Shoreline response to variations in waves and water levels

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Title:
Shoreline response to variations in waves and water levels an engineering scale approach
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xvi, 173 leaves : ill. ; 29 cm.
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English
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Miller, Jonathan K., 1976-
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Civil and Coastal Engineering thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Civil and Coastal Engineering -- UF   ( lcsh )
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theses   ( marcgt )
non-fiction   ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 2004.
Bibliography:
Includes bibliographical references.
Statement of Responsibility:
by Jonathan K. Miller.
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Printout.
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Vita.

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University of Florida
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Full Text











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 One-Line/N-Line M odels .................................................... 12
2.2 Cross-shore (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

3-1. Established erosion/accretion criteria.............................................................33

3-2. Categorical assessment procedure score matrix developed for this study ..............44

3-3. Subjective rating system based upon model performance statistics.......................44

4-1. Summary of data sources. ................................................................................... 47

4-2. Relevant site characteristics ................................................ ......................... 48

4-3. Data analysis techniques applied at each site.......................... ...................65

4-4. Summary of time domain analysis results ..........................................................66

4-5. Summary of EOF analysis results.......................................................................73

5-1. Summary of SLMOD results............................................................................76

5-2. NMSE associated with various rate parameter combinations at Long Beach, WA,
and typical of the NMSE tables presented in Appendix B.....................................80

5-3. CAP associated with various rate parameter combinations at Long Beach, WA, and
typical of the CAP tables presented in Appendix B...............................................81

5-4. Calibration coefficients for Long Beach, WA, and typical of the coefficient tables
presented in Appendix B.....................................................................................83

5-5. NMSE associated with various rate parameter combinations at Crescent Beach,
FL. ....................................................................................................................... 88

5-6. CAP associated with various rate parameter combinations at Crescent Beach,
FL. ....................................................................................................................... 89

6-1. Best performing rate parameters for each site...................................................1..04

6-2. Rate coefficient statistics according to geographic region...................................108

6-3. Percent change in NMSE values at Torrey Pines, CA for Case 1 (A(Q)) ...........116








6-4. Percent change in NMSE values at Torrey Pines, CA for Case 2 (minimum W.
im posed). .......................................................................................................... 116

6-5. Percent change in NMSE values at Torrey Pines, CA for Case 3 (A(Q) and
minimum W imposed). ...................................................................................117

6-6. Percent change in NMSE values when only the longshore uniform EOF modes are
considered at the Gold Coast, Australia..........................................................118

B-1. Calculated NMSE values for model hindcasts at Island Beach, NJ.....................136

B-2. Calculated CAP scores for model hindcasts at Island Beach, NJ ........................137

B-3. Calibration coefficients ka, ke, and Ayo for Island Beach, NJ ..............................137

B-4. Calculated NMSE values for model hindcasts at Wildwood, NJ...................... 138

B-5. Calculated CAP scores for model hindcasts at Wildwood, NJ ............................139

B-6. Calibration coefficients ka, ke, and Ayo for Wildwood, NJ ..................................139

B-7. Calculated NMSE values for model hindcasts at St. Augustine, FL....................140

B-8. Calculated CAP scores for model hindcasts at St. Augustine, FL.......................141

B-9. Calibration coefficients ka, ke, and Ayo for St. Augustine, FL.............................141

B-10. Calculated NMSE values for model hindcasts at Crescent Beach, FL ................142

B-11. Calculated CAP scores for model hindcasts at Crescent Beach, FL....................143

B-12. Calibration coefficients ka, ke, and Ayo for Crescent Beach, FL..........................143

B-13. Calculated NMSE values for model hindcasts at Daytona Beach, FL................ 144

B-14. Calculated CAP scores for model hindcasts at Daytona Beach, FL ....................145

B-15. Calibration coefficients ka, ke, and Ayo for Daytona Beach, FL ..........................145

B-16. Calculated NMSE values for model hindcasts at New Smyrna Beach, FL. ..........146

