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Proudman Oceanographic Laboratorylnternal Document No. 65

EXTREME SEA-LEVELS AT THE UK A-CLASSSITES: SITE-BY-SITE ANALYSES.

M.J. DIXON & J.A. TAWN

March 1994

P R O U D M A NO C E A N O G R A P H I C

L A B O R A T O R Y

PROUDMAN OCEANOGRAPHIC LABORATORY

Bidston ObservatoryBirkenhead, Merseyside, L43 7RA, UK

Tel: 0151 653 8633Telex: 628591 Ocean B

Fax: 0151 653 6269

Director: Dr. B.S. McCartney

Natural Environment Research Council

ESTIMATES OF EXTREME SEA CONDITIONSFinal Report

EXTREME SEA-LEVELS AT THE UK A-CLASSSITES: SITE-BY-SITE ANALYSES.

Mark J. Dixon and Jonathan A. Tawn

Department of Mathematics and Statistics,Lancaster University,Lancaster LA1 4YF.

In collaboration with

The Proudman Oceanographic Laboratory,

Bidston Observatory,Birkenhead,

Merseyside L43 7RA.

March 1994

1

CONTRACT

This report describes work funded by the Ministry of Agriculture, Fisheries and Food(Flood and Coastal Defence Division) under contract CSA 1782 and Commission FD0303 with the NERC Proudman Oceanographic laboratory (POL). POL’s Nominated Of-ficer was Mr G Alcock. The Ministry’s Project Officer was Mr A C Polson. Publicationdoes not imply endorsement by the Ministry of the report’s conclusions or recommenda-tions.

@Copyright Ministry of Agriculture, Fisheries and Food 1994.

2 CONTENTS

Contents1 Introduction 19

2 Data and the Sea-Level Process 21

2.1 Description of the Data and Location of the Sites . . . . . . . . . . . . . . 21

2.2 The Sea-level Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 The Extreme Sea-level Process . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Data Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 Illustration Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Concepts used in the Design Stage 45

3.1 Return Levels and Return Periods . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Future Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Existing Separate Site Extreme Value Methods 49

4.1 Annual Maxima Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 .1 .1 NoTrend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.2 With Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.3 Previous Applications . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 r-largest Annual Events Method . . . . . . . . . . . . . . . . . . . . . . . . 52

4 .2 .1 NoTrend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.2 With Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


4.3 Joint Probabilities Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 .3 .1 NoTrend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 .3 .2 WithTrend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


4.4 Revised Joint Probabilities Method . . . . . . . . . . . . . . . . . . . . . . 55

4.4.1 Obtaining the Distribution of the Annual Maximum. . . . . . . . . 57

4.4.2 Obtaining F, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.3 Estimating Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4.4 Obtaining Return Level Estimates . . . . . . . . . . . . . . . . . . . 59

4 .4 .5 WithTrend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60


4.5 Maximum Likelihood Estimation and Measures of Precision . . . . . . . . 60

CONTENTS 3

5 Comparison of Methods 63

5.1 Assumptions of the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Relationships Between Methods . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Additional Considerations for Comparison . . . . . . . . . . . . . . . . . . 66

5.4 Recommendations for Use . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Refinements to the Separate Site Methods 70

6.1 Tide-Surge Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

. . . . . . . . . . . . . 73

6.2 Extremal Index Estimation in the RJPM . . . . . . . . . . . . . . . . . . . 86

6.3 Declustering Series: Sensitivity Approach . . . . . . . . . . . . . . . . . . . 88

7 Applications and Results for A-Class Sites 89

7.1 r-largest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.1.1 Choice of Number of Order Statistics . . . . . . . . . . . . . . . . . . 89

7.1.2 Choice of Storm Length in Declustering . . . . . . . . . . . . . . . . 94

7.1.3 Results for the r-Largest Method . . . . . . . . . . . . . . . . . . . 96

7.2 Joint Probabilities Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2.1 Choice of the Number of Tidal Bands . . . . . . . . . . . . . . . . . 102

7.2.2 Results for the JPM . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3 Revised Joint Probabilities Method . . . . . . . . . . . . . . . . . . . . . . 110

7.3.1 Obtaining the Annual Maximum Distribution of Surges . . . . . . . 110

7.3.2 Estimation of the Extremal Index Ratio . . . . . . . . . . . . . . . 111

7.3.3 Tide-surge Interaction Modelling . . . . . . . . . . . . . . . . . . . 117

7.3.4 Results for the RJPM . . . . . . . . . . . . . . . . . . . . . . . . . 117

8 Comparison of Results 1288.1 Comparison of Return Levels . . . . . . . . . . . . . . . . . . . . . . . . . 1288.2 Comparison of Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.3 Comparison of Return Levels with Previous Studies . . . . . . . . . . . . . 1418.4 Recommendations Site-by-Site . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.5 Design levels for Future Years . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.6 Conclusions of Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9 New methods 1489.1 Wave-Surge Joint Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.2 Within-Storm modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.3 Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9.4 Annual Maxima Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

CONTENTS

9.4.1 Inclusion of Trends . . .I . . . . . . . . . . . . . . . . . . . . . . . . 1569.4.2 Inclusion of Tide-Surge Interaction . . . . . . . . . . . . . . . . . . 157

9.4.3 Inclusion of Trends and Tide-Surge Interaction . . . . . . . . . . . . 157

10 Spatial Extensions 158

11 Acknowledgements 161

12 References 162

13 Appendices 165

LIST OF FIGURES

List of Figures

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Map showing the locations of all of the UK A-class gauges. . . . . . . . . . 22

Tidal density plots for Wick to Felixstowe. . . . . . . . . . . . . . . . . . 26

Tidal density plots for Southend to Fishguard. . . . . . . . . . . . . . . . 27

Tidal density plots for Holyhead to Lerwick. . . . . . . . . . . . . . . . . . 28

Surge densities obtained using kernel density estimation, for Wick to Fe-lixstowe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Surge densities obtained using kernel density estimation, for Southend toFishguard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Surge densities obtained using kernel density estimation, for Holyhead toLerwick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Tide against surge data, with lines of equal tide plus surge to show extremecombinations of tide and surge, for Wick to Felixstowe. . . . . . . . . . . 33

Tide against surge data, with lines of equal tide plus surge to show extremecombinations of tide and surge, for Southend to Fishguard. . . . . . . . . 34

Tide against surge data, with lines of equal tide plus surge to show extremecombinations of tide and surge, for Holyhead to Lerwick. . . . . . . . . . 35

Example of erroneous hourly surge data: Newlyn 1944. . . . . . . . . . . . 37

Map showing the edited subset of UK A-class gauges and the additionalSouthend site which are suitable to be used in the site-by-site extremesea-level analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Information on the spans of available hourly data from each A-class site. 44

Tide against surge data, and tide against transformed surge data for Wickto Whitby. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Tide against surge data, and tide against transformed surge data for Im-mingham to Felixstowe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Tide against surge data, and tide against transformed surge data for Southendto Newlyn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Tide against surge data, and tide against transformed surge data for Ilfra-combe to Fishguard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Tide against surge data, and tide against transformed surge data for Holy-head to Stornoway. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Tide against surge data, and tide against transformed surge data for Ul-lapool to Lerwick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

83

8 LIST OF FIGURES

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Declustered surge data: the 15 largest independent events in each year forPortpatrick to Ullapool. The heights of the lines have been scaled so that

the maximum observed surge is the height of three years on the year-axis. . 172

Declustered surge data: the 15 largest independent events in each year forLerwick. The heights of the lines have been scaled so that the maximumobserved surge is the height of three years on the year-axis. . . . . . . . . . 173

Annual maximum intercept parameter obtained by the r-largest methodat Wick to Felixstowe, plotted against the number of order statistics used. 174

Annual maximum intercept parameter obtained by the r-largest method atSouthend to Fishguard, plotted against the number of order statistics used. 175

Annual maximum intercept parameter obtained by the r-largest method atHolyhead to Lerwick, plotted against the number of order statistics used. 176

Annual maximum trend parameter obtained by the r-largest method at

Wick to Felixstowe, plotted against the number of order statistics used. . 177

Annual maximum trend parameter obtained by the r-largest method atSouthend to Fishguard, plotted against the number of order statistics used. 178

Annual maximum trend parameter obtained by the r-largest method atHolyhead to Lerwick, plotted against the number of order statistics used. . 179

Annual maximum scale parameter obtained by the r-largest method atWick to Felixstowe, plotted against the number of order statistics used. . 180

Annual maximum scale parameter obtained by the r-largest method atSouthend to Fishguard, plotted against the number of order statistics used. 181

Annual maximum scale parameter obtained by the r-largest method atHolyhead to Lerwick, plotted against the number of order statistics used. 182

Annual maximum shape parameter obtained by the r-largest method atWick to Felixstowe, plotted against the number of order statistics used. . 183

Annual maximum shape parameter obtained by the r-largest method atSouthend to Fishguard, plotted against the number of order statistics used. 184

Annual maximum shape parameter obtained by the r-largest method atHolyhead to Lerwick, plotted against the number of order statistics used. 185

100 year return level against number of order statistics in the r-largestsea-level analysis for Wick to to Felixstowe. . . . . . . , . . . . . , . , . . . 186

100 year return level against number of order statistics in the r-largestsea-level analysis for Southend to Fishguard. . . . . . . . . . . . . . . . . . 187

100 year return level against number of order statistics in the r-largestsea-level analysis for Holyhead to Lerwick. . . . . . . . . . . . . . . . . . . 188

LIST OF FIGURES

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100 year return level against storm length in the r-largest sea-level analysisfor Wick to to Felixstowe. . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

100 year return level against storm length in the r-largest sea-level analysisfor Southend to Fishguard. . . . . . . . . . . . . . . . . . . . . . . , . . . 190100 year return level against storm length in the r-largest sea-level analysisfor Holyhead to Lerwick. . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

100 year return level against the number of tidal bands used in the JPMfor Wick to to Felixstowe. . . . . . . . . . . . . . . . . . . , . . . . . . . . 192

100 year return level against the number of tidal bands used in the JPMfor Southend to Fishguard. . . . . . . . . . . . . . . . . . . . . . . . . . . 193

100 year return level against the number of tidal bands used in the JPMfor Holyhead to Lerwick. . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Annual maximum surge intercept parameter against the number of orderstatistics in the surge r-largest analysis, for the RJPM: Wick to to Felixstowe195

Annual maximum surge intercept parameter against the number of orderstatistics in the surge r-largest analysis, for the RJPM: Southend to Fishguard

Annual maximum surge intercept parameter against the number of orderstatistics in the surge r-largest analysis, for the RJPM: Holyhead to Lerwick197

Annual maximumsurge trend parameter against the number of order statis-tics in the surge r-largest analysis, for the RJPM: Wick to to Felixstowe . . 198

Annual maximumsurge trend parameter against the number of order statis-tics in the surge r-largest analysis, for the RJPM: Southend to Fishguard . 199

Annual maximum surge trend parameter against the number of order statis-tics in the surge r-largest analysis, for the RJPM: Holyhead to Lerwick . . 200

Annual maximum surge scale parameter against the number of order statis-tics in the surge r-largest analysis, for the RJPM: Wick to to Felixstowe . , 201

Annual maximum surge scale parameter against the number of order statis-tics in the surge r-largest analysis, for the RJPM: Southend to Fishguard , 202

Annual maximum surge scale parameter against the number of order statis-tics in the surge r-largest analysis, for the RJPM: Holyhead to Lerwick . . 203

Annual maximum surge shape parameter against the number of orderstatistics in the surge r-largest analysis, for the RJPM: Wick to to Felixstowe204

Annual maximum surge shape parameter against the number of orderstatistics in the surge r-largest analysis, for the RJPM: Southend to Fish-guard. . . . , . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . 205

Annual maximum surge shape parameter against the number of orderstatistics in the surge r-largest analysis, for the RJPM: Holyhead to Lerwick206

LIST OF FIGURES

90scribed in Section 6.2 for Wick to to Felixstowe. . . . . . . . . . . . . . . . 207

91scribed in Section 6.2 for Southend to Fishguard. . . . . . . . . . . . . . . 208

92scribed in Section 6.2 for Holyhead to Lerwick. . . . . . . . . . . . . . . . .

93 Variation of extremal index ratio with storm length for Wick to Felixstowe. 21094 Variation of extremal index ratio with storm length for Southend to Fish-

guard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

95 Variation of extremal index ratio with storm length for Holyhead to Lerwick.212

96 Variation in the 100 year return level in the RJPM with extremal indexratio for Wick to Felixstowe. . . . . . . . . . . . . . . . . . . . . . . . . . . 213

97 Variation in the 100 year return level in the RJPM with extremal indexratio for Southend to Fishguard. . . . . . . . . . . . . . . . . . . . . . . . . 214

98 Variation in the 100 year return level in the RJPM with extremal indexratio for Holyhead to Lerwick. . . . . . . . . . . . . . . . . . . . . . . . . . 215

99 100 year return level against number of tidal bands used in the RJPM forWick to to Felixstowe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

100 100 year return level against number of tidal bands used in the RJPM forSouthend to Fishguard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

101 100 year return level against number of tidal bands used in the RJPM forHolyhead to Lerwick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

102 Port diagrams obtained using each of the 3 methods described in Section 4,the three curves are: ( - -

and (- - - - ) the RJPM. For Wick and Aberdeen. . . . . . . . . . . . . . . 219

103 Port diagrams obtained using each of the 3 methods described in Section 4,the three curves are: ( - - -) the r-largest method, (......) the JPM method,and (- - - - ) the RJPM. For North Shields and Whitby. . . . . . . . . . . 220

104 Port diagrams obtained using each of the 3 methods described in Section 4,the three curves are: ( - - ) the r-largest method, (......) the JPM method,and (- - - - ) the RJPM. For Immingham and Cromer. . . . . . . . . . . . . 221

105 Port diagrams obtained using each of the 3 methods described in Section 4,the three curves are: ( - - ) the r-largest method, (......) the JPM method,and (- - - - ) the RJPM. For Lowestoft and Felixstowe. . . . . . . . . . . . 222

106 Port diagrams obtained using each of the 3 methods described in Section 4,

and (- - - - ) the RJPM. For Southend and Sheerness. . . . . . . . . . . . . 223

LIST OF FIGURES 11

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Port diagrams obtained using each of the 3 methods described in Section 4,the three curves are: ( - - ) the r-largest method, (......) the JPM method,and (- - - - ) the RJPM. For Dover and Newlyn. . . . . . . . . . . . . . . . 224Port diagrams obtained using each of the 3 methods described in Section 4,the three curves are: ( - - ) the r-largest method, (......) the JPM method,and (- - - - ) the RJPM. For Ilfracombe and Avonmouth. . . . . . . . . . . 225Port diagrams obtained using each of the 3 methods described in Section 4,

the three curves are:and (- - - - ) the RJPM. For Milford Haven and Fishguard. . . . . . . . . 226Port diagrams obtained using each of the 3 methods described in Section 4,the three curves are: ( - -and (- - - - ) the RJPM. For Holyhead and Heysham. . . . . . . . . . . . . 227Port diagrams obtained using each of the 3 methods described in Section 4,

and (- - - - ) the RJPM. For Portpatrick and Stornoway. . . . . . . . . . . 228

Port diagrams obtained using each of the 3 methods described in Section 4,the three curves are: ( - - -and (- - - - ) the RJPM. For Ullapool and Lerwick. . . . . . . . . . . . . . 229

