Statistical Resampling Methods for Coastal Flood Risk Management

Transcription

1 Super Work Package 2 Work Package 2.2 Research Report 3 Deliverable 2.6 Statistical Resampling Methods for Coastal Flood Risk Management Prof. Dominic E. Reeve Dr. Jose M. Horrillo-Caraballo University of Plymouth University of Plymouth July 2

2 FRMRC SWP2 WP2.2 Research Report 3 Project Website:

3 Document Details Document History Version Date June 2 5 June 2 3 June 2 8 July 2 Lead Authors Dominic E. Reeve Dominic E. Reeve Dominic E. Reeve Dominic E. Reeve Institution Joint Authors Comments UoP UoP UoP UoP Dominic E. Reeve, Jose M. Horrillo-Caraballo Dominic E. Reeve, Jose M. Horrillo-Caraballo Dominic E. Reeve, Jose M. Horrillo-Caraballo Dominic E. Reeve, Jose M. Horrillo-Caraballo First Draft Second Draft addressing comments following review Version addressing comments following review Response to internal review, final version Statement of Use This report is intended to be used by researchers working on flood risk management and coastal engineering. This report is intended to provide an overview of the research undertaken, akin to an Executive Summary. For detailed descriptions of methods, algorithms, data sets and so on the reader is referred to the papers cited at the end of this report. The deliverable comprises this short technical report, plus a flash card that describes some key outputs from the case studies. Acknowledgements This research was performed as part of a multi-disciplinary programme undertaken by the Flood Risk Management Research Consortium. The Consortium is funded by the UK Engineering and Physical Sciences Research Council under grant GR/S7634/, with cofounders including the Environment Agency, Rivers Agency Northern Ireland and Office of Public Works, Ireland. Disclaimer This document reflects only the authors views and not those of the FRMRC Funders. The information in this document is provided as is and no guarantee or warranty is given that the information is fit for any particular purpose. The user thereof uses the information at its sole risk and neither the FRMRC Funders nor any FRMRC Partners is liable for any use that may be made of the information. Statistical Resampling Methods for i

4 Copyright 2 The content of this report remains the copyright of the FRMRC Partners, unless specifically acknowledged in the text below or as ceded to the Funders under the FRMRC contract by the Partners. Statistical Resampling Methods for ii

5 Summary This report, together with a flash card, forms Deliverable 2.6 of Super Work Package 2 of the FRMRC 2 project. The deliverable provides guidance on the use and application of ensemble re-sampling methods for coastal flood risk management. It has been developed in conjunction with users through a series of meetings throughout the project. The methods are explained in general terms in this report, with the reader being referred to relevant books and publications for further technical details. The methods are illustrated through application to several case studies, most in the UK but also from overseas. The methodology is a data-driven approach which means it relies on existing measurements and statistical analysis to provide the means to forecast the evolution of beaches and other coastal forms. The examples show applications to different forms of beach (sandy, mixed sand/gravel and gravel); natural and defended beaches, and nearshore sandbanks. The sensitivity of the forecast quality in relation to the forecast period and the duration and frequency of historical measurements has been assessed. The quantification of uncertainty in the forecasts has been described using standard statistical techniques (jackknife and bootstrap methods) to provide ensemble forecasts from which forecast statistics can be determined. The primary results are: that the methods outlined in this report can provide a useful degree of forecast quality for coastal management; the current frequency of beach measuring undertaken by New Forest District Council seems to be sufficient for this purpose, although the duration of the measurement record at this site is important in this regard; forecasts of beach profile, beach plan shape and sand bank configuration have been performed over periods of months to decades and have shown a useful degree of accuracy. Caveats for the application of the methods are provided and are essentially that a good historical record of beaches and waves is required (typically years of 6-monthly beach surveys, time series of measured or hindcast wave conditions), the method is unlikely to forecast changes in the beach that are not contained in the historical dataset well. Statistical Resampling Methods for iii

6 Contents Summary... iii Contents... iv INTRODUCTION.... Introduction....2 Scope....3 Background... 2 TECHNIQUES USED / METHODS USED Introduction Empirical Orthogonal Functions (EOF) method Canonical Correlation Analysis (CCA) method Jack-knife method Bootstrap method CASE STUDIES Duck, USA / Milford-on-Sea, UK Beach profiles Wave data Methodology Analysis of the data Results Conclusions Slapton Sands Field Site Shorelines Wave data Methodology Results Conclusions Great Yarmouth Sandbanks Field site Survey data Methodology Results Statistical Resampling Methods for iv

7 3.3.5 Conclusion Walcott Case study area Beach profiles Wave Data Methodology Results Conclusions GENERAL CONCLUSIONS RESEARCH PUBLICATIONS Journals Book Chapters Conferences ACKNOWLEDGEMENTS REFERENCES APPENDIX A A.. Empirical Orthogonal Function (EOF) method APPENDIX B B.. Canonical Correlation Analysis (CCA) method APPENDIX C... 5 C.. Resampling methods... 5 C... Jack-knife... 5 C..2. Bootstrap... 5 Statistical Resampling Methods for v

8 FRMRC Research Report SWP2.2 INTRODUCTION. Introduction This report, together with the summary flash card, comprises deliverable D2.6 of Super Work Package 2 within the Flood Risk Management Research Consortium Phase 2 project..2 Scope The original scope of the report was to provide guidance on the use and application of ensemble re-sampling methods in the area of coastal flood risk management, including Case studies, working towards a Code of Practice in this area. Following consultation with endusers in this Super Work Package a slight modification of the scope was agreed, so that the deliverable comprises this short technical report, plus a flash card that describes some key outputs from the case studies. This report is intended to provide an overview of the research undertaken, akin to an Executive Summary. For detailed descriptions of methods, algorithms, data sets and so on the reader is referred to the papers cited at the end of this report..3 Background In the aftermath of implementing coastal and flood management plans, Cooper and Hutchinson (22) expressed the widespread acknowledgement that there is a need for more robust methodologies to assess risks in coastal engineering design. A large proportion of coastal areas depend upon the characteristics of the shoreline to protect them from flooding and erosion. The move towards adopting soft engineering solutions has changed the emphasis from prevention of flooding and erosion towards management of flood and erosion risks and has increased the importance of understanding how beaches and shorelines respond to the prevailing tide and wave regimes. Coastal flooding occurs predominantly in storm conditions typically associated with high tides and storm surge. It results from waves overtopping sea defences or the natural shoreline barrier. Severe or extended overtopping can lead to localised damage to the defence and thence to breaching, which markedly increases inundation. The size of the waves reaching the shoreline is controlled by the depth of water, which is a combination of the water level and the beach level. Retaining a healthy (high) beach is one means of reducing the wave heights. Some of the newer flood defence schemes being constructed around the UK use this principle to good effect (Figure ). Statistical Resampling Methods for

