• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 1. Introduction
 2. Literature review
 3. Development of joint probability...
 4. Examples of application
 5. Conclusions
 Appendix. Numerical examples of...
 Reference






Group Title: UFLCOEL-95015
Title: Joint distribution function of significant wave height and average zero-crossing period
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00085023/00001
 Material Information
Title: Joint distribution function of significant wave height and average zero-crossing period
Series Title: UFLCOEL-95015
Physical Description: ix, 55 p. : ill. ; 28 cm.
Language: English
Creator: Pasiliao, Eduardo Lewis, 1969-
University of Florida -- Coastal and Oceanographic Engineering Dept
Publisher: Coastal & Oceanographic Engineering Dept., University of Florida
Place of Publication: Gainesville Fla
Publication Date: 1995
 Subjects
Subject: Offshore structures -- Hydrodynamics -- Mathematical models   ( lcsh )
Ocean waves -- Mathematical models   ( lcsh )
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Statement of Responsibility: by Eduardo Lewis Pasiliao, Jr.
Thesis: Thesis (M.E.)--University of Florida, 1995.
Bibliography: Includes bibliographical references (p. 54-55).
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Bibliographic ID: UF00085023
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 34531829

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
    List of Tables
        Page iv
        Page v
    List of Figures
        Page vi
        Page vii
    Abstract
        Page viii
        Page ix
    1. Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
    2. Literature review
        Page 6
        Page 7
        Page 8
        Page 9
    3. Development of joint probability distribution
        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
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
    4. Examples of application
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
    5. Conclusions
        Page 48
        Page 49
    Appendix. Numerical examples of the bivariate generalized Gamma distribution function
        Page 50
        Page 51
        Page 52
        Page 53
    Reference
        Page 54
        Page 55
Full Text



UFL/COEL-95/015


JOINT DISTRIBUTION FUNCTION OF SIGNIFICANT
WAVE HEIGHT AND AVERAGE ZERO-CROSSING
PERIOD






by



Eduardo Lewis Pasiliao, Jr.






Thesis


1995


















JOINT DISTRIBUTION FUNCTION
OF SIGNIFICANT WAVE HEIGHT
AND AVERAGE ZERO-CROSSING PERIOD



















By

EDUARDO LEWIS PASILIAO, JR.


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


1995

















ACKNOWLEDGMENTS


Eduardo Lewis Pasiliao, Jr. wishes to express thanks to

graduate committee members Dr. Michel K. Ochi, Dr. D. Max

Sheppard, and Dr. Robert J. Thieke for their guidance during

this study. Special thanks are extended to Dr. Ochi for his

patience and understanding.

The author would also like to thank his parents, Mr. and

Mrs. Eduardo and Helen Pasiliao, and Ms. Kari A. Svetic for

their love and support.

















TABLE OF CONTENTS



page

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

LIST OF TABLES ........................................... iv

LIST OF FIGURES ........................................ vi

ABSTRACT ................................................ viii

CHAPTER 1 INTRODUCTION ................................... 1

CHAPTER 2 LITERATURE REVIEW ............................ 6

CHAPTER 3 DEVELOPMENT OF JOINT PROBABILITY DISTRIBUTION
3.1 Basic Concepts ............................ 10
3.2 Bivariate Generalized Gamma
Distribution Function .................. 16
3.3 Joint Generalized Gamma and Conditional
Log-Normal Distribution Function ........ 24

CHAPTER 4 EXAMPLES OF APPLICATION
4.1 Joint Probability Density Function ...... 31
4.2 Estimation of Extreme Sea State for
Design Consideration ....................... 38

CHAPTER 5 CONCLUSIONS .................................... 48

APPENDIX NUMERICAL EXAMPLES OF THE BIVARIATE
GENERALIZED GAMMA DISTRIBUTION FUNCTION ...... 50

REFERENCE LIST ........................................... 54

BIOGRAPHICAL SKETCH ....................................... 56
















LIST OF TABLES


Table page

1-1 Contingency Table of Tromsoflaket Data .............. 3

1-2 Contingency Table of Buoy 46001 Data ................ 4

3-1 Parameters for the Generalized
Distribution Function, f(H ) (Tromsoflaket) ......... 13

3-2 Parameters for the Generalized Gamma
Distribution Function, f(i) (Buoy 46001) ........... 14

3-3 Parameters for the Generalized Gamma
Distribution Function, f(H,) (Tromsoflaket) ......... 17

3-4 Parameters for the Generalized Gamma
Distribution Function, ff() (Tromsoflaket) ......... 18

4-1 Parameters for the Joint Generalized Gamma and
Conditional Log-Normal Probability Distribution
Function (Buoy 46001) .............................. 32

4-2 Probability of Occurrence of the Wave Climate
Within a Given Contour Line (Buoy 46001) ........... 33

4-3 Parameters for the Joint Generalized Gamma and
Conditional Log-Normal Probability Distribution
Function (Tromsoflaket) ............................ 34

4-4 Probability of Occurrence of the Wave Climate
Within a Given Contour Line (Tromsoflaket) ......... 35

4-5 Extreme Hs for given 1=7.5 s (Buoy 46001) ......... 41

4-6 Extreme Hs for given %=8.5 s (Buoy 46001) ......... 42

4-7 Extreme H, for given Z=9.5 s (Buoy 46001) ......... 43

4-8 Extreme H, for given Z=7.5 s (Tromsoflaket) ....... 45












4-9 Extreme Hs for given %=8.5 s (Tromsoflaket) ....... 46

4-10 Extreme H, for given %=9.5 s (Tromsoflaket) ....... 47

A-i Parameters for the Bivariate Generalized Gamma
Distribution Function (Buoy 46001) ................ 51

A-2 Parameters for the Bivariate Generalized Gamma
Distribution Function (Tromsoflaket) .............. 52
















LIST OF FIGURES


Figure page

3-1 Comparison of Generalized Gamma
Distribution Function with H. (Tromsoflaket) ....... 13

3-2 Comparison of Generalized Gamma
Distribution Function with Hs (Buoy 46001) ......... 14

3-3 Comparison of Generalized Gamma
Distribution Function with Z (Tromsoflaket) ....... 17

3-4 Comparison of Generalized Gamma
Distribution Function with X (Buoy 46001) ......... 18

3-5 Parameter g as a function of H. (Tromsoflaket) ..... 27

3-6 Parameter (T as a function of H. (Tromsoflaket) ..... 27

3-7 Parameter p as a function of Hs (Buoy 46001) ....... 28

3-8 Parameter a as a function of H. (Buoy 46001) ....... 28

4-1 Comparison of the Joint Generalized Gamma
and Conditional Log-Normal Distribution Function
with Buoy 46001 Data ................................ 32