12 LIST OF TABLES

List of Tables1

2

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4

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Table of maximum tide, surge and sea-level at each site over the observationperiod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Data span information for sites used in the analyses, for Aberdeen toHeysham. ODN factor is the additive adjustment for conversion from ChartDatum to ODN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Data span information for sites used in the analyses, for Holyhead to New-lyn. * denotes the conversion for OD Local. ODN factor is the additiveadjustment for conversion from Chart Datum to ODN. . . . . . . . . . . . 41Data span information for sites used in the analyses, for North Shields toWick. * denotes the conversion for OD Local. ODN factor is the additiveadjustment for conversion from Chart Datum to ODN. . . . . . . . . . . . 42

. . . 76

Return levels for 1990, in metres to chart datum, using the r-largest method

Return levels for 1990, in metres to chart datum, using the r-largest method250, 500, 1000, and 10000 years. . . . . . . . . . 98

Return levels for 1990 using the joint probabilities method for return peri-= 10, 25, 50, and 100 years. . . . . . . . . . . . . . . . . . . . . . . 105

Return levels for 1990 using the joint probabilities method for return peri-

Return levels for 1990 using the revised joint probabilities method for return10, 25, 50, and 100 years. . . . . . . . . . . . . . . . . . . . 123

Return levels for 1990 using the revised joint probabilities method for return

Estimated return periods for the maximum observed still water level ateach site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Trend estimates and standard errors for: the r-largest method, the extremesurges and the transformed surge series in the highest tidal band. The NAentries in the table are when no fit could be obtained in that case. . . . . 140

Data sites, data spans and site-by-site estimates of return level for 1990

are given, in metres, relative to Chart Datum. . . . . . . . . . . . . . . . . 143Data sites, data spans and spatial estimates of return level for 1990 with

given, in metres, relative to chart datum. . . . . . . . . . . . . . . . . . . . 144

SUMMARY 13

SummaryThis report describes the first stage in a full scale statistical extreme value analysis ofhourly sea-level data from 21 ‘A-class’ sites around the UK coastline. This first stage isrestricted to methods of analysis where only the data from the site of interest are usedin the analysis of extremes for that site. The second stage of the work extends thesemethods and develops spatial procedures which simultaneously exploit all the data from

a coastline in the estimation. Throughout we term these two approaches site-by-site andspatial methods respectively. Explicit numerical results are provided for estimates ofextreme levels at each A-class site obtained using site-by site methods. Also presentedare the associated details, and advances, in the statistical methodology for the estimationof extreme still water levels and the combination of extreme still water levels with waves.Topics covered

l A summary of existing site-by-site based methods:

- Annual maximum,

- Joint probabilities (JPM),

- Revised joint probabilities (RJPM).

l A discussion of the comparative properties of these site-by-site methods.

l A clarification of the concepts of return levels and return periods when trends arepresent.

l Details of statistical methodology, developed in this project, which have been usedto extend and refine the site-by-site methods and enable them to have wider appli-cability:

- Inclusion of trends in each of the methods,

- Modelling tide-surge interaction in the JPM and the RJPM,

- Estimation of the extremal index in the RJPM.

l Return level estimates, with an associated measure of precision, at each A-class sitefor each of the refined site-by-site methods of estimation.

l A comparison of resulting estimates, leading to the selection of the most suitablemethod for each site and corresponding best estimates.

SUMMARY

l The development of new models and methods which exploit additional historicaldata and enable waves, and all aspects of the events, to be incorporated into theanalysis:

- Modelling wave-surge dependence and its resulting impact on design levels,

- Inclusion of all historical annual maximum sea-level data into the RJPM.

- Inclusion of within-extreme events.

l A discussion of the benefits of using spatial methods based on the site-by-site esti-mates of parameters and return levels found here.

Findings and Conclusions

l At sites where the surge has a large variability at high tidal levels relative to thevariability between high tides, then each of the methods of analysis give broadlysimilar results. Generally, this is the case for sites in the South-East.

l At sites where the variability of high tides is large relative to the variability ofthe surge at high tidal levels, then both the annual maxima and r-largest methodsunderestimate return levels, that is overestimate the return period of observed levels.Generally, this is the case for sites on the west coast. The degree of underestimationis reduced as the length of annual maxima records is increased, but in some casesdata from many nodal cycles of the tide are required before the bias is adequatelyreduced. This feature was unknown prior to this work and it has potentially largerepercussions for existing designs on the west coast which have been based on returnlevels estimated using annual maximum data, such as in Graff (1981), as these couldhave been under designed.

l Estimates of return levels, and associated standard errors, are given in Tables 6-11for each method applied to each site in the study.

l Trends in extreme sea-level data have been estimated, but the estimates are highlyvariable owing to the short period of hourly observation. At present we recommendthe use of estimates of mean sea-level trend, which incorporate the vertical landlevel trend for the site, as a surrogate for the trend in extreme levels at that site.

l Estimates of trends at extreme levels calculated in this study using each of themethods are given in Table 13 for all sites.

l The JPM and RJPM provide accurate estimates from much shorter series thaneither the annual maxima or the r-largest methods.

SUMMARY 15

l Of the JPM and the RJPM, the RJPM is the better provided there are at least 5-10years of hourly observation, or there are supplementary annual maxima available.

We recommend that estimates of return levels based on a spatial analysis (to be givenin the second stage of the work) will provide the most accurate design levels. However,based just on the site-by-site methods, the RJPM should generally be used provided thatsufficient hourly data are available.

16

Glossary

GLOSSARY

Terms and Definitions

ACDAdmirality Chart Datum. This is referred to throughout as Chart Datum.BODCBritish Oceanographic Data Centre which holds all the hourly data for the UK sites.Direct methods.Methods of extreme sea-level analyses in which extremes of the sea-level process are anal-ysed.Extremal index ratio.

HATHighest Astronomical tide. The highest tide, at a particular site, which occurs over the18.61 year nodal tidal cycle.LATLowest Astronomical tide. The lowest tide, at a particular site, which occurs over the18.61 year nodal tidal cycle.Indirect methods.Methods of extreme sea-level analyses in which the constituent processes (i.e. tides andsurges) are modelled separately and extremes of the sea-level inferred.JPMJoint probabilities method.OD LocalLocal datum for non-mainland sites.

ODNOrdnance Datum Newlyn. A common datum, relative to which sea-levels are often re-ferred to for UK mainland sites. Here Chart Datum is used and conversions to ODN aregiven in Tables 2-4.POLThe Proudman Oceanographic Laboratory.Quantile

RJPMRevised joint probabilities method.

GLOSSARY 17

Mathematical notation

GEV

the location, scale, and shape parameters respectively, if the distribution function of X is

(see Section 4)

The distribution function of the annual maximum sea-level, and of the annual maximum

These are the location, scale and shape parameters of the annual maximum sea-level dis-tribution, when it is taken to be GEV.

These are the location, scale and shape parameters of the annual maximum surge distri-bution.

X t

The meteorologically induced surge level

The annual maximum sea-level in year i

The GEV location parameter for the sea-level in year i.

The base year for trends.

18 GLOSSARY

NThe number of hours in a year, N = 8766.

The marginal distribution function of hourly surges.

T

Extremal indices for the sea-level and surge processes respectively.

19

1 IntroductionAn important component of the design of a coastal flood defence is the statistical analysisof the extreme sea-levels for the site of interest. In essence there are two approaches tothis analysis:

1. site-by-site, using data from the site of interest only,

2. spatial, where all the data from the coastline of concern are used to infer the extremecharacteristics at the relevant site.

This report, which describes the first stage of the Estimates of Extreme Sea Conditionsproject, provides descriptions, comparisons, and refinements of the various site-by-siteprocedures and their application to A-class sites. The aims of the overall project are

Stage 1. To provide the first systematically obtained design results for the UK A-classsites, results which can be used to improve the design of flood prevention schemes.

Stage 2. To extend the site-by-site procedures to spatial methods and apply the resultingtechniques to provide both improved estimates for the A-class sites and estimatesat intermediate sites along the coastline.

In response to considerable coastal flooding arising from a number of historical ex-treme storms, most notably the 1953 extreme storm surge, statistical analysis of extremesea-levels has been recognised as forming a significant element of the design phase. Con-sequently, since 1953 there has been large scale upgrading of flood defences, and in eachcase some form of statistical analysis has been used in the assessment of a suitable designlevel. Although attempts have been made to develop procedures for systematic applica-tion (Lennon, 1963; Suthons, 1963; Graff, 1981; Pugh and Vassie, 1979, 1980) these havenot been widely adopted mainly because there was concern that the physical processes areinconsistent with the assumptions of the models used. The lack of universally acceptedmethods means that practising coastal engineers apply a variety of ad hoc methods toobtain results for particular studies. ,4 consequence of this is that it is difficult to assessthe regional level of protection offered by the resulting flood defences or even to identifysites at high risk due to inadequate protection.

Over recent decades many new tide gauges have been installed, so extensive data,in the form of hourly sea-levels, are now available for a network of UK sites. Particularemphasis has been given to data collection for a set of sites, the so called A-class sites, andthese are the focus of attention here. The increased access to data together with recentdevelopments in statistical methods of analysis for extreme sea-levels (Tawn, 1988; Tawnand Vassie, 1989, 1991; Tawn 1992; Coles and Tawn, 1990) now enables the systematic

1 INTRODUCTION

analysis of extreme sea-levels for all A-class sites. Furthermore, the A-class data providesufficient spatial coverage upon which to base estimates for intermediate sites along certaincoastal regions, which is the focus of the second stage of this work.

The main objectives of the first stage of the work are:

l To describe in detail the existing statistical methodology for the site-by-site tech-niques and to compare these method. These are given in Sections 4 and 5.

l Based on the wide experience gained through systematic application of the existingmethods to a range of data sites, where the sea-level processes exhibit a variety ofphysical characteristics, to refine these methods so that they have better propertiesand more extensive applicability. These refinements are described in Section 6.

l To obtain the first systematically derived set of return level estimates, with associ-ated measures of precision, at each A-class site for which site-by-site methods givereliable estimates. Such results, displayed in both graphical and tabular form, aregiven for each site in Sections 7 and 8. Also in Section 8 sample results show howtrends in extreme sea-levels can be included into return level estimates.

In Section 2, the data are described, and a basic summary of the sea-level process isgiven. The new substantive methodology, developed within this project, is detailed inSection 9, whilst in Section 10, a discussion of the benefits of adopting a spatial approachto analysing extreme sea-levels is given, forming a forward look to the second stage of the

work.

21

2 Data and the Sea-Level Process

2.1 Description of the Data and Location of the Sites

The data used in this study are hourly measurements of still water level, which is definedas the observed sea-level after surface waves have been averaged out. These data are ob-served for a network of sites around the UK, and are measured relative to Chart Datum.For the mainland sites these can be converted to a common datum, Ordnance DatumNewlyn (ODN). Through BODC, POL hold these data for each of the sites now classifiedas A-class. In addition to these sites, Southend has also been included in the data-base asit provides a long period record at a coastal location with a history of flooding. Althoughearly records for the current A-class sites may be of poorer quality than current data, thedata used for each of the sites in this study are records which are relatively easily availableand which have been previously screened for quality. Figure 1 shows a map of the UKwith positions of the A-class sites indicated. Screening the data for their suitability forextreme level analyses (see Section 2.4) led to the removal of a number of sites and ofyearly records from the analysis.Sites removedLeith, Newhaven, Portsmouth, Weymouth, Devonport, Hinkley, Newport, Mumbles, Bar-mouth, Llandudno, Liverpool, Port Erin, Workington, Millport, Islay, Tobermory andKinlochbervie.Sites remaining in the analysisWick, Aberdeen, North Shields, Whitby, Immingham, Cromer, Lowestoft, Felixstowe,Southend, Sheerness, Dover, Newlyn, Ilfracombe, Avonmouth, Milford Haven, Fishguard,Holyhead, Heysham, Portpatrick, Ullapool, Stornoway and Lerwick.

Historical data in the form of annual maximum sea-levels are available for many ofthese sites. These data are not included in these analysis since the main purpose of thisstage of the project is to compare the existing site-by-site methods on an equal basis.Inclusion of these extra data makes comparison difficult as these data cannot be directlyincluded into some of the methods. However, it is important that these data are includedin a final analysis, so the additional annual maximum data are used in the second stageof the project.

2.2 The Sea-level Process

2.2 The Sea-level Process 23

The tidal level is the component of the observed sea-level which is determined by astro-nomical forcing. It can be shown (Pugh, 1987) that the tidal level is

24 2 DATA AND THE SEA-LEVEL PROCESS

The Surge SeriesThe surge series is defined as the residual of the tidal series, i.e. the difference between

the observed still water series and the predicted tide. The difference between observedlevels and tidal predictions is due to meteorological forcing in the form of changes in airpressure and winds which cause surges to be generated (Pugh, 1987). Again it is helpful toconsider the distribution of these levels for each of the sites to be studied, these are givenin Figures 5-7. Here, kernel density estimation (Silverman, 1986), a standard statisticaltechnique of smoothing the probability density estimate, is used. As a consequence ofthe tidal analysis the surge sample has a zero average although may contain trends. Themost notable feature in these figures is the difference in variability of the surge from coastto coast. On the east coast variability generally increases with distance down the coast,reaching a maximum at Southend, while on the west coast the surge is much less variableexcept at shallow water sites. The surge distribution is uni-modal, with the mode close tozero, and is positively skewed, leading to large positive surge levels being more likely thanlarge negative surges. This is particularly apparent on the east coast where the maximumsurges observed over the study period for each site, given in Table 1, are considerablylarger than typical levels shown in Figures 5-7.


2.3 The Extreme Sea-level Process

This study focuses on extreme sea-levels, which are a combination of extreme still waterlevels and extreme waves. We concentrate on extreme still water levels which arise asa large combination of tides and surges. Extreme tidal levels (the highest astronomicaltide, HAT, and high tides which are close in level to HAT) arise relatively frequently.Consequently for flood defences, interest is in levels which are only attainable from alarge positive surge occuring with a high tide. To illustrate this, Figures 8-10 show largepositive surge levels (i.e. all surge levels greater than the 0.995 quantile for each site)plotted against the associated tide level for each site. The lines on the plots are contoursof equal still water level, so any combination of tide and surge along a line gives the samestill water level. Moving to a higher line corresponds to combinations of tide and surgewhich give larger still water levels. Some immediate conclusions can be drawn from thesefigures, which have considerable relevance later.

l For Lerwick to Immingham extreme still water levels typically have arisen from hightidal levels combined with moderately extreme surges;

l for Cromer to Dover and Heysham to Portpatrick extreme still water levels typicallyhave arisen from extreme surges occuring on a high tide;

l for Newlyn to Holyhead and Ullapool to Stornoway extreme still water levels typi-cally have arisen from a high tidal level combined with a moderate surge.

Thus for some sites extreme still water levels are determined primarily by extremetides and moderate surges whereas for other sites it is primarily the extreme surges andmoderate tides that are important. This is in accord with the ideas of Middleton andThompson (1986) in which sites were classified as either tide or surge dominated. Thereason for this variation is explained by the relative variabilities of the tide and surgeprocesses and by aspects of interaction between them at each site. Clearly results forextreme still water levels are highly influenced by the characteristics of large surges. Asthe surge process is non-deterministic, estimation of the distribution of extreme still waterlevels is a statistical problem. This problem is addressed in this report and substantialanalysis is undertaken.