B-17. Calculated CAP scores for model hindcasts at New Smyrna Beach, FL.............147

B-18. Calibration coefficients ka, ke, and Ayo for New Smyrna Beach, FL.....................147

B-19. Calculated NMSE values for model hindcasts at North Beach, WA.....................148

B-20. Calculated CAP scores for model hindcasts at North Beach, WA.......................149


I_ _








B-21. Calibration coefficients ka, ke, and Ayo for North Beach, WA.............................149

B-22. Calculated NMSE values for model hindcasts at Long Beach, WA....................150

B-23. Calculated CAP scores for model hindcasts at Long Beach, WA .......................151

B-24. Calibration coefficients ka, ke, and Ayo for Long Beach, WA..............................151

B-25. Calculated NMSE values for model hindcasts at Clatsop Plains, OR .................152

B-26. Calculated CAP scores for model hindcasts at Clatsop Plains, OR.....................153

B-27. Calibration coefficients ka, ke, and Ayo for Clatsop Plains, OR...........................153

B-28. Calculated NMSE values for model hindcasts at Torrey Pines, CA....................154

B-29. Calculated CAP scores for model hindcasts at Torrey Pines, CA........................155

B-30. Calibration coefficients ka, ke, and Ayo for Torrey Pines, CA..............................155

B-31. Calculated NMSE values for model hindcasts at Brighton Beach, AS..................156

B-32. Calculated CAP scores for model hindcasts at Brighton Beach, AS ...................157

B-33. Calibration coefficients ka, ke, and Ayo for Brighton Beach, AS .........................157

B-34. Calculated NMSE values for model hindcasts at Leighton Beach, AS ...............158

B-35. Calculated CAP scores for model hindcasts at Leighton Beach, AS...................159

B-36. Calibration coefficients ka, ke, and Ayo for Leighton Beach, AS.........................159

B-37. Calculated NMSE values for model hindcasts at the Gold Coast, AS.................160

B-38. Calculated CAP scores for model hindcasts at the Gold Coast, AS ....................161

B-39. Calibration coefficients ka, ke, and Ayo for the Gold Coast, AS...........................161

B-40. Calculated NMSE values for model hindcasts at the Gold Coast, AS, using filtered
(fc = 0.033 days') data. ........................................................................................ 162

B-41. Calculated CAP scores for model hindcasts at the Gold Coast, AS, using filtered (fc
= 0.033 days-') data............................................................................................. 163

B-42. Calibration coefficients ka, ke, and Ayo for the Gold Coast, AS, using filtered (fc =
0.033 days-') data ................................................................................................. 163















LIST OF FIGURES


Figure page

2-1. Profile schematization in Swart model.............................. ...................................14

2-2. SBEACH profile schematization .........................................................................17

2-3. Typical process based model schematic..............................................................19

3-1. Beach recession due to a combination of an increased water level, S and wave
induced setup, b(y). ...........................................................................................28

3-2. Example illustrating the role of Ayo in correcting for differences in the baseline
conditions of y(t) and yob(t) ................................................................................. 29

3-3. Schematic of model calibration routine. ............................................................39

4-1. Location of data sets from the East Coast of the United States .............................49

4-2. Improvement in the consistency of the Duck shoreline data after adjusting for the
volume change between subsequent profiles .................................... ............. 52

4-3. Location of available shoreline data along the west coast of the United States.......53

4-4. Location of Australian shoreline data sets .........................................................55

4-5. Calculation of the mean correlation profile including r. and ravg........................67

4-6. Spectral analyses of Gold Coast shoreline data where the thick line represents the
mean spectra .......................................................................................................70

4-7. Mean coherence and phase for three selected shorelines at the Gold Coast...........70

4-8. Spatial eigenfunctions el(x)-e3(x) for Duck, NC............................................ ....72

4-9. Spatial eigenfunctions el(x)-e3(x) for Washington State.......................................72

4-10. Spatial eigenfunctions el(x)-e3(x) for the Gold Coast, QLD .................................73