4-2 Comparison of the Joint Generalized Gamma
and Conditional Log-Normal Distribution Function
with Tromsoflaket Data .............................. 34

4-3 Comparison of f(T) Represented by the Generalized
Gamma Distribution Function with that Evaluated
from the Joint Density Function (Buoy 46001) ....... 36

4-4 Comparison of f(3) Represented by the Generalized
Gamma Distribution Function with that Evaluated
from the Joint Density Function (Tromsoflaket) ..... 37

4-5 Extreme Hs for given Z=7.5 s (Buoy 46001) ......... 41












4-6 Extreme H. for given ,=8.5 s (Buoy 46001) ......... 42

4-7 Extreme Hs for given '=9.5 s (Buoy 46001) ......... 43

4-8 Extreme Hs for given ==7.5 s (Tromsoflaket) ....... 45

4-9 Extreme Hs for given 'Z=8.5 s (Tromsoflaket) ....... 46

4-10 Extreme H, for given '=9.5 s (Tromsoflaket) ....... 47

A-i Comparison of the Bivariate Generalized Gamma
Distribution Function with Buoy 46001 Data .......... 51

A-2 Comparison of the Bivariate Generalized Gamma
Distribution Function with Tromsoflaket Data ........ 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

JOINT DISTRIBUTION FUNCTION
OF SIGNIFICANT WAVE HEIGHT
AND AVERAGE ZERO-CROSSING PERIOD

By

Eduardo Lewis Pasiliao, Jr.

December 1995

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

This paper addresses the probabilistic description of

long-term sea severity and presents the estimation of extreme

sea severity for the design of ocean and coastal structures.

Joint probability distribution functions representing

significant wave height and average zero-crossing period data

are developed through two different approaches. One approach

develops the joint distribution by applying the Laguerre

polynomials including the correlation between wave height and

period. The other approach develops the joint distribution

based on the product of the marginal probability distribution of

significant wave height and the conditional probability

distribution of average zero-crossing period for a given

significant wave height. The results of the study show that the

joint distribution developed through the latter approach has

great promise. In the present study, the generalized gamma and


viii












the log-normal probability distributions are considered for the

marginal and the conditional distributions, respectively.

Comparisons between the newly developed joint probability

distribution and data obtained in the North Pacific and in the

North Sea both show excellent agreement. Based on this joint

probability distribution function, a method to estimate extreme

sea severity with period critical for floating offshore and

coastal structures is demonstrated through numerical examples.
















CHAPTER 1
INTRODUCTION



For the design and assessment of ocean and coastal

structures, a reliable knowledge of long-term wave climate is

necessary not only for evaluating a possible structural failure

associated with the extreme sea state but also for estimating

the wave-induced loading which may cause a possible fatigue

failure of the structure. The development of a reliable means

of predicting long-term wave conditions is made difficult by the

great variations in environmental factors between different

geographic locations.

Description and prediction of long-term wave climate is

commonly achieved through joint probability distribution

functions with significant wave height and average zero-crossing

period as its parameters. Due to the lack of theoretical

evidence as to the probability structures of these two

parameters, several joint distribution functions have been

proposed to represent the statistical distribution of wave data.

Unfortunately, the application of these distribution functions

are limited to specific geographic locations. The difficulty is

in finding a function that is capable of well representing the

wave data distribution at any given geographic location.












It is the purpose of this study to develop a joint

probability distribution function of significant wave height and

average zero-crossing period which reliably predicts long-term

wave conditions. Two different approaches in the derivation of

joint distribution functions are carried out. The functions are

then applied to wave data obtained from different geographic

locations. One joint distribution function is derived from the

combination of the marginal distribution function of significant

wave height with that of average zero-crossing period while

taking the correlation coefficient between the two random

variables into consideration. A second joint distribution

function is derived from the combination of the marginal

distribution function of significant wave height with the

conditional distribution function of average zero-crossing

period for a given significant wave height. Consideration of

the generalized gamma distribution function for the marginal

distribution function is based on Ochi's (1992) findings which

show that this particular function well represents significant

wave height data at any given geographic location.

The two joint distribution functions are applied to the

wave data obtained from Tromsoflaket and Buoy 46001 as given in

Tables 1-1 and 1-2, respectively. The contour curves of the

joint distribution functions are compared directly with the

observed wave data. As an application of joint density

functions, a method of estimating extreme significant wave

height for design consideration is presented.












Table 1-1.


Contingency Table of Tromsoflaket Data
Source: Mathisen and Bitner-Gregerson (1990)


Sign. Interval of zero-up-crossing period [s]
Wave 0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0
Ilt.rml 4.0 4.9 5.9 6.9 7.9 8.9 9.9 10.9 11.9 12.9 13.9
11.5-11.9 0 0 0 0 0 0 0 0 1 1 0
11.0-11.4 0 0 0 0 0 0 0 0 0 0 0
10.5 10.9 0 0 0 0 0 0 0 1 0 0 0
10.0- 10.4 0 0 0 0 0 0 0 1 0 1 0
9.5 9.9 0 0 0 0 0 0 1 0 1 0 0
9.0-9.4 0 0 0 0 0 0 1 7 1 0 0
8.5-8.9 0 0 0 0 0 0 2 8 0 0 0
8.0-8.4 0 0 0 0 0 1 11 3 0 0 0
7.5-7.9 0 0 0 2 0 4 11 6 1 1 0
7.0-7.4 0 0 0 1 0 12 22 4 3 0 0
6.5-6.9 0 0 0 0 0 35 21 3 1 0 0
6.0-6.4 0 0 0 0 7 53 19 3 0 0 0
5.5-5.9 0 0 0 0 29 90 13 0 0 0 1
5.0-5.4 0 0 0 2 107 104 18 2 0 2 0
4.5-4.9 0 0 0 24 259 95 22 2 2 1 0
4.0-4.4 0 0 1 151 279 81 20 1 3 0 0
3.5 3.9 0 0 14 436 286 59 12 3 0 0 0
3.0-3.4 0 1 187 776 234 61 13 1 1 0 0
2.5-2.9 0 20 688 804 204 43 8 3 1 0 0
2.0-2.4 3 188 1505 700 181 42 4 1 0 0 0
1.5-1.9 12 899 1341 605 117 27 2 1 1 0 0
1.0- 1.4 161 1456 1025 325 37 3 3 1 0 0 0
0.5-0.9 337 591 453 124 12 2 0 0 0 0 0
0-0.5 4 17 6 4 0 1 0 0 0 0 0
















