Citation
Joint distribution function of significant wave height and average zero-crossing period

Material Information

Title:
Joint distribution function of significant wave height and average zero-crossing period
Series Title:
UFLCOEL-95015
Creator:
Pasiliao, Eduardo Lewis, 1969-
University of Florida -- Coastal and Oceanographic Engineering Dept
Place of Publication:
Gainesville Fla
Publisher:
Coastal & Oceanographic Engineering Dept., University of Florida
Publication Date:
Language:
English
Physical Description:
ix, 55 p. : ill. ; 28 cm.

Subjects

Subjects / Keywords:
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

Thesis:
Thesis (M.E.)--University of Florida, 1995.
Bibliography:
Includes bibliographical references (p. 54-55).
Statement of Responsibility:
by Eduardo Lewis Pasiliao, Jr.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
34531829 ( oclc )

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(Hs) (Tromsoflaket) .........13 3-2 Parameters for the Generalized Gamma
Distribution Function, fft) (Buoy 46001) ........... 14
3-3 Parameters for the Generalized Gamma
Distribution Function, f(Hs) (Tromsoflaket) .........17 3-4 Parameters for the Generalized Gamma
Distribution Function, fft) (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 =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 T0=7.5 s (Tromsoflaket) .......45




4-9 Extreme Hs for given %=8.5 s (Tromsoflaket) ......... 46
4-10 Extreme H. for given Z=9.5 s (Tromsoflaket) ......... 47
A-1 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 H, (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 ( as a function of H. (Tromsoflaket) ..... 27 3-7 Parameter g as a function of H. (Buoy 46001) ....... 28 3-8 Parameter ( 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(TI) Represented by the Generalized
Gamma Distribution Function with that Evaluated
from the Joint Density Function (Buoy 46001) ....... 36 4-4 Comparison of f( ) Represented by the Generalized
Gamma Distribution Function with that Evaluated
from the Joint Density Function (Tromsoflaket) 37
4-5 Extreme H, for given 'Z=7.5 s (Buoy 46001) ......... 41




4-6 Extreme Hs for given -Z=8.5 s (Buoy 46001) ......... 42
4-7 Extreme Hs for given 'Z=9.5 s (Buoy 46001) ......... 43
4-8 Extreme Hs for given T%=7.5 s (Tromsoflaket) ....... 45 4-9 Extreme H. for given 'Z=8.5 s (Tromsoflaket) ....... 46 4-10 Extreme Hs for given Z=9.5 s (Tromsoflaket) ....... 47 A-1 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 Ochils (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
1Itjm] 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




(D
rt, Cr
rt,
0 a




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, 'Z, 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 -Z. 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 1, 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(H.), and the conditional probability density function of the average zero-crossing period for a given significant wave height, f(_T1Hs. 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(JHs); 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(H3), and the log-normal distribution for the conditional distribution f(%IHs). Here, the distribution of significant wave height f(Hs) 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(%IH.), is a serious shortcoming of the proposed method.
Burrows and Salih (1986) compare measured data with
various combinations of distributions of f(H5) and f(TIH), and




conclude that the combination of the 3-parameter Weibull distribution for both f(Hs) and f( IH,) yields the best fit to the data. Mathisen and Bitner-Gregersen (1990) also compare
measured data with various combinations of f(Hs) and f(@IH.) 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, < -, 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 zerocrossing 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 zerocrossing 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




c(1x).
f(x) = c(x) exp x)Cj}
F(m)x (3-1)
0 5 x < o
and its cumulative distribution function as
F(x) = (m (x)c) (3-2)
r (m)
F(m), the gamma function, is defined as
F(m) = tm-1 exp{-t} dt (3-3)
0
and f(m, (ux)C), the incomplete gamma function, is defined as
(XX
F(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(H.) (Tromsoflaket)
Source: Ochi (1992)

0.9999! 17

0.000
0.95
0.90
0.80 1

2 4 6 8 10
SIGNIFICANT WAVE HEIGHT IN M.

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

+

#% #%#%1

0.050




14
Table 3-2. Parameters for the Generalized Gamma
Distribution Function, f(Hs) (Buoy 46001) mH= 3.8881
CH = 1.0318 X= 1.3194

r~

0.999 .
0.99
0.90
F ____

U.OU 0.50
0.20 0.10 0.01,

2 4 6 8
SIGNIFICANT WAVE HEIGHT IN M.