5-1. Complete hindcast shoreline time series for Daytona Beach, FL..........................78








5-2. 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

5-3. 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

5-4. 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

5-5. 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

5-6. 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

5-7. 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

5-8. Complete hindcast shoreline time series for the Gold Coast, QLD .......................96

5-9. 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

5-10. 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

6-1. Variability of the amplitude response function, IF(o),k1 with forcing frequency, o,
and rate coefficient, k ........................................................................................ 102

6-2. Variability of the phase response function, <(o,ka) with forcing frequency, o, and
rate coefficient, k ............................................................................................... 102

6-3. Histograms of accretion coefficients, ka, determined by the procedure detailed in
Chapter 3. ......................................................................................................... 106

6-4. Histograms of erosion coefficients, ke, determined by the procedure detailed in
Chapter 3. ......................................................................................................... 107








6-5. Comparison of "best" modified NMSE predictions with the standard NMSE
prediction and the measured data at Torrey Pines, CA........................................111

6-6. 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
non-dimensional fall velocity parameter, Q ..........................................................113

6-7. Variation of active surfzone width, W., with A(Q). ............................................113

6-8. Effect of A(Q) on calculated Ay, values at Torrey Pines, CA.............................. 114

B-1. 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

B-2. 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

B-3. 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

B-4. 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

B-5. 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

B-6. 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

B-7. 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

B-8. 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

B-9. 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








B-10. 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

B-11. 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

B-12. 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

B-13. 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

B-14. 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 half-century, 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 one-quarter 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 80-90% 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 three-dimensional, time-dependent, 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 so-called 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 50-100 years, corresponding to relevant time and space scales of hours-decades, and

meters-hundreds 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 three-dimensional, 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 3-D 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 cost-prohibitive for most

applications. One-line 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 long-term predictions are often based upon the assumption

that the effects of cross-shore 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 cross-shore 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 cross-shore processes is not a trivial task.








Numerous cross-shore 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 cross-shore models

inadequate for long-term 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 site-specific 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 event-based 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 state-of-

the-art 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 cross-shore 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) process-based models have been able to reproduce.

High-quality 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 cross-shore 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 multi-decadal.

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 long-term

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 cross-shore model to work

in concert with the existing simple longshore models (e.g., Hanson and Larson, 1998) to

obtain a robust, quasi-two-dimensional 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

real-time 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 first-approximation

predictions of the erosive potential of approaching storms. The efficiency of the new

model will make it particularly useful for long-term 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 cross-shore

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 cross-shore 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 cross-shore 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 cross-shore 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 cross-shore 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 Pelnard-Considere (1956) for predicting shoreline evolution due

to gradients in the longshore sediment transport. The key assumption of the model is that

the cross-shore profile remains in equilibrium, and does not change along the extent of

the shoreline being studied. The "one-line" 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 one-line 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 cross-shore 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 One-Line/N-Line Models

In order to obtain more realistic one-line 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, cross-shore 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, one-line models such as

GENESIS (Hanson and Kraus, 1989) have proven to be fairly successful, despite their

inability to simulate cross shore effects.

N-line models are an extension of typical one-line models where the profile is

divided into a series of N mutually interacting layers. In these quasi-3D 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 N-line models to

inappropriate representations of both the cross-shore sediment transport and the cross-

shore distribution of longshore transport, and suggest innovative approaches are required.

2.2 Cross-shore (Profile) Models

Cross-shore or profile models are generally used to describe the nearshore response

to events over limited temporal (hours-years) and spatial (meters-hundreds 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 post-storm recovery, and therefore medium to long term profile

evolution. Schoonees and Theron (1995) reviewed ten different cross-shore 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 cross-shore models have been developed, none have

proven nearly as useful as the Pelnard-Considere equation. Larson and Ebersole (1999)

used a simple diffusion equation to describe the evolution in time of an offshore mound

placed in the x-z 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 2-1, 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 cross-shore