Table 1-2. Contingency Table of Buoy 46001 Data











T avg Is) 0 I 2.01 3.01 4.01 5.01 6.01 7.01 8.01 9.01 10.01 11.01 12.01 13.01 14.01 15.01 16.01|
.... ...... I I I -I -I I -I -I -I -I -I -I I I -I -I S I
Hi ll 2.01 3;01 4.01 5.01 6.01 7. 61 901 61 0 o |00 11.01 12.01 13.01 14.01 15.01 1 .6 01 > I I

.0- .5 0 4 45 69 30 12 0 0 0 0 0 0 0 0 0 0 160
.5- 1.0 0 6 246 1643 1923 894 223 40 9 0 0 0 0 0 0 0 4984
1.0- 1.5 0 0 131 3555 5731 2774 551 99 33 1 0 0 0 0 0 0 12875
1.5- 2.0 0 0 2 1955 8010 5070 1564 273 35 0 0 0 0 0 0 0 16909
2.0- 2.5 0 0 0 501 6442 6478 2728 650 124 10 0 0 0 0 0 0 16933
2.5- 3.0 0 0 0 17 2694 6612 3508 975 148 16 0 0 0 0 0 0 13970
3.0- 3.5 0 0 0 0 733 5167 3818 1096 233 26 0 0 0 0 0 0 11073
3.5- 4.0 0 0 0 0 65 2877 3597 1379 281 45 6 0 0 0 0 0 8250
4.0- 4.5 0 0 0 0 3 1046 3212 1443 347 42 5 2 0 0 0 0 6100
4.5- 5.0 0 0 0 0 0 229 2219 1340 404 85 12 4 2 0 0 0 4295
5.0- 5.5 0 0 0 0 0 33 1263 1202 411 84 9 0 0 0 0 0 3062
5.5- 6.0 0 0 0 0 0 1 424 973 379 70 10 1 0 0 0 0 1858
6.0- 6.5 0 0 0 0 0 0 82 744 313 82 16 0 0 0 0 0 1237
6.5- 7.0 0 0 0 0 0 0 9 465 292 79 13 1 0 0 0 0 859
7.0- 7.5 0 0 0 0 0 0 1 196 238 64 17 1 0 0 0 0 517
7.5- 8.0 0 0 0 0 0 0 0 67 197 55 12 3 0 0 0 0 334
8.0- 8.5 0 0 0 0 0 0 0 15 131 48 14 0 0 0 0 0 208
8.5- 9.0 0 0 0 0 0 0 0 1 77 33 11 3 0 0 0 0 125
9.0- 9.5 0 0 0 0 0 0 0 0 25 29 9 4 1 0 0 0 68
9.5-10.0 0 0 0 0 0 0 0 0 2 23 4 4 0 0 0 0 33
10.0-10.5 0 0 0 0 0 0 0 0 0 7 4 5 0 0 0 0 16
10.5-11.0 0 0 0 0 0 0 0 0 0 2 3 2 0 0 0 0 7
11.0-11.5 0 0 0 0 0 0 0 0 0 2 1 0 2 0 0 0 5
11.5-12.0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2
12.0-12.5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1
12.5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1


SUM 0 10 424 7740 25631 31193 23199 10958 3739 803 149 31


5 0 0 0 103882





5





This paper consists of five chapters including the

introduction, literature review, and conclusion. Chapter 3

outlines the basic concepts and presents the joint distribution

functions following two different approaches. Chapter 4

presents numerical examples of application for a joint

distribution function including the estimation of probable and

design extreme sea state conditions.
















CHAPTER 2
LITERATURE REVIEW



Several joint probability density functions representing a

contingency table consisting of significant wave height, H,, and

average zero-crossing period, %,, have been proposed to date.

One approach to obtain the joint probability distribution

function is given in the form of a bivariate probability

distribution including the correlation between H, and _%. Ochi

(1978) applied the bivariate log-normal distribution to several

contingency tables from data obtained at various locations in

the world. Although the joint distribution function represents

well the overall distribution of H, and Z, the marginal

probability distribution of significant wave height, f(H,),

deviates from the sample distribution for large values of

significant wave height, and this appears to be a drawback of

this joint probability distribution.

Another approach to obtain the joint probability

distribution is given in the form of the product of the marginal

probability density function of significant wave height, f(Hs),

and the conditional probability density function of the average

zero-crossing period for a given significant wave height,

f(%-IH). A variety of combinations of the probability











distributions have been proposed for this category. Houmb and

Overick (1976) consider the Weibull probability distribution for

both f(H.) and f(l|IH); however, a comparison between the

proposed model and measured data does not appear promising.

Harver (1985) proposes a consideration of the log-normal

and Weibull distributions for f(H,), and the log-normal

distribution for the conditional distribution f(lIH). Here,

the distribution of significant wave height f(H,) is divided into

two parts; the lower values of the significant wave height data

are represented by the log-normal distribution, while the higher

values follow the Weibull distribution. The separating point,

however, depends on the data which may negate use of the

distribution in general.

Fang et al. (1989) proposes a consideration of either the

log-normal or normal probability distribution for the

conditional distribution with the mean value and variance

evaluated by applying a statistical linear regression model to

the data. Although the proposed method may be used for analysis

of individual sets of data, the choice of different probability

distributions, whichever fits the data for the conditional

distribution f(rIH,), is a serious shortcoming of the proposed

method.

Burrows and Salih (1986) compare measured data with

various combinations of distributions of f(H,) and f(*|IH), and











conclude that the combination of the 3-parameter Weibull

distribution for both f(H,) and f(~IH,) yields the best fit to

the data. Mathisen and Bitner-Gregersen (1990) also compare

measured data with various combinations of f(H,) and f(TIH,) as

well as with bivariate joint distributions, and conclude that a

combination of the 3-parameter Weibull distribution for f(H,) and

the log-normal distribution for f(IH,) appears to be the best

representation of the measured data.

As stated in the previous paragraphs, both Burrows and

Salih and Mathisen and Bitner-Gregersen claim the 3-parameter

Weibull probability distribution best represents the significant

wave height data. This implies that the sample space of the

significant wave height is given by a < H, < co, where a # 0 and

is determined by fitting data by the Weibull distribution. For

instance, Mathisen and Bitner-Gregersen give the value of 'a' as

0.98 meters for the Tromsoflaket data. Thus, the 3-parameter

Weibull distribution assumes that sea states of significant wave

height less than 0.98 meters do not exist. It is difficult to

explain the physical meaning of this minimum significant wave

height in the distribution. The sample space of the significant

wave height has to be chosen between zero and infinity, where a

significant wave height of zero represents calm water. For this

reason, the 3-parameter Weibull distribution cannot be

considered rational for the analysis of significant wave height

data.