10 12

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

-- 4

U




in Figure 3-1 while the parameter values are listed in Table 31. 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(T.), 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(TIH.), 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 zerocrossing 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)

mT = 12.3566 C = 1.2190 T = 1.3017

0.9999 0.999
0.99
0.95
0.90
0.80
i
0.50i
0.20
0.10- /
/0
0.01
3 4 5 6 8 10 12 15
AVERAGE ZERO-CRDSSING 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(T) (Buoy 46001) mT = 15.7159 cT = 1.3210 XT = 1.1886

0.9999, 1 1 1 1 1

0.c
0

999
.99 K/

0.95 /
0.90-0.80
0.50
0.20_0.10
0.01 / I n _1
0.001)

A AJ v S NT I
AVERAGE ZERO-CROSSING TIME IN SEC.

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

J




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
E[xJ] = M F(m) (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)Y2fIm + 3/C) E[x3]
r'm + 2c)- E [x 2] 2 (3- 6 )




F(m)F(m + 4/c) E[x']
= (3-7)
F(m + 2/C)2 E[x2]2
1 F(m + 2/c)12
- X2]/ r(m)2 (3-8)
E[x22 m
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 ) 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(Y1, Y21 (Yl -exp{-(YI)} (Y2)}
f(y1, y2) =(1f1 Brpm-?y2 eXp{-(y2)}(9
S I~mJLy)l ~2Y 2 (3-9)
{i + a, [(yI )] .L'-[(y2)] + a2 -1 (1 ) 2 [ 2)]}




m > m2
0 > X < i
where L, and L2 are the Laguerre polynomials and parameters a, and a2 are obtained from the following relationships given by Sarmanov.
IF(mi)F m2 +i
a, = ~1(m)m2 + i) (3-10)
FTm2)F m1 + 1)
a2 = m1)Fm2 + 2) (3-11)
2 m2)1m1 + 2)
In general, the Laguerre polynomials have the following orthogonal property
0 for i j
Sya exp{-y}. L(a)[Y] L(a)[y] dy = F( + i + 1) = (3-12)
0 for i
By writing the significant wave height and the average zerocrossing period as H and T, respectively, and letting the random variables

yl = (( H)'

(3-13)




and
Y2 = (TT)' (3-14)
a joint generalized gamma probability function can be transformed from the joint gamma function as follows
f(H, T) = c H) exp-(H)cs} c ( exp{-(,T) (3-15)
1-(M H)H XF-XH'(MT)T T) 1 3j5
-1 ', .Hy o. CT]L
* + a La. [(n, ,H)a] ,-1[(XT) ] + a2 L.-1 H c2T T)T
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
I-1[(x)c] = m (Xx)c (3-16)
L-1[(?x)_] = (m + 1)m (m + 1)()x)c + (Xx)2c (3-17)
2
and have the following property
J c( exp{-(x)c} L -1[(x)] dx M 0 (3-18)
0 r(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(H)m e Xp-(XHH)C'} exp-(T T) (3-19)
+.1 + a L'- [(HH)ca ] L -1[( TT)c
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 Z by taking the expected mean of the bivariate generalized gamma distribution which results in
a
E[HT] = E[H] E[T] + --- (3-20)
CHCT




Then by using the following expression for the correlation coefficient
P E[HT] E[H] E[T] (3-21)
p E[H 2 E[H1 -E[T] E[T1
and after some algebraic manipulation, the parameter 'a' can be expressed as a function of the correlation coefficient as follows
j rrtH)FMH + 2/cH) [mT)Fmr + /CT)
a mP). mH 1. cT . ..T (3-22)
a =/ + cm cH)' T + c)
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 lognormal paper may be approximately represented by straight lines. The conditional distribution may, therefore, be assumed to follow the log-normal probability law. The conditional lognormal probability density function, in this case, is given as
f(TH) 1 {1 (In T-)}
f (TI H) = 1 expf (i
271T 2 2 (3-23)
0 T and its theoretical conditional moments are given by
E[TJI H] = exp + 1 j202 (3-24)
The parameters, g and G, are functions of significant wave height and are obtained by plotting the cumulative conditional distributions on log-normal paper. The i and G parameters can then be evaluated by the following formulae

= in x0.50

(3-25)




a ln 1(XO.5 + X.4(3-2 6) 2\ X016 X0.50
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 L 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 C as a function of significant wave height. These are given as,
= a + b. H" (3-27)
G= a. + bo ecG (3-28)
The p. 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




2 3 4 5
Significant Wave Height in meters

Figure 3-5.

