UFL/COEL95/015
JOINT DISTRIBUTION FUNCTION OF SIGNIFICANT
WAVE HEIGHT AND AVERAGE ZEROCROSSING
PERIOD
by
Eduardo Lewis Pasiliao, Jr.
Thesis
1995
JOINT DISTRIBUTION FUNCTION
OF SIGNIFICANT WAVE HEIGHT
AND AVERAGE ZEROCROSSING 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
LogNormal 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
11 Contingency Table of Tromsoflaket Data .............. 3
12 Contingency Table of Buoy 46001 Data ................ 4
31 Parameters for the Generalized
Distribution Function, f(H ) (Tromsoflaket) ......... 13
32 Parameters for the Generalized Gamma
Distribution Function, f(i) (Buoy 46001) ........... 14
33 Parameters for the Generalized Gamma
Distribution Function, f(H,) (Tromsoflaket) ......... 17
34 Parameters for the Generalized Gamma
Distribution Function, ff() (Tromsoflaket) ......... 18
41 Parameters for the Joint Generalized Gamma and
Conditional LogNormal Probability Distribution
Function (Buoy 46001) .............................. 32
42 Probability of Occurrence of the Wave Climate
Within a Given Contour Line (Buoy 46001) ........... 33
43 Parameters for the Joint Generalized Gamma and
Conditional LogNormal Probability Distribution
Function (Tromsoflaket) ............................ 34
44 Probability of Occurrence of the Wave Climate
Within a Given Contour Line (Tromsoflaket) ......... 35
45 Extreme Hs for given 1=7.5 s (Buoy 46001) ......... 41
46 Extreme Hs for given %=8.5 s (Buoy 46001) ......... 42
47 Extreme H, for given Z=9.5 s (Buoy 46001) ......... 43
48 Extreme H, for given Z=7.5 s (Tromsoflaket) ....... 45
49 Extreme Hs for given %=8.5 s (Tromsoflaket) ....... 46
410 Extreme H, for given %=9.5 s (Tromsoflaket) ....... 47
Ai Parameters for the Bivariate Generalized Gamma
Distribution Function (Buoy 46001) ................ 51
A2 Parameters for the Bivariate Generalized Gamma
Distribution Function (Tromsoflaket) .............. 52
LIST OF FIGURES
Figure page
31 Comparison of Generalized Gamma
Distribution Function with H. (Tromsoflaket) ....... 13
32 Comparison of Generalized Gamma
Distribution Function with Hs (Buoy 46001) ......... 14
33 Comparison of Generalized Gamma
Distribution Function with Z (Tromsoflaket) ....... 17
34 Comparison of Generalized Gamma
Distribution Function with X (Buoy 46001) ......... 18
35 Parameter g as a function of H. (Tromsoflaket) ..... 27
36 Parameter (T as a function of H. (Tromsoflaket) ..... 27
37 Parameter p as a function of Hs (Buoy 46001) ....... 28
38 Parameter a as a function of H. (Buoy 46001) ....... 28
41 Comparison of the Joint Generalized Gamma
and Conditional LogNormal Distribution Function
with Buoy 46001 Data ................................ 32
42 Comparison of the Joint Generalized Gamma
and Conditional LogNormal Distribution Function
with Tromsoflaket Data .............................. 34
43 Comparison of f(T) Represented by the Generalized
Gamma Distribution Function with that Evaluated
from the Joint Density Function (Buoy 46001) ....... 36
44 Comparison of f(3) Represented by the Generalized
Gamma Distribution Function with that Evaluated
from the Joint Density Function (Tromsoflaket) ..... 37
45 Extreme Hs for given Z=7.5 s (Buoy 46001) ......... 41
46 Extreme H. for given ,=8.5 s (Buoy 46001) ......... 42
47 Extreme Hs for given '=9.5 s (Buoy 46001) ......... 43
48 Extreme Hs for given ==7.5 s (Tromsoflaket) ....... 45
49 Extreme Hs for given 'Z=8.5 s (Tromsoflaket) ....... 46
410 Extreme H, for given '=9.5 s (Tromsoflaket) ....... 47
Ai Comparison of the Bivariate Generalized Gamma
Distribution Function with Buoy 46001 Data .......... 51
A2 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 ZEROCROSSING 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
longterm 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 zerocrossing 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 zerocrossing 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 lognormal 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 longterm wave climate is
necessary not only for evaluating a possible structural failure
associated with the extreme sea state but also for estimating
the waveinduced loading which may cause a possible fatigue
failure of the structure. The development of a reliable means
of predicting longterm wave conditions is made difficult by the
great variations in environmental factors between different
geographic locations.