The problem of analysing combinations of extreme still water levels and waves isaddressed theoretically in Section 9, and illustrated with a simple but artificial example.More general application is not possible at present as wave data are of more limitedtemporal and spatial extent.

36

2.4 Data Pre-processing

2 DATA AND THE SEA-LEVEL PROCESS

Although the still water level data had already been screened for errors by POL, beforebeing stored by BODC, further simple checks are carried out prior to our statisticalanalysis of extremes. The estimated surges are plotted against time on a month bymonth basis for each site. Since problems with gauge malfunctioning or timing errorstypically lead to spurious large positive or negative surges, examination of these plotsenables erroneous data to be identified. Our approach is to identify these based on

l within series comparisons - questioning whether the suspect extreme event is con-sistent with other extreme events at the site, or whether there is periodicity in theextremes at the site.

l between series comparisons - if a surge appears erroneous at one site does it appearconsistent with the surge series at the corresponding times for neighbouring sites.

An example of a site and a year in which timing errors are present is Newlyn 1944.Figure 11 shows this surge series which as a result of a timing error during the middle ofthat year has the characteristic spurious periodicity caused by residual tide contaminatingthe surge. Often though, errors are less obvious than this.

Although quality control should identify most errors any such procedure will not detectall errors. How important these remaining errors are depends largely on the subsequentuse of the data. Provided the errors are small and the series are relatively long, the extremevalue models developed and applied in this work will be reasonably robust to such errors.As we shall consider methods in which extremes of the still water levels are both directlyand indirectly analysed, different forms of error will be reduced by each method. Thusour methods can be assessed for error sensitivity and comparative inferences made.

Nevertheless to reduce the influence of remaining errors in the data on the extremevalue analyses, the following procedure was adopted: if a year was suspected of contain-ing errors or contained less than 75% of data over the winter period (January to Marchcombined with October to December), then data from that year was excluded from theextreme value analyses. Our reason for excluding observed data from years with substan-tial missing data in the storm season was that extreme levels in that year may not havebeen recorded and subsequent inclusion of the data in the analysis would lead to potentialbias in return level estimation.

Several years (typically more than three) of good quality hourly data are requiredbefore the site-by-site based methods, described in Section 4, can be reliably used forestimation of extremes. There are two reasons for this

l data are required to fit the probability distribution tail models and in some caseswith limited data (even with more than three years) fits cannot be obtained.


l using only a few years of data can lead to bias due to longer term variations in stormclimate: observations may have been taken over a period with few or many storms.

l return level estimates have associated measures of precision. Even if estimates canbe obtained from very short series they will be highly imprecise and of little value.

For these reasons, results for the sites with very short series, and in particular those withrecently installed gauges in the expansion of the A-class network, cannot be obtainedusing these models when restricted in their application to a single site at a time. Hencefor this report, data pre-processing involves effectively removing those sites with less thanfour years of hourly data. The spatial coverage of the resulting edited data-set is given,in the form of a map, in Figure 12. The poor spatial coverage along the west and southcoasts is now clearly evident, especially when contrasted with Figure 1.

Tables 2-4 list these sites: from site to site the data records vary in start date and timespan, and have periods of missing values which differ over sites. A graphical summaryindicating the pattern of available data records is shown in Figure 13 where it is seen thatthe longest and nearly continuous record is at Newlyn; several sites have recently installedgauges, and so only have data for a few years. The majority of data covers the period

around 1960 to the present, this being due to the installing of many gauges after the 1953storm event. Note that nearly all the sites have data available up to and including 1993.These are not included in the analysis at present since it was not feasible to keep updatingthe data sets throughout the analysis stage.

In addition to using plots of the derived surge time series to identify poor qualitydata throughout the error control stage of the work plots of the largest independent peaksurge levels per year were used, as this feature is the most relevant for extreme valueanalysis. Figures 49-56, in the appendix, show these plots for all the sites and yearsthat remained in the study after the pre-processing stage. As well as showing the spatialdependence of extreme events they also show the strong Seasonality of the process, andin particular a definite storm season in the winter during which extreme events occurreasonably homogeneously over time.

2.5 Illustration Sites

All the methods discussed and developed in Sections 4 and 6 will be applied to each of theremaining 22 sites (21 A-class sites and Southend). Rather than present results for eachof these sites in the main body of the text we restrict detailed discussion to two sites inthe main text and give results for all other sites in the appendix. For detailed discussionwe have selected Immingham and Lowestoft owing to

1. continuous long records of high quality data,

2.5 Illustration Sites 39

2. different tidal characteristics: Lowestoft and Immingham have tidal ranges of ap-proximately 2.5m and 7.5m respectively,

3. variation of tide-surge interaction: at Lowestoft it is weak, at Immingham it ismoderate,

4. previous analyses exist for comparison: Tawn (1988, 1992) and Tawn and Vassie(1989) apply the existing methods to data up to and including 1983.

2.5 Illustration Sites 43

Figure 12: Map showing the edited subset of UK A-class gauges and the additionalSouthend site which are suitable to be used in the site-by-site extreme sea-level analyses.


Wick

Aberdeen

North Shields

Whitby

Immingham

Cromer

Lowestoft

Felixstowe

Southend

Sheerness

Dover

Newlyn

Ilfracombe

Avonmouth

Milford Haven

Fishguard

Holyhead

Heysham

Portpatrick

Stornoway

Ullapool

Lerwick

Figure 13: Information on the spans of available hourly data from each A-class site.

45

3 Concepts used in the Design StageConsider the standard design question for sea-walls: to what height should the wall bebuilt to offer a specified level of protection? In this section we address this question byconsidering suitable design criteria. In particular, after describing the standard terminol-ogy of return levels and return periods, three design criteria are discussed and theirrelationships with return levels are identified. In each case the sea-wall is deemed to havefailed if the still water level exceeds the sea wall height, w, at any time over the period ofinterest.

3.1 Return Levels and Return Periods

The standard concepts used by coastal engineers when quantifying the risk of failure ofsome structure are return levels and return periods. These concepts however are implicitlytied to assessing risk for independent and identically distributed processes, and hence haveto be considered carefully in sea-level applications where trends and temporal dependenceare common features. First we clarify what return levels and periods are for independent

Suppose the common distribution

to the first failure we have

46 3 CONCEPTS USED IN THE DESIGN STAGE

Since the process is independent this is also the expected waiting time between failures.So the expected waiting time between failures is the reciprocal of the probability of afailure.

but differ significantly for return periods of less than 15 years. Throughout we use

3.2 Design Criteria

The three possible design criteria are

0 Criterion 1The height of the sea wall should be such that the probability of failure in year i is

This level, termed the p design level for year i, and denoted by z;(p), satisfies

Unlike the return level this design level concept applies to processes with trendsas well as stationary sequences. To clarify the connections between the concepts

48 3 CONCEPTS USED IN THE DESIGN STAGE

3.3 Future Terminology

So that the terminology for the remainder of the report is consistent with common

49

4 Existing Separate Site Extreme Value MethodsThe existing methods for estimating the distribution of extreme sea-levels at a site, usingdata from that site only, are described in this section. These methods can be characterisedas either

l direct methods - in which extremes of the observed still water level are analysed,

l indirect methods - in which the constituent processes (i.e. tides and surges) aremodelled separately and extremes of still water level inferred.

The direct methods, summarised in Sections 4.1 and 4.2, are the classical annual max-ima approach and its extension to the r-largest annual events method respectively. Theexisting indirect approaches, the joint probabilities method (JPM) and the revised jointprobabilities method (RJPM), are described in Sections 4.3 and 4.4 respectively. Foreach method we discuss the cases of analysis with and without incorporating a trend.These form simple extensions of the methods as previously published. The four meth-ods are compared on general theoretical grounds in Section 5, providing insight into thesubsequent performance of the methods. A general discussion of maximum likelihoodestimation, the method of estimation used throughout, is given in Section 4.5.

4.1 Annual Maxima Method

The best known, simplest, and most widely used method of analysis of extreme sea-levelsis the annual maxima method (Gumbel, 1958). First consider the case where there is notrend and the series of annual maxima is stationary.

4.1.1 No Trend

function

influence the location, scale, and shape of the distribution respectively; we term

50 4 EXISTING SEPARATE SITE EXTREME VALUE METHODS

The annual maxima method takes the limiting GEV distribution to be the exact distri-

The justification for using the GEV as an approximation to the true

of linearly normalised maxima of stationary sequences (Leadbetter et al., 1983). Conse-quently the quality of the approximation depends on both the form of the upper tail of the

approximation should be reasonably good. The importance of stationarity is less clearbut results in Sections 7 and 8 clarify this issue for UK sea-levels.

The GEV and return level parameters can be estimated by many methods, we usemaximum likelihood estimation throughout as this is most flexible in its handling ofcovariates, dependence, and for providing a measure of precision of return levels. Detailsabout maximum likelihood estimation are given in Section 4.5. In this case maximumlikelihood is used to fit the model (4.1) to the observed series of annual maximumassumingthe annual maxima in separate years are independent. Maximum likelihood involvesmaximising, with respect to the GEV parameters, the likelihood

4.1 Annual Maxima Method 51

Thus a positive shape parameter leads to an upperendpoint and easier inferences concerning design questions, whereas a negative shapeparameter leads to more difficult statistical problems concerning estimation of returnlevels.

4.1.2 With Trend

For many sites, the series of annual maximumdata appear to have non-stationary featuressuch as trends. Examination of the data shows that it is adequate to consider the simplestcase of non-stationarity, namely a linear trend with identically distributed errors. Hence,in this case, model (4.1) is extended to

(4.5)

Again maximum likelihood can be used to fit this model under the assumption that

the annual maxima in separate years are independent. Here the likelihood is


4.1.3 Previous Applications

This method of analysing extreme sea-levels has been extensively applied to UK sea-level data in numerous studies. However, the most accessible studies which are also wideranging in terms of sites covered are Lennon (1963), Suthons (1963), Graff (1981), andColes and Tawn (1990).

Proposals to extend the annual maximum method to incorporate all independent extremesea-level observations into the estimation of the annual maximum distribution have beenmade in terms of

l the peaks over thresholds method (Davison and Smith, 1990)

l the r-largest events method (Smith, 1986; Tawn, 1988)

l the point process method (Smith, 1989).

These methods are in essence equivalent when the process is stationary, so in the context ofextreme sea-level analyses the choice of adopted method depends on convenience as muchas anything. We use the r-largest annual events method for a combination of reasons:to be consistent with historical analyses, simplicity, and due to the equivalence of the

= 1. Again consider the stationary casefirst:

4.2.1 No Trend

a

4.2 r-largest Annual Events Method 53

it corresponds to the class of all possible limiting distributions of the r largest linearlynormalised extreme values of a stationary sequence with independent extreme events(Leadbetter et al., 1983). Here convergence assumptions similar to the annual maximamethod are required to justify the use of (4.6).

are also the parameters of the annual maximum distribution. In this case, assumingindependence of the r largest annual events in separate years, the likelihood function is

used in year i. the same number of extreme annualevents are used per year. The maximum likelihood estimates obtained in this case can besubstituted into (4.2) to give the maximum likelihood estimate of the return level z(p).

4.2.2 With Trend

Extension of the r-largest annual events method to the non-stationary case of trendsfollows identically the approach for annual maxima in Section 4.1.2, thus only brief detailsare given. In particular, the joint density for year i becomes


Application of the r-largest annual events method has been more limited than the an-nual maxima method itself. Alcock et al. (1987) and Tawn (1988) give a substantialexploratory analysis for Lowestoft, Tawn and Vassie (1989, 1991) also analyse data fromImmingham, whilst Tawn (1992) extends that application to Sheerness. The methodshave also been applied by Alcock and Blackman (1987) to Avonmouth.


4.3 Joint Probabilities Method

The simplest indirect method of extreme value analysis of sea-level data is the joint prob-abilities method introduced by Pugh and Vassie (1979, 1980). In essence the method in-volves empirical evaluation of the probability density functions of tide and surge separately(similar to Figures 2-7), together with a step which combines these under assumptionsof independence between the two processes and temporal independence of each process,giving a non-parametric estimate of the distribution function of the annual maximumhourly sea-level. Although non-parametric, this approach differs from using the empiricaldistribution of annual maximum sea-levels since the method exploits the fact that hourlysea-levels are available. Furthermore, in an intermediate step the non-parametrically

estimated distribution of hourly sea-levels differs from the empirical estimate as the com-positional structure of the sea-level process is exploited in the indirect analysis of surgeand tide levels. Again we first consider the case with no trend.

4.3.1 No Trend

hourly observations with the nodal tidal cycle of 18.61 years. Then assuming that hourlysurge levels are independent, we have

the same from year to year, the annual maximum sea-level can be taken to be approxi-mately independent and identically distributed over years. So, with N = 8766 denotingthe number of hourly observations within a year, we have

where G is the distribution function of the annual maximum still water level. From (4.7)and (4.8), we get

4.4 Revised Joint Probabilities Method 55

can be calculated by numerical

surge levels irrespective of the concomitant tidal level. Pugh and Vassie (1980) briefly dis-cussed the situation where the surge sequence depends on the tidal level, termed here theinteraction case. For interaction cases the empirical conditional surge distribution given

distribution function estimate based on separating the tide into equal width bands. Morerobust alternative estimators exist which are described in Section 6.

4.3.2 With Trend

An extension of the method to incorporate trends is not covered in the literature althoughit can be shown that to a good approximation the p design level, z;(p), in year i is givenby the solution of

where m; is the mean sea-level in year i. This extension is not used in the analysis asextreme sea-level trends rather than mean sea-level trends are of interest.


In illustrating their method, Pugh and Vassie (1979, 1980) applied the JPM to Aberdeen,Newlyn, Fishguard, Malin Head, Stornoway and Lerwick assuming that tide and surgeare independent, and to Southend by modelling interaction via splitting the tide intobands of equal width. Subsequent published examples are for the east coast sites Im-mingham, Lowestoft, Harwich, Walton and Sheerness in Alcock and Blackman (1985)and for Avonmouth and Ilfracombe in Alcock and Blackman (1987).

4.4 Revised Joint Probabilities Method

Here the RJPM is described in the case where the surge is independent of the tidal state,that is the tide and surge are assumed to be independent. We begin by also assuming thatthe surge distribution does not depend on time, i.e. that there are no trends in the surge

56

series. Although the case where the distribution of the surge depends on the tidal stateis outlined by Tawn (1992), those methods need extending and revising before they canbe applied routinely to all sites in this study. Discussion of these methods for handlinginteraction is therefore deferred until Section 6.1.