9





As this review of the literature reflects, it may be

concluded that an appropriate joint probability distribution

function representing significant wave height and average zero-

crossing period does not exist at the present time in either of

the two approaches for obtaining the joint distribution.
















CHAPTER 3
DEVELOPMENT OF JOINT PROBABILITY DISTRIBUTION



3.1 Basic Concepts



Prior to presenting the derivation of joint probability

distribution of significant wave height and average zero-

crossing period, it may be well to outline the probability

distribution of the significant wave height alone since it is

the basis for developing the joint probability distribution

function. In addition, the two different approaches in deriving

a joint density function based on this marginal distribution

function of significant wave height are introduced.

A large number of different models have been proposed to

represent the statistical distribution of significant wave

height. These include the log-normal distribution (Ochi 1978),

the modified log-normal distribution (Fang and Hogben 1982), the

three-parameter Weibull distribution (Burrows and Salih 1986)

(Mathisen and Bitner-Gregersen 1990), the combined exponential

and power of significant wave height (Ochi and Whalen 1980), and

the modified exponential distribution (Thompson and Harris

1972), and so on. Unfortunately, the application of the

distribution functions is extremely limited in that none of them

can represent all significant wave height data adequately.












Because there is no scientific basis that exists for selecting a

distribution function to represent sea state conditions at a

specific geographic location, the selection of a distribution is

largely subjective. Frequency of storm occurrence, water depth,

and fetch length are some of the crucial factors that have

strong and varying effects on sea severity. Because of these

unpredictable effects, a probability distribution function that

well represents a set of data from a particular geographic

location will probably not be adequate in representing a set of

data from another location. While the statistical behavior of

significant wave height may vary considerably from one

geographic location to another, nonetheless, it would be

desirable to have a distribution function that can represent

significant wave height data independent of the geographic

location and, therefore, environmental factors. Rather than

having to choose from the many types of distribution functions

to represent a set of significant wave height data, only the

parameters of the chosen distribution function need to be

determined.

Through the study of various distribution functions, only

the generalized gamma distribution function was found to be able

to represent significant wave height data at any geographic

location. The generalized gamma probability density function is

given as











f(x) = e xp{-(Xx)c}
f(x) =
F(m)x (3-1)



and its cumulative distribution function as



rF(m, (hx)')
F(x) = (3-2)
r(m)



r(m), the gamma function, is defined as



r(m) = j t-1 exp{-t} dt (3-3)
0



and F(m, (Xx)C), the incomplete gamma function, is defined as




r(m, (Xx)C) = tm-1 exp{-t} dt (3-4)
0


Ochi (1992) has shown seven examples of comparison between

the cumulative distribution function of significant wave height
data and the generalized gamma distribution in which all data
are well represented by the generalized gamma distribution

function. One of these examples is the Tromsoflaket data. This

example of comparison of significant wave height data is shown













Table 3-1. Parameters for the Generalized Gamma
Distribution Function, f(Hs) (Tromsoflaket)
Source: Ochi (1992)


0.9999 -


0.99



0.9 T


0.80 -




0.50




0.20 --

0.10--

n 005


8 10


SIGNIFICANT WAVE HEIGHT IN M.






Figure 3-1. Comparison of the Generalized Gamma
Distribution Function with H, (Tromsoflaket)
Source: Ochi (1992)


v.vJ





14





Table 3-2. Parameters for the Generalized Gamma
Distribution Function, f(H,) (Buoy 46001)

m = 3.8881

CH = 1.0318
S= 1.3194


Snnnn


U


0.999






0.95 -

0.90


U.OU



0.50



0.20

0.10




0.01


2 4 6 8
SIGNIFICANT WAVE HEIGHT IN M.


1012


Figure 3-2. Comparison of the Generalized Gamma
Distribution Function with H. (Buoy 46001)


r


I I ;











in Figure 3-1 while the parameter values are listed in Table 3-

1. In addition, a comparison of significant wave height data

obtained from Buoy 46001 and the generalized gamma distribution

is newly made and the result is shown in Figure 3-2, and the

parameter values are listed in Table 3-2. As can be seen in the

figures, the agreement between the cumulative distribution of

significant wave height and the generalized gamma distribution

function is excellent. This result confirms Ochi's previous

findings; the long-term distribution of significant wave height

data at any geographic location is well represented by the

generalized gamma distribution function.

In order to derive a joint distribution function for

significant wave height and average zero-crossing period, two

approaches based on the marginal distribution function of

significant wave height, f(H,), may be considered. The

generalized gamma distribution function has already been shown

to well represent significant wave height data. One approach is

to also establish the marginal distribution function of the

average zero-crossing period, f( ), and then derive the joint

probability density function from the combination of the

marginal distributions of significant wave height and average

zero-crossing period while taking the correlation coefficient

into consideration. Another approach is to develop the

conditional distribution of the zero-crossing period for a given

significant wave height, f(TjH,), and then derive the joint












probability density function from the product of the marginal

distribution of significant wave height and the conditional

distribution of zero-crossing period. The joint probability

distribution functions of significant wave height and average

zero-crossing period developed by these two approaches are

presented in the following two sections.





3.2 Bivariate Generalized Gamma Distribution Function



Since the marginal distribution functions of significant

wave height data at Buoy 46001 and Tromsoflaket have already

been shown to be well represented by the generalized gamma

distribution, it may be possible to apply the same distribution

function to average zero-crossing period data. Hence, a joint

distribution function from the combination of the marginal

distributions of significant wave height and average zero-

crossing period can be developed. The comparison plot for

Tromsoflaket is shown in Figure 3-3, and the parameter values

are listed in Table 3-3. The figure indicates that the average

zero-crossing period may be approximated by the generalized

gamma distribution function. The same may be said of the

comparison plot for Buoy 46001 as shown in Figure 3-4 with the

parameter values listed in Table 3-4. Average zero-crossing

period may, therefore, be assumed to be represented by the













Table 3-3. Parameters for the Generalized Gamma
Distribution Function, f(T) (Tromsoflaket)


m, = 12.3566

c, = 1.2190

XT = 1.3017


0.9999


0.999 9



0.99


0.95

0.90-- -

0.80 -



) 0.50



0.20

0.10-




0.01
3 4 5 6 8 10 12 15
AVERAGE ZERO-CROSSING TIME IN SEC.


