UFL/COELTR/100
PROBABILITY DISTRIBUTIONS OF PEAKS
AND TROUGHS FOR RESPONSES OF
NONLINEAR SYSTEMS
by
M.K. Ochi
May 1994
Naval Facilities Engineering Service Center
Contract N4740891C1204
(Phase III Study)
PROBABILITY DISTRIBUTIONS OF PEAKS AND TROUGHS
FOR RESPONSES OF NONLINEAR SYSTEMS
by
M. K. Ochi
Naval Facilities Engineering Service Center
Contract N4740891C1204
(Phase III Study)
May 1994
TABLE OF CONTENTS
Page
INTRODUCTION 1
DERIVATION OF PROBABILITY DISTRIBUTION OF PEAKS 3
DERIVATION OF PROBABILITY DISTRIBUTION OF TROUGHS 10
CONCLUSIONS 11
ACKNOWLEDGEMENTS 11
APPENDIX: EVALUATION OF VARIANCES ASSOCIATED WITH
PEAKS AND TROUGHS ENVELOPE PROCESS 12
REFERENCES 16
I
ABSTRACT
This paper deals with the development of probability density functions
applicable for peaks and troughs of the responses of a nonlinear system in
closed form. The derivation is based on the probability density function
representing a nonGaussian random process developed by the author in 1994.
A comparison of the newly developed probability function and the histogram
constructed from a record indicating strong nonGaussian characteristics shows
satisfactory agreement for both peaks and troughs of the random process.
INTRODUCTION
This study presents the analytical development of probability density
functions for peaks and troughs of the responses of a nonlinear system.
In developing the probability density function of the response of a
nonlinear system, in general, it is highly desirable that probability
distribution of peaks and troughs are presented separately. This is because
the statistical information of peaks (or troughs) only is necessary for the
design of many nonlinear systems. For example, for estimating the wave
induced extreme force working on cables of a moored system, the magnitude of
tension forces (peaks) provides information vital for the design.
Only a limited number of studies has been carried out on the probability
distribution of peaks and troughs of a nonGaussian random process. In these
studies, the probability density functions are derived numerically based on
the joint probability density function of displacement and velocity of the
response, f(y, y), assuming that the expected number of peaks is equal to
that of upcrossing a specified level. Ochi and Malakar (1984) applied the
FokkerPlanck equation to evaluate the nonlinear response of a tensionleg
platform in a sea and obtained the solution f(y, y) numerically, and therefrom
evaluated the probability density function of peaks. In applying the Fokker
Planck equation, the input to the nonlinear system is a white noise spectrum.
If we assume that displacement y and velocity y are statistically
independent, then the probability distribution of peaks and troughs may be
obtained from the probability density function of y only. Kato et al. (1987,
1
1989) evaluated the probability distribution of peaks and troughs of the
responses on a moored ship in random seas based on the probability density
function f(y) obtained by the KacSiegert solution.
On the other hand, several studies have been made on the probability
distribution of peaks and troughs of nonGaussian random waves observed in
finite water depth assuming that wave profiles follow the Stokes expansion to
the 2nd order exponents; Tayfun (1980), Arhan and Plaisted (1981), among
others. Although these distributions are given in closed form, the
distributions are developed by imposing a preliminary form on a random process
which appears to be inappropriate, in general.
As this short review of the literature reflects, the probability density
functions of peaks and troughs have not been developed in closed form, and
thereby the mathematical relationship between the probability density function
representing a nonGaussian random process and that of peaks (or troughs)
cannot be clearly identified, and prediction formula for evaluating extreme
values cannot be obtained.
In the present study, the probability density functions of peaks and
troughs of the responses of a nonlinear system are analytically derived in
closed form based on the probability density function applicable for a non
Gaussian random process developed by the author (Ochi and Ahn 1994). The
probability density function of peaks (and troughs) reduces to the Rayleigh
probability distribution for a Gaussian random process.
DERIVATION OF PROBABILITY DISTRIBUTION OF PEAKS
Prior to the derivation of a probability distribution, it may be well to
outline the approach considered in the present study.
The statistical properties of the peaks of a nonGaussian random process
are significantly different from that of the troughs. Therefore, the
derivation of the probability distributions applicable to peaks and troughs
will be considered independently. It is assumed that the random process is
narrowbanded and hence there exists an envelope. However, we consider
envelopes for peaks and troughs separately. In other words, the statistical
properties of peaks are represented by a peak envelope process, C(t), while
those of troughs are represented by a trough envelope process, n(t). As an
example, Figure 1 shows a pictorial sketch of the peak envelope process ((t).