9 Figure. Seawall at Walcott, Norfolk (left). Detached breakwaters at Sea Palling, Norfolk, UK (right). From the shoreline manager s perspective, predicting beach behaviour with some level of confidence is important, not only for amenity value but also for flood and erosion protection. One way to do this is by using numerical models that solve the hydrodynamic and sediment transport equations. However they suffer from several disadvantages such as: they can be difficult to operate; they require large amounts of computing time for medium and long-term changes predictions; they can suffer from numerical instability; the predictions can be very sensitive to initial and boundary conditions. Although such models are relatively good for predicting beach evolution over the period of a storm; they have some problems in predicting medium to long-term evolution with the same level of accuracy (Southgate et al., 23). A substitute to numerical modelling techniques is data-driven modelling techniques, a term given to statistically based analysis of patterns in observed measurements (eg. Gunawardena et al., 28; Larson et al., 23; Ró y ski, 23; Southgate et al., 23). Forecasts are made on the basis of extrapolating past patterns of behaviour into the future. This approach has had some success for medium and long-term prediction, and thus provides some complementarity to process-based modelling. Data driven technique models are based on a statistical analysis of observations and extrapolation into the future. The mixture of new statistical techniques, duration and value of coastal observations and computer power has made the use of data driven techniques viable. However these techniques have a disadvantage in that if a particular event is not encapsulated within the historical observations, it is uncertain that an extrapolation technique can reproduce this event accurately. Nevertheless, this approach has the benefit that it is ideally suited for adaptation to quantify the uncertainty of the forecasts (Larson et al., 23). Using data-driven techniques (Habib and Mesellie, 2; Ró y ski et al., 2; Ró y ski and Jansen, 22) on datasets from around the world it is possible to characterise these complex relationships. By understanding the relationships between the beach shape and the prevailing wave conditions, beach material and structures; it is possible to quantify the variability in the beach profile. A clear relationship between wave conditions and beach levels will allow us to use wave forecasts to predict the corresponding beach conditions and hence flood Statistical Resampling Methods for 2

10 potential. This is important because forecasts of wave conditions are made on a routine basis and are generally acknowledged to be easier to make and are more accurate than direct forecasts of beach levels. 2 TECHNIQUES USED / METHODS USED 2. Introduction There are both advantages and disadvantages with data-driven techniques, as there are for all forms of prediction techniques. The disadvantages include the need for a series of observations over a long period of time and the assumption that past behaviour is a good indicator of future evolution. The advantages are that time-stepping is not necessary, thus avoiding issues of numerical stability, accumulation of rounding errors and excessive computational time. The primary techniques that we have investigated for data analysis are the Empirical Orthogonal Function expansion, (EOF), and the Canonical Correlation Analysis, (CCA). EOFs provide a means of analysing a time history of observations of a quantity along a line or over a two-dimensional grid so that the variability within the observations is split into those in space and those in time. A forecasting scheme can be based on extrapolating the time variations into the future and then combining these with the spatial information. CCA on the other hand provides a method of analysing the correlation between the changes in time of two separate sets of data, such as waves and beach profiles. In this case, if good quality forecasts of one of the data sets can be made, the CCA provides the means to forecast the second variable from the first through the correlation properties. The approaches for ensemble forecasting are the established statistical techniques known as the jack-knife and bootstrap methods. In the former, one observation in a collection of N observations is removed, in turn. The chosen analysis is performed on the remaining N- observations. The procedure is repeated N times, in each case with a different observation excluded from the analysis. This yields N sets of results. The spread in the results provides a measure of the uncertainty inherent in the data analysis process and permits sample statistics such as the mean and variance to be determined. Bootstrapping is similar, with the difference that rather than excluding one observation in turn, the set of observations is sampled randomly to create equivalent samples of N observations from the original data set. This can lead to repeated values but it usually leads to more robust assessments of uncertainty than the jack-knife method. 2.2 Empirical Orthogonal Functions (EOF) method The Empirical Orthogonal Function (EOF) technique is used to define the patterns of spatial and temporal behaviour in a set of data. It is also frequently called Principal Component Analysis (PCA). The EOF method is a decomposition of a data set in terms of orthogonal basis functions which are determined from the data. Pearson (9) established the method, but, being computationally demanding, has only been used since the arrival of computers. Statistical Resampling Methods for 3

11 The EOF technique has been used in coastal disciplines by Winant et al. (975) to analyse beach profile measurements at Torrey Pines Beach, California (USA), and by Aranuvachapun and Johnson (979) to study beach erosion at Gorleston, Norfolk (UK). A brief description of the technique is provided in Appendix A. The EOF procedure is not dissimilar to Fourier analysis. However, it differs in two important aspects. First, EOF analysis can be used on data that is irregularly spaced in time. Secondly, the shape of the functions is not specified, but is determined by the data. 2.3 Canonical Correlation Analysis (CCA) method CCA is one of a family of correlation techniques, such as product moment correlation and multiple regression analysis. CCA may be used to investigate the presence of any patterns that tend to occur simultaneously in two different data sets (predictor and predictand) and the correlation between associated patterns (Clark, 975). In other words, CCA can find the optimum linear combination of one of the variables the predictor - that will explain most of the variance in the other variable the predictand, (Horrillo-Caraballo and Reeve, 2). Barnett and Preisendorfer (987) have used the CCA method in the field of meteorology and climatology. In Coastal Engineering, this method has good potential due to the fact that wave conditions are, in general, much easier to measure than beach profiles. Moreover, wave conditions are frequently forecast for many oceans and coastal waters by national weather institutions. Thus, beach profiles can be forecast from wave conditions if a strong link between beach profiles and waves can be established. This method has been used in coastal engineering already by Larson et al. (2), with data from the FRF (Field Research Facility in North Carolina, USA), in order to find patterns in the wave and profile data and to investigate the use of CCA to predict the profile response due to waves. Horrillo-Caraballo and Reeve (28) extended this study to investigate the sensitivity of the quality of predictions on the choice of distribution function used to describe the wave heights. Also, CCA has been used by Ró y ski (23) to analyse the evolution patterns of multiple longshore bars and the interactions amongst them. The main disadvantage of using this method has been the lack of good quality simultaneous measurements of waves and beach levels over sufficiently extended periods. As a result there has been little investigation into the accuracy of data-driven methods as a function of the data sampling rate, or a function of the length of the data record, or as a function of beach type. Details of the CCA method are given in Appendix B. As wave records are currently more widely available than beach measurements, equation (5), in Appendix B, may be used to predict beach profiles from forecast wave conditions, as proposed by Larson et al. (2). Such predictions are considered data-driven because no dynamical equations are being solved. Clearly, the forecast will depend on the quality of the data used in the statistical analysis. The forecasts are also restricted by the range of past conditions and are unlikely to accurately presage changes in dynamical regime unless such changes are contained within the data series used to define the regression matrix (Horrillo- Caraballo and Reeve, 2). Statistical Resampling Methods for 4

12 2.4 Jack-knife method Assuming we have a sequence of beach profile or bathymetry measurements which have already been analysed using EOF, an ensemble of EOFs with which to make predictions can be created using a resampling technique. The jack-knife method is used because it is a wellestablished and robust technique, and also requires less intensive computational effort than other methods. Although the jack-knife was introduced by Quenouille (949), it only became popular with the availability of readily accessible high speed computing. The motivation for introducing the jack-knife was to construct an estimator of bias that could be used in a wide range of situations. Tukey (958) subsequently suggested that the jack-knife could be used to estimate the variance of an estimator that was a function of the sample data. Efron and Tibshirani (993) describe in details the jack-knife and other resampling methods but a brief description of the technique is given in Appendix C. In the case of the bathymetry forecasts here, the jack-knife is used to produce an ensemble of equally possible sequences of bathymetry. Each member of the ensemble consists of the sequence of bathymetries between certain period with one bathymetry left out. This thus produces N equally possible datasets (based on the N- bathymetries left), each of which is analysed using EOFs, and each of which is extrapolated to a certain year in order to make the ensemble of forecasts. Then the mean of the ensemble of forecasts is computed and taken to be the best forecast for that particular year. The confidence interval around this mean may then be computed using Equation (24), in Appendix C, but with the standard error being the normal standard error, that is Appendix B). nvar JK, and with ˆ* θ being the standard mean, (see 2.5 Bootstrap method Bootstrap is one of the computationally very expensive techniques. It was introduced by Efron (979) as a computer-based method for estimating the standard error of an estimate. This technique is able to estimate the sample distribution of almost any statistical property using simple methods. Generally, it is considered a resampling method. The idea behind this method is to evaluate the properties of an estimator (such as variance, standard deviation, etc.) by measuring those properties when sampling from an empirical distribution (standard choice) of the observed data. If a data set population is assumed to be independent and identically distributed, then it is possible to construct a number of resamples of the observed data (equal size to the observed dataset), each of which is obtained by random sampling with replacement from the original dataset. In other words, this method involves randomly resampling the dataset a very large number of times (B times), similar to Monte Carlo simulation (Li et al., 28). This method can be used to construct hypothesis tests. Also, it is often used as an alternative method for inference based on parametric suppositions when those suppositions are in Statistical Resampling Methods for 5

