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SHORELINE RESPONSE TO VARIATIONS IN WAVES AND WATER LEVELS: AN ENGINEERING SCALE APPROACH By JONATHAN K. MILLER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 ___ __ This dissertation is dedicated to my beautiful wife Diana, who has always been there for me when I needed it the most. During this long journey, she has patiently followed me across the globe, sacrificing much of herself, now it is finally time to go home. ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor, Robert Dean, for his inspiration and guidance. Many a time I came to him frustrated and dejected, only to be reinvigorated by his infectious enthusiasm during an enlightening Sunday morning meeting. Sharing in his passion for coastal engineering over the past five years has been an extraordinary experience. I would also like to thank the other members of the Coastal Engineering Department for their insightful seminars and thought provoking discussions. In particular I would like to thank Becky Hudson and Robert Thieke for making sure everything ran smoothly, and Dan Hanes, Ashish Mehta, Joann Mossa, and Robert Thieke once again, for agreeing to serve on my supervisory committee. It has been a pleasure working with all of them. For welcoming Diana and I to a foreign land with wide open arms, I would like to express my appreciation to Peter Nielsen and everyone at the University of Queensland. From the moment we arrived in Brisbane we were treated like family, making our transition "down under" much easier. Having the chance to work with Peter, Tom, Ling, and all of the coastal students was an honor and sincere pleasure. Our time in Australia was truly unforgettable thanks to the many wonderful people we encountered during our journeys. Several teachers that I was fortunate enough to encounter prior to beginning my graduate studies have also played an integral role in my development as a student and as a person. Br. Paul Joseph and K.Y. Billah always believed in me and encouraged me to learn for the love of learning and not for the grade attached to it. Both epitomize the true meaning of the word teacher. I need to thank Dimitris Dermatas and Michael Bruno for encouraging me to follow my heart and study coastal engineering when others told me I would be better off studying something more practical. I would be remiss in not thanking the Florida Sea Grant, the American Society for Engineering Education, the United States Department of Defense, and the Australian American Fulbright Commission, all of whom provided financial support for various stages of this project. Their contributions have been greatly appreciated. Along the way, I made many friends who have had a profound impact on my life and whom I will never forget. To Kristen, Justin, Chris, Jamie, Sean, Al, Dave, Nick, Ian, Finney, Carlos and everyone else who has helped me get through the last five years, I truly value each of their friendships. Cliff deserves a special mention for taking a leap of faith and moving to Gainesville with me at the start of this unforgettable journey. Most importantly, I need to thank my family for their continued love and support, without whom this would not have been possible. They have always been there to back me and encourage me in whatever I have chosen to do, and I am much indebted to them. Finally, I need to thank my wonderful wife Diana who inspires me each and every day. She has shown me the true meaning of love, and without her encouragement and emotional support over the past five years I would not have made it through the past five years. She has always been there for me and I can never thank her enough for her infinite love and patience. TABLE OF CONTENTS page ACKNOW LEDGM ENTS ................................................................................................. iii LIST OF TABLES ............................................................................................................ vii LIST OF FIGURES ...................................................................................................... x ABSTRACT..................................................................................................................... xiv 1 INTRODUCTION .................................................................................................. 1 2 BACKGROUND .................................................................................................... 8 2.1 Longshore (Planform) M odels............................................. ......................... 10 2.1.1 Analytical M odels .................................................... .......................... 11 2.1.2 Numerical OneLine/NLine M odels .................................................... 12 2.2 Crossshore (Profile) M odels...................................................... .................. 12 2.2.1 Analytical M odels .................................................... .......................... 13 2.2.2 Empirical M odels ............................................................ ...................13 2.2.3 Energy Dissipation M odels .....................................................................14 2.2.4 Process Based M odels ................................... ........................................17 2.2.5 Alternative M odels ................................................................................ 20 2.3 Need for Innovative Approaches .....................................................................20 3 M ODEL DEVELOPM ENT..................................................................................22 3.1. Theoretical Background................................. ................. .............................22 3.2. Defining the Equilibrium Shoreline, yq(t) ............................................. .....25 3.3. Defining the Rate Parameter, k ................................ ........................................ 29 3.4 Solution Technique .........................................................................................36 3.4.1 Numerical Scheme.................................... .............. ............................36 3.4.2 Forcing Data........................................... ................ ...........................37 3.4.3 Shoreline Data. ................................................................... .....................38 3.4.4 M odel Calibration................................................. ..............................39 3.4.5 M odel Evaluation .................................................................................. 40 4 FIELD DATA AND SITE SUITABILITY .................................................. .........46 4.1 East Coast Data............................................................................................... 48 4.2 W est Coast Data ............................................................................................. 53 4.3 Australian Data ............................................................................................... 55 4.4 Evaluation Tools................... .............. ........................................................57 4.4.1 Time Dom ain Based Statistics......................... .......................................58 4.4.2 Frequency Dom ain Based Statistics ............................................... .... 59 4.4.3 Method of Empirical Orthogonal Functions.......................................60 4.5 Site Suitability ..................................................................................................64 5 RESULTS .............................................................................................................75 5.1 New Jersey ...................................................................................................... 84 5.2 Florida............................................................................................................ 86 5.3 W ashington and Oregon ....................................................................................92 5.4 California........................................................................................................ 93 5.5 Australia.......................................................................................................... 94 6 DISCUSSION ..................................................................................................... 100 6.1 Timescale of Response ............................................................. ..................101 6.2 Selection of Appropriate Rate Parameters.......................................................... 103 6.3 M odified Error Estim ates / Cost Functions ...................................................... 109 6.4 Time Varying Sediment Scale Parameter, A(Q(t)) ..........................................111 6.5 Application to EOF Filtered Data....................................................................... 17 7 SUMMARY, CONCLUSIONS, AND FUTURE DIRECTIONS..........................19 7.1 Summ ary...................................................................................................... 119 7.2 Conclusions...................................................................................................121 7.2 Future Directions .......................................................................................... 123 APPENDIX A M ODEL SOURCE CODE ....................................... ..............................................126 B COMPLETE SET OF MODEL RESULTS ....................................... .............. 135 LIST OF REFERENCES ............................................................................................ 164 BIOGRAPHICAL SKETCH ..................................................................................... 172 3 LIST OF TABLES Table pag 31. Established erosion/accretion criteria.............................................................33 32. Categorical assessment procedure score matrix developed for this study ..............44 33. Subjective rating system based upon model performance statistics.......................44 41. Summary of data sources. ................................................................................... 47 42. Relevant site characteristics ................................................ ......................... 48 43. Data analysis techniques applied at each site.......................... ...................65 44. Summary of time domain analysis results ..........................................................66 45. Summary of EOF analysis results.......................................................................73 51. Summary of SLMOD results............................................................................76 52. NMSE associated with various rate parameter combinations at Long Beach, WA, and typical of the NMSE tables presented in Appendix B.....................................80 53. CAP associated with various rate parameter combinations at Long Beach, WA, and typical of the CAP tables presented in Appendix B...............................................81 54. Calibration coefficients for Long Beach, WA, and typical of the coefficient tables presented in Appendix B.....................................................................................83 55. NMSE associated with various rate parameter combinations at Crescent Beach, FL. ....................................................................................................................... 88 56. CAP associated with various rate parameter combinations at Crescent Beach, FL. ....................................................................................................................... 89 61. Best performing rate parameters for each site...................................................1..04 62. Rate coefficient statistics according to geographic region...................................108 63. Percent change in NMSE values at Torrey Pines, CA for Case 1 (A(Q)) ...........116 64. Percent change in NMSE values at Torrey Pines, CA for Case 2 (minimum W. im posed). .......................................................................................................... 116 65. Percent change in NMSE values at Torrey Pines, CA for Case 3 (A(Q) and minimum W imposed). ...................................................................................117 66. Percent change in NMSE values when only the longshore uniform EOF modes are considered at the Gold Coast, Australia..........................................................118 B1. Calculated NMSE values for model hindcasts at Island Beach, NJ.....................136 B2. Calculated CAP scores for model hindcasts at Island Beach, NJ ........................137 B3. Calibration coefficients ka, ke, and Ayo for Island Beach, NJ ..............................137 B4. Calculated NMSE values for model hindcasts at Wildwood, NJ...................... 138 B5. Calculated CAP scores for model hindcasts at Wildwood, NJ ............................139 B6. Calibration coefficients ka, ke, and Ayo for Wildwood, NJ ..................................139 B7. Calculated NMSE values for model hindcasts at St. Augustine, FL....................140 B8. Calculated CAP scores for model hindcasts at St. Augustine, FL.......................141 B9. Calibration coefficients ka, ke, and Ayo for St. Augustine, FL.............................141 B10. Calculated NMSE values for model hindcasts at Crescent Beach, FL ................142 B11. Calculated CAP scores for model hindcasts at Crescent Beach, FL....................143 B12. Calibration coefficients ka, ke, and Ayo for Crescent Beach, FL..........................143 B13. Calculated NMSE values for model hindcasts at Daytona Beach, FL................ 144 B14. Calculated CAP scores for model hindcasts at Daytona Beach, FL ....................145 B15. Calibration coefficients ka, ke, and Ayo for Daytona Beach, FL ..........................145 B16. Calculated NMSE values for model hindcasts at New Smyrna Beach, FL. ..........146 B17. Calculated CAP scores for model hindcasts at New Smyrna Beach, FL.............147 B18. Calibration coefficients ka, ke, and Ayo for New Smyrna Beach, FL.....................147 B19. Calculated NMSE values for model hindcasts at North Beach, WA.....................148 B20. Calculated CAP scores for model hindcasts at North Beach, WA.......................149 I_ _ B21. Calibration coefficients ka, ke, and Ayo for North Beach, WA.............................149 B22. Calculated NMSE values for model hindcasts at Long Beach, WA....................150 B23. Calculated CAP scores for model hindcasts at Long Beach, WA .......................151 B24. Calibration coefficients ka, ke, and Ayo for Long Beach, WA..............................151 B25. Calculated NMSE values for model hindcasts at Clatsop Plains, OR .................152 B26. Calculated CAP scores for model hindcasts at Clatsop Plains, OR.....................153 B27. Calibration coefficients ka, ke, and Ayo for Clatsop Plains, OR...........................153 B28. Calculated NMSE values for model hindcasts at Torrey Pines, CA....................154 B29. Calculated CAP scores for model hindcasts at Torrey Pines, CA........................155 B30. Calibration coefficients ka, ke, and Ayo for Torrey Pines, CA..............................155 B31. Calculated NMSE values for model hindcasts at Brighton Beach, AS..................156 B32. Calculated CAP scores for model hindcasts at Brighton Beach, AS ...................157 B33. Calibration coefficients ka, ke, and Ayo for Brighton Beach, AS .........................157 B34. Calculated NMSE values for model hindcasts at Leighton Beach, AS ...............158 B35. Calculated CAP scores for model hindcasts at Leighton Beach, AS...................159 B36. Calibration coefficients ka, ke, and Ayo for Leighton Beach, AS.........................159 B37. Calculated NMSE values for model hindcasts at the Gold Coast, AS.................160 B38. Calculated CAP scores for model hindcasts at the Gold Coast, AS ....................161 B39. Calibration coefficients ka, ke, and Ayo for the Gold Coast, AS...........................161 B40. Calculated NMSE values for model hindcasts at the Gold Coast, AS, using filtered (fc = 0.033 days') data. ........................................................................................ 162 B41. Calculated CAP scores for model hindcasts at the Gold Coast, AS, using filtered (fc = 0.033 days') data............................................................................................. 163 B42. Calibration coefficients ka, ke, and Ayo for the Gold Coast, AS, using filtered (fc = 0.033 days') data ................................................................................................. 163 LIST OF FIGURES Figure page 21. Profile schematization in Swart model.............................. ...................................14 22. SBEACH profile schematization .........................................................................17 23. Typical process based model schematic..............................................................19 31. Beach recession due to a combination of an increased water level, S and wave induced setup, b(y). ...........................................................................................28 32. Example illustrating the role of Ayo in correcting for differences in the baseline conditions of y(t) and yob(t) ................................................................................. 29 33. Schematic of model calibration routine. ............................................................39 41. Location of data sets from the East Coast of the United States .............................49 42. Improvement in the consistency of the Duck shoreline data after adjusting for the volume change between subsequent profiles .................................... ............. 52 43. Location of available shoreline data along the west coast of the United States.......53 44. Location of Australian shoreline data sets .........................................................55 45. Calculation of the mean correlation profile including r. and ravg........................67 46. Spectral analyses of Gold Coast shoreline data where the thick line represents the mean spectra .......................................................................................................70 47. Mean coherence and phase for three selected shorelines at the Gold Coast...........70 48. Spatial eigenfunctions el(x)e3(x) for Duck, NC............................................ ....72 49. Spatial eigenfunctions el(x)e3(x) for Washington State.......................................72 410. Spatial eigenfunctions el(x)e3(x) for the Gold Coast, QLD .................................73 51. Complete hindcast shoreline time series for Daytona Beach, FL..........................78 52. Example hindcast plot of observed and predicted shorelines at Long Beach, WA, and typical of those presented in Appendix B. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations. .................79 53. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Crescent Beach, FL. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ................88 54. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Daytona Beach, FL. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations. .................90 55. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at New Smyrna Beach, FL. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ................91 56. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Torrey Pines, CA. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ................................94 57. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Brighton Beach, AS. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations. .................96 58. Complete hindcast shoreline time series for the Gold Coast, QLD .......................96 59. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at the Gold Coast, AS, using filtered (fc = 0.033 days) data. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations. ..................................................................................... 98 510. Comparison of the extreme values of the measured shorelines and "best" simulations according to the NMSE and CAP criteria at the Gold Coast, AS, using filtered (fc = 0.033 days') data. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ................................99 61. Variability of the amplitude response function, IF(o),k1 with forcing frequency, o, and rate coefficient, k ........................................................................................ 