Before describing the RJPM, we state two results from the theory of extreme valueswhich are used in this section:

1. is stationary, and

in the extremes of the sequence,

(see Leadbetter et al. (1983) for explicit definitions of these two conditions) then

process, and is determined by the amount of dependence amongst the extremes of

for coastal engineering as it gives the mean duration of overtopping events.

is stationary, satisfies the weak mixing

(4.11)

quency when compared with the length of the series. Then


profile of the deterministic component at its peak and the temporal dependence of

Obtaining the Distribution of the Annual Maximum

In the development of the JPM in Section 4.3, an expression for the distribution of the an-

that hourly surges levels are independent. Here we assume the weaker condition that thesurge series is stationary. Although the sea-level series is non-stationary, due to the cyclictidal sequence, it should satisfy the conditions of Result 2, and under this assumption,

obtain (4.8) we have, again, that

Equating (4.13) and (4.14) gives that

which is identical to (4.9) except that here the dependence between hourly sea-levels is

In order to exploit equation (4.15) to obtain an estimate of G(z), a model for the surge


well by non-parametric methods similar to those used in the JPM. In particular, the em-is used in this region. The observed

data then completely determine the distribution of surges below some level. For the tailof the surge distribution, a non-parametric model is inadequate, so above some level anasymptotically justified parametric model is used to smooth and extrapolate the surgedistribution tail as follows:

Applying approximation (4.10) to hourly surges from a year gives

(4.16)

realisation of a stationary sequence, similar arguments to those used in obtaining (4.1)

annual maximum surge location, scale, and shape parameters respectively. This gives

(4.17)

Equation (4.17) is a limiting result, but is taken to be exact for any large surges level above

to a parametric model is undertaken must be chosen to be large enough so that (4.17)is a good approximation yet small enough so that there are enough data above u to givegood parametric estimation of the tail of the distribution. Combining (4.16) and (4.17)

(4.18)

Thus the RJPM model for the surge distribution, given by equation (4.18), gives a the-oretically justified parametric family for the tail of the surge distribution above u, anduses a non-parametric estimate below u, where the data are relatively dense.

4.4.3 Estimating Parameters

There are two separate types of parameters which require different methods of estimation.

dependence in the still water levels and surge processes.

mated using the r-largest approach applied to surge levels, as described in Section


4.4.4 Obtaining Return Level Estimates


where

imately GEV with the same scale and shape parameters as the annual maximum surgedistribution, but with a location parameter which depends primarily on the tidal seriesthrough the variability of the tide. Unfortunately, when there is tide-surge interaction thesimplicity of this interpretation is lost. However, to a reasonable approximation (4.22)

occur at times of high tidal levels (for clarification see Section 6.1 ).

4.4.5 With Trend

method so only briefest details are given. In particular, the equivalent result to (4.20) is,


The only published examples of this method are covered in Tawn and Vassie (1989) andTawn (1992) hw ere Lowestoft, Immingham and Sheerness are analysed.

4.5 Maximum Likelihood Estimation and Measures of Preci-

sion

In the preceeding sections we have referred to estimating parameters of statistical modelsby maximum likelihood. This is a standard statistical procedure which has many optimaland flexible features including evaluation of the precision of parameter estimates, andfunctions of parameters, within this framework. The general theory of maximum likeli-hood is justified by asymptotic arguments as the sample size increases. This procedure isnow briefly described.

4.5 Maximum Likelihood Estimation and Measures of Precision 61

The precision of the maximum likelihood estimate is determined by the associatedstandard errors. For a single parameter problem, i.e. d =

(4.25)

(4.26)


(4.28)

63

5 Comparison of MethodsTo assess the merits and deficiencies of each of the existing methods we must first lookat the assumptions made in developing the approaches described in Sections 4.1 to 4.4.These are considered in Section 5.1. Subsequently, in Section 5.2 the relationships betweenthe methods are outlined so that the associations between the assumptions are clarified.In Section 5.3 the various features of the methods are assessed and finally in Section 5.4general recommendations of usage are made. The comparative assessments here are basedon theoretical properties of the methods; additional comparisons, made in Section 8, arebased on findings for the sites in the study. In places these differ as the extensive analysishas significantly increased our knowledge of application of these methods.

only three methods are considered here, these being the r-largest, the JPM and the RJPM.

5.1 Assumptions of the Methods

The T-largest method is based on extreme value limit theory for stationary randomsequences. Assumptions made are

l the limiting generalised extreme value distribution is a good approximation to thedistribution of the annual maximum still water level.

The first assumption is the most critical since, due to the tide, the still water level seriesis highly non-stationary. The argument which has previously been used to over-ride thisobjection is that as most high tidal levels are similar in value all variability arises fromvariation in the surge series, which can be treated as stationary over the winter stormseason. For this argument to be viable the variability of surges in extreme events mustbe greater than the variation in high tides from which extreme still water levels areproduced. If high tide variations are a significant component of variations in extreme stillwater levels then extrapolation of the still water level variable is likely to be complex, and

the estimation of return levels poor. Previously, this has not been fully appreciated andso the annual maximum method has been widely used without regard to this issue. Theconsequences of this stationarity assumption were not known before this work but hereclear conclusions arise from a study of a range of sites, and these findings are discussedin Section 8. The second assumption is reasonable as the generalised extreme valuedistribution is quite flexible and so provides an adequate model for the annual maximum.

The joint probabilities method is totally empirical so the only assumptions arethat

64 5 COMPARISON OF METHODS

l an empirical probability density function for the surge component is a good repre-sentation of the true distribution,

l extreme hourly still water levels are independent,

are discussed in Section 6.1.

If the raw empirical surge probability density estimate is used then the JPM imposes

the artificial upper bound to the still water level, of the sum of HAT and the maximumobserved surge. For this reason, if the series is short, so the observed surge maximum ismuch below the upper endpoint of the surge distribution, or if the tidal range is small,so little extrapolation into the tail of the still water level distribution is obtained by thedecomposition into tide and surge, this is quite a restrictive assumption. In applicationa smoothed non-parametric estimate of the surge distribution could be used, so thisrestriction can be reduced but not eliminated. Whichever non-parametric estimate isused the consequence of the assumption is that return levels for long return periods areunderestimated.

Clearly hourly still water levels are not independent, but extreme levels may be only

is close to one. Tawn and Vassie (1989) studied this feature and found that the assumptionwas false but led only to small overestimation of return levels.

Accurately modelling interaction is most important for high tidal levels, and this isdiscussed in Section 6.1. Even with long records however, there is very little data in theregions of both extreme tides and surges, and any site-by-site based method will giverelatively poor estimates of the conditional surge distribution given the tide. This is apotentially important issue, so the sensitivity of results to various interaction models mustbe tested.

The revised joint probabilities method has characteristics which were developedto overcome the known deficiencies of the joint probabilities method as applied by Pughand Vassie (1980). The assumptions of the method are that

l the limiting generalised extreme value distribution is a good approximation to thedistribution of the annual maximum surge,

l limiting extreme value theory results hold for temporal dependence,

are discussed in Section 6.1.

5.2 Relationships Between Methods 65

The use of extreme value models for the tail of the surge distribution is reasonable as thesurge is relatively stationary over the winter storm season and the limiting result should bea good approximation to the annual maximum distribution. Experience suggests that theextremal dependence, modelled through the extremal indices, is also of sufficient flexibilityso as not to be a restrictive assumption. The interaction model is the same as for theJPM so comments discussed above apply here in the same way.

5.2 Relationships Between Methods

There is a clear relationship between the two joint probabilities methods, but what is lessclear is the relationship between these and the r-largest method.

Considering the RJPM (in the independent tide-surge case) with distribution functiongiven by equation (4.20), if there were no tide, i.e.

becomes

the transition between the parametric and non-parametric components of the surge dis-tribution, is taken to be larger than the maximum observed surge then the RJPM isidentical to the JPM.

and the still water level is the tidal level. In this case the distribution function of the

This form is due to the deterministic nature of the tide, HAT is the largest level to occur,and from the cyclic pattern of the tide it will occur every nodal cycle. This distribution


is exactly the form given by both the JPM and RJPM in this situation. By comparison,

Consequently, using independence of annual maxima, the maximum still water level overa nodal cyclic follows a

distribution which is quite different from the true form. In using such an incorrect model,extrapolation to long return periods is bound to be very poor. Consequently, if the surgevariability is very small the JPM and RJPM should give very similar results but ther-largest method is subject to bias.

5.3 Additional Considerations for Comparison

The key considerations when comparing the three methods of analysis are the validity ofthe assumptions made. However other considerations are relevant in wide scale applica-tions and these are discussed here.

Knowledge of tidal sequenceThe JPM and RJPM use knowledge of the constituent physical processes of tideand surge which produce the still water levels. By failing to exploit this knowledgethe r-largest method omits valuable information for the analysis at the site. Inparticular it can be theoretically shown that standard errors are smaller for theRJPM than the r-largest method due to the averaging effect of the tide combining

with the surge. This aspect is of some value here but has more relevance whenviewing the problem on the spatial scale. Exploiting the coherence of the separatesurge and tide processes is feasible as individually they change in a simple fashionalong a coast, whereas for a spatial r-largest method it is less clear how parametersvary along a coastline due to changes in tide and surge. This aspect is pursued inthe second stage of the work.

Measures of precisionThe RJPM and the r-largest method, but not the JPM, as applied by Pugh andVassie, have associated standard errors.

Erroneous DataThe sensitivity of the methods to erroneous data is of much importance for dataanalysis. Erroneous data are typically of one of two forms: data recorded with atiming error, and from tide gauge malfunction. The way the data are used in theanalysis is quite different for direct and indirect methods so erroneous data influencethe results differently.

5.3 Additional Considerations for Comparison 67

If there is a timing error in the measurements this will not affect the r-largest methodas observations are not taken to occur at high tides. However, with the JPM andRJPM, when the predicted tides are removed, some tide is left as a contaminant inthe surge series. Since the tide typically has a greater variability than the surge,such a period is likely to give the largest annual derived surges, so timing errorshave a large effect on any indirect method of extreme sea level estimation. Thereare two possible actions in such cases

l remove the erroneous data: this removes the problems for the indirect methodsbut leads to inefficiency for the direct methods,

l leave the data as they are: which is optimal for the direct methods but canlead to significant overestimation of return levels for the indirect methods.

If there is a gauge malfunction then both approaches will give biased estimates butwhich is the more sensitive method to the erroneous data depends on the nature ofthe malfunction.

To conclude the discussion of this feature we believe that high quality data, whichhave been carefully screened for errors are necessary for application of the JPMand RJPM. When differences arise between the indirect methods and the r-largestmethod the data quality must be reassessed for errors.

Historical data.Throughout the description of the existing methods, and our later analysis, it isassumed that the data are hourly observations only. However often historical datasuch as annual maxima, as collected by Lennon (1963), Suthons (1963) and GrafF(1981), are available. These data should be incorporated into the analyses as theyprovide additional information on historical events, such as the 1953 event, and longterm trends, which may not be well represented in the period of hourly observation.Inclusion of such data into the r-largest method is straightforward: the number ofannual extreme events can be varied from year to year (see equation (4.6) and the

incorporate historical annual maxima still water level data as the occurrence timesof these historical observations are often unknown so the corresponding surge valuescannot be derived. More critical though is that we do not know whether other largersurges occurred, which happened not to produce as large a still water level becausethe corresponding tide was much less. In fact for the JPM these data cannot be

included, so this is quite a weakness with the method. For the RJPM a method to


incorporate such data has been developed within the project and details of this arereported in Section 9.4.

Subjective parameter choicesThe methods vary in the degree of subjective choice required at the estimation stage

constitute extreme independent events have to be made. For the JPM a model forinteraction of tide and surge needs to be chosen, whilst for the RJPM each of theabove features need to be selected together with the thresholds for the extremalindices and the threshold joining the non-parametric and parametric components ofthe surge distribution. Hence the RJPM requires much experience in its application.

TrendsThe handling of trends is different in each of the three methods. The estimated trendcorresponds to the trend in still water level extremes for the r-largest method, to thetrend in mean sea-level for the JPM, and to the trend in extreme surges (or extreme

surges at high tide) in the RJPM. Therefore the r-largest and RJPM estimateessentially the same trend but the JPM focusses on mean sea-levels. For this reportwe ignore the trend component in the JPM but return to this issue in Section 8.

Simplicity of applicationThe r-largest method is simple to apply. The JPM is also relatively simple toapply but the requirement to model the interaction complicates its use, so only intide-surge independent cases can the JPM be routinely applied.

Forms of dataDirect methods are the only possible approach when the data are purely annualmaxima, peaks over thresholds, or r-largest still water levels observations.

5.4 Recommendations for Use

In conclusion, for short data sets (of l-8 years length) the only viable option of the threemethods under study is the JPM although this is likely to underestimate return levelscorresponding to long return periods and is highly dependent on the observation period.If long data records are available the RJPM should be the chosen method. Selectionbetween these methods must be based on a comparison of each, using the confidenceintervals as a guide, and also should depend on whether there is access to historical data.

The r-largest method, and hence the annual maximum method, is subject to bias dueto the violation of the stationarity assumption so it must be used with care. Situationsin which it can be helpful for comparative purposes are when the variation in high tides

5.4 Recommendations for Use 69

is small compared to the surge variation; when there is large interaction between the tideand surge as modelling this is a potentially vital component of the indirect methods; andwhen the data quality, with respect to timing, is low.

70 6 REFINEMENTS TO THE SEPARATE SITE METHODS

6 Refinements to the Separate Site Methods

6.1 Tide-Surge Interaction

In this section, we describe how we have revised the handling of tide-surge interactionin both and the JPM and the RJPM within this stage of the project. The methodsused here are modifications of those developed by Pugh and Vassie (1980) and Tawn(1992) respectively. The methods need revision to give additional flexibility required forapplication to the wide range of interaction characteristics observed at the 22 sites, andto improve their numerical stability.

A key feature for successful application of the JPM and the RJPM is the accurateestimation of the tail of the surge distribution. As described in Section 4, if the tideand surge are independent processes, the distribution of the surge conditional on tidallevel is the same for all tidal levels so standard estimation techniques for extremes ofstationary sequences, such as the r-largest met hod can be used. However, in shallow waterareas, dynamic processes such as bottom friction cause the tidal and surge componentsto interact; in particular, surge values that occur at the time of high tide tend to bedamped, whereas surge values on the rising tide typically are amplified. Accountingfor such interaction in the modelling of extreme surges is important, since ignoring thisfeature and proceeding as if the processes were independent is liable to result in significantoverestimation of return levels of the still water level.

Interaction characteristics vary from site to site due mainly to variations in waterdepth. As we are interested only in interaction between extreme surges and the associatedtides then results from dynamical studies, for example Wolf (1978), cannot be applied.Consequently, the form of the interaction at any site must be estimated directly from theobserved hourly data. Using data from each site separately, as is the case in this report ofsite-by-site methods, estimation of interaction is problematic for sites with short recordlengths. The reason for this difficulty is that if only a few large surges occur at high tidallevels this could be due to either

1. the presence of interaction between the tide and surge, or

2. the tide and surge being independent, but by chance, many of the largest surgelevels having occurred at times other than those of high tidal levels.

Distinguishing between the two possibilities is critical in determining whether or notinteraction exists for any given data site. As longer records become available for a site itwill become increasingly feasible to distinguish between these possibilities, although some

questions are still likely to remain about the exact form of interaction at the very highest

6.1 Tide-Surge Interaction 71

tidal levels (that is those very close to HAT) as purely by chance few extreme surges willoccur at these rare tidal levels within an observational period.

Consequently most problems in this first stage of the work will be encountered atsites with short record lengths. However as interaction varies smoothly over spatial scalesestimation of interaction should generally be improved by viewing the estimation problemas spatial; details of this are covered in the second stage of the work.

