concrete, asphalt 1.0 Closed, smooth, block revetment 1.0 Grass (3 cm) 0 . 9 - 1 . 0 1 rubble layer (HJD = 1.5-3) 0 . 5 5 - 0 . 6 2 or more rubble layers (HJD = 1.5-6) 0 . 5 0 - 0 . 5 5 Table 3.5 Reduct ion Factor yf for Rough Slope f r o m Headland et al. (2000) Tvpe of slope Y f •J " ML . — One laver of rock 0 . 5 5 - 0 . 6 0 •s J • Two or more rock layers 0 . 5 0 - 0 . 5 5 Table 3.6 Empi r i ca l Coeff ic ients for Smooth Impermeable Simple S lop ing Coastal Structures f r o m Besley and A l l s o p (2000) Slope A B 1:1 7.94 x 10" J 20.1 1:1.5 8.84 x 10" J 19.9 1:2 9.39 x 10" j 21.6 1:2.5 1.03 x 1 0 ' j 24.5 1:3 1.09 x 10" J 28.7 1:3.5 1.12 x 10" J 34.1 1:4 1.16 x 10" J 41.0 1:4.5 1.20 x 10" J 47.7 1:5 1.31 x 10"' 55.6 86 Table 3.7 Tolerable M e a n Discharge (m 3 / s per meter run) f r o m Besley and A l l s o p (2000) Buildines Q < 1 x 10'b N o damage Q<3x W M i n o r damage to f i t t ings, etc. 0> 3x10'' Structural damage Embankment sea walls Q< 0.002 N o damage Q<0.02 Damage i f crest not protected Q<0.05 Damage i f back slope not protected 0> 0.05 Damage even i f f u l l y protected Revetment sea walls 0< 0.05 N o damage O<0.2 Damage i f promenade not paved O>0.2 Damage even i f promenade paved 87 Tides, i = 1, m Intervals Xi x2 Xm Relative Frequencies f(X2) f{Xm) Storm Surges, j = 1, n Intervals Y, Y2 Yn Relative Frequencies /a,) f{Y2) ; / ( > ; ) Figure 3.1 Example o f Two-D imens iona l H is togram for the J P M Deep Water Shal low Water Figure 3.2 Waves in Deep Water and Waves i n Shal low Water F r o m Ochi (1998) 88 Po in t o f m a x i m u m wave runup Design $ W L H„ Figure 3.3 Wave R u n - U p F r o m C E R C (1984) 89 Figure 3.4 Wave R u n - U p on Smooth, Impermeable Slopes on a 1:10 B o t t o m when d s / H 0 = 2.0 F r o m C E R C (1984) 90 Ol ! J i—il Li I : i >•• ' i • ' ? < ' * i i — J J I 1.00 1,04 1.08 1.12 I I 6 ; M.20 1.24 : Runup Correction Factor r,k 1 - , \ . - : ; Figure 3.5 Wave R u n - U p Correct ion for Scale Factor F r o m C E R C (1984) 91 I-1 dike Figure 3.6 De f in i t i on o f the Ang le o f the Wave A t tack F r o m Pi larczyk (2000) ;ure 3.7 Over topp ing Parameters a and Qo* for Riprapped and 1:1.5 Structure Slope F r o m C E R C (1984) 93 lOOO lOO e te 1:0 • o 6 O.OOI O.OOOI-F u n c t l o o a l S o f e t y U n s o f o , o t : a n y ' s p n « d •Horizontal c o m p o s i t e , wall u n s u l e l a pork V e r t i c a l wait u n s a f e to p a r k U n i o f « a t h igh s p e e d Sa fe a t a l l 3 p a « d 9 V a r y d a n g e r o u s . 'C raaa dike ' . • d a n g e r o u s H o r i z o n t a l c o m p o s i t e wall - .Ver t ical wall.-vdongaroua- . ' . :Un c o m for t obi o bu t not... s ' d a n g a r o u s Wet, bu t no t u n c o m f o r t a b l e P e d e s t r i a n s S t r u c t u r a l d a m a g e : Minor, d a m a g e , t o f i t t ings N O d a m a g e B u i l d i n g s S t r u c t u r a l S a f e t y D a m a g * If fu l ly p r o t e c t e d v - ' - b o e k : -*.v n o t . protmcttmiw : O a m a g e If • cree*. h'6t::: p r o t e c t e d N o d a m a g « :: E m b o n k m e n t: " ' ieaWaltar D a m a g e even ' fo r p a v e d p r o m e n a d e g « ' If - . n o t p « v c d -SO r-26 N o d a m a g e R e v e t m e n t : s e a w a l l s • 200 -o.e - 0 . 3 O O 3 r-O.004 Figure 3.8 Suggested L i m i t s for Over topp ing Discharges F r o m A l l sop and M c C o n n e l l (2000) 94 4 DIRECT JOINT PROBABILITY METHOD (DJPM) The limitations under in the JPM or RJPM, imposed by the requirement of an independent relationship between variables, such as for tides and storm surges, or for sea levels and wave run-up, inspired the development of the Direct Joint Probability Method (DJPM). The major difference between the DJPM and JPM or RJPM is that in the former the sample cumulative density function for a given sea level is determined directly based on the collection of joint relative frequency values for combinations of contributing factors that result in this sea level and all sea levels of smaller magnitude. The sample cumulative density function is then used along with Equation 3.2 to determine the sample joint probability of exceedance. Finally, the instantaneous (or hourly, etc., depending on the recording frequency of the data) and annual joint probabilities of exceedance are determined based on the sample joint probability of exceedance. While the assumptions and data requirements of the DJPM are similar to those of the JPM or RJPM, the direct determination of the sample cumulative density function values for a range of sea levels without application of Equation 3.7, allows for the accommodation of two or more contributing factors and of dependence among these factors. This chapter provides a description of the DJPM in Section 4.1, and discussions of the data requirements and assumptions of the method in Sections 4.2, and 4.3, respectively. 4.1 D e s c r i p t i o n o f the D J P M Assuming the availability of hourly data, application of the DJPM involves two steps: obtaining the sample cumulative density function values for a range of sea levels using multivariate histograms that compile the joint relative frequency of combinations of contributing 95 factors, and determining the sample, hourly, and annual joint probabilities of exceedance for the range of sea levels based on these values. The corresponding return period can then be determined using Equation 3.3. The process of obtaining the sample cumulative density function values is the major difference between the D J P M and the J P M or R J P M , and is the feature that provides the additional advantage relative to the J P M or R J P M of facilitating the inclusion of multiple dependent variables. The number of components that affect the extreme sea levels dictates the number of dimensions required for the histograms that are used to determine the joint relative frequency of combinations of contributing factors, and thereby the sample cumulative density values for sea levels based on these. Figure 4.1 and Figure 4.2 are examples of two- and three-dimensional histograms, that represent cases where two and three factors contribute to resulting sea levels, respectively. Each element of the histogram represents the joint relative frequency, flztj) or f\zij,k), for the specified intervals of a set of conditions, X,•, Yj•, Wk , where /= l,...m, and j = I, ...n, and k= \,...o that affect the sea level, Zy or z v > In any case, each element of the multi -dimensional histogram represents the joint relative frequencies of the contributing factors and these are used to determine the sample cumulative density function for sea levels directly (i.e., Equation 3.7 and 3.8 are not applied). For a given sea level, all elements of the histogram that result in sea levels of smaller magnitude are identified and the sum of the joint relative frequencies in these elements comprise the sample cumulative density function value for the specified sea level. Thereafter, Equation 3.2 can be applied to determine the sample joint probability of exceedance using the sample cumulative density function value for a given threshold sea level, and Equations 3.10, 3.13, and 3.3 can be applied to.determine the hourly and annual joint probabilities of exceedance, and return period as described in Sections 3.1.1 and 3.1.5. 96 A n example appl icat ion o f the D J P M , where the three major cont r ibut ing factors are t ides, s torm surges, and wave run-up, is described here w i t h reference to the f l o w chart shown in Figure 4.3. The required data, equations employed to convert the data into tides, storm surges, and wave run-up, and the overal l equation that integrates these factors for p roduc ing an extreme sea level are represented in the f l o w chart. Storm surge m a y be s imp ly estimated as the dif ference between observed sea levels and t ide levels or s imulated us ing numer ica l models. Wave run-up m a y be estimated using a range o f methods described i n Sect ion 3.4.2. I f the S P M M e t h o d is selected, Equat ions 3.16 and 3.17 are appl ied to estimate wave run-up w i t h the requirement o f data regarding wave heights and periods, data record ing intervals, and deep-water wave height, as w e l l as structural design in fo rmat ion such as the slope o f the dyke, depth o f the f ront ing slope, and roughness and porosi ty o f the dyke materials. The h is togram o f j o i n t relat ive frequencies o f this mul t ivar ia te p rob lem is three dimensional w i t h elements representing three-dimensional j o i n t intervals o f the data, and the j o i n t relat ive frequency in each element is the l i ke l ihood o f each o f the cont r ibut ing factors being i i n these j o i n t intervals. Here, combinat ions o f values for t ides, s torm surges, and wave run-up (based on wave height data) that occur s imultaneously over the h is tory o f record are used to evaluate the j o i n t relat ive frequency associated w i t h a g iven j o i n t data interval . The j o i n t relat ive frequencies are the number o f data points i n each j o i n t data interval d iv ided b y the total number o f data points. The sum o f g iven t idal level , s torm surge value, and wave run-up def ine the sea level , and therefore, each combinat ion o f di f ferent values o f these cont r ibut ing factors and the j o i n t relat ive f requency o f the combinat ion is associated w i t h a g iven sea level . For a g iven sea level , the sum o f the j o i n t relat ive frequencies associated w i t h a l l sea levels o f equal or lower magni tude is the estimate o f the sample cumulat ive density func t ion value for that sea level . 97 The sample joint probability of exceedance is found using Equation 3.2 and the determination of the hourly and annual joint probabilities of exceedance proceed as described above for the general application of the DJPM. The relationship between sea levels and annual probabilities of exceedance may be obtained by examining a range of sea level-probability of exceedance pairs and fitting an appropriate function to these results. Figure 4.4 and Figure 4.5 show examples such functions. Figure 4.4 demonstrates the shape of the curve generated by two variables, whereas Figure 4.5 demonstrates the shape of the curve generated by three variables. In both cases, the data for these curves are obtained from the Point Atkinson station near Richmond, B.C., where the data are tides and storm surges for Figure 4.4, and tides, storm surges, and wave run-up for Figure 4.5. Further details regarding these examples and others are provided in Chapter 6 and 7. The curve in Figure 4.5 exhibits lumpiness; thus, it would be difficult to find an appropriate function to represent these results. 4.2 D a t a R e q u i r e m e n t s The data required in the application of the DJPM depend on the number of variables considered as critical in the estimate of extreme sea levels. For extreme sea level estimates based on a number of variables, the length of data required for a reasonable estimate increases with the number of variables. The range of sea levels estimated may depend on the length and quality of data collected as well as the intensity of collection. The availability of more data may help to capture greater extremes, but this outcome does not necessarily occur. Nonetheless the availability of more data should increase the accuracy of estimation. As the number of contributing factors increases, the irregularity in the results (e.g., lumpiness) may increase, because of the larger amount of data required to adequately characterize the system. The 98 ef f icacy o f the D J P M for est imating extreme sea levels for the C i t y o f R i chmond , B.C. is evaluated i n Chapter 7. The t i m i n g o f each data series should be as accurate as possible. 4.3 Assumptions W h i l e the D J P M does not require the assumption o f independent variables, i t st i l l requires that hour l y sea levels be independent f r o m one t ime step to the next, and that the data are representative o f the system being analyzed. The D J P M does not require the independence between the data for the various contr ibut ing factors, because the estimated probab i l i t y o f a g iven sea level is not obtained through the product ion o f the ind iv idua l d is t r ibut ion funct ions for each o f the variables as in the J P M and R J P M . Instead, the D J P M determines the j o i n t p robab i l i t y o f the variables d i rect ly and uses this in fo rmat ion to estimate the probab i l i t y o f exceedance and return per iod o f the corresponding extreme sea level . 4.4 Advantages and Disadvantages The D J P M can be appl ied broadly i n the case o f f i nd ing extreme sea levels based on di f ferent combinat ions, such as tides, storm surges, and wave run-up, or observed sea levels and wave run-up, because i t does not require the assumption o f independent variables. The extreme sea levels are composed o f several factors, and such factors are rare ly independent o f each other. Thus, the D J P M m a y be very pract ical especial ly i n regions where sea levels are a func t ion o f more than t w o variables. 