102 62. Variability of the phase response function, <(o,ka) with forcing frequency, o, and rate coefficient, k ............................................................................................... 102 63. Histograms of accretion coefficients, ka, determined by the procedure detailed in Chapter 3. ......................................................................................................... 106 64. Histograms of erosion coefficients, ke, determined by the procedure detailed in Chapter 3. ......................................................................................................... 107 65. Comparison of "best" modified NMSE predictions with the standard NMSE prediction and the measured data at Torrey Pines, CA........................................111 66. New relationship for A proposed by Wang (2004), where Afit/A, is the ratio of the new A value to that given by Moore (1982), and Hb/wT is the breaking form of the nondimensional fall velocity parameter, Q ..........................................................113 67. Variation of active surfzone width, W., with A(Q). ............................................113 68. Effect of A(Q) on calculated Ay, values at Torrey Pines, CA.............................. 114 B1. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Island Beach, NJ. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............................136 B2. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Wildwood, NJ. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............................138 B3. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at St. Augustine, FL. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............................140 B4. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Crescent Beach, FL. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............142 B5. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Daytona Beach, FL. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations. ...............144 B6. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at New Smyrna Beach, FL. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............146 B7. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at North Beach, WA. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............148 B8. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Long Beach, WA. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............................150 B9. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Clatsop Plains, OR. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............152 B10. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Torrey Pines, CA. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............................154 B11. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Brighton Beach, AS. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............156 B12. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at Leighton Beach, AS. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations ..............158 B13. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at the Gold Coast, AS. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations. ...............160 B14. Comparison of measured shorelines and "best" simulations according to the NMSE and CAP criteria at the Gold Coast, AS, using filtered (fc = 0.033 days"') data. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations. .................................................................................. 162 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SHORELINE RESPONSE TO VARIATIONS IN WAVES AND WATER LEVELS: AN ENGINEERING SCALE APPROACH By Jonathan K. Miller December 2004 Chair: Robert G. Dean Major Department: Civil and Coastal Engineering A simple new shoreline change model has been developed, calibrated and evaluated with several sets of high quality field data. The model is based upon previous research, which indicates that the shoreline will approach an equilibrium position exponentially with time, when subjected to constant forcing in the form of waves and water levels. The engineering scale approach used here simulates the shoreline response to these cross shore processes in an extremely efficient and practical manner, while requiring only readily obtainable wave and water level data as input. The equilibrium shoreline is defined by using a conservation of volume argument and equilibrium beach profile theory to derive an expression for the equilibrium shoreline change due to a combination of local tide, storm surge, and wave induced setup. The rate at which the equilibrium condition is approached is governed by a rate coefficient that can either be taken as a constant, or parameterized in terms of the local wave and sediment properties. A total of eight physically based rate parameters are evaluated, where the erosion and accretion are parameterized separately. According to the results, the most effective parameterization of the accretion rate is obtained using a surf zone Froude number, while the erosion rate is best parameterized by either the surf similarity parameter or the breaking wave height cubed. Three calibration coefficients representing a baseline for converting the equilibrium shoreline changes into equilibrium shoreline positions, and separate erosion and accretion constants, are evaluated by minimizing the error between model hindcasts and historical shoreline data. The extensive set of shoreline data used to calibrate and evaluate the model was compiled from a variety of sources, and consists of shoreline measurements from a total of thirteen sites within the United States and Australia. Overall the model successfully simulates the shoreline changes at 11 of the 13 study sites with an average normalized mean square error of 0.643. Other tools designed to help evaluate the model, such as a categorical assessment procedure and a model performance index, also indicate a similar high degree of success. CHAPTER 1 INTRODUCTION In the past halfcentury, coastal populations worldwide have swelled as more and more people have begun to recognize the recreational benefits and economic potential associated with beaches. Eleven of the world's fifteen largest cities lie within the coastal region, and over 400 million people live within twenty meters of sea level and twenty kilometers of the coast (Small et al., 2000). In 1990, 133 million people or approximately 54% of the U.S. population lived in one of 673 coastal counties (Culliton, 1990). In Australia, nearly 85% of the population lives within fifty kilometers of the coast and nearly onequarter of the population growth between 1991 and 1996 occurred within three kilometers of the coastline (CSIRO Atmospheric Research, 2002; Australian State of the Environment Committee, Coasts and Oceans, 2001). The economic investment in these coastal regions is substantial. In the U.S., nearly 75% of the gross domestic product is generated in coastal states (Colgan, 2003). According to Houston (2002), travel and tourism is America's leading industry and employer, of which beaches are the primary component contributing an estimated $257 billion to the national economy in 1999. The economic impact of tourism is not limited to the United States as similar statistics are reported in Australia where nearly 50% of international tourists and 42% of domestic tourists visit the coast, contributing over $15 billion annually to the marine tourism industry (Australian State of the Environment Committee, 2001). Australia and the U.S. are not alone in recognizing the economic value of the coastal region as countries such as Japan, Germany, and Spain have been known to spend as much $1.5 billion on shore protection and restoration in a single year (Marine Facilities Panel, 1991). In the U.S. and Australia, the most significant threat to this substantial investment comes in the form of beach erosion due to a combination of natural and anthropogenic factors. Galgano (1998) recently estimated that as much as 8090% of the non engineered U.S. Atlantic coastline was experiencing net erosion, while a 1994 report by the U.S. Army Corps of Engineers classified 33,000 km of the U.S. coastline as erosional, 4,300 km of which was classified as critical. While the exact numbers are often the subject of intense debate, the importance of understanding the processes leading to beach erosion (and accretion) is indisputable. Unfortunately, beaches are extraordinarily complex, dynamic systems and describing the governing physical process over the wide range of relevant spatial and temporal scales is an extremely difficult task. Individual swash events alter the beach topography with spatial and temporal scales on the order of millimeters and seconds, while sporadic storms can cause tens of meters of erosion in only a few hours. Some natural processes altering the shoreline such as sand waves exhibit both temporal and spatial periodicity, while others such as storm related erosion occur randomly. The sheer number and complexity of the physical processes responsible for inducing coastal change make representing them all in a fully threedimensional, timedependent, process based numerical model impossible given our current level of understanding. Fortunately for engineering and planning purposes, the spatial and temporal scales are related, which allows us to simplify the problem somewhat by considering only those scales important to a specific problem. Arguably, the most relevant scale is the socalled engineering scale, which refers to the range of temporal and spatial changes expected to impact a structure during its lifetime. Typically, the expected lifespan of a structure is on the order of 50100 years, corresponding to relevant time and space scales of hoursdecades, and metershundreds of meters. Even when the analysis is limited to the engineering time scale however, modeling shoreline changes remains a difficult task, therefore a number of different approaches have been developed. These range from simple extrapolations based on historical data to highly detailed, fully threedimensional, process based models. The fact that rudimentary extrapolation techniques are still used despite the potential for considerable inaccuracies is testament to the need for improved models. State of the art 3D models have been shown to be fairly accurate over shorter time scales after significant calibration; however they tend to break down near the shoreline and remain costprohibitive for most applications. Oneline models provide simple yet accurate solutions for predicting shoreline changes adjacent to structures related to longshore processes; however no comparable technique exists for accurately modeling the shoreline response to cross shore processes. Although longterm predictions are often based upon the assumption that the effects of crossshore processes will cancel over the long run, the most significant changes likely to impact a structure, particularly on a natural coastline, are in fact related to these neglected processes. Accurately representing the potential shoreline change due to crossshore processes, such as those related to seasonal variations in wave energy or extreme storms, must be included as an essential component of any complete shoreline model. Unfortunately, it is much easier to diagnose the problem than to fix it, as modeling the shoreline response to these crossshore processes is not a trivial task. Numerous crossshore models have been developed with a variety of different goals; however none have proven particularly successful at modeling shoreline changes at the engineering scale. Although the specific capabilities of each model vary widely, there are some common factors that make the majority of existing crossshore models inadequate for longterm predictions. Most conspicuous is the fact that nearly all models predict erosion more accurately than accretion, and while this inability to accurately model recovery processes has long been recognized, it remains a significant limitation. The general applicability of most process based models is often restricted by the extensive data required for calibration, as the paucity of available data, combined with the need for sitespecific information, means extensive costly field work is often required. More disheartening is the fact that even the most detailed models tend to break down in the vicinity of the shoreline, which for engineering purposes is nearly always the region of greatest interest. In many studies, model performance is either not evaluated near the shoreline or evaluated separately so as not to negatively impact the otherwise "good" results. Furthermore, even if these eventbased models were able to successfully handle accretion, there is no guarantee that the results could be integrated up to yield reliable predictions over longer timescales (Hanson et al. 2004). The above factors, when combined with the extensive computational resources often required, make most stateof theart models extremely inefficient, cost prohibitive, and ultimately impractical for many engineering applications. The objective of the present work is to present a new shoreline change model which is capable of reproducing the shoreline response to crossshore forcing over a variety of temporal scales. In order to provide the widest possible range of applicability, simplicity and efficiency along with a high degree of accuracy, were primary considerations. Rather than approach the problem from a purely process based standpoint, simple physical concepts were used in combination with empirical evidence to create a new tool capable of fulfilling the proposed objectives. The result is a simple model, which can be of immediate use to the engineering profession. Details of the model are discussed in Chapter 3, however the basic concept borrows from classical equilibrium theory, where the shoreline strives to reach an equilibrium state which continuously changes in response to the dynamic conditions of the nearshore environment. In accordance with physical observations, the rate at which this equilibrium is approached is proportional to the degree of disequilibrium between the instantaneous shoreline position and that suggested by the local forcing as a result of time varying wave and water level conditions. Consistent with nature, such a model predicts the strongest shoreline recovery immediately after the passage of a major storm, a result which few (if any) processbased models have been able to reproduce. Highquality data from both coasts of the United States as well as Australia were collected to calibrate and evaluate the model. Unlike previous model studies that may have been hampered by a lack of available data, recent emphasis on field data collection and dissemination has resulted in an abundance of suitable data for this project. Instead of haphazardly applying the model at each site for which sufficient data were available, several criteria were used to eliminate those locations for which a crossshore model was considered inappropriate. These criteria helped identify and eliminate several sites where the shoreline behavior exhibited significant longshore variations, potentially indicating the predominance of longshore processes. The geographical diversity of the data sets provided an interesting platform for examining the natural variability in the nearshore system, and for evaluating the model over a wide range of wave, tide, and geologic conditions. By incorporating shoreline measurements made using a number of different techniques, the skill of the model could be evaluated over a variety of timescales ranging from daily to multidecadal. Undoubtedly, process based models containing full detailed descriptions of the governing hydrodynamics and resulting sediment transport will eventually yield the most accurate predictions of shoreline change; however our present knowledge of the complex relationships and feedback mechanisms is insufficient to justify their use in longterm shoreline studies. The shoreline model developed and discussed herein is significant in that it is able to accurately predict shoreline changes, while requiring only minimal, readily available forcing and calibration data. The simplicity, efficiency, and adaptability of the new model make it a useful tool for a variety of engineering applications. With additional research, it should be possible to adapt this simple crossshore model to work in concert with the existing simple longshore models (e.g., Hanson and Larson, 1998) to obtain a robust, quasitwodimensional shoreline model. Although the emphasis here has been placed upon maintaining the simplicity of the model, it can readily be adapted and used with more detailed wave transformation models to analyze the potential implications of alongshore variations in the incident wave field. The current analysis has been limited to the comparison of model hindcasts with measured data; however the exhibited skill suggests it should be possible to apply the model in either a predictive sense using statistical descriptions of the forcing parameters and Monte Carlo simulations, or in a realtime sense using instantaneous measurements or storm forecasts of the forcing parameters. In the first case the results would represent the probabilities associated with various magnitudes of shoreline change based upon the statistical characteristics of the forcing parameters, while in the second case the model could provide firstapproximation predictions of the erosive potential of approaching storms. The efficiency of the new model will make it particularly useful for longterm studies ranging from the prediction of seasonal shoreline changes, to the prediction of decadal shoreline migration patterns for coastal management applications. In order to help the reader navigate through the remainder of this document, it is useful to provide a roadmap detailing its layout. In Chapter 2, some background information is provided including a more detailed discussion of the problem, as well as some of the more popular techniques for modeling the shoreline response to crossshore forcing. Details of the new model including a description of the numerical approach are presented in Chapter 3, while the available field data and the tools used to help eliminate the inappropriate sites for a crossshore model are discussed in Chapter 4. The results are presented in Chapter 5, followed by a detailed discussion of some of the key aspects of the model in Chapter 6. Finally, Chapter 7 summarizes the results and presents some suggestions for future work. CHAPTER 2 BACKGROUND The complexity of the extremely dynamic nearshore environment makes accurate predictions of morphological evolution in this region over even limited temporal and spatial scales extremely difficult. Unfortunately, practical design requirements demand that a wide range of scales be taken into consideration, as the relevant engineering timescale ranges from hours to decades and encompasses spatial scales ranging from meters to hundreds of meters. The societal relevance of understanding and predicting changes in the nearshore region is illustrated by the long and varied history of attempts to model it using physical, analytical, and numerical techniques. Hanson et al. (2004) reviewed in detail some of the conventional and less conventional modeling approaches that have been used to predict coastal evolution over yearly to decadal timescales. Despite considering over twenty different types of models, the authors were unable to Th identify any capable of reproducing adequate results over the full range of time scales considered. The remainder of this chapter is devoted to a discussion of some of these conventional modeling techniques, which through their inadequacy stress the need for innovative approaches. The four basic tools available to coastal engineers consist of experience/empirical models, physical models, analytical models, and numerical models. In some respects, local experience constitutes the best model, as a thorough understanding of the local processes (waves, tides, currents, sediment transport) and geomorphology are essential tools in understanding a coastal system. Similar projects on adjacent beaches often provide invaluable information regarding unexpected results attributed to localized phenomena. Relying on previous experience alone however, is insufficient for a number of reasons, including the inability to consider innovative approaches or optimize design. Although useful, experience or empirical models are nearly always best when applied in combination with either physical, analytical, or numerical models. Coastal physical models typically consist of scaled down versions of a natural system and are often constructed in a laboratory. The primary advantage of a physical model is the ability to control the ambient conditions so that specific design scenarios can be isolated and evaluated more precisely. Although physical models play a critical role in understanding coastal processes, they do have several significant disadvantages as well. In order to ensure similar behavior between the model and prototype, both scaling effects as well as laboratory effects must be considered and accounted for. Scaling problems can occur when the correct balance of forces is not preserved in the model, while laboratory effects can be equally detrimental, and include the generation of higher harmonics and presence of boundary induced reflections. These and other considerations combine to make physical models quite labor intensive and often extremely expensive, as highly skilled labor and specialized facilities are frequently required. Analytical models consist of closed form mathematical solutions to simplified versions of the equations governing shoreline and profile change, and are often derived for schematized geometries and basic input and boundary conditions. The objective of an analytical model is to capture the essential physics of the problem in a simplified manner that allows the fundamental features of the beach response to be derived, isolated, and more readily comprehended. Unfortunately, these types of models are generally too crude for design purposes; however they can provide a means to identify characteristic trends and investigate the basic dependencies of the shoreline response to different combinations of input and boundary conditions. Increasingly numerical models are being used to study complex coastal systems as advancements in our ability to represent the dominant physical processes, combined with rapid advancements in computational capability, make them ever more efficient. Numerical models provide greater flexibility in the selection of boundary conditions and allow for the representation of arbitrary forcing. In addition, numerical models are extremely dynamic in the sense that recent scientific advances are easily incorporated due to their typically modular nature. The model presented here falls into this category; therefore the majority of this chapter is devoted to a discussion of some of the conventional numerical modeling approaches that have thus far failed to produce a generally accepted crossshore model, applicable at the engineering timescale. 