Description and prediction of longterm wave climate is
commonly achieved through joint probability distribution
functions with significant wave height and average zerocrossing
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 zerocrossing period which reliably predicts longterm
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 zerocrossing 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 zerocrossing
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 11 and 12, 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 11.
Contingency Table of Tromsoflaket Data
Source: Mathisen and BitnerGregerson (1990)
Sign. Interval of zeroupcrossing 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
11.511.9 0 0 0 0 0 0 0 0 1 1 0
11.011.4 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
9.09.4 0 0 0 0 0 0 1 7 1 0 0
8.58.9 0 0 0 0 0 0 2 8 0 0 0
8.08.4 0 0 0 0 0 1 11 3 0 0 0
7.57.9 0 0 0 2 0 4 11 6 1 1 0
7.07.4 0 0 0 1 0 12 22 4 3 0 0
6.56.9 0 0 0 0 0 35 21 3 1 0 0
6.06.4 0 0 0 0 7 53 19 3 0 0 0
5.55.9 0 0 0 0 29 90 13 0 0 0 1
5.05.4 0 0 0 2 107 104 18 2 0 2 0
4.54.9 0 0 0 24 259 95 22 2 2 1 0
4.04.4 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
3.03.4 0 1 187 776 234 61 13 1 1 0 0
2.52.9 0 20 688 804 204 43 8 3 1 0 0
2.02.4 3 188 1505 700 181 42 4 1 0 0 0
1.51.9 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
0.50.9 337 591 453 124 12 2 0 0 0 0 0
00.5 4 17 6 4 0 1 0 0 0 0 0
Table 12. 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
9.510.0 0 0 0 0 0 0 0 0 2 23 4 4 0 0 0 0 33
10.010.5 0 0 0 0 0 0 0 0 0 7 4 5 0 0 0 0 16
10.511.0 0 0 0 0 0 0 0 0 0 2 3 2 0 0 0 0 7
11.011.5 0 0 0 0 0 0 0 0 0 2 1 0 2 0 0 0 5
11.512.0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2
12.012.5 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 zerocrossing 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 lognormal 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
zerocrossing 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(lIH); however, a comparison between the
proposed model and measured data does not appear promising.
Harver (1985) proposes a consideration of the lognormal
and Weibull distributions for f(H,), and the lognormal
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 lognormal 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
lognormal 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 3parameter Weibull
distribution for both f(H,) and f(~IH,) yields the best fit to
the data. Mathisen and BitnerGregersen (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 3parameter Weibull distribution for f(H,) and
the lognormal 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 BitnerGregersen claim the 3parameter
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 BitnerGregersen give the value of 'a' as
0.98 meters for the Tromsoflaket data. Thus, the 3parameter
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 3parameter 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 lognormal distribution (Ochi 1978),
the modified lognormal distribution (Fang and Hogben 1982), the
threeparameter Weibull distribution (Burrows and Salih 1986)
(Mathisen and BitnerGregersen 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 (31)
and its cumulative distribution function as
rF(m, (hx)')
F(x) = (32)
r(m)
r(m), the gamma function, is defined as
r(m) = j t1 exp{t} dt (33)
0
and F(m, (Xx)C), the incomplete gamma function, is defined as
r(m, (Xx)C) = tm1 exp{t} dt (34)
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 31. 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 31. Comparison of the Generalized Gamma
Distribution Function with H, (Tromsoflaket)
Source: Ochi (1992)
v.vJ
14
Table 32. 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 32. Comparison of the Generalized Gamma
Distribution Function with H. (Buoy 46001)
r
I I ;
in Figure 31 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 32, and the
parameter values are listed in Table 32. 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 longterm 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 zerocrossing 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 zerocrossing period, f( ), and then derive the joint
probability density function from the combination of the
marginal distributions of significant wave height and average
zerocrossing period while taking the correlation coefficient
into consideration. Another approach is to develop the
conditional distribution of the zerocrossing 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 zerocrossing period. The joint probability
distribution functions of significant wave height and average
zerocrossing 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 zerocrossing 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 33, and the parameter values
are listed in Table 33. The figure indicates that the average
zerocrossing 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 34 with the
parameter values listed in Table 34. Average zerocrossing
period may, therefore, be assumed to be represented by the
Table 33. 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 ZEROCROSSING TIME IN SEC.
