• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 List of Tables
 Abstract
 Introduction
 Literature review
 Background/study approach
 Probability density function...
 Trend analysis of parameters
 Summary and conclusions
 Comparison of original and smoothed...
 List of references
 Biographical sketch






Group Title: UFLCOEL-96011
Title: Non-Gaussian properties of waves in finite water depth
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00084992/00001
 Material Information
Title: Non-Gaussian properties of waves in finite water depth
Series Title: UFLCOEL-96011
Physical Description: x, 68 leaves : ill. ; 29 cm.
Language: English
Creator: Robillard, David J., 1966-
University of Florida -- Coastal and Oceanographic Engineering Dept
Publication Date: 1996
 Subjects
Subject: Coastal and Oceanographic Engineering thesis, M.E   ( lcsh )
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (M.E.)--University of Florida, 1996.
Bibliography: Includes bibliographical references (leaves 66-67).
Statement of Responsibility: by David J. Robillard.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00084992
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 35128646

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Figures
        Page v
        Page vi
        Page vii
    List of Tables
        Page viii
    Abstract
        Page ix
        Page x
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
    Literature review
        Page 5
        Page 6
        Page 7
    Background/study approach
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
    Probability density function analysis
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
    Trend analysis of parameters
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
    Summary and conclusions
        Page 56
        Page 57
    Comparison of original and smoothed "a" values
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
    List of references
        Page 66
        Page 67
    Biographical sketch
        Page 68
Full Text



UFL/COEL-96/011


NON-GAUSSIAN PROPERTIES OF WAVES IN
FINITE WATER DEPTH





by



David J. Robillard






Thesis


1996













NON-GAUSSIAN PROPERTIES OF WAVES IN FINITE WATER DEPTH


By


DAVID J. ROBILLARD













A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ENGINEERING



UNIVERSITY OF FLORIDA


1996


















ACKNOWLEDGMENTS


David J. Robillard wishes to express sincere thanks to his wife and children for
their understanding and support. The author would also like to thank the members of the

graduate committee, Dr. Michel K Ochi, Dr. Robert G. Dean and Dr. Ashish Mehta, for

their input and guidance during this study. Special thanks are extended to the committee
chairman, Dr. Ochi, whose instruction and insight enabled me to learn a great deal on the

topic of this study.













TABLE OF CONTENTS


ACKNOWLEDGMENTS ......................................................................................... ii

LIST OF FIGURES................................................................................................... v

LIST OF TABLES ....................................................................................................... viii

ABSTRACT........................................ ..................... .............................................. ix

1 INTRODUCTION ............................................................................................... 1
1.1 Purpose of Study.......................................................................................... 2

2 LITERATURE REVIEW ..........................................................................................5

3 BACKGROUND/STUDY APPROACH........................................................... 8
3.1 Wave Record Data...................................... ............................................... 8
3.1.1 Data Acquisition.................................................................................. 8
3.1.2 Wave Profile Transformations ............................................ ............... 9
3.2 Nearshore Zone Characteristics................................................................ 16
3.2.1 Discussion of Water Depth Regions During ARSLOE......................... 16
3.2.2 Discussion of the Surf Zone During ARSLOE ........................................ 17
3.3 Non-Gaussian Probability Density Function................................ ............ .. 19
3.3.1 Non-Gaussian Probability Density Function Background ..................... 19
3.3.2 Probability Density Function Application to Wave Data......................... 21
3.3.3 Dimensional Analysis of PDF Parameters.......................... ........... .. 22

4 PROBABILITY DENSITY FUNCTION ANALYSIS.............................................. 24
4.1 Probability Density Function Applicability Verification................................... 24
4.2 Gaussian/Non-Gaussian Boundary Definition .............................................. 33

5 TREND ANALYSIS OF PARAMETERS...........................................................42
5.1 Relationship Between a and Water Depth.................................... ............ 42
5.2 Non-Dimensional PDF Parameter Analysis................................. ............ .. 44
5.2.1 Spatial Analysis of Data...................................................................... 44
5.2.2 Relationship Between aa, .,/o and ./ ............................................... 52
5.2.3 Relationship Between ao and a/d........................................ ............ 54

6 SUMMARY AND CONCLUSIONS .............................................. ........ 56







APPENDIX

COMPARISON OF ORIGINAL AND SMOOTHED "a" VALUES.............................. 58

LIST OF REFERENCES....................................... ................................................ 66

BIOGRAPHICAL SKETCH .................................................................................. 68













LIST OF FIGURES


Figure page

1.1. Wave Profile Examples: a) Gage 615, Severe Sea Condition, and b) Gage
625, M ild Sea Condition .................................................. ................................. 1

3.1. Coastal Engineering Research Center Field Research Facility ......................... 10

3.2. Gage 615 10/23 @ 2155 Wave Profile................................. ......................... 12

3.3. Gage 635 10/23 @ 2155 Wave Profile............................................... 12

3.4. Gage 645 10/23 @ 2155 Wave Profile..................... ......................... 12

3.5. Gage 655 10/23 @ 2155 Wave Profile..................... ............................ 12

3.6. Gage 665 10/23 @ 2155 Wave Profile..................... ................................ 13

3.7. Gage 675 10/23 @ 2155 Wave Profile..................... ......................... 13

3.8. Gage 625 10/23 @ 2155 Wave Profile...................................... ............ 13

3.9. Gage 615 10/25 @ 1315 Wave Profile......................................................... 14

3.10. Gage 635 10/25 @ 1315 Wave Profile....................................................... 14

3.11. Gage 645 10/25 @ 1315 Wave Profile...................................... ............ 14

3.12. Gage 655 10/25 @ 1315 Wave Profile..................... ......................... 14

3.13. Gage 665 10/25 @ 1315 Wave Profile..................... ................................ 15

3.14. Gage 675 10/25 @ 1315 Wave Profile..................... ......................... 15

3.15. Gage 625 10/25 @ 1315 Wave Profile.......................................................... ... 15

4.1. Gage 615 10/23 @ 2155 Histogram Data .................................................. 25

4.2. Gage 635 10/23 @ 2155 Histogram Data ........................................................26

4.3. Gage 645 10/23 @ 2155 Histogram Data ........................................... ............. 26







4.4. Gage 655 10/23 @ 2155 Histogram Data .................................. ............. 27

4.5. Gage 665 10/23 @ 2155 Histogram Data .................................. ............. 27

4.6. Gage 675 10/23 @ 2155 Histogram Data .................................. ............. 28

4.7. Gage 625 10/23 @ 2155 Histogram Data .................................. ............. 28

4.8. Gage 615 10/25 @ 1315 Histogram Data ................................................ 29

4.9. Gage 635 10/25 @ 1315 Histogram Data .......................................................29

4.10. Gage 645 10/25 @ 1315 Histogram Data .................................. ............. 30

4.11. Gage 655 10/25 @ 1315 Histogram Data .................................. ............. 30

4.12. Gage 665 10/25 @ 1315 Histogram Data .......................................................31

4.13. Gage 675 10/25 @ 1315 Histogram Data ................................................ 31

4.14. Gage 625 10/25 @ 1315 Histogram Data ........................................................32

4.15. Sample Data Points to Analyze Gaussian/non-Gaussian Boundary.................... 34

4.16. Probability Density Function of POINT A of Figure 4.15 ................................. 35

4.17. Probability Density Function of POINT B of Figure 4.15 .................................36

4.18. Probability Density Function of POINT C of Figure 4.15 .................................36

4.19. Probability Density Function of POINT D of Figure 4.15 ................................. 37

4.20. Probability Density Function of POINT E of Figure 4.15.................................. 37

4.21. Probability Density Function of POINT F of Figure 4.15.................................. 38

4.22. Probability Density Function of POINT G of Figure 4.15................................. 38

4.23. Probability Density Function of POINT H of Figure 4.15................................. 39

4.24. Gaussian/Non-Gaussian Boundary Defined................................................41

5.1. Spatial Comparison of a at Mild and Severe Sea Conditions ............................43

5.2. All a Data Plotted as a Function of Water Depth........................................... 43

5.3. Spatial Comparison ofaa at Mild and Severe Sea Conditions........................... 45

5.4. Spatial Comparison of p./o at Mild and Severe Sea Conditions........................ 46







5.5. Spatial Comparison of a./a at Mild and Severe Sea Conditions........................ 47

5.6. aa as a Function of Water Depth................................................................ 49

5.7. p./a as a Function of Water Depth.................................................................. 50

5.8. c./a as a Function of Water Depth.......................................................... 51

5.9. Plot of p*/o as a Function ofaa for Gages 615 and 625 ................................. 53

5.10. Plot of y/o as a Function ofaa for Gages 615 and 625 ................................... 53

5.11. Plot of aa as a Function of c/d for Mild and Severe Sea Conditions ................. 55

A.1. Plot of Gage 615 Original and Smoothed "a" Values........................................ 60

A.2. Plot of Gage 635 Original and Smoothed "a" Values....................................... 61

A.3. Plot of Gage 645 Original and Smoothed "a" Values....................................... 62

A.4. Plot of Gage 655 Original and Smoothed "a" Values......................................... 63

A.5. Plot of Gage 665 Original and Smoothed "a" Values .................................... 64