14

sediment transport rate, Qs, is driven by the disequilibrium of profile characteristics, and

is given by

Q. = s, W (W ) (2.3)

where (Li-L2)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 ----M-M1



Figure 2-1. 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 cross-shore

sediment transport based upon the difference between the actual and equilibrium levels of

wave energy dissipation in the surfzone,

Q, = K(D-D.) (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 vice-versa. 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 cross-shore 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 post-storm 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 2-2, each having its own sediment transport relationship. In the

breaking zone (Zone II), the magnitude of the cross-shore sediment transport, Q,, is

calculated based upon energy dissipation arguments, with an extra term added to account

for down slope transport,

K D-D.+ 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 non-dimensional 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 2-2. 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 state-of-the-art 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 2-3

illustrates the general solution procedure. Gradients in the time averaged cross-shore

sediment transport rate drive bed level changes according to the continuity equation,


(1-n) 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 cross-shore 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 process-based

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 intra-wave 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 2-3. Typical process based model schematic.

Although invaluable in terms of understanding the complex physical relationships

between hydrodynamic forcing and sediment response at the micro-scale, 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 long-term 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 cross-shore 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 micro-scale 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 semi-empirical cross-shore transport to

extend the capabilities of N-line 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 N-line model by schematizing the cross-shore 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 long-term shoreline changes in the vicinity of structures; however attempts

to extend their capabilities by parameterizing the cross-shore 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 cross-shore

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, cross-shore 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 erosion-recovery sequence, and

coastal area models which ignore cross-shore 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 cross-shore 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 large-scale

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 large-scale 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

non-dimensional 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)e-a(-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 3-1 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.(3/5-K) S -7b
W. 5B W.) B (1-K) 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 3-2, 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 3-1. 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 3-2. 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, non-dimensional

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 non-dimensional 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 3-1 lists some of the more common beach change discriminators.

Although some of the criteria listed in Table 3-1 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 3-1 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 non-dimensional 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 3-1 as

guidance, numerous non-dimensional 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 3-1. 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.

"Ci-e refer to empirical constants that vary depending upon scale effects in the data, i.e. small-large scale or lab-field.

Adapted from:
Larson, M. and Kraus, N.C., 1989. SBEACH: Numerical Model to Simulate Storm-Induced Beach Change. Technical
Report No. 89-9, 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 non-dimensional 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, run-up, 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 semi-implicit, 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 Crank-Nicholson 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 non-directional 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 cross-shore model. Cross-shore 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 small-scale spatial irregularities that the model is not designed to reproduce

and which may have an impact on the perceived accuracy of simulations. Persistent

long-term 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 cross-shore 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 multi-decadal.
























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 3-3. 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 3-3 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 3-3 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, non-subjective 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 non-analytic 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 -- =1-BSS, 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 3-3 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 0-1 is assigned to each possible combination of

conditions as indicated in Table 3-2, 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 3-3.

Table 3-2. 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 3-3. Subjective rating system based upon model performance statistics.
Rating Range of Values
NMSE CAP MPI

Excellent <0.3 >0.8 5
Good 0.3-0.6 0.6-0.8 4
Reasonable 0.6-0.8 0.4-0.6 3
Poor 0.8-1.0 0.2-0.4 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 1-5 and is presented in Table 3-3.













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 cross-shore 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 cross-shore model is not intended to reproduce

small-scale 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

4-1 and 4-2. With so many good, high-quality 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 cross-shore model.

Table 4-1. 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


Buoy-44025 8531680
WISII-79 (Sandy Hook)
Buoy-44025 8531680
WISII-69 (Sandy Hook)
Buoy-44025 8531680
WISII-69 (Sandy Hook)
Buoy-44009 8534720
WISII-66 (Atlantic City)