2.1 Longshore (Planform) Models Because of the complexity of the nearshore system, it is common to separate longshore and crossshore processes, and to treat planform and profile evolution separately. In planform models, shoreline changes are assumed to result from gradients in the longshore sediment transport, while crossshore effects such as storm induced erosion, or seasonal shoreline fluctuations, are either assumed to cancel over the length of the simulation, or are accounted for separately. These assumptions make planform models much more appropriate when applied over longer periods at segmented coastlines with systematic long term trends, and less applicable over shorter periods on more natural uninterrupted coastlines without dominant trends. 2.1.1 Analytical Models Perhaps the most often utilized analytical model in coastal engineering is the one line model developed by PelnardConsidere (1956) for predicting shoreline evolution due to gradients in the longshore sediment transport. The key assumption of the model is that the crossshore profile remains in equilibrium, and does not change along the extent of the shoreline being studied. The "oneline" moniker relates to the assumption that the movement of the entire profile can be represented by the translation of a single contour, usually the shoreline. If the oneline assumption holds, then the principle of mass conservation in the longshore direction must apply at all times, +D =0 (2.1) ax at where x and y are the longshore and crossshore coordinates, respectively, Qi is the longshore sediment transport rate, t is time, and D is the vertical extent of the active profile, defined as the sum of the depth of closure, h., and the berm height, B. The longshore sediment transport rate is given by Qs1 = Qosinab, where Qo is the amplitude of the longshore sediment transport rate, and ab is the angle between the breaking wave crests and the shoreline. Under the assumptions of constant longshore forcing, and small constant breaking wave angles, Equation 2.1 reduces to the classic heat conduction equation for which numerous analytical solutions exist, ay a'y aye a2y (2.2) at ax, In Equation 2.2, the diffusion coefficient e is given by e = 2QdD. Larson et al. (1997) provide a collection of analytical solutions to Equation 2.2 describing the basic shoreline behavior under various combinations of simplified forcing and boundary conditions. 2.1.2 Numerical OneLine/NLine Models In order to obtain more realistic oneline solutions, Equation 2.1 can also be solved using numerical techniques capable of handling more realistic forcing and boundary conditions. Similar to the analytical solutions, crossshore effects are assumed to cancel, as these models are typically applied over periods of years to decades and at sites with dominant long term trends. In terms of predictive skill, oneline models such as GENESIS (Hanson and Kraus, 1989) have proven to be fairly successful, despite their inability to simulate cross shore effects. Nline models are an extension of typical oneline models where the profile is divided into a series of N mutually interacting layers. In these quasi3D models, cross shore effects are included in a highly schematized sense through interaction terms. Models in this category include those of Bakker (1968) and Perlin and Dean (1983). Hanson and Larson (2000) attribute the lack of success of conventional Nline models to inappropriate representations of both the crossshore sediment transport and the cross shore distribution of longshore transport, and suggest innovative approaches are required. 2.2 Crossshore (Profile) Models Crossshore or profile models are generally used to describe the nearshore response to events over limited temporal (hoursyears) and spatial (metershundreds of meters) scales. In contrast to planform models where gradients in the longshore sediment transport drive the bathymetric evolution, profile models are most readily applied along coastlines dominated by the influence of cross shore processes. In general, these models have been most successful at simulating storm induced erosion, and have been less successful at reproducing poststorm recovery, and therefore medium to long term profile evolution. Schoonees and Theron (1995) reviewed ten different crossshore models with respect to theoretical merit and validation criteria, grouping the models into "best," "acceptable," and "less suitable" categories. According to their conclusions, none of the models could be identified as clearly superior, as each potentially performs the best under certain conditions. 2.2.1 Analytical Models Although several analytical crossshore models have been developed, none have proven nearly as useful as the PelnardConsidere equation. Larson and Ebersole (1999) used a simple diffusion equation to describe the evolution in time of an offshore mound placed in the xz plane, as direct physical analogies exist between several of the analytical solutions to the classical diffusion equation and the filling of dredged holes and spreading of offshore mounds. Bender and Dean (2003) reviewed many of the analytical solutions for wave transformation over bathymetric anomalies, including the potential shoreline impacts. Another useful analytical model developed by Kriebel and Dean (1993) for describing the time dependent evolution of the nearshore profile is discussed in some detail in the next chapter. 2.2.2 Empirical Models Most empirical models are based upon observations of morphological evolution made in the laboratory under controlled circumstances, and as such are subject to certain limitations. The empirical model of Swart (1974) was developed based upon observations of profile recession under monotonic waves, in both small and large scale tests. Swart schematized the profile into three separate zones as shown in Figure 21, and developed empirical relations for the equilibrium profile as well as the onshore and offshore limits of the active profile. In the Swart model, the time dependent crossshore 14 sediment transport rate, Qs, is driven by the disequilibrium of profile characteristics, and is given by Q. = s, W (W ) (2.3) where (LiL2)t is a time dependent profile width, W is the profile width at equilibrium, and Sy is an empirical constant for a given set of boundary conditions. Although the model was subsequently applied to field data with some success (Swain and Houston, 1984; Swart, 1986), the intensive calibration and complicated empirical formulas involved make the method too complex for widespread application. 1,ItM0 uNE UPPCR UN SOUat 0 OFIL *I 0W 11 SCHtATIZATIC A  ~   L_ N A r MM1 Figure 21. Profile schematization in Swart model (from Swart, 1974). 2.2.3 Energy Dissipation Models Based on an analysis of over 500 beach profiles from the Atlantic and Gulf Coasts of the United States, Dean (1977) derived an equilibrium beach profile of the form h = Ay2, using linear wave theory and the premise of uniform wave energy dissipation per unit volume due to breaking. The key assumption in the derivation is that sediment of a certain size will be stable for a certain level of wave energy dissipation per unit volume, I d(EC,) D.(d) = (2.4) h dy where h is the total depth, E is the wave energy density, and Cg is the group velocity. In this case, the y coordinate is shore normal and increases offshore. Using equilibrium beach profile theory, Kriebel and Dean (1985) proposed an expression for crossshore sediment transport based upon the difference between the actual and equilibrium levels of wave energy dissipation in the surfzone, Q, = K(DD.) (2.5) where D is the actual time dependent energy dissipation per unit volume, and K is an empirical sediment transport parameter. The bracketed term represents the degree of disequilibrium and suggests that for steep profiles sediment will be transported offshore to restore equilibrium and viceversa. Zheng and Dean (1997) subsequently modified Equation 2.5 by raising the disequilibrium term to the third power, in order to satisfy the appropriate scaling relationship given by Qr = Lr3n, where Qr is the sediment transport ratio, and L, is the length ratio. In both models, profile adjustments occur in response to gradients in the crossshore sediment transport according to the continuity equation, y =_, (2.6) at oh Equations 2.5 and 2.6 form a closed system of equations which can be evaluated numerically. On a storm time scale, both models are capable of adequately representing the storm induced erosion but are less successful at reproducing poststorm recovery. Inspired by the success of the EDUNE model (Kriebel and Dean, 1985), Larson and Kraus (1989) attempted to extend the capabilities of energy dissipation based models with SBEACH. In SBEACH, the nearshore region is separated into four distinct zones as illustrated in Figure 22, each having its own sediment transport relationship. In the breaking zone (Zone II), the magnitude of the crossshore sediment transport, Q,, is calculated based upon energy dissipation arguments, with an extra term added to account for down slope transport, K DD.+ E D>D. A Q. Kdy K dy (2.7) VK = (2.7) e dh 0 D< D. K dy where dh/dy is the local beach slope, and K and e are sediment transport coefficients for the energy dissipation and slope dependent terms, respectively. In SBEACH, the direction of transport is calculated separately, and is based upon an empirical criterion relating the deep water wave steepness, HoLo, to the nondimensional fall velocity parameter, Ho/w,T, according to O < 0.00070 2 Q (2.8) H > 0.00070 + Q where Ho, Lo, and T are deepwater wave parameters representing the wave height, wave length, and wave period, and w, is the sediment fall velocity. Sediment transport magnitudes in the remaining zones are calculated based upon empirical relationships for which the energy dissipation based transport (as calculated from Equations 2.7 and 2.8) serves as a boundary condition. In these regions, the magnitude of the sediment transport typically follows an exponential decay, with coefficients that vary from region to region. At both the onshore and offshore boundary, the magnitude of the sediment transport reduces to zero, such that there is no bathymetric change. The SBEACH model has been widely applied in numerical studies of storm related erosion and beach nourishment equilibration with adequate results; however the inability of the model to accurately simulate accretional events and onshore bar migration limits its usefulness for long term studies. pp j I Wave Height SI1 SWASH BREAKER TRANSITION ZONE BROKEN WAVE ZONE ZONE PREBREAKING ZONE Figure 22. SBEACH profile schematization (from Larson and Kraus, 1989). 2.2.4 Process Based Models Process based models aim to reproduce profile evolution on the basis of first physical principles. Roelvink and Broker (1993), and van Rijn et al. (2003) provide in depth reviews of several stateoftheart process based models including: UNIBEST (Delft Hydraulics), LITCROSS (Danish Hydraulics Institute), SEDITEL (Lab Nationale d'Hydraulique), WATAN3 and BEACH (University of Liverpool), COSMOS (H. R. Wallingford), CROSMOR (University of Utrecht), and CIRC (University of Catalunya). Although the exact details of the methods used to calculate the hydrodynamics, sediment transport, and bed evolution vary from model to model, the schematic in Figure 23 illustrates the general solution procedure. Gradients in the time averaged crossshore sediment transport rate drive bed level changes according to the continuity equation, (1n) A = (2.9) at ay where n is the sediment porosity, Zb is the local bed elevation, and y is once again the shore normal coordinate. In general terms, the time averaged crossshore sediment transport rate is given by 1 t2 Z, Q(y () = lu(y, z, t) c(y, z, t)dzdt (2.10) t2 ti Z4 where u is a horizontal velocity, and c is the sediment concentration. From a practical standpoint, a complete time dependent solution of these equations is virtually impossible, as specification of the velocity and concentration fields down to the scales associated with turbulence is required. In order to arrive at workable solutions, most processbased models distinguish between four different process scales, of which only those scales relevant to a particular application may be considered. The turbulent scale is the smallest scale and is usually not considered due to its relatively minor influence on the horizontal flow field. The intrawave time scale includes processes such as time lag effects within the wave period and wave asymmetry, which can be particularly important for onshore transport. Processes related to long waves and wave groups, such as variations in sediment concentration, make up a third scale. The fourth and final scale consists of mean variations of the wave field over time scales associated with the tidal period, and includes tidal currents and time averaged return flows. >Hydrodynamic Module 1) Waves across the profile 2) Currents across the profile Sediment Transport Module 1) Many different formulations Morphologic Module 1) Conservation of sediment volume Figure 23. Typical process based model schematic. Although invaluable in terms of understanding the complex physical relationships between hydrodynamic forcing and sediment response at the microscale, process based models are inadequate for modeling long term profile development, particularly in the vicinity of the shoreline. The combination of computational effort and the extensive data required to calibrate these models makes them extremely inefficient and expensive to run, especially for longterm studies. Stive and DeVriend (1995), Kobayashi and Johnson (2001), and van Rijn et al. (2003) all reached the same conclusion, that given our current "rudimentary" understanding of crossshore sediment transport processes in the surf and swash zones, the prediction of long term shoreline change using any of the existing models based on first physical principles was virtually impossible. Even as our understanding of microscale hydrodynamics and sediment transport processes improve, Input 1) Initial bathymetry 2) Waves and water levels at the boundary 3) Sediment parameters uncertainties such as those associated with prediction windows and the effects of storm sequencing, bring into question whether the results of these small scale models can be integrated up to the relevant engineering scales. 2.2.5 Alternative Models Alternative models can be broadly described as those which do not follow any of the traditional approaches, and which often combine elements of proven techniques in an attempt to fill specific voids in our modeling capabilities. Some examples of alternate models include Steetzel (1995) who added a semiempirical crossshore transport to extend the capabilities of Nline models, and both Inman (1987) and Larson and Kraus (1991) who modified Equation 2.2, adding an advective term to simulate sand wave propagation. Hanson and Larson (1998) attempted to incorporate seasonal effects into a traditional Nline model by schematizing the crossshore sediment transport, but were only moderately successful. Plant et al. (1999) and Madsen and Plant (2001) used alternative methods which closely parallel the proposed shoreline model to successfully simulate bar evolution and nearshore beach slope changes at Duck, NC. These equilibrium based techniques are discussed in more detail in Chapter 3. 