13 doubt, or when parametric inference is not possible or requires very complicated formulas for the calculation of standard errors. In a very simple form it works as follows. For bootstraps, a dataset of size n is used to create another set of size n, comprised of elements that are selected at random from the original dataset. After each value is selected it is replaced so that the same value can be selected again. This process of creating new datasets, each of size n, is repeated many times over, (say B times), to create what are termed bootstrap samples. In the bootstrap method, it is possible that a sample of n data points can contain repeated values (twice or more) or some of the original values may not appear at all. The length of all bootstrap samples (replicates) are the same as the original samples and each of the B bootstrap replicates can provide an estimator!!. The spread of the estimator can be calculated from the resampled datasets and it provides information on the stability of the estimator with respect to different possible outcomes represented by the bootstrap replicates (Li et al., 28). The disadvantage of resampling with replacement is that it can lead to unrealistic bootstrap samples. Then, it is important to perform a large number of replicates. Further details of bootstrapping techniques can be found in, Efron and Tibshirani (993) and Davison and Hinkley (997). Also, a brief description of the technique is given in Appendix C. 3 CASE STUDIES 3. Duck, USA / Milford-on-Sea, UK Here we apply the CCA technique to a series of historical beach profiles surveys in two different locations. The first location is the Duck site, located in North Carolina, USA where historical data of wave records and beach profiles covering a period of 24 years have been obtained from the Field Research Facility (FRF) website (FRF, 27). The FRF is located on the Atlantic Ocean in Duck, North Carolina, USA (Figure 2). According to FRF (27), this facility is managed by the U.S. Army Corps of Engineers. Since its creation, a long-term measurement campaign (waves, tides, currents, local meteorology and beach response) has been maintained. Christchurch Bay, New Forest District Council (NFDC), UK is the second location. The surveys of beach profiles and wave conditions cover a period of over 5 years. Christchurch Bay contains a shallow embayment delimited by Hengistbury Head to the West and Hurst Spit to the East (Figure 3). According to SCOPAC (24), an unstable configuration relative to a wave climate characterised by dominant waves from the south-west is characteristic of Christchurch Bay, which is considered to be in an immature form, still evolving to achieve an equilibrium plan shape. Most of Christchurch Bay is exposed to swell waves, and has a high energy wave climate, despite the effects of wave refraction (SCOPAC, 24). Statistical Resampling Methods for 6

14 Portsmouth Norfolk Virginia Beach ATLANTIC OCEAN VIRGINIA NORTH CAROLINA FIELD RESEARCH FACILITY Duck Kitty Hawk ALBEMARLE SOUND Roanoke Island Oregon Inlet Elevation (m) PAMLICO SOUND Cross-shore distance (m) 25 KM Cape Hatteras Figure 2. Field Research Facility and profile surveys at Duck, NC, USA. Christchurch Highcliffe Barton-on-Sea Milford-on-Sea Hengistbury Head Edinburgh Christchurch Bay Hurst Spit Cardiff Birmingham London km Figure 3. Study site in Milford-on-Sea and Christchurch Bay, New Forest District Council, UK and profile surveys. 3.. Beach profiles The Field Research Facility (FRF) part of the US Army Corps of Engineers has been monitoring the nearshore area of Duck, NC, USA for more than 27 years, and has collected hundreds of surveys. Regular shore-parallel contours, a moderate slope and bars in the surf zone are characteristics of the nearshore bathymetry in this area (Figure 2). Elevation changes occur just seaward of the shoreline. The sediment consists of medium to fine sand mix with a grain size decreasing from mm on the foreshore to.-.2 mm offshore. The Statistical Resampling Methods for 7

15 area is micro-tidal with a tidal range between.7 m on neap tides and.5 m on spring tides (FRF, 27). Twenty six transects normal to the shore are surveyed at Duck site from the dune to approximately km offshore. Four of these transects (58, 62, 88, 9) are monitored every two weeks and the others every month (FRF, 27). The data used for this case were from the Profile 62 obtained from the FRF webpage (FRF, 27) and covered the period from July 98 to December 25 (226 profiles). Using the interpolation described by Li et al. (25), each profile was interpolated to a regular spacing with 2 m resolution. The interpolation was carried out from 7 m of the main baseline (approximately the dune level) until 9 m offshore (around 8 m water depth), to take into account the maximum number of usable surveys and to include the depth of closure (estimated to be between 4 and 6 m water depth (e.g., Larson and Kraus, 994)), and as much offshore information as possible (Figures 2). All profile elevation data are referred to the US 929 National Geodetic Vertical Datum (NGVD29). For Milford-on-Sea, this area has been monitored since 987 (Bradbury et al., 23), as a part of the shoreline management plan and a long-term monitoring beach survey programme. The data sets for Profile 5f7 were obtained from the Channel Coastal Observatory database (CCO, 26) and covered the period from November 988 to October 24 (5 profiles). Only measurement surveys that extended from the dune region to a water depth of the MLWL (Mean Low Water Level) were included in the analysis (Figure 3). The measurements were interpolated to a regular spacing with.5 m resolution following the same procedure as the measurements for Duck. All profile elevation measurements are referenced to the Ordnance Survey Datum Newlyn (ODN) Wave data The wave data available, for Duck, consists of significant wave height (H m) and peak spectral wave period (T p) obtained from the instruments deployed in the FRF (FRF, 27) and are also accessible from the FRF webpage. The data was collected from a directional wave rider buoy located approximately 3 km offshore at a water depth of 7.4 m from which the offshore wave data sets were obtained. For this instrument data were available from 98 (FRF, 27). Wave data is typically recorded every 6 hours but more frequently during some parts of the observation period, for which hourly values were recorded (Figure 4, top left panel). At Milford-on-Sea, the wave data has been monitored since 23 by a long-term waverider buoy deployment, which is located in approximately -2 m water depth. The predominant wave direction is from the southwest. The data obtained from the wave buoy have been compared with synthetic offshore wave data from the UK Meteorological Office, which has later been transformed to the wave buoy site, using a spectral wave refraction model that accounts for spectral saturation (Bradbury et al. 24). In the comparison of a one-month sample of measured and synthetic data, Bradbury et al. (24) found a strong correlation between measured and modelled conditions concluding that confidence in the offshore Statistical Resampling Methods for 8

16 synthetic wave data, the numerical wave transformation process used and the wave buoy measurements is extremely high. This was done in order to use data outside of the period of the wave data measurement. Since, the wave data measured have been available from 23. Figure 4. (a) Raw wave data (H s) and (b) empirical probability density function at Duck, NC, USA and (c) raw wave data (H s) and (d) empirical probability density function at Milfordon-Sea, UK. Some differences exist between the profiles at Duck and at Milford-on-Sea: The beach material at Duck is sand while at Milford-on-Sea the sediment is a combination of sand and gravel. In Milford, the gravel is predominantly found on the upper part of the beach and berm while below Low Water Level the beach is predominantly sand. Because of this, the differences in slope of the beaches at the two sites differs; a beach foreshore slope of in 2 is characteristic of the Duck profile whereas at Milford-on-Sea, the foreshore slope is steeper, ( in 6). There is a presence of a sandbar in the beach profile at Duck, and in stormy seasons, sporadically an offshore sandbar is created. At Milford-on-Sea there is a shore-parallel sandy bar with distinct breaks in it along the shoreline. Secondary bars appear occasionally offshore. Further, small amplitude beach cusp features are regularly present along the gravel-sand interface. Statistical Resampling Methods for 9