Figure 3-3. Comparison of the Generalized Gamma
Distribution Function with Z, (Tromsoflaket)






18






Table 3-4. Parameters for the Generalized Gamma
Distribution Function, f(() (Buoy 46001)


mT = 15.7159

c = 1.3210

S= 1.1886






0.9999


0.999



0.99


0.95

0.90

0.80

I--

< 0.50--
0


0.20

0.10




0.01



0.001
3 4 5 6 8 10 12 15
AVERAGE ZERO-CROSStNG TIME IN SEC.







Figure 3-4. Comparison of the Generalized Gamma
Distribution Function with T (Buoy 46001)











generalized gamma distribution function for the Tromsoflaket and

Buoy 46001 wave data.

In regards to the estimation of the three parameters

associated with the generalized gamma distribution function,

Ochi (1992) has suggested a method which directly equates the

theoretical moments to those of the sample. Since the sample

size of significant wave height data is usually very large, the

error incurred in this method is minimal. Here, the j-th moment

of the generalized gamma distribution function is given by



1 r(m + /c)
E[xj]- (rTm) (3-5)



In general, any three moments may be used to estimate the three

parameters. However, results of trial and error computations

show that a set of three moments consisting of the 2nd, 3rd, and

4th moments yields the parameters which give the cumulative

distribution functions closest to that of the samples. The

actual set of equations used to estimate the generalized gamma

parameters are given by



F(m)2 m + 3/c) E[x3]
m + /c)2 E[x22 (3-6)










F(m)r(m + 4/c) E[x4]
- (3-7)
(m + c)2 -E[x212



1 (m + 2/c)2
SEx21/2 (m)2 (3-8)


The first equation comes from extracting the X parameter from

the second moment and third moment equation and then equating

the two expressions. The second equation is similarly obtained

except that the second and fourth moment equations are used.
After calculating the m and c parameters from the first two

equations, the X parameter is calculated from the second moment

to give the third equation.

Thus, from these two marginal distributions represented by

the generalized gamma distribution function, a joint probability
density function can be developed. By performing a
transformation of random variables on the bivariate gamma

distribution developed by Sarmanov (1968), a bivariate

generalized gamma function may be derived. The bivariate gamma
probability distribution is given by




f(y, 2 (y exp{-(y1)} y2 eXp -(y2)
y Imi)y1 Em2)y2 (3-9)
.{1+ a, L-'[(y1)] L--1[(y2)] + a2 L41-[(y)] L- ',)










mi > m2

0 > x < 1


where L1 and L2 are the Laguerre polynomials and parameters a1

and a2 are obtained from the following relationships given by

Sarmanov.



a r(mJ)r(m2 + )(3-0)
a, = L (3-10)
r(m2,)(m, + 1)



r(m)r(m2 + 2)
a2 = m2 m2 + 2) (3-11)
r(mr(m,1 + 2)


In general, the Laguerre polynomials have the following

orthogonal property



0 for i j
yU exp--y} L(a)y] La)[y] dy = (a + i + 1) (3-12)
o for i = j
Si!


By writing the significant wave height and the average zero-

crossing period as H and T, respectively, and letting the

random variables


yl = (HH)'A


(3-13)










and

y, = (TT) (3-14)


a joint generalized gamma probability function can be
transformed from the joint gamma function as follows



f(H, T) = H exp{-(HH)c} C TC exp {-(T) } (3-15)


S1 + a1 L- '(H)] L,-1 [(XT)] + a, L1 H)"] L2 T[(TT)]}



Here, the parameters al and a2 given in the above equation have
no relation to those given by Sarmanov. The Laguerre
polynomials are explicitly expressed as


L-1[(x)c] = m (Xx) (3-16)



L-[(lx)C] = 1 (m + l)m (m + 1)(Xx)c + (X)2c (3-17)
2


and have the following property



J c(x) p-(x)} Lm-1[(x)] dx = 0 (3-18)
0 F(m)x











The bivariate generalized gamma probability distribution reduces

to the desired marginal distribution when integrated with

respect to either significant wave height or zero-crossing

period.

Although Equation 3-15 involves first and second order

Laguerre polynomials, it is found from the results of

computation that the contribution of the second order term to

the probability density function is not appreciable. Hence, the

following bivariate generalized gamma with only the first order

Laguerre polynomials is considered in the present study.




f(H, T) = CH X-H)m ex"p (IH)c} C ex -(XTT) (3-19)}
Sr(mT)T (3-19)
.1 + a L',-1 [(H)c] L-1[,H]T)cy]}



The parameter 'a', which has no relationship with any of

the previously defined parameters, is expressed in terms of the

correlation coefficient between H, and by taking the expected

mean of the bivariate generalized gamma distribution which

results in



a
E[HT] = E[H] E[T 1 + -- (3-20)
cycT










Then by using the following expression for the correlation

coefficient


E[HT] E[H] E[T] (3
P (3-21)
E[H2] E[H]2 E T2 E[T2



and after some algebraic manipulation, the parameter 'a' can be

expressed as a function of the correlation coefficient as

follows



rim r m, + 2c,) lm,) m, + 2/c,}
a = p* cH, 2 1 cT 2 1 (3-22)
a mH + /cH) Tm, + 1c,)T



Unfortunately, when applied to the Buoy 46001 data and

Tromsoflaket wave data, the bivariate generalized gamma

distribution function provides a very poor representation of

sea-state conditions. The results of numerical computations of

this joint density function is presented in the appendix.




3.3 Joint Generalized Gamma
and Conditional Log-normal Distribution Function


Another approach in developing a joint distribution

function is from the product of the marginal distribution

function of significant wave height and the conditional











distribution function of zero-crossing period for a given

significant wave height. The results of analysis show that the

conditional cumulative distributions of the zero-crossing period

for various values of significant wave heights plotted on log-

normal paper may be approximately represented by straight lines.

The conditional distribution may, therefore, be assumed to

follow the log-normal probability law. The conditional log-

normal probability density function, in this case, is given as



f(TH) 1 1 (InT l)2
f(TI H) = exp (3n23)
=VT 1xp 2 (3-23)
0 T


and its theoretical conditional moments are given by




E[TI H] = exp + j2f (3-24)




The parameters, g and 0, are functions of significant wave

height and are obtained by plotting the cumulative conditional

distributions on log-normal paper. The g and 0 parameters can

then be evaluated by the following formulae


g = In x0.50


(3-25)













2 IXo. xo.0



where the subscripts indicate the value of the cumulative

distribution in percentage. A good approximation of the

cumulative distribution function is obtained by applying a least

square fit method on the data plotted on log-normal paper.

To determine the relationship between significant wave

height and the conditional log-normal parameters, the JL and G

values are plotted as functions of significant wave height.