2.3 Need for Innovative Approaches The models mentioned in the preceding two sections represent only a small subset of those available, yet they illustrate an important point. In spite of intensive efforts to develop robust numerical models of the nearshore region and the shoreline in particular, conventional modeling approaches have failed to produce a generally accepted engineering scale model. In general, planform models have proven somewhat successful at describing longterm shoreline changes in the vicinity of structures; however attempts to extend their capabilities by parameterizing the crossshore sediment transport have been much less successful. As a result, these models still cannot reproduce changes at the storm and seasonal time scales, and are inappropriate on long, straight, natural coastlines. Profile models, whether based on energy dissipation arguments or first physical principles, have typically been unable to adequately simulate beach recovery; therefore their applicability over longer time scales related to sequences of storms, or seasons is limited. Despite intensive research, there still exists a range of relevant scales of practical importance to engineering design, for which conventional crossshore modeling approaches have proven unsuitable. The model described herein represents an attempt to fill this void using an innovative engineering scale approach. The new model combines empirical evidence with basic theory to produce a simple, yet effective, crossshore shoreline change model, applicable at the engineering scale. The primary objective is to create a robust, model capable of simulating shoreline changes over a variety of different time scales, under a variety of conditions, in an accurate and efficient manner, which is considered suitable for practical engineering applications. The model, is described in detail in the following chapter, and takes the form of a classic equilibrium equation as suggested by previous empirical studies of the shoreline response to variations in waves and water levels, where tides, storm surges, and wave induced setup have been included. Unlike many of the conventional approaches, this innovative model requires only readily available wave and water level information, and is extremely computationally efficient. Because of its simplicity, the new approach has many conceivable applications, providing a potentially crucial link between profile models which are unable to accurately reproduce the erosionrecovery sequence, and coastal area models which ignore crossshore processes completely. CHAPTER 3 MODEL DEVELOPMENT 3.1. Theoretical Background The shoreline model presented here differs from the conventional models discussed in the previous chapter which thus far have been unable to produce a robust, generally applicable shoreline change model based on crossshore processes. Instead, an innovative approach is proposed that uses empirical evidence based upon previous laboratory and numerical investigations of shoreline change, to guide the development of a simple new shoreline change model. Both small (Swart, 1974) and largescale laboratory experiments (as reported by Dette and Uliczka, 1987; Sunamura and Maruyama, 1987; and Larson and Kraus, 1989) have suggested that an initially plane beach subjected to steady erosional forcing in the form of a fixed elevated water level and constant wave action, will evolve towards an equilibrium state with an approximately exponential time scale. Numerical simulations performed by Kriebel and Dean (1985) and Larson and Kraus (1989) support these observations, suggesting that shoreline change can be modeled heuristically using an equation of the form, dy (t) t= k (y (t) y(t)) (3.1) dt where y(t) and yq(t) are the instantaneous actual and equilibrium shoreline positions at time t, and ka is an empirical rate coefficient. Equation 3.1 is a classical equilibrium equation, and implies that the rate of shoreline change is proportional to the degree of shoreline disequilibrium. In previous studies, analogous relations of this form have been used successfully to describe largescale coastal phenomena where detailed knowledge of the complex physical mechanisms producing the phenomena were lacking. Wright et al. (1985) suggested that the rate at which a beach transitioned between the various beach states in the morphodynamic classification scheme of Wright and Short (1984), could be described by an equation similar in form to Equation 3.1, st)= a + b ((t)) (S(t) S (t)) (3.2) where the term (S(t)Sq(t)) represents the disequilibrium between the actual and predicted beach state at any given time, and b((t))P is a rate parameter dependent on the nondimensional fall velocity parameter, Q(t). The combination of these two terms was described by the authors as representing the disequilibrium stress. Unfortunately, a lack of available data outside the stable region meant the empirical coefficients a, b, and p were left undetermined pending the collection of more field data. More recently, the equilibrium concept has been used by Plant et al. (1999), and Madsen and Plant (2001), to describe bar and beach slope evolution at Duck, NC. In the earlier study, Plant et al. showed that bar morphology, as represented by the time varying location of the bar crest, could be described by an equilibrium equation of the form, dX (t) = ( (t))'(X (t) X, (t)) (3.3) where X(t) and X,(t) are the time dependent actual and equilibrium bar positions, respectively, and a(H(t))3 is a parameter influencing the rate at which equilibrium is approached. By comparing the model to observations Plant et al. were able to show that this relatively simple model had significant predictive capability over periods of nearly a decade. Madsen and Plant used a very similar model to describe beach slope evolution, where the form of the model is identical to that of Plant et al., with X(t) and Xq(t), replaced by D(t) and qB(t), the time varying shore normal and equilibrium beach slopes, respectively. While Plant et al. found that the rate parameter was proportional to the wave height cubed, Madsen and Plant determined the corresponding dependence for the beach slope change was closer to (H(t))42, where the range of exponents produced similar, acceptable results. The Madsen and Plant model was found to explain between 30 and 40 percent of the observed beach slope changes at Duck. Kriebel and Dean (1993) used an analytical approach to solve Equation 3.1 for an idealized case. Sensitivity studies performed on the EDUNE model (Kriebel and Dean, 1985), indicated that the equilibrium shoreline response varied nearly linearly with changes in the water level, and that the water level did not affect the rate of shoreline response. This information allowed decomposition of the equilibrium response, yq(t), into a term associated with the magnitude of the response, Yq, and a unit amplitude function of time containing the temporal dependence, f(t). The simplified differential equation, d(t)= a(Yf (t) y(t)) (3.4) was then solved in terms of the convolution of the time dependent forcing and a characteristic solution for steady input conditions, I y(t) = aY, f (r)ea(T)d (3.5) 0 where T was a time lag. The analytical solution given in Equation 3.5 possesses several attractive characteristics that suggest the method is worthy of further study. The convolution solution indicates that antecedent conditions are important, and that the actual shoreline response occurring in nature will be damped and lagged with respect to the maximum or equilibrium state. The analytic model is also consistent with nature in that it predicts the maximum rate of shoreline recovery will occur immediately after the passage of a storm, and even provides an analytic justification for the different time scales associated with beach erosion and accretion. As discussed in the previous chapter, the analytic solution is limited however, due to its simplified nature and inability to handle complex, realistic forcing conditions. Even the simplified Equation 3.4, can only be solved analytically for a limited number of cases where the time dependence of the equilibrium response is known and can be represented by a simple analytical function. Although extensions of the analytical solution are possible, only a numerical approach will be able to provide a realistic representation of the time dependent forcing function, y q(t). 3.2. Defining the Equilibrium Shoreline, yq(t) Equilibrium beach profile methodology and a modified version of the Bruun (1962) rule, which considers increases in the local water surface elevation due to a combination of tide, storm surge, and wave induced setup, are used to calculate the equilibrium shoreline response, yq(t), for a given set of forcing conditions. Although the applicability of equilibrium concepts in the nearshore environment remains a controversial issue (see for example Thieler et al. (2000)), Bruun (1954) and Dean (1977) have illustrated the ability of a single empirical equilibrium beach profile relationship, h = Ay3, to adequately describe the nearshore bathymetry at numerous sites throughout the United States and Denmark. When applied to field data where conditions are constantly changing, the empirical relationship refers to a dynamic equilibrium state and average profile conditions. Dean even showed the equilibrium beach profile could also be derived analytically based on the assumption of uniform wave energy dissipation due to breaking waves through the surfzone. Subsequently, Moore (1982) and Dean (1991) were able to develop graphical, empirical relationships between the profile scale parameter, A, and sediment characteristics such as median diameter and fall velocity. The Bruun rule was originally developed to describe shoreline changes resulting from an increase in the local water surface elevation, S. If the assumption is made that the entire profile (not necessarily an equilibrium beach profile) shifts landward and upward without changing form with respect to the new water line, and that sediment volume is conserved, the resulting shoreline recession, Ay, is Ay = S ( (3.6) (h4 + B) where h. and W* are the vertical and horizontal extents of the active profile, and B is the berm height. This expression has proven adequate in the absence of waves; however previous studies have indicated that the most significant shoreline changes occur when increased water levels are accompanied by large waves. Figure 31 illustrates this modified situation, where the wave induced setup, q (y), alters the water surface elevation across the profile. The assumptions required in order to derive an analytical solution remain the same, namely the volume of sediment eroded from the foreshore equals that deposited offshore, and the form of the equilibrium profile remains unchanged with respect to the increased water level. In order to simplify the resulting expression, the common volume can be added to both sides of the conservation of volume equation, (B S (y))d+ A(y Ay)dy (3.7) = Ay"dy + r (S + (y))y which after integration simplifies to Ay 3h. Ay)3 =h.(3/5K) S 7b W. 5B W.) B (1K) B B(3.8) K 3x/8 1+ 3 /8 where K is the depth limited breaking coefficient. Equation 3.8 relates the non dimensional shoreline recession, Ay/W., to the dimensionless berm height, B/h*, storm surge, S/B, and wave setup, jdbB. In general, the non dimensional recession will be small and with K = 0.78, Equation 3.8 can be simplified even further, .0( .068H, (t)+ S(t) Ayeq(t) = W, (t).068H) (39) A B+1.28Hb Wt where Hb(t) is the breaking wave height, B is the berm height, and W.(t) is the width of the active surf zone. Here W.(t) is defined as the distance to the break point, such that it may be represented in terms of the breaking wave height, as W. = (HI/KA)3/2. The wave height and breaking index used in the derivation of the above equation assumes constant or average wave conditions. An alternate form of Equation 3.9 may also be derived for significant wave conditions, where K is taken as 0.5 and the coefficients in the numerator and denominator are replaced by 0.106 and 2.0, respectively. Although all of the quantities in Equation 3.9 exhibit some degree of temporal dependence, the berm height is taken as a constant, as information regarding its variability is sparse. In the field, where conditions are constantly changing, Equation 3.9 represents a theoretical condition which will hardly, if ever, be reached. The quantity Ayeq(t) calculated from Equation 3.9 gives the shoreline change from a stable or baseline condition; therefore in order to convert this time series of equilibrium shoreline change into a time series of equilibrium shoreline positions, this baseline condition must be identified. If the assumption is made a priori that the baseline condition corresponds to the average measured shoreline, the Ayq(t) calculated by Equation 3.9 are identically equal to the equilibrium shoreline positions (since the data are detrended). However in general this assumption is incorrect, as the baseline conditions for yq(t) and yob(t) are not necessarily the same. In fact, it has been argued (Wright, 1995) that the average shoreline position actually represents an average disequilibrium condition. In order to account for any potential offset in the baseline conditions, a constant calibration parameter, Ayo, is introduced which provides an additional degree of freedom. The role of Ayo is illustrated in Figure 32, where in the example provided, a shift of Ayo = 25 m is required to align the reference frames for yeq(t) and yb(t). The calibration routine discussed in Section 3.4.4 is used to determine the values of Ayo and ka for each simulation. Once the baseline condition has been identified, the equilibrium shoreline displacement, ycq(t), is given by y(t)Ay + Ayeq(t) (3.10) Figure 31. Beach recession due to a combination of an increased water level, S and wave induced setup, ?(y) (from Dean 1991). 29 E 40 I 20 0 MVyo = 25 20 Yob , y,, 40 0 100 200 300 400 500 600 700 800 900 1000 40Days E c 20 2 ba Fc s of 40 0 100 200 300 400 500 600 700 800 900 1000 Days Figure 32. Example illustrating the role of Ayo in correcting for differences in the baseline conditions of yq(t) and yob(t). 3.3. Defining the Rate Parameter, ka The coefficient ka, governing the rate of shoreline response in Equation 3.1, can either be taken as an empirically determined constant or parameterized to incorporate some measure of the local conditions. Both alternatives have been considered here. In the simplest case, ka is assumed to be a locally determined constant, where the subscript a is used to signify that ka may be double valued, with one value, ke, representing erosion and a second, ka, associated with accretion. For most situations, it is assumed that ke will be much larger than ka, as in nature the time scales of erosion are generally much shorter than those of accretion. Although convenient, this representation is perhaps overly simplified, as it is more logical to assume the shoreline response rate depends in some manner on the local wave and sediment properties. Two different approaches were considered to determine effective parameterizations of the rate function. In the first approach, the rate parameter was related to a measure of the local wave energy, while in the second, nondimensional parameterizations involving measures of both the wave and sediment properties, were considered. In both cases, the final form of ka(t) is given by ka(t) = kaf(t), where ka is the previously mentioned double valued coefficient, and f(t) is the time dependent parameterization. By parameterizing the rate function, the spread of the empirical coefficients is expected to reduce, as f(t) explicitly includes a measure of some of the important differences between the sites. The exact values of ka for each simulation are obtained by calibrating the model against historical data according to the procedure described in Section 3.4.4. Both Plant et al (1999) and Madsen and Plant (2001) based their rate coefficients on parameterizations of the local wave energy, adopting relationships of the form a(t)=a(H(t))P, where H(t) was a representative wave height and p was determined through an empirical fit to the data. Rather than explore an infinite range of possible values for p (p need not necessarily be an integer), a similar dependence is considered here, where the potential parameterizations are limited to ka(t) = kali(t)2 and lk(t) = kaHb(t)3. While Hb2(t) is obviously related to the wave energy, Hb3(t) can be thought of as approximating the wave energy flux into the surfzone which actually has an HbI5 dependence. One of the major disadvantages of the assumed wave energy relationship however, is that it contains no dependency on either sediment size, beach slope or wave period. In addition, in order for Equation 3.1 to remain dimensionally consistent, wave energy based parameterizations impose increasingly complex units of time'length1p on the empirical coefficients. Kraus et al. (1991) reviewed many of the nondimensional parameters often used to separate erosional and accretional conditions and found several, that when plotted together with the deep water wave steepness, Ho/Lo, were capable of differentiating between the two. Table 31 lists some of the more common beach change discriminators. Although some of the criteria listed in Table 31 were based upon consideration of profile type, bar bermm) profiles are generally considered to be representative of erosive accretivee) conditions, and here the assumption is made that these conditions will have an in kind impact on the shoreline. The various parameter combinations listed in the table typically incorporate measures of both the wave environment (Ho, T, Lo), as well as morphologic and sediment properties (d50, ws, tan 3). Although some of the parameters in Table 31 have thus far only been used to distinguish between the expected type of change, the assumption made here is that the magnitude of several of these parameters can potentially be related to the shoreline change rate through the parameter ka(t). One of the advantages of using nondimensional parameterizations is that the rate coefficient, ka, retains the units of inverse time, which is more appealing from a physical perspective, as the inverse of this coefficient can be interpreted as the time scale of the shoreline response (see Section 6.2 for a more complete discussion). Using Table 31 as guidance, numerous nondimensional parameterizations were considered; however the following five were deemed most appropriate: * Fall velocity parameter, Q(t) = H(t) Gourlay (1968), Dean (1973) w,T(t) * Froude number, F,(t) = Kraus et al. (1991), Dalrymple (1992) S FroudenumberF F() (t) * Inverse Froude number, IF,(t) = F,(t)' gH, (t) * Profile parameter, P(t) = gH(t) Dalrymple (1992) wT(t) HU (t) * Surf similarity parameter, (t) = ,(t) Battjes (1974) L (t)(tan s)2 where Hb is the breaking wave height, T the period, w, the sediment fall velocity, g gravity, tan B the local beach slope, and Lo the deep water wave length (LI=gT2I2n). Each parameter contains a description of both the wave (Hb, T, Lo) and sediment (either w, or tan 0) properties and has a sound physical interpretation. The fall velocity parameter has been used extensively in sediment transport and profile evolution studies (Dean 1973, Wright and Short 1984, Kraus et al. 1991), and provides a measure of the ratio of the amount of time a particle with a settling velocity ,ws, remains suspended relative to the wave period under a wave of height Hb. Under a breaking wave crest, the wave particle velocities are directed onshore, so if the particle manages to settle during the first half wave period the net displacement is onshore resulting in accretion. If the particle takes longer than half a wave period to settle, the net displacement is offshore and erosion occurs. Kraus et al. (1991) showed that Q(t) is related to the wave energy dissipation in the surfzone, and suggested that as the magnitude of the fall velocity parameter increases, so should the transport magnitude. This alternate derivation has a similar physical interpretation, where above some critical value of Q(t), sediment is suspended and transported seaward by the near bottom return flow, while below this value accretion will occur as wave asymmetry moves sediment onshore as bedload. Table 31. Established erosion/accretion criteria. Reference Parameters* Waters (1939), Johnson (1949) Rector (1954) Iwagaki & Noda (1962) Nayak (1970) Dean (1973), Kriebel et al. (1987) Sunamura & Horikawa (1975), Sunamura (1980) Hattori & Kawamata (1981) Wright & Short (1984) Larson & Kraus (1989) Kraus et al. (1991) Dalrymple (1992) MacMahan & Thieke (2000) Ho/Lo, Ho/dso Ho/Lo, HodsoS Ho/Lo, inw8/gT Ho/Lo, dso/Lo, tanl (Ho/Lo)tanp, w,/gT Hb/w.T Ho/Lo, Holw,T, rw,/gT Ho/Lo, Ho/wT, w/(gHo)5 gHo2/w83T, gHo2/w3T Cb, TH/H, 2nUb/Tg, Ub/uca, UblUbl 42/T2g Erosion : Ho/Lo > 0.025 Accretion : Ho/Lo < 0.025 Erosion : do/Lo < 0.0146(Ho/Lo)1.25 Accretion : do/Lo < 0.0146(Ho/L)1.25 Graphical Method Graphical Method Erosion: Ho/Lo > CI[nw,/gT] Accretion: Ho/Lo < C [nw./gT] Erosion: H/Lo > C2[tanp (dso/Lo)067] Accretion : Ho/L > C2[tan (dW/L)0 67] Erosion : tanp(Ho/Lo) > 0.5w,/gT Accretion : tanp(HJ/L) > 0.5w./gT Erosion : Hb/wT > 6 Accretion : Ht/w,T < 1 Erosion : Ho/L > Cs[(lew/gT)15] Accretion : Ho/Lo < Ca[(nw/gT)15] Erosion : Ho/Lo > C4[(Ho/wT)3] Accretion : Ho/Lo < C4[(Ho/wT)] Erosion: Ho/Lo > C5{[wJ(gHo)06} Accretion : Ho/Lo < Cs{[w/(gHo)6} Erosion : Ho/wT >C6[w2/gHo] Accretion : H/w,T > Cs[w,2/gHo] Erosion; gHo0/w,3T >10,000 Accretion : gHo2/w 3T < 10,000 Erosion : gHb2/wsT >~22,400 Accretion : gHb2/wT < 22,400 Graphical Method * H. = deepwater wave height, Lo = deep water wave length, do = median sediment size, S = sediment specific gravity, w = sediment fall velocity, g = gravity, T = wave period, tano = beach slope, Hb = breaking wave height, U = wave orbital velocity under a trough, u, = critical velocity required to initiate sediment motion according to Hallermeler (1980), ;b = surf similarity parameter, Ti, = water surface displacement at the wave trough, Ub = uniform seaward directed return flow, ub = near bed wave orbital velocity at breaking. "Cie refer to empirical constants that vary depending upon scale effects in the data, i.e. smalllarge scale or labfield. Adapted from: Larson, M. and Kraus, N.C., 1989. SBEACH: Numerical Model to Simulate StormInduced Beach Change. Technical Report No. 899, Coastal Engineering Research Center, U. S. Army Corps of Engineers, Vicksburg, MS. Criteria** The surf zone Froude number was used in combination with the deepwater wave steepness by both Kraus et al. (1991) and Dalrymple (1992) to distinguish between erosional and accretional conditions. As formulated, the surf zone Froude number is a ratio of competing forces, where Hb represents an upward suspending force, while w, and g are related to particle settling. Although the Froude number contains measures of both the wave and sediment properties, it is a potentially less accurate discriminator than the fall velocity parameter since it does not include the wave period, and thus the wave steepness, in its formulation. Kraus et al. presented two derivations where the Froude number was shown to