99 Yj,j = l,...,n Y, Y2 Xi / ( * . . , ) / ( * . . 2 ) s x2 / K . ) / ( * « ) / K J . I  Xm / ( * „ . . ) /<X ,J ' . Whk = l,...,o Figure 4.1 Example o f Two-D imens iona l H is togram J/- : V::T ; M)^')P.! ' ' : Yj,j = l,...,n Xj, i = l,...,m Figure 4.2 Example o f Three-Dimensional H is tog ram 100 Required Data Equation Critical Factors 3-D Histogram Observed sea levels (0) Tides (T,) StormSurge = 0-Tt Wave Run-up Tides Tides Tides Significant wave height Significant wave period (T, s) Duration (D, h) Slope of dyke ( 0) Depth fronting the slope (ds) Deep-water wave height (H0) Roughness and porosity of dyke H =H. ln (3600y) -storm Surges Overall Equation Extreme Sea Levels = Tides + Storm Surges+ Wave Run-up Figure 4.3 F l o w D iag ram o f D J P M for C o m b i n i n g o f Tides, Surges, and Wave Run-up 101 5.00 Dt -1.00 jjl-R e t u r n P e r i o d (year) Figure 4.4 Example of the Result of the Direct Joint Probability Method for Two Variables -2.00 • R e t u r n P e r i o d (year) Figure 4.5 Example of the Result of the Direct Joint Probability Method for Three Variables 102 5 SEA FLOOD PROTECTION IN RICHMOND, B.C. The C i t y o f R i chmond , located in the southeast o f the Province o f B r i t i sh Co lumbia , is compr ised o f a series o f islands, inc lud ing most o f L u l u Is land, Sea Is land, and f i f teen other smal l islands. The islands are surrounded b y the Fraser River , but are also adjacent to the Strait o f Georgia i n the Paci f ic Ocean. The format ion, diverse industr ies, r i ch cultures, and f lood issues o f this unique is land c i ty are al l related to its locat ion. The g r o w i n g populat ion and developments i n R i c h m o n d have mot ivated the government to imp lement a long- term strategic c i t yw ide p lan for f l ood protect ion and management. The 2021 F lood Protect ion Strategy is a long- term comprehensive strategy inc lud ing p lanning o f f l ood-p roo f ing designs and techniques, land use p lann ing, f l ood protect ion works , emergency preparedness, and disaster f inancial assistance. The strategy is implemented i n two phases: the analysis and strategy development i n 2002 - 2003, and the f l ood strategy implementat ion in the subsequent years (Manager, Po l i cy Planning, 2002). I n 2002, the C i t y o f R i chmond engaged the services o f U M A Engineer ing L td . to assist i n the f irst phase o f the R i c h m o n d F lood Protect ion and Management Strategy. One o f the objectives o f the p lan is to main ta in and upgrade the per imeter dyke system on L u l u and Sea Islands (R ichmond , 2002). U M A Engineer ing L t d . raised the concern o f adequacy o f the current crest height o f the sea dyke. W h i l e most attention is paid to the r iver dykes and r iver f loods, U M A chose to also examine the current sea dyke elevat ion level . A p r imary goal o f this thesis is to determine more accurate, updated design extreme sea levels and to conclude whether the crest elevat ion is adequate to wi thstand these condit ions. Th is chapter discusses the cr i t ical factors cont r ibut ing to sea level variat ions and the data that are necessary for determin ing the extreme sea levels for R ichmond . Sections 5.1 and 5.2 introduce the surroundings and the background o f R i c h m o n d , respectively. Section 5.3 discusses the f l ood concerns, Sect ion 5.4 describes the 103 current dyke system, Sect ion 5.5 discusses the variat ions i n sea levels, and Sect ion 5.6 describes the data avai labi l i ty . 5.1 Surroundings The C i t y o f R i c h m o n d is surrounded b y the Strait o f Georg ia on the west and the Fraser R iver on the nor th and south. The format ion o f the Strait o f Georgia began around 150 m i l l i o n years ago b y a depression o f the Earth crust along the Paci f ic Coast. The depression is accompanied b y an up l i f t o f the adjo in ing land, w h i c h forms the Vancouver Is land Range and O lymp ic Mounta ins to the west, and the Coast Mounta ins and Cascade Mounta ins to the east. A l t h o u g h the Strait o f Georgia has the m a x i m u m depth o f 3940 m, water depths inside this protected coastal domain vary considerably over short distances. A r o u n d 10,000 years ago, an ice sheet w i t h a thickness o f 1200 m at the present Fraser R ive r delta started me l t ing , and the decreased load on the Earth caused an 80 to 140 m upward heave (Thomson, 1981). A postglacial up l i f t o f approximately 15 m in the L o w e r Fraser R ive r V a l l e y also diverted the Fraser R iver to its present course into the Strait o f Georgia. Since the d ivers ion o f the Fraser River , large sediments and deposits have been carr ied to the ocean. The r iver deposits have gradual ly bu i l t the Fraser R ive r De l ta inc lud ing L u l u Is land and Sea Is land. The delta extends to the near shore zone o f the coast, mak ing the near shore areas shal low due to sedimentat ion. 5.2 Settlements The large quantit ies o f fresh water from the Fraser R ive r i n the coastal basin have created an estuary w h i c h acts as a nutr ient trap a l low ing river4oorne organic and inorganic materials to 104 col lect i n concentrated amounts. L u l u Is land's association w i t h estuaries fo rmed the nuclei for settlement and industr ia l izat ion. The r i ch soils i n the delta area create the oppor tun i ty for agriculture. Europeans are among the f irst pioneers to settle here fo r agr icul tural purposes, and the d ivers i ty o f agr icul ture inc luded da i ry ing and berry g r o w i n g w h i c h remain important to this day. I n addi t ion, the Paci f ic Ocean coast a long the west side o f the C i t y provides abundant fisheries, w h i c h have become another major industry. Agr icu l tu re and fisheries have been the basis o f the economy and industr ia l development. Since R i c h m o n d was incorporated as a mun ic ipa l i t y on November 10, 1879, and designated as a c i ty on December 3, 1990, the c o m m u n i t y has seen cont inued g rowth , gradual ly expanding into a mul t i - indust r ia l area boast ing agriculture, f isheries, transportat ion services, manufactur ing, and technologies. G r o w t h o f the C i t y is also apparent i n an ever increasing populat ion, w i t h a steady annual popula t ion g rowth o f 2 % it has become a dynamic urban center o f fer ing a unique m i x o f residential and commerc ia l propert ies, agr icul tural lands, industr ia l parks, waterways, and natural areas. 5.3 F l o o d C o n c e r n s Unfor tunate ly , the topographic and c l imat ic characteristics indicate that R i chmond current ly faces potent ia l threats o f f lood ing. The sources o f f l ood ing inc lude r iver f loods, sea f loods, and an excessive amount o f rain. The f o l l o w i n g three sections discuss the background, histor ical data, and magni tude o f hazards for each o f these sources. 105 5.3.1 River Floods The potent ia l r iver f loods i n R i c h m o n d are the result o f r u n o f f in to the Fraser R iver from the me l t ing snow on the R o c k y Mounta ins dur ing early spr ing and late summer. The amount o f r u n o f f varies each year and depends on the precip i tat ion i n the previous winter , temperature change f r o m win ter to spr ing, and t idal var iat ion. W h e n the snowfa l l i n the previous win ter is heavy and the temperature changes suddenly f r o m co ld to w a r m , the amount o f r u n o f f for that year is increased and the possib i l i ty o f f l ood ing is h igh . The h igh var ia t ion o f water level is closely moni tored. R i c h m o n d relies on records at a gauge at Hope, B C . A c c o r d i n g to the f lood data col lected at this station, two o f the largest recorded f loods i n the Fraser R iver occurred i n 1894 and 1948. It is estimated that the f lood reached a peak o f 17,000 m 3 / s i n 1894, the largest f l ood o f record. F igure 5.1 shows an example o f f l o w data at the Hope gauge o f 15,000 m 3 / s i n 1948. The 200-year r iver f l ood has a f l o w rate o f 17,000 m 3 / s , and is adopted as the design f low. A m o n g al l the three sources o f f loods, H A Y C O (1989) ident i f ies the r iver f loods as the worst source o f f l ood ing , because the possible water levels are highest and the f lood durat ion is longest. The r iver water levels can remain h igh for 12 to 18 days dur ing a major f lood. Thus, the prevent ion o f r iver f loods has been the histor ical focus o f f l ood p lann ing i n R ichmond . 5.3.2 Sea Floods R i c h m o n d is also confronted w i t h possible f l ood threats from the sea, because i t has a l o w elevat ion relat ive to the sea level. H A Y C O (1989) ident i f ies the three ma jo r components cont r ibut ing to changes i n sea level as astronomical t ides, storm surges, and long- term sea level rise due to c l imato log ic and geologic effects. A t R i chmond , s torm surges occur dur ing winter , 106 w i t h the largest s torm surge recorded at 0.9 m above the GSC datum. Storm surges in the area can typ ica l l y last for three consecutive days. W h e n storm surges coincide w i t h h igh tides, the m a x i m u m sea level o f 1 m cou ld be added to the storm surges and cause potent ia l sea f loods. The sea level i n R i c h m o n d is moni to red b y the t ide gauges at Point A tk inson , B.C ( H A Y C O , 1989). The highest observed sea level at this station since 1914 is 5.6 m above the C H A R datum or 2.56 m above the GSC datum on December 16, 1982. Based on this value, the 200-year design sea f lood level estimated b y the Fraser R iver F lood Cont ro l Program is 2.8 m above the GSC datum. The inf luence o f sea f loods is not as strong as r iver f loods, but R i c h m o n d has a l o w elevat ion relat ive to sea level . The average elevat ion o f R i c h m o n d is 1 m above the GSC datum, but the h igh t ide at Point A t k i n s o n is 1.4 m above the GSC datum. D u e to tectonic movements and greenhouse effects, the r is ing o f g lobal sea levels can ef fect ive ly result i n local ized f lood ing in port ions o f the Ci ty . I t is estimated that the global average rate o f sea level r is ing dur ing the 2 0 t h Century is near 2 m m per year (Church, 2002). H A Y C O (1989) suggests that the sea level rise w i l l range f r o m 0.2 to 1.17 m b y the year 2050 and from 0.5 to 3.68 m b y 2100. A l t h o u g h the change is invar iab ly s low and gradual this needs to be taken in to considerat ion w h e n determin ing the design sea f lood level. 5.3.3 Excessive A m o u n t o f R a i n F lood ing b y ra infa l l i n R i c h m o n d has received less attent ion than r iver or sea f loods in most f l ood management plans for the Ci ty . A l t h o u g h ra in m a y not a lways cause f lood ing , a certain strength and durat ion o f ra in can ef fect ively f l ood the Ci ty . A heavy ra in fa l l persist ing for several days w i t h intensities reaching f r o m 6 to 7 m m / h o u r per 24 hours w i l l cause f lood ing in the lower areas o f R ichmond . A c o m m o n prob lem w i t h the per imeter dyke system is that the 107 dyke b locks the normal passage o f ra infa l l into r ivers and seas. For example, be ing surrounded b y a dyke, L u l u Is land requires an extensive drainage system i n order to prevent f loods due to excessive ra in. The current drainage system includes 32 p u m p stations and canals around the C i t y per imeter (C i t y o f R i c h m o n d Websi te) . The pump stations serve as the p r imary system o f drainage, whereas the canals operate as a secondary system w h e n the p u m p stations are at f u l l capacity. 5.4 C u r r e n t D y k e Sys tem Sea and r iver f loods can be mi t igated w i t h a dyke system. The construct ion o f the dyke system was in i t iated b y residents w h o real ized that a norma l l i fe was impossible w i thou t the dykes. The residents o f R ichmond , m a i n l y farmers, cont inued bu i l d i ng their o w n dykes un t i l 1905. I n 1905, a local pet i t ion cal led for the fo rmat ion o f a commiss ion to adequately mainta in the dyke and drainage system in R ichmond. Since then the dyke system has been gradual ly constructed around R ichmond . The current dyke system in R i c h m o n d surrounds L u l u and Sea Islands. The f o l l o w i n g section presents the legis lat ion for the dyke system. The current dyke system is then described in Section 5.4.2, and the maintenance p rogram executed b y R ichmond is presented in Section 5.4.3. 