The standard oceanographic approach used to assess the level of tide-surge interactionfrom observations is to examine of the standard deviation of the surge conditional on thetidal level (Prandle and Wolf, 1978; Pugh 1987; Walden et al., 1982). Since interest hereis in the extreme surges a more relevant aspect of interaction is based on the extremesof the distribution of surges on tidal level. We develop tests and models for interactionbased on these characteristics. First consider the test for interaction between extremesurges and the associated tidal level.A Test for InteractionBy splitting the tidal range into five equi-probable bands there are then an equal numberof surge observations with associated tides in each band, and so a surge observationhas an equal probability of falling in any one of the five bands. If the surge and tidewere independent processes the number of surges per tidal band expected to exceed acommon level, u, would be the same. If the surge and tide interact then the largest surgeswould occur at mid-tidal bands and the smallest in the highest tidal band. Consequentlythe number of surges per tidal band which exceed a high level u would be expected todiffer from tidal band to tidal band, with most occuring in the mid-tidal bands and least

To illustrate this test the left hand columns of Figures 14-19 show all the surgeswhich exceed the 99.75% quantile against the associated tidal levels. The lines on theplots determine the five tidal bands, with the lowest and highest lines at LAT (lowestastronomical tide) and HAT respectively. Around the coast there are varying degrees


of interaction however where interaction clearly occurs, for example at Southend, thereare significantly more extreme surges in the lower tidal bands. The second column in

Clearly only a few sites show no significant evidence for interaction, these being Newlyn,Ilfracombe, Stornoway, and Felixstowe, however these are generally of short record length.It is assumed that there is interaction present at all sites and this is modelled thoughoutrather than assuming independence and potentially over simplifying results.

of interaction at each site which is approximately independent of record length. Thesevalues are also given in Table 5 and show a smooth increase in interaction down the eastcoast to Immingham, followed by a decrease to near independence at Lowestoft and largeinteraction levels at Southend and Sheerness. At the sites with least data this interactionstatistic is most variable, i.e. at Cromer and Felixstowe, but the basic smoothness canbe fully exploited by spatial analyses, as discussed in the second stage of the work. Onthe west coast there is notably less spatial coherence in the interaction statistic, with thelargest degree of interaction at Heysham and Avonmouth.

Modelling Interaction at each SiteNow consider modelling this observed interaction site-by-site. Section 4.4 describes howto model the surge distribution when tides and surges do not interact. One way ofextending this approach to incorporate interaction is to model the upper tail of the surgedistribution conditional on the tide, in other words extending the model of Section 4.4 sothat the extreme surge parameters depend on the concomitant tidal level. This can be

applied this approach to data from some east coast sites; and found that the location andscale parameters could be suitably modelled as a function of tidal level using low-orderpolynomial regression functions, and that there was no evidence for dependence of theshape parameters on the tide. A more flexible approach for routine application is to usecovariate forms obtained from the empirical joint distribution of the tide and surge. Thishas the benefit that it is particularly relevant when interaction is viewed on a spatialscale. Suppose that the location-scale normalisation of the surge series,

is a stationary series for some functions of tidal level, a(X), and b(X) yet to be specified.Then the methods for estimating the surge distribution in the tide-surge independencecase can be applied on this transformed series to give estimates of the parameters of the


The approach differs from Pugh and Vassie (1980) as all surge levels, irrespective of theassociated tidal level, are used in the evaluation of the empirical estimator.

6.1.1

and b(X) from the empirical conditional distribution of surges given

tidal bands,

j74 6 REFINEMENTS TO THE SEPARATE SITE METHODS

tidal band then our estimates of u(X) and b(X) are

site using five and also ten tidal bands. Each curve is piecewise linear, taking a constantlevel separately above and below the upper and lower tidal band mid-points. Deviations

largest surges occur, typically at mid and low tidal levels. Sites with largest interaction

Avonmouth, and Heysham. Furthermore there are clear spatial patterns in the formof interaction. Generally interaction appears to be modelled well with five tidal bands,with the possible exceptions being Southend, Sheerness, and Ilfracombe where the curvesfor the ten tidal bands are reasonably different, particularly in the top band. In thesecases, ten or more tidal bands may be required but otherwise five bands seems adequate.This form of interaction modelling has some limitations due to the piecewise nature ofthe estimation and since spatial changes in interaction are not incorporated. These arediscussed further in the second stage of the work.

large if the interaction is of a complex nature. Alternatively, if too many tidal bands areused the variability in estimation of the quantiles of the conditional surge distribution is

to be small, so a compromise is required for the choice of the number of tidal bands. Inpractice many choices for the number of tidal bands can be considered, and the sensitivityof results assessed.

with five tidal bands, for all sites. Withone or two exceptions there is notably less interaction than between tide and surge,and generally the transformation appears to remove all interaction. In particular see


now occur equally thoughout the tidal range. We can assess whether interaction remains

Cromer, Felixstowe, Stornoway, each with short series, and Ullapool. For short recordsites a possible explanation for this is that the conditional quantiles are poorly estimated,i.e. u(X) and b(X) are badly estimated, so additional data or spatial analyses should

improve this estimation, The situation is less clear for Ullapool, either more tidal bands

is little interaction to remove. In conclusion the proposed method of modelling interactionappears to work adequately.

86 6 REFINEMENTSTO THE SEPARATE SITE METHODS

6 . 2 Extremal Index Estimation in the RJPM

the number of extreme values within an event for the respective processes. In particular,

independent event which exceeds the level x then

As discussed in Section 4.4.3, Tawn and Vassie (1989) and Tawn (1992) propose that inor x in the range of the data using the respective

sample mean size of clusters of exceedances of x. There x is chosen to be large enough so

parameters, rather than the parameters individually, that is important in obtaining return

the level x to be the same quantile of the hourly still water level and surge distributions

The procedure for estimating the extremal index parameters at each site is to examine a

variability versus limiting value compromise discussed above.

the ensuing discussion. The storm length, which determines what are considered to be

extremal index ratio. Similar findings were obtained for other sites but details are omitted.Consequently this approach is used in applications of the RJPM in this report. Clearly

variations in the estimate of the extremal index ratio will be examined, see Section 7.3.2.


6.3 Declustering Series: Sensitivity Approach

The first step in application of the r-largest method applied to any series is the identifi-cation of independent extreme events. Various methods of declustering a series to removedependence have been proposed in the literature. Results are not sensitive to the formof the procedure used; so here a simple method is used. We assess the sensitivity of thereturn level estimates to various choices of declustering parameters in order to determinean effective declustering procedure. The method of declustering assumes that two eventmaxima are independent if they are greater than some pre-determined time separationapart. This separation, which we term the storm length declustering parameter, needs tobe chosen in some way. If the storm length is too small the r-largest events will be de-pendent whereas if it is too large independent events will be excluded. Further discussionof this aspect is given in Section 7.1.2 where the sensitivity to this choice is assessed foreach of the data-sites.

89

7 Applications and Results for A-Class SitesIn this section the methods described in Sections 4 and 6 are applied to each site given

in Tables 2-4. First, in Section 7.1, the r-largest method is applied. As the special case

Sections 7.2 and 7.3, the indirect methods of the JPM and the RJPM respectively areapplied, and results given for each site.

A common way to display the results of extreme value analyses is via the graphicalpresentation of port diagram curves. A port diagram curve is a plot of the return level,

i.e. the return level plotted against the returnperiod, with the latter on a log scale. This scale of plotting the relationship betweenreturn level and return period is convenient because if the tail of the annual maximumdistribution is Gumbel (i.e. k = 0) the port diagram curve is a straight line, whereas

There are a number of features in each method which require some element of subjectiveselection. For this method the choice of how many extreme events to use per year andthe choice of what constitute independent extreme events within a year must be made.In accord with Section 6.3 rather than arbitrarily adopting a selection we try all possiblechoices and select from these, this provides an in-built sensitivity assessment into theseselections.

7.1.1 Choice of Number of Order Statistics

The first step of the r-largest method is the selection of a suitable number of independentextreme events from each year of data, or equivalently the choice of a suitable threshold.

and shape respectively for Immingham and Lowestoft. These estimates were obtained by

7 APPLICATIONS AND RESULTS FOR A-CLASS SITES

A notable feature of this plot is the significant improvement in the estimation ofthe trend and shape parameters, as measured by the 95% confidence intervals for each

the inclusion of additional information about extreme events leads to a positive trendestimate that is in accord with previous studies of long term annual maximum data (Graff,1981; Dixon and Tawn, 1992) and the neighbouring Lowestoft estimate. Furthermore theconfidence interval suggests, in fact, the trend really is positive. The difference in interceptparameter between the two sites is a consequence of the tidal range being much greater for

shape parameter is due to the surge extremes being more variable at Lowestoft, as shownby Figure 5, Table 1 and the discussion concerning approximation (4.22).

Similar plots for all the other sites are given in Figures 57-68 in the appendix. Figures

tidal variability, and estimated trend and shape parameters have greater precision as r

increases. The range of scale parameter estimates is much greater than for Imminghamand Lowestoft.

reduces the associated 95% confidence intervals. However selection based solely on param-eter estimates which exhibit inter-relationships (e.g. the intercept and trend parametersat Immingham) is problematic. Instead we also consider the function of the parameterswhich is of most interest, namely the return level:

concentrating on the 100 year return level. Figures 25 and 69-71 show the 100 year

the uncertainty of the estimate, although again an improved precision, i.e. reduced 95%

for the other plots, with notable increases in precision for Wick, North Shields, Whitby,Southend, Dover, Newlyn, Fishguard, Holyhead, Heysham, Portpatrick, Stornoway andLerwick as r is increased, i.e. in moving from the annual maxima method to the r-largestmethod. For some sites, e.g. Aberdeen, Sheerness, Ilfracombe and Avonmouth there is a

7.1 r-largest 91

loss of estimated precision. However, we believe the poor fit of the annual maxima datahas led to estimates of an overly short distribution tail for each of these sites, so in eachcase the precision was underestimated when r = 1, and the results for larger r are a moreaccurate reflect ion.

One final consideration when selecting r is that if too large a value is taken, the asymp-totic theory which justifies the selection of the statistical extreme value model may beviolated. As a compromise the best value of r appears to be around r = 8. This is a good

choice for all sites around the coastline as there are no apparent spatial characteristics toinfluence this choice, so r = 8 is the chosen number of extreme events per year in the finalr-largest analysis at each site.

7.1 r-largest 93

Immingham

2 4 6 8 10 12 14

number of order statistics

Lowestoft

2 4 6 8 10 12 14

number of order statistics

largest method, plotted against the number of order statistics used.

94 7 APPLICATIONS AND RESULTS FOR A-CLASS SITES

7.1.2 Choice of Storm Length in Declustering

for the two example sites and all sites respectively. Excluding the sites with very short

the associated uncertainty of the estimates given by the 95% confidence intervals and it isclear that storm length has relatively little impact so long as an intuitively sensible stormlength is taken. There appears to be an insignificant systematic increase in return levelswith storm length and no obvious coastal variation in relationship with storm length soagain a common storm length is adopted. Consequently, as with published examples, we

7.1 r-largest 95

Immingham

20 30 40 50 60 70

Storm length (hours)

Lowestoft

20 30 40 50 60 70

Storm length (hours)

largest method, plotted against the storm length used.

j96 7 APPLICATIONS AND RESULTS FOR A-CLASS SITES

7.1.3 Results for the r-Largest Method

More generally port diagram curves, shown in Figures 27-29 give return levels, andalso show the associated 95% confidence intervals, for all return periods. Since the shape

levels. The sites with poor estimates, e.g. Cromer and Felixstowe, exhibit either largeconfidence intervals rendering the estimates of little value or are so dissimilar to neigh-bouring sites so that the estimates are doubtful. Excluding these sites, the remaining sitesdo exhibit considerable spatial coherence which provides further basis for the subsequentspatial analysis of the second stage of the work.


7.2 Joint Probabilities Method

The application of the JPM involves the least subjective choice in the modelling of extremesea-levels. In the tide-surge independent case no choices are required, but more generallywhen the tide and surge interaction is modelled, a selection has to be made in terms ofthe number of tidal bands that need to be used. Again the sensitivity of the analysisto this aspect can be assessed by considering a range of choices for the numbers of tidalbands. Given that empirical methods are used for estimating the distribution of hourlysurges (or hourly transformed surges) then strictly no element of modelling is required atthat stage.

Once the number of tidal bands has been selected, intermediate calculations to evalu-

Thus the impact of the choice of the number of tidal

tions of the tidal level, we use piecewise linear estimates, discussed in Section 6.1, whichare crude approximations. As the number of tidal bands is increased, the number of line

bands improves the approximation allowing potentially complex forms of interaction to bemodelled. At the same time increasing the number of tidal bands requires more aspects ofthe joint distribution of tide and surge to be estimated. Again there is a trade-off betweenbias and precision which must be accounted for in the choice of the number of tidal bandsto be used.

7.2.1 Choice of the Number of Tidal Bands

Recall that taking one tidal band is equivalent to assuming the tide and surge are indepen-dent processes. By incorporating independence as a special form of interaction then theimpact of assuming independence or interaction can also be assessed within the frameworkof varying the number of tidal bands.

The sensitivity of the 100 year level to the selected number of tidal bands is assessedusing Figure 30 which shows the estimates for Immingham and Lowestoft. These plotsare similar in nature: for each site there is a large reduction in return level in movingfrom independence (1 tidal band) to crude models for interaction (2-3 tidal bands). Morerefined models for interaction, obtained by increasing the number of tidal bands used, leadto much more limited changes, with slight increases/decreases at Lowestoft/Immingham

respectively.Figures 75-77, for the other sites, show a similar pattern to Immingham and Lowestoft,

103

with variation in the degree of interaction found from site-to-site being consistent withthe results of Section 6.1. For all east coast sites (other than Cromer and Felixstowe) thedegree of interaction: shown by the drop in return level in moving from one to six tidalbands say, varies smoothly along the coast following the pattern previously found. In

respectively. Elsewhere on the west coast. the reductions are 1Ocm or less.

In conclusion it can be seen that, the variation in interaction is relatively smooth onthe east coast. whilst having a large effect, but generally is less important for the west

coast sites. Again, these are important observations for the subsequent spatial study.Furthermore: the precise number of tidal bands does not seem to have much impact onthe return level estimates, provided that more than 3 tidal bands are used: although atthis stage this is difficult to assess without confidence intervals despite similar findings at

Immingham

number of bands in the JPM

Lowestoft

number of bands in the JPM

110

7.3 Revised Joint Probabilities Method

7.3.1 Obtaining the Annual Maximum Distribution of Surges

jjjj7.3 Revised Joint Probabilities Method 111

and suggests that surges amplify as they travel south. Due to tide-surge interactionvarying spatially this does not necessarily imply Southend and Sheerness will necessarilyhave the largest. surges at high tide. On the west coast the intercept parameter is typicallymuch less than on the east coast., i.e. the west coast sites have smaller extreme surges:with the notable exceptions being Avonmouth and Heysham. Estimation of the trend

long distributed tails.