be related to both the wave energy dissipation in the surfzone, and the power per unit volume expended by the waves via the bottom shear stress on suspending the sediment. The relationship between wave energy dissipation and the surfzone Froude number indicates that the magnitude of the shoreline response might be expected to vary with the magnitude of Fr(t). A rate parameter based upon an inverse Froude number, IFr(t), is also considered here in order to remain consistent with the expectation that larger waves will result in an increased erosion rate. The profile parameter derived by Dalrymple (1992) is essentially a rearrangement of the empirical relationship between H/L, and (tcws/gT)3a and Ho/L and (Ho/wT)3 presented in Kraus and Larson (1988) and Larson and Kraus (1989). By taking the ratio of each set of terms and canceling common factors, Dalrymple showed that a single parameter, P(t), was extremely effective at separating the erosional and accretional events described by Larson and Kraus (1989). Furthermore, he illustrated that the resulting parameter was composed of a combination of two of the nondimensional functions discussed earlier, namely P(t) = IFr(t)2Q(t). By extension of the results presented in Kraus et al. (1991), the magnitude of P(t) is also potentially related to the rate of shoreline change. Although first introduced by Irabarren and Nogales (1949), Battjes (1974) is generally credited with illustrating the ability of a single parameter, C(t), to describe a variety of surfzone characteristics. The surf similarity parameter, ,(t), has been related to breaker type, breaking index, runup, reflection coefficient, and beach type, all of which have the potential to strongly influence the shoreline change rate. The inverted form of the surf similarity parameter used here closely resembles the surf zone interference index used by Wang and Yang (1980), who interpret the parameter as the ratio of the natural swash period, to the period of the incoming waves. Using this interpretation, larger values correspond to an increasing degree of interference from successive waves, which manifests itself as a stronger return flow in the main water column resulting in enhanced offshore transport. Compared to the other parameterizations, the one glaring weakness of the surf similarity parameter is that it requires knowledge of the beach slope a priori. This is problematic in that even if the initial profile shape is known, there is no consensus as to which slope (mean? foreshore? active profile?) to use. Here, an average nearshore beach slope was used, where both the surf zone and subaerial beach up to and including the berm were considered. A total of eight different rate parameters were evaluated, one where ka was assumed to be a locally determined constant, two where ka(t) was assumed to scale with the wave height alone, and five where both sediment and wave properties were used to parameterize ka(t). The possibility that the same parameterization may not apply for both erosion and accretion has been addressed by considering separate parameterizations for ka(t) and ke(t) resulting in a total of 64 (8x8) possible rate parameter combinations. The quantities w, and tan 0 involved in some of the parameterizations have been considered constant, since information regarding the temporal variability of these properties is sparse. The fall velocity, ws, used in each of the proposed parameterizations is that calculated from the median sediment size using the Hallermeier (1981) relation, while the beach slope, tan 0, is typically determined based upon a visual analysis of several nearshore profiles. 3.4 Solution Technique 3.4.1 Numerical Scheme Equation 3.1 can be discretized using a semiimplicit, finite difference scheme according to kcAf/ t+i y7 l 2 (3.11) 1+ ka~t 2 where n is a time index. The unconditionally stable CrankNicholson scheme used provides order two accuracy along with computational efficiency, while the oscillatory nature of the forcing function yq(t) limits the buildup of numerical error, as errors tend to cancel rather than perpetually increase. The maximum response or equilibrium shoreline position, yq(t), is defined as that which would be attained if the forcing conditions were held constant indefinitely, and may be calculated from either observed or simulated data. In reality the equilibrium shoreline is a dynamic quantity, changing significantly with time scales on the order of hours; however over a single time step the forcing is assumed to remain constant. As with the analytical solution, the actual shoreline response will be lagged and damped with respect to the equilibrium shoreline. The particular time step used in model simulations varies with the temporal density of the input data, but is generally on the order of several hours. In all cases, the resolution of the forcing data is sufficient to capture the shortest (storm related) time scales intended to be reproduced by the model. 3.4.2 Forcing Data The model is forced by a combination of increased water levels due to tides, storm surge, and wave induced setup. Water levels used as input to the model have been obtained from tide gauges located near the sites of interest. In some cases, local tide information was unavailable or inadequate during the period of analysis and the nearest tide gauge with a complete record was used as a surrogate for the local water levels. Where necessary, tide factors (both in time and space) based upon a comparison of the local and surrogate data were applied in an effort to more closely match the local conditions. A visual comparison of regions of overlap in the records indicates that the application of tide factors improves the agreement between the local and surrogate tide records. Wave data were obtained from a combination of buoy measurements and statistical Wave Information Study (WIS) hindcasts made by the U.S. Army Corps of Engineers. Wherever possible an attempt was made to use buoy data; however in many cases a combination of data sources was needed to help eliminate significant gaps in the wave record. In these cases the secondary source was related to the primary source (both Hso and Tp) through a linear regression analysis performed on overlapping sections of the record. Directional information was added to nondirectional wave data sets by assuming all waves approached from the median deep water wave direction reported in the WIS statistical summaries. The median angle was adopted after preliminary results using wind direction as a proxy for wave direction proved unsatisfactory. These preliminary results agreed with observations made by Masselink and Pattiaratchi (2001) in Western Australia, where the wind and wave direction tended to become decoupled during the falling leg of a storm. Once the offshore wave conditions were determined, linear wave theory was used to convert the offshore conditions to breaking values for input to the shoreline change model. 3.43 Shoreline Data The individual shoreline data sets used to calibrate and evaluate the model are discussed in detail in Chapter 4; however some general information is provided here. An attempt was made to identify and utilize only data that exhibited characteristics likely to be modeled well by a purely crossshore model. Crossshore processes tend to be most important on long, straight, natural coastlines, and wherever possible adjacent shorelines have been compared with one another to assess the degree of longshore uniformity. Longshore averaged shorelines were used for model comparisons to help minimize the influence of smallscale spatial irregularities that the model is not designed to reproduce and which may have an impact on the perceived accuracy of simulations. Persistent longterm shoreline trends were assumed to be related to gradients in the longshore sediment transport rate and were removed prior to applying the model. While a portion of these removed trends may in fact be the result of crossshore processes (long term increases in wave energy, sea level rise), it has been assumed that they play a subservient role in comparison to the aforementioned longshore processes. The temporal resolution of the shoreline data sets varies, which allows the skill of the model to be evaluated over a range of time scales from daily to multidecadal. Yes An Initial range, R, Is selected for each parameter Iterations are performed using a user specified step (e.g. R/20) The value of the cost function, J is evaluated at each Iteration and compared to Jm Does Jn lie on an extreme of the range Ra? No Select k, k, and Ay used to calculate J.in Figure 33. Schematic of model calibration routine. 3.4.4 Model Calibration The completely specified model contains three empirical coefficients, ka, ke, and Ayo, which are evaluated based upon a comparison between model hindcasts and historical data. This is achieved by minimizing an objective or cost function, J, J (k.,ke,,y) = E(yo,(W y,,p(ka, k,Ay., t)) (3.12) where yob(t) and yp(ka,ke,Ayo,t) refer to the observed and predicted shorelines respectively. Several numerical procedures were considered to help identify the most appropriate values of ka, ke, and Ayo, where in the end, the simple numerical routine illustrated in Figure 33 was used to locate the minimum of J. In general, the error minimization procedure gives satisfactory results in terms of both computational time, as well as accuracy, with the infrequent exception of a few cases where local minima in the cost function are misidentified as the global minimum. This occurrence is rare, and while other numerical routines, specifically simulated annealing (Bohachevsky et al., 1986), were considered to attempt to correct this deficiency, none were satisfactory due to the poorly defined constraints on ka and ke. At present, the method illustrated in Figure 33 is deemed acceptable, as the misidentification of the global cost function minimum is extremely uncommon. 3.4.5 Model Evaluation Unfortunately, model evaluation remains as much an art as a scientific technique. Despite recent calls for the development of a set of standardized, nonsubjective model evaluation criteria, this is simply not feasible. It is impossible to evaluate a model in a completely objective sense and effectively consider all of the factors which have contributed to its success or failure. While objective measures of model performance are required to help quantify model skill, a number of factors including an appreciation of the model's objectives, an analysis of the quality of data used with the model, and a subjective interpretation of the quantitative measures are all required to accurately judge a model. Although objective measures may be useful in identifying problems, ultimately it is a subjective analysis of the objective criteria that identifies what the problem is and how to fix it. Three criteria, two objective and one subjective, have been used to help evaluate the new shoreline change model. Various quantitative measures including the normalized mean square error (NMSE), and several related criteria were used to objectively measure the prediction skill of the model. Since none of these are capable of evaluating the ability of the model to discriminate between erosional, accretional, and stable conditions, a separate objective categorical assessment procedure (CAP) is used for this purpose. Finally, a subjective measure of model performance, the Model Performance Index (MPI), was used to summarize all of the pertinent information, including both objective skill measures, in order to provide a single composite measure of model performance. Three of the more popular parameters used to evaluate model accuracy are the relative mean absolute error (RMAE), the normalized mean square error (NMSE), and the Brier Skills Score (BSS). The RMAE is defined as RMAE= 0, O < RMAE< w (3.13) t where means is the measurement error associated with yob. The RMAE has been used extensively by the European community (Sutherland et al., 2004, van Rijn et al., 2003) and is favored over the NMSE in part because it is less sensitive to outliers, as the difference term in the numerator is not squared as it is in the NMSE. One of the primary disadvantages of the RMAE however, is that the modulus appearing in the numerator makes the statistic nonanalytic and thus more difficult to work with than the NMSE. Plant et al. (2004) advocate the use of a skill measure based loosely upon the NMSE because it allows the computation of confidence limits. They define skill as [(Y yp2] 2 C w6 mW eas Skill = 1 0o > Skill > 1 (3.14) t where w is a weighting function, and yobs,t=o is the initial observation made at time t = 0. Although potentially appropriate for profile data, the skill as defined in Equation 3.14 is inadequate for shoreline studies, as the skill becomes a strong function of yobs,t= due to the significant shoreline changes that are routinely observed between successive surveys. One of the primary advantages of using this skill statistic is the capability of quantifying its significance; however this advantage is negated in shoreline studies, as even the most extensive shoreline data sets are too short (not enough degrees of freedom) for this to become meaningful. Sutherland and Soulsby (2003) and Sutherland et al. (2004) advocate the use of a similar skill measure for the evaluation of morphodynamic models. The Brier Skills Score (BSS) is defined as :(Yp, y.)2 BSS =1 ', oo< BSS <1 (3.15) Z (Yb Yob)2 t where yb refers to a baseline condition. The BSS is slightly preferable to the previous measure of skill in that the baseline condition is not specified a priori. In fact, if detrended shoreline data are used, and yb is specified as the mean of yob, the BSS reduces to the complement of the traditional NMSE, E(Y,r Yb)2 NMSE = Y  =1BSS, 0 < NMSE < (3.16) Careful consideration of the aforementioned measures of model skill reveals that each has its distinct strengths and weaknesses. Because of its relative simplicity and compatibility with the often utilized BSS, the NMSE was selected as the most appropriate tool for evaluating the new model. The NMSE has a direct physical interpretation as the ratio of the error variance to the measured shoreline variance, and unlike the RMAE it can be evaluated analytically. A perfect model in which the predictions exactly match the observations is characterized by a NMSE of zero. Although unbounded at its upper limit, errors on the order of one indicate model predictions with mean square deviations from the measured data approximately equal to the variance of the data. A subjective rating of the objective NMSE criterion is given in Table 33 In general, the NMSE is an extremely effective measure of model performance; however, since the difference term in the numerator is squared, it has the unfortunate property of being oversensitive to large deviations. The consequences of this are discussed in more detail in relation to model predictions in Chapters 5 and 6, along with several techniques that were used to attempt to overcome this shortcoming. None of the aforementioned measures of model skill are capable of evaluating the ability of the model to predict the correct type of shoreline change, i.e. erosion or accretion. This is illustrated clearly by considering two separate cases. In the first case, the correct type of change is predicted, but the magnitude is severely over predicted. In the second case, the wrong type is predicted, but the difference between yob and yp, is small. All of the measures discussed previously will assign less skill to the model that over predicts the magnitude of shoreline change even though the direction of change was predicted correctly. In order to assess the ability of the model to accurately distinguish between erosion and accretion, a separate categorical assessment technique was applied. Predictions and measurements were divided into three general categories: accretion, erosion, and no change (or stable), where a shoreline is defined as stable if the change in shoreline position between two successive surveys is less than five percent of the maximum range over the entire data set. This relative definition provides a useful sliding scale whereby energetic coastlines can undergo more significant changes and still be considered stable. A score from 01 is assigned to each possible combination of conditions as indicated in Table 32, where 1 represents a match between the prediction and observation, and 0 represents a complete mismatch. Since stable events are hardest to predict, they are assigned values reflecting the seriousness of the mismatch. For example miscasting an erosional period as stable has more serious potential consequences than miscasting an accretional period as stable hence the lower score (0.3 vs. 0.6). The categorical assessment procedure score, or CAP, is simply the average of all the individual scores, where higher values indicate more accurate predictions as reflected in Table 33. Table 32. Categorical assessment procedure score matrix developed for this study. Pr" ted Erosion Stable Accretion Measured ___________ Erosion 1.0 0.3 0.0 Stable 0.4 1.0 0.5 Accretion 0.0 0.6 1.0 Table 33. Subjective rating system based upon model performance statistics. Rating Range of Values NMSE CAP MPI Excellent <0.3 >0.8 5 Good 0.30.6 0.60.8 4 Reasonable 0.60.8 0.40.6 3 Poor 0.81.0 0.20.4 2 Bad >1.0 <0.2 1 The Model Performance Index (MPI) provides a holistic evaluation of model skill by incorporating the aforementioned objective measures of model performance with a subjective analysis of some of the more subtle aspects. The MPI takes into account many factors including: the quality and completeness of the input data, the characteristics of the modeled shoreline, the character of the NMSE (is it unduly influenced by a single data point), and the CAP. Higher expectations are placed on model performance when it is applied at sites with better data. For example, the availability of local tide data and measured directional wave information is expected to significantly improve the accuracy of predictions. The subjective analysis allows for the consideration of the character of the NMSE. Is it providing an accurate measure of model performance, or is a disproportionate amount of error resulting from a single outlier? These are important questions relating to the skill of the model that cannot be answered using purely objective methods. The MPI classification system ranges from 15 and is presented in Table 33. CHAPTER 4 FIELD DATA AND SITE SUITABILITY In order to accurately evaluate the model, data were collected from numerous sources encompassing a variety of geographical locations and typical beach conditions. Broadly, the field sites may be separated into three groups, one each representing the East and West coast of the United States, and a third representing Australia. Since the model only simulates shoreline changes due to crossshore processes an attempt was made to select data from long uninterrupted natural coastlines. The format of the available shoreline information ranges from beach width measurements obtained from aerial photography and video analysis, to several sets of complete profiles, some surveyed to depths of nearly twenty meters. The crossshore model is not intended to reproduce smallscale features with alongshore wavelengths less than several hundred meters; therefore generally alongshore averaged shorelines have been used. The varying temporal resolution of the available shoreline measurements is such that it allows the performance of the model to be evaluated over a number of time scales ranging from days to decades. Wherever possible actual wave and tide data were used; however statistical hindcasts were substituted for physical measurements where required. Summaries of the available data and relevant site characteristics are provided in Tables 41 and 42. With so many good, highquality data sets to choose from, it is essential to develop tools capable of identifying potentially inappropriate sites. The remainder of this chapter is devoted to providing a brief description of the collected data as well as 47 describing the techniques which were used to help select the most suitable locations for the evaluation of the proposed crossshore model. Table 41. Summary of data sources. Shoreline Latitude Longitude Orientation Orientation Wave Data Nearest Tide Shoreline Source Gauge ID Data East Coast Sites East Hampton, NY 40.93N Harvey Cedars, NJ 39.70N Island Beach, NJ 39.83N Wildwood, NJ 38.98N Duck, NC 36.18N St. Augustine, FL 29.95N Crescent Beach, FL 29.75N Daytona Beach, FL 29.17N New Smyma Beach, FL 28.88N West Coast Sites North Beach, WA 47.20N Grayland Plains, WA 46.80N Long Beach, WA 46.50N Clatsop Plains, OR 46.100N Torrey Pines, CA 32.87N Australian Sites Brighton Beach, WA 31.92S Leighton Beach, WA 32.08S Gold Coast, QLD 27.97S 72.20MW 74.1 3W 74.100W 74.80W 75.75OW 81.33W 81.25W 81.05W 80.93W 124.05W 124.05W 124.05W 124.05W 117.26W 115.750E 115.75E 153.42E 1510 1270 98 1420 680 750 75" 600 600 2700 2700 2700 270" 2700 2700 2700 900 Buoy44025 8531680 WISII79 (Sandy Hook) Buoy44025 8531680 WISII69 (Sandy Hook) Buoy44025 8531680 WISII69 (Sandy Hook) Buoy44009 8534720 WISII66 (Atlantic City) FRF Gauge WISII23 WISII23 WISII22 WISII22 Buoy46029 CDIP036 Buoy46029 CDIP036 Buoy46029 CDIP036 Buoy46029 CDIP036 WISSC002 Buoy38 Buoy38 Buoy23 Buoy13 8651370 (Duck) 8720220 (Mayport) 8720220 (Mayport) 8720220 (Mayport) 8720220 (Mayport) 9440910 (Willapa Bay) 9440910 (Willapa Bay) 9440910 (Willapa Bay) 9440910 (Willapa Bay) 9410660 (Los Angeles) Freemantle Freemantle 100035 (Gold Coast) Profiles Profiles Profiles Profiles Profiles Profiles & Aerials Profiles & Aerials Profiles & Aerials Profiles & Aerials Profiles Profiles Profiles Profiles Profiles Profiles Profiles Video " Direction is the approximate azimuth of the outward shoreline normal. b Data from the nearby Cape May tide gauge, ID# 8536110, was also used to fill in missing data. Table 42. Relevant site characteristics Approximate Median Sediment site Tal Range a Significant Grain Size Fal Vc\ Nearhore Bermnn Site Tidal Range Hb (m) Beach Slope Height (m) (m) (mm) (cm/s) East Coast Sites East Hampton, NY 1.14 1.24 0.375 4.66 2.45 Harvey Cedars, NJ 1.38 1.24 0.305 3.72 2.60 Island Beach, NJ 1.38 1.21 0.370 4.57 1:30 2.30 Wildwood, NJ 1.50 1.30 0.200 2.33 1:65 2.30 Duck, NC 1.12 1.13 0.200b 2.33 2.50 St. Augustine, FL 1.57 1.27 0.149 1.64 1:25 1.85 Crescent Beach, FL 1.40 1.27 0.139 1.56 1:40 1.85 Daytona Beach, FL 1.35 1.29 0.153 1.74 1:55 1.85 New Smyrna Beach, FL 1.25 1.29 0.138 1.55 1:55 1.85 West Coast Sites North Beach, WA 2.72 2.46 0.135 1.37 1:70 3.00 Grayland Plains, WA 2.72 2.46 0.178 2.05 1:45 3.00 Long Beach, WA 2.92 2.46 0.193 224 1:55 3.00 Clatsop Plains, OR 2.92 2.46 0.160 1.82 1:45 3.00 Torrey Pines, CA 1.62 1.76 0.194 2.27 1:50 2.40 Australian Sites Brighton Beach, WA 0.6c 1.10 0.574 7.92 1:15 2.00 Leighton Beach, WA 0.60 1.10 0.375 5.00 1:15 2.00 Gold Coast, QLD 1.50 1.25 0.290 3.51 1:20 2.20 * Defined as MHHWMLLW. b Calculated from the median grain size using the Hallermeier (1981) relation. Mean spring tide range. 