5.4.1 L e g i s l a t i o n D y k e systems in B.C. are bui l t , maintained, and operated b y local authorit ies under c o m m o n law and i n accordance w i t h pert inent legislat ion and agreements. The Fraser River F lood Cont ro l Agreement (1973) provides the means to improve the dyke system and 108 rehabil i tate the internal drainage system w i t h i n R ichmond . I n addi t ion, the D y k e Maintenance A c t (1995) is the legislat ive basis for operat ion and maintenance o f pub l ic dykes in B. C. Legis la t ion relevant to the fo rmat ion and operat ion o f dyke authorit ies includes the Drainage D i t c h and D y k e A c t (1996) and the Loca l Government A c t (1996). The Loca l Government A c t provides guidance for new dyke authorit ies. Other pert inent p rov inc ia l legis lat ion includes the Emergency Management A c t (1996), Water A c t (1996), and Env i ronmenta l Assessment A c t (1996). Relevant federal legislat ion includes the Canada Fisheries A c t (1985) and the Navigable Waters Protect ion A c t (1985). 5.4.2 D e s c r i p t i o n o f the D y k e Sys tem The dyke system i n R i c h m o n d includes the Sea D y k e at the west end o f L u l u Is land, R iver Dykes at the nor th and south o f L u l u Is land, the dyke surrounding M i t c h e l l Is land, and the dyke surrounding Sea Island. The Sea Island dyke is under the ju r i sd ic t i on o f the Vancouver A i rpor t . Under the D y k e Ac t , the design and construct ion o f the dyke system in R i c h m o n d is regulated b y the D y k e Des ign and Construct ion Guide o f the M i n i s t r y o f Water, Land , and A i r Protect ion. The standard design o f the cross section o f the dyke system is shown in Figure 5.2. R i c h m o n d adopts the revetment pavements on the outer slope, where the slope is 1:2. The crest elevat ion o f the dyke system is set to prevent the 200-year f l ood w i t h a m i n i m u m freeboard o f 0.6 m ( G A and A E , 2003) . The R iver D y k e and Sea D y k e surround L u l u Is land and create the perimeter dyke system for the Is land, as shown in Figure 5.3. The per imeter dyke system includes 10 pumps and 17 f lood boxes, w h i c h prov ide protect ion for over 34,000 bui ld ings. The R iver D y k e includes t w o segments, the N o r t h A r m and South A r m . B o t h segments are used to prevent the 109 r iver f l ood caused by spr ing freshets. The current R iver D y k e elevat ion is on average at 3.5 m above the GSC datum, a l though the elevations o f the R iver D y k e vary f r o m west to east w i t h elevations at the east end o f L u l u Is land being greater than elevations at the west end (Brownlee, Personal Commun ica t ion , September 23, 2004). This value was obtained by convert ing the f lood rate into the water level and adding a freeboard o f 0.6 m. The Sea D y k e or West D y k e is the ma in defence system against sea f loods. The current Sea D y k e elevat ion is at 3.35 m above the GSC datum, that is, the 200-year sea level , w i t h a freeboard o f 0.6 m. The total length o f the Sea D y k e is 6 k m f r o m N o . 1 Road to Garry Point, over look ing the t ida l f lats, w h i c h extend 1.6 k m westward into the Georgia Strait. A l t h o u g h Figure 5.2 shows the standard design for the dyke systems adopted in R i chmond , the standard design is on ly implemented at two locations o f the Sea Dyke , Terra N o v a and Garry Point. Figure 5:4 shows the actual condi t ion o f the major i ty o f the sea dyke. The major i t y o f the Sea D y k e has grass mat on the slope. A n interesting observat ion is the vegetal g rowth on the t ida l f lats. Some areas have very dense vegetat ion to the degree that water cannot be v is ib ly ident i f ied beneath this vegetation (Figure 5.4 (a)), wh i le other areas exh ib i t less vegetat ion, mak ing the water v is ib le (Figure 5.4 (b)). The Sea Dyke has become the popular West D y k e Tra i l for the residents o f R ichmond . 5.4.3 D y k e M a i n t e n a n c e P r o g r a m R ichmond undertakes the operat ion and maintenance o f the dyke system under the D y k e Maintenance Ac t . Under the author i ty o f the D ike Maintenance A c t , the M in i s t r y o f Water, Land and A i r Protect ion ( fo rmer ly the M in i s t r y o f Env i ronment , Land , and Park or M E L P ) regulates the operat ion and maintenance o f f l ood protect ion structures. Thus, R ichmond implements the maintenance program described in the M E L P D y k e Operat ion and Maintenance 110 Manua l . R i c h m o n d performs the regular inspect ion and maintenance o f the dyke. Regular inspect ion is essential for ident i f icat ion o f areas requi r ing maintenance before major problems develop. The dyke system requires annual inspect ion, w h i c h is completed p r io r to the h igh f l o w season and suf f ic ient ly early to a l low for adequate t ime for any required w o r k to be completed pr io r to possible f l ood events. The inspect ion o f the dyke system includes the f o l l o w i n g ( M E L P , 2001) : 1) Check crest, slopes, and toe for settlement, depressions, sinkholes, cracking, slides, s loughing, erosion, seepage, p ip ing , boi ls , loss o f freeboard, and l o w spots; 2) L o o k fo r unauthor ized act iv i ty , such as construct ion, excavat ion, etc.; 3) Check for areas where vegetat ion hampers inspect ion and m a y weaken the dyke; 4) L o o k fo r rodent act iv i ty ; 5) Check for unauthor ized excavat ion or construct ion on or adjacent to dyke; 6) Check r iver f l o w pattern for changes, deposit ion, scour, debris jams, etc.; and 7) Check condi t ions exh ib i t ing scour or erosion around bridges or other structures i n the v ic in i t y . I f there are obvious problems and damages, repair w o r k is in i t ia ted immediate ly . The Maintenance Program is carr ied out b y the C i t y ' s Publ ic W o r k Department. Acco rd ing to the foreman for the dyke system, the inspect ion is per formed every three months to include seasons o f h igh and l o w tides (M inns , Personal Communica t ion , December 23, 2003) . I t is important to have inspect ion dur ing l o w tides i n order that inspect ion o f the toe and repair, i f necessary, can be executed. The sections f r o m N o . 1 Road to B lunde l l Road and Garry Point are of ten attacked b y stronger waves. The section from N o . 1 Road and B lunde l l Road is subject to bo th r iver discharges and ocean waves, and the currents f r o m both systems can cause the waves to overtop the dyke. A n incident o f over topping was not iced on December, 2002, but there was no apparent damage, on l y concerned phone calls from the local residents ( M i n n s , Personal Communica t ion , 111 December 23, 2003). Gar ry Point is another t rouble spot, where wave attacks are generated b y boats t rave l l ing th rough this area. Bu t to date there has been no damage reported. The ma in reason for dyke fai lure i n R i c h m o n d is general ly due to rodent act iv i ty , w h i c h has usual ly led to p ip ing (G i l f i l l an , Personal Communica t ion , February, 2003). 5.5 Sea Level Variations The generations o f w i n d waves are a funct ion o f w i n d speed, w i n d durat ion, and fetch length as prev ious ly stated. Nevertheless, w i n d waves at R i c h m o n d are m a i n l y l im i ted b y the w i n d strength and fetch length, because the Strait o f Georgia is located to the east o f Vancouver Island. The Is land restricts the w i n d act iv i ty b y reducing the w i n d speed and the fetch length, m a k i n g the m a x i m u m wave heights i n this area 0.6 m on average and approx imate ly 2.7 m dur ing the winter . The m a x i m u m waves occurr ing dur ing w in te r are usual ly from the northwest and southwest d i rect ion (Thomson, 1981). The w i n d waves do not d i rec t ly impact the coastal structures i n R i c h m o n d , however wave run-up is an important factor to be considered as a wave approaches the shore. Because the west coast o f R i c h m o n d has a f lat and long shoreline, wave run-up occurr ing at the near shore area is signi f icant (Isaacson, Personal Communica t ion , A p r i l 11 , 2003). The long extent o f sedimentation deposits on the west coast o f R i chmond also impl ies that the m a x i m u m wave m a y break before the wave reaches the dykes, so the extreme sea condi t ions can be caused b y smaller waves. The calculat ion o f wave run-up w i l l determine the m a x i m u m impact o f waves. Even though storms are general ly not a p rob lem on the Paci f ic Coast, the l o w elevat ion o f R i c h m o n d requires considerat ion o f storm events. The t ime series o f s torm surges can be estimated b y subtract ing the t ide signal f r o m the observed sea levels. S torm surges can also be 112 predicted us ing a numer ica l model . A numer ica l mode l for the area o f the Juan de Fuca Strait and the northern end o f Johnston Strait, bo th near R i chmond , the GF7 mode l , was developed b y the Inst i tute o f Ocean Sciences (Hodgins, 1985). Ano ther essential factor contr ibut ing to sea level variat ions for R i c h m o n d is variat ions in astronomical t ides, because the west coast o f B.C. has a large t ida l range. The coast o f B.C. usual ly experiences h igh tides twice a year, and the t idal range fo r the area is from 3 to 5 m (Thomson, 1981). The B.C. coast usual ly experiences the highest tides on June 21 and December 22, and experiences the lowest tides on September 22 and M a r c h 2 1 . I t should be noted that the s torm surges also occur i n winter , w i t h the extreme condi t ions at this locat ion l i ke l y to constitute h igh tides and storm surges. Fortunately, the probab i l i t y that a large tsunami f r o m the ocean w i l l propagate far into the Strait o f Georgia is very low. I t is also un l i ke ly that the occurrence o f a ma jo r earthquake o f f the coast w i l l generate a tsunami because fau l t ing in this geographical locat ion is most ly o f the sideway sl ip type. T w o histor ical events o f tsunami affected Vancouver Is land and caused damages (see Table 2.1). B o t h events suggest that R i c h m o n d is shielded b y Vancouver Is land and is un l i ke l y to be affected b y tsunami even i f tsunami occur i n close p rox im i t y . Therefore, tsunami are not a concern for R ichmond. F ina l ly , the long- term sea level change is also an important factor for R ichmond . A long-term sea level rise is attr ibuted to the global w a r m i n g and the movement o f tectonic plates, w i t h the c l imate factor dominat ing the rate o f r is ing sea levels. A s stated earlier, the sea level rise is predicted to range f r o m 0.2 to 1.17 m b y 2050 and 0.5 to 3.68 m b y 2100 ( H A Y C O , 1989). The average rate o f sea level r is ing is predicted to be 2 m m per year (Church, 2002) . For a l o w - l y i n g area such as R i c h m o n d , the gradual increase in sea level has to be considered in long- term p lanning. 113 I n conclusion, the cr i t ica l factors cont r ibut ing to extreme sea levels i n R i c h m o n d are tides, storm surges, wave run-up, and long- term sea level changes. 5.6 Data Availability The three types o f data necessary to ident i fy the tides, s torm surges, wave run-up, and long- term sea level changes are t ide, wave, and water level data. R i c h m o n d current ly has no gauge station o f its o w n to col lect any data. Therefore, est imat ing extreme sea levels for R i c h m o n d requires a search among exist ing data at nearby stations. The three locations closest to R i c h m o n d are Point A tk inson , Ha l ibu t Bank, and Roberts Bank, and the relat ive locations o f the stations are shown in Figure 5.5 . The Point A t k i n s o n Stat ion has been used h is tor ica l ly to estimate sea f l ood for R i c h m o n d as prev ious ly ment ioned. Th is locat ion also has a complete set o f data inc lud ing t ide, wave, and water level data. Since i t is a permanent gauge station, hour ly t ide and water level data have been cont inuously moni tored. Unfor tunate ly , among the three stations, the locat ion o f Point A t k i n s o n is furthest f r o m R i c h m o n d and does not share s imi lar geological condi t ions as R ichmond. The wave data col lected at Point A t k i n s o n are on ly for a short per iod o f t ime f r o m December 19, 1972 to M a y 18, 1974 and are simple point measurements w h i c h do not include direct ional in format ion. The instrument invo lved i n the data col lect ion is the non-di rect ional Waver ider b y Datawel l . A l t h o u g h i t nomina l l y consists o f hour l y data, there are several miss ing data, and therefore the actual sampl ing frequency is on ly 2.3 hours per data point . The Ha l ibu t Bank Station is located close to the Point A t k i n s o n Stat ion. A l t h o u g h i t does not have a record o f t ide and sea level data, the wave data col lected at this station are recent and 114 are col lected over a re la t ive ly long per iod o f t ime from M a r c h 13, 1992 to June 25, 2003. I n addi t ion, the w i n d di rect ion is col lected here wh i l e measur ing waves. Roberts Bank is of ten used for estimates related to R i c h m o n d due to its s imi lar oceanographic characteristics. The Roberts Bank Stat ion is the on l y locat ion that has simi lar geological characteristics to R i chmond , because i t is also in the delta region. W i t h the absence o f data at R i c h m o n d , this locat ion provides an ideal cond i t ion for a complementary data set. However , there are on ly two sets o f data available here, that o f t ide and signi f icant wave data. The in fo rmat ion regarding the data avai labi l i ty at the three stations is summarized in Table 5.1 w i t h their station numbers, coordinates, and summary remarks. 