A.6. Plot of Gage 675 Original and Smoothed "a" Values .................................... 65

A.7. Plot of Gage 625 Original and Smoothed "a" Values ........................................ 66







LIST OF TABLES


Table page

3.1. Deep Water Wave Statistics for Mild and Severe Sea Conditions........................ 16

3.2. Surf Zone Locations for Mild and Severe Sea Conditions................................... 19

4.1. Wave Profile Statistics for Gage 615 10/23 @ 2155........................................ 25

4.2. Wave Profile Statistics for Gage 635 10/23 @ 2155........................................ 26

4.3. Wave Profile Statistics for Gage 645 10/23 @ 2155........................................ 26

4.4. Wave Profile Statistics for Gage 655 10/23 @ 2155........................................ 27

4.5. Wave Profile Statistics for Gage 665 10/23 @ 2155........................................ 27

4.6. Wave Profile Statistics for Gage 675 10/23 @ 2155........................................ 28

4.7. Wave Profile Statistics for Gage 625 10/23 @ 2155....................................... 28

4.8. Wave Profile Statistics for Gage 615 10/25 @ 1315......................................... 29

4.9. Wave Profile Statistics for Gage 635 10/25 @ 1315..........................................29

4.10. Wave Profile Statistics for Gage 645 10/25 @ 1315......................................... 30

4.11. Wave Profile Statistics for Gage 655 10/25 @ 1315.........................................30

4.12. Wave Profile Statistics for Gage 665 10/25 @ 1315.........................................31

4.13. Wave Profile Statistics for Gage 675 10/25 @ 1315.........................................31

4.14. Wave Profile Statistics for Gage 625 10/25 @ 1315................................ 32

4.15. Statistical Properties of Data Points A-H in Figure 4.15..................................... 35

5.1. General Shallow Water Limits of Parameters ............................................... 52













Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Engineering


NON-GAUSSIAN PROPERTIES OF WAVES IN FINITE WATER DEPTH

By

David J. Robillard

May 1996

Chairman: Michel K. Ochi
Major Department: Coastal and Oceanographic Engineering

As waves propagate into shallower waters, their profile changes from a Gaussian

distribution to an increasingly non-Gaussian profile. The non-Gaussian properties of

waves in finite water depths are analyzed through the use of a probability density function

defined in closed form and that can be solved by the application of wave displacement

data. Initially, the applicability of the probability density function is verified using wave

data records obtained from the Coastal Engineering Research Center (CERC) at Duck,

North Carolina. A broad range of sea conditions and water depths are represented to

ensure the probability density function correlates well with actual histogram data over

varied conditions. With the applicability verified, the probability density function is used

to define criteria for the boundary where the wave field can no longer be considered

Gaussian. The ability to determine when a shoaling wave profile can no longer be

considered Gaussian is of paramount importance in order to ensure proper wave theories

are being applied. This boundary between Gaussian and non-Gaussian wave profiles will

be determined in terms of significant wave height and water depth.








Presently, wave data records are required to define the probability density function

used in this study. Trend analyses were conducted on the parameters of the probability

density function to determine if simplifications can be made to its form and to see if it is

possible to define the probability density function in terms of the local conditions, sea
severity and water depth, without the need for wave data records.












CHAPTER 1
INTRODUCTION
As wind generated waves transition from deep water to shallow water, a significant
change is evident in the wave profile. As an example, Figure 1-1 shows portions of wave
records obtained by the Coastal Engineering Research Center, (CERC), during the
ARSLOE Project. Figure 1-1(a) is the wave profile recorded at a distance of 60 meters
offshore in a shallow intermediate water depth condition. Figure 1-1 (b) is the wave
profile taken at the same time but at a distance of 12 km offshore in a deep water
condition. Deep water wave profiles in the sea characteristically reflect a Gaussian (linear)
random process. The height of the wave crests are probabilistically equal to the height of
the wave troughs. As waves move from deep to shallow water, the wave profile
transforms from a Gaussian random process to a non-Gaussian (nonlinear) random
process in which the wave profile shows a definite excess of high crests and shallow
troughs.




u AJ i 4.AJJ ,Ij,,o. AN ..,! A Al A Lv AA 4gA
we j^JJ^J WIy IV Y-b 'Mr V V



a) Gage 615, Severe Sea Condition b) Gage 625, Mild Sea Condition


Figure 1.1 Wave Profile Examples









This remarkable transformation in the wave profile, as an irregular sea transitions

from deep to shallow water, has also been evidenced in laboratory experiments (Doering

and Donelan 1993). The transformation of wave profiles can be attributed to the non-

linear wave-wave interactions augmented primarily by decreasing water depth. The
"degree" of non-linearity a wave field has at a particular site of interest in the coastal

region appears to depend largely on sea severity and water depth.
Although there has been awareness that Gaussian-based solutions do not always

represent accurate results in the coastal region, clarification of wave characteristics from

deep to shallow water has been a subject of considerable interest in ocean and coastal

engineering. There have been few studies which have offered non-Gaussian solutions

describing the probability density function of waves in coastal regions. A commonly

known study was by Longuet-Higgins (1963), which statistically predicts coastal wave

properties using a power series expansion. This probability distribution is based essentially

on the Gram-Charlier series distribution. A probability density function developed by Ochi
and Ahn (1994a) appears to be applicable throughout the coastal region from deep to

shallow water and is defined in closed form Equation (1).
Ax) = -exp 1 ) (1 ya p. exp(-yax))2 yax (1)
2) 7r-. 2(rao-.)
28 for: x 0
where, y = {
3.00for:x <0

This is the probability density function that will be used as the basis to analyze

coastal wave profiles for this study.


1.1 Purpose of the Study

The overall objective of this study involves two components. The first component

will be to validate the applicability of the probability density function developed for waves

in finite water depth (Ochi and Ahn 1994a) and to analyze the transition point where wave








profiles can no longer be considered Gaussian. In order to accomplish the first objective of

this study, a statistical analysis was carried out on wave time histories obtained during a

storm by the Coastal Engineering Research Center (CERC) during the Atlantic Ocean

Remote Sensing Land-Ocean Experiment (ARSLOE) Project in 1980. To interpret the

transformation of waves from a Gaussian to a non-Gaussian random process, wave
records at seven locations were taken where the water depths ranged from two meters to

twenty-one meters. In total, approximately 1000 wave records were analyzed with each
wave record consisting of 4800 water displacement measurements. The individual wave
displacement measurements were recorded at a frequency of 4 Hertz in twenty minute
intervals.

Upon confirmation of the probability density functions' applicability, the histogram

data will be compared closely with the corresponding normal distribution to determine the

conditions under which the wave profile can no longer be considered Gaussian. In

particular, general relationships to sea severity and water depth will be determined. A
criteria will be established which will define the limit to when the Gaussian wave profile

becomes non-Gaussian. This limit will be based on the nonlinearity parameters analyzed
from the wave profile data. The second objective of the study will then be to analyze the
parameters of the probability density function. The three parameters, a, p. and o. will be
statistically analyzed in both dimensional and non-dimensional form. Trends based on the
analysis will be described and relationships linking these parameters of the probability
density function to site conditions will be developed.
This study consists of six chapters. Chapter 2 provides a general review on the
development of probability density functions applicable to coastal regions. Chapter 3
describes the ARSLOE Project of 1980 and addresses the statistical approach taken in this

study. The data used as the basis for this study, were obtained from the ARSLOE Project.
This chapter will also describe the local field conditions at CERC during the time the data
were collected. Chapter 4 addresses the probability density function applicability and








analysis. This chapter has two distinct sections. The first section reviews the finite water

depth probability density function (Ochi and Ahn 1994b) and verifies the applicability of

the probability density function. The second section discusses sea severity and water

depth and the transition between Gaussian and non-Gaussian wave profiles. Chapter 5

relates the non-dimensionalized parameters of the probability density function, aa, 1A./o

and c /o, to sea severity, (a), and water depth in both dimensional and non-dimensional

form. The parameters will also be compared to each other and relationships determined.

This chapter describes trends evident based on statistical analysis. These trends are

qualitative and are intended to give a better understanding of the interaction the wave
profile has with its environment as it propagates in shallow waters. Finally, Chapter 6

provides the summary and conclusion for this study.













CHAPTER 2
LITERATURE REVIEW


As stated in the Introduction, the profile of waves in finite water depths is quite

different from that observed in deep water. From the stochastic view point, waves in finite

water depth are categorized as a non-Gaussian random process for which a precise

description of the process does not exist. This is in contrast to the Gaussian random

process which can be defined by a rigorous mathematical formulation. Although

considerable attention has been given to Gaussian waves in deep water, little information

is available on the non-Gaussian characteristics of shallow water waves.

One of the most commonly known presentations describing non-Gaussian waves is

the Gram-Charlier series probability density function. This density function is given in a

series consisting of an infinite number of terms. Longuet-Higgins (1963) theoretically

showed that the statistical properties of nonlinear waves can be presented by the Gram-
Charlier series distribution. Bitner (1980) found reasonable agreement in a comparison

between the Gram-Charlier series distribution and the histogram constructed from wave

data obtained in a relatively shallow water area.