4.1 East Coast Data In general terms, much of the Atlantic and Gulf coast of the United States is characterized by lowlying barrier island topography. Most of the coastline experiences lowmoderate wave energy, with larger waves occurring most frequently during the winter months from November to March, and in association with isolated tropical weather systems. In all, shoreline data from nine east coast sites spread amongst four different states was considered. Figure 41 shows the location of each site as well as the approximate location of nearby wave and tide gauges that were used in the analysis. DA185WD BUhC. XMW SNYA SIHI v w WS R SATIC Figure 41. Location of data sets from the East Coast of the United States. The northernmost site on the Atlantic Coast is East Hampton, NY. Located along the southern shore of Long Island, the coastline has been monitored extensively since at least 1979 by a variety of state agencies, with the current monitoring being performed by the Marine Sciences Research Center of Stony Brook University. Profiles collected on inconsistent intervals over the past 25 years have indicated that the East Hampton shoreline is relatively stable. The dominant mode of variability along this stretch of coastline is a strong annual fluctuation, which corresponds to a distinct seasonal pattern in the wave climate. The mean annual significant wave height recorded by NOAA buoy 44025 in 40 m of water is 1.2 m, although waves as large as 9.2 m have been recorded during the winter months. The tidal range at the site is approximately 1.1 m and contains a dominant semidiurnal component. The Richard J. Stockton College of New Jersey and the New Jersey Department of Environmental Protection maintain an extensive set of beach profile data for the State of New Jersey. Wading depth profiles were performed at over 100 sites along the Atlantic and Delaware Bay coastlines annually between 1986 and 1994, with the frequency increasing to biannually beginning in 1995. Shoreline data from three sites, Island Beach State Park, Harvey Cedars, and Wildwood were used in the current study. Conditions along the coast vary from location to location; however typical offshore wave heights as measured by NOAA buoy 44025 in 40 m of water range from 0.91.6 m, with a reported mean annual significant wave height of 1.2 m. Tides along the coast are mainly semidiurnal, with a range on the order of 1.4 m. Although occasionally impacted by hurricanes and tropical storms, the most significant threats to the New Jersey shoreline are large waves and storm surges produced by strong winter storms and northeasters. The Duck, NC shoreline data were extracted from profiles collected by U. S. Army Corps of Engineers Field Research Facility (FRF) staff, and form only a small subset of the available data along one of the most intensively studied coastlines in the world. Detailed profiles have been collected monthly (biweekly at four selected sites), along the onekilometer stretch of coastline since 1981. A variety of instruments simultaneously collect additional data ranging from wave heights and tide information to air temperature and wind direction. Despite a recognized seasonality in the wave climate in both height and direction, the dominant shoreline fluctuations at Duck occur with periods greater than one year (Plant and Holman, 1996; Miller and Dean, 2003). The mean annual significant wave height recorded at NOAA buoy 44014 in 47.5 m of water was 1.4 m, while the mean tidal range recorded by a gauge mounted on the FRF pier is 1.12 m. Despite the quantity and quality of the available data at Duck, the presence of the research pier significantly alters the nearshore environment, making it a potentially inappropriate site for the evaluation of a crossshore shoreline model. Bathymetric changes related to the disruption of the natural longshore sediment transport by the pier have been found to be particularly pronounced in the vicinity of the shoreline (Plant et al., 1999; Miller and Dean, 2003). Miller and Dean (2003, 2004) discussed a method for attempting to isolate the shoreline changes due to crossshore processes, by using a simple conservation of volume argument. According to the procedure, the change in sediment volume within a profile between two successive surveys, AV(t), is presumed to be the result of longshore processes. Under the assumptions that the profile translates without changing form and that the volume change is distributed evenly over the vertical dimension of the active profile, (h.+B), the shoreline change due to longshore processes, or the shoreline adjustment, Ay(t), can be obtained, Ay( t) = (4.1) ( h + B) Since the model only considers crossshore forcing, shorelines that have been adjusted by Ay(t) may potentially be more appropriate for evaluation purposes. Figure 42 illustrates the qualitative improvement in the consistency of shoreline changes between two profiles located on opposite sides of the Duck pier, suggesting that longshore effects are at least partially responsible for some of the initial nonuniform shoreline behavior. The same procedure was applied to the Torrey Pines profile data discussed in the next section; however the absence of any significant disturbances on the relatively straight, natural coastline resulted in only small shoreline adjustments. Interpolated Raw Shorelines, Duck, NC 60 Profile 2 4E 0I Profile 188 40 S 20 I . 20 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year Interpolated Adjusted Shorelines, Duck, NC 60 Profile 62 E  Profile188 t : : iI i " 20 20C I I I I I I I 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year Figure 42. Improvement in the consistency of the Duck shoreline data after adjusting for the volume change between subsequent profiles. The Florida shoreline data consist of a combination of beach profile data collected by the Florida Department of Environmental Protection Bureau of Beaches and Coastal Systems, and beach width measurements obtained through the analysis of aerial photographs from a variety of sources (Miller, 2001). Based on the availability of WIS hindcasts in the region and a lack of reliable buoy data, the duration of the analysis in this region was restricted to the period between 1956 and 1995 even though additional shoreline data exist. Shoreline measurements during this fortyyear interval were highly irregular due to the combination of data sources used but generally increased in frequency with time. All four Florida sites experience similar wave conditions characterized by a mean annual significant wave height on the order of 1.2 m as measured by NOAA Buoy 41009 in 42 m of water. The tidal range increases slightly from south to north along the coastline, with a mean value of 1.35 m reported at Daytona Beach, and a slightly larger value of 1.57 m reported for St. Augustine Beach. Typical threats to this coastline include the catastrophic effects of both hurricanes and severe winter storms. 4.2 West Coast Data In contrast with much of the Atlantic coast, compartmentalized beaches interrupted by numerous rocky headlands characterize a majority of the Pacific coastline of the United States. The wave climate ranges from moderate to severe, with the largest waves typically occurring in the Pacific Northwest. Global weather systems have a considerable impact along the Pacific coast, where El Nifio/La Nifia weather patterns significantly alter the typical conditions. The West Coast data set is composed of a total of five sites, one located in Southern California, with the rest concentrated around the WashingtonOregon border. The approximate location of each site, along with nearby wave and water level gauges is indicated in Figure 43 F WeS oasTZOn n BlWa 3nT T T=D Geat Figure 43. Location of available shoreline data along the west coast of the United States. An excellent set of shoreline data for the Pacific Northwest has been collected as a part of the Washington State Coastal Erosion Study, which is being conducted jointly by the United States Geological Survey and the Washington Department of Ecology. Profile data have been collected quarterly since 1997 (biannually prior to 1999) at more than fifty sites along a 160 km stretch of coast between Tillamook Head, Oregon and Point Grenville, Washington, the approximate boundaries of the Columbia River Littoral Cell (CRLC). The CRLC contains four subregions, separated by three inlets, known as the North Beach, Grayland Plains, Long Beach, and Clatsop Plains subcells. Each subcell consists of long (>30 km) straight sections of coastline bound by natural headlands and navigational entrances. Shorelines in the CRLC are subjected to a strongly seasonal, highenergy wave climate where offshore significant wave heights measured in 228 m of water average 2.3 m and increase to over 3.0 m during the winter months. Local tides are semidiurnal with an average range of approximately 2.8 m. In addition to storm and seasonal related changes, the shoreline also responds to El Nifio/La Nifia weather patterns, and longterm events including geologically frequent subduction zone earthquakes. The two years of profile data collected at Torrey Pines, California, by Nordstrom and Inman (1975), represents one of the first complete sets of nearshore bathymetric measurements ever obtained. Surveys were conducted to depths of nearly 18 m on a monthly basis along three shoreperpendicular transects between June 1972 and April 1974. Nearshore wave conditions along this section of the California coast vary significantly due to the extremely irregular offshore bathymetry, including numerous submarine canyons. The mean annual significant wave height reported by Scripps Buoy 101 in 549 m of water over the period 20012003 was approximately 1.1 m. The wave climate exhibits a distinct seasonal periodicity similar to the Pacific Northwest; however the range of variability (0.81.3 m) is reduced considerably. The local tide is semi diurnal with an average range of 1.6 m, and like most Pacific coast locations, has a large diurnal inequality. Shoreline changes in Southern California occur over a variety of time scales; however the short duration of the data set limits the current analysis to changes with periods of two years or less. Figure 44. Location of Australian shoreline data sets. 4.3 Australian Data A total of three data sets were obtained from Australia, with two sites located near Perth in Western Australia, and the third located near Brisbane along the east coast. Each of the three Australian sites provides a unique test for the model. The two west coast sites are composed of the coarsest sand and experience the smallest tides of any site studied. The unique aspect of the east coast site is related to the daily shoreline sampling interval, which allows the model to be evaluated over a wider range of timescales. The three Australian sites are depicted in Figure 44, along with the location of nearby wave and tide gauges. Masselink and Pattiaratchi (2001) collected shoreline data from a number of beaches in the Perth region of Western Australia. Although the focus of their study was on beach changes dominated by the seasonal reversal in longshore sediment transport, data were collected at several long, uninterrupted, natural beaches where the shoreline responded primarily to changes in the incident wave energy. The available shoreline data consisted of a combination of beach width measurements and nearshore profiles collected over weekly to biweekly intervals between November 1995 and November 1998. Perth beaches are sheltered by a series of submerged shore parallel sand ridges that significantly reduce the incident wave energy reaching the coast. The average mean significant wave height and peak period measured by a buoy inside the outermost sand banks in 17 m of water were 0.9 m and 10.1 s, respectively. The sheltering effect of the sand banks can be seen when the inshore wave height is compared to that observed at a buoy just offshore in 48 m of water, where H, averages 2.2 m. Perth experiences a primarily diurnal tide with a maximum spring tide range of only 0.6 m. Unique factors influencing beach changes in the region include passing weather systems that can overwhelm the local tide and extremely energetic alongshore seabreezes that can average up to 8 m/s during the summer (Masselink and Pattiaratchi, 2000). Shoreline position data from Narrowneck located at the northern end of the famed Gold Coast have been collected since mid1999 as part of the postconstruction monitoring program of the Northern Gold Coast Beach Protection Strategy. A beach nourishment was carried out between February 1999 and June 2000 in conjunction with the construction of an artificial surfing reef completed in December of 2000, just to the north of the site. An ARGUS video monitoring system (Holman et al., 1993) was set up to monitor the resulting shoreline changes with both a high temporal and spatial resolution. Daily shoreline measurements along the southernmost 1500 m of the site, where the effects of the nourishment and reef construction are least pronounced, have been extracted from video images using a technique described in Turner and Leyden (2000) and Turner (2003). The wave climate at Narrowneck as determined from a waverider buoy located offshore of the site in 16 m of water is characterized by a mean significant wave height of 1.1 m with an associated peak period of 9.2 s. Tides at the site are mainly semidiurnal with a maximum spring tide range of approximately 1.5 m. Beach changes at the site have typically been described as "event driven;" however recent work by Turner (2004) has identified a significant seasonal component. 4.4 Evaluation Tools Only recently has the selection of appropriate data for the evaluation of new long term models become an issue, as multiple highquality data sets simply did not exist. With the increased availability of good quality data, comes a new responsibility and challenge to choose the data sets most appropriate for the intended application. In order to do this, a variety of statistical techniques and data analysis tools can be applied to help reveal some of the relevant characteristics of the data sets. The nature of the proposed model is such that it is expected to reproduce only those changes related to crossshore processes; therefore longshore uniformity is an extremely important characteristic. By utilizing simple statistical techniques to analyze the shoreline data at several adjacent locations, the impact of the unpredictable (for this model), nonuniform shoreline movements can be quantified and compared to the more uniform, predictable, largescale changes. An appropriate data set will be one in which the ratio of the uniform behavior to the nonuniform behavior is high. 4.4.1 Time Domain Based Statistics Traditional time domain based statistics provide valuable insight into the relevant characteristics of a data set with minimal computational effort. The linear association between two data sets x and y is given by the correlation coefficient, rxy, r = n 1 r,5 1 (4.2) SxS where overbars denote mean quantities, and s, and Sy are the standard deviations. A correlation coefficient of 1 indicates perfect correlation (a negative indicates the variables change inversely to one another), while 0 indicates the lack of any linear relationship between x and y. If the spacing between values of x and y is constant, a lagged correlation coefficient can be calculated which may provide useful information as to the phase relationship of x and y (i.e. it is possible for x to either lag or lead y). If x and y are beach width measurements taken at adjacent locations along a uniform beach, a large positive correlation is expected. Although the correlation coefficient is useful for comparing two variables with each other, it is less useful for comparing large numbers of variables. For this purpose, the longshore uniformity index provides a meaningful, nondimensional measure of the degree of shoreline homogeneity. The twodimensional method applied here is a simplification of a threedimensional method used by Plant et al. (1999). The non dimensional longshore uniformity index, ILs, is simply the ratio of the longshore uniform 2 2 portion of the shoreline variance, shu2, to the total variance, sto, in the system. In the two dimensional case, the longshore uniform portion of the variance is essentially the temporal variance of the longshore averaged deviations from the time mean shoreline location, y'(x,t)= y(x,t)y,(x) (4.3) s = Yt'(t ,) (4.4) nt j=i where x and y represent the longshore and crossshore coordinates respectively, and the subscripts t and x refer to temporally and spatially averaged quantities. In Equation 4.3, the shoreline data are separated into the time mean component, yt(x), and a time and space dependent deviation from this mean, y'(x,t). The temporal variance of the alongshore average of these deviations defines the uniform component of the total shoreline variance. The longshore uniformity index, Ilsu, is then S2 =I, = (4.5) 4.4.2 Frequency Domain Based Statistics In some cases it is possible to look at the relationship between adjacent shoreline fluctuations in the frequency domain. If shoreline'data are available over constant intervals (or can be interpolated to constant intervals with out too much loss of accuracy) spectral analysis can be used to examine relationships between the various frequency components of the overall signal. In particular, the coherence (or squared coherence or coherency), Coh, indicates the degree of linear correlation between the various frequency components of two signals, while the phase, <(f), indicates the lag or lead. The coherence and phase are given by IG, (f f) 2 Coh(f) = (4.6) G (f)G,(f) (f tan (f (4.7) C, ( f ) where Gx and Gyy are the autospectra of x and y respectively, and Gxy is the cross spectrum of x and y, which is made up of both a real, Cxy (coincident), and an imaginary, Qxy (quadrature), part. In general, if x and y are two adjacent shoreline data sets a high coherence would be expected, particularly in the low frequency domain. Unfortunately, when dealing with relatively small data