Machines and Bitner-Gregerson carried out the conditional

probability distribution of zero-crossing period for given

significant wave heights by similarly applying the log-normal

probability distribution function, and presented the parameters

g and G as a function of significant wave height. These are

given as,



S= a1 + b H (3-27)



S= a, + b, ecH (3-28)



The g and G values evaluated from various significant wave

heights from the log-normal plots and the least-square fitted

curves are shown in Figures 3-5 and 3-6 for Tromsoflaket and in

Figures 3-7 and 3-8 for Buoy 46001. The plots include the




























E
1.7

1.6

1.5

1.4

1.3

0





Figure 3-5.


1 2 3 4 5 6 7
Significant Wave Height in meters


Parameter L as a Function of Hs


(Tromsoflaket)


0

I I I I I I


1 2 3 4 5
Significant Wave Height in meters



Parameter G as a Function of Hs


0.25




0.2




E0.15
(n9


0.1 F


0.05
0


0
o


Figure 3-6.





(Tromsoflaket)







































0 1 2 3 4 5 6 7
Significant Wave Height in meters


Figure 3-7.


M
E
.)0.15
W,


Parameter g as a Function of H.


(Buoy 46001)


1 2 3 4 5
Significant Wave Height in meters



Parameter G as a Function of H,


(Buoy 46001)


Figure 3-8.












least-square curves representing Equations 3-27 and 3-28. For

Buoy 46001, the equations give excellent representations of the

statistical behaviors of both parameters. The Tromsoflaket

curves, on the other hand, are not as good in characterizing the

R and G parameters. The reason for this discrepancy could be

attributed to the difference in the size of the data. The Buoy

46001 data spans thirteen years during which 103,886

observations were recorded, while the Tromsoflaket data spans

seven years during which only 15,605 observations were recorded.

The number of observations from Buoy 46001 is, therefore,

significantly larger than Tromsoflaket. As the number of

observations increase, so does the reliability of the set of

data.

Since the marginal distributions of significant wave

height data for both Buoy 46001 and Tromsoflaket have already

been shown to be well represented by the generalized gamma

function and the conditional distributions of average zero-

crossing period for a given significant wave height have been

established, a joint distribution function may then be derived

by multiplying the marginal distribution function of significant

wave height and the conditional distribution function of average

zero-crossing period. That is,


f(H, T) = f(H) f(TI H)


(3-29)









When the generalized gamma distribution function and the
conditional log-normal distribution function are substituted

into Equation 3-29, the resulting joint probability density

function takes the following form.



c(%H). 1 1 (In T y)2
f(H, T) = ) exp-(H)c I (exp n T (3-30)
F(m)H J27T 2 02



When applied to the Buoy 46001 data and the Tromsoflaket

data, the joint generalized gamma and conditional log-normal
distribution function provides a very good representation of

sea-state conditions as presented in the following chapter.















CHAPTER 4
EXAMPLES OF APPLICATION



4.1 Joint Probability Density Function



Numerical computations of the joint generalized gamma and

conditional log-normal distribution function developed in

Section 3.3 are carried out by comparing its contour curves

directly to the observed wave data from Buoy 46001 and

Tromsoflaket. An analysis of the marginal distribution function

of average zero-crossing period evaluated from the joint density

function is also presented.

Table 4-1 lists the parameter values for the joint density

function as applied to the Buoy 46001 data while the contour

plot of the density function is shown in Figure 4-1. The

function's contour curves closely follows the same general trend

as the observed data and, therefore, provides an overall

excellent representation of the statistical distribution of wave

climate data.

The probability of occurrence of the two variables, Hs and

T,, within a given contour line of the joint density function is

compared with the observed data. This probability is calculated

numerically by finding the volume encompassed by the contour

line of the joint density function. Several volumes











Table 4-1. Parameters for the Joint Generalized Gamma and
Conditional Log-Normal Probability Distribution Function
(Buoy 46001)

f(H) my = 3.8881 C, = 1.0318 h = 1.3194

f(|j H) a, = 1.2605 b, = 0.4286 c, = 0.4161

a1 = 0.0994 ba = 0.1326 CG = -0.6596


AVERAGE ZERO-CROSSING PERIOD IN SEC.

Figure 4-1. Comparison of the Joint Generalized Gamma and
Conditional Log-Normal Distribution Function
with Buoy 46001 Data











Table 4-2. Probability of Occurrence of the Wave
Climate Within a Given Contour Line (Buoy 46001)

Contour Probability of
Line Value Occurrence
0.000001 99.9987
0.00001 99.9897
0.0001 99.9087
0.001 99.1825
0.01 92.6713
0.05 66.0600
0.1 34.1982


corresponding to different contour lines are listed in Table 4-2

for the Buoy 46001 data. As can be seen from this table, the

contour line of 1 X 10-6 covers more than 99.99 percent of the

data. The result of comparison with the observed Buoy 46001

data shows that the estimated probability of occurrence of the

wave climate agrees very well with the observed data.

Table 4-3 lists the parameter values for the joint density

function as applied to the Tromsoflaket data while the contour

plot is shown in Figure 4-2. Results from the plot are similar

to the Buoy 46001 plot. The joint density function has the same

general trend as the Tromsoflaket data distribution. Volumes

corresponding to different contour lines are listed in Table 4-

4. Estimated probability of occurrence of the wave climate

within the contour line of 1 x 106 also show good agreement with

the observed data.











Table 4-3. Parameters for the Joint Generalized Gamma and
Conditional Log-Normal Probability Distribution Function
(Tromsoflaket)

f(H,) mH = 8.71 c, = 0.612 = 15.77

f (I Hs) a = 1.3974 b = 0.2499 c. = 0.6026

a. = 0.0856 bG = 0.1260 C. = -0.3332


1 2 3 4 5 6 7 8 9 10 11 12
AVERAGE ZERO-CROSSING PERIOD IN SEC.


13 14 15


Figure 4-2. Comparison of the Joint Generalized Gamma and
Conditional Log-Normal Distribution Function
with Tromsoflaket Data











Table 4-4. Probability of Occurrence of the Wave
Climate Within a Given Contour Line (Tromsoflaket)

Contour Probability of
Line Value Occurrence
0.000001 99.9985
0.00001 99.9893
0.0001 99.9086
0.001 99.1988
0.01 99.9105
0.05 67.2445
0.1 36.7000


Included also in Figures 4-1 and 4-2 are the peak values

as well as its location with respect to significant wave height

and average zero-crossing period. The peak location of the

joint density function is less than one meter of significant

wave height and less than one second of average zero-crossing

period from the Buoy 46001 data's peak. The result of

comparison to the Tromsoflaket data's peak location shows an

even better agreement. The joint density function is,

therefore, able to yield the appropriate predictions of the

significant wave height and the average zero-crossing period

which is most likely to occur.