On the other hand, Ochi and Wang (1984) analyzed more than 500 wave records

obtained by the Coastal Engineering Research Center (CERC) at Duck, North Carolina

during the ARSLOE Project (1980). They found that while the statistical properties of

waves at any water depth can be well represented by the Gram-Charlier series distribution,
there is a drawback to applying this density function in practice. The probability density

function becomes negative for large negative wave displacements and this negative portion

does not vanish with increasing number of terms in the series distribution. Because of this








difficulty, the transition property of wave characteristics from gaussian to non-gaussian

distribution can not be well clarified. They derived, however, a limiting sea severity for a

specified water depth below which wind-generated coastal waves may be considered to be

a Gaussian random process.

Another approach for describing the surface elevation ofnon-Gaussian random

waves is to assume that individual waves can be expressed as a Stokes wave. There

appears to be some reservation to applying this approach for representing the statistical

distribution of the wave profile. This method presupposes a preliminary form of the wave

profile before establishing its probability density. This supposition may impose

unfavorable bias on the outcome.

Tayfun (1980) derived the probability density function based on the Stokes profile

approximation up to the second order. Huang et al (1983) developed the probability

density function considering all terms up to the third order. These probability density

functions are superior to the Gram-Charlier series distribution in that they do not have any

negative portion over the entire range of wave displacement. Comparison of these

probability density functions and wave data show good agreement.

Langley (1987) applied a probability density function representing the response of

a non-linear system developed by Kac and Siegert (1947), to nonlinear random waves.

This probability density function is given in the form of a series and can only be evaluated

numerically. Ochi and Ahn (1994a) modified the Kac Siegert density function so that it is

presented in closed form with parameters determined through spectral analysis. They

applied the probability density function to non-Gaussian coastal waves. In the application

process, they decomposed the spectra into linear and nonlinear components to clarify the

degree of non-linearity involved in shallow water waves (Ochi and Ahn 1994b). The

probability density function does not have any negative portion in its entire domain and

appears to represent the statistical distribution of the wave profile in any sea state during a

storm in deep, intermediate and shallow water.








Studies concerned with the non-Gaussian properties of waves in finite water depth

include Thornton and Guza (1983) who claimed that the results of their analysis of field

data obtained at Torrey Pines Beach, California, show that the Rayleigh probability

distribution can be applied to represent wave heights observed from shallow to deep (10.8

meter) water. This implies that all observed waves can be considered a Gaussian random

process. However, the sea severity in relation to water depth at all locations in the

nearshore zone where their measurements were taken was too mild and below the limit for

waves to be a non-Gaussian random process. This subject will be discussed in detail in

this present study.

In closing this chapter, it is important to note that there still remains a significant

amount of work to do in refining our ability to accurately describe irregular seas in finite

water depths and over a wide range of sea severity.













CHAPTER 3
BACKGROUND / STUDY APPROACH


3.1 Wave Record Data


An experiment jointly hosted by the U.S. Army Coastal Engineering Research

Center (CERC) and the National Ocean Survey (NOS) was conducted from 06 October

1980 to 30 November 1980. An aspect of this Atlantic Ocean Remote Sensing Land-

Ocean Experiment (ARSLOE) involved ocean wave studies implemented near the

CERC's Field Research Facility (FRF) at Duck, North Carolina. In situ wave measuring

devices were deployed in mean water depths ranging from 1.35 meters to 24.4 meters and

were placed in a linear array extending from 60 meters offshore to 12,000 meters offshore,

Figure 3.1.


3.1.1 Data Acquisition

The wave displacement data analyzed in this study came from a total of eight wave

measuring devices. The FRF has a 550 meter research pier oriented perpendicular to the

local shore which was equipped with seven Baylor resistance-type wave staffs. These

Baylor wave staffs function by determining the change in electrical resistance experienced

by changes in water elevations. Round-trip travel time of an electrical current from an

above-water emitter to the water surface is measured. Following along the line of the

pier off-shore was a Waverider Buoy Gauge. This is a surface-following class buoy. It

functions by double-integrating the output of an accelerometer held vertical by a pendulum

to produce the time history of buoy heave motion. The statistics of the buoy heave then

represent the wave displacements. In the Waverider buoy, the pendulum is contained









entirely within the spherical float so the buoy does not have to be stabilized. These

devices recorded both wave displacements and wave periods but only the displacement

data were analyzed for this study. The wave gauges recorded water surface fluctuations

simultaneously at all eight locations at a frequency of 4 Hertz over twenty minute

intervals. This resulted in 4800 individual wave displacements recorded for each twenty

minute segments continuously. Due to the significant amount of data obtained during this

unique experiment, only a sub-set of the data was analyzed. This chosen data sub-set

began at 1200 on 23 October 1980 and ended at 1315 on 25 October 1980. This two day

segment was unique in that a severe storm occurred from 23-25 October which was the

most significant meteorological event recorded during the experiment. Wind speeds on

the order of fifteen meters per second were recorded during the severe storm. This two

day sub-set of data was used as the basis for this study since it included the entire range of

wave profiles experienced at each of the eight wave gauge locations. From this two day

sub-set, a total of nearly 5 million individual wave displacements were analyzed and the

wave profile data covered the entire range of sea conditions present during the complete

56 day ARSLOE Project.


3.1.2 Wave Profile Transformations

The data collected at the FRF during ARSLOE is unique and impressive. It is very

well suited for analyzing changes in the wave field as the waves propagate through

shoaling depths. The bathymetry at the experiment location is relatively simple showing

straight and parallel contours. This provided favorable conditions which maximized
control over the environmental impact on the quality of the data received. The wave

profile data provide a matrix of wave profiles allowing both spatial and temporal analyses

to be performed.





























-



S .\
I P



p .
9 ^.* '


i 0
4
54

FIELD IESIARCH FACILITY
10'S4.6"N1 75'4S'S.2"W



S ? ~


.#\


I., I %
(15
m.i l


\\
t '


GAGE
I

12 K 710










456 625









310- 675




200-665





151 655

115-645

90--635
60-615





ZERO-MSL


Figure 3.1 Coastal Engineering Research Center Field Research Facility


a








A wide range of wave profiles was observed during the experiment. Figures 3.2-

3.8 represent a mild sea condition recorded on 10/23 @ 2155 for all finite water depth

gage locations shown in Figure 3.1. Figures 3.9-3.15 represent a severe sea condition

recorded at the height of a storm on 10/25 @ 1315 for the same gage locations. It is

evident from the sample wave data that the wave field transitions as it shoals from deep to

shallow water. The wave data representing the mild sea condition at Gage 625, Figure

3.8, are characterized by an almost symmetrical distribution of positive and negative wave

profile displacements. This is characteristic of Gaussian random processes. As the wave
shoals into shallower water, the wave profile transforms showing progressively higher

peaks and shallower troughs. This process is clearly non-Gaussian and must be analyzed

differently. It is also clear from observing these wave records that the degree of non-

linearity increases as the relative water depth decreases and as sea severity increases. This

implies that analysis of this nonlinearity must include a dependence on water depth as it

relates to the magnitude of the waves' displacement (sea severity).










j a I 0 A A l A'" 1 n a' A / ' 4 " J -A A A


Figure 3-2 Gage 615 10/23 @ 2155 (d=1.74m)



IV v "Y %.vw %d ,.. -wwy wv TV -v v w" - "vv -vl% IV Yv -V ,kr 4 wv lv "


Figure 3-3 Gage 635 10/23 @ 2155 (d=2.83m)








Figure 3-4 Gage 645 10/23 @ 2155 (d=4.31m)





v^ ^M "4v^^ '1 ^\S v VW ,vy\ ^^ v1- ,


Figure 3-5 Gage 655 10/23 @ 2155 (d=6.0m)














Figure 3-6 Gage 665 10/23 @ 2155 (d=6.08m)







Figure 3-7 Gage 675 10/23 @ 2155 (d=6.82m)



L A kA 1 k A A


Figure 3-8 Gage 625 10/23 @ 2155 (d=9.66m)














Figure 3-9 Gage 615 10/25 @ 1315 (d=2.23m)






Figure 3-10 Gage 635 10/25 @ 1315 (d=3.32m)







Figure 3-11 Gage 645 10/25 @ 1315 (d=4.66m)





VV V v vvvvV vvJ vv -' ^ LUv AjvVN VUN V6-V-


Figure 3-12 Gage 655 10/25 @ 1315 (d=6.44m)










Figure 3-13 Gage 665 10/25 @ 1315 (d=6.52m)


Figure 3-14 Gage 675 10/25 @ 1315 (d-7.22m)


Figure 3-15 Gage 625 10/25 @ 1315 (d=10.04m)


A k L L A .A 6-a 4 pA q A A








3.2 Nearshore Zone Characteristics

All wave fields propagating in finite water depths are influenced by the attributes

of the local sea environment. These attributes include water depth and beach slope. An

awareness of the characteristics of these attributes is important to understand the type of

influence they have on the wave field.


3.2.1 Discussion of Water Depth Regions During ARSLOE

Based on linear theory, there are three water depth regions which are defined using

the velocity potential for a progressive wave. The velocity potential is defined as
HgTcosh(k(d +z)) 2n 2* t.
(x,z,t)= sin(kx ).
4r cosh(kd) T

The distinct water depth regions are determined by the asymptotes of the hyperbolic

function. Deep water wave data obtained at gage location 710 and shown in Table 3.1

allow these regions to be defined for the mild sea condition and the severe sea condition.

