Application of a Statistical Method to investigate patterns of beach evolution in the vicinity of a seawall

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1 Joural of Coastal Research SI ICS (Proceedigs) Polad ISSN 79-8 Applicatio of a Statistical Method to ivestigate patters of beach evolutio i the viciity of a seawall J. M. Horrillo-Caraballo ad D.E. Reeve School of Marie Sciece ad Egieerig, Uiversity of Plymouth, Reyolds Buildig Plymouth PL 8AA, UK Jose.Horrillo-Caraballo@plymouth.ac.uk Domiic. Reeve@plymouth.ac.uk ABSRAC Horrillo-Caraballo, J.M. ad Reeve, D.E.,. Applicatio of a statistical method to ivestigate patters of beach evolutio i the viciity of a seawall. Joural of Coastal Research, SI (Proceedigs of the th Iteratioal Coastal Symposium),. Szczeci, Polad, ISSN 79-8 I this paper we describe the applicatio of Caoical Correlatio Aalysis (CCA) to a historical dataset of seabed elevatios withi a coastal segmet of the Eglish south easter coast. he study site is located alog the frotage of Walcott (Norfolk, East coast of Eglad). he beaches alog this segmet of coast are maily composite, mixed sad ad shigle. he dataset comprises detailed bathymetric surveys of beach profiles coverig a period of over 7 years (~ beach profile surveys every year). he structure of the dataset ad the data hadlig methods are described. he applicatio of the CCA method is discussed as well as the ability of the CCA to provide useful forecast of the beach profile at Walcott. Some results ad iterpretatio of the aalysis are preseted. Beach profiles predictios agreed well with the measured profiles for up to five years of forecast at the site. he level of error usig CCA is proportioate with that foud with dyamical modellig. It is expected that this relatioship will be used by coastal plaers to predict profile evolutio based o waves ad storm surges. ADDIIONAL INDEX WORDS: Beach profile, Caoical Correlatio Aalysis, Empirical Orthogoal Fuctios, Forecast, Statistical models. INRODUCION I recet years, as a result of the implemetatio of a strategic approach to flood ad coastal maagemet, the ecessity of more robust methodologies for icorporatig risk assessmets withi coastal egieerig desig has bee clearly idetified (Cooper ad Hutchiso, ). Climate chage will modify wave ad water level coditios ad for this reaso affect the vulerability of coastal defece structures (Sutherlad ad Gouldby, ). here are a large umber of coastal structures whose stability depeds upo the characteristics of the beach where they are located. From this perspective, it is ecessary to uderstad how seawalls ad beaches iteract durig storms ad how beach profiles iteract with waves, tides ad storms. Without this, it is difficult to provide realistic estimates of iteractio betwee the structure ad the beach profiles ad how they are likely to chage over time. Sea walls are ofte part of a larger scheme, operatig i combiatio with beach ad other structures (e.g. beach ourishmet, groyes, etc.) to cotrol wave eergy, thereby improvig the resistace to coastal erosio or limitig wave overtoppig. herefore, coastal egieerig has dedicated sigificat efforts to study the iteractio betwee wave coditios ad morphological chages o the beaches. his pheomeo is of great complexity because the morphology of a beach at a certai time is the result of several hydrodyamic processes that act over a vast rage of space ad temporal scales (Larso ad Kraus, 995). I order to improve predictios of likely wave overtoppig ad thus flood risk, improvemets i both wave forecastig ad beach movemet are required. Wave forecastig models have see sigificat improvemets over the last few decades, but predictig beach levels remais difficult. he importace of beaches for coastal flood risk is due to the process of wave breakig i shallow water. Breakig dissipates much of the eergy i a wave ad ca materially reduce overtoppig. Beach predictio models come i essetially two forms: a) formulae that describe a equilibrium shape as a result of fixed wave coditios (e.g. Dea et al., 99); b) detailed process models that ca provide good results over the