Figure 33. Comparison of the Generalized Gamma
Distribution Function with Z, (Tromsoflaket)
18
Table 34. 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 ZEROCROSStNG TIME IN SEC.
Figure 34. 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 jth moment
of the generalized gamma distribution function is given by
1 r(m + /c)
E[xj] (rTm) (35)
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 (36)
F(m)r(m + 4/c) E[x4]
 (37)
(m + c)2 E[x212
1 (m + 2/c)2
SEx21/2 (m)2 (38)
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 (39)
.{1+ a, L'[(y1)] L1[(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 + )(30)
a, = L (310)
r(m2,)(m, + 1)
r(m)r(m2 + 2)
a2 = m2 m2 + 2) (311)
r(mr(m,1 + 2)
In general, the Laguerre polynomials have the following
orthogonal property
0 for i j
yU expy} L(a)y] La)[y] dy = (a + i + 1) (312)
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
(313)
and
y, = (TT) (314)
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) } (315)
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
L1[(x)c] = m (Xx) (316)
L[(lx)C] = 1 (m + l)m (m + 1)(Xx)c + (X)2c (317)
2
and have the following property
J c(x) p(x)} Lm1[(x)] dx = 0 (318)
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 zerocrossing
period.
Although Equation 315 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 XH)m ex"p (IH)c} C ex (XTT) (319)}
Sr(mT)T (319)
.1 + a L',1 [(H)c] L1[,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 +  (320)
cycT
Then by using the following expression for the correlation
coefficient
E[HT] E[H] E[T] (3
P (321)
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 (322)
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
seastate conditions. The results of numerical computations of
this joint density function is presented in the appendix.
3.3 Joint Generalized Gamma
and Conditional Lognormal 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 zerocrossing period for a given
significant wave height. The results of analysis show that the
conditional cumulative distributions of the zerocrossing 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 lognormal 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 (323)
0 T
and its theoretical conditional moments are given by
E[TI H] = exp + j2f (324)
The parameters, g and 0, are functions of significant wave
height and are obtained by plotting the cumulative conditional
distributions on lognormal paper. The g and 0 parameters can
then be evaluated by the following formulae
g = In x0.50
(325)
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 lognormal paper.
To determine the relationship between significant wave
height and the conditional lognormal parameters, the JL and G
values are plotted as functions of significant wave height.
Machines and BitnerGregerson carried out the conditional
probability distribution of zerocrossing period for given
significant wave heights by similarly applying the lognormal
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 (327)
S= a, + b, ecH (328)
The g and G values evaluated from various significant wave
heights from the lognormal plots and the leastsquare fitted
curves are shown in Figures 35 and 36 for Tromsoflaket and in
Figures 37 and 38 for Buoy 46001. The plots include the
E
1.7
1.6
1.5
1.4
1.3
0
Figure 35.
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 36.
(Tromsoflaket)
0 1 2 3 4 5 6 7
Significant Wave Height in meters
Figure 37.
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 38.
leastsquare curves representing Equations 327 and 328. 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
zerocrossing period. That is,
f(H, T) = f(H) f(TI H)
(329)
When the generalized gamma distribution function and the
conditional lognormal distribution function are substituted
into Equation 329, 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 (330)
F(m)H J27T 2 02
When applied to the Buoy 46001 data and the Tromsoflaket
data, the joint generalized gamma and conditional lognormal
distribution function provides a very good representation of
seastate 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 lognormal 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 zerocrossing period evaluated from the joint density
function is also presented.
Table 41 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 41. 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 41. Parameters for the Joint Generalized Gamma and
Conditional LogNormal 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 ZEROCROSSING PERIOD IN SEC.
Figure 41. Comparison of the Joint Generalized Gamma and
Conditional LogNormal Distribution Function
with Buoy 46001 Data
Table 42. 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 42
for the Buoy 46001 data. As can be seen from this table, the
contour line of 1 X 106 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 43 lists the parameter values for the joint density
function as applied to the Tromsoflaket data while the contour
plot is shown in Figure 42. 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 43. Parameters for the Joint Generalized Gamma and
Conditional LogNormal 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 ZEROCROSSING PERIOD IN SEC.