0.25
0.2
Ca
E 0.15
U,

0.1 -

Parameter g as a Function of Hs

(Tromsoflaket)

12 3 4 5
Significant Wave Height in meters
Parameter G as a Function of Hs

0
~~0"0 ~.0
0
0
za

6 7

I I I I I I

0

0 o
o

U

0

(Tromsoflaket)

Figure 3-6.

5 l




0 12 3 4 5 6 7
Significant Wave Height in meters

Figure 3-7.

Ca.1
E,

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 g 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 zerocrossing 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.
ex,% 1~ 1_____T(3-30)
f(H, T) =c(H)m exp-(H)} 1/ {- ( nT( 30)2}
F(m)H 27ET 2 2 3
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, H., 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.) M,= 3.8881 CH = 1.0318 X = 1.3194
fQ IjH) a, = 1.2605 b = 0.4286 c, = 0.4161 a. = 0.0994 bo = 0.1326 Ca = -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 44. Estimated probability of occurrence of the wave climate within the contour line of 1 x 10-6 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 CH = 0.612 X = 15.77
f(TIH.) 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




-f F (HsTo d, /
< 0.50 i
0 /
0.20
O~ I /I
0.01 t4
0.0011
3 4 5 6 8 10 12 15 AVERAGE ZERO-CROSSING TIME IN SEC.
Figure 4-3. Comparison of F(q) Represented by the
Generalized Gamma Distribution Function with that
Evaluated from the Joint Density Function (Buoy 46001)




0.80 (HST H
- fF (HsT0) dHs
0.20
0.1c
0.01
3 4 5 6 8 10 12 15
AVERAGE ZERO-CROSSING TIME IN SEC.
Figure 4-4. Comparison of F(%) 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, F(T.), 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, fF(H.I _%) -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 zerocrossing 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.1 T). 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 F(H.I'%) :-- N (4-1)
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-1 = 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, C, is evaluated from the cumulative distribution function by the following expression
F(H.I t) = (1 (4-2)
where a is also referred to as the risk parameter. If the following approximation is made
1 (i a)11V = aN (4-3)
the design extreme value with a specified risk parameter may be found from the following expression
1 s =) N/a (4-4)
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




Extreme H. for given Z=7.5 s

15k ---------------

14 --------------

131 - - - -

E-P 12 11 10
9
8

I_ . . .-,-. . .

~ij liii IL i~ ~Zy~ pic

-- 100 year design
- - --- 50yearTdesign- -
0 yea pobable -------------

't --
-f -

7
/

3 9 10 11 12 13
SIGNIFICANT WAVE HEIGHT IN METERS

Extreme H, for given Z=7.5 s

--' 50 mar roable

17.

Table 4-5.

(Buoy 46001)

I

II

I

Figure 4-5.

(Buoy 46001)




Table 4-6.

Extreme Hs for given Z=8.5 s

(Buoy 46001)

16-----15 -

14 13
12

I.,P
10
r-4 9
8 7

, ,- ---- 10p eardepign"
' -- _5Q ar design
.. :" r

SI f I
------------ -- ------------ --Sa t-- 100 year probable |
4 4 - - -4
..<--50 ydar probate
a t a/

a 3 year probable
- - - - .-
a .
ea
- -- - - -

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

Extreme Hs for given =8.5 s (Buoy 46001)

- - -

Figure 4-6.




Table 4-7.

Extreme H. for given 1=9.5 s

(Buoy 46001)

:<-41'
500y(rdsg

- - - - -
i ..' "
I I..
- ---- --- ---- --' -- -
k- 501ea pO~erobabe I
a a / I .. a
a a 13 ye ar p~ab e
r- -
/,
' --:- 10y rI~oal
a.a a a
a' a'1: yea a abl
II
a a a a
a a"ar
.;aa
..:" I a

00 year design aardeign.r

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

Extreme H. for given 1=9.5 s

1 r

14 F -

13
12 11 10

E-p

Figure 4-7.

(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 H. 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 (=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 Hs 6.0-6.5 m
15 ,-- ob year design
14--- --- -s ---
14, 7 erpoal ',"
13--- - - ---
* a /" a
12- - -----------------
5I 1 1 11
11 - - ,---- -
10--------------- --- IOa.poba~e4 ,
7 ye 50 yearpb '
* I i
7 - "- -
6- -., - -1 III
67 8 9 10 11 12
SIGNIFICANT WAVE HEIGHT IN METERS
I II
I I Ii
a "T"" I a
. a: a a a a
6 7 8 9 10 11 12
SIGNIFICANT WAVE HEIGHT INl METERS

Extreme H. for given =7.5 s

(Tromsoflaket)

Figure 4-8.