period of a storm but have difficulties i terms of accuracy ad stability for loger term predictios. As a cosequece of the limitatios metioed for both types of models, recet studies have highlighted the ecessity of use of robust statistical models (Różyński, ; Li et al., 5). his type of approach uses correlatio techiques of series of data to idetify relatioships ad structures of two simultaeous series of time. he objective of these techiques is to associate the chages cotaied i the data sets to physical processes i the area of study. I essece, this relies o a statistical aalysis of historical data ad a process of extrapolatio. Give the growig amout of beach survey data ad earshore wave forecasts that are ow almost routiely available, i this paper we examie the use of data-drive techique for predictig beach chages. Caoical Correlatio Aalysis (CCA) is oe of these methods that has bee used i areas such as meteorology ad climatology (Glah, 98; Graham et al., 987). his method has bee used with data from the FRF (Field Research Facility i North Carolia, USA) by Larso et al. () Joural of Coastal Research, Special Issue, 95

2 Short Ruig Head (e.g. Beach Maagemet ad Safety. Use imes New Roma 9 fot ormal) i order to detect occurrig patters i the wave ad profile data ad to ivestigate the use of CCA to predict the profile respose due to waves. Horrillo-Caraballo ad Reeve (8) exteded this study to ivestigate the sesitivity of the quality of predictios o the choice of distributio fuctio used to describe the wave heights. I additio, Rozyski () used the CCA method to aalyse the evolutio patters of multiple log-shore bars ad the iteractios amog them. I this paper, CCA techiques are applied to a series of historical beach profile survey of the Walcott area (east coast of Eglad) coverig a period of over 7 years. he data were aalysed i order to determie the covariability betwee waves ad profile respose. he coastlie i Walcott is orietated approximately orthwest - southeast ad is proe to attack from the North Sea, especially whe the wid blows from the ortheast (Figure ). he dager icrease whe this wid is combied with a North Sea surge, which ca raise the still water level of the sea by up to m above the predicted sea level. hese coditios have caused disastrous floodig i the tow, the worst i recet memory beig i 95. (HR Walligford, ). Ediburgh Wave hidcasts ad beach profiles from the Eviromet Agecy (EA) were used to ivestigate the relatioship betwee the waves ad the profile respose over a loger time period. he data covered beach profile surveys take alog Profile NC8 (Figure ) over a period of 7 years ad spectral wave properties (sigificat wave height, peak period ad wave directio) calculated at least every h durig the same period. Wid measured at Weyboure, UK (Chii, pers. comm.) were used to estimate wave parameters usig the so-called CERC formulas (USACE, 98) he wave data poit (WDP), where wave parameters were calculated, is located at positio N, 59. E (Figure ).he waves were hidcasted at mcd (Chart Datum) water depth ad these data were used to derive the wave properties employed i the CCA. Iitially, the aalysis focused o directly likig the profile respose to the sigificat wave heights (H s ) ad to wave steepess (H/L). I order to represet the sigificat wave heights ad wave steepess, probability desity fuctios (pdf) were derived for the two wave parameters. Secodly, the theory behid the CCA ad some descriptio of how it was applied i this study is give. he results of the CCA betwee the profiles ad the two Cardiff Birmigham Lodo Figure. Locatio of the study area ad photo from the seawall i Walcott (top right) ad wave data locatio (WDP). wave parameters are preseted. A discussio ad coclusive remarks are provided o the potetial for usig these results to predict profile evolutio based o the waves. CANONICAL CORRELAION ANALYSIS (CCA) MEHOD CCA was proposed by Hotellig (9) ad ca be see as the problem of fidig basis vectors for two sets of variables such that the correlatio betwee the projectios of the variables oto these basis vectors are mutually maximised. CCA