115 Table 5.1 Data A v a i l a b i l i t y Location Tide Data Significant Wave Data Water Level Remarks Point A t k i n s o n 49 .35N 123.74W Yes • Stat ion#7795 Yes • M E D S 1 0 6 , • W i thou t d i rect ion, • 19/12/1972-18/05/1974 Yes • Stat ion#7795 • 01/05/1914-06/08/0997 • h is tor ica l ly used to estimate sea f lood level for R i chmond • most complete set o f data col lect ion Ha l ibu t Bank 49 .34N 123.74W N o Yes • C46146 • W i t h w i n d direct ion • 13/03/1992-25/06/2003 N o • most recent and longest wave data • station is very close to Point A t k i n s o n Roberts Bank 49 .02N 123.27W Yes • Stat ion#7592 Yes • M E D S 1 0 8 / • W i thou t d i rect ion • 07/02/1974-03/04/1976 N o • delta area • s imi lar geological characteristics to Sturgeon Bank 116 Hydrograph". Fraser River at Hope 14D00 12000 Normal A Previous peaks Flood Yeer (194B) Drought Year (1941) 1999 flows 2002 flows 1- 6- 11- 1B- 21- 26- 1- 6- 11- 113- 21- 26- 31- S- 10- 15- 20- 25- 30- 5-Jul 10- 15- 20- 25-Apr Apr Apr Apr Apr 'Apr May May May May May May May Jun Jun Jun Jun Jun Jun Jul Jul .Jul Jul Date Figure 5.1 Hydrograph at Hope Gauge Existing: 8aiik?L me •. Flood .Ctinsmictionlicvetj Scour Protection (Riprap mi«I TocTiciKh) Prepared Hank Grade luistiog-HcdJicvclfe Grave] Fiiter Layer ::: Maximum ;Scour<Dcp its*-Laiifidrtrd lie our Protcciion , Figure 5.2 Standard Cross Section o f the D y k e System 117 Figure 5.3 F lood Protect ion Features in R i c h m o n d F r o m C i t y o f R i chmond Websi te Figure 5.4 Sea D y k e o f R i c h m o n d a) L o o k i n g N o r t h b) L o o k i n g toward West 118 Figure 5.5 M a p o f Gauge Stations near R i chmond 119 6 METHODOLOGY The D J P M is demonstrated herein for est imating extreme f l ood and sea levels for R ichmond , B.C. A s a pre l im inary assessment o f the effectiveness o f the method, f l ood levels estimated w i t h the Annua l M a x i m a and Simple A d d i t i o n M e t h o d are compared w i t h those estimated w i t h the D J P M . Due to the geologic condi t ions i n the R i c h m o n d area, the data for tides and storm surges are part ial ly-dependent, and therefore appl icat ion o f the J P M to this case is not possible. Then, the est imation o f extreme sea levels is undertaken w i t h an approach that accounts for the cr i t ica l factors af fect ing sea levels at R i c h m o n d and accommodates the l im i ted available data. A s ident i f ied i n Section 5.5, the important factors cont r ibu t ing to the f luctuat ions i n sea levels at R i c h m o n d include storm surges, tides, wave run-up, and long- term sea level variat ions. A n appl icat ion o f the D J P M is used to ident i fy extreme sea level estimations affected b y combinat ions o f s torm surges, t ides, and wave run-up (as i n the example described w i t h Figure 4.3), and o f a hyb r id o f the Di rect Joint Probabi l i ty M e t h o d and the Simple A d d i t i o n M e t h o d to account for these three factors plus long- term sea level var iat ions. Because the 200-year extreme sea level is the standard regulated under the B.C. D y k e A c t , and the 1250-year extreme sea level is an alternative design standard that the C i t y o f R i c h m o n d is current ly consider ing, the 200- and 1250-year extreme sea levels are estimated. 6.1 Input Parameters Since there are no data col lected i n R i c h m o n d as indicated i n Sect ion 5.6, the analyses undertaken are based on the data col lected at Point A tk inson . The t w o reasons for selecting the data col lected at this station are that the data are complete and compr ised o f observed sea levels, 120 t ides, and waves, and that the data are h is tor ica l ly used to determine design f l o o d levels i n R i chmond and therefore prov ide consistency w i t h respect to the present design approach fo r the dyke. The input parameters necessary for the est imat ion o f the extreme f l o o d levels inc lude the tides and storm surges, whereas the input parameters used to predict extreme sea levels include the observed sea levels, t ides, storm surges, waves, and dyke characteristics. A s described i n Section 3.4.2, the wave data and dyke characteristics are employed in the computat ion o f wave run-up. The observed sea level data are prov ided by the Mar ine Env i ronment Data Service ( M E D S ) , w h i c h is a branch o f the Canadian Department o f Fisheries and Oceans ( D F O ) . The observed sea levels are col lected hour ly f r o m 1:00 on M a y 1, 1914 to 23:00 o n December 1, 1997. The hour l y sea levels measured in meters above the C H A R datum are taken di rect ly f r o m the d ig i ta l readings w i thou t interpolat ion. The water levels, i n meters above the C H A R datum, can be converted to the water levels above the G S C datum by : GSC = CHAR- 3.04 6.1 The observed data undergo t w o levels o f qual i ty contro l and are adjusted i f there is a change in reference da tum at the station ( M E D S Websi te) . The qual i ty cont ro l is per formed b y c o n f i r m i n g that the t ime series o f the observed sea and t ide levels f o l l o w the same pattern (Thoml inson , Personal Communica t ion , October 2 1 , 2003). Observed sea levels can also be used to obtain a long- term trend o f sea levels by per fo rming a l inear regression, as shown in Figure 6 . 1 . A l t h o u g h the length o f observed sea level data is insuf f ic ient to faci l i tate a def in i te conclus ion regarding the long- te rm trend, w h i c h requires at least 100 years o f data, F igure 6.1 indicates an increase i n sea levels at the average rate o f 1.3 m m per year based on the exist ing data. The t ide data prov ided by M E D S are predicted f r o m 0:00 on M a y 1, 1914 to 23:00 on August 6, 1997. The t ide data are predicted hour ly apply ing the procedure described in Sect ion 121 2.3.5.4 w i t h the constituents established b y the Inst i tute o f Ocean Science dated June 4, 1997. The level o f accuracy for predicted t ide data is h igh. G iven the hour l y observed sea levels and predicted tides at Point A tk inson , the hour ly storm surges can be estimated b y the Indirect Measurement M e t h o d , w h i c h involves a simple subtract ion to obtain the dif ference between the observed sea levels and the corresponding t ide levels, and the m a x i m u m wave heights are calculated based on the s igni f icant wave heights and peak wave periods using Equat ion 2.5. G iven that the observed sea levels undergo a qual i ty check, the level o f accuracy for the estimated storm surges is also h igh. The wave data obtained from the M E D S website are col lected hou r l y f r o m 17:00 on December 19, 1972 to 18:00 M a y 18, 1974, a l though there are some miss ing data f r o m t ime to t ime. The average sampl ing frequency is t w o hours. Each data po in t contains the f o l l o w i n g i tems: Date, QC_Flag , Lat i tude, Longi tude, Depth , V C A R , and V T P K . QC_F lag is the qual i ty code assigned subject ively to ind iv idua l spectral records based on the shape o f the spectral curve, and the descript ions o f qual i ty codes are summarized i n Table 6.1 ( M E D S Websi te) . The depth for co l lect ing the wave data is 40 m, so the waves col lected here are considered as deep-water waves. A s def ined in Section 2 .3 .1 , for deep-water waves, water depth is greater than 2 5 % o f the wavelength. Wavelengths can be expressed as a func t ion o f wave periods, and described as: ZT2 L = ^— 6.2 In where T is the wave per iod in seconds. Wave periods are measured and denoted as V T P K . The waves have an average wave per iod o f 4.112 sec, and this can be represented as a ratio o f water depth to wavelength o f 1.52, w h i c h is greater than 0.25. V C A R denotes the characteristic s igni f icant wave height in metres, and i t is calculated b y emp loy ing Equat ion 2.7 (Toml inson, e-ma i l , Ju ly 16, 2003) . 122 The characteristics o f the Sea D y k e at R i c h m o n d are also essential to the est imation o f wave run-up, and the characteristics required are the slope o f the dyke, the mater ia l o f the dyke surface, and the elevat ion at the toe o f the dyke. The Sea D y k e i n R i c h m o n d as described i n Section 5.4 has the slope o f 1:2. W h i l e some sections have r ip-rap slopes, most sections have grass mat on the slope. The elevat ion at the toe o f the dyke is estimated based on a f ie ld survey conducted i n 1974 b y the C i t y o f R i c h m o n d as shown in Figure 6.2. The survey, per formed for the purpose o f improv ing the dyke system in 1974, is the on ly construct ion d raw ing available for the area west end o f the B lunde l l Road in R ichmond. The elevat ion at the toe o f the dyke here is 5.0 ft or 1.52 m above the GSC datum. 6.2 Assessment of the D J P M for Estimating Extreme Flood Levels The ef f icacy o f the D J P M is evaluated b y compar ison w i t h the A n n u a l M a x i m a and Simple A d d i t i o n Methods. The latter two methods can on l y be used to determine the extreme f lood levels, w h i c h are composed o f tides and storm surges, and thus the compar ison is based on estimated extreme f lood levels. 6.2.1 Application of the D J P M for Estimating Extreme Flood Levels The 200- and 1250-year estimates o f the f lood level based on tides and storm surges is determined. The data used i n apply ing the D J P M include data for tides and observed sea levels col lected at Point A tk inson . Since the wave data col lected at Point A t k i n s o n are on ly for the years from 1972 to 1974 and these data are used i n the est imat ion o f extreme sea levels as w e l l , the t ide and observed sea level data for the match ing per iod o f t ime are used to estimate extreme 123 f l ood levels. S torm surges are estimated b y subtract ing tides f r o m the corresponding observed sea levels. The t ime series o f tides and storm surges are f irst d iv ided into equal intervals, and the t ime series are used to construct the two-d imensional h is togram o f relat ive frequencies for the determinat ion o f hour l y and annual p robab i l i t y o f exceedance for extreme f l ood levels based on the selected intervals. The number o f equal intervals for al l data was selected to m in im ize the sensi t iv i ty o f the results; results for 5, 10, 15, 20, and 25 equal intervals were examined, and 20 equal intervals were selected. A funct ional f o r m o f the relat ionship between the annual probabi l i t ies o f exceedance and the corresponding extreme f l ood levels is then determined, and the 200-year and 1250-year f l ood levels are estimated using this funct ion. I t should be noted here that the adjustment factor o f \ - P x ^ in Equat ion 3.12 or 6n''(z) i n Equat ion 3.14 is not used because the dependency o f sea levels i n this case study cannot be determined due to the inconsistent sampl ing frequency for wave measurements. 6.2.2 Application of the Annual Maxima Method for Estimating Extreme Flood Levels The A n n u a l M a x i m a M e t h o d is used to estimate the 200-year and 1250-year f l ood levels. The data required for this method are the long- term data o f observed sea levels, and the record o f observed sea levels at Point A tk inson from 1914 to 1997 is used. The f o l l o w i n g steps are undertaken: 1. The m a x i m u m value o f the observed sea levels is determined for each year, and these values comprise the annual m a x i m a series. 2. The annual m a x i m a series is screened for miss ing data and outl iers. 3. The annual m a x i m a series is f i t to various d is t r ibut ion types. T w o software packages are used i n this step to determine the parameters o f d ist r ibut ions, and they are the 124 Consol idated Frequency Analys is Package ( C F A ) and Re l iab i l i t y Analys is Package ( R E L A N ) . C F A is developed b y the Surveys and In fo rma t ion Systems Branch, Env i ronment Canada (1994), and R E L A N is developed b y Dr . R. Foschi at the Un ivers i t y o f B r i t i sh Co lumb ia (1994). 4. Goodness-of- f i t tests, standard error estimates, and engineer ing judgement are used to determine the performance o f the d is t r ibut ion types. 