112

year return level for still water level against the extremal index ratio taken over a rangeof l-3, which covers that observed over all sites. This is shown in Figures 40 and 96-98 for Immingham and Lowestoft: and all sites respectively. The influence is largest at

8cm when only variation of the extremal index ratio with respect to storm length atthat. site is considered. The results for Avonmouth, and all other sites with significant.

interaction, overestimate the importance of precise estimation of the extremal index ratioas the tide and surge are taken as independent. Concentrating on sites with weakestinteraction the site most sensitive to variations in the extremal index ratio is Lowestoft:with variations of 25cm in the 100 year return level as the extremal index ratio variesfrom l-3, but the variation is only 5cm for variation with respect to storm length at that,site.


7.3.3 Tide-surge Interaction Modelling

7.3.4 Results for the RJPM

five tidal bands for tide-surge interaction, and throughout adopting a declustering storm


Immingham

1 .oo

percentage of data below threshold

Lowestoft

0.95 0.96 0.97 0.98 0.99

percentage of data below threshold

1 .oo

Figure 38: Extremal index ratio against quantile threshold, 4, for Immingham and Low-

estoft.


Immingham

20 30 40 50 60 70

Storm length (Hours)

Lowestoft

20 30 40 50 60 70

Storm length (Hours)

Figure 39: Variation of the extremal index ratio with storm length for Immingham and

Lowestoft.

7 APPLICATIONS AND RESULTS FOR A-CLASS SITES

using the RJPM. These are shown in Table 10 for return periods of 10, 25, 50 and 100years, and Table 11 for return periods of 250, 500, 1000 and 10000 years. In each caseestimates are to’ Chart Datum and are in metres. Confidence intervals can be constructedfor the return level for any given return period using the results for maximum likelihoodestimators (discussed in Section 4.5). These take the form of the return level estimateplus or minus 1.96 times the associated standard error.

Port diagram curves, shown in Figures 42-45, give return levels, and also show theassociated 95% confidence intervals, for all return periods. The sites with poor estimates,e.g. Cromer and Felixstowe, exhibit either large confidence intervals, rendering the esti-mates of little value, or are so dissimilar to neighbouring sites that we do not trust theestimates. The remaining sites do exhibit considerable spatial coherence in form, whichprovides further basis for the subsequent spatial analysis of the second stage of the work.For some sites the estimates are of high precision even at long return periods, this is gen-erally the case for the west coast sites, and others have large confidence intervals despitehaving long data sets, for example Lowestoft and Southend, this being due to the largevariability of extreme surges at these sites.

128

8 Comparison of Results

8 COMPARISON OF RESULTS

In this section, estimates for the three methods obtained in Section 7, are compared.Specifically, return level and trend estimates are discussed in Sections 8.1 and 8.2 re-spectively. In Section 8.3 these values are compared with both site-by-site and spatialestimates obtained by Coles and Tawn (1990) who used long term historical annual max-ima. In Section 8.4 recommendations are made as to the best of the existing methodsfor each of the sites. In Section 8.5 we illustrate the calculation of design levels for yearsother than 1990, the year to which return level estimates in this report are given.

8.1 Comparison of Return Levels

Figures 45-47 give port diagram curves (discussed in Sections 7.1-7.3) obtained by appli-cation of the three methods to the 22 sites. In each case the methods have been appliedusing the uniformly chosen ‘optimal’ selections for the subjective elements of the methods.The plots presented here aid comparison between sites; larger versions of these plots forexplicit comparison of methods at a site are given in Figures 102-112 in the appendix.

Cromer and Felixstowe have very large confidence intervals for at least one of themethods at each site. These sites have previously been shown to have insufficient datato obtain a reasonable fit based on site-by-site analyses. Therefore we exclude them fromthe present discussion.

Of the sites with sufficient data to base conclusions there appear to be three types ofsite:

l For Whitby, Immingham, Lowestoft, Southend, Sheerness, Heysham andLerwick the agreement of return levels between methods is very good for all returnperiods, particularly once the associated confidence intervals are accounted for.

largest method gives lower return levels for long return periods than given by theindirect methods. In particular, the upper limit of the confidence interval for ther-largest estimate is close to the lower limit of the confidence interval for the RJPM.For these sites the RJPM and JPM agree well for return periods over 50 years,

l For Wick, Newlyn, Ilfracombe, Avonmouth, Milford Haven, Fishguard,Holyhead, Stornoway and Ullapool the RJPM and JPM agree well for returnperiods over 50 years but the r-largest method gives significantly lower return levelsparticularly for return periods longer than ten years.

These classifications are approximate due to variations in the lengths of data records

between sites; notably Whitby and Avonmouth have short records, and hence reasonably

8.1 Comparison of Return Levels 129

large confidence intervals. The lack of agreement may be masked in such cases because ofthese large uncertainties. More data should clarify these classifications, though broadly

the current classifications appear reasonably valid. The significant discrepancies betweenthe methods at some sites, but not others, were unexpected since all previous analysesfound ‘good’ agreement between the three methods. There are two reasons for this:

1. Previously return periods of up to only 100 years were considered. As seen hereagreement is generally reasonable at these levels and discrepancies observed werebelieved to be explained by associated confidence intervals, which were not evaluated(Pugh and Vassie, 1979, 1980; Alcock and Blackman, 1985).

2. All three methods had previously been applied to Lowestoft, Immingham and Sheer-ness only (Tawn and Vassie, 1991; Tawn, 1992) and these are sites for which a goodagreement is found here.

To be able to make recommendations for the use of these methods we need to decide whichmethod is the most suitable for each site, and more generally need to develop a diagnosticprocedure to identify in advance the classification of a site. To address these issues itis helpful to recall the assumptions that are made for each of the methods since bothestimates and confidence intervals are obtained subject to these assumptions. Figures 45-47 show that for some sites these assumptions are reasonable for each method yet for othersites the assumptions appear valid for only a direct or an indirect approach. Assumptionswhich possibly lead to the observed discrepancy, i.e. the r-largest being lower, are

1. poor quality data which introduces artificially large surges. In this case the JPMand the RJPM overestimate return levels;

2. incorrect modelling of the tide-surge interaction, In this case the JPM and theRJPM overestimate return levels;

3. the non-stationarity of the sea-level extremes being critical to the application of ther-largest method, so predictable variations in the tide are not being exploited in theextrapolation.

As both the first two possibilities lead to the observed findings these seem the most likelyreasons.Reasons for discrepancy: (a) Poor quality data.The data were carefully checked by carrying out iterative cycles of data analysis and re-moval of suspect erroneous data. This improved the comparisons only slightly, so dataquality does not appear to be the factor causing the discrepancy.

130 8 COMPARISON OF RESULTS

Reasons for discrepancy: (b) Inadequate interaction modelling.Interaction modelling does not appear to explain the discrepancy either, since the siteswith the worst interaction models (judged by variations in the tidal functions in movingfrom five to ten tidal bands) are Southend and Sheerness, yet these have a good agreementbetween the methods. Generally, we found that increasing the number of tidal bandsreduced the 100 year return level by at most 1Ocm (see Figures 75-77 and 99-101) andmore usually this reduction was less than 5cm. These reductions though are much less

than the discrepancy between the direct and indirect methods for some sites. Thereforewithin our modelling framework for interaction this aspect does not appear to explainthe discrepancy, however there is still a small possibility that at some sites there is stronginteraction at the very highest tidal levels but this seems unlikely as

l interaction is present throughout the tidal range, and sudden changes in form seemunrealistic from physical oceanographic reasons,

l the sites where the methods differ are not atypical in any aspect of interaction,covering sites with no apparent interaction (e.g. Newlyn) through to sites withlarge observed interaction (e.g. Avonmouth).

To address this point further spatial assessment of interaction is required, and this isdeveloped in the second stage of the work, but it should make only limited changes forthe above reasons.

largest method.Another possible explanation for the discrepancy in the observed estimates is the inap-propriate application of the r-largest method to sea-levels due to the non-stationarity ofextreme sea-levels. In Section 5 it was argued that if there was no tide at a site thenthe assumptions would be entirely valid, but if, at the other extreme, there was no surgethen the assumptions of stationarity and that the generalised extreme value distributiondescribes the distribution of the annual maximum would be strictly inappropriate. Con-sequently the validity of the assumptions is most influenced by the relative variability oflarge surges by comparison to variability of high tides, i.e. an extreme value version of thetide or surge dominant classification of a site given by Middleton and Thompson (1986).In particular, if the principal variation in extreme combinations of tide and surge, i.e.extreme still water levels, is due to variations in high tides then the assumptions are moresignificantly violated than if the variation is principally due to the surges. As extremeexamples, the tide is smallest in range for Lowestoft and Lerwick, and for both sites thereis a good agreement between methods; whereas the tidal range is largest for Avonmouthand Ilfracombe and these sites have poor agreement between methods.

8.1 Comparison of Return Levels 131

Further Investigation into the stationarity assumption.Using the interaction function b(X) in the top tidal band, i.e. the difference in the 99%and 98% quantiles of surges with concomitant tides in the top tidal band, to measure thevariability of extreme surges at a site (see Section 6.1) and taking the difference betweenHAT and the 98% tidal quantile as a measure of high tide variation, surge variation againsttide variation is plotted in Figure 48. All sites in the top region of the plot are characterisedas giving a good agreement between the methods; the sites in the middle region correspond

to those sites for which the return levels for the r-largest method are slightly lower thanthose for the indirect methods (Whitby being an exception); and the bottom regioncorresponds to those sites for which there is a large discrepancy between return levelsfrom the r-largest and the indirect methods (here Lerwick is an exception). Thus, withour selected measures of variation in large surges and tides, the level of agreement ofestimates depends mainly on the surge variability with some evidence of dependence alsoon the tide.

Assuming that our reasons for discrepancies in the methods are correct, then on thebasis of Figure 48, Whitby appears to have been misclassified. If more data were availablesome discrepancy, possibly similar to that at North Shields in Figure 45 may be observed.This is less clear at Lerwick as there are sufficient data to trust the original classification,so one possibility is that the high tide variation measure is not the most suitable and thatother forms may be more appropriate. Possibilities for alternative tidal measures includethe difference between HAT and mean high water springs, or mean high water springsand mean high water neaps. The possibilities are large and the current choice of measureseems adequate for all other sites so this issue is not considered further.

We address the possibility of underestimation of return levels by the r-largest methodnow in an alternative way. In particular, we examine the difference between the returnperiods for the observed maximum level, estimated using each method at each site, tosee if similar conclusions are drawn. Table 12 lists the number of years of observed dataand the estimated return periods for the maximum observed level for each method. Thecalculations did not account for the year in which the maximum occurred, so exclude thetrend, but our conclusions drawn are reasonably invariant to this. For each of the siteswith the poorest agreement between the methods, Table 12 shows the r-largest methodpredicts the return period associated with the observed maximum to be much larger thanthe observed number of years of data, i.e. the r-largest method appears to be consistentlyunderestimating the return levels at these sites. On the other hand the JPM and RJPMgive estimates which are generally much closer to the observed data length (e.g. Wickand Newlyn) and show no systematic bias. This seems to strongly support the previous

findings that the r-largest method underestimates return levels in these cases.

j132 8 COMPARISON OF RESULTS

From Figures 45-47, Figure 48, and the findings in Table 12 we must conclude for sites

largest method, and hence also the annual maxima method, underestimates return levelsfor long return periods. Due to the spatial variations in tide and surge these conclusionscould be stated as, on the east coast the agreement is good but on the west coast it ispoor. These are new findings but, as shown, do have a strong theoretical and practicalbasis.

Potentially there are large repercussions due to this finding as traditionally all extremesea-level analyses have been undertaken using the annual maxima met hod. However, thereis one caveat associated with this conclusion, namely that the data to which the r-largestmethod has been applied are only those from hourly observations, i.e. are restricted induration and exclude historical maxima which are included in typical applications of theannual maxima method (Graff, 1981; Coles and Tawn, 1990). We shall return to thisissue in Section 8.3.

138 8 C O M P A R I S O N O F R E S U L T S

upper bound on transformed surge distribution being taken as the maximum observedtransformed surge. This is particularly restrictive at Lowestoft where, from Figure 45, itis seen that the estimates for the JPM are much less than those from the RJPM for longreturn periods. There is some evidence of similar features for Southend and Sheerness butelsewhere this causes no problems. Hence the sites with discrepancies between indirectand direct methods are the most surge dominant, as seen from Figure 48.

The second element concerns the extremal index ratio. From Figures 93-98 it can be

seen that this element influences the 100 year level by up to 6cm only. Generally this effectis minimal at long return periods but can be quite important for short return periods.Consequently the JPM appears to overestimate relative to the RJPM for short returnperiods. It is a notable feature for Wick to North Shields, Dover to Newlyn, Avonmouthto Heysham, and Stornoway to Lerwick, although this effect rarely causes a significantdifference for return periods longer than 50 years. Consequently the JPM and RJPM areconsistent for return periods in the range 50-10000 years, other than for sites which arestrongly surge dominant.

Finally, returning to Cromer and Felixstowe, none of the estimates are likely to bereliable, however the only estimate which is viable is that due to the JPM. As thesesites are strongly surge dominant, even the JPM estimates will be biased for most returnperiods due to the reasons discussed above.

8.2 Comparison of Trends 139

8.2 Comparison of Trends

In the application of the three methods of extreme sea-level analysis different estimatesof the trend have been obtained. These are

i.e. in extreme surges,

e trend in the RJPM applied assuming interaction, i.e. in the surgesfor which the concomitant tide, X, is in the top tidal band, see equations (6.3) and(6.4). The notation here is as in Sections 4 and 6.1.

These estimates are given in Table 13. Owing to the short duration of data all trendestimates have relatively large associated standard errors. Once these errors are accountedfor, there are no significant differences between the three trend estimates, so any of the

independent still water levels event peaks are not necessarily the observations with thelargest surges) these estimates are reasonably independent and could be pooled to improveprecision, that is

However, the potential gain is limited by comparison with that from viewing the estima-tion as a spatial problem as discussed below.

As expected the trends vary spatially due to changes in land level trends from siteto site, and the observed pattern is consistent with that found in Shennan (1989). Inparticular, large positive trends occur in the South-East and the North-West, but smallor negative trends occur in southern Wales.

As there exists some spatial pattern in the trends, primarily due to varying verticalland movements from site to site, trend estimation could be improved by incorporatingknowledge of this spatial pattern. There are two possibilities:

l use estimates of land level trends obtained by Shennan (1989), or some other source,

l assume only that there is a smooth spatial pattern to the trend, without taking aspecific form.

trends, and in the second stage of the project similar methods are assessed for extremesea-level trends.

8.3 Comparison of Return Levels with Previous Studies 141

8.3 Comparison of Return Levels with Previous Studies

Tables 14 and 15 give the return level estimates obtained by Coles and Tawn (1990) fromanalyses of annual maximum data. In Table 14 the estimates are obtained using theannual maxima method, as described in Section 4.1, and in Table 15 have been obtainedusing the spatial annual maxima method developed in that paper. The results in Graff(1981) are contained within Table 14.

In comparing the estimates with those from Coles and Tawn (1990) three aspects mustbe considered:

l uncertainty in estimates, as given by standard errors,

l differences between the site-by-site and spatial estimates from Coles and Tawn

(1990),

l Coles and Tawn (1990) estimates are based on annual maxima analyses so are sub-ject to the same biases as the r-largest method.