is used to ivestigate the itercorrelatio betwee two sets of variables, whereas factor/pricipal compoet aalysis or EOF (Empirical Orthogoal Fuctios) idetifies the patter of relatioship withi oe set of data (Clark, 975). CCA may be used to ivestigate the presece of ay patters that ted to occur simultaeously i two differet data sets ad what the correlatio is betwee associated patters (Graham et al., 987). If the two origial data sets are deoted Y (wave parameter data matrix with size t y) ad Z (profile data matrix with size t z), we deote ew trasformed variables U ad V that have maximally correlated colum vectors for the same idex ad zero correlatio for differig idices. U ad V are orthoormal. he desired weights to trasform Y ito U (Graham et al., 987), [( Y Y ) ( Y Z )( Z Z ) ( Z Y ) I ] = µ () where µ deotes the eigevalues (squared correlatio betwee the correspodig temporal amplitudes of the caoical modes - the colums vectors i U ad V - ad the associated eigevectors R), yields the trasformatio U = YR. he same to trasform Z ito V, [( Z Z ) ( Z Y )( Y Y ) ( Y Z ) I ] = µ () havig the same µ eigevalues ad the associated eigevectors Q, yields the other trasformatio V = ZQ. he spatial amplitudes (G ad H) of the caoical mode are obtaied as: G = Y U () H = Z V () he origial data set are expressed as: Y = UG (5) Z = VH () Noise i the data ca adversely affect the accuracy of these matrix maipulatios. hus some filterig of the data is ofte performed prior to the aalysis. EOF is commoly used i the filterig. he data set (Y ad Z) are expressed i terms of EOFs Y = AE (7) Z = BF (8) where A ad B cotais the temporal EOF (or pricipal scores) ad E ad F the spatial EOF for Y ad Z respectively. Filterig is performed simply by trucatig the expasio of EOFs. Joural of Coastal Research, Special Issue, 9

3 Author s last ame (e.g. Smith or Smith ad Joes or Smith et al. Use imes New Roma 9 fot ormal) he regressio matrix is origiated i the CCA aalysis ad relates the profiles to the wave proper-ties based o the correlatio betwee the domiat patters i the profile ad wave data. his meas that the regressio matrix ca be used to predict the profile respose, if the wave properties are kow (Larso et al., ). he associated profile data matrix Z p, havig a wave matrix Y p (measured or simulated) is give by: where Z p = Y p ψ (9) ψ = GSF () S = U B () he ecessary matrix algebra required i these cases is ow available i FORRAN or MALAB routies. SURVEY DAA Beach Profile Data he mixed shigle-sad beaches of Walcott (Figure ) have bee moitored sice 99 alog 8 shore ormal profiles, as part of a log-term beach maagemet survey programme by North Norfolk District Coucil (HR Walligford, ). he surveys were carried out i the summer ad witer moths so that the seasoal variatios i beach morphology ca be examied. I this study, Profile NC8 has bee used for this study, due to that this profile takes ito accout the seawall. Figure displays the subset of measured profiles alog Profile NC8 icluded i this study that were used i the CCA (5 profiles show). Elevatio (m) Cross-shore distace (m) Figure. ime series of beach profile NC8. Oly surveys that exteded from the due regio out to a water depth of approximately the Mea Low Water Level (MLWL) were icluded i the aalysis. Measured levels are referred to the UK Referece System - Ordace Survey Datum Newly (ODN). Wave Data he wave dataset available for this area cosisted of sigificat wave height (H m ), peak spectral wave period ( p ) ad wave directio (θ) obtaied from Nicolas Chii (Chii, pers. comm.). he waves were obtaied every h. he hidcasted wave poit is located i approximately mcd water depth ad.km offshore. he predomiat wave directios are from the orth, orth-east, ad east ad the largest surges are associated with wids from the orth-west ad orth. he tidal rages are.5m ad.7m for sprig ad eap respectively. Wave coditios obtaied from the hidcast will be used to calculate the empirical probability desity fuctios that were used i the CCA aalysis. MEHODOLOGY Decompositio of the Data he data used for this study have bee detailed i the sectio above. he