5. The d is t r ibut ion types w h i c h best represent the data series are p lo t ted i n the f o r m o f a peak water level versus return per iod plot . The plots also show the data series used to conduct the analysis, the 95 % Conf idence Interval , and the estimations o f the 200-year and 1250-year f l ood levels. A s suggested b y Ts impl is and B lackman (1997), an appropriate d is t r ibut ion type is important , so this approach includes several tr ials o f various types o f distr ibut ions. 6.2.3 Application of the Simple Addition Method for Estimating Extreme Flood Levels Results o f this approach can prov ide a conservative benchmark o f the extreme f lood level. This method estimates the 200-year and 1250-year f l ood levels b y adding estimates o f the corresponding quanti les o f two components, the tides and storm surges. The quantiles o f the tides and storm surges are estimated b y the A n n u a l M a x i m a M e t h o d , and the procedure invo lved i n the A n n u a l M a x i m a M e t h o d is presented i n Section 6.2.2. A p p l y i n g the Annua l M a x i m a M e t h o d precludes the use o f wave data, because the record o f wave data at Point A tk inson , col lected for less than t w o years, is too short to prov ide rel iable estimates o f the quantiles. Thus, the data used i n the method include the long- term data o f observed sea levels and tides at the Point A t k i n s o n station from 1914 to 1997. A s prev ious ly stated, the series o f storm surges is 125 estimated b y subtract ing the t ida l signals from the observed sea levels. The est imat ion o f f lood levels is obtained b y s imp ly adding the tides and storm surges w i t h the same return per iod together after app ly ing the Annua l M a x i m a Method . 6.3 Estimating Extreme Sea Levels for Richmond The cr i t ica l factors cont r ibut ing to the f luctuat ions i n sea levels at R i c h m o n d include storm surges, t ides, wave run-up, and long- term sea level variat ions. The Annua l M a x i m a M e t h o d cannot consider the effect o f wave run-up and long- term change in sea levels, and the Simple A d d i t i o n M e t h o d cannot evaluate the actual j o i n t p robab i l i t y o f al l the factors. The J P M and R J P M cannot consider the wave component i n the est imat ion o f extreme sea levels, because the waves and sea levels i n shal low regions, such as that at R i c h m o n d , are general ly dependent. The D J P M is the most appropriate method to estimate extreme sea levels i n this case and al lows for the incorporat ion o f wave run-up. F ina l ly , a hyb r id o f the D J P M and Simple A d d i t i o n M e t h o d m a y be used to estimate the extreme sea levels w i t h the long- te rm sea level variat ions. These approaches are described i n Sections 6.3.1 and 6.3.2. 6.3.1 Application of the D J P M for Estimating Extreme Sea Levels The D J P M is f irst appl ied to estimate the extreme sea level composed o f t ides, storm surges, and wave run-up, as described w i t h respect to Figure 4.3. The data used in apply ing the D J P M include data for t ides, storm surges, and waves col lected at Point A tk inson . Since the wave data col lected at Point A tk inson are on ly from 1972 to 1974, the t ide and storm surge data 126 for the match ing per iod o f t ime are used. A s stated earlier, the adjustment factor o f 1 - i n Equat ion 3.12 or d„~'(z) i n Equat ion 3.14 is not used i n the est imat ion o f the hou r l y probab i l i t y o f exceedance. The D J P M is appl ied f irst to obtain the extreme f l ood levels composed o f observed sea levels and wave run-up as described i n Section 6.3.1.1, and then i t is appl ied to determine the extreme sea levels composed o f the tides, storm surges, and wave run-up as described i n Section 6.3.1.2. 6.3.1.1 App l i ca t i on o f D J P M for C o m b i n i n g Observed Sea Levels and W a v e Run-up The data for this case includes the observed sea levels and waves from 1972 to 1974. The f l o w d iagram fo r this appl icat ion is shown i n Figure 6.3, w h i c h fo l l ows the same logic as Figure 4.3. A s shown i n Figure 6.3, the extreme sea level is estimated b y the adding observed sea levels and the wave run-up, and j o i n t relat ive frequencies o f these data are the elements o f the two-d imens iona l h istogram for these in format ion . W h i l e the observed sea levels are prov ided, the wave run-up is calculated according to the S P M M e t h o d int roduced i n Section 3.4.2.1. The S P M M e t h o d is chosen because the waves at this locat ion are deep-water waves. Data required for est imat ion o f wave run-up using the S P M M e t h o d include the observed water levels, wave in fo rmat ion , and dyke characteristics. Recal l f r o m Equat ion 3.16, wave run-up on smooth and impermeable surfaces is a funct ion o f bo t tom slope (6), relat ive depth (ds/H0), and relat ive steepness (H0/gT2). Before comput ing the value o f wave run-up on smooth and impermeable surfaces using Equat ion 3.16, 6, djH0, and H0/gT2 are estimated. The slope o f 1:2, g iven b y the standard design o f the dyke system, ident i f ies cot# as 2. The value o f ds/HQ is determined b y f irst iden t i f y ing the water depth f ron t ing the toe (ds) and the wave heights (Ho ). 127 W h i l e ds is obtained b y f ind ing the di f ference between the observed sea levels and the elevation at the toe o f the dyke, Ho can be interpreted as the m a x i m u m wave height calculated using Equat ion 2.5 and the signi f icant wave height and peak wave per iod from the wave data. W i t h the m a x i m u m wave height and corresponding wave per iod, H0/gT2 can also be calculated. The value o f dj HQ f i rst identi f ies the appropriate wave run-up curve(s) from the range o f wave run-up curves for a bo t tom slope o f 1:10 i n Append ix 1. The computat ion carr ied out i n this w o r k ut i l izes the wave run-up curves for a bo t tom slope o f 1:10, rather than those for a hor izontal bo t tom slope, because the wave run-up curves for a bo t tom slope o f 1:10 are for the ds/H'0 ranging from zero to three, w h i c h is a more suitable range o f dj'H0 for R ichmond . The wave run-up estimated f r o m the curves are for smooth and impermeable surfaces i n exper imental condi t ions, thus adjustments are then made for di f ferent slope mater ia l and scale factor w i t h the roughness (yf) and scale (ys) reduct ion factors. The mater ial used on the seaward surface o f the dyke is the govern ing factor for jy , and j y i s 0.8 for f i t ted quarrystone or grass according to Table 3.2. I n addi t ion, ys is 1.13 for Ho ranging from 0.45 to 1.37 m and 1.19 for Ho ranging from 3.66 to 1.22 m as indicated i n Figure 3.5. G iven the wave run-up on smooth and impermeable surfaces, yf, and ys, the actual wave run-up is estimated emp loy ing Equat ion 3.17. The j o i n t probabi l i t ies o f exceedance associated the observed sea levels and wave run-up are represented b y the j o i n t relat ive frequencies o f the observed sea levels and waves, and are estimated using the D J P M . 128 6.3.1.2 App l i ca t i on o f D J P M for Comb in ing Tides, S torm surges, and Wave run-up The D J P M is also appl ied for the three d imensional input condi t ions, t ides, storm surges, and wave run-up. The data required include the observed sea levels, t ides, and waves. As shown i n F igure 6.4 ( w h i c h is ident ical to Figure 4.3), the extreme sea levels are estimated b y adding the tides, storm surges, and wave run-up, and the corresponding j o i n t p robab i l i t y o f exceedance is based on the j o i n t relat ive frequencies o f three events. S torm surges are estimated b y subtract ing the t ida l signals f r o m the observed sea levels. Wave run-up is predicted using the S P M method w i t h a s imi lar procedure as that described in Section 6.3.1.1. The associated probab i l i t y o f exceedance is obtained b y construct ing a three-dimensional h is togram o f j o i n t relat ive frequencies. Since there is no direct ional in fo rmat ion for the wave data, the approach taken is to f i nd the range for wave run-up b y assuming that waves are at tacking the dykes w i t h i n the angle o f propagat ion o f between 0° and 30°. I n order to include the angle o f propagat ion i n the predic t ion o f wave run-up, Equat ion 3.17 i n the S P M method is m o d i f i e d as: D n smooth,tab TT' £ O K= „ • H0yfyjp 6.3 where yp is the reduct ion factor for angle o f propagat ion and the value o f yp is estimated b y Equat ion 3 .21. 129 6.3.2 Application of the Hybrid of the DJPM and Simple Addition Method for Combining Tides, Storm surges, Wave Run-up, and Long-term Sea Level Rises The long- term rate o f sea level rise for R i c h m o n d is considered to be a constant, so the H y b r i d o f the D J P M and the Simple A d d i t i o n Me thod , the H y b r i d DJP-S imple A d d i t i o n Me thod , is designed for the inc lus ion o f a constant long- term trend i n sea levels. A l t h o u g h the component o f long- term sea level change is incorporated i n the est imat ion o f extreme sea levels, the effect o f us ing o ld data to project the future changes is not analyzed. Th is w o r k not on ly estimates the extreme sea level inc lud ing the long- term sea level changes, but also demonstrates the t i m i n g issues that must be addressed when o ld data are used to project future changes. A s shown in Figure 6.5, the goal is to estimate the extreme sea level b y adding tides, s torm surges, wave run-up, and long- term sea level rises. T w o majo r steps invo lved i n this approach are the estimate o f extreme sea levels composed o f t ides, s torm surges, and wave run-up using the D J P M as discussed in Section 6.3.1.2, and the addi t ion o f the long- term sea level increases and these extreme sea level estimates. The rate o f long- te rm sea level change adopted herein in this approach is 1.3 m m per year, w h i c h is the rate experienced i n R ichmond as discussed in Section 6 .1 . The rate o f the long- term sea level changes suggests that the cumulat ive effect o f rising sea levels f r o m 1972 to 2010 is approx imate ly 0.05 m. This means that when data col lected i n 1972 are used to estimate extreme sea levels, the estimations are under-estimated b y 0.05 m due to long- term sea level rises. A l t h o u g h 0.05 m is a smal l value, i t m a y make a big^dif ference i n the est imation o f extreme sea levels. Th is approach demonstrates the effect o f the long- term sea level increases on the est imat ion o f extreme sea levels b y pro ject ing the extreme sea levels for 2000 and 2010 (recal l that the exis t ing data are for the years 1972 to 1974). I n each scenario, a constant value o f sea level increase is added to the est imation 130 o f extreme sea level based on the D J P M , and the constant value is obtained b y m u l t i p l y i n g the dif ference between the projected and data year b y the constant rate o f 1.3 m m per year. 131 Table 6.1 Descr ip t ion o f the Qua l i t y Codes f r o m Mar ine Env i ronment Da ta Service Websi te QC Flag Description 0 B lank - N o qual i ty contro l (QC) has been per fo rmed 1 G o o d - Q C has been per formed: record appears correct 2 D o u b t f u l - Q C has been per formed: record appears doubt fu l 3 Erroneous - Q C has been per formed: record appears erroneous 4 Changes - The record has been changed as a result o f Q C 5 Acceptable - Q C has been per formed: record seems inconsistent w i t h other record 6 O f f Posi t ion - There is a p rob lem w i t h the b u o y pos i t ion or moor ing . Data m a y st i l l be useful 7 Reserved 8 Reserved - indicates miss ing elements 132 Annual Mean y -11.0013x + 3.0404 FT = 0.2617 > A A n N A A , > A {.N A A A A rTl A ^ A k ^ ^ A A * e . s a!" oi> ( J c& tk^  ( & of> ^ ^ > N o> scf> so<> so<» NoJ> sc?> scJ= ,c£ ^ ^ ^ ^ ^ sc?> ^ ^ sc?> NcJ> N c? ^ N o? N o? Year Figure 6.1 Long-term Trend of Sea Levels at Point Atkinson W di4ch(259#feiricl before :Mmi\nq d&ibarqe irrudune j piping a m - ' SECTION. &m>r line \ -Wdtitte-'m (c < 1.""""' . L - , y -<Basc line • For effseid'fri 6 6 ! 4735-1-3. • E L 11,0 - E L 9.0 .