In general, the deep water wave condition is defined as kd > n, where k is the wave

number and h is the water depth. This condition occurs at d = 21m for the mild sea

condition on 10/23 @ 2155 and d = 43m for the severe sea condition on 10/25 @ 1315.
The shallow water condition is defined as kd < x/10. This condition occurs at d = 0.6m for

the mild sea condition and d = 1.3m for the severe sea condition on 25 October. The

intermediate water condition occurs between these bounds. All wave profile data

obtained from ARSLOE are located in the intermediate water depth region.

Table 3.1 Deep Water Wave Statistics for Mild and Severe Sea Conditions



Date/Time Zero Crossing Period, T Deep Water Wavelength, Xo

10/23 @ 2155 5.17 sec. 41.715 m

10/25 @ 1315 7.43 sec. 86.157 m







3.2.2 Discussion of the Surf Zone During ARSLOE

An integral aspect of any study which covers the nearshore region must analyze

and consider the surf zone. The surf zone is a transient entity which varies in width off-

shore depending on the local sea conditions. These local conditions include the beach

profile, sea severity and direction of wave field. In order to define the surf zone in this

study, wave field data were analyzed which represented the range of sea conditions

experienced over the two day period. The first condition, representing the mildest sea
condition, was taken as 2155, 23 October 1980. The second condition, representing the

most severe sea condition was taken as 1315, 25 October 1980.
Linear wave theory will be used to determine the effects of shoaling as the wave field

enters shallow water. Refraction will not be considered since this process tends to reduce

the shoaling wave heights and thus move the surf zone shoreward. This is shown by the

definition of the refraction coefficient, K, = cos- where Oo and 0 are the deep water
V cos0

wave angle and the finite water depth wave angle respectively. This ratio will always be

less than one causing a reduction in the resulting wave height. In order to ensure that

conservative results are obtained, refraction will be neglected in this study. The beach
profile at the ARSLOE project site shows slightly varying beach slopes so a wave

breaking criteria which considers the beach slope will be used. Weggel (1972)

reinterpreted laboratory results and showed a dependency of breaker height on beach

slope. The following equations were used to define the surfzone locations:


CCEgl
Conservation of Energy: H 2 = H C g
CE2








Breaker Criterion: H, = Kdb, where K is defined by Weggel as follows:
K = b(m) a(m) H with a(m) = 43.8(10 e-")
gT
b(m) = 156(L0 + e-9m)-

As the beach slope, m approaches zero, this equation reduces to K = 0.78.
The Group Velocity, C, = (I 1+)( 2kd will be determined based on the
2 \ 2
dispersion relation, = gk tanh(kd), where L = wave length, T = wave period

(assumed constant), g = gravitational acceleration, k = the wave number and d = local
water depth.
In order to standardize the location of the surf zone, the root-mean-square wave

height, Hn,, will be used to represent the irregular wave field. H,,,,, = -V H

where Hi is defined as the wave height of the i' wave of the irregular wave field. The surf
zone which results from the use of HI,. will represent the approximate "center of mass"
location where the most significant set of individual waves in the wave field begin to
break. The surf zone calculations were based on a root-mean-square wave height that was
determined from the deep water wave displacement data. This single wave height was
used to represent the irregular sea and shoaled landward. Since the real sea will have a
significant number of waves with heights larger and smaller than the root-mean-square, the
actual breaking zone is an obscure entity. This technique for determining the surf zone
locations is adequate to make general comparisons when analyzing the spatial changes in
the nonlinearity of the irregular sea in finite water depths. The surf zone locations during
the mildest and most severe sea conditions over the two day period are shown in Table
3.2.
In Table 3.2 it is shown that during the mild sea condition, the breakpoint, indicating
the beginning of the surf zone, is located 70 meters from the shoreline in 1.51 meters of
water. This breakpoint is the location where a wave having a height equivalent to the









root-mean-square wave height of the wave field will break. During the severe sea

condition, this breakpoint occurs 110 meters from the shoreline in 3.32 meters of water.

Realistically, waves would be observed breaking considerably seaward of the breakpoints

determined in the respective sea conditions due to the spectrum of wave heights present in

an irregular sea. The root-mean-square wave height breakpoint is used in order to

standardize the position of the surf zone for comparison purposes.


Table 3.2 Surf Zone Locations for Mild and Severe Sea Conditions


Sea Condition Depth where a wave with Start of Surf Zone
height H,, Breaks -Dist. from Shore-
Mild 10/23 @ 2155 1.51 m 70 m
Severe 10/25 @ 1315 3.32 m 110 m


3.3 Non-Gaussian Probability Density Function


The probabilistic approach used in this study will make use of the probability

density function applicable to non-gaussian random processes developed by Ochi and Ahn

(1994a). This probability density function appears to represent non-Gaussian waves in

shallow and intermediate water depths as well as Gaussian waves in deep water.


3.3.1 Non-Gaussian Probability Density Function Background

The probability density function has a background of the Kac-Siegert solution

developed for the response of non-linear systems. The density function, y(t), is given as

the sum of the standardized normal variates and its squares as follows
N
y(t)= Z (pZ, + A ZJ)
j=l









The parameters Pj and j are determined by finding the eigenfunction and eigenvalues

of the integral equation given by
f K(col,co2)y (c.2)dco2 = yj,(i,)

where K(co 2 )= H(oo, o2 ) S(acI)S(wo2)
j(o) = orthogonal eigenfunction = { fy (co)yl (k)dco = 1 forj = k

otherwise, Wj(co) = 0

H(mc 02 ) = second order frequency response function

S(co) = output spectral density function

As can be seen from the form of this solution, the probability density function ofy(t),

can not be found without knowledge of the second order frequency response function,

H(co c2 ) and there is no way to evaluate this response function for irregular seas. Ochi

and Ahn showed however, that Kac-Siegerts' solution can be obtained from the time

history of the wave record without any knowledge of the second order frequency response

function, H(o, oh ). The probability density function thus obtained for describing non-

gaussian random processes is

f(x) = exp (1 ya. exp(-ax))2 Tax
2;cra. [ 2(yao-.)
28for:x 2 0
where, y =
3.00for: x <0

The parameters, a, p. and o2. are determined from the cumulants evaluated from the wave

data. These cumulants are related to the 3j and Xj values of the Kac-Siegert solution by

the following relationships
2N
ki = Z-'A 2
j=
2N 2N
k2 = p 2 + Z 2
J=1 J-1
2N 2 N
k3=61/ p21J + 8 2 l3J etc.
ji- j=-








In the present study, ki represents the mean water level and is therefore defined by,

kl = 0. k2 and k3 are equal to the data sample moments, E[x2] and E[x3] respectively.

Based on this relationship, the parameters, a, p. and a2. can be determined through the

iterative solution of three equations along with the analysis of a data record for the non-

Gaussian random process being analyzed. The three equations are

a oa2 + a .2 + p. = E[x] = 0

2. 2a a = E[x2]

2a 4.(3 8a202.)= E[x3]



3.3.2 Probability Density Function Application to Wave Data

In order to determine values for the three parameters, a, 1,. and o2., which are

used to define the degree ofnon-linearity a shoaling wave field has at a given location and

sea state, data records must be analyzed. With the applicability of the probability density

function verified based on the wave data collected during ARSLOE, the approximately

1000 data records from 1200 on 23 October to 1315 on 25 October will be analyzed to

determine values for the three parameters defining the probability density function. The

procedure used during this study to perform this task is as follows:

Compute the standard deviation of each twenty minute wave record consisting of
4800 discrete wave displacement data points E(x2) = x2
n ,

(E[x] does not need to be computed since it is defined as zero and the data are
already standardized).


Compute the third moment of the same twenty minute wave record
E(x'3)= xY .
n ,








Define the standard deviation of the data record as o" = E(x) .

Evaluate the three parameters, a, p. and o2., through iterative solution of the

following three equations:

a a2. + a A.2 + p.= E[x] = 0

02. 2a2a. = E[x2]

2a a4.(3 8a22*) = E[x3].

It is important to note that the standard deviation of the data record is used in the

analysis as an indicator of the sea severity and not the significant wave height. This is

because the definition of significant wave height is not applicable to non-Gaussian waves.
All available data sets covering the seven finite water depth gage locations

between 1200, 23 October 1980 and 1400, 25 October 1980 were analyzed and a, p. and

a2* were determined.

Based on a plot of the individual parameters in the space domain, it was

determined that a smoothing technique should be used to refine the "a" parameter values.

This smoothing was done in order to remove the sporadic tertiary effects while ensuring

the smoothed "a" value does not lose any authenticity. An exponential smoothing

technique was used which resulted in the lowest mean absolute deviation (MAD) between

the raw "a" parameter value and the smoothed "a" parameter. The remaining two

parameters, p. and o2* did not show the same susceptibility to tertiary effects so no

smoothing was done. Comparisons between the actual and smoothed "a" parameters are

shown in Appendix A for all Gage positions.