17 The coverage of the survey data is another important difference between the sites. At Duck, the profile measurements extend much further offshore, to what is arguably the depth of closure. The profiles at Milford-on-Sea do not extend to the corresponding point on the profile. The main focus of this case is on the active section of the upper beach and the performance of the CCA in predicting this zone; which is covered by the measurements of both sites Methodology For this case study, the data have been described in the sections above and have been manipulated in order to obtain time series with the same sampling rate for profiles and wave conditions. The profile dates were fixed in accordance with the Milford-on-Sea dataset. That is the reason why at the Duck site 55 profiles were used to perform the CCA analysis and the rest (7 profiles) were used to compare with the predictions obtained on the basis of the regression. In the case of Milford-on-Sea, 29 beach profiles were used in the CCA and 2 beach profiles were used for comparison with the predictions on the basis of the regression (Horrillo-Caraballo and Reeve, 2). Wave conditions between two selected profiles were used to characterise the changes in each subsequent beach profile. Wave observation measurements are more common than beach profiles measurements and to apply the CCA method, it is required to generate two series of equal length. For this reason, probability density functions (pdfs) are required to compile the wave conditions. A parametric form of pdf was proposed by Larson and Kraus (995). However, more recent studies have suggested that an empirical distribution gives better performance (Horrillo-Caraballo and Reeve, 27, 28). The empirical pdfs of H s were calculated from wave heights between two consecutive beach profile measurements. The resulting pdf was then assigned to the later of the two profile measurements. The empirical distribution is a cumulative probability distribution function that concentrates probability /n at each of the n numbers in a sample. A combined pdf (p n) may then be derived by superimposing the individual pdfs available for the measurement period between surveys, p n n = n ( H ) I( H H ) i= i (3.) where H is the wave height, n the number of individual wave measurements between surveys and i an index. Figure 4 (right panels) shows the wave height raw data and the empirical distribution for that data for each site. More detailed information can be found in Horrillo-Caraballo and Reeve (2) Analysis of the data Statistical Resampling Methods for

18 For the purpose of comparison the CCA is applied to both the Duck and the Milford-on-Sea datasets using years of records and then performing predictions 8 years into the future (until 26). In this case study, four cases were performed: Baseline case: All data at Duck from 2/7/98to 3//998 (55 profiles and 55 empirical wave distributions) were used in the CCA to determine the regression matrices and then predictions from 3//998 to 3//26 (7 profiles) were computed using the measured wave data (see Table ). First test (T): Used Duck data set from 2//988 to 3//998, having the same date and the same amount of data as for Milford-on-Sea (29 profiles) and then predicting from 3//998 to 3//26 (7 profiles). Predictions were made for each date coinciding with a profile measurement at Duck. Thus both the duration and sampling density of the Duck data is reduced in comparison to the baseline case (see Table ). Second test (T2): Used exactly the same input data is used as in the first test. However, in this case the output is calculated for those dates for which there are observations at Milford. In practice because the two sets of observations at Duck and Milford were not coincident we chose the measurement at Duck which was closest in time to that at Milford. The purpose of this is to a) degrade the sampling of the Duck data to closely match that at Milford-on-Sea and b) compare the quality of the predictions over the same window and at the same intervals in both cases. Third test (NFDC): Milford-on-Sea data set were used and the CCA analysis was performed using beach profile and significant wave height from 2//988 to 3//998 for regression purposes (29 profiles) and for prediction purposes (2 profiles). The data for the prediction goes from 3//998 to 3//26 (see Table ). The first and second test (T and T2) were performed with the aim of seeing how the degradation of the data affects the error in the prediction. The degradation of the data consists of using the Duck data with the same length of the data of Milford-on-Sea to compare the results of Duck and Milford-on-Sea. These analyses were made to attempt a like-with-like comparison, that is to say, to base the prediction on the same length of data at each site. Table. Summary of the data used for the CCA analysis (Horrillo-Caraballo and Reeve, 2) Test Name Period No. of Prediction No. of Data set No. of considered waves Period profiles for used profiles (dd/mm/yyyy) obs. (dd/mm/yyyy) prediction Baseline Duck 2/7/ // //998 3//26 First Test (T) Duck 2// // //998 3//26 Second Test Duck 2// //998-2 Statistical Resampling Methods for

19 (T2) 3//998 3//26 Third Test Milford 2// //998- (NFDC) 3//998 3//26 2 The aim of the different tests was to investigate: The importance of sampling rate over the same period, The differences in performance of CCA on sandy and mixed beaches, The variability of forecast as a function of forecast window The importance of the length of sampling record Results Figure 5 shows the comparison of the profile averaged RMSE for each of the four cases over the 8 year forecast window and the time series of H s at Duck and at Milford. A lower level of error is present at Milford and this can be in part explained by the less natural variation of the profile at Milford. The profile averaged RMSE of ~.25 m is of a similar order of magnitude to the errors in surveying a steep uneven beach. As the sampling of beach profiles is being more frequent at Duck than at Milford, a large range of combinations of wave conditions and behaviour will be present at Duck records; which would suggest that predictions at Duck would be based on a better quality of regression. Also, Figure 5 shows the impact of severe degradation of the data on forecasting quality where, for about 2 months at the end of 23 there was a gap in the wave records due to equipment failure at Milford. In the forecast immediately after this episode, the RMSE almost doubles and then just as rapidly falls again as normal wave data supply is resumed (Horrillo-Caraballo and Reeve, 2). Statistical Resampling Methods for 2

20 Figure 5. Spatial averaged RMSE using 3 modes for the predicted CCA beach profiles at Duck and Milford-on-Sea, New Forest District Council (NFDC) (top panel), measured significant wave height (H s) at Duck (middle panel) and measured significant wave height (H s) at Milford-on-Sea (bottom panel). Figures 6 and 7 show examples of the comparison between measured and predicted shorelines and the associated absolute error for both sites. At Duck (Figure 6), there is a large discrepancy between the measured and predicted profiles at the baseline of the measurements, but generally in the swash zone the changes are between and m. Sometimes the CCA method overpredicts or underpredicts the position and the height of the bar, mostly due to the changeable nature of the beach profile at Duck (movement of the sandbars). In the top and the bottom profiles the height of the bar shows a good agreement with the measurements but in the middle profile is underestimated by the prediction and it is shifted slightly to the coast. Also, the CCA method overpredicts where the offshore height of the end of the profile is. The variability in this section of the profile is less so it might have been thought that prediction would be easier. However, the predictive method is based solely on wave height. The site at Duck has some quite complex morphological behaviour, including sandbars aligned obliquely to the shoreline which propagate in a longshore sense (Lippmann et al., 993). The movement of these features depends on wave direction and period as well as just height (Lippmann et al., 993, Plant et al., 999). A similar trend is evident in the results of Van Dorengen et al. (28) and a likely reason for the Statistical Resampling Methods for 3