*lope*^ ;«i6votion9 egcqvok install WW drainage discharge piping before canolsiina earmorks •, ,'. Figure 6.2 Construction Drawing at West End of Blundell Road in 1974 New cfaiwqe ctechorqg piping \o be insfolld j for details -see 47#>-l -12 133 Required Data Observed sea levels Equation Critical Factors Observed Sea Levels Significant wave height (//,) Significant wave period (T, s) Duration (D, h) Slope of dyke ( 0) Depth fronting the slope (ds) Deep-water wave height (H0) Roughness and porosity of dyke ln(3600—) H =H "o ST1 Hj Wave Run-up 3-D Histogram Wave Run-up Observed Sea Levels Relative Frequencies Overall Equation Extreme Sea Levels = Observed Sea Levels+ Wave Run-up Figure 6.3 F l o w D iag ram o f the D J P M for Comb in ing Observed Sea Levels and Wave Run-up 134 Required Data Equation Critical Factors 3-D Histogram Observed sea level (O) Tide (T,) StormSurge = 0-Ti Storm Surges Wave Run-up Overall Equation Tides (T^ Tides Tides Significant wave height (Hs) Significant wave period (T, s) Duration (D, h) Slope of dyke ( 0) Depth fronting the slope (ds) Deep-water wave height (Ho) Roughness and porosity of dyke H =H ln(3600y) 'Storm Surges Extreme Sea Levels = Tides + Storm Surges+ Wave Run-up Figure 6.4 F l o w D iag ram o f the D J P M for Comb in ing Tides, S torm Surges, and Wave Run-up 135 Required Data Projected Year (Y P ) Data Year (YD) Rate of sea level rising (0 Observed Sea Levels (O) Tides (7,) Tides Equation SeaLevelRise = (Yp -YD)r StormSurge = o-T, Critical Factors Long-Term Sea Level Rises 3-D Histogram Wave Run-up Tides Overall Significant wave height Significant wave period (T, s) Duration (D, h) Slope of dyke ( 9) Depth fronting the slope (ds) Deep-water wave height (H0) Roughness and porosity of dyke H fcn(Q,-—,—) Storm Surges Extreme Sea Levels = Tides + Storm Surges+ Wave Run-up+ Long-Term Sea Level Rises Figure 6.5 F l o w D iagram o f the H y b r i d o f the D J P M and Simple A d d i t i o n Me thod for C o m b i n i n g Tides, Storm Surges, Wave Run-up, and L o n g - T e r m Sea Leve l Rises 136 7 RESULTS AND DISCUSSION Sections 7.1 and 7.2 present and discuss the results o f tests w h i c h assess the ef f icacy o f the D J P M for est imat ing extreme f lood and sea levels for R i chmond , B.C., respect ively. Section 7.3 concludes this chapter w i t h a discussion o f the sources o f error fo r the case study. 7.1 Assessment of the Direct Joint Probability Method The assessment o f the D J P M is achieved b y compar ing the estimates o f extreme f lood levels for specif ied return periods using three approaches: the D J P M , A n n u a l M a x i m a Method , and Simple A d d i t i o n Me thod . Extreme f lood levels on ly depend o n the components o f tides and storm surges, and therefore, al l three methods are capable o f achiev ing the f l ood level estimates. The results us ing the D J P M , Annua l M a x i m a Me thod , and Simple A d d i t i o n M e t h o d are presented i n Sections 7 .1 .1 , 7.1.2, and 7.1.3, respectively. Section 7.1.4 summaries and discusses these results. 7.1.1 Estimation of Extreme Flood Levels Using the Direct Joint Probability Method The est imat ion o f extreme f lood levels is achieved b y app ly ing the D J P M and the data o f tides and storm surges col lected at Point A tk inson from 1972 to 1974. The D J P M first constructs the two-d imensional h istogram using the t ime series o f tides and storm surges as shown in Table 7 .1 . Here the co lumn and r o w tit les indicate the m a x i m u m values o f the intervals. The cells i n Table 7.1 contain the relat ive frequencies associated w i t h the 137 corresponding t idal and storm surge intervals. The relat ive frequencies are sorted according to the descending order o f f l ood levels and the annual p robab i l i t y o f exceedance associated w i t h each f l ood level is calculated. The results are summarized i n F igure 7 .1 . Here the smoothed p lot d rawn represents the relat ionship between extreme sea level and return per iod and can be used to estimate the extreme sea levels for specif ied return periods. W h e n the numbers o f intervals used for construct ing the his togram increase, the curve representing the relat ionship between extreme sea level and return per iod becomes smoother. A s indicated i n Figure 7 . 1 , the estimated 200-year and 1250-year extreme f lood levels are approx imate ly 2.25 m and 2.45 m, respectively. 7.1.2 Estimation of Extreme Flood Levels Using the Annual Maxima Method The section presents the o f extreme f lood levels estimated b y app ly ing the Annua l M a x i m a M e t h o d and using the observed sea level data col lected at Point A t k i n s o n from 1914 to 1997. The A n n u a l M a x i m a M e t h o d f irst extracts the annual m a x i m a series. W h i l e Table 7.2 presents the annual m a x i m a series, i t should be noted that the or ig ina l data from Point A tk inson for the f o l l o w i n g years are miss ing entirely: 1923, 1924, 1925, 1926, 1928, 1929, 1930, 1931, 1934, 1935, 1936, 1937, 1938, 1940, 1941, 1942, 1943, 1945, 1946, 1949, 1959, and 1960. These miss ing data are def ined as a broken record, so the remain ing data are used and treated as a cont inuous record. I n addi t ion, a smal l out l ier w i t h the water level o f 3.500 m above the C H A R datum is detected i n the year 920. Therefore, 1920 is also removed from the annual m a x i m u m series. The annual m a x i m a series is f i t ted, and the three-parameter L o g - N o r m a l D is t r ibu t ion ( L N ( 3 ) ) is determined b y bo th the C F A and R E L A N software packages to be the most appropriate d is t r ibut ion type for representing the annual m a x i m u m series. The estimated L N ( 3 ) is shown in Figure 7.2 along w i t h the data series, where the theoret ical d is t r ibut ion is the 138 sol id l ine, and the 9 5 % confidence intervals are marked b y the dashed l ines. F r o m Figure 7.2, the 200 year extreme f l ood level is estimated as 2.60 m above the G S C datum, and the 1250-year extreme f l ood level is estimated as 2.68 m above the GSC datum. 7.1.3 Estimation of Extreme Flood Levels Using the Simple Addition Method The Simple A d d i t i o n M e t h o d is also appl ied to estimate f l ood levels b y us ing data for tides and storm surges col lected at Point A t k i n s o n from 1914 to 1997. I t f i rst requires the determinat ion o f the frequency distr ibut ions for tides and storm surges, and a s imi lar procedure as described for the A n n u a l M a x i m a M e t h o d is undertaken for the t ime series o f tides and storm surges. The results o f the frequency analyses o f tides and storm surges are presented i n Figure 7.3, where the estimated frequencies for the t ime series o f tides are p lo t ted on the dashed l ine and those for s torm surges they are p lot ted on the dotted l ine. The estimated f lood levels are the sol id l ine they are determined b y adding the tides and storm surges for the corresponding return periods. The 200-year f l ood level is estimated to be 3.096 m above the G S C datum, calculated b y adding the 200-year t ide (2.023 m ) and the 200-year s torm surge (1.073 m ) . The 1250-year f l ood level is 3.196 m above the GSC datum, obtained b y adding the 1250-year t ide (2 .040,m) and the 1250-year storm surge (1.156 m ) . 7.1.4 Assessment of the Direct Joint Probability Method for Estimating Extreme Flood Levels The general ef f icacy o f D J P M m a y be assessed b y comparisons o f the extreme f lood level estimates from applications o f the D J P M , A n n u a l M a x i m a M e t h o d , and Simple A d d i t i o n 139 Method . The estimated extreme f lood levels resul t ing from these three methods are summarized in Table 7.3. A s expected, the appl icat ion o f the Simple A d d i t i o n M e t h o d produces the highest est imat ion o f extreme f lood levels among al l approaches due to its conservative nature, and the results obtained for the Simple A d d i t i o n M e t h o d are 23 to 2 8 % higher than the results obtained for the D J P M , and 19% higher than those for the A n n u a l M a x i m a Me thod . Furthermore, the results obtained for the Annua l M a x i m a M e t h o d are closer to the results obtained for the D J P M . The percent di f ference o f the 200-year and 1250-year f l ood levels estimated b y these two methods are 13.46% and 8.58%), respectively, relat ive to the D J P M values, w i t h the dif ference in the number o f year o f required data be ing 59 years. The reason fo r the lower f l ood level values determined i n the appl icat ion o f the D J P M is that the sea levels for the years 1972, 1973, and 1974 are general ly l o w f lows years compared w i t h those i n the 62 years o f data used for the other methods o f analysis. The observed sea levels for 1972, 1973, and 1974 rank 44, 37, and 62 i n descending order o f observed sea levels, for the 62 year h is tory o f record used. Therefore, the results o f this method are sensitive to the choice in years used for the estimations, because they are largely dominated b y the m a x i m u m tides and m a x i m u m storm surges captured dur ing the years o f data used. The D J P M also exhibi ts the disadvantage o f not be ing able to prov ide an extreme f l ood level greater than the data available. Consider ing the data used i n this method, the results demonstrate that the D J P M is able to prov ide a reasonable estimate o f the f lood level us ing data w i t h short periods o f record. Under l im i ted data avai lab i l i ty , the D J P M can st i l l p rov ide a reasonable est imat ion o f extreme f lood levels. I t should also be noted that the estimations resul t ing for the D J P M and Annua l M a x i m a M e t h o d are smaller than the current 200-year f l ood level o f 2.8 m used b y R ichmond. Thus, the current design f l ood is w e l l w i t h i n the estimations computed using al l o f these approaches. 140 7.2 Estimation of Extreme Sea Levels using the DJPM and Hybrid DJP-SimpIe Addition Method The D J P M and the H y b r i d DJP-Simple A d d i t i o n M e t h o d are appl ied to estimate extreme sea levels for R i chmond , because the extreme sea levels for R i c h m o n d include the effects o f tides, s torm surges, wave run-up, and long- term sea level increases. Sect ion 7.2.1 presents the results o f the appl icat ion o f the D J P M for est imating extreme sea levels based on data for either a combinat ion o f observed sea levels, t ides, and waves or a combina t ion o f t ides, storm surges, and waves. Section 7.2.2 presents the results o f the appl icat ion o f the H y b r i d DJP-Simple A d d i t i o n M e t h o d for est imat ing extreme sea levels based on tides, s torm surges, wave run-up, and long- term sea level increases. Section 7.2.3 summarizes and discusses the results. 7.2.1 Estimations of Extreme Sea Levels Using the Direct Joint Probability Method Section 7.2.1.1 presents the estimations o f extreme sea levels us ing observed sea levels and wave run-up, and Section 7.2.1.2 presents the estimations o f extreme sea levels us ing tides, storm surges, and wave run-up. 7.2.1.1 D J P M App l i ca t ion to Observed Sea Levels and W a v e Run-up The resul t ing extreme sea levels, w h i c h include the effect o f observed sea levels and wave run-up, estimated using the D J P M and data for observed sea levels and waves f r o m 1972 to 1974 are shown i n Figure 7.4. Recal l that wave run-up is a func t ion o f sea levels and wave in fo rmat ion , based on Equat ion 3.16. Figure 7.4 is obtained based on t w o years o f continuous 141 data; thus, the p lo t o f results starts f r o m the m i n i m u m sea level for the t w o years o f data record. For the purpose o f f i nd ing extreme sea level values, the focus is on re turn periods o f 100 years or more. The p lo t shown i n Figure 7.4 has a dist inct cont inu i ty where the wave run-up begins at the m i n i m u m sea level o f 1.5 m. Wave run-up increases gradual ly w i t h the increasing sea levels above 1.5 m. The estimated 200- and 1250-year extreme sea level i nc lud ing wave run-up are 3.25 m and 3.69 m above the GSC datum, respectively. 7.2.1.2 D J P M App l i ca t ion to Tides, Storm Surges, and W a v e Run-up The results o f the appl icat ion o f the D J P M for the data for t ides, s torm surges, and waves f r o m 1972 to 1974 are summarized i n Figure 7.5. S imi lar to the observations for Figure 7.4, Figure 7.5 demonstrates that wave run-up is generated at the m i n i m u m water depth o f 1.5 m. The estimated 200- and 1250-year extreme sea levels are 3.78 m and 3.90 m above the GSC datum, respectively. The effect o f the angle o f propagat ion on the est imat ion o f wave run-up and thus the est imat ion o f extreme sea level is summarized i n Figure 7.6. A s shown i n the f igure, wave run-up decreases as the propagat ion angle increases. Thus, wave run-up and extreme sea levels are at a m a x i m u m w h e n the propagat ion angle is 0° , and wave run-up and extreme sea levels decrease b y 2 % as the wave propagat ion angle increases t o : 3 0 ° . The 200- and 1250-year extreme sea levels w i t h a propagat ion angle o f 30° are estimated as 3.67 m and 3.79 m above the GSC datum, respectively. 142 7.2.2 Hybrid DJP-Simple Addition Method Application to Tides, Storm Surges, Wave Run-up, and Long-term Sea Level Changes The H y b r i d DJP-Simple A d d i t i o n M e t h o d is appl ied to the est imat ion o f extreme sea levels for R i c h m o n d so as to include the effects o f t ides, s torm surges, wave run-up, and long-term sea level increases. Figure 7.7 shows that the effect o f the long- term sea level changes on the extreme sea levels increases as the return per iod increases. For return periods greater than 200 years, the extreme sea levels predicted increase considerably for the years 2000 and 2010, because the added affect o f long- term sea level rise generates increased wave run-up. The 200-year extreme sea levels for the years 1972, 2000, and 2010 are 3.78 m, 3.81 m, and 3.82 m above the GSC datum, respectively. The 1250-year sea levels for the years 1972, 2000, and 2010 are 3.90 m, 4.15 m, and 4.16 m above the GSC datum, respectively. 