Of the sites in Tables 14 and 15 only North Shields, Lowestoft, Heysham and Ullapool giveinconsistent estimates in the Coles and Tawn (1990) analyses. This is generally owing tothe record lengths being short so the site-by-site estimate has a large confidence interval.The spatial estimate falls within this confidence interval, but itself has a much smallerconfidence interval. We base comparisons solely on the spatial estimates in these cases.

l For Immingham, Lowestoft, Southend, and Sheerness all the methods agreewell with the Coles and Tawn (1990) estimates.

l For Aberdeen, North Shields and Heysham the indirect methods agree wellwith the Coles and Tawn (1990) estimates, however the r-largest method appearsto underestimate return levels.

l Dover, Avonmouth and Ullapool all have poor agreement in estimates but, ow-ing to the large confidence intervals, there is no apparent discrepancy. In particular,the Coles and Tawn (1990) estimates for Ullapool are poor due to an extremely badtrend estimate.

l For Newlyn the Coles and Tawn (1990) site-by-site method agrees with the r-largestestimate here, whereas the spatial estimate agrees with the indirect methods. Eachanalysis is based on just over 60 years of data, but the data years available are notidentical. To draw conclusions from these results an assessment needs to be made onwhich of the Coles and Tawn (1990) estimates is the better. We believe the spatialestimate is best and this therefore supports the RJPM and the JPM.

142 8 COMPARISON OF RESULTS

l For Milford Haven and Fishguard the r-largest method has best agreement withthe Coles and Tawn (1990) estimates, although for Fishguard it gives estimateswhich are much lower. The annual maximum data records are of roughly the samelength as the hourly data, so provide little extra information in these cases. There-fore the previous conclusions of the r-largest method underestimating seems reason-able.

largest method, particularly via spatial analyses, then agreement with the indirect meth-ods is improved. As the r-largest method fails to exploit the tidal variations in theextrapolation to long return periods the ability to accurately extrapolate critically hingeson the observed data, which in turn depends on the random occurrence of large surges onlow or high tides. To partially remove this random element in the extrapolation, data overa few nodal tides cycles are required in model fitting when the site is tide dominant. Thisis the case for Aberdeen, Heysham and Newlyn, where improvement in the agreement isfound from using longer records of annual maxima. For Milford Haven and Fishguardlonger records do not exist so there remains a discrepancy.

Now, for the sites in Tables 2-4, consider return level estimates other than thoseobtained by Coles and Tawn (1990).

l For Newlyn, the difference between the Pugh and Vassie (1979) estimates for the50, and 100 year return levels obtained using the annual maxima method and theJPM are 24cm, and 27cm respectively. Using the three methods here the largestdiscrepancy for these return periods at Newlyn are 6cm, and llcm, respectively.Thus refined inference and increased data much improve the agreement even thoughthe discrepancy remains at the 1000 and 10000 year levels.

l The annual maxima and JPM estimates in Pugh and Vassie (1980) exhibit thefeature found here: slight underestimation of return levels by the annual maximamethod.

l The return level estimates, obtained separately by Pugh and Vassie (1979, 1980),using the annual maxima method and the JPM, agree well with those from theequivalent met hods here.

l Alcock and Blackman (1985) obtain very similar estimates to those here for returnperiods of up to 250 years at east coast sites. Lowestoft was identified as a problem-atic site as the inclusion, or not, of the 1953 value had a large influence, changingthe 100 year level by 46cm, from 4.83m to 4.37m. Here the r-largest method gives

jj8.4 Recommendations Site-by-Site 145

8.4 Recommendations Site-by-Site

Based on the findings in Sections 8.1 and 8.2 we make the following recommendations forapplication of the site-by-site analyses:

l Wick: use the RJPM and a trend of 3.9mm/yr,

l Aberdeen: use the RJPM and a trend of l.Gmm/yr,

l North Shields: use the RJPM and a trend of 3.3mm/yr,

l Whitby: use the JPM and a trend of approximately 3.0mm/yr,

l Immingham: use the RJPM and a trend of 2.6mm/yr,

l Cromer: use the JPM and a trend of approximately 2.5mm/yr,

l Lowestoft: use the RJPM and a trend of 2.4mm/yr,

l Felixstowe: use the JPM and a trend of approximately 3.Omm/yr,

l Southend: use the RJPM and a trend of 3.7mm/yr,

l Sheerness: use the RJPM and a trend of 7.lmm/yr,

l Dover: use the RJPM and a trend of 2.4mm/yr,

l Newlyn: use the RJPM and a trend of 3.3mm/yr,

l Ilfracombe: use the JPM and a trend of 5.4mm/yr,

l Avonmouth: use the RJPM and a trend of 2.9mm/yr,

l Milford Haven: use the RJPM and a trend of -2.2mm/yr,

l Fishguard: use the RJPM and a trend of O.lmm/yr,

l Holyhead: use the RJPM and a trend of 4.8mm/yr,

l Heysham: use the RJPM and a trend of 8.4mm/yr,

l Portpatrick: use the RJPM and a trend of ll.=Zmm/yr,

l Stornoway: use the RJPM and the mean-sea level trend at that site,

l Ullapool: use the RJPM and a trend of 4.2mm/yr,

l Lerwick: use the RJPM and a trend of 0.3mm/yr,

8 COMPARISON OF RESULTS

In each case the trend has been estimated using the pooled estimate, discussed in Sec-tion 8.2, except for Whitby, Cromer and Felixstowe where an average of estimates atneighbouring sites is used. Generally these trend estimates may be poor as they are sub-ject to large variability, consequently mean sea-level trend estimates may be preferred.These trend estimates are the weakest element of the recommendations. Of the recom-mendations for the method of evaluating return levels, the sites for which we have leastconviction are:Dover, as the estimates are low relative to the Coles and Tawn (1990) estimates.Avonmouth, due to worries over whether all the interaction has been accounted for, andthe poor agreement with the Coles and Tawn (1990) analysis.Stornoway, as the precision of the estimates may be too small as the observation periodmay be unrepresentative.

8.5 Design levels for Future Years 147

8.5 Design levels for Future Years

So far all return levels have been given relative to 1990. These levels are inappropriatefor other years as the trend will have increased or decreased the level. In Section 3 theconcept of a design level was introduced and here we illustrate how this may be evaluatedusing the estimates obtained from the site-by-site analyses.

8.6 Conclusions of Comparison

The findings of Section 8.1 appear to be substantiated by previous estimates and theo-retical arguments. In summary these are:

l if the site is surge dominant all methods agree,

l if the site is tide dominant then the r-largest method underestimates return levels.The amount of underestimation is reduced as the length of annual maxima recordsis increased,

l the indirect methods provide more accurate estimates from shorter series than ther-largest method does,

l of the indirect methods the RJPM is the better provided there are at least 5-10years of hourly observations, or there are supplementary annual maxima available(see Section 9.4).

148

9 New methods

9 NEW METHODS

As well as refining and implementing the existing methods various other aspects of theanalysis have been examined. All the existing methods are for extreme still water levels yetit is extreme sea-levels which cause flooding, and these incorporate waves. Accordingly,in Section 9.1 methods to incorporate waves are outlined. Another minor deficiency inthe existing methods is that only event maxima are used in the analysis and informationfrom other extreme values within events is not exploited. Methods which allow theseadditional data to be incorporated in the analysis are outlined in Section 9.2. Throughoutthe analysis the surge series has been assumed to be stationary all through the year, or atleast through the winter storm season. This is clearly an approximation, and the resultingimpact of this false assumption together with methods to overcome this are discussed inSection 9.3. Finally, the analyses in this report have been restricted to hourly data,however, long term historical annual maxima exist for many of these sites. Incorporatingthese into the analyses would significantly improve the trend estimates. This is describedin Section 9.4.

9.1 Wave-Surge Joint Modelling

Throughout the report the presence of surface waves has been ignored, and design resultshave been based on extremes of still water levels. In the design of a sea wall, it is clearthat waves can have a considerable influence on the degree of flooding, since flooding canoccur when the still water level remains below the sea wall but waves cause overtopping.The most important function of a sea wall is therefore to prevent overtopping, or wherethat is economically infeasible, to reduce the volume of water overtopping within a stormevent. Overtopping results when some critical combination of still water level and wavesexceeds the flood defence: this can occur when the still water level is extreme and thewaves are small; at a spring tide with no surge but with extreme waves; or when bothstill water level and waves are extreme. In Section 4.3 the JPM for still water levels wasdeveloped, the problem for still water level and waves is much harder as:

1. there are two stochastic variables, surge and waves, to be modelled as opposed tojust the surge in the JPM.

2. Extrapolation of the distribution tail is required for both the surge and the waves,and the dependence between extremes of these variables must be accounted for.Thus the problem is one of bivariate extremes.

3. The combinations of tide and surge that lead to failure in the JPM are obvious, i.e.(tide, surge) pairs such that the still water level exceeds some given level. However,

9.1 Wave-Surge Joint Modelling 149

the combinations of tide, surge, and waves that lead to failure could be based on

the combined level = tide + surge + significant wave height

or based on the overtopping discharge rate, which is a function of tide, surge, sig-nificant wave height, and wave period.

4. Wave data are limited in temporal and spatial coverage, so exploratory data analysisstudies are restricted.

On the third point Hydraulics Research Station (1980) have argued for overtopping dis-charge rate to be used, and, from experimentation, have derived an empirical functionalrelationship with the still water level, significant wave height, and wave period.

In order to base design criteria on extremes of this overtopping rate variable ratherthan still water levels, modelling of the joint distribution of extreme surges, significantwave heights and wave periods is required. The extremes of the individual variables canbe addressed simply using standard extreme value models, which are adequately describedin the flood defence design literature. However the dependence between extremes of theindividual variables is fundamental to the assessment of the risk since

l if the processes were independent then extreme overtopping rates would typ-ically occur when the still water level is extreme and the waves are small, or whenthe tide is high, the surge is small, but the waves are large,

l if the processes were strongly dependent then extreme overtopping rates wouldtypically occur when both the still water level and waves are extreme.

So dependence between the processes is critical in determining the risk of overtopping.Existing methods account for this aspect of dependence in a number of ways:

1. assuming that waves and still water level are independent,

2. assuming that waves and still water level are completely dependent,

3. opting out of modelling waves all together by making an ad hoc additive adjustmentto the still water sea-level design level to account for the omission of waves from theanalysis.

Adopting any of these approximations is likely to produce biased design level estimates.A further concern with existing methods is that the probability of overtopping dischargerate exceeding a given level has been calculated as the probability of the still water levelexceeding some level and the significant wave height simultaneously exceeding some other

j150 9 NEW METHODS

level. This excludes combinations, discussed above, where only one component is extremebut the overtopping discharge rate is extreme, so again these approaches are liable toproduce biased results.

The problem of incorporating waves into extreme sea-level analyses can be split intothree steps:

1. Estimating the joint distribution of the surges and the waves, given that the tidal

2. Estimating the probability of the overtopping discharge rate exceeding some design

3. Estimating the probability of the overtopping discharge rate exceeding some designlevel irrespective of the tidal level.

As yet no full analysis involving all three steps has been undertaken, although Colesand Tawn (1994) h developed methods for the first two steps of the estimation. Inparticular, they have developed methods for multivariate extremes which enable extrap-olation of the variables into each tail accounting for the observed dependence in the jointextremes of the variables. Fundamental to their methods is the specification of thresholds,

these thresholds the dependence between the variables is taken to be of a multivariateextreme value form. Thus the thresholds have to be chosen with both these aspects takeninto consideration. If the marginal and dependence aspects converge to the relevant ex-treme value model at different rates then these methods lead to inefficient use of the data,as the higher of the two separate thresholds has to be adopted to avoid bias.

Dixon and Tawn (1994) extend the methods of Coles and Tawn (1994) by proposinga semi-parametric model for multivariate extremes. This model has a separate thresholdfor the marginal and dependence aspects and thus overcomes the inefficiency present inthe Coles and Tawn proposal. Both of these methods have been applied to surge datafrom Newlyn with wave data from Seven Stones.

Both the Coles and Tawn (1994) and the Dixon and Tawn (1994) methods were devel-oped within this project, but details are not given within this report as the methods arestatistically intensive, and practical application has so far been restricted to one, rather ar-tificial example. However, each of these papers is given in its original form accompanyingthe report.

9.2 Within-Storm modelling 151

9.2 Within-Storm modelling

In the analysis of extreme values of a stationary sequence, the standard approach is basedon the joint distribution of the largest independent observed values. Such an approachis adopted in both the r-largest method, and the estimation of the surge distributionupper tail in the RJPM, with a suitable adjustment to incorporate trends. An importantaspect of the method is that the observed extreme data, to which the model is fitted,must be independent. This means that independent extreme events must be identified,using storm length declustering procedures discussed in Section 6.3, and from each of thethese ‘independent extreme events’ in a year the r-largest event maxima form the datafor that year. This selection of data has two distinct weaknesses:Arbitrary choice of independent extreme events.Independent events are not identifiable exactly from the data so a single choice of stormlength is arbitrary. Here we have applied the analysis to data from a range of stormlength choices to overcome this problem. This has shown that return level estimates arereasonably insensitive to the choice of storm length, and thus although arbitrary, thechoice it is not a critical one.Inefficient use of data.Using only event maxima is inefficient, a point which has not been addressed within ouranalyses. Clearly all large values within an extreme event, as well as the event maximum,provide information about the extremes of the process. At present these data are excluded

hence the adopted methods are inefficient.Prior to the project neither the extreme value theory, nor the necessary statistical

methods existed to exploit these data. Here we outline the progress made in extendingthis feature, whereas full details are given in the accompanying paper by Smith, Tawnand Coles (1994). H owing to the additional modelling required to capture thetemporal dependence within each event, we believe that the additional gain in efficiencyis outweighed by the potential for bias if such an analysis is performed on a routine basisover many sites. So, we conclude that at present the current procedures should not beupdated to incorporate these additional extreme event data.

which the sequence rarely exceeds, i.e. exceedances of u correspond to extreme values ofthe sequence, then exceedances will be clustered in time, in the form of separate extremeevents. The traditional form of analysis is based on modelling the maximum excess of thethreshold within each event by a generalised Pareto distribution, which has distributionfunction

152 9 NEW METHODS

This distribution is closely related to the generalisedextreme value distribution, i.e. the distribution of the annual maxima of a stationarysequence, for which the shape parameter is also k, and the scale parameter is a function

threshold u. Recall that accurate estimation of the scale and shape parameters of theGEV is vital to obtain accurate return level estimates, see (4.2).

It can be shown that the distribution of an arbitrary excess over the threshold alsofollows the generalised Pareto distribution with the same parameters as that for the eventmaxima. Thus all excesses could be used to estimate the parameters of the distribution,and hence also the parameters of the associated GEV. Obtaining ad hoc estimates forthese parameters from the data of all excesses is not a problem, however, owing to thedependence between data from the same events, obtaining the maximum likelihood esti-mates and the associated standard errors is much more difficult. In particular, a modelfor the dependence between observations within an extreme event is required. The Smith,Tawn and Coles (1994) approach is to use a Markov model, that is to assume that condi-

appears to be

l adequate for the surge series

These require consideration for each application,

generalised Pareto distribution.

Benefits of such modelling include the ability to address questions of interest otherthan that of return levels. For example, the volume of water overtopping the sea wallduring an extreme event may be of interest, in which case it is an aggregate of the excessesof the sea wall that is of interest. As the Markov model provides a joint description ofall observations in the extreme event the distribution of this aggregate measure is easilyevaluated. Hence although these extensions to the extreme value methodology do notprovide a significant advance for obtaining extreme still water level return values, theymay be valuable when it comes to the final stages of incorporating waves into the designstage.