3.3.3 Dimensional Analysis of PDF Parameters

For each of the approximately one-hundred-fifty data sets per gage position, the

probability density function parameters, a, t o* were determined iteratively. Each of

the parameters contain discrete dimensions. Based on dimensional analysis, these








dimensions are as follows: It is known that k2, representing the data sets' variance has

dimensions of [L2]. It is also known that k3, representing the data sets' third moment has

dimensions of[L3]. It follows from the knowledge of these dimensions that the

parameters, a, p* a. must have dimensions [1/L], [L] and [L] respectively.

In order to minimize any dependence on external conditions and to ensure the

solutions are universally applicable, these parameters, a, .,, a. must be made non-

dimensional. The non-dimensional quantities that will be used in this study are, aa, p./o,

c./J and o/d. o is defined as the standard deviation of that wave displacement data and d
is defined as the water depth.













CHAPTER 4
PROBABILITY DENSITY FUNCTION ANALYSIS


4.1 Probability Density Function Applicability Verification

As stated in the introduction, one of the objectives of this study is to determine

whether or not the probability, density function developed by Ochi and Ahn is valid for

representing waves in finite water depths. To verify the applicability of this probability

density function, wave displacement data sets obtained from the ARSLOE experiment

were used in comparing the histogram of the data record to the probability density

function. Each data set contained 4800 wave displacement readings taken over a twenty

minute period. To ensure a thorough verification of the probability density function, the

data records used for the analysis were from all seven "finite" water depth gage locations

and analysis was done over a wide range of sea state conditions. One analysis was done

on data from 23 October 1980 at 2155. This represented a mild sea state. The other

analysis was done on data from 25 October 1980 at 1315. This data set was at the peak

of the most severe storm experienced during the entire ARSLOE study.

Figures 4.1-4.14 show comparisons between the histogram data and the non-

Gaussian probability density function. Included also in the figures is the Gaussian

(normal) probability density function for comparison. Figures 4.1-4.7 pertain to the mild

sea condition while Figures 4.8-4.14 represent the severe sea condition. As can be seen
from these figures, the probability density function agrees very well with the histogram

generated from the data record in all cases. This implies that the probability density

function derived by Ochi and Ahn can be used with confidence when analyzing non-

Gaussian random processes such as wave displacements in shallow and intermediate










waters. It can also be seen that the probability density function approximates the normal

distribution when the wave field is in the deeper intermediate water condition under mild

sea conditions.


Gage 615 -10123 @ 2155


Di9 9 0t
OlDsplacm Mnt (m)


4 -


Table 4.1 Gage 615 10/23 @ 2155


615

a (1/m)

p.(m)

a. (m)

a (m)

d (m)

cr/d


WAVE PROPERTIES

10/23 2155

.572645

-.04948

.275683

.267219

1.74

.153574


IStHistogramData ----- PDF *** Gaussian

Figure 4.1 Histogram Data












Gage 635- 10IZ 21z' Table 4.2 Gage 635 10/23 @ 2155

1.60
WAVE PROPERTIES
1.40

1.20 ', 635 10/23 a 2155

1.00
a (1/m) .481441

0. (m) -.0547

S0.40 a .(m) .318871

020
a (m) .30989
0.00
S a "9 d (m) 2.83
Displacment (m)
c/d .109502
SIg l Histogram Dta PP ....... Gaussian


Figure 4.2 Histogram Data





Gage 645 10123 @ 2155
Table 4.3 Gage 645 10/23 @ 2155
1.2
WAVE PROPERTIES
1
6 645 10/23 ( 2155

S0.8 a (1/m) .200988

S06 (m) -.03084

C F. (m) .343446
S0.4
IaC ((m) .340708
0.2 .
d (m) 4.31

o /d .079051
-1 -0.6 -0.2 02 0.6 1 1.4
Displacement (m)

I gnm ~lt stogramData ---PF **** Gaussan


Figure 4.3 Histogram Data











Table 4.4 Gage 655 10/23 @ 2155


I


-1 -0.8 -0.6 -0.4 -02 0 0.2 0.4 0.6 0.8 1 1.2 1.4
[Dspcaement (m)


I HMstogram I~at FF ....... Gauss n


655

a (1/m)

*(m)

P. (m)

a (m)

d (m)

cy/d


WAVE PROPERTIES

10/23 (, 2155

.104437

-.01771

.340056

.339137

6.0

.056523


Table 4.5 Gage 665 10/23 @ 2155

WAVE PROPERTIES

665 10/23 ( 2155

a (l/m) .126266

p.(m) -.01835

C. (m) .365155

a (m) .364236

d(m) 6.08

a/d .059907


Figure 4.5 Histogram Data


0.8

0.6


2 0.4
A.
0.2

0


0age 665 10/23 2l5












Gage g76 -10/23 @ 215 Table 4.6 Gage 675 10/23 @ 2155

12
WAVE PROPERTIES

1 675 10/23 1 2155

U
0.8 / a (1/m) .094349

0. p. *(m) -.01014

S a.(m) .375221
0.4

k (m) .374948
0.2
d (m) 6.82
0
99 C o 0 0 a/d .054978

Olsplacm ent (m)

a stgram Data PF -....... -Guss


Figure 4.6 Histogram Data




Gage 626 -10W23 @ 2166
Table 4.7 Gage 625 10/23 @ 2155
1

0.9 WAVE PROPERTIES

0.8
- 625 10/23 (a 2155
0
t 0.7

u 0.6 a (l/m) .069914

S0.5 f. (m) -.00819
- 0.4
(. (m) .394571
S0.3

02 a (m) .394401

0.1 d(m) 9.66

-1.5 -1 -o.s 0 o.s 1 1.5 a/d .040828


Displacement (m)


I m IsamlHstogram Dta PCF *.... Gaussian


Figure 4.7 Histogram Data














Table 4.8 Gage 615 10/25 @ 1315


WAVE PROPERTIES

615 10/25 s 1315

a (1/m) .372027

p. (m) -.09863

(m) .498230

a (m) .47981

d (m) 2.23

a/d .215161


Figure 4.8 Histogram Data


Gage 636 10126 @ 1316


-1.8 -1.2 -0.6 0 0.6 12
Displacement(m)


1.8 2.4


Table 4.9 Gage 635 10/25 @ 1315


635

a (1/m)

g. (m)

a. (m)

a (m)

d (m)

o/d


WAVE PROPERTIES

10/25 1315

.318011

-.10497

.591069

.573397

3.32

.17271


I M Hstogram Data PF ...**** Gaussian


Figure 4.9 Histogram Data














Oage 64 -10126 @ 1316 Table 4.10 Gage 645-10/25 @ 1315

0.8
WAVE PROPERTIES
0.7

0.- 645 10/25 a 1315
0
0.5 a (1/m) .256391

0.4 j p.t (m) -. 10245

Sa. (m) .628243
0.2
Sa (m) .612186
0.1
Sd (m) 4.66
0- dm
7 99; q 0 0 ar/d .13137
DIsplacement (m)
mm= Histogram Data PF .... Gaussan


Figure 4.10 Histogram Data





Gage 566 10128 @ 1316
Table 4.11 Gage 655 -10/25 @ 1315
0.8

0.7 WAVE PROPERTIES


S0.6 655 10/25 ( 1315

S0.5 a (l/m) .269927

0.4 't. (m) -.12356

J 0.3 a".(m) .658813

oa a (m) .636821
0.1
d(m) 6.44
0
0 N o c /d ..098885


Figure 4.11 Histogram Data












Table 4.12 Gage 665 -10/25 @ 1315


WAVE PROPERTIES

665 10/25 Q 1315


a (1/m)

p. (m)

a. (m)

a (m)

d (m)

a/d


.269743

-.12978

.67434

.650698


6.52

.09980


Figure 4.12 Histogram Data


Gage 76 10126 @ 1316


0.4 +


-2 -1.2


-0.4 0.4 12 2 2.8 3.6
Displacement (m)


Table 4.13 Gage 675 -10/25 @ 1315


675

a (1/m)






a (in)
t. (m)





d (m)

a/d


WAVE PROPERTIES

10/25 1315

.220143

-.12062

.727669

.708477

7.22

.098127


Figure 4.13 Histogram Data


-m HIstogram Data PF ....... Gaussian















Gage 62 -10/26 @ 1316 Table 4.14 Gage 625 -10/25 @ 1315

0.6
WAVE PROPERTIES

0.5 625 10/25 (, 1315

S.4 a (1/m) .140009

sc. (t) -.10961
S0.3
S0o. r. (m) .872244


0.2 o(m) .858792

d (m) 10.04
0.1
c/d .085537


-2.5 -1.5 -0.5 0.5 1.5 2.5 3.5

EiiB Hstogram Data PF .... Gaussian

Figure 4.14 Histogram Data



Based on a comparison with the water depth regions defined in Section 3.2.1, it

can be seen that the histogram data and the probability density function closely resemble

the normal distribution even after the wave field enters intermediate water for milder sea

conditions. This is shown in Figures 4.6 and 4.7. This implies that the characteristics of

the wave profile are dependent on the sea condition as well as the water depth.

The histogram data for both the mild and severe sea conditions in Figures 4.1-4.14

show the distribution of the wave profiles progressively becomes more non-Gaussian as

the waves propagate toward shallower depths. The non-Gaussian trend becomes more

extreme as the sea condition becomes more severe. It is significant to note also that the








non-Gaussian characteristics are most pronounced at the shallower gage locations, Gages

615 and 635.