7.2.3 Discussion of the Estimations of Extreme Sea Levels for Richmond I n this thesis, the D J P M is not on ly appl ied to estimate the extreme sea levels using di f ferent combinat ions o f data that capture the effects o f t ides, s torm surges, and wave run-up, but also to estimate the effect o f the angle o f propagat ion on wave run-up and resul t ing extreme sea levels. The H y b r i d DJP-Simple A d d i t i o n M e t h o d demonstrates the effect o f long- term sea level increases. The f o l l o w i n g discussion reviews the results o f the est imat ion o f extreme sea levels w i thou t the long- term sea level increase, the est imat ion o f extreme sea levels w i t h di f ferent angles o f propagat ion, the est imation o f extreme sea levels w i t h the long- term sea level increase, and the relevance o f these results for R ichmond . 143 Wave impacts at the contact point o f coastal structures are characterized b y wave run-up. The extreme sea levels, exc lud ing the long- term sea level changes, are estimated using the D J P M appl ied for analysing observed sea levels and wave run-up, and tides, s torm surges and wave run-up. A s shown i n Table 7.4, the results obtained for the D J P M appl icat ion based on on ly two data types are smaller than those for the D J P M appl icat ion based on three data types. B y breaking d o w n the observed sea levels into tides and storm surges, more combinat ions o f observed sea levels are produced to be used i n the estimations o f extreme sea levels. This is a def in i te advantage i n the case o f short data sets. I t also demonstrates that the wave run-up can make a dramatic di f ference i n the extreme sea levels as shown i n F igure 7.8, where the two higher-valued plots are for computat ions inc lud ing wave run-up. Analys is o f the propagat ion angle is also undertaken for the case i n w h i c h extreme sea levels are estimated based on tides, storm surges, and wave run-up. T w o propagat ion angles are examined, 0° and 30°, and the resul t ing extreme sea level estimates are summar ized i n Table 7.5. A s shown in the table, the 200- and 1250-year extreme sea levels are bo th reduced for the 30° propagat ion angle relat ive to the 0° angle. Thus, the ranges o f 200- and 1250-year sea level estimates for R i c h m o n d are f r o m 3.78 to 3.67 m and from 3.90 to 3.79 m above the GSC datum, respectively. I n addi t ion, the predicted long- term rise in sea levels is inc luded i n the est imation o f extreme sea levels for di f ferent points i n t ime and these estimates are summar ized in Table 7.6. The results demonstrate the importance o f inc lud ing the long- te rm sea level changes in these estimates. For R i chmond , where long- term sea rises are prevalent, the extreme sea level estimates w i t h higher return periods show dramatic differences between cases that incorporate long- term sea level change, and those that do not. These results indicate the importance o f inc lud ing wave run-up i n the est imat ion o f extreme sea levels for locations where long- term sea level increases are signif icant. These results also i m p l y that for such cases, the standard 144 approach o f us ing short data records that are re lat ive ly o ld i n appl icat ions o f the D J P M , J P M , and R J P M m a y be mis leading because the data m a y be insuf f ic ient to characterize current situations. The degree o f impact o f the propagat ion angles is re la t ive ly smal l compared w i t h the degree o f impact o f the long- term sea level changes. Compar ing the results w i t h the current f l ood construct ion level o f 3.35 m for R ichmond , the current f l ood construct ion is w i t h i n i n the 200-year estimates resul t ing from al l o f the aforement ioned methods. A l t h o u g h the effects o f vegetat ion as observed at the west coast o f R i c h m o n d are not considered i n any method, the vegetat ion can certa in ly prov ide a damping effect on waves approaching the shore. However , there is no research or exper imental results regarding the effect o f vegetat ion on the propagat ion o f waves. Therefore, the estimations prov ided here are more conservative than the actual sea level cond i t ion , since no adjustment for the damping effect o f vegetat ion is incorporated. Nevertheless, i n the long- term, when the sea levels cont inue to rise, the damping effect m a y be weakened. 7.3 Sources of Error for Estimates of Extreme Flood and Sea Levels The errors invo lved i n the est imation o f extreme f l ood and sea levels at R ichmond , B.C. include the errors that are inherent for the D J P M , and the errors caused b y the assumptions due to the lack o f data i n R ichmond . One o f the major errors associated w i t h the appl icat ion o f the D J P M is that w h i c h can results from the construct ion o f the plots that approximate the relat ionship between extreme sea level and return per iod. These plots are used to estimate the sea level for specif ied return periods, and the smoothness o f them depends on the number o f data available and the intervals o f the histograms developed. However , the data used i n this case study are not suff ic ient for construct ing smooth plots, even w i t h a large number o f intervals i n 145 the histogram. Thus, the plots cannot be represented b y a s imple equat ion for the relat ionship between extreme sea level and return per iod. The assumptions made due to lack o f data in R i c h m o n d and the errors that m a y occur due to these assumptions are summarized as fo l lows: 1. Tides, s torm surges, and waves are the govern ing factors o f sea level f luctuat ions. This assumption is va l id for most locations in R i chmond , but as prev ious ly stated, for two problemat ic locations in R ichmond , other factors that are not analyzed here contr ibute to extreme sea levels. These factors are the effects o f r iver currents (e.g., near Terra Nova) and the effects o f water transport (e.g., near Garry Point ) . Ignor ing these factors could lead to underestimates o f sea levels i n these locations, but inc lud ing them w o u l d lead to overestimates at most locations along the dyke. 2. The data col lected at Point A tk inson represent the data at R i chmond . Th is assumption m a y be weak for R i chmond since the oceanographic characteristics between Point A t k i n s o n and R i c h m o n d have the least s imi lar i t ies among al l stations considered. There is no evidence support ing or re fu t ing the va l id i t y o f this assumption, because R ichmond does not have any data for comparison. 3. The wave data contain ing miss ing data points are treated as cont inuous. Th is assumption is weak as the wave data at Point A tk inson exhib i t miss ing data f r o m t ime to t ime, and the average f requency o f sampl ing is 2.3 hours. There is no record o f an extreme event for the per iod o f data col lect ion, so treating the wave data as a cont inuous series is considered a standard procedure. The disadvantage o f re l y ing on this assumption is that some values o f tides and storm surges are excluded in order to accommodate the miss ing wave data, so some extreme values m a y be excluded at the same t ime. 4. The elevat ion at the toe o f the dyke along the coastl ine is constant at 1.52 m. The elevat ion at the toe o f the dyke is not constant a long the w h o l e sea dyke due to shoal ing 146 and settlement o f the dyke structures. Because there is on l y one record o f measurement avai lable at B lunde l l Road, this assumption is necessary, but is weak. Shoal ing increases the sedimentat ion at the toe o f the dyke and increases the elevat ion at the toe, whereas settlement reduces the elevat ion at the toe. Var ious locations can display various degrees o f shoal ing and settlement, thus the elevations at the toe o f the dyke are not on ly changing w i t h t ime but are also di f ferent at d i f ferent locations. These elevations are important for the est imation o f wave run-up as wave run-up increases w i t h decreases in the toe elevat ion, see Equat ion 3.16. 5. The slope o f dyke is assumed to be a constant 1:2 throughout the dyke system. A s shown in Figure 5.4, the ma jo r i t y o f the dyke is not constructed according to the standard design. A n actual f ie ld survey conducted b y the author showed that the slope varies f r o m 1:2 to 1:5 along the sea dyke. 6. The bo t tom slope is 1:10. The wave run-up is calculated based on the wave run-up curves fo r a bo t tom slope o f 1:10, w h e n i n fact R i c h m o n d has an almost hor izontal bo t tom slope. A qu ick compar ison o f the wave run-up calculated us ing the bo t tom slope o f 1:10 and that for the hor izontal bo t tom slope demonstrates that the wave run-up calculated for the 1:10 slope case underestimates the actual wave run-up and therefore the extreme sea levels that include wave run-up estimated in this w o r k l i ke ly underestimate the actual extreme sea level. 7. Vegetat ion does not affect the extreme sea levels. A s discussed earlier, the effect o f vegetat ion is not quant i f ied yet, so this assumption is necessary. U n t i l better in fo rmat ion is p rov ided, the approach taken i n this thesis m a y be used to prov ide a conservative extreme sea level . 8. The long- term rise i n sea levels at R i c h m o n d is at a rate o f 1.3 m m per year. This is an average value obtained f r o m 61 years o f data, based on actual data col lected at Point 147 Atk inson . 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O T © d d • © • • © •:d':: •:d'' vP:! !::dr !'d!:! o o o o o o O o O O © C) o o o o o o O © o o © o o o o o © © © O © o vo o o o © © © © © © © © Ci o o o o o o O o o o o d d d d d d d d d d d c> c> •n Ov Os <CJ C i c> C) Os o CJ C i C i n C5 >n c c> c> c> Ci co • • Ci i ci 1 Ci (uiinep 3 S O aq) SAoqe 'ui) sap;x 149 Table 7.2 Screened A n n u a l M a x i m a Series o f Observed Sea Levels Year Month Day Time Annual Max (m, above C H A R datum) Year Month Day Time Annual Max (m, above C H A R datum) 1914 11 22 10 4.930 1968 12 24 10 5.440 1915 12 8 7 5.070 1969 12 11 8 5.310 1916 1 6 7 5.240 1970 12 7 12 5.350 1917 12 31 8 5.070 1971 1 15 9 5.210 1918 1 14 7 5.120 1972 12 22 8 5.330 1919 12 24 7 5.050 1973 12 13 9 5.280 1921 12 21 11 5.120 1974 1 14 10 5.380 1922 2 16 8 4.900 1975 12 26 11 5.220 1927 12 12 9 4.970 1976 2 18 7 5.060 1932 12 22 12 5.370 1977 12 15 10 5.450 1933 12 19 8 5.340 1978 2 9 7 5.460 1939 12 14 9 5.150 1979 12 24 10 5.320 1944 1 1 10 5.440 1980 12 26 9 5.220 1947 12 18 10 5.100 1981 11 14 8 5.280 1948 1 1 10 5.370 1982 12 16 7 5.600 1950 1 10 10 5.220 1983 1 27 5 5.490 1951 12 1 8 5.400 1984 11 27 10 5.230 1952 12 30 6 5.430 1985 2 11 10 5.040 1953 1 20 9 5.340 1986 11 18 8 5.050 1954 1 7 8 5.210 1987 1 3 9 5.520 1955 12 6 11 5.120 1988 11 22 15 5.200 1956 1 5 11 5.210 1989 3 11 7 4.970 1957 12 24 8 5.430 1990 12 4 8 5.250 1958 1 10 9 5.250 1991 2 2 8 5.250 1961 2 15 6 5.060 1992 1 25 9 5.280 1962 2 8 8 5.230 1993 12 13 6 5.240 1963 1 1 10 5.150 1994 12 19 7 5.200 1964 12 22 9 5.330 1995 11 29 11 5.240 1965 12 28 10 5.150 1996 2 20 7 5.340 1966 12 4 11 5.330 1997 1 1 10 5.240 1967 12 5 9 5.570 150 Table 7.3 Summary o f Estimates o f Extreme F lood Levels Direct Joint Probability Method Return Period (year) Extreme Flood Level (m, above the GSC datum) % Difference relative to DJPM 200 2.25 0 .00% 1250 2.45 0 .00% Annual Maxima Method Return Period (year) Extreme Flood Level (m, above the GSC datum) % Difference relative to DJPM 200 2.60 13.46% 1250 2.68 8.58% Simple Addition Method Return Period (year) Extreme Flood Level (m, above the GSC datum) % Difference relative to DJPM 200 3.10 27 .42% 1250 3.20 23 .44% Table 7.4 Summary o f Estimates o f Extreme Sea levels Joint Probability of Observed Sea Levels and Wave Run-up Return Period (year) Extreme Sea Level (m, above the GSC datum) 200 3.25 1250 3.69 Joint Probability of Tides, Storm surges, and Wave Run-up Return Period (year) Extreme Sea Level (m, above GSC datum) 200 3.78 1250 3.90 151 Table 7.5 Summary o f Estimates o f Ext reme Sea Levels under D i f fe ren t Propagat ion Angles P ropaga t i on A n g l e = 0° Return Period (year) Extreme Sea Level (m, above the GSC datum) 200 3.78 1250 3.90 P r o p a g a t i o n A n g l e = 30° Return Period (year) Extreme Sea Level (m, above the GSC datum) 200 3.67 1250 3.79 Table 7.6 Summary o f the Estimates o f Ext reme Sea Levels Projected fo r D i f fe rent Years based on L o n g - T e r m Sea Leve l Changes 1974 . Return Period (year) Extreme Sea Level (m, above the GSC datum) 200 3.78 1250 3.90 2000 Return Period (year) Extreme