Finally, the marginal distribution function of average

zero-crossing period calculated directly from the wave data is

compared with the marginal distribution function calculated from

the joint distribution function. One marginal distribution




































< 0.50 I i --




o .10

0.01
/1





0.001
3 4 5 6 8 10 12 15
AVERAGE ZERO-CROSSING TIME IN SEC.







Figure 4-3. Comparison of FQ() Represented by the
Generalized Gamma Distribution Function with that
Evaluated from the Joint Density Function (Buoy 46001)































0.999



0.99 -







.80 F (Hs,To) dHs-



0.50- -


0.20 /


0.10 /




0.01
3 4 5 6 8 10 12 15
AVERAGE ZERO-CROSSING TIME IN SEC.


Figure 4-4. Comparison of F(T) Represented by the
Generalized Gamma Distribution Function with that
Evaluated from the Joint Density Function (Tromsoflaket)


^ ^^^^











function is evaluated directly from the average zero-crossing

period data, Fk(), and is represented by the generalized gamma

distribution function as shown in Figures 3-3 and 3-4. Another

marginal distribution function is evaluated from integrating the

joint generalized gamma and conditional log-normal distribution

function with respect to significant wave height, IF(HS, ) dH,.

Comparisons of the two marginal distributions are shown in

Figure 4-3 for the Buoy 46001 data and Figure 4-4 for the

Tromsoflaket data. Overall, the difference between the marginal

distribution functions of average zero-crossing period

calculated from the wave data directly and from the integration

of the joint density function is small in both sets of data.

This difference decreases as zero-crossing period increase so

that the two marginal distribution functions are almost

identical at large extreme values of zero-crossing period. The

joint density function is, therefore, able to provide a good

representation of the marginal distribution of average zero-

crossing period data.





4.2 Estimation of Extreme Sea State
for Design Consideration



As an example of application of the joint probability

density function, estimation of the extreme significant wave












heights for design application of floating marine structures may

be considered. It is extremely important for the design of

floating marine structures to estimate the extreme sea condition

with wave periods close to a natural motion period. In this

situation, a structure will experience resonance and oscillate

with an unpredictably large amplitude causing a critical

situation of the system.

The extreme sea state which occurs at the zero-crossing

period close to a natural frequency may be evaluated from the

conditional distribution of significant wave height for a given

average zero-crossing period, F(H ,IT). The extreme significant

wave height is approximately the value which satisfies the

equation of return period being equal to the number of

significant wave heights in a specified time if the number is

large. That is,



1
= N (4-1)
1 F(Hj )_



where N is the expected number of observations within the time

frame of interest, 50 or 100 years for example. The extreme H,

estimated is called the probable significant wave height, and

the probability that the observed extreme value exceeds the

estimated probable H, is theoretically 1 e = 0.632 In











practice, the computation of the Equation 4-1 is carried out by

taking the natural logarithm of both sides of equation.

Given the very large probability of exceedance of the

probable significant wave height, the extreme value calculated

from Equation (4-1) should not be used from marine system

design. A design extreme value with a small specified

probability of exceedance, CC, is evaluated from the cumulative

distribution function by the following expression



F(H |) = (1 aV) (4-2)



where X0 is also referred to as the risk parameter.

If the following approximation is made



1 (1 a)2N = /N (4-3)



the design extreme value with a specified risk parameter may be

found from the following expression



1
F( ) = N/ (4-4)
1 F(Hj ,)



Several calculations of both probable and design extreme

value of significant wave height for given zero-crossing periods

at Buoy 46001 are listed in Tables 4-6, 4-7, and 4-8 while the


















Table 4-5.


Extreme H, for given '=7.5 s


(Buoy 46001)


|-P12









S10
*~i


- --- - - <-- 100 yoa probably - -I- - -


I I
I


S :"-- 100 year design

* --- 50iyea-design- -

i I

i- -


.


13 - - --


I


k-


/<- 50 year probable
I / -- --
/ I I
* I


---13 yea probable


S- - - ---


/

/ I



/
I'




.9"
/*
/
2


3 9 10 11 12 13
SIGNIFICANT WAVE HEIGHT IN METERS


Extreme H, for given '=7.5 s (Buoy 46001)


17


I-


15 - - - - -- -



14 - - --- - - -


I

I


I
I

I



I
I
1
I
I
I
I


I I I
- - - - - - - -
I I I


Figure 4-5.
















Table 4-6.


Extreme H, for given Z=8.5 s


(Buoy 46001)


161- -

15 - - -


14


13


12


11


- - - -1opyearleign -

, <- 50 yar desiLr
- r


- - - - --- - - - -
SL* /







S-- 100 year probable
----- -----.: -- -

.<-<- 50 ar probable


10 - - - - - - - - -
: I I
1- 3 year probable
9 - - - ..........



I : a
a--- - - a - -

8-1




7 -



/*I


9 10 11 12 13 14 15
SIGNIFICANT WAVE HEIGHT N METERS


Extreme H, for given =8.5 s (Buoy 46001)


I


* a


a a


-P


- -' -- -


: : I:


I


I


I I
I I
I I
I I

I I



I


Figure 4-6.






43







Table 4-7. Extreme H, for given %=9.5 s (Buoy 46001)


13 years 50 years 100 years

probable H. 10.70 m 11.36 m 11.68 m

design H. 12.82 m 13.40 m 13.68 m

observed H. 10.0-10.5 m















15 '-4 100 yer design

14-- - - - ---- ----- 5-- year design .

13 --- -------- - -



II
12 ----------- ---- --.. ---------



I I I .1


.<-- 100 year Probable
10 - - - - - - - -

"I ;<- Syear probable





.-- 13 year probable
8 . . .-. .
,-
I




I /I
/*







9 10 11 12 13 14 15 16
SIGNIFICANT WAVE HEIGHT N METERS
*'/ II






9 10 11 12 13 14 15 16
SIGNIFICANT WAVE HEIGHT IN METERS


Figure 4-7.


Extreme H, for given =9.5 s


(Buoy 46001)












corresponding plots are shown in Figures 4-5, 4-6, and 4-7. The

examples are for average zero-crossing period values of 7.5,

8.5, and 9.5 seconds while a = 0.01 for the design extreme

values. Estimates of the extreme values at Tromsoflaket are

listed in Tables 4-8, 4-9, and 4-10 with the corresponding plots

shown in Figures 4-8, 4-9, and 4-10.