9.3 Seasonality 153

9.3 Seasonality

Throughout the report, for the application of extremevalue methods, both the high watersof the still water level series, and the surge series were each assumed to be stationary.This is clearly a false assumption as there is a definite winter storm season when nearly allof the largest positive surges occur. This is reflected in Figures 49-56 which show that fewof the 15 largest surges in each year occur in the May to September period. If the year issplit into two separate periods, of storm and calm periods, such that within each periodthe process was stationary and all the extreme values of the entire process occurred inthe storm period, then, as currently applied, the methods would introduce no bias. Thisis the standpoint we have adopted, although in this section we question this and discussan alternative. As in Section 9.2, the alternative requires additional modelling whichintroduces the possibility of bias from routine application, so again we recommend thatthe simpler existing approach continues to be used. Below we briefly outline the progressmade in extending this feature, with full details being given in Coles, Tawn and Smith(1994) which accompanies the report.

As seen from Figures 49-56 the Seasonality of the surge process involves a smoothtransition from a stormy period to a calm period and most extreme surges occur in theDecember to February period. This indicates that the process is continuously changingform throughout the year, and is in fact not exactly stationary for any period. Con-sequently, for this and other seasonal environmental processes, Coles, Tawn and Smith(1994) propose modelling the extremes with a distribution which depends on the time ofyear of each extreme value. In particular, the extremes on day j are taken to follow a

Since the process changes smoothly in the form of non-stationarity, the GEV parameters

that,

for all j. Selection of suitably flexible, well fitting, forms for the parametric models isdifficult, so errors are likely to occur if routinely applied. Since errors may lead to asignificant bias in return level estimates, this aspect is not pursued for the analysis ofextreme still water levels, as the Seasonality is not of intrinsic interest, and the two seasonargument which justifies the current approach is a reasonable approximation.

j154 9 NEW METHODS

9.4 Annual Maxima Inclusion

A feature of many of the UK coastal sites in Tables 2-4 is the availability of historicaltide gauge records, stored as annual maximum levels. These data can provide valuableextra information about the distribution of the annual maximum sea-level to that fromthe hourly data. In particular, they provide significant further information about thetrend in extreme sea-levels and past extreme events such as that in 1953. Using the an-nual maximum, or r-largest method, these extra data can easily be included and used toimprove results (see Sections 4.2 and 5), however, as they stand, the indirect methodscannot include these historical data, since the tidal level associated with the annual max-imum of a particular year is not recorded. As discussed in Section 5.3, these data cannotbe included in the JPM, which is a weakness with that method. Extensions of the RJPMto enable these historical annual maximum data to be included have been developed inthe project, and these are described in this section.

Assume first, for simplicity, that there are no trends present, that the surge is inde-pendent of the tidal level and that the tidal sequence is known. Consider the situation

that these two data types are such that the years of annual maxima do not overlap theyears of hourly data. The following notation will be used throughout the section:

as described in Section 4.2. Assuming that the annual maximum levels and the surge lev-els are independent, the joint likelihood for surges and annual maximum levels is obtained

9.4 Annual Maxima Inclusion 155

sponding methods for estimating the distribution of annual maximum sea-levels whichinclude the historical maxima.

Method 1.The most direct method, which requires the least assumptions, is to use a sepa-rate parametric model for the annual maximum data. Specifically, following theannual maximum approach by using the GEV distribution and incorporating three

The joint likelihood, which now involves six parameters, is given by

these parameters are estimated by maximum likelihood. Return level estimates canbe obtained by combining the estimated distribution functions for the hourly andannual maximum parts of the likelihood. A suitable way of combining the twodistribution functions is to weight each proportionally, according to the length ofthe corresponding data series. This leads to the distribution function of the annualmaximum sea-level being given by

Method 2.Following the discussion in Section 4.4 (equation 4.22), the annual maximum sea-level parameters are related in a simple way to the annual maximum surge param-eters by

If this approximation is reasonable, an improvement over Method 1 will result. It

level parameters replaced with the expressions in (9.6). The distribution of annualmaximum sea-levels is then obtained by direct use of (4.15).

156 9 NEW METHODS

Method 3.The third method is to base the likelihood on only the surge parameters, by makinguse of the expression for the annual maximum sea-level distribution derived in theRJPM. The second part of the likelihood is taken to be

This derived form of the distribution function effectively integrates out the tidallevel of the annual maximum still water level, to obtain the likelihood of annualmaximum observations for which no tidal level has been recorded.

9.4.1 Inclusion of Trends

For the design of coastal flood defences, accurate estimation of trends in extreme sea-levels is clearly important. For the estimation of the trend at a site, it is probably moreimportant to have data over a long span, even if it is limited in extent, than it is to havea large amount of data over a short span. For this reason, including annual maximumdata, which is often available for longer spans, can considerably improve the precision oftrend estimates.

the intercept parameter, the trend parameter, the year, and the base year respectively.In the presence of trends, it is not clear how to weight the likelihood in Method 1, but

incorporated as follows.

9.4 Annual Maxima Inclusion 157

9.4.2 Inclusion of Tide-Surge Interaction

In practice tide-surge interaction can significantly affect results at many UK sites, asshown throughout the report, so incorporating it in these methods is important. Thereis no obvious way in which tide-surge interaction can be included in Method 2, butit is implicitly contained in Method 1. Method 3 can easily be extended to cover the

with notation as in Section 6.1. Return levels are then obtained by direct use of equation(6.4), as described in Section 6.1.

9.4.3 Inclusion of Trends and Tide-Surge Interaction

with notation as in Section 6.1. Again, return levels are then obtained by direct use of(6.4), as described in Section 6.1.

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10 Spatial Extensions

10 SPATIAL EXTENSIONS

The next stage of the work is the development and application of new methods of extremesea-level analysis that use information from neighbouring sites along a coastline, as wellas data from the site of interest, for obtaining return level estimates. Such an approach istermed a spatial method. As spatial methods use additional data in estimating extremecharacteristics, the precision of return level estimates will be improved. This is clearlybeneficial for sites such as Cromer and Felixstowe with poor site-by-site estimates. Toderive relationships between levels at different sites spatial methods necessarily providea continuous model for extreme sea-levels along the coastline. Therefore, an additionalconsequence is that estimates of return levels are obtained at intermediate sites along thecoastline, such as sites with insufficient data for a site-by-site analysis, for example Leith.

Spatial extensions will be considered for each of the three site-by-site methods ofanalysis studied here. However, based on the site-by-site findings most emphasis willbe given to the development of the spatial RJPM as the RJPM has been found to besuperior to other methods. Extending the site-by-site methods to spatial procedures, andto achieve successful application of these, requires the extreme sea-level process to havespatially coherent characteristics along a coastline, and in particular that this be evidentfrom the data at the limited spatial coverage given by A-class sites. Throughout thisstudy we have identified parameters which have spatial coherence over sites and it is onthese features that the spatial methods are to be based. Here it has been seen that spatialcoherence, as observed from the data sites, is much greater along the east coast than thewest coast, and that on the south coast little can be inferred owing to the absence of siteswith sufficient data. There are two reasons for this difference between the east and westcoasts:

l the spatial coverage of A-class sites, with sufficient data for extreme value consid-erations, is extensive on the east coast and sparse on the west coast,

l the surge process is more coherent on the east coast, as typically surges propagatedown that coast gradually amplifying in level whereas on the west coast surgesare more localised in extent as they approach from offshore rather than along thecoast line.

A possible source of additional data on the west and south coasts, which would signif-icantly increase the temporal and spatial extent of the available data from that given bythe data sites, is from hydrodynamical models. These additional data form the basis ofthe third phase of the work, in which a 30 year run of the hydrodynamical model will beused, and these data interfaced with historical annual maxima and hourly observations

159

at the analysis stage. However, at present it appears that, from the data in this study, aspatial model for extreme sea-levels can only be developed for the east coast.

Here we outline the aspects observed in this report where a spatial approach willimprove the estimation of extreme sea-levels. We focus on the RJPM to illustrate theextension. A spatial version of the RJPM will involve spatial estimation of the parametersin the site-by-site model. A brief summary of the way these are handled, and potentialproblems with the estimation, are discussed in this section as a lead in to the second stageof this project.

Tidal sequenceIn order to give estimates of return levels at any point along the east coast, the tidalsequence is required for each site of the application of a spatial version of the RJPM.Some form of spatial interpolation of the tidal sequences at each data-site will berequired. The tidal characteristics change smoothly along the coast so interpolationof this aspect is feasible, however despite the spatial coherence of the tides we shallnot use spatial analysis to improve tidal estimation at each site.

Tide-surge InteractionHere we have found that interaction is an important feature of the extreme sea-levelprocess for the entire length of the east coast, and that the degree of interactionvaries from site to site in a coherent manner. We have modelled interaction through

functions are influenced by changes in amplification of the surge as well as interac-tion. Therefore, we will separate these functions into amplification and interactioncomponents and model these separately along the coast. Thus, in combination, theamplification and interaction models will provide spatial analogues of the presentinteraction functions, i.e. they will be defined at any point along the coast. Thesechange smoothly along the coast and with tidal state.

Extremal IndexFrom Figures 90-92, there is clear spatial coherence in the extremal index ratiofor sites along the east coast, with some relationship on tidal range at the site.This suggests that it will be relatively easy to develop a spatial model for this ratioparameter.

The Surge DistributionOnce spatial versions of the interaction functions have been developed then spa-tially modelling the surge distribution is reduced to spatially estimating the annual

160 10 SPATIAL EXTENSIONS

This transformed surge series is stationary in time for each site. What is requiredfor spatial modelling of this distribution of S* is generally that it is coherent oversites, and in particular, that the upper tail of its distribution exhibits strong spatialcoherence. From Figures 14-19, it is seen that this series exhibits a high degree ofspatial smoothness, and even a spatially homogeneous model for the upper tail ofthe distribution may be a good approximation over sections of the coastline. Thiswill allow the use of simple spatial models for the extreme value modelling.

By separating the modelling of extreme surges into the two stages of the interactionfunctions and the distribution of the S* variable, the complex spatial modelling isrestricted to the interaction functions which are estimable from the bulk of the data,not just the extremes. Consequently, sites excluded from the site-by-site analysescan be used to estimate the interaction functions therefore improving the spatialinterpolation by increasing the spatial density of sites used.

Extreme trend parameterAccurate estimation of the trend parameter is difficult using hourly data separatelyfrom site to site. However, in Section 8.2 the trend was seen to exhibit spatialvariations consistent with the pattern of land trends. Estimates of these trendscan be improved by exploiting additional data on mean sea-level trends, historicalmaxima, land trends, and spatial variations.

Historical DataAs has been discussed in Section 9.4, inclusion of historical data can considerablyimprove estimation. The available historical data will be included in the spatialanalysis of the next stage of the work.

Clearly in developing, and using, a spatial model in effect all that is being used is theadditional physical knowledge about the sea-level process, in that the distribution ofextreme sea-levels must vary smoothly along a coastline as the sea-level processes of tideand surge themselves vary smoothly over such a spatial scale. Therefore, in essence,extension of the site-by-site methods to spatial procedures corresponds to the use ofimproved oceanographic modelling within the statistical analysis.

161

11 AcknowledgementsWe thank the Ministry of Agriculture Fisheries and Food for funding this work. We havefound considerable support from colleagues and POL staff. In particular at POL, wethank Ian Vassie who made a large contribution in the development of the r-largest andRJPM methods originally, provided a significant input of knowledge of oceanographicprocesses and been enthusiastic about the work, all of which have helped produce resultsof practical meaning; Sheila Shaw, who has been most helpful in supplying data andanswering repeated questions concerning datum, data quality, and availability; and bothDavid Blackman and Graham Alcock who have always been very helpful in advice andsupport.

Stuart Coles, Richard Smith and Clive Anderson have all made valuable contributionsthrough joint work and discussions of the viability of methods, and their support has beengratefully received.

Finally at Lancaster University we gratefully acknowledge the help that Barry Rowl-ingson has provided through advice in computing throughout the project and aspectsconcerning the preparation of the report, and that Anthony Ledford has given in hisproof reading of the final report.

j162 12 REFERENCES

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Lennon, G. W. (1963). A frequency investigation of abnormally high tidal levels at certainwest coast ports. Proc. Inst. Civ. Engrs., 25, 451-484.Middleton, J. F. and Thompson, K. R. (1986). Return periods of extreme sea-levels fromshort records. J. Geophys. Res., 91, 11707-11716.Prandle, D. and Wolf, J. (1978). The interaction of surge and tide in the North Sea andRiver Thames. Geophys. J. R. astr. SOC., 55, 203-216.Pugh, D. T. (1987). T-dz es, surges and Mean Sea-Level. Chichester: Wiley.Pugh, D. T. and Vassie, J. M. (1979). Extreme sea-levels from tide and surge probability.Proceedings 16th Coastal Engineering Conference, 1978, Hamburg. American Society ofCivil Engineers, New York, 1, 911-930.Pugh, D. T. and Vassie, J. M. (1980). Applications of the joint probability method forextreme sea-level computations. Proc. Instn. Civ. Engrs., Part 2, 69, 959-975.Shennan, I. (1989). Holocene crustal movements and sea-level changes in Great Britain.J. Quart. Sci., 4, 77-89.Silverman, B. (1986). Density Estimation for Statistics and Data Analysis, Chapman andHall, London.Smith, R. L. (1986). Ex reme value theory based on the r largest annual events, J. Hy-tdrol., 86, 27-43.Smith, R. L. (1989). Extreme value analysis of environmental time series: an applicationto trend detection in ground level ozone. Statist. Sci., 4, 367-393.Smith, R. L., Tawn, J. A. and Coles, S. G. (1994). Markov chain models for thresholdexceedances. Accepted for Biometrika.Suthons, C. T. (1963). Frequency of occurrence of abnormally high sea-levels on the eastand south coasts of England. Proc. Instn. Civ. Eng., 25, 433-450.Tawn J. A. (1988). An extreme value theory model for dependent observations, J. Hy-drol., 101, 227-250.Tawn, J. A. (1992). E ts imating probabilities of extreme sea-levels. Appl. Statist., 41,77-93.Tawn, J. A., Dixon, M. J. and Woodworth, P. L. (1994). Trends in sea-levels. To appearin Statistics for Water, eds. V. Barnett and F. Turkman, Wiley.Tawn, J. A. and Vassie, J. M. (1989). Extreme sea-levels: the joint probabilities methodrevisited and revised. Proc. Instn. Civ. Engrs. Part 2, 87, 429-442.Tawn, J. A. and Vassie, J. M. (1991). Recent improvements in the joint probabilitymethod for estimating extreme sea-levels. In Tidal Hydrodynamics, 813-828, ed. B. B.Parker, New York: Wiley.Walden, A. T., Prescott, P. and Webber, N. B. (1982). The examination of surge-tideinteraction at two ports on the central south coast of England. Coastal Eng., 6, 59-70.

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13 AppendicesSites for which estimation proved infeasible are included in the arrays of figures throughoutthe appendix, but the plots are given as a single continuous line (e.g. Cromer in Figure57).

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