data obtaied from Walcott have bee maipulated i order to acquire time series with the same samplig rate for profiles ad wave coditios. he profile dates were fixed i accordace with the Walcott dataset. I geeral terms, aroud 5 beach profiles were used i the CCA ad aroud beach profiles were used for compariso with the predictios o the basis of the regressio matrix. he beach profile is highly depedet o the prevailig wave coditios. herefore, the wave coditios betwee two selected profiles were used to characterise the chages i each subsequet beach profile. Noetheless, there are about 559 wave data records for Walcott to characterise aroud beach profiles. At each profile take for this study there are may more wave observatios tha beach profile measuremets. With the aim of geerate two series of equal legth, required to apply the CCA method, the wave coditios are compiled i probability desity fuctios (pdfs). A parametric form of pdf was proposed by Larso ad Kraus (995). However, more recet studies (Horrillo- Caraballo ad Reeve, 8; ) have suggested that a better performace is obtaied usig a empirical distributio. he empirical distributio is a cumulative probability distributio fuctio that cocetrates probability / at each of the umbers i a sample. A combied pdf (p ) may the be derived by superimposig the idividual pdfs available for the measuremet period betwee surveys, p = ( H ) I( H H ) i= i () where H is the wave height, the umber of idividual wave measuremets betwee surveys ad i a idex. he superpositio carried out i Equatio () implies that all the idividual pdfs derived from the wave measuremets (H s ) have the same weight. he composite pdfs used i the CCA aalysis were discretised ito about itervals, sufficiet to provide a good resolutio over the probability iterval of sigificace. Figure shows the correspodig composite pdf for the offshore sigificat wave height valid for the time period betwee surveys ad obtaied by addig over a large umber of Empirical pdfs accordig to Equatio (). Data for the area of Walcott is quite scarce i compariso with other field data sets (e.g. Duck); eve that profiles have bee measured from the eighties, there is ot eough wave data to correlate waves ad profile from this time. Wave data is oly available from the early ietee-ieties ad profiles i this site are measured o average twice a year. At Walcott, the higher probabilities are located i the area of smaller waves (Figure ). he Empirical distributio is cosidered to be a reliable descriptor Joural of Coastal Research, Special Issue, 97

4 Short Ruig Head (e.g. Beach Maagemet ad Safety. Use imes New Roma 9 fot ormal) of the wave height distributio i deep waters ad i shallow waters (Horrillo-Caraballo ad Reeve, 8; ). Pdf of sigificat wave height Pdf of wave steepess Sigificat wave height (m) Wave steepess (H/L) Figure. ime series of empirical pdfs of sigificat wave height (top pael) ad time series of empirical pdfs of wave steepess (bottom pael) at Walcott I this paper, the offshore distributios have bee used to correlate the waves ad the beach profile usig CCA ad the results are discussed below. Aalysis of the Data For the purpose of compariso, the CCA is applied to a profile at Walcott datasets usig years of records ad the performig predictios 5 years, util 8. wo differet CCA aalysis were performed (oe for each of the wave parameters sigificat wave height ad wave steepess) usig beach profiles, sigificat wave heights ad peak period from /8/99 to /8/ (~ 5 profiles ad 5 empirical pdf distributios) to determie the regressio matrices ad the predictios from 9// to 5/9/8 (~ profiles) were computed usig the hidcasted wave ad period data. he approach used here with the differet profiles is to ivestigate: the variatio i profile predictios usig differet regressio matrices (empirical pdfs with sigificat wave height ad wave steepess) For each case, EOF expasio ad trucatio were used i order to pre-filter the data sets before performig the CCA aalysis (e.g., Clark, 975). he rule of thumb of North et al. (98) was applied to determie the appropriate trucatio poit i the EOF