Table 4-9.

Extreme Hs for given %=8.5 s

(Tromsoflaket)

13- -------- -- ------ -- -12

3 9 10 11
SIGNIFICANT WAVE HEIGHT IN METERS

Extreme H. for given Z=8.5 s (Tromsoflaket)

11 10
9

- 100 year design
- 50.year.deign -

I I' I
* I
- - - - --- -- - - - ---- - -.. 4- 100 yearprobable
-- 50 yeat probable '
. -A- L. .
I .; a a a a
a

* .-r-- 7 y r probable a
i i i a
i .: *---------------

-

12 13

4 1'1'

Figure 4-9.




Table 4-10.

Extreme H. for given Z=9.5 s

(Tromsoflaket)

100 tar desi n S- -- a desig r -
a

10 . . . . . . . -.-.-. .
8 - - - - - -1 -10 year probable -1
-50 ye probable
7 - - - --- --
Sa I II
6 ------ -- - --
- 7 year probable a probab
I I
i a a ai ia/ :< 50 N~ai
7---------r-proab.e
a a
a a a a,
6-----2 -----.---.

10 1 1 1
a I I

9 10 11 12 13
SIGNIFICANT WAVE HEIGHT IN METERS
Extreme H, for given "=9.5 s

- - r - -

Figure 4-10.

14 15
(Tromsoflaket)

~~~~1

E-I
I

1".4 .,

2 ~ .

12 - -
11 . .




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.) mH = 3.8881 cH = 1.0318 XH = 1.3194
fQ ) mT = 15.7159 CT = 1.3210 XT = 1.1886 a = 0.0937

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
AVERAGE ZERO-CROSSING PERIOD IN SEC.
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 XH = 15.77 mT = 12.3566 CT = 1.2190 XT = 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 (?x)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.




REFERENCE LIST

Burrows, R. and Salih, B.A. (1986), "Statistical Modeling
of Long-Term Wave Climates", Proc. 20th Int. Conf.
on Coastal Engineering, Vol. 1, 42-56.
Fang, Z., Dai, S. and Jin, C. (1989), "On the Long-Term Joint
Distribution of Characteristic Wave Height
and Period and Its Application", Acta Oceanologica
Sinica, Vol. 8, No. 3, 315-325.
Fang, Z.S. and Hogben, N. (1982), "Analysis and Prediction
of Long Term Probability Distributions of Wave Heights
and Periods", Nat. Mar. Inst. Report R146.
Harver, S. (1985), "Wave Climate off Northern Norway",
Applied Ocean Research, Vol. 1, No. 2, 85-92.
Houmb, O.G. and Overvik, T. (1976), "Parametrization
of Wave Spectra", Proc. Conf. on Behavior Ship
& Offshore Structures, Vol. 1, 144-169.
Mathisen, J. and Bitner-Gregersen, E. (1990), "Joint
Distribution for Significant Wave Height
and Wave Zero-Crossing Period", Applied Ocean Research,
Vol. 12, No. 2, 93-103.
Ochi, M.K. (1978), "On Long-Term Statistics for Ocean
and Coastal Waves", Proc. 16th Int. Conf. on Coastal
Engineering, Vol. 1, 59-75.
Ochi, M.K. (1992), "New Approach for Estimating the
Severest Sea State from Statistical Data",
Proc. 23rd Int. Conf. on Coastal Engineering.
Ochi, M.K. and Whalen, J.E. (1980), "Prediction of the Severest
Significant Wave Height", Proc. 17th Int. Conf. on
Coastal Engineering, Vol. 1, 587-599.
Sarmnaov, 1.0. (1968), "A Generalized Symmetric Gamma
Correlation", Soviet Mathematics Doklady, Vol. 9,
No. 2, 547-550.
Stacy, E.W. and Mihram, G.A. (1965), "Parameter Estimation
for a Generalized Gamma Distribution", Technometrics,




55
Vol. 7, No. 3, 349-358.
Thompson, E.F. and Harris, D.L. (1972), "A Wave Climatology
for U.S. Coastal Waters", Proc. Offshore Tech. Conf.,
Vol. 2, 675-688.