4.2 Gaussian/non-Gaussian Boundary Definition


The histogram data shown in Figures 4.1 4.14 represent the same wave profiles

depicted in Figures 3.2 3.15. The transformation from Gaussian to non-Gaussian

profiles begin with subtle changes in the amplitude of the negative displacements and the

width of the troughs. This initial transformation can be seen be comparing Figures 4.4 and

4.6. Figure 4.6 shows a nearly normal distribution. Figure 4.4 shows a slight elevation of

the apex and a narrowing of the negative tail. It is also significant to note that the positive

part of the distribution remains relatively unchanged. This indicates that the initial changes

affect the trough region of the profile more drastically than the positive region. Figure 4.4

demonstrates that the trough region becomes shallower and wider. As the profile

continues to shoal, changes begin to occur in the positive displacement region of the

distribution. The peaks become narrower and higher. This can be seen graphically by

comparing Figures 4.2 and 4.4. The positive region of the wave profile is not as full when

compared with the normal distribution and the positive tail extends farther than the normal

curve.

As described above, significant transformations occur in the wave profile as the

wave train propagates into shallower water. There is a point in this transformation when

the wave profile deviates significantly enough from the normal distribution that the waves

can no longer be considered a Gaussian random process. Identification of this boundary is

important to ensure assumptions made regarding the waves remain valid. Analysis of the

histogram data shows that the non-Gaussian characteristics of coastal waves depends on

water depth and sea severity. Based initially on analysis of the histogram data shown in

Figures 4.5 and 4.6, it appears this transition boundary occurs when the non-dimensional,









wave standard deviation, o/d, is approximately 0.06. The non-dimensional parameter,

a/d, is used because it considers both sea severity and water depth. Eight random sea

conditions were selected having c/d values between 0.04 and 0.10. These eight data sets

are depicted in Figure 4.15 in terms of ac as a function of /d. The data points labeled A-

H in Figure 4.23, each correspond to a distribution shown in Figures 4.15-4.23. Data

points A-G were intentionally grouped around the 10 meter water depth. This minimized

any impact water depth and/or beach slope may have on the analysis results. The

statistical data for these points are provided in Table 4.15.


0.12
0.1
0.08
0.06
0.04
0.02
0
0


.02


0.04


0.06
o/d


0.08


A Non-Gaussian Profiles A Gaussian Profiles


Figure 4.15 Sample Data Points to Analyze Gaussian/Non-Gaussian Boundary


A


B


G F^

H^










Table 4.15 Statistical Properties of Data Points A-H of Figure 4.15

POINT a/d H. Water Depth, d

A 0.1000 3.08m 9.51m

_B 0.0800 2.74 m 9.67 m

C 0.0710 2.47m 9.64m

D 0.0616 2.47m 10.03 m

E 0.0600 2.43 m 10.13 m

F 0.0520 1.78 m 8.90 m

G 0.0398 1.57 m 9.88 m

H 0.0590 4.77 m 20.20 m


Comparison with Normal Distribution
old = .100


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Displacement (m)

-- Prob. Density Fcn. - --- Normal Distribution


Figure 4.16 Point A of Figure 4.15































Figure 4.17 Point B of Figure 4.15


Comparison with Normal Distribution
l/d = .071
0.6-

0.5
0.4
2 0.3
0.2
C. 0.1
0--
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Displacement (m)

S Prob. Density Fcn. ----.... Normal Distribution


Figure 4.18 Point C of Figure 4.15































Figure 4.19 Point D of Figure 4.15


Figure 4.20 Point E of Figure 4.15


Comparison with Normal Distribution
o/d = .0616
0.7
0.6
0.5 "
S0.4
S0.3
0. 2 0.2
o.
0. 0.1
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Displacement (m)

Rob. Density Fcn. ...... Normal Distribution









Comparison with Normal Distribution
aod = .062
0.9
0.8
u 0.7
S0.6
7' 0.5
5 0.4
S0.3
2 0.2
S0.1
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Displacement (m)

Prob. Density Fcn. -..--... Normal Distribution


Figure 4.21 Point F of Figure 4.15


Comparison with Normal Distribution
old = .0398


1.2


.0.8

c 0.6
0.4

. 0.2
0 ---- ,~ -----,-------, --1 ---
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Displacement (m)

SRob. Density Fcn. -- ---- Normal Distribution


Figure 4.22 Point G of Figure 4.15


i










Comparison with Normal Distribution
aid = .069
0.35
0.3
u 0.25
S 0.2
0.15
.d 0.1 -
L. 0.05

-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
Displacement (m)

Prob. Density Fcn. -- ---- Normal Dstribution


Figure 4.23 Point H of Figure 4.15



Figures 4.16 4.23 confirm that the region between a/d = 0.06 and a/d = 0.08,

shown in Figures 4.20 and 4.17 respectively, appears to be when the wave profile

transforms from Gaussian to non-Gaussian. A closer inspection of this region is shown in

Figures 4.18 4.20. The distribution shows a considerable increase in deviation from the

normal between a/d = 0.06 and a/d = 0.0616 as shown in Figures 4.20 and 4.19

respectively.

Due to the subjective nature of this boundary, a general boundary can be defined

based on probabilistic analysis. This boundary, defining the conditions under which the

wave profile can no longer be considered Gaussian, occurs when o/d =0.06.

A similar study was conducted by Ochi and Wang (1984); however, they defined

the boundary condition between Gaussian and non-Gaussian wave profiles in terms of the

wave record parameter, .3. X3 represents the skewness of the wave profile which they

determined to be the dominant parameter affecting the non-Gaussian characteristics of








coastal waves. Ochi and Wang concluded that coastal waves in seas for which X3 is less

than 0.2 can be considered as a Gaussian random process. For comparison purposes, this

boundary condition is plotted along with the findings of this present study in Figure 4.24.

In terms compatible with the approach of this present study, their boundary condition can

be defined as a /d = 0.0543. It is encouraging that there shows very close agreement

between both findings. Additional analysis was contributed by Ochi and Ahn (1994b)

whose data points are shown in Figure 4.24 as white circles. Ochi and Ahn (1994b)
defined the transition criterion between Gaussian and non-Gaussian as occurring when a <
0.01 where "a" is one of the parameters in the non-Gaussian probability density function.

The white circle data points define the limiting cases based on his criterion. This third

method for determining the boundary between Gaussian and non-Gaussian wave profiles
agrees very well with the other two further verifying the validity of this finding.





I


6


?5


<4





2


1


0


I~ I


Water Depth (m)


- - - Ochi & Wang (1984) --- Present Study (1996)



Figure 4.24 Gaussian/Non-Gaussian Boundary Defined


I-.


A


Non-Gaussiai Wave Profile


B A

--Gaussia i WaY Phofile
o F" A Region



o


H


~e~0'













CHAPTER 5
TREND ANALYSIS OF PARAMETERS


5.1 Relationship between a and Water Depth

As was discussed earlier, the distribution of the wave profile is highly dependent

on sea severity and water depth with sea severity being defined in terms of the wave

profiles' standard deviation, a.' Since these two terms will be used considerably in the

following trend analyses, their relationship with each other will be examined.

Figure 5.1 shows examples of simultaneous measurements of a plotted as a

function of distance offshore for two cases. It is evident from Figure 5.1, that the

standard deviation of the wave profile is influenced by water depth. As water depth

decreases, the relative magnitude of the standard deviation decreases. It is also seen that

for more severe sea conditions, represented by 10/25 @ 1315 data in Figure 5.1, the

overall change in a as the wave field propagates into shallower waters is greater as

compared with milder sea conditions (10/23 @ 2155). Based on theoretical

considerations of a, it can be assumed that a approaches zero as distance offshore

approaches zero. This trend is observed in Figure 5.1 since both cases shown appear to

converge to the origin. From the information on the surf zone locations for these two

cases computed in Chapter 3, it appears that a begins to decrease rapidly towards zero

once the break point is reached.

The information shown in Figure 5.1 based on two cases is shown in Figure 5.2 for

all data sets analyzed in this study. Instead of a being plotted as a function of distance
offshore, Figure 5.2 plots a as a function of water depth. This will take into consideration

any local changes in mean sea level.






























Figure 5.1 Spatial Comparison of a at Mild and Severe Sea Conditions


The trends evident in Figure 5.2 reinforce the analysis done for the two cases in

Figure 5.1. The upper bounds of the discrete data represent the severe sea conditions

while the lower bounds represents the mild sea conditions.


Relationip Hn a and D


2 3 4 5
Depth(m)


6 7 8 9 10


*Ce~3615 g96 A g8a xg6es aGBogB oCas .C65]

Figure 5.2 All a Data Plotted as a Function of Water Depth








It can also be inferred from Figure 5.1 and Figure 5.2 that as the sea state becomes

milder, the waves' standard deviation approaches a constant value and hence the sea

severity approaches a constant before decreasing towards the origin.