expasio. RESULS he aalyses were performed as described above. he motivatio for the tests was to ascertai how the choose of pdfs might affect the quality of predictios. Such iformatio could have direct applicatio i the plaig ad collectio of beach moitorig measuremets for coastal maagemet ad predictio of the coditio (erosio or accretio) of the coastlie. he first three spatial EOFs obtaied from the profile data sets at Profile NC8 explaied 97% of the variatio i the data(e : 7%; E :%; E :5%). For the empirical sigificat wave pdfs, the first three spatial EOFs explaied 5% of the variatio i the data (E : %; E :%; E :9%) ad for the empirical wave steepess distributio, % of the variatio is explaied (E : %; E :%; E :9%). I all cases, the time mea was subtracted before aalysis i all data sets. For the two CCA aalyses, the EOFs describig the profile shape are quite complex, but the EOF shapes ca be used to quatify mea properties of the ifluece of the berm over the profile (Larso et al., ). Moreover, the temporal EOFs ca be used to determie treds of profile chages ad oscillatory cycles. CCA was the performed at profile NC8 as described i the sectio above. he regressio matrices obtaied were used to forecast the beach profiles usig the wave data set for the predictio period. he wave data used goes from /8/ - 5/9/8. he wave data was treated usig the EOF method ad the replaced i Equatio 9 with the regressio matrix obtaied with the CCA aalysis. he beach profile is obtaied replacig the two i Equatio 9. he RMSE (Root Mea Square Error) obtaied betwee the measured beach profiles ad the predicted profiles are used to measure the quality of the predictios. Values of RMSE are show as a measure of the error betwee predictio ad measuremet as a fuctio of cross-shore positio, averaged over the forecast widow. Figure shows the time averaged RMSE for the two cases; the CCA was performed usig differet modes (show i the figure leged). Overall, the best performaces for Profile NC8 are: the use of two caoical modes if the empirical wave sigificat pdfs are applied ad also two caoical modes if the wave steepess pdfs are used. he geeral behaviour of the differet modes i the two cases are very similar, the RMSE icreases where major chages i the elevatios are preset. his is due largely to the movemet ad chage of positio of the berm i the beach profile. RMSE (m) RMSE (m) cross-shore distace (m) cross-shore distace (m) Figure. ime averaged RMSE for predictios made with the umber of CCA modes idicated by the label key, usig empirical wave sigificat pdfs (top pael) ad usig empirical wave steepess pdfs (bottom pael). he major chages are preset betwee 7m ad 5m chaiage i the profile ad the behaviour of the error is very similar usig or modes. CCA performed with modes obtaied the miimum RMSE. he error i the profile shows similar behaviour i their cross shore distace, with this it is possible to assume that the wave sigificat pdf ad the wave steepess preset similar Joural of Coastal Research, Special Issue, 98

5 Author s last ame (e.g. Smith or Smith ad Joes or Smith et al. Use imes New Roma 9 fot ormal) performace. I this case, it is possible to use ay of the two methods to predict the profiles. Figure 5 shows the measured beach profile at the profile NC8 for February 5 th ad the predictio made from Jauary st usig the two differet empirical pdfs distributios. Satisfactory agreemet is obtaied i the area where the profile shows evidece of cosiderable chage. I the offshore area, the error is over predicted. I cotrast, at the begiig of the profile (betwee 8m ad 5m) there is a uder predictio of the beach levels. elevatio (m) elevatio (m) 5// cross-shore (m) RMSE =.8m RMSE =.m Predicted Measured cross-shore (m) Figure 5. Measured ad predicted profiles at Walcott (date 5//), usig empirical wave sigificat pdfs (top pael) ad usig empirical wave steepess pdfs (bottom pael). CONCLUSIONS Coastal flood risk due to wave overtoppig is closely liked to the health of beaches frotig the sea defeces ad hiterlad. A full beach will dissipate much of the icomig wave eergy before it reaches the