As seen, one side of the envelope covers peaks. The statistical properties of
peaks are evaluated for the amplitude of this envelope process. Therefore,
the variance for this envelope process should be that affiliated with the peak
side only of a nonGaussian random process y(t). The variance, denoted by a,
can be evaluated by taking the conditional second moment of y(t), truncated at
y Q. That is, assuming that the peaks are on the positive side,
S y2 f(y) dy
a2
J f(y) dy
0
= { 1 (Ac.)2 +VTh (a.) (A//u,)4 } 3
2^ i/. J^ 7 2
where A, ay, 7 1.28 for the positive y
S cumulative distribution function of standardized normal
distribution.
The details for evaluating u2 are given in Appendix.
The probability distribution applicable to deviations from the mean
value, y, of a nonGaussian random process can be expressed as a function of a
single normal random variable and its squared quantity (Ochi and Ahn 1994).
That is,
Y = U + aU2 (2)
where "a" is a constant and U is a normal random variable with mean p* and
variance a2; all of these parameters can be evaluated from a knowledge of the
2nd and 3rd moments of a nonGaussian record. Then, the functional
relationship given in Eq.(2) is inversely expressed as
1 yaY 1 ,Y (3)
U (1e ) = (1 e (3)
ya T,
Subtracting the mean value p. from the random variable U, we may write
1 X1Y (4)
V =Up. = ( le )
For the derivation of the probability distribution applicable to peaks
of a nonGaussian random process, let us consider a narrowband random process
V(t) which is a Gaussian random process with zero mean and variance o. For
the Gaussian random process, we may write
V(t) = E (acos not + bnsin nwt)
a = V(t).cos nut
an==
bn=.2 IV(t).sin nw't
Here, the coefficients an and bn are normally distributed with zero mean and
variance a2.
Further, the assumption of a narrowband random process with frequency
.o, V(t) can be written in terms of amplitude A(t) and phase E(t) as
V(t) = A(t) cos{ wt + s(t)}. (6)
We may write Eq.(6) as
where
Vs(t) = A(t) sinE(t).
Since V. and V, are statistically independent normal variates for a given
time, the joint probability density function of Vc and Vs is given by
f(vcvs) exp 2 2)
2ir1 2u1
V(t) = Vo(t) cos ,t V,(t) sinUo(t)
V,(t) =A(t) cos (t)
On the other hand, from Eq.(4), we may write the components Vc and V, as
follows:
Ve(t) = {(1A.) e }
(10)
V,(t) = {(1 .)
By applying the change
probability density function
(10) as
e c }.
of random variables technique, the joint
of YC and YS can be obtained from Eqs.(9) and
1(1 Al'z.)2 1 Alp..( yo A1ys
f(y.,y.) 2 exp A ( + y) + __ e + e
12r1 11 l
 1 (e2AY + e2ls) },
2\i 1 1
m < yc, < .
For the random process y(t), we may write
Y = ( cos r
(12)
Ys = E sin r
where ( amplitude (peak), r phase.
From Eqs.(ll) and (12), the joint probability density function of ( and
r becomes,
(11)
f(, 7) = exp { A1 (cosr + sinr)
1 AI.* (eA1cosr
+ 2 (e
+A1Csinr)
+ e )
2 A cos
 (e
2 2
2i(
+2AZ1sinr)
+ e ) ,T
O < O < r 5 2 r .
(13)
The probability density function of peaks can now be obtained as the
marginal density function of Eq.(13). In order to carry out the integration,
the exponential part of the 3rd term of Eq.(13) is expanded in a series of
A1C. That is,
A1Ccos
e
A\ sinr 1 + } +
+ e s = 2 + (A()2 A\I + (A)3 (sinr + cosr)
(14)
+ (AC)3 (sin3r cos3r) +
As a first order approximation, we may take terms up to (AC)2. This
yields
AM1cosr A1Csinr
e + e
= 2 A1(sinr + cos (Ar)2
= 2 52 A\. cos(r ) + (A )2
(15)
Similarly, by expanding the exponential part of the 4th term of Eq(13)
and by taking terms up to (AC)2, we have
2 X cosT
2X1 sinr
(16)
=2 2V2 A\1 cos(r 7) + 2 (A)2.