sets such as even the most comprehensive bathymetric data, spectral confidence limits become difficult to apply and are not presented here. 4.43 Method of Empirical Orthogonal Functions Although technically a time domain based statistic, the method of empirical orthogonal functions (EOFs) is discussed separately due to some unique characteristics. Pearson (1901) and Hotelling (1933) originally developed the method in the early 1900's as a means of extracting the dominant behavioral patterns from a set of data. Winant et al. (1975), Vincent et al. (1976), and Dolan et al. (1977) were among the first to apply the technique to geophysical data sets in the coastal environment. Although much of the subsequent work with EOFs has centered around applications of the method to profile data, it can also be used to analyze the longshore variations in a data set. Here they are used to examine the longshore variability of data sets consisting of beach width measurements. When applied in this manner, EOF's are able to extract the dominant modes of variability, which may correspond to either longshore uniform behavior, which can potentially be modeled with a crossshore model, or nonhomogenous behavior, which cannot. In simple terms, the EOF method exploits the properties of matrices to identify patterns of standing oscillations within the data. These patterns allow individual modes of variability to be analyzed separately, which can then used for a variety of purposes, among them simplifying the representation of the original data. The first step in an EOF analysis is to separate the spatial and temporal variability of the data by representing the original data set, y(x,t), as a series of linear combinations of functions of time and space, y(x,t)= c, (t)ek (x) (4.8) kI where ek(x) are referred to as the spatial eigenfunctions, and Ck(t) are referred to as either weighting functions or the temporal eigenfunctions. The summation is carried out from k = 1 to n, where n is the lesser of nx or nt, the number of spatial and temporal samples respectively. In some derivations, the temporal coefficient ck(t) is given as the product of a unit amplitude function of time ck(t) and a normalizing factor ak given by jnn,, where Xk is the eigenvalue associated with the kth eigenfunction. The requirement, Se, (x)e (x) = 8 (4.9) x where S. is the Kronecker delta, ensures that the eigenfunctions, ek(x), form a set of statistically independent, or uncorrelated vectors, which are normalized to unity. So far, the derivation has remained fairly general, and an infinite number of functions ek(x) may be specified that satisfy the conditions of Equations 4.8 and 4.9. What separates the EOF method from other series decomposition techniques such as Fourier analysis, is the fact 62 that the data are used to select the eigenfunctions rather than specifying them a priori. The selection is made such that the eigenfunctions bestfit the data in a least squares sense, with the first eigenfunction representing the bulk of the variability in the data set, and each subsequent eigenfunction accounting for the majority of the remaining variability. Mathematically, the Lagrange multiplier approach is used to formalize this requirement, resulting in a solvable eigenvalue problem, Aek(x)= ek(x) or AE=AE (4.10) where E is a matrix containing the spatial eigenfunctions, ek(x), and A is a diagonal matrix containing the eigenvalues, Xk. The matrix A represents some measure of the spatial covariability of the original data set y(x,t). Winant et al. (1975) defined A as a correlation or sum of squares and crossproducts matrix, while Aubrey and Ross (1985) utilized a demeaned version of y(x,t) and defined A as the covariance matrix. Either method is correct as long as the results are interpreted in the context of the frame of reference from which they were derived. Here A is taken as the correlation or sum of squares and crossproducts matrix, A= (YYT) [n,n.] (4.11) nxn, where the bracketed term, [n, nx], indicates the dimensions of A. Two methods exist for determining the temporal eigenfunctions, Ck(t). The first method is directly analogous to the technique set forth for calculating the spatial eigenfunctions and involves solving the set of equations, Bc (t)= (t) or BC=AC (4.12) B=  (YTY) [n,,nt (4.13) nfxn "r' Where A provided a measure of the spatial covariability within the data set, B measures the temporal covariabilty. Analogous to Equations 4.10 and 4.11, [nt, nd defines the dimensions of B, and C contains the temporal eigenfunctions ck(t). In comparing Equations 4.11 and 4.13, it should be obvious that in general, matrices A and B have different dimensions, and thus the A's calculated from Equations 4.10 and 4.12 also must have different dimensions. Although disconcerting at first glance, it can be proven that only the first k values of X are nontrivial (non zero), and in fact that the first k values from either equation will be identical. Alternatively, the temporal eigenfunctions may be calculated directly as ck t) y(x,, t)ek x) (4.14) ak i=1 where ak (=J nn, ) is the normalizing factor mentioned previously. Square matrices have many interesting properties, some of which can be exploited to help explain the physical significance of the calculated quantities. Given the definitions of A and B, the trace, or sum of the diagonal elements of both A and B is simply the mean square value of the data. From Equations 4.10 and 4.12 it can be shown that the sum of eigenvalues must equal the mean square value of the data, thus each individual eigenvalue, Xk, can be thought of as representing the relative contribution of mode k to the overall variability of the data set. The percent contribution of level k is given by S x100 (4.15) k As mentioned previously, the first few modes will contain the bulk of the variability, and the significance of each mode will decrease. In the present context, the usefulness of the EOF technique is rooted in the longshore variability of the primary mode, ei(x). Extrema in el(x) define regions of maximum variability, while nodes indicate regions of zero variability. With rare exception, nodes separate eroding and accreting regions; and therefore shoreline changes are said to be out of phase across nodes. The presence of nodal points usually indicates the influence of longshore processes, as they provide the mechanism for transferring sediment across the node from eroding to accreting regions. Here, multiple nodal points are used to identify those data sets which are not likely to be well represented by a crossshore model. In cases where some modes are longshore uniform and others are not, it is possible to use the EOF method as a means to filter out the nonuniform behavior by reconstructing the data set according to K y,(x,t) = ack(t)ek(x) (4.16) k=1 where k refers to the uniform modes, and yrK(X,t) refers to a reconstructed data set which retains only the longshore uniform information. 4.5 Site Suitability Although intuitively obvious, the fact that not every data set is appropriate for every application is rarely discussed. In the past there has been an overabundance of models calibrated and evaluated with whatever data was available, regardless of the synergy between the data and the model. A good example is the fact that data from Duck, has indiscriminately been used to validate many models regardless of whether the influence of the pier has been accounted for. Because of the abundance of good, high quality data sets available here, inappropriate data can be eliminated using some of the 65 methods discussed in section 4.4. Table 43 summarizes the methods used at each site as only certain techniques are applicable at each location. Table 43. Data analysis techniques applied at each site. Site Length Sampling Duration Interval Time Frequency EOF (km) Locations Domain Domain East Coast Sites East Hampton, NY 1.50 3 19791997 Variable Yes No No Harvey Cedars, NJ 9.00 3 19862002 Biannually Yes No No Island Beach, NJ 4.00 3 19862002 Biannually Yes No No Wildwood, NJ 3.50 2 19862002 Biannually Yes No No Duck, NC 1.00 20 19802002 Monthly Yes Yes Yes St. Augustine, FL 2.50 3 19551995 Variable Yes No No Crescent Beach, FL 3.25 2 19551996 Variable Yes No No Daytona Beach, FL 3.50 3 19551997 Variable Yes No No New Smyrna Beach, FL 2.75 3 19551998 Variable Yes No No West Coast Sites North Beach, WA 41.00 12 19982002 Quarterly Yes No Yes Grayland Plains, WA 17.00 8 19982003 Quarterly Yes No Yes Long Beach, WA 38.00 16 19982004 Quarterly Yes No Yes Clatsop Plains, OR 25.00 6 19982005 Quarterly Yes No Yes Torrey Pines, CA 1.00 3 19721974 Monthly Yes Yes No Australian Sites Brighton Beach, WA NA 1 19951998 Weekly Yes No No Leighton Beach, WA NA 1 19971998 Weekly Yes No No Gold Coast, QLD 1.50 300 20002003 Daily Yes Yes Yes In the time domain, the appropriateness of each data set was evaluated by the combination of the correlation coefficient, rxy, and the longshore uniformity index, I,. Large positive correlations are characteristic of the type of homogenous shoreline change desired for this particular application. At sites with fewer than three sampling points the correlation between each individual sampling location and each of the others was calculated according to Equation 4.2. In Table 44, both the mean and maximum correlation is reported, along with the 95% significance level as determined from a standard ttest (Davis, 1986). In most cases, the calculated correlation coefficient is larger than the 95% significance level, meaning the null hypothesis (that the correlations may have occurred by chance) can be rejected. Bold, italic values denote cases where insufficient evidence exists to reject the null hypothesis. Of the sites with less than three sampling locations, the East Hampton, Harvey Cedars, and Wildwood data sets appear to be nonuniform and hence inappropriate for the model. Table 44. Summary of time domain analysis results. Number of Maximum Average Significant ite Correlations Correlation Correlation Correlation li (95%) East Coast Sites East Hampton, NY 3 0.356 0.231 0.325 0.588 Harvey Cedars, NJ 3 0.211 0.072 0.404 0.325 Island Beach, NJ 3 0.677 0.564 0.433 0.608 Wildwood, NJ 2 0.238 0.238 0.413 0.629 Duck, NC 20 0.401 0.315 0.122 0.433& Adjusted Data 20 0.305 0.222 0.122 0.355b St. Augustine, FL 3 0.767 0.693 0.482 0.766 Crescent Beach, FL 2 0.709 0.709 0.532 0.928 Daytona Beach, FL 3 0.871 0.778 0.576 0.868 New Smyma Beach, FL 3 0.887 0.817 0.468 0.888 West Coast Sites North Beach, WA 12 0.643 0.491 0.468 0.4160 Grayland Plains, WA 8 0.530 0.385 0.482 0.1540 Long Beach, WA 16 0.606 0.489 0.482 0.597c Clatsop Plains, OR 6 0.623 0.497 0.482 0.4990 Torrey Pines, CA 3 0.869 0.775 0.288 0.827 Adjusted Data 3 0.777 0.715 0.288 0.834 Australian Sites Brighton Beach, WAd 1 0.519 0.519 0.279 0.796 Leighton Beach, WAd 1 0.519 0.519 0.279 0.796 Gold Coast, OLD 300 0.888 0.867 0.081 0.877 a Considering subsections on either side of the pier: lw =0.821 north of the pier, 6u =0.704 south of the pier. b Considering subsections on either side of the pier: lIu =0.586 north of the pier, lbu =0.597 south of the pier. When only the data from the center of the site which was used to calculate the longshore averaged shorelines is considered, I.u increases to at least 0.835. d For the purposes of the time domain analysis shoreline data from the two Western Australian sites were compared. Correlation Analysis Duck, NC (Adjusted) . ., .. . Mean r: =0.2634 Xy FRF ler Mean r = 0.412 ... Correlation with yx. Correlation with Yxgl. 1 0.5 . 0 0.5 1 1 0.8 Aj.6 0.4 0.2 1000 0.3049 0.2224 0.1221 0 200 400 600 800 1000 Longshore Distance (m) Figure 45. Calculation of the mean correlation profile including rm. and ravg. At sites with more than three sampling locations, the average and maximum correlation coefficients were calculated in a slightly different manner. As with the smaller data sets, Equation 4.2 was used to calculate the correlation between each individual data set and all of the others. The result is a symmetrical nx x nx matrix of correlation coefficients describing the covariability of the data. Each column of this matrix was then averaged to obtain a mean correlation coefficient for each sampling location. In Figure 45, an example is provided using the adjusted data set from Duck, NC. The upper panel shows correlation profiles for two specific profiles (yx=183 and y,=777), where the correlation of data set with itself is identically one. Similar profiles exist at each shoreline measurement location. The average correlation coefficient can then be calculated at each of the sites as was done for the two profiles in the upper panel 600 800 FRF 01W '_ _5% Signlficanoealenl~ II . = ' 'I I 0 200 400 . . . . . (rxy=0.263 and ry=0.241). In the lower panel, these average r.y values are plotted along with lines representing the mean and maximum values, and the 5% statistical significance level. The data reported in Table 44 are these average and maximum correlation coefficients. From the results of the correlation analysis of the larger data sets, the shorelines at both Grayland Plains, WA and Duck, NC behave nonuniformly and are therefore considered inappropriate for evaluating the proposed model. The longshore uniformity index, ILa, provides an additional useful measure of shoreline homogeneity. Larger values of Isu indicate stretches of coastline where cross shore processes are most likely dominant and the entire shoreline tends to translate in unison. Calculated values of the longshore uniformity index are presented in the last column of Table 44, where a value of 0.6 is used to identify potentially inappropriate data sets. Although the results suggest that all of the Washington and Oregon data are unsuitable, information gained from other analyses, particularly the EOF method, clearly show that the majority of the nonuniform behavior at these sites is due to isolated end effects. In fact when Iau is recalculated using the subset of data used to calculate the longshore averaged shoreline (taken at the center of each site), all the values increase to at least 0.835. Although in general the longshore uniformity index supports the results of the correlation analysis, there is one glaring discrepancy. While the correlation analysis suggests the Wildwood data set be eliminated, a longshore uniformity index above the cutoff criterion was calculated (0.629). Closer examination reveals that while the two shorelines comprising this data set generally move in unison with one another, a rather large discrepancy is observed in the data point collected immediately after the infamous "Perfect Storm" occurring in late October 1991. The profile located closest to the nearest down drift inlet exhibits significant accretion while the majority of the coastline experienced severe erosion. The hypothesis is that the unusual severity of the storm extended the typical region of influence associated with the inlet to encompass the southernmost profile location. Rather than eliminate a potentially good data set on the basis of a single inconsistent data point, the decision was made to retain the Wildwood site after eliminating the data set closest to the inlet. Spectral analysis, coherence, and phase were used to analyze the frequency domain behavior of those data sets that were either sampled at a constant interval, or were sampled frequently enough to allow the original data to be interpolated with an acceptable degree of accuracy. Spectra similar to those presented for the Gold Coast in Figure 46 were calculated for the Duck and Torrey Pines data as well. Figure 46 clearly shows that the periodic trends at each longshore location are consistent, as would be expected. Similar behavior is observed at Torrey Pines; however the Duck spectra are much more scattered indicating nonuniform behavior. The coherence and phase plotted in Figure 47 are the average values using three selected shorelines as the basis for comparison. As expected for uniform shorelines, the coherence is high and the phase oscillates slightly about zero in the highenergy region (f = 0.0010.04 cycles/day). Similar behavior is exhibited at Torrey Pines; however low coherence values and wildly fluctuating phase estimates once again illustrate the inconsistent behavior of adjacent shorelines at Duck. In general, the conclusions of the frequency domain based analysis are consistent with the previous results, and support the assertion that Duck is an inappropriate location for the evaluation of the proposed model. Shoreline Change Spectra Gold Coast, QLD f (cycles/day) Figure 46. Spectral analyses of Gold Coast shoreline data where the thick line represents the mean spectra. Mean Coherence & Phase Gold Coast, QLD 0.8 i 0.6 ~0.4 :E Coh & with Yx17so e Coh & O with y125 S Coh & with y7 03 1/yr 2/yr 10 f (cycles/day) Figure 47. Mean coherence and phase for three selected shorelines at the Gold Coast. _ It~P_~1P~4P~L;rd~a~pl~ The EOF method was used to analyze the dominant spatial and temporal modes of variability at sites with multiple (>3) longshore sampling locations. The dominant modes of spatial variability are of primary interest here, and are plotted in Figures 48 to 410. A summary of the nature and the amount of variability explained by the first two modes is given in Table 45. In every case, the first two modes account for nearly 90% of the total variability of the data, where the majority of modes can be classified as uniform. Nonuniform behavior is indicated by the presence of nodal points in Figures 48 to 410, where they tend to represent transition points that separate eroding and accreting regions. Typically, shoreline changes are referred to as out of phase across the nodes. The absence of nodal points in a given mode is a reflection of longshore uniform behavior, as the entire coastline tends to advance and retreat in unison. In terms of the principal modes of transport, crossshore processes tend to dominate on uniform coastlines, while longshore processes provide the primary mechanism for transporting sand across nodal points from eroding to accreting regions. As indicated in Table 45, most of the primary eigenfunctions, el(x), exhibit some form of uniform behavior. Inlets and rocky headlands tend to have a significant impact on the Washington data as pronounced "end effects" are present. Uniform regions are most difficult to identify at Duck and Grayland Plains. At most sites, the second and third spatial eigenfunctions, e2(x) and es(x), begin to describe the deviations from the dominant longshore uniform behavior described by ei(x). Based upon the form of the primary and secondary eigenfunctions and their relative importance, the EOF analysis suggests that potentially neither the Duck nor the Grayland Plains data are appropriate for the evaluating a crossshore model. First Three Spatial Eigenfunctions Duck, NC 0 200 400 Longshore C First Three Spatial Eigenfun 0.5 ... 00 0 200 400 Longshore D Figure 48. Spatial eigenfunctions ei(x)e3(x) f _1 First Three Spatial Eigenfunctions Washington State e,(x) 76.41% e e2(x) 16.61% 0.5 e3(x) 3.71% .... ... . 0.5 .... .. ......... ... Clatsop Plains 1 L nahoeDstacekin 5 10 15 20 25 30 35 Longshore Distance (km) Long Beach 0 .5................ ....... e,(x)77.09% 0.5 e ex) 14.34 .... e3(x) 5.63% 1  40 50 60 70 80 Longshore Distance (km) 0.5 ....... .......... ... .... ... : e,(x) .n % .t :/ .y. e e~(x) 7.03% ...... rayla i Plain e3(x) 2.93% 1 1  95 100 105 110 115 120 130 140 150 160 Longshore Distance (km) Longshore Distance (km) Figure 49. Spatial eigenfunctions el(x)e3(x) for Washington State. FRF Per N .. ... ... .. ...... ... .... .. .. ...... . S....... ...... e (x) 81.98% Se e(x) 7.45% Se(x) 3.63% 600 800 1000 1200 stance (m) actions Duck, NC (Adjusted) FRF Pier VOi ......... .... e,(x)55.25% e e2(x) 16.37% 1 .. e (x) 9.68% 600 800 1000 1200 distance (m) or Duck, NC. 200 0 I I I I I I ........................................ 0.5 73 First Three Spatial Eigenfunctions Gold Coast, QLD 0.1 0 0 1 .. ...... .. ... ... ......... ............... ... ..... ............ ...... 