21 overprediction is the exclusion of key processes and variables. In the case of our CCA forecasts this includes wave period and direction, as well as tidal flows. elevation (m) 5-5 /5/2 Predicted Measured RMSE=.49 error (m) /5/ elevation (m) 5-5 RMSE=.42 error (m) /6/ elevation (m) 5-5 RMSE=.53 error (m) cross-shore distance (m) cross-shore distance (m) Figure 6. Measured and predicted profiles for three different dates (left panels) and associated RMSE absolute error (right panels) at Duck. RMSE values refer to averages across the profile length (Horrillo-Caraballo and Reeve, 2). As at Duck, the CCA method also overestimates the elevation of the beach profile at Milford-on-Sea (Figure 7) in the lower area of the profile, but to a much smaller degree. The first m are well predicted. In the top panel, the prediction of the position of the berm is under estimated by the CCA as in the middle panel, but in the bottom panel, the position of the berm is well predicted by the method. According to figures 6 and 7, the profile at Milford-on-Sea was more stable than the profile at Duck. The results are not directly comparable because the length of the measured profile was different in both cases as is the beach material. This has been mitigated to a degree by performing tests with the Duck data degraded to match the sampling rates in the Milford data (Horrillo-Caraballo and Reeve, 2). Statistical Resampling Methods for 4

22 5 9/5/2 elevation (m) RMSE=.37 Predicted Measured /5/22 5 error (m) elevation (m) RMSE=.26 error (m) /6/ elevation (m) RMSE=.39 error (m) cross-shore distance (m) cross-shore distance (m) Figure 7. Measured and predicted profiles for three different dates (left panels) and associated RMSE absolute error (right panels) at Milford-on-Sea. RMSE values refer to averages across the profile length (Horrillo-Caraballo and Reeve, 2). The results of the study indicated that data-driven statistical analysis, such as described here, can be a valuable method for analysing profile response to waves, complementing process based prediction techniques. It also has some promise as a beach prediction tool where forecast wave conditions are available Conclusions CCA was applied at two different sites at which there are coincident records of beach profiles and wave conditions for over a decade. The first site, Duck, has gently sloping sandy beaches while the second, Milford-on-Sea, has a steep gravel upper beach. The analysis of forecasts performed for the two sites demonstrates that there is no clear degradation in prediction accuracy over an 8 year forecast period. The levels of error are such that this form of modelling is a viable complement to dynamical forecasting, particularly for periods beyond one year, and further that the predictions could be of practical use for engineers and planners. The magnitude of errors is generally smaller for the gravel beach, reflecting the smaller range of beach level variation measured at this site. As Statistical Resampling Methods for 5

23 anticipated, degrading the duration and density of the records used to calculate the regression matrix leads to an increase in forecast errors. The CCA method may be expected to provide good predictions to forecast beach profiles if there is strong correlation between the two variables (beach profiles and wave height distributions), taking into account that the future changes have similarities with past behaviour. Beach profiles are conditioned primarily by wave height, but other factors such as tidal currents, wave period and wave direction will have an impact. 3.2 Slapton Sands This is another case study using CCA analysis, in which the purpose was to investigate the relationship between wave forcing and shoreline response (shorelines obtained from the ARGUS system) on a gravel beach using CCA analysis and in particular to determine shorelines from measured wave records through CCA analysis. As we have seen in the sections before, this technique has the potential to be a forecasting method if future wave conditions can be predicted with sufficient accuracy Field Site According to Ruiz de Alegria-Arzaburu et al. (2), Slapton Sands barrier is aligned roughly north-south and faces east towards the English Channel. Argus Cameras Hs (m) N E 2% W 6% % 4% 8% S Figure 8. Location of Slapton Sands gravel beach (UK) and wave climate in the area (Ruiz de Alegria-Arzaburu et al., 2). Statistical Resampling Methods for 6

24 It is a macro-tidal gravel barrier, 4.5 km long and 4 m wide, located between two cliff outcrops in Start Bay, southwest England (Figure 8). The slope of the beach is steep, approximately :8 and the sediment size (D 5) fluctuates between 2 and 2 mm. The mean significant wave (H s) obtained in the barrier varies between m and 3m (medium-to-high energy wave conditions) and the mean spring and neap tidal ranges are 4.3 m and.5 m, respectively. Slapton Sands is affected by a bimodal wave climate dominated by southerly swell and easterly wind waves (Figure 8) Shorelines The shorelines obtained for this case study were obtained from video images from the ARGUS system. For more detailed information of the extraction of the shoreline from videoimages refers to the paper by Ruiz de Alegria-Arzaburu et al. (2). The video-derived shorelines dataset cover the period between April 27 and September 28 (~ 6 months). One shoreline per week was extracted at a certain day and time at which the tidal elevation was ±. m ODN; with this procedure a total of 76 shorelines were obtained (Figure 9). These shorelines extend approximately 2.5 km along the northern half of the barrier. These shorelines are referred to a local coordinate system established for the setup of the video-cameras. The origin of the system is roughly located in the middle of the barrier, as such, positive values will refer to the northern half of the barrier, and negative values will be given to the southern half. 8 6 cross-shore distance along-shore distance (m) -5 Figure 9. Video-derived shorelines used for the CCA analysis. According to Ruiz de Alegria-Arzaburu et al. (2) the -m shoreline contour located in the middle of the intertidal beach will reflect the intertidal beach change depending of the variability of the shoreline. Storms that take place in the study site are expected to cause Statistical Resampling Methods for 7

25 cross-shore and alongshore exchange of sediment between the supertidal and intertidal beach Wave data The wave data used for this case study was collected by a directional wave buoy located 2 km offshore from Slapton at an approximate water depth of m CD, and managed by the Plymouth Coastal Observatory ( Wave measurements from April 27 to September 28 were analysed. The mean significant wave height (H s) obtained was.65 m with a zero-crossing period (T z) of 4.2 s (Ruiz de Alegria-Arzaburu et al., 2). Being an area where there is a bimodal wave climate, it is important to include the direction of the waves in the calculation of the predictor for the analysis of the CCA. For this, the alongshore wave energy flux (E f) for intermediate waters is derived using the significant wave heights (H s) and directions collected every 3 minutes, where, E f = E sin(2α ) (3.2) E 2 ρ gh s C g 6 = (3.3) and, C g gl d tanh(2π ) 2π L = (3.4) where α and H s are wave direction and significant wave heights measured at the buoy position, respectively; g the gravity, the water density, C g the group velocity for intermediate waters; L the wavelength and d the depth at which the buoy is located. Waves direction are usually given relative to north, consequently a conversion was performed to transform these directions normal to the shoreline. Thus, waves coming from the northeast to the southeast will mean positive alongshore energy fluxes, and negative fluxes will correspond to waves coming from the southeast to the southwest (Figure top panels). Statistical Resampling Methods for 8

26 3.5 4 H s (m) Energy flux (N s - ) Jun/7 Dec/7 Jul/8 Time (month/year) -4 Jun/7 Dec/7 Jul/8 Time (month/year).25.8 Probability of Hs Probability of energy flux H s (m) Energy flux (N s - ) Figure. Measured significant wave heights (H s) top left panel, and corresponding alongshore energy fluxes for intermediate waters top right panel. Pdfs for significant wave height bottom left panel, and pdfs for energy fluxes (positive flux for waves coming from the east to the southeast, and negative for waves from the southeast to the southwest (Ruiz de Alegria-Arzaburu et al., 2) Methodology Here we also applied the CCA method to predict shorelines from a certain number of historic video-derived shorelines. The wave conditions measured during two consecutives shorelines will characterise the last shoreline position at the end of that time. Therefore, H s and wave directions measured within a period of time will be used to characterise the shoreline positions at the end of that period, and generate the canonical correlation matrix. However, there are about 24, wave data points to characterise 6 shorelines (the ones used to create the regression matrix), thus in order to generate two series of equal length, a unique representative value needs to be obtained for each shoreline. Miller and Dean (27) demonstrated that using monthly or daily averaged nearshore parameters and correlating them to the EOF modes showed the largest shoreline variability. In the study carried out by Ruiz de Alegria-Arzaburu et al. (2) it was considered more appropriated to use probability density functions (pdfs), because the pdfs maintain the statistical information of the variability of the waves and also accounts for the variation of the probability. For this reason, pdfs of significant wave heights (H s) and energy flux (E f a function of significant wave height and wave direction) were calculated for the time between video-derived shorelines to characterise the wave forcing and to calculate the regression matrix of the CCA Statistical Resampling Methods for 9

27 (Figure bottom panels). Similar to the previous case study, an empirical distribution was used to derive the pdfs Results Following the work of North et al. (982), and their rule of thumb, it is possible to determine the appropriate number of modes to be used in the CCA in order to obtain a significant correlation between variables. In this case study, the best numbers of modes were three, which accounts for the 96% of the variation in the dataset with respect to the mean. The first, second and third modes represent the 8%, % and 5% of the variation, respectively. The spatial and temporal variations become increasingly complex as the amount of variation described by each mode decreases; eventually, these variations become noise, and contain no recognisable patterns (Miller and Dean, 27a). The majority of the shoreline variations are explained by the first two modes (9%). In this case study, the CCA methodology was applied to examine the statistical relation between the significant wave height H s and the alongshore energy flux E f, and the shoreline shape. Using sixty shorelines (April 27 to June 28) to generate variability regression matrixes ( ), the medium-term performance of the CCA has been evaluated for the pdfs of significant wave height (H s) and alongshore energy flux (E f). The variability regression matrixes ( ) obtained applying the CCA technique were then used with the pdfs of H s and E f for the following 6 weeks (June to September 28), to determine the corresponding shoreline positions (see equations 5 7, Appendix B). Thus, calculations of 6 shorelines (one per week, shorelines 6 to shoreline 76) were made for two cases: first, using pdfs of H s and second, using pdfs of E f. Figure shows the comparison between video-derived and predicted or forecast shorelines for the first week, the th week and the 5 th week, using pdfs of H s and E f, respectively, and the associated RMSE. The mean relative error was calculated for each of the shorelines, presented in figure, from the absolute cross-shore difference between real and predicted shorelines, and dividing it by the real position times one hundred. Generally, the performance of the CCA method is good, especially when pdfs of E f are used, with RMSE generally less than 3 m. The use of E f to determine the alongshore ends of the study area accounted for higher errors than only considering H s, whereas in the middle north of the barrier (,3 to 2, m alongshore) the shoreline estimations were significantly improved when the wave direction was taken into account (Ruiz de Alegria-Arzaburu et al., 2). Statistical Resampling Methods for 2

28 Figure. Predicted and video-derived shorelines for three different dates (weeks 65, 7 and 76 - top to bottom) using H s PDFs (left panels) and E f PDFs (right panels) on the CCA. In each panel, the predicted shoreline (solid) and the video-derived shoreline (dashed) together with the alongshore distribution of the relative error of the estimations, and the average RMSE in metres are presented (adapted from Ruiz de Alegria-Arzaburu et al., 2) Conclusions Canonical Correlation Analysis (CCA) was employed, in this case study, to determine the relationship between the wave conditions and beach plan shape evolution on a gravel barrier. Measurements of more than 8 months have been used for this purpose, which comprise wave conditions measured with a directional wave buoy and video-derived shorelines (Argus video system) covering 2.5 km of the northern barrier. As it was said before, the main constraint of the CCA method is the requirement of large morphological and hydrodynamic datasets. Therefore, the validated Argus video system was proven to be a useful tool to derive shorelines and monitor shoreline variability at different temporal and spatial scales. In general, the predictions derived from the pdfs of E f energy flux are considered better than the predictions obtained from the pdfs of H s, which indicate that the inclusion of the wave direction into the CCA improves its performance. In the middle of the study area the determined shoreline shape was significantly improved using pdfs of E f, whereas at both ends these calculations were less successful. The medium-term CCA calculations are considered satisfactory and of practical use for the quantification of shoreline variability given the characteristics of the remote data acquisition Statistical Resampling Methods for 2

29 and the uncertainty within the image analysis. On the other hand, the CCA technique could be used to test different scenarios to assess possible movements of the shoreline in response to a projected wave climate. 3.3 Great Yarmouth Sandbanks For this case study a data-driven model has been used to analyse the long term evolution of a sandbank system and to make ensemble predictions over a period of 8 years (Reeve et al., 28). The method is a combination of EOF analysis, used to define spatial and temporal patterns of variability; jack-knife resampling, in order to generate an ensemble of EOFs; a causal auto-regression technique, used to extrapolate the temporal eigenfunctions, and basic statistical analysis of the resulting ensemble of predictions to determine a forecast and associated uncertainty. The methodology has been applied to the Great Yarmouth sandbanks (Norfolk, UK) a site which includes a group of mobile nearshore sandbanks. The site is on the eastern coast of the UK and includes the Great Yarmouth sandbanks and neighbouring shoreline. The aim of this case study was to analyse the evolution of nearshore sandbanks over an 8- year period, and in particular to investigate the strength of the predictions with respect to two different prediction techniques. The dataset used for the study is a group of historical survey charts of the area immediately offshore from Great Yarmouth, on the east coast of the UK. This area contains a well-documented system of sandbanks and channels, and has been subjected to regular surveys for navigational purposes over the period of many decades Field site Great Yarmouth is located on a thin spit between the River Yare and the North Sea in which a curved shoreline and a group of mobile sandbanks can be found. The key features in this area are (Figure 2): Caister Shoal Scroby Sands (North, Middle and South Scroby) Cross Sands (North, Middle and South Cross Sands) Corton Sands Newcombe Sands Additionally, seven channels between the sandbanks may be identified (Figure 2): Barley Pickle Caister Channel Caister Road Yarmouth Road Gorleston Road Holm Channel Statistical Resampling Methods for 22

30 Corton Road The sandbanks have major environmental, commercial, leisure and physical importance as well as providing protection to the coast from wave action (Reeve et al., 28). 32 North Cross Sand Winterton Caister Channel Barley Pickle 35 Newport Caister Shoal North Scroby Middle Cross Sand 3 35 California Caister-on-Sea Great Yarmouth Caister Road Yarmouth Road Middle Scroby South Scroby South Cross Sand Northing (m) 3 Gorlestonon-Sea Gorleston Road Corton Sand Holm Channel Hopton-on-Sea Corton Road 295 Corton Holm Sand Lowestoft 29 Pakefield Newcombe Sand Edinburgh Kessingland 285 Benacre Cardiff Birmingham London m 25 m 5 m Easting (m) Figure 2. Morphology of the Great Yarmouth Sandbanks (adapted from Horrillo-Caraballo and Reeve, 28). One of the main sandbanks of this group is Scroby Sands which plays a particularly important role in that it is a breeding ground for seals and, since 25, has been the site for 3 turbines in an offshore wind farm. Significant changes in the location, height and width of the individual banks have been observed during more than 5 years (nautical charts from UKHO). This means that navigable channels through the sandbanks have also changed over time. According to HR Wallingford, (22a) the sediments around the Great Yarmouth Sandbanks are dominated by transport of coarser material in a southerly direction due to littoral drift, Statistical Resampling Methods for 23

31 with finer materials generally transported offshore. A protected zone is formed between Winterton to Benacre due to the sandbank system in which a complex circulation of sediment between the shore, the inner banks and the outer banks can be developed. An anticlockwise movement of sediments around the banks is typically seen in this area. Tides found in this area are commonly microtidal at Lowestoft and low-meso tidal at Winterton-on-Sea. The mean spring and neap tidal range at Winterton-on-Sea are 2.6 and.4 m, respectively; while at Lowestoft are.9 and. m, respectively (UKHO, 2). The wave climate in the Great Yarmouth area is subject to the storm wave climate of the North Sea. Most waves approach from the North and Northeast or South and Southeast; almost 4% of all waves are less than m high and 76% less than 2 m high; being the waves approaching from the North and Northeast the largest (Reeve et al., 28) Survey data The Great Yarmouth area is well-documented thorough nautical charts from the United Kingdom Hydrographic Office (UKHO) and bathymetric surveys. The nautical charts used for this case study are listed in Table. Old charts have been constructed with different datum levels and orientation. Also, the new technology has improved the acquisition and accuracy of the data. As part of the map digitisation post-processing, all bathymetry was reduced to the equivalent of Chart Datum and the chart was oriented to grid North. For the final analysis a common area for all charts was determined. Table 2. Set of Nautical charts used (Reeve et al., 28). Year Chart Date Year Chart Date Year Chart Date Apr Jul Nov Apr Jun Nov Apr Dec Jan May Sep May Nov Dec Mar Feb Dec Feb Dec Jul Feb Dec Oct Jun Sep Jan Jan Aug Apr Jul Dec Feb Sep May-6 The area obtained from the historical nautical charts is delimited by the UK national grid coordinates of 65, me, 287, mn and 663,5 me, 32,3 mn (Figure 2). Digitised data were checked thoroughly in order to exclude any errors introduced during the digitising processes. After that, the digitised charts were interpolated into regular grids (m x m). In total, 36 bathymetry grids were created covering the area depicted in Statistical Resampling Methods for 24

32 Figure 2 over the period between 848 and 26 (see Table 2), with 33 of them covering the period from 848 to Methodology The methodology adopted in this case study brings together several different techniques. This methodology combines EOF analysis, jack-knife resampling, autoregression techniques (Burg s and Least Square Auto-Regressive LSAR -algorithms) and ensemble statistics to determine the forecasts and uncertainties. Figure 3 shows the methodology that was followed to obtain the different prediction for the Great Yarmouth area. Figure 3. Scheme of the methodology. This methodology provides a means of predicting the future bathymetry as well as yielding a measure of the uncertainty in the predictions, provided by the statistical sampling. More detailed description of the methodology can be found in the paper by Reeve et al., (28). The main of this case study was to investigate the sensitivity of the results to the choice of method used to perform the extrapolation of the temporal modes. Statistical Resampling Methods for 25

33 3.3.4 Results EOF Analysis As a benchmark an EOF analysis was performed on the 33 bathymetries from 848 to 998. Table 3 summarises the results of the EOF analysis for the eight most significant functions. It demonstrates that over 96% of the mean square of the data is contained in the first function and that 99.% of the mean square of the data is captured by only 8 functions. This is a large proportion of the variance for analyses of this kind and suggests that the EOFs describe the variability in the sandbank morphology extremely efficiently. Table 3. EOF results Eigenfunction number Normalised eigenvalue % Mean square (Cumulative) % Variance % Variance (Cumulative) The temporal variation in the bathymetry is described by the temporal eigenfunctions. There is one temporal eigenfunction corresponding to each spatial eigenfunction. The temporal eigenfunctions are normalised, and so describe how the amplitude of the corresponding spatial eigenfunction varies throughout the period covered by the surveys. The temporal eigenfunctions for the first three eigenfunctions are shown in Figure 4..6 C C2 C3.4.2 C Time (Years) Figure 4. Plots of the temporal eigenfunctions Statistical Resampling Methods for 26

34 Figure 5 shows a contour plot of the first three spatial eigenfunctions. It shows strong spatial structure with areas of maxima and minima elongated in the area of the sandbanks and channels. At locations where a spatial eigenfunction has the same sign a similarity in morphological behaviour can be anticipated. The first spatial eigenfunction corresponds to the time mean bathymetry, and therefore has a smoothed appearance, although the main elements of the sandbanks are evident (e.g., Scroby Sands, Holm Channel, Barley Pickle, etc.). The other eigenfunctions also have a coherent spatial structure, with maxima and minima at different locations EOF EOF 2 Figure 5. Bathymetric reconstruction in shading map elevations (in metres) for the first three spatial eigenfunctions The second and third spatial EOF modes have their peak values near to the flanks of the sandbanks. This can be understood when it is considered that small changes in the channel position will coincide with relatively large depth changes at the margins of the sandbanks. The second spatial EOF mode tends to reflect larger scale variations than the third spatial EOF mode, which describes larger scale variation than the third, and so on. Thus the relatively small importance of the other EOFs is reflected in the fact that they describe small scale changes Ensemble of EOFs Each of the 33 jack-knife samples is analysed to generate EOFs for that sample. This produces an ensemble of spatial and temporal EOFs. All the EOFs thus produced exhibit similar general behaviour as the benchmark EOF but have a spread that arises from the differences in the sets of data used to compute the EOFs in each jack-knife sample. Figure 6 shows the 2 nd temporal EOF modes for the jack-knife ensembles from the dataset. EOF 3 Statistical Resampling Methods for 27

35 .4.3 C 2 (t).2. C Time (Years) Figure 6. Second temporal EOF modes for the ensembles Bathymetric reconstruction in shading map elevations (in metres) for the first three spatial eigenfunctions Forecast of temporal EOFs The bathymetric measurements up to and including 998 were used to perform an EOF analysis. After that, the temporal eigenvectors obtained from this analysis were interpolated onto regular intervals and then forecasts to the year 26 were made. The absolute RMSE of forecasts relative to measures obtained with Burg s algorithm is shown in Figure 7a. If three EOF modes are considered (left panel), the best order corresponds to r=5. Burg s algorithm with order r=5 provides better forecasts up to 24 if eight EOF modes are considered (right panel). For 25 and 26 there is not much difference in forecasting capabilities for the three cases considered. a) Burg Algorithm b) LSAR algorithm RMSE.5.5 RMSE Years Years Years Years Statistical Resampling Methods for 28

36 Figure 7. Absolute RMSE of the forecasts relative to the measured EOFs for (a) Burg s and (b) LSAR algorithms of order r=3 (solid line), r=5 (dashed line) and r=5 (line and markers). Taken from Horrillo-Caraballo et al. (29). Figure 7b shows the RMSE for the LSAR case. If three EOFs modes are considered, the best order is r=3. The same order when choosing the first eight EOFs for the forecasts up to 25. Nevertheless, for the 26 prediction, the best forecast is obtained when r=5 is used considering the first eight EOFs (Horrillo-Caraballo et al., 29). Observing the figure, the RMSE obtained is smaller with three eigenvectors than with eight in the case of 8 year forecasts. Then, a compromise should be made between the RMSE of the forecast and desirable percentage of resolved variance. When there is no clear indication of which techniques have to be used for the extrapolation, then a calculation of the RMSE between the measured bathymetry and the forecasted can be calculated. In this case, not only the temporal but the spatial distribution of the errors can be defined. The RMSE was calculated doing a reconstruction of the forecasted bathymetry using the extrapolated eight EOFs modes (Figure 7a and 7b right panels).then, the bathymetry for 26 is reconstructed from the extrapolation of the temporal eigenvectors and the spatial eigenvalues at each point of the grid (Equation, Appendix A). The predicted bathymetry for 26 was then compared with the original bathymetry from 26. The variance contours were minimal when r = 5, and therefore this value was chosen to make the predictions from year 998 up to 26 with the two AR techniques (Horrillo- Caraballo et al., 29). Statistical Resampling Methods for 29