FRF Gauge

WISII-23

WISII-23

WISII-22

WISII-22


Buoy-46029
CDIP-036
Buoy-46029
CDIP-036
Buoy-46029
CDIP-036
Buoy-46029
CDIP-036

WISSC-002



Buoy-38

Buoy-38

Buoy-23
Buoy-13


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 4-2. 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 MHHW-MLLW.
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 low-lying barrier island topography. Most of the coastline experiences

low-moderate 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 4-1 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 4-1. 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 semi-diurnal 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 bi-annually 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 0.9-1.6 m, with

a reported mean annual significant wave height of 1.2 m. Tides along the coast are

mainly semi-diurnal, 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 (bi-weekly at four selected sites), along the

one-kilometer 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 cross-shore 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 cross-shore 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 cross-shore forcing, shorelines that have been adjusted by

Ay(t) may potentially be more appropriate for evaluation purposes. Figure 4-2 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 non-uniform 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 4-2. 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 forty-year 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 Washington-Oregon

border. The approximate location of each site, along with nearby wave and water level

gauges is indicated in Figure 4-3


F WeS oasTZOn
n BlWa 3nT
T T=D Geat
















Figure 4-3. 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 (bi-annually 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 sub-regions, 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,

high-energy 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

semi-diurnal 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 long-term 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 shore-perpendicular 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 2001-2003 was approximately 1.1 m. The wave

climate exhibits a distinct seasonal periodicity similar to the Pacific Northwest; however

the range of variability (0.8-1.3 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 4-4. 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 4-4, 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 bi-weekly 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 mid-1999 as part of the post-construction








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 high-quality 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 cross-shore

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), non-uniform shoreline

movements can be quantified and compared to the more uniform, predictable, large-scale

changes. An appropriate data set will be one in which the ratio of the uniform behavior

to the non-uniform 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, non-dimensional measure of the

degree of shoreline homogeneity. The two-dimensional method applied here is a

simplification of a three-dimensional 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 cross-shore 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 auto-spectra 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 cross-shore model, or non-homogenous 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)
k-I

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 jn-n,,

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 best-fit 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 cross-products 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 cross-products 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 non-trivial (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 cross-shore 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 non-uniform 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 4-3 summarizes the methods used at each site as

only certain techniques are applicable at each location.

Table 4-3. 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 1979-1997 Variable Yes No No
Harvey Cedars, NJ 9.00 3 1986-2002 Biannually Yes No No
Island Beach, NJ 4.00 3 1986-2002 Biannually Yes No No
Wildwood, NJ 3.50 2 1986-2002 Biannually Yes No No
Duck, NC 1.00 20 1980-2002 Monthly Yes Yes Yes
St. Augustine, FL 2.50 3 1955-1995 Variable Yes No No
Crescent Beach, FL 3.25 2 1955-1996 Variable Yes No No
Daytona Beach, FL 3.50 3 1955-1997 Variable Yes No No
New Smyrna Beach, FL 2.75 3 1955-1998 Variable Yes No No

West Coast Sites
North Beach, WA 41.00 12 1998-2002 Quarterly Yes No Yes
Grayland Plains, WA 17.00 8 1998-2003 Quarterly Yes No Yes
Long Beach, WA 38.00 16 1998-2004 Quarterly Yes No Yes
Clatsop Plains, OR 25.00 6 1998-2005 Quarterly Yes No Yes
Torrey Pines, CA 1.00 3 1972-1974 Monthly Yes Yes No

Australian Sites
Brighton Beach, WA NA 1 1995-1998 Weekly Yes No No
Leighton Beach, WA NA 1 1997-1998 Weekly Yes No No
Gold Coast, QLD 1.50 300 2000-2003 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 4-4, both the mean and maximum

correlation is reported, along with the 95% significance level as determined from a

standard t-test (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 non-uniform and hence inappropriate for the model.

Table 4-4. 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 4-5. 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 4-5, 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 4-4 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 non-uniformly 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 4-4, 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 non-uniform 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 4-6 were calculated for the Duck and Torrey Pines data as well. Figure 4-6 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 non-uniform behavior. The coherence and phase plotted

in Figure 4-7 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 high-energy region (f = 0.001-0.04 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 4-6. 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 Yx-17so
-e- Coh & O with y-125
S Coh & with y-7


0-3 1/yr 2/yr 10-


f (cycles/day)
Figure 4-7. 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 4-8 to 4-10.