From Eqs.(15) and (16), the joint probability density function given
Eq(13) can be written as
(17)
f(C r) = L exp + 1, (A C. cos 
2a > *L 2x a1 I+ OS
TIrr' a 2ar [T
where L is a normalization factor which is introduced because of the
assumptions involved in Eqs.(15) and (16). The normalization factor will be
evaluated later.
By integrating Eq.(17) with respect to r, the probability density
function of C becomes,
f()=L exp I + 2 2 1 O < 18)<
al a, 21 I (18)
where, Io is the modified Bessel function of the 1st kind of order zero.
The normalization factor L can be evaluated by equating the integration
of the probability density function over the sample space (0, m) to be unity.
It is obtained as
S{ (Ala A.)2 } (19)
L = (1 + AXi.) exp IA^ 2
a, (1 + Xl/',)al
Thus, the probability density function of peaks, f(), can be derived as
2 2
f (A1 g.)2 1 + A1. )2
f(+) e + (i A.)2 2a2 io ) }
0 < (20)
In case of a Gaussian random process, p* as well as Ai are both zero,
and the variance a2 reduces to a2, the variance of the Gaussian random
process. Hence, Eq.(20) becomes
202 (21)
f(0) = e
which is the Rayleigh probability density function applicable for amplitudes
of a narrowband Gaussian random process.
Figure 2 shows a comparison of the probability density function
applicable for the peaks of a nonGaussian random process derived in Eq.(20)
and the histogram constructed from wave data obtained in water of finite water
depth indicating typical features of nonGaussian random process. Good
agreement can be seen between the histogram and the probability density
function.
DERIVATION OF PROBABILITY DISTRIBUTION OF TROUGHS
The probability density function applicable to troughs of a nonGaussian
random process, denoted by f(n), is essentially the same as Eq.(20). However,
the parameters A, and variance o2 in Eq.(20) should be substituted for those
appropriate for troughs. That is,
f (A2a .)2 +1 + A2~
f(77) = Az\1)f2 e (1 2 +A A ) }
(22)
where parameter A2 ay with 7 3.00. The variance a2 may be evaluated by
the conditional second moment of a nonGaussian random process given in
Eq.(l), with integration over the negative domain. Unfortunately, however,
the approximation used in evaluating a cannot be applied for y < 0. Hence,
a2 may be evaluated by the following formula using the variance of the non
Gaussian record and the variance a2 evaluated by Eq. ().
2 y (23)
a2
where 2 V y is the second moment of the nonGaussian process. The details
n
for evaluating a2 is also given in Appendix.
Figure 3 shows a comparison of Eq.(22) and the histogram constructed
from the data. Although a slight shift in the peak frequency can be seen, the
overall agreement between the histogram and probability density function is
satisfactory.
CONCLUSIONS
Probability density functions applicable for peaks and troughs of the
response of a nonlinear system are derived based on the probability density
function representing a nonGaussian random process developed by the author in
1994. In the development, peak and trough envelope processes are considered
separately, and the statistical properties of peaks and troughs are evaluated
for the amplitude of each narrowband envelope process. The variance of this
envelope process is affiliated with the peak side (or trough side) only of the
nonGaussian random process. The derived probability density functions reduce
to the Rayleigh probability distribution for a Gaussian random process. A
comparison of the newly developed probability function and the histogram
constructed from a record indicating strong nonGaussian characteristics shows
satisfactory agreement for both peaks and troughs of the random process.
ACKNOWLEDGEMENTS
This study was carried out as Phase III of the project "Probability
Functions for Maxima/Minima Excursions of Nonlinear Systems" sponsored by the
Naval Facilities Engineering Service Center through contract N 4740891C1204
to University of Florida. The author would like to express his appreciation
to Mr. Paul Palo for his valuable discussions received during the course of
this project. The author is also grateful to Ms. Laura Dickinson for typing
the manuscript.
APPENDIX:
EVALUATION OF VARIANCES ASSOCIATED WITH
PEAK AND TROUGH ENVELOPE PROCESSES
The peak envelope process ((t) of a nonGaussian random process is shown
in Figure 1. The variance applicable for e(t), denoted by a2, can be
evaluated by the conditional second moment given in Eq.(1). That is,
f y2 f(y) dy
2 y) d
f f(y) dy
"0
(A.1)
where f(y) is
process. The
the probability density function of a nonGaussian random
density function f(y) is derived as (Ochi and Ahn, 1994)
1
 (1 ayp, eavy)2 a. y
2(ayj.)
1
f(y) = 1.