As can be seen from these tables and figures, some of the

observed extreme significant wave height exceed the probable Hs

but not the design values. It is exactly for this reason that

the probable extreme values, as given in Equation (4-1), should

not be considered in marine structure design; the design extreme

values, as given in Equation (4-4), with a small specified risk

parameter provide the appropriate value for design

consideration.














Table 4-8. Extreme H, for given t=7.5 s (Tromsoflaket)


7 years 50 years 100 years

probable H. 7.26 m 7.91 m 8.12 m

design H. 8.67 m 9.19 m 9.36 m

observed H, 6.0-6.5 m















15 : .<-- 1 b year design


i I I
i i / I I
123 - - - ----------- - -- - -



11 - - - - -
120--- -- --- ----- yeapoablej - -








1 - - - - -------- ----- - - -- - --
I I I








7 7 10 1 prob le










SIGNIFICANT WAVE HEIIGT I METERS
5!~~~------I------------------l----1-----1
SI I M


Figure 4-8.


Extreme H, for given T=7.5 s


(Tromsoflaket)















Table 4-9.


Extreme H, for given %=8.5 s


(Tromsoflaket)


13 -------------------

12 --------------- - -


11


10


I-'c

'-1


4L


3 9 10 11
SIGNIFICANT WAVE HEIGHT IN METERS


12 13


Extreme H, for given 'Z=8.5 s (Tromsoflaket)


:*-t 100 yeardesign
--- 50year.design.


/' '- - ..


I I ~ I..- I


*:I
t I



- - - - - - -




A L
S- 0 year probably e




- - I



? /
I I I I

S' 7ye probable I I
I ,: e r -rI






d
I
/I
." i i i


Figure 4-9.















Table 4-10.


Extreme H, for given =9.5 s


(Tromsoflaket)


12------------------r---------- --

12 --- -- - - --5 desig -








7 y r
11 . "




61------ -- -- ------ --- --- -----

,- .4.
S 7 /<:-l year probable




I I I I
I IJ


SI I I I I I
6 ---L-- ~ --L-~ -~

i' i I I

a < 7 year probable, a a






5" ,


9 10 11 12 13
SIGNIFICANT WAVE HEIGHT IN METERS



Extreme H, for given Z=9.5 s


14 15


~-----rr-~



x
rl





c:
-1


Figure 4-10.


(Tromsoflaket)
















CHAPTER 5
CONCLUSION



This study discusses the development of a joint

probability distribution function of significant wave height and

average zero-crossing period to reliably describe long-term sea

severity. Two joint density functions are derived through

different approaches. One approach combines the marginal

distribution function of significant wave height and average

zero-crossing period which produces the bivariate generalized

gamma distribution function. However, this joint density

function provides a poor representation of observed wave data.

Another approach combines the marginal distribution function of

significant wave height with the conditional distribution

function of average zero-crossing period which produces the

joint generalized gamma and conditional log-normal distribution

function. This joint density function provides an excellent

description of sea severity.

Numerical examples of the joint generalized gamma and

conditional log-normal distribution function are applied to the

North Pacific and North Sea wave data. The joint density

function's contour curves are compared directly to the observed

wave data. As shown by the contour plots, the joint density

function closely follows the general trend of the data's












statistical distribution. Analysis of the contour curves of the

joint density function show the estimated probability of

occurrence of the wave climate within given contour lines agrees

well with the observed wave data. The joint density function

also yields the appropriate predictions of the significant wave

height and the average zero-crossing period which is most likely

to occur. From the comparison of the two methods of obtaining

the marginal distribution function of average zero-crossing

period, the function evaluated from integrating the joint

density function with respect to significant wave height is

shown to accurately describe the statistical distribution of

average zero-crossing period.

Examples of estimating extreme sea severity through the

conditional distribution function of significant wave height for

a given average zero-crossing period are presented. With a

small specified risk parameter, the design extreme significant

wave heights obtained from this method is appropriate for the

design of coastal structures. Overall, the joint generalized

gamma and conditional log-normal probability distribution

function reliably describes and predicts long term wave climate.
















APPENDIX
NUMERICAL EXAMPLES OF THE BIVARIATE GENERALIZED GAMMA
DISTRIBUTION FUNCTION



Numerical computations of the bivariate generalized gamma

distribution function developed in Section 3.2 is carried out by

comparing the function's contour curves directly to the observed

wave data from Buoy 46001 and Tromsoflaket.

Table A-i lists the parameter values for the joint density

function as applied to the Buoy 46001 data while the function's

contour plot is shown in Figure A-I. The dotted lines in the

figure indicate the joint probability density function is zero,

and the density function becomes negative in the domain outside

this line. The parameter values as applied to the Tromsoflaket

data are listed in Table A-2 while the contour plot of the joint

density function is shown in Figure A-2. Results from these

figures show regions of negative probability density values.

Due to these negative values, the bivariate generalized gamma

distribution function, as is considered in Equation 3-19,

provides an overall poor representation of the statistical

distribution of the Buoy 46001 and Tromsoflaket wave data.

The negative probability values from the bivariate

generalized gamma distribution function are caused by the

Laguerre polynomials. A first order Laguerre polynomial has











Table A-i. Parameters for the Bivariate Generalized Gamma
Distribution Function (Buoy 46001)

f(H) .m, = 3.8881 cH = 1.0318 H = 1.3194

f() mT = 15.7159 cT = 1.3210 T = 1.1886
a = 0.0937


1 2 3 4 5 6 7 8 9 10 11 12
AVERAGE ZERO-CROSSING PERIOD IN SEC.


13 14 15


Figure A-i. Comparison of the Bivariate Generalized Gamma
Distribution Function with Buoy 46001 Data












Table A-2. Parameters for the Bivariate Generalized Gamma
Distribution Function (Tromsoflaket)


mH = 8.71 CH = 0.612 X, = 15.77

m, = 12.3566 c, = 1.2190 = 1.3017

a = 0.0935


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
AVERAGE ZERO-CROSSING PERIOD IN SEC.

Figure A-2. Comparison of the Bivariate Generalized Gamma
Distribution Function with Tromsoflaket Data












values less than zero whenever (xx)c > m; which results in some

of the observed data being inside the negative probability

regions. Adding the second order Laguerre polynomial, as is

considered in Equation 3-15, shows no appreciable contribution

in improving the fit. In general, a probability density

function given in series form has negative values at some part

of the distribution which is caused by the finite number of

terms in computing the density function. The Gram-Charlier

probability distribution which is often considered for the

probability distribution of a non-Gaussian random process is a

typical example. The bivariate generalized gamma distribution

function can be considered as another example of this type of

distribution.
















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