A summary of the nature and the amount of variability explained by the first two modes

is given in Table 4-5. 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.

Non-uniform behavior is indicated by the presence of nodal points in Figures 4-8 to 4-10,

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, cross-shore 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 4-5, 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 cross-shore 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 4-8. 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- nahoe-Dstace-kin
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 4-9. Spatial eigenfunctions el(x)-e3(x) for Washington State.


FRF Per
N
.. ... ... .. ...... ... .... .. .. ...... .




S....... ...... e (x)- 81.98%
S-e- 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%
9-e3(x) 2.22%
no-I-I


-2000 -1500 -1000
Longshore Distance (m)
Figure 4-10. Spatial eigenfunctions ez(x)-e3(x) for the Gold Coast, QLD.

Table 4-5. 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 cross-shore 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 cross-shore 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 5-1, 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 5-1. 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) P-E
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 G-E

Classification according to Table 3-3, 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 1-3 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 5-1, 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 semi-diurnal 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 5-1, 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 (11/22/1984-11/25/1984). 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 5-1. Although it is somewhat difficult to tell based upon Figure 5-1 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 -40-40
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 5-1. 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 5-2, 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 5-2 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 ............. : .......
0-5 .. .... .........
_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 5-2 to 5-4.

In Table 5-2, 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 5-2,

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 5-2. 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 5-3 is an example of the second type of table generated for each site, where

the format is very similar to that of Table 5-2, 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 5-3. 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 5-4, 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 5-4, 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 5-4, 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 5-4. 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


3.0E-04
5.0E-04
24.384


3.0E-05
5.5E-04
27.432


f(Q)
[hr'1


4.5E-04
2.5E-05
12.192


4.0E-05
3.0E-05


f(Hb2)
hr'1m'21


5.0E-04
1.6E-05
6.096


1.0E-03
1.6E-05


f(Hb3)
Ihr-1m4


4.5E-04
2.5E-06
3.048


4.5E-05
2.8E-06


f(Fr)
[hr'1


3.5E-04
1.5E-01
24.384


3.0E-05
1.5E-01


f(P)

4.5E-04
2.5E-10
0.144
1.OE-03
2.5E-10


Avg

3.9E-04

15.240
2.8E-04


3.4E-05 4.6E-04
5.5E-10
IiR An 9A nCn


It fH 3)k, 1.8E-05 2.1E-05 5.3E-05 3.5E-0311.8E-05
[h(1&] k. 1.5E-03 7.OE-05 3.8E-05 5.3E-06 5.OE-01
A 42.672 33.528 6.096 -36.576 45.720


00

ka 1.5E-06 1.5E-06 2.5E-06 2.OE-06 1.5E-06
k 6 E


[hr'] ]
Av.


.u -4 t
27.432


J3.Uc-0u
21.336


I .6O96UO
6.096


2., E-u0
6096


2.UE-U I
30.480


k, 2.5E-06 2.0E-06 3.OE-06 3.OE-06 2.5E-06
[h( ke 5.5E-04 3.OE-05 1.6E-05 2.7E-06 1.5E-01
AyI 27.432 27.432 12.192 9.144 27.432
ka 1.0E-08 1.0E.OE 1.0E-08 1.0E-08 6.OE-10
[hr ] ke 9.0E-04 4.5E-05 2.2E-05 3.2E-06 3.0E-01
AYo 27.432 21.336 12.192 6.096 39.624


ka
Avg k.
Ayo


1.9E-03
29.337


3.8E-05
22.479


2.SE-06
2.5E-10
6.096
4.5E-06
2.5E-10
6.096
1.5E-09
3.SE-10
6.096

3.1E-10
7.239


2.1E-05 7.1E-06 2.4E-01
8.001 0.381 33.147


1.8E-06
-
18.669
2.8E-06

19.431
6.6E-09

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 3-3, 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 pre-1995 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