shorelie, ad is a very effective part of the defece. Predictig the evolutio of beaches remais difficult. However, good improvemets i wave forecastig have bee made i recet years. Wave forecasts ad measuremets are, i geeral, much more easily available tha the correspodig iformatio o beaches levels. Fidig a reliable lik betwee wave coditios ad beach levels would yield a ecoomic meas of forecastig beach levels, ad thus a crucial factor i assessig coastal flood risk. I this paper we have applied CCA at Walcott, at which there are coicidet records of beach profiles ad wave coditios for more tha a decade, i order to ivestigate the correlatio betwee wave climate ad beach profile respose i the viciity of a seawall. he results of the preset study idicate that data-drive statistical aalysis, such as CCA, are useful for aalysig profile respose to waves if there is strog correlatio betwee the two variables (beach profiles ad wave height distributios). Used i cojuctio with EOF to reduce oise of the data, CCA was foud to be well suited for idetifyig combied patters of variatio i the wave ad profile data. Also, CCA showed potetial as a method for forecastig profile respose usig differet wave parameters for the empirical distributios. LIERAURE CIED Clark, D., 975. Uderstadig Caoical Correlatio Aalysis. Cocepts ad techiques i moder geography, No.. Norwich, UK: Geo Abstracts Ltd. Cooper, N.J. ad Hutchiso, J.. Strategic approach to flood ad coastal maagemet i Eglad. Proceedigs 8th Iteratioal Coferece o Coastal Egieerig. (Cardiff, Wales, UK. ASCE), pp Dea, R.G.; Healy,.R. ad Dommerholt, A.P., 99. A 'blidfolded' test of equilibrium beach profile cocepts with New Zealad data. Marie Geology, 9 (-), 5-. Glah, H.R., 98. Caoical correlatio aalysis ad its relatioship to discrimiat aalysis ad multiple regressio. Joural of Atmospheric Scieces, 5,. Graham, N.E.; Michaelse, J. ad Barett,.P., 987. A ivestigatio of the El Niño-souther oscillatio cycle with statistical models:. Predictor field characteristics. Joural of Geophysical Research, 9 (C), 5 7. Horrillo-Caraballo, J.M. ad Reeve, D.E., 8. A ivestigatio of the lik betwee beach morphology ad wave climate at Duck, North Carolia, USA. Joural of Flood Risk Maagemet, (), -. Horrillo-Caraballo, J.M. ad Reeve, D.E.,. A ivestigatio of the performace of a data-drive model o sad ad shigle beaches. Marie Geology, 7 (-), -. Hotellig. H., 9. Relatios betwee two sets of variates. Biometrika, 8, 77. HR Walligford,. Overstrad to Walcott Strategy Study. Hydrodyamics. Part II: echical Support Iformatio. Report EX 9. Larso, M. ad Kraus, N.C., 995. Predictio of cross-shore sedimet trasport at differet spatial ad temporal scales. Marie Geology,, -7. Larso, M.; Capobiaco, M. ad Haso, H.,. Relatioship betwee beach profiles ad waves at Duck, North Carolia, determied by caoical correlatio aalysis. Marie Geology,, Li, Y.; Lark, M. ad Reeve, D.E., 5. he multi-scale variability of beach profiles at Duck, N. C., USA, 98-. Coastal Egieerig, 5 (), -5. North, G.R.; Bell,.L.; Cahala, R.F. ad Moeg, F.J., 98. Samplig errors i the estimatio of empirical orthogoal fuctios. Mothly Weather Review,, Różyński, G.,. Data-drive modellig of multiple log-shore bars ad their iteractio. Coastal Egieerig, 8, 5-7. USACE, 98. Shore Protectio Maual. U.S. Army Waterways Experimet Statio. Coastal Egieerig Research Ceter, U.S. Govermet Pritig Office, Washigto D.C. Sutherlad, J. ad Gouldby, B.,. Vulerability of coastal defeces to climate chages. Proceedigs of the Istitutio of Civil egieers, Water ad Maritime egieerig, 5, 7-5. ACKNOWLEDGEMEN his research was supported uder the Flood Risk Maagemet Research Cosortium (FRMRC) via the UK Egieerig ad Physical Scieces Research Coucil grats No. GR/S7/ ad EP/F5/. I additio, the support of the Europea Commissio through FP7.9-, Cotract - HESEUS ( Iovative techologies for safer Europea coasts i a chagig climate ), is gratefully ackowledged. he authors would also like to thak to Prof. Peter Stasby ad Dr. Nicolas Chii from the Uiversity of Machester for the wave ad beach profile data. Joural of Coastal Research, Special Issue, 99