In the absence of elaborate wave data, the value for a can be readily determined

with a knowledge of the local and breaking wave heights and the relative water depth,
d
defined by where g = gravitational constant and T = wave period, using a method
gT

such as stream function theory (Dean 1974). To ensure the use of accurate wave height

values in applying the stream function theory, nonlinear shoaling methods should be used

in finite water depths to determine the breaking and local wave heights.


5.2 Non-Gaussian PDF Parameter Analysis

The non-Gaussian probability density function parameters, a, p. and a., will be

analyzed in non-dimensional form spatially as a function of distance offshore, against each

other in non-dimensional form, and with aa as a function of a/d. The first analysis of the
non-Gaussian probability density function parameters will involve comparing the non-

dimensional parameters against the distance the gage was located offshore. This will

provide a spatial orientation of the parameters which will disregard any local fluctuations

in water depth due to tidal cycle of storm surge. The results are shown in Figures 5.3-5.5.


5.2.1 Spatial Analysis of Parameters

All three parameters shown in Figures 5.3-5.5 appear to gradually increase in

absolute magnitude until approximately 150 meters to 175 meters offshore. The mild

condition represented by 10/23 @ 2155 consistently yields lower values when compared

with the more severe sea condition on 10/25 @ 1315. As the sea condition becomes more

mild, the wave profile resembles the normal distribution. When the wave profile

represents exactly the normal distribution, "a" equals zero and ar equals a. It can be seen







Plot of Bottom Profile and "acr" Comparison
Distance Offshore (m)
0 100 200 300 400 500 600
0 0.3

.2 ~- -----------^ .- ..- --- ------.--- ---------- ------- -------------------------.0.2

-2 -------------0.25



E -4 -0.2


a)

6 0.15 b



I -8 0.1



-10 .05



-12 n


-x-Gage Positions ---10/23 @2155 -e-10/25@ 1315



Figure 5.3 Spatial Comparison of aa at Mild and Severe Sea Conditions






Plot of Bottom Profile and "p.lo" Comparison
Distance Offshore (m)


Figure 5.4 Spatial Comparison of t./Co at Mild and Severe Sea Conditions


.15 ^
:L


-x-Depth Profile -a-10/23 @ 2155 ---10/25 @ 1315






Plot of Bottom Profile and "r. la" Comparison
Distance Offshore (m)


Figure 5.5 Spatial Comparison of a./o at Mild and Severe Sea Conditions


.0o
--t1.05


E-4






-10.
a)




(0




-10


-x-Bottom Profile -A-10123 @ 2155 -o-10/25 @ 1315








from Figures 5.3 and 5.5 that as approaches zero under mild conditions and st/s

approaches one. Figure 5.4 shows that m./s approaches zero under these same conditions

which is also consistent with the theoretical limit of the non-Gaussian probability density

function.

There is also a significant perturbation evident in Figures 5.3-5.5 around 150

meters offshore. The mild sea condition shows this abrupt change occurring seaward

when compared with its location under severe sea conditions and the perturbation is more

pronounced in the severe sea condition case. In general, all trends analyzed for the non-

dimensional parameter, m./s, are almost identical results to those found for as and s./s but

in the mirror image. This is due to the fact that the value of m. is negative. The negative

sign is supported by the theory used to develop the probability density functions' three

governing equations.

The next spatial analysis will consist of comparing the non-dimensional parameters

with water depth. This comparison will be sensitive to any influence of local depth

changes occurring from tidal cycles and/or storm surges. Figures 5.6 5.8 show the
results of this analysis.
The trends from this analysis are the same as those determined by Figures 5.3-5.5.

Due to the difference in scale of the abscissa the curves based on water depth appear to

show much smoother transitions. Figures 5.6-5.8 include two additional cases

representing intermediate sea conditions between those represented by 10/23 @ 2155 and
10/25 @ 1315. These two additional cases approximate the trends displayed by the mild

sea condition of 10/23 @ 2155. This is consistent with the data and time of these cases

since they occurred prior to the severe storm so the sea condition was not yet impacted

significantly.
As was discussed earlier, in the deeper water conditions all three parameters tend
to show convergence towards their respective values for a Gaussian wave profile. In

extremely shallow water conditions, the parameters also tend to converge towards a






0.2

0.18

0.16

0.14

0.12

o 0.1

0.08

0.06

0.04

0.02

0


1 2 3 4 5 6 7 8 9

Water Depth (m)


-- 10/23 2155 - 10/25 1315
S- -10/24 2135 - -10/24 1315


Figure 5.6 ao as a Function of Water Depth







0 1


0




-0.05




-0.1




-0.15




-0.2




-0.25


Water Depth (m)
4 5


--- 10/23 @ 2155 - 10/25 1315
-- -10/24 2135 ----10/24 @.1315


Figure 5.7 tL./o as a Function of Water Depth





1.04

1.035

1.03

1.025

1.02

1.015

1.01

1.005

1


0 1 2


3 4 5 6
Water Depth (m)

- 10/23 @ 2155 - 10/25 @ 1315
- -10/24 @ 2135 -10/24 @ 1315


Figure 5.8 a./o as a Function of Water Depth


I]








constant value which is different depending on the sea condition. For the milder sea

conditions, the parameters tend to converge to the values shown in Table 5.1.




Table 5.1 General Shallow Water Limits of Parameters

Non-Dimensional Parameter Mild 10/23 @ 2155 Severe 10/25 @ 1315

ao 0.15 0.18

A,/o -0.18 -0.22 to -0.25

Q./a 1.033 1.04 to 1.045


When the separate plots for each non-dimensional parameter are compared in both

spatial cases, it is apparent that similar trend descriptions fit all three parameters. For this

reason, a closer analysis will be made among the three variables to determine if this

similarity can be quantified. The next trend analysis consists of comparing the three

parameters with each other.


5.2.2 Relationship Between the three non-dimensional parameters, ar., p./o and a./o

As discussed earlier, the three non-dimensional parameters appear to be highly

correlated with each other. A closer look will be taken to see if specific relations can be

developed connecting the three non-dimensional parameters together. Figure 5.9 and

Figure 5.10 show plots of the raw data representing t./o and c6./ compared with aa

respectively at the extreme water depth locations, gage 615 and gage 625.

All data sets from Gages 615 and 625 were plotted since they represent the

shallowest and deepest gage locations. There are very definite relationships which hold

constant between the extreme range of water depths and sea conditions, that link aa to

ji./a and a./o. The functional relationship can be determined only after considering all










the data points from all water depth and sea severity conditions to ensure it remains

consistently valid. A deterministic relationship between these non-dimensional parameters

is a topic of active research.


Gage 615o Gage 625

Figure 5.9 Plot of pj/a as a function ofac for Gages 615 and 625


Gage 615 0 Gage 625

Figure 5.10 Plot of a./c as a function ofacy for Gages 615 and 625


Trend of ac vs pJa

5 aa 0.1 0.15 0.2


0 0.0


0

-0.05

-0.1

~Ja-0.15
-0.2-

-0.25-


Trend of aa vs a./a


1.04
1.035
1.03
1.025
1.02
1.015
a.ra 1.01
1.005
1


10 0*



,- ---------- ,,ow't
-----&Wf^te-


0.05 0.1

ac


d '9 C~ P~_)


0


0.15 0.2








5.2.3 Relationship Between aa and a/d

Based on the apparent correlation between the parameters of the non-Gaussian

probability density function, aa will be used to represent the general trends of the other

two parameters when plotted as a function ofo/d. This relationship is provided in Figure

5.11 to demonstrate general trends between the wave ao, and the local environment

represented in terms of sea severity/water depth in non-dimensional form. The lower

values of /d correspond to the deeper gage locations and the trend is for these values to

approach the values defined by wave profiles having a normal distribution. This plot also

reinforces the analysis done in Chapter 4 which determined the boundary values of the

Gaussian distribution. At c/d = 0.06, the aa values appear to begin converging towards

the Gaussian limit located at the origin of Figure 5.11.

The higher values of o/d represent the shallower water depths. It appears that at

these values aa tends to approach a finite asymptotic limit. This is consistent with the

findings from the spatial analysis figures. In between these two limits of Figure 5.11 there

is a depression in the severe sea condition case. This phenomenon must be analyzed more

thoroughly to ascertain a reasonable explanation for its existence and will be the subject of

a later study.

































0.05


0.15


o/ld
-- 10/23 @ 2155 .-.-..10/25 A 1315


Figure 5.11 Plot of aa as a Function of old for Mild and Severe Sea Conditions


0.2

0.18

0.16

0.14

0.12

m 0.1

0.08

0.06

0.04

0.02

0


*1'''"


0.25













CHAPTER 6
SUMMARY AND CONCLUSIONS


It is shown, based on the results of this study, that wind generated waves exhibit

significant changes in their profile forms as they propagate into finite water depths. The

actual nature of this transformation has been a subject that has not been covered in any

significant detail. The unique wave data records collected during the ARSLOE Project of

1980 enabled a detailed study on this topic to be produced.

One objective of this study was to verify the applicability of the non-Gaussian

probability density function developed by Ochi and Ahn (1994b) defined as
f(x) = exp (1 yap. exp(-yax))2 yax
f) ar-. I 2(yact.)2
L28for: x 0
where, =<
3.00for:x < 0

Through statistical analysis of the ARSLOE data, it has been confirmed that this

probability density function can be used with confidence to represent wave profiles in

finite water depths. It correlates very well with histogram data over an extensive range of

sea conditions. The distribution of the wave profile in finite water depths shows a
dependence on water depth and sea severity. In particular, as water depth decreases, the

distribution becomes increasingly non-Gaussian. Also, as sea severity increases the same

trend occurs.