0.1  0.15 e1(x) 88.6% e e(x) 2.58% 9e3(x) 2.22% noII 2000 1500 1000 Longshore Distance (m) Figure 410. Spatial eigenfunctions ez(x)e3(x) for the Gold Coast, QLD. Table 45. Summary of EOF analysis results. e (x) e2(x) Site % Variance m % Variance Fo Explained Form Explain Form Explained Explained e2.n(x) % Variance Remaining East Coast Sites Duck, NC 81.98% Uniform 7.45% Variable 10.57% Duck, NC (adjusted) 55.25% Variable 16.37% Variable 28.38% West Coast Sites North Beach, WA 82.11% Uniform 7.03% Uniform 10.86% Grayland Plains, WA 90.09% Variable 8.89% Uniform 1.02% Long Beach, WA 77.09% Uniform 14.34% Uniform 8.57% Clatsop Plains, OR 76.41% Uniform 16.61% Uniform 6.98% Australian Site Gold Coast, QLD 88.59% Uniform 2.58% Variable 8.83% As a result of the site suitability analysis performed using the tools described in Section 4.4, several data sets have been determined to be inappropriate for evaluating the new model. The crossshore nature of the model makes longshore uniformity an N important characteristic of suitable sites. A combination of inconsistent survey data along with poor performance in the correlation analysis resulted in the elimination of the East Hampton data. The Harvey Cedars site was included primarily as a check of the site evaluation technique since its location within a groin field makes it extremely unsuitable for analysis with a crossshore model. Surprisingly, the majority of the Washington and Oregon State data exhibits enough longshore uniform behavior to be considered suitable, with the lone exception of the Grayland Plains data set. A low average correlation coefficient, and an extremely small longshore uniformity index, combined with the EOF analysis that failed to identify any significant longshore uniform behavior, all support the decision to eliminate the Grayland Plains site. Last but not least is the revered Duck data set. The significant impact of the pier on the adjacent shorelines has long been recognized, but often ignored when selecting appropriate data for the evaluation of new models. The Duck data performed poorly in all of the suitability tests illustrating its inappropriateness for the proposed application. Fortunately, the number of high quality, readily accessible data sets is constantly increasing, providing plenty of more suitable alternatives. CHAPTER 5 RESULTS At most of the selected sites, the model is able to reproduce the historical shoreline changes with a degree of accuracy that is on par with or better than most traditional approaches, but at a fraction of the computational cost and effort. A total of 64 simulations representing all possible rate parameter combinations were performed at each of the 13 sites, for a total of 832 separate hindcasts. As expected, the results varied from site to site, and even at a given location depending upon the parameterizations selected for ka(t) and ke(t). A succinct summary of the results is provided in Table 51, where the columns from left to right represent: the average NMSE and classification (from Table 3.2), the minimum NMSE and classification, the mean CAP and classification, the maximum CAP and classification, and the MPI and associated classification. The column averages given in the last row indicate that overall the model is successful according to all three criteria, particularly when only the best simulations (columns 4 & 8) corresponding to the most suitable rate parameters are considered. There are however, some cases where the model does not perform nearly as well (Island Beach, NJ for example), although at least in some cases this poor performance can be partially explained by the unexpected, and somewhat anomalous behavior of the observed shoreline. A complete tabulation of the results at each of the thirteen sites is presented in Appendix B. The remainder of this chapter is devoted to a description of the typical model performance using the Daytona Beach, FL data, examples of the types of results presented in Appendix B using the Long Beach, WA data, and a general description of the results in New Jersey, Florida, Washington, California, and Australia. Table 51. Summary of SLMOD results. NMSE CAP MPI Site Mean Minimum Mean Maximum Score Rating Island Beach, NJ 0.932 P* 0.885 P 0.632 G 0.705 G 1 B Wildwood, NJ 0.686 R 0.596 G 0.659 G 0.779 G 3 R St. Augustine, FL 0.782 R 0.668 R 0.640 E 0.805 E 4 G Crescent Beach, FL 0.849 P 0.259 E 0.663 G 0.835 E 2(5) PE Daytona Beach, FL 0.703 R 0.619 R 0.748 G 0.841 E 4 G New Smyrna Beach, FL 0.765 R 0.595 G 0.684 G 0.800 E 4 G North Beach, WA 0.628 R 0.537 R 0.828 E 0.917 E 4 G Long Beach, WA 0.363 G 0.281 E 0.873 E 0.926 E 5 E Clatsop Plains, OR 0.423 G 0.312 G 0.902 E 0.974 E 5 E Torrey Pines, CA 0.745 R 0.48 G 0.596 G 0.779 G 4 G Brighton Beach, AS 0.615 R 0.524 G 0.647 G 0.656 G 3 R Leighton Beach, AS 0.624 R 0.522 G 0.655 G 0.680 G 3 R Gold Coast, AS 0.521 G 0.470 G 0.715 G 0.718 G 4 G Gold Coast (filt), AS 0.367 G 0.298 G 0.985 G 0.987 G 4 G Average 0.643 R 0.503 G 0.731 G 0.814 E 3.79 GE Classification according to Table 33, where B=Bad, P=Poor, R=Reasonable, G=Good, E=Excellent The unconditionally stable nature of the numerical technique employed in the model allows for simulations to be performed with an arbitrary time step. Since one of the objectives of the model is to encompass as much of the broad engineering scale as possible, the shortest time step, corresponding to the temporal density of the input data, is used. Depending upon the source of the wave data, this time step varies, but is generally on the order of 13 hours. This is short enough to capture the smallest scale intended to be reproduced by the model corresponding to storms, and also allows for realistic simulations to be completed in a reasonable amount of time. For clarity, only predictions for those days where a corresponding measured data point was recorded are plotted in the figures appearing in Appendix B; however for each simulation a complete time series of equilibrium and hindcast shoreline positions are calculated at each time step. An example of these time series is plotted in Figure 51, where dissecting the figure helps to illustrate the typical model behavior. The specific simulation presented corresponds to the hindcast with the lowest NMSE at Daytona Beach, FL, where the accretion rate parameter is a function of the breaking wave height squared, namely ka(t) = 1.08x10 4H2(t) hr1, and the erosion rate parameter is a function of the Froude number, specifically ke(t) = 0.45Fr(t) hr'1. All three plotted time series contain dominant seasonal signals, corresponding to strong annual periodicities in both the wave and water level forcing. The noise or variability in yq(t) reflects the fact that the equilibrium shoreline represents a complete and instantaneous response to the forcing, which includes high frequency phenomena such as the semidiurnal tidal signal. The predicted shoreline on the other hand, responds with a much longer timescale that is highly dependent on the value of ka. The seasonal trend predicted for Daytona Beach is generally consistent with previous observations that have been made along the Florida coastline (DeWall, 1977). The inset chart in Figure 51, provides a close up view of the typical annual cycle that occurs in nature and is well predicted by the model. The inset plot shows an initially eroded winter shoreline, which recovers gradually over the spring and summer months, only to be eroded by a succession of storms, in this case Hurricane Diana (9/8/1984 9/16/1984) and the Thanksgiving Day Storm (11/22/198411/25/1984). The significant difference between the predicted erosion and accretion time scales, exhibited in nearly all of the simulations is characteristic of the natural response, and is particularly pronounced in Figure 51. Although it is somewhat difficult to tell based upon Figure 51 alone, a more complete analysis including an evaluation of the NMSE, CAP, and MPI indicates that the model performs reasonably well at Daytona Beach. Shoreline Hindcast Time Series Daytona Beach, FL 80 y (t) eq Y y (t) 60 pr Mt x ox Y(t 40  .G 20 20 N (N.S T 1 , C 4040 Hu DIan 60 O ; 4 . 20 1964) 80 ...... ........... ":40 100 0 1984 1984.5 1985 1960 1965 1970 1975 1980 1985 1990 1995 Date Figure 51. Complete hindcast shoreline time series for Daytona Beach, FL. The figures and tables presented in Appendix B are designed to provide a concise yet informative description of the model results at each site. A typical hindcast generated for the Long Beach, WA site is shown in Figure 52, where only those model predictions corresponding to the dates for which measured data were available have been plotted. The two hindcasts in each figure represent those generated using the parameter combinations resulting in the best predictions as selected objectively using the NMSE and CAP criteria. In some cases, both criteria suggest the same parameter combination and the two hindcasts overlie one another. The error bars appearing in Figure 52 are used to indicate the range of shoreline predictions produced by alternate parameter combinations. In other words, all shoreline predictions for a given time fall within the bounds defined by the error bars. Although the model is sensitive to the particular form of the rate parameter, the error bands typically show that regardless of the parameterization, the predictions fall within an acceptable range of one another. Long Beach, Washington 20 15 10. ... .. 25 ................ C.......... ....... CAP) 0 5 ............. : ....... 05 .. .... ......... _Il I" !II 30' ' 1997 1998 1999 2000 2001 2002 2003 WA, and typical of those presented in Appendix B. Error bars indicate the variation in predicted shoreline position for different rate parameter combinations. Accompanying each figure is a series of three tables similar to Tables 52 to 54. In Table 52, the NMSE for each possible parameter combination at a given site is presented, where each column corresponds to a different form of ke(t), while each row represents a different form of ka(t). In addition, column and row averages have been included, which give the mean NMSE for the various forms of ka(t) and ke(t), respectively. Two separate criteria were used to select the "best" rate parameters at a given site. The first criterion is based upon the average performance of the model for each form of ka(t) and ke(t) taken over all eight forms of the opposing parameter. Gray shading has been used to identify the "best" parameters according to the first NMSE criterion. The second criterion uses the best individual simulation (lowest overall NMSE), identified by the bold outlined value, to select the most appropriate parameter combination. There is a subtle difference in the two criteria, in that the first criterion identifies the best rate parameters, ka(t) and ke(t) independently of one another, while the second criterion identifies a parameter combination. In the example given in Table 52, the most appropriate individual parameterizations are given by ka(t) = kaFr(t), and ke(t) = ke4b(t), while the best parameter combination also happens to correspond to ka(t) = kaFr(t), ke(t) = keb(t). The fact that both methods suggest the same parameter set is not surprising since the criteria are related; however a quick glance at the results in Appendix B will confirm that this is frequently not the case. Table 52. NMSE associated with various rate parameter combinations at Long Beach, WA, and typical of the NMSE tables presented in Appendix B. Erosion Parameter ke(t) = Con f(Q) f(Hb2) f(Hb3) f(Fr) f(IF,) f(P) Avg Con 0.307 0.311 0.335 0.338 0.308 0.314 0.321 0.316 S f(Q) 0.332 0.330 0.383 0.369 0.337 0.341 0.383 0.349 f(Hb2) 0.390 0.400 0.441 0.471 0.375 0.400 0.426 0.411 SfHb) 0.436 0.463 0.493 0.486 0.426 0.455 0.493 0.463 c f(IFr) 0.322 0.333 0.355 0.365 0.320 0.337 0.343 0.336 f f(b) 0.327 0.322 0.350 0.348 0.319 0.333 0.333 0.330 f(P) 0.396 0.390 0.426 0.445 0.368 0.392 0.413 0.401 Avg 0.351 0.355 0.388 0.393 0.343 0.360 0.377 0.363 Table 53 is an example of the second type of table generated for each site, where the format is very similar to that of Table 52, but rather than containing NMSE values, it contains CAP values. Once again row and column averages have been calculated and have been used to help select the "best" parameters according to the first criterion, which in the CAP tables are shaded with a diagonal striped pattern to distinguish them from the previous table. Using the CAP criterion, the "best" rate parameters at Long Beach are given by ka(t) = kalb2(t) and kI(t) = kcIFr(t). Similar to the NMSE table, the maximum CAP for an individual simulation is outlined (only italicized rather than boldfaced) in order to indicate the "best" parameter combination based on the second CAP criterion. Because the CAP value is essentially an average of a finite number of specified weights, it is not unusual for multiple hindcasts to receive the same CAP score. This is the case at Long Beach, as multiple parameter combinations result in the same maximum CAP score of 0.926, indicating several simulations that perform equally well according to the second criterion. It should be noted that several of these combinations do not correspond to either of the optimal rate parameters based upon the row and column average CAP. Table 53. CAP associated with various rate parameter combinations at Long Beach, WA, and typical of the CAP tables presented in Appendix B. Erosion Parameter k, = Con f(Q) f((Hb) f(Hb3) f(Fr) f() f(P) Avg Con 0.858 0.858 0.853 0.853 0.858 0.858 0.853 0.856 f(Q) 0.895 0.890 0.853 0.853 0.895 0.890 0.853 0.882 F(Hb) 0.926 0.874 0.868 0.816 0.926 0.874 0.853 0.883 S f(Fr) 0.858 0.858 0.821 0.853 0.858 0.858 0.821 0.848 f(IF,) 0.890 0.853 0.853 0.853 0.890 0.890 0.853 0.871 Sf(&) 0.895 0.858 0.858 0.890 0.895 0.821 0.890 0.875 f(P) 0.911 0.911 0.853 0.816 0.890 0.911 0.874 0.882 Avg 0.893 0.874 0.854 0.846 0.890 0.874 0.859 0.873 The last type of table created for each site is illustrated by Table 54, and gives the specific coefficient values for each simulation as determined by the procedure discussed in Section 3.4.4. Once again, the columns represent different forms of ke(t), while the rows indicate different forms of ka(t). The set of coefficients, ka, k1, Ayo, for each parameter combination are listed vertically. Row and column averages are also calculated, where ka and ke can only be averaged horizontally and vertically, respectively, as different forms of the rate coefficients have different units. The shading patterns used in the NMSE and CAP tables to indicate the best parameterizations of ka(t) and ke(t) are repeated in Table 54, in order to make cross referencing the tables easier. In the example presented, the row and column corresponding to ka(t) = kaFr(t) and ke(t) = ke b(t) are shaded, indicating the best parameters according to the NMSE, while the optimum parameterization based on the CAP score, ka(t) = kaHb2(t), ke(t) = keFr(t), is represented by the striped row and column. The individual cells resulting in the best simulations according to the second criterion are also identified in Table 54, where the bold outlined text denotes the best individual simulation based upon the NMSE, and the italicized outlined text indicates the simulations receiving the maximum CAP score. At Long Beach, the best individual simulation according to the NMSE occurs for the specific parameter combination, ka(t) = 0.05Fr(t) hri, ke(t) = 0.000003(b(t) hr"', Ayo = 24.38 m, while several parameter combinations result in the same maximum CAP value. While the availability of numerous sets of field data has advantages in terms of allowing the model to be evaluated over a wide range of conditions, it makes the presentation of the results, even in a compact form, quite cumbersome. In order to streamline the process, the remainder of this chapter is devoted to qualitative descriptions 83 of the overall model performance in each of five geographic regions: New Jersey, Florida, Washington, California, and Australia. Where necessary, figures or tables such as those previously discussed are presented within the text to illustrate specific points; however the complete set of figures and tables for all thirteen sites can be found in Appendix B. Table 54. Calibration coefficients for Long Beach, WA, and typical of the coefficient tables presented in Appendix B. Con ka [hr1] k. Av. f(Q) [hr"] Con [hr1il 3.0E04 5.0E04 24.384 3.0E05 5.5E04 27.432 f(Q) [hr'1 4.5E04 2.5E05 12.192 4.0E05 3.0E05 f(Hb2) hr'1m'21 5.0E04 1.6E05 6.096 1.0E03 1.6E05 f(Hb3) Ihr1m4 4.5E04 2.5E06 3.048 4.5E05 2.8E06 f(Fr) [hr'1 3.5E04 1.5E01 24.384 3.0E05 1.5E01 f(P) 4.5E04 2.5E10 0.144 1.OE03 2.5E10 Avg 3.9E04 15.240 2.8E04 3.4E05 4.6E04 5.5E10 IiR An 9A nCn It fH 3)k, 1.8E05 2.1E05 5.3E05 3.5E0311.8E05 [h(1&] k. 1.5E03 7.OE05 3.8E05 5.3E06 5.OE01 A 42.672 33.528 6.096 36.576 45.720 00 ka 1.5E06 1.5E06 2.5E06 2.OE06 1.5E06 k 6 E [hr'] ] Av. .u 4 t 27.432 J3.Uc0u 21.336 I .6O96UO 6.096 2., Eu0 6096 2.UEU I 30.480 k, 2.5E06 2.0E06 3.OE06 3.OE06 2.5E06 [h( ke 5.5E04 3.OE05 1.6E05 2.7E06 1.5E01 AyI 27.432 27.432 12.192 9.144 27.432 ka 1.0E08 1.0E.OE 1.0E08 1.0E08 6.OE10 [hr ] ke 9.0E04 4.5E05 2.2E05 3.2E06 3.0E01 AYo 27.432 21.336 12.192 6.096 39.624 ka Avg k. Ayo 1.9E03 29.337 3.8E05 22.479 2.SE06 2.5E10 6.096 4.5E06 2.5E10 6.096 1.5E09 3.SE10 6.096 3.1E10 7.239 2.1E05 7.1E06 2.4E01 8.001 0.381 33.147 1.8E06  18.669 2.8E06 19.431 6.6E09 20.193 19.574 I I I I 1 I I I , , I I I . . 5.1 New Jersey The New Jersey region consists of two different sites, Island Beach along the central New Jersey coastline, and Wildwood located in the southern portion of the state. Hydrodynamic conditions at both sites are similar however, both the shoreline orientation (Ono = 98 Island Beach, non = 142 Wildwood) and sediment size (dso = 0.370 mm Island Beach, d50 = 0.200 mm Wildwood) vary considerably. Model performance also varies significantly between the two sites, performing poorly at Island Beach, but reasonably well at Wildwood. While the model successfully simulates a majority of the large shoreline changes at Wildwood, including an extreme erosional event in 1991 related to the "Perfect Storm", it fails to reproduce most of the changes at Island Beach. Closer examination of the data at Island Beach however, reveals that the measured shoreline exhibits some characteristics that would make simulating it with any model extremely difficult. Given the similarities in the forcing conditions for both New Jersey simulations, it is reasonable to assume that if the shoreline observations at each site were fairly consistent, the results should be as well. Unfortunately, this is not the case, mainly due to two glaring differences involving the magnitude of the erosion related to the "Perfect Storm", and the sudden increase in the severity of shoreline changes experienced at Island Beach after 1995. The infamous "Perfect Storm" occurring in late October of 1991 impacted a majority of the east coast of the United States, causing widespread damage and significant beach erosion. According to the data points spanning this event, the storm resulted in relatively minor erosion at Island Beach (~ 5 m), which was not well predicted by the model, and significant erosion at Wildwood (~ 60 m ), which was successfully predicted. Since the forcing data for both hindcasts are similar, it is not surprising that the model was only able to reproduce one of the two scenarios, and that the result that was reproduced corresponded to the severe erosion that was more typical of the situation along the majority of the east coast. The sudden sharp increase in the magnitude of shoreline changes after 1995 at Island Beach is also considered somewhat abnormal. Neither the Wildwood shoreline data, nor the forcing data exhibit a similar trend, indicating that perhaps some outside factor not considered by the model is influencing that particular stretch of shoreline. Both objective evaluation criteria support these qualitative observations, as according to the classification system set forth in Table 33, the model performance with respect to the NMSE is poor at Island Beach and reasonable to good at Wildwood. Based upon the average NMSE for each form of the rate parameters, the best parameterizations are given by ka(t) = ka (constant), ke(t) = keHb3(t) at Island Beach, and ka(t) = kaP(t), ke(t) = keP(t) at Wildwood. The CAP scores are somewhat misleading as the performance of the model at both sites can be classified as good according to this criterion. The abrupt jump in the magnitude of the shoreline fluctuations at Island Beach after 1995, minimizes the influence of the pre1995 data on the overall CAP score as most of these data points correspond to a stable shoreline as defined by the sliding scale. The optimum parameters based upon the average CAP scores for Island Beach and Wildwood are, ka(t) = ka, ke(t) = kHb2(t) and ka(t) = kaP(t), ke(t) = keCb(t), respectively. The inability of one of the two objective measures of model skill to correctly characterize the performance of the model at Island Beach, presents a clear indication of the necessity of incorporating some subjectivity into the analysis of numerical models. The previously described Model Performance Index (MPI) provides a useful measure of 