37 Statistical Resampling Methods for b) a) d) c) e) f) g) Figure 8. Forecast to year 26. a) Bathymetry from 26 survey data. Mean bathymetry of ensemble forecast using: b) Burg s and c) LSAR algorithms. Standard deviation between predictions and measurements: d) Burg s and e) LSAR predictions. Skewness between predictions and measurements: f) Burg s and g) LSAR predictions (Horrillo-Caraballo et al., 29). Figure 8 shows the bathymetry for 26 (Figure 8a) obtained from the digitised nautical chart as well as the mean from the ensemble forecast for 26 (Figure 8b and 8c), the standard deviation (Figure 8d and 8e), and the skewness (Figure 8f and 8g), between the ensemble forecasts and the survey data comparing the skill of the predictions for the two AR techniques for the first eight spatial EOF modes. A better skill for all three variables is obtained with Burg s algorithm,. For example, if the means are compared (Figure 8b and 8c), the prediction using the LSAR algorithm suggested that the depths in Barley Pickle are Statistical Resampling Methods for 3

38 too large in relation to the original data, while the prediction from Burg s algorithm are in good agreement with the original data. An ensemble forecast provides a sampling of the probability distribution for the future bathymetry. There is no conventional method to select the best forecast of an ensemble. Taking into account the characteristics of the jack-knife method; the mean is used to select the best forecast. To assess the accuracy of the ensemble comparing the forecast and the actual measurements, two methods were proposed. The first approach was to use the standard deviation (Figures 8d and 8e) and the second was to use the skewness (Figures 8f and 8g). To detect areas of large ensemble changes (if any), the standard deviation ( ) was calculated at each position for each of the AR techniques (Figures 8d and 8e). Figure 8d shows a region with large forecast uncertainties around Cross Sands. This may be explained from the dynamic nature of the area. Other areas of large changes are the south-western and northeastern flanks of Scroby Sands and the region between Corton Sand and Holm Sand, which coincide with each end of Holm Channel. On the contrary, large areas of uncertainty are found over the whole area, with fewer uncertainties in Yarmouth Road, Gorleston Road, Newcombe Sand and the area seawards of Holm Sands. In Holm Channel, South Cross Sand, flanks of Scroby Sands and Corton Sands large areas of discrepancies are found. The LSAR extrapolation technique provides a greater spread in the behaviour of the ensemble than the Burg algorithm. The skewness between the 26 bathymetry and the mean of the 26 forecast ensembles (Figure 8f and 8g) was used as the second approach. The underpredicted depth areas are shown by the green areas and the overpredicted areas are showm by the yellow areas. Major areas of discrepancies are located at the seaward side of Scroby Sands and the northern tip of South Cross Sand, as well as in the entrance to Holm channel (Figure 8f). Also, there are some areas at the flanks of Barley Pickle that are overpredicted. Figure 5g shows, on the contrary, that the LSAR method gives larger values of the skewness (both positive and negative). These large values are located throughout the whole area (both in the channels and on the sandbanks). Therefore, Burg s algorithm gives a narrower range of behaviour while LSAR gives a more diverse ensemble with different statistical properties. In general, the mean ensemble forecasts for Burg s and LSAR algorithms produce similar results but one of the statistical properties of this forecast (skewness) distinguishes some differences Conclusion The methods presented in this case study use a combination of EOF analysis, jack-knife resampling, causal auto-regression technique (Burg s algorithm and LSAR algorithm) and basic statistical analysis of the resulting prediction ensembles to determine a forecast and associated uncertainty. The methodology has been used to analyse the patterns of morphological evolution of the sandbank system and to predict morphological changes over an eight year period. Statistical Resampling Methods for 3

39 The mean bathymetry is explained by the first spatial EOF mode and clearly exhibits the inner and outer set of sandbanks, as well as the main channel system. The second and third EOFs explain the intermittent changes in Holm Channel and Caister Channel, as well as the large fluctuations observed in the configuration of the Cross Sands. The combination of different techniques ( EOF, jack-knife and AR extrapolation) was used to predict the evolution of the bathymetry from 999 to 26, and predictions compared with the 26 survey measurements. Burg s algorithm produces a narrower range of behaviour while LSAR gives a more diverse ensemble with different statistical properties. In summary, an alternative method to simple equilibrium or behavioural models of morphological evolution and an additional mechanism for trying to quantify uncertainties inherent in the predictions have been developed. While the application described here is site specific the data-driven ensemble forecasting methodology has broad applicability and could be deployed on any coastal regions with good historical hydrographic data. 3.4 Walcott We applied Canonical Correlation Analysis (CCA), in this case study, to an historical dataset of seabed elevations within a coastal segment of the English south eastern coast. The study site is located along the frontage of Walcott (Norfolk, East coast of England). The beaches along this segment of coast are mainly composite, mixed sand and shingle. The dataset comprises detailed bathymetric surveys of beach profiles covering a period of over 7 years (~ 2 beach profile surveys every year). The coastline in Walcott is orientated approximately such that it runs NW to SE and is prone to the attack of waves and winds from the North Sea, especially when the wind blows from the northeast. If this wind is combined with a storm surge then the threat grows, which can raise the still water level of the sea by up to 2m above the predicted sea level. According to HR Wallingford (22b) conditions like this have triggered terrible flooding in the town, as in 953, and more recently in 27. The health of the beach in front of a coastal structure influences the performance of the structure, as a healthy beach will induce more wave breaking, dissipating energy before it reaches the structure. The water depth in front of the structure is a crucial parameter which depends upon the level of the beach and the still water level (this being the sum of the astronomical tide and the meteorological surge component). Astronomical tides can be forecast quite accurately (from Admiralty tide tables, etc.) while surges are more difficult to forecast and will depend upon surface pressure and winds. Without a doubt the highest uncertainty in knowing the water depth rests with specifying the level and shape of the beach. From this perspective, it is necessary to understand how seawalls and beaches interact during storms and how beach profiles interact with waves, tides and storms. In this case study, CCA techniques are applied to a series of historical beach profile survey from the Walcott area covering a period of over 7 years, provided by the Environment Statistical Resampling Methods for 32

40 Agency. The data were analysed in order to determine the covariability between waves and profile response Case study area The Walcott site is a sandy beach in a meso-tidal environment with a hard structure at the top of the beach. Wave parameters were estimated with wind measured at Weybourne, UK (Chini, pers. comm.) using the CERC formulae (USACE, 984). The wave data point (WDP), where wave parameters were calculated, is located at position N, E (Figure 9). Independent wave predictions provided by the Environment Agency were used to validate the wave hindcasts. Initially, the analysis focused on directly linking the profile response to wave parameters (significant wave heights - H s, wave steepness - H/L). In order to represent the significant wave heights and wave steepness, probability density functions (pdfs) were derived for the two wave parameters. The regression matrixes obtained from the CCA analysis were then used to predict the beach profile for the day of the storm as well as for the date of a poststorm survey. Edinburgh Birmingham Cardiff London Figure 9. Location of the study area and photo from the seawall in Walcott (top right) and wave data location (WDP) Beach profiles The mixed shingle-sand beaches of Walcott have been monitored since 99 along 8 shore normal profiles, as part of a long-term beach management survey programme by North Norfolk District Council (HR Wallingford, 22b). The surveys were carried out in the summer and winter months so that the seasonal variations in beach morphology can be examined. For this case study, Profile N3C8 has been used as it includes the seawall (Figure 2). Beach profiles measured along Profile N3C8 were used in the CCA analysis (25 profiles shown). Only surveys that extended from the dune region out to a water depth of Statistical Resampling Methods for 33