With the applicability of the probability density function verified, an analysis of the
boundary condition where the wave profile can no longer be considered Gaussian was

conducted. A general relationship has been proven to exist between the largest significant

wave height and the local water depth where the wave profile can still be considered







HS a
Gaussian. This relationship is defined as s = 0.24 or = 0.06. Both forms of this
d d

relationship are identical since the significant wave height, Hs is defined as four times the

standard deviation, a, in deep water conditions. This relationship has been verified using

three different approaches with each yielding similar results. In shallower water

conditions, this boundary, defining the transition from Gaussian to non-Gaussian wave

profiles, appears to shows a dependence on beach slope. This is a qualitative observation

from the present study and additional research must be done to quantify this finding. This

dependence on beach slope may be similar to the influence of beach slope on breaking

waves. The linear result defined by this study is akin to the earliest breaking criterion

based on Solitary Wave Theory.

At present, the only way this probability density function can be defined is through

knowledge of statistical properties of the wave field of interest. In particular, the mean,

variance and third moment of the wave profile must be known. This requires that wave

profile data be obtained in order to evaluate the parameters of the probability density

function.

Another objective of this study was to determine relationships linking the

parameters of the probability density function to local field conditions. If the wave profile

statistics can be defined in terms of local environmental conditions such as sea state and

water depth, the probability density function can be defined without wave data. This study

has revealed qualitative trends between the parameters of the probability density function
and local field conditions based on the extensive data complied from ARSLOE. From

these qualitative trends, it is evident that the specific relationships between the parameters

and environmental conditions are complex and may require consideration of additional

parameters besides sea severity, a, and water depth. The pursuit of a functional

relationship between the parameters and the local environment continues to be a topic of

considerable interest and active research.













APPENDIX
COMPARISON OF ORIGINAL AND SMOOTHED "a" VALUES


It can be seen that the nonlinearity parameter, "a" is characterized by a significant

amount of local fluctuations superimposed on the overall trend of the parmeter. Based on

the definition of"a" as a nonlinearity parameter used to describe the probability density

function of a wave profile, these small fluctuations can be smoothed without any noticable
loss of accuracy. It is highly unlikely that the probability density function of a wave profile

changes so sporadically over such short periods of time as indicated by the minor

variations of"a". There was not any one smoothing technique that was used in the

smoothing of"a" at all gage positions. Instead, several smoothing techniques were
evaluated and the technique which produced the lowest mean absolute deviation (MAD)

was the technique used for that particular gage position.







Gage 615


0.9


0.8 -----


0.7


0.6 -


0.5


0.4 -


0.3 ---


0.2


0.1


0
10/23/80 12:00


I. r 1.


1


10/24/80 0:00


10/24/80 12:00


10/25/80 0:00


10/25/80 12:00


10/26/80 0:00


I--- Actual "a" -- - Smoothed "a"


Figure A. I. Plot of Guge 615 Original ami Smoollied "a" Values


;,h I




u er
1-





---- -'-`---- ~~-'--- ~---'


--


i/


"'






Gage 635


0.9

0.8 ----

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
10/23/80 12:00


10/24/80 0:00


10/24/80 12:00


10/25/80 0:00


10/25/80 12:00


10/26/80 0:00


S Actual "a" ...... --- .moothed "'a"


Figure A.2. Plot of Gage 635 Original and Smoothed "a" Values


-11_1


I I


~-I


w.P-

--_~.~... _


"--~1 `I


1-~





Gage 645


0.9

0.8

0.7 -

0.6

0.5 -

0.4

0.3

0.2

0.1

0
10/23/80 12:00


10/24/80 0:00 10/24/80 12:00 10/25/80 0:00 10/25/80 12:00

|-- Actual "a" I-Smoothed "a" 1


10/26/80 0:00


Figurei. A. 3. I'lot of'G(agte 04-5 OrigililiI md Smioodlwd "a" Values


5vw


''


i_- ..


~ii~v







Gage 655


0.9


0.8 -


0.7


0.6 --


0.5


0.4 -


0.3


0.2


0.1 -


0
10/23/80 12:00


10/24/80 0:00 10/24/80 12:00 10/25/80 0:00 10/25/80 12:00



Actual "a" -. --. -Smoothed ."a"


10/26/80 0:00


Figure A.4. Plot of Gage 655 Original and Smootled "a" Values


4






Gage 665


0.9

0.8 ---

0.7

0.6 ---

0.5

0.4

0.3

0.2



0.1

10/23/80 -12:00
10/23/80 12:00


10/24/80 0:00


10/24/80 12:00


10/25/80 0:00


10/25/80 12:00


10/26/80 0:00


-I ." Actual "a" -....: Smoothed "a;'"


Figure A.5. Plot ol'Gage 665 Original and Smioothed "a" Values


4 4


. ~~(- -~u\~A 7\ Nlft(le --L'


ii' _


I





Gage 675


0.9

0.8

0.7

0.6

0.5 -

0.4

0.3 -

0.2

0.1

0
10/23/80 12:00


10/24/80 0:00 10/24/80 12:00

---Actual "a"


K- r


10/25/80 0:00 10/25/80 12:00

- ---Smoothed "a"


10/26/80 0:00


Figure A.6. Plot ol'Gage 675 Original and Smoothed "a" Values


__II


U( wk


IKMt





Gage 625


0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10/23/8


10/24/80 12:00


1~


10/25/80 0:00


10/25/80 12:00


10/26/80 0:00


S--Actual "a"


. - -Smoothed "a"


I igu re A. 7. Plot of ( ge 02 5 Or)iginail andic s iool lied "ai" Valuies


10/24/80 0:00


0 1
0 12:00


I


ITM"


EL


.. .. .


.j













LIST OF REFERENCES


Bitner, E.M. (1980), "Nonlinear Effect of the Statistical Model of Shallow Water Wind
Waves," Applied Ocean Resources VoL 2,No.2, pp. 63-73.

Dean, R.G. (1974), "Evaluation and Development of Water Wave Theories for
Engineering Application," Vols. 1 and 2, Spec. Rep. 1, U.S. Army, Coastal
Engineering Research Center, Fort Belvoir, VA.

Doering, J. C., Donelan, M. A (1993), "The Joint Distribution of Heights and Periods of
Shoaling Waves," Journal Geophy. Res., Vol 98, No. C7, pp. 12,543-12,555.

Huang, N.E., Long, S.R., Tung, G.G. and Yuan, Y. (1983), "A Non-Gaussian Statistical
Model for Surface Elevation of Nonlinear Random Waves," Journal Geophy. Res.,
Vol.88, No.C12, pp.7597-7606.

Kac, M. and Siegert, A.J.F. (1947), "On the Theory of Noise in Radio Receivers with
Square Law Detectors," J. Applied Physics, VoL8, pp.383-397.

Langley, R S. (1987), "A Statistical Analysis of Non-Linear Random Waves," Ocean
Engineering, Vol.14, No.5, pp.389-407.

Longuet-Higgins, M.S. (1963), "The Effect of Non-Linearities on Statistical Distributions
in the Theory of Sea Waves," J. Fluid Mech., Vol.17, Part 3, pp. 459-480.

Ochi, M.K and Wang, W.C. (1984), "Non-Gaussian Characteristics of Coastal Waves,"
Proc. 19th Coastal Eng. Conf., Vol.1, pp. 836-854.

Ochi, M.K and Ahn, K. (1994a), "Probability Distribution Applicable to Non-Gaussian
Random Process," J. Probabilistic Eng. Mech., Vol. 9, pp.255-264.

Ochi, M.K and Amn, K (1994b), "Non-Gaussian Probability Distribution of Coastal
Waves," Proc. 24th Coastal Engineering Conf., Vol.1, pp. 482-496.

Tayfni, M.A. (1980), "Narrow-Band Non-Linear Sea Waves," J. Geophy. Res. Vol.85,
No.C3, pp. 1548-1552.

Thornton, E.B. and Guza, R.T.(1983), "Transformation of Wave Height Distribution," J.
Geophy. Res., Vol.88, No.C10, pp. 5925-5938.









Weggel, J.R. (1972), "Maximum Breaker Height," J. Waterways, Harbors Coastal Eng.
Div., ASCE, Vol 98, No.WW4, pp. 1245-1267.













BIOGRAPHICAL SKETCH


The author was born on September third, 1966, in Albany, New York. His affinity
toward the ocean was evident in high school Upon graduation from high school, he

attended the U. S. Naval Academy and graduated in 1988 with a B.S. degree in Ocean

Engineering. He began his naval career as a Navy Diver serving on a minesweeper. In

October 1991, he transferred into the U. S. Navy Civil Engineer Corps.

He arrived in Gainesville in January 1995 to begin graduate work in the field of
coastal engineering under Dr. Michel K. Ochi A short one and one-half years later, in

May 1996, he is on his way to Panama City, Florida, to serve as the Engineering Officer at

the Navy Experimental Diving Unit.




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