Sea Level (m, above the GSC datum) 200 3.81 1250 4.15 2010 Return Period (year) Extreme Sea Level (m, above the GSC datum) 200 3.82 1250 4.17 152 o o o o o o o o o o o o p p p p CO C\i i-1 O ^ fN (OSD 3 A o q e 'UJ ) |3A3~| eas aiuaijxg 153 o o o co o o IT) O O O CN O O IT) ( 0 S 9 9Aoqe tu) |aAa-| J3)ey\y\ pe/uasqo 154 155 o o o o o o o o o o o o q q o o q q o (OSO a A o q e ui) \OAST\ B O S aiuajjxg 156 o o o o o o o o o o o o o o o o o o o o o p (OSO 3 A o q e 'ui) |3A3-| eas auiajpcg 157 o o o o o o o o 00 CO CM q o q LO •<fr CO o o o o CO •>* CNJ o CO CO CO CO o o o o C3 fl O (0S9 9"Oqe 'ui) |aAan eas aiuajjxg 158 159 (0S9 aAoqe 'ui) |3Aan eas 160 8 CONCLUSIONS AND RECOMMENDATIONS A key element i n the design o f coastal structures is the crest elevat ion, and the crest elevat ion is general ly determined b y the extreme sea condi t ions. Est imat ing extreme sea condi t ions requires an understanding o f oceanographic science and coastal engineering. B y determin ing the local oceanographic environment and emp loy ing coastal engineering knowledge, the process o f predic t ing extreme sea condi t ions is demonstrated for a case study in R ichmond , B.C., Canada. Th is study provides an example o f the c o m m o n problems facing coastal engineers. Several recommendat ions m a y be appl ied to estimate extreme sea condit ions for R i c h m o n d in the future, and some m a y be appl ied in general practice. W h i l e oceanographic science explains numerous factors causing sea level variat ions, coastal engineering addresses the concerns regarding the funct iona l i ty o f coastal structures and leads to methodologies for predic t ing extreme sea condit ions. The factors cont r ibut ing to sea level f luctuat ions include w i n d waves, storm surges, astronomical t ides, tsunami, E l N i n o , and long- term sea level increases due to c l imato log ic and geologic effects. These components generate di f ferent degrees o f impact on sea levels at di f ferent locat ions, so the f irst step in determin ing extreme sea condi t ions is to define the cr i t ica l components causing extreme condi t ions at the site o f interest w i t h the understanding o f the fa i lure modes o f coastal structures. Failures such as o v e r f l o w i n g and over topping occur w h e n coastal structures have insuf f ic ient crest elevations. T o prevent ove r f l ow ing , the crest elevat ion should be at least at the extreme f lood level , w h i c h is the sea level w i thou t the inf luence o f waves. T o m i n i m i z e overtopping, the crest elevat ion should be at least at the extreme sea level , w h i c h includes the effect o f waves. The increase i n estimates o f extreme sea levels due to the inf luence o f waves m a y be determined b y the wave run-up or wave over topping discharges. Wave run-up and over topping discharges are impacts o f waves on coastal structures, and they can be computed based on exist ing empir ical 161 methods. However , there is no standard regarding the a l lowable over topp ing discharges, so wave run-up is st i l l the govern ing factor i n the est imat ion o f extreme sea levels. Methodologies are then constructed in the second step to combine these components. Methods for est imating extreme f lood levels include the A n n u a l M a x i m a , ^ -Largest M a x i m a , S imple A d d i t i o n Methods, and the J P M and R J P M . These methods are incapable o f est imat ing extreme sea levels due to the deficiencies in each method, such as the inab i l i ty to represent the di f ferent components o f extreme sea condi t ions, to ident i fy the actual j o in t p robab i l i t y o f comb in ing factors, or to estimate the j o i n t p robab i l i t y o f any number o f dependent factors. I n order to overcome these deficiencies and estimate extreme sea levels, the D J P M is developed i n this thesis, w h i c h is capable o f combin ing any number o f dependent variables in the est imat ion o f extreme sea levels. The results o f extreme sea levels estimated b y the D J P M for R i c h m o n d are reasonable based on on ly two years o f data. The D J P M can also accurately port ray the relat ionship between sea levels and wave run-up, i n that wave run-up is on ly generated at suff ic ient sea levels (e.g., 1.5 m i n the R i c h m o n d case), and higher wave run-up is generated w i t h higher sea levels. The H y b r i d DJP-Simple A d d i t i o n M e t h o d is also developed to account for the inf luence o f long- term sea level changes on extreme sea levels. The results o f the H y b r i d M e t h o d suggest the importance o f inc lud ing long- term sea level increases and o f account ing for data records that are re lat ively o ld . The long- term sea level increases amp l i f y the effect o f wave run-up, and the estimated extreme sea levels increase dramat ica l ly under such condit ions. For this reason, o ld data used in the est imat ion o f extreme sea levels can be mis leading w h e n used to represent future condit ions. A l t h o u g h this thesis demonstrates successful appl ications o f the D J P M , more tests should be per formed to determine the required length o f data for a reasonable est imat ion o f extreme sea levels. Future tests should include a sensi t iv i ty analysis regarding h o w the degree o f dependency between variables affects the length o f data required. Other tests should include sensi t iv i ty 162 analysis o f variables that m a y affect the accuracy o f the D J P M , such as number o f components invo lved in the est imat ion o f extreme sea levels. A demonstrat ion o f the decision process for est imating extreme sea levels is prov ided for a case study based on R ichmond , B.C., Canada. The results o f the case study indicate that the current standard for the Sea D y k e in R i c h m o n d is w i t h i n the range o f the 200-year sea levels estimated w i t h al l methods. However , as discussed in Chapter 7, there are several l imi tat ions invo lved in the computat ion due to the poor avai lab i l i ty o f data and the necessary assumptions. T o prov ide a better est imat ion o f extreme sea levels, recommendat ions for R i c h m o n d include: 1. Co l lec t ing data o f observed sea levels, t ides, and waves at R i chmond . Acco rd ing to previous studies and results i n Chapter 7, the D J P M , J P M , and R J P M can prov ide reasonable estimates o f design sea levels w i t h short te rm data sets. Thus, i f R i chmond starts to col lect the observed sea level , t ide, and wave data n o w , i n a short t ime, the C i t y w i l l achieve more representative estimates o f design sea levels. 2. Co l lec t ing direct ional wave data. The direct ional i n fo rmat ion o f waves is important for est imat ing wave run-up. Th is thesis makes an assumption that al l waves are attacking the shore at angles ranging f r o m 0° to 30°. The actual data w o u l d p rov ide better estimations o f wave run-up and extreme sea levels. 3. Conduct ing a f ie ld survey at the toe o f the dyke. The assumption o f a constant elevation at the toe along the entire dyke is weak because factors, such as shoal ing and settlement o f the dyke structure, can inf luence the actual e levat ion at the toe o f the dyke. The value o f the elevations at the toe inf luences the water depth f ron t ing the slope o f the dyke and the computat ion o f wave run-up. 4. Inc lud ing the effects o f r iver current and water transport i n the est imat ion o f extreme sea levels at applicable locations. M o r e research should be conducted to estimate extreme sea levels consider ing the effects o f river current and water transport. A s ment ioned 163 earlier, one o f the locations i n R i chmond that has experienced over topp ing is at the nor th end o f the Sea D y k e due to the currents caused b y the discharges f r o m the Fraser R iver so r iver effects certa in ly p lay an important role here. S imi la r ly , the disturbance caused b y water transport i n the Garry Point area o f R i c h m o n d can also be a potent ia l cause o f over topping. 5. M o n i t o r i n g long- term sea level changes. This thesis demonstrates that long- term sea level changes increase the amount o f wave run-up. Thus, the standard o f the crest elevat ion o f the dyke system in R ichmond w i l l need to be raised due to the long- term change i n sea levels. A mon i to r ing program should be implemented i n R i c h m o n d to have an accurate annual rate o f increasing sea levels. 6. Iden t i f y ing a standard for a l lowable over topping discharges. The controversy regarding the use o f wave run-up to estimate the crest elevat ion is va l i d , but a standard for a l lowable over topping discharges that is suitable for R i c h m o n d or the site o f interest should be constructed f i rst before app ly ing this approach. 7. Upda t ing the extreme sea levels regular ly. 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Wave forecasting and hindcast ing i n deep and shal low water, i n Handbook of Coastal and Ocean Engineering, Chapter 5, G u l f Publ ish ing Co., Houston, T X , U S A . Wat t , W . E., La them, K. W . , N e i l l , C. R., Richards, T. L., and Rousselle, J. (1989). Hydrology of floods in Canada: a Guide to Planning and Design, Nat iona l Research Counc i l o f Canada, Ot tawa, O N , Canada. 169 APPENDICES 170 Appendix A: Wave Run-up Curves A 1. Wave Run-up Curve for djH'0 = Oon a 1:10 Bottom Slope from CERC (1984) O . I 5 h O. I ' — O..I 0.15 0 2 0.3 0.4 0 .50 .6 O S I.O 1.5 2.0 Slope ( C O t 6) 3 0 4 0 S O 6.0 8.0 IO.O 171 A 2. Wave Run-up Curve for ds/H0 » 0.45 on a 1:10 Bottom Slope from CERC (1984) Slope ( cot 0 ) 172 A 3. Wave Run-up Curve for ds/H0 « 0.8 on a 1:10 B o t t o m Slope f r o m C E R C (1984) 173 A 4. Wave Run-up Curve for and ds/H0 > 3.0 on a 1:10 B o t t o m Slope from C E R C (1984) SlOpe (C0t 8) I S w i l K , 1958.0.1 174 A 5. Wave Run-up Curve for djH0 = 3 on a Hor izonta l B o t t o m f r o m C E R C (1984) Struc ture Slope iiotB) 175 A 6. Wave Run-up Curve o f ds/H0 = 5 on a Hor izonta l B o t t o m f r o m C E R C (1984) Structure Slope (cot 9) 176 A 7. Wave Run-up Curve for ds/H'0 = 8 on a Hor izon ta l B o t t o m f r o m C E R C (1984) S t r u c t u r e S l o p e {Cot6) 177 Appendix B: Overtopping Parameters B 1. Over topp ing Parameters for a Smooth Ver t ica l W a l l on a 1:10 B o t t o m Slope f r o m C E R C (1984) 4a. 178 B 2. Over topp ing Parameters for a 1:10 Structure Slope on a 1:10 B o t t o m Slope from C E R C (1984) 179 Appendix C: List of Notations S y m b o l D e s c r i p t i o n A Over topp ing parameter Ai A m p l i t u d e o f constituent i a Wave ampl i tude a Empi r i ca l coeff ic ient B Over topp ing parameter Bw B e r m w i d t h P Ang le o f the propagat ion d i rect ion D Tota l water depth D Dura t ion dh Wave depth over the b e r m ds Water depth at structure toe 8 Ext remal index Fix) Cumulat ive d is t r ibut ion funct ion o f x f Relat ive frequency ft Noda l correct ion o f constituent i M Probabi l i ty density func t ion o f x m Probabi l i ty density func t ion o f s torm surges MC-y) probabi l i t y density funct ion o f tides g- Phase lag for constituent i yp Reduct ion factor for an obl ique wave attack 7b Reduct ion factor for a be rm Reduct ion factor for slope roughness and poros i ty jh Reduct ion factor for a shal low foreshore 7s Adjus tment factor for scale effects H Wave height Ho Incident deep-water wave height Hmax M a x i m u m wave height Hrms Root mean square height Hs Signi f icant wave height L Wave lengths Lberm B e r m length m N u m b e r o f storm surge intervals m0 Area under the spectral density func t ion mn Rank M M e a n N Tota l number o f data points ND Design l i fe n Number o f t ide intervals k Parameter o f the d is t r ibut ion P Probabi l i ty o f exceedance Pi H o u r l y probab i l i t y o f exceedance Pe Encounter p robab i l i t y Pm Empi r i ca l p robab i l i t y o f exceedance or p lo t t i ng pos i t ion 180 PN Sample probab i l i t y o f exceedance py Annua l p robab i l i t y o f exceedance Q Over topp ing rate Qo Empi r i ca l coeff ic ient R N u m b e r o f extreme events Rb Dimensionless crest height Smooth lab Wave run-up on smooth and impermeable slopes Rw Wave run-up Ru,2% Wave run-up level above the st i l l water level w h i c h exceeds 2 % o f the incoming waves cr, Angu la r speed o f constituent i q Standard deviat ion T Wave per iod Tp Peak wave per iod TR Return per iod T S torm length 6 B o t t o m slope in front o f a structure ut N o d a l correct ion for constituent i Vt E q u i l i b r i u m t idal phase at r=0 o f consti tuent / Ui N o d a l correct ion for constituent i <!;eq Equivalent breaker parameter for a slope w i t h a be rm. £0p Breaker parameter Y(t) Meteoro log ica l l y induced level Zo(t) M e a n sea level z Ext reme sea level C(t) Sea level 181 

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