13 14 15
Figure 42. Comparison of the Joint Generalized Gamma and
Conditional LogNormal Distribution Function
with Tromsoflaket Data
Table 44. 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 41 and 42 are the peak values
as well as its location with respect to significant wave height
and average zerocrossing 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 zerocrossing
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 zerocrossing period
which is most likely to occur.
Finally, the marginal distribution function of average
zerocrossing 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 ZEROCROSSING TIME IN SEC.
Figure 43. 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 ZEROCROSSING TIME IN SEC.
Figure 44. 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 zerocrossing
period data, Fk(), and is represented by the generalized gamma
distribution function as shown in Figures 33 and 34. Another
marginal distribution function is evaluated from integrating the
joint generalized gamma and conditional lognormal distribution
function with respect to significant wave height, IF(HS, ) dH,.
Comparisons of the two marginal distributions are shown in
Figure 43 for the Buoy 46001 data and Figure 44 for the
Tromsoflaket data. Overall, the difference between the marginal
distribution functions of average zerocrossing 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 zerocrossing period increase so
that the two marginal distribution functions are almost
identical at large extreme values of zerocrossing 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 zerocrossing
period close to a natural frequency may be evaluated from the
conditional distribution of significant wave height for a given
average zerocrossing 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 (41)
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 41 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 (41) 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) (42)
where X0 is also referred to as the risk parameter.
If the following approximation is made
1 (1 a)2N = /N (43)
the design extreme value with a specified risk parameter may be
found from the following expression
1
F( ) = N/ (44)
1 F(Hj ,)
Several calculations of both probable and design extreme
value of significant wave height for given zerocrossing periods
at Buoy 46001 are listed in Tables 46, 47, and 48 while the
Table 45.
Extreme H, for given '=7.5 s
(Buoy 46001)
P12
S10
*~i
    < 100 yoa probably  I  
I I
I
S :" 100 year design
*  50iyeadesign 
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 45.
Table 46.
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  
81
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 46.
43
Table 47. 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. 10.010.5 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 47.
Extreme H, for given =9.5 s
(Buoy 46001)
corresponding plots are shown in Figures 45, 46, and 47. The
examples are for average zerocrossing 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 48, 49, and 410 with the corresponding plots
shown in Figures 48, 49, and 410.
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 (41), should
not be considered in marine structure design; the design extreme
values, as given in Equation (44), with a small specified risk
parameter provide the appropriate value for design
consideration.
Table 48. 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, 6.06.5 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!~~~Il11
SI I M
Figure 48.
Extreme H, for given T=7.5 s
(Tromsoflaket)
Table 49.
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 49.
Table 410.
Extreme H, for given =9.5 s
(Tromsoflaket)
12r 
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 410.
(Tromsoflaket)
CHAPTER 5
CONCLUSION
This study discusses the development of a joint
probability distribution function of significant wave height and
average zerocrossing period to reliably describe longterm sea
severity. Two joint density functions are derived through
different approaches. One approach combines the marginal
distribution function of significant wave height and average
zerocrossing 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 zerocrossing period which produces the
joint generalized gamma and conditional lognormal distribution
function. This joint density function provides an excellent
description of sea severity.
Numerical examples of the joint generalized gamma and
conditional lognormal 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 zerocrossing period which is most likely
to occur. From the comparison of the two methods of obtaining
the marginal distribution function of average zerocrossing
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 zerocrossing period.
Examples of estimating extreme sea severity through the
conditional distribution function of significant wave height for
a given average zerocrossing 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 lognormal 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 Ai 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 AI. 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 A2 while the contour plot of the joint
density function is shown in Figure A2. 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 319,
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 Ai. 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 ZEROCROSSING PERIOD IN SEC.
13 14 15
Figure Ai. Comparison of the Bivariate Generalized Gamma
Distribution Function with Buoy 46001 Data
Table A2. 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 ZEROCROSSING PERIOD IN SEC.
Figure A2. 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 315, 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 GramCharlier
probability distribution which is often considered for the
probability distribution of a nonGaussian 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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