Parameters a, p* and
and
7 1.28 for
{ 3.00 for
(A.2)
e
a2 are evaluated from the 2nd and 3rd moments of data,
y 2 0 (defined here as the peakside of record)
y < 0 (defined here as the troughside of record).
In order to evaluate the denominator of Eq.(A.1), let us write \A ay
where 7 = 1.28, and let us define ej1Y =u. Then, integration of the
denominator can be reduced to that associated with the normal probability
distribution. That is,
I
f(y) dy = exp( I 1)du
(A.3)
= (A./Ia.) I(.\(,a.)).
where D = cumulative distribution function of the standardized normal
distribution.
Since the second term of Eq.(A.3) is nearly equal to zero, in practice,
we may approximately write
0O
o f(y) dy = D(A./7.). (A.4)
Note that for a Gaussian random process p* 0 and thereby D(p,/o,.)
becomes 1/2.
For integration of the numerator, let us write again e"Iy = u. Then
we have
a1
fy2 f(y) dy 1J (n u)2 exp ( 21 {u(l ) }2)du (A.5)
(AXi*) in the above equation is a dimensionless value and is 0.1 at most, in
general; hence, we may neglect it for the evaluation of the second moment.
Since y in Eq.(A.5) is positive value, we have ju 11 < 1. Therefore, we may
expand (n u)2 in the following series:
(in u)2 = (u1) (u1)2 + (u 1)3 3 (u1)4 + .
= (u1)2 (u1)3 + (u1)4 (1)5 + (A.6)
By taking the first three terms of Eq.(A.6), Eq.(A.5) may be written as
follows:
1 u1 )2
y2 f(y) dy {(u1)2 (u1)3 + (u1)4 } e du
A1
It is noted that the series converges fast, therefore the first three
terms appear to yield the second moment with sufficient accuracy. From
Eqs.(A.4) and (A.7), we have the variance a2 applicable for the peak envelope
process as
a02 1 (Xla*1)2 + /27(A.)3 + ( a.) (A.8)
The variance, a2 applicable for the trough envelope process n(t) may
be evaluated by the equation similar to Eq.(l). That is,
0
f y2 f(y) dy
a2 = (A.9)
2 0
f f(y) dy
Since Eq.(A.9) deals with negative yvalues, the approximation given in
Eq.(A.6) cannot be applied in this case. The variance au, however, can be
evaluated through the following procedure:
The second moment of a nonGaussian random process can be written as
y2 f(y) dy = y2 f(y) dy + y2 f(y) dy
0 0
= i f(y) dy + a f (y) dy
2 2 (A.10)
= a2 (p,/a,.) + al (p,/a.) (A.10)
Since the second moment (the left side of the equation) is known from
22
data, and since ,*, ao, and a, are also known, the variance applicable to the
trough envelope process r(t) can be obtained by
2 (A.11)
a2
REFERENCES
Arhan, M.K. and Plaisted, R.O. (1981); "Nonlinear Deformation of SeaWave
Profiles in Intermediate and Shallow Water", Oceanol. Acta, Vol.2, pp.107115.
Kato, S., Ando, S. and Kinoshita, T. (1987); "On the Statistical Theory of
Total SecondOrder Responses of Moored Floating Structures", Proc. Offshore
Tech. Vol.4, OTC 5579, pp.243257.
Kato, S., Kinoshita, T. and Takase, S. (1990); "Statistical Theory of Total
Second Order Responses of Moored Vessels in Ramdom Seas", J. Applied Ocean
Res., Vol.12, No.l, pp.213.
Ochi, M.K. and Malakar, S.B. (1984); "Nonlinear Response of Ocean Platforms
with Single Degree of Freedom in A Seaway", Tech. Rep. UKL/COEL/TR/050,
University of Florida.
Ochi, M.K. and Ahn, K. (1994); "Probability Distribution Applicable to Non
Gaussian Random Process", Journal of Prob. Eng. Mechanics (in printing).
Tayfun, M.A. (1980); "NarrowBand Nonlinear Sea Waves", Joural Geophy. Res.,
Vol.85, No.C3, pp.15481552.
~
'S
Figure 1 Definition of peak envelope process
0 0.4 0.8 1.2 1.6 2.0
PEAK IN METERS
Figure 2 Comparison of probability density function
applicable for peaks of nonGaussian random
process and histogram constructed from data
1.1
z
S1.;
z
LU
w
0.
>
co
O
cr
0
C
6
2
8
0 0.2 0.4 0.6 0.8 1.0 1.2
TROUGH IN METERS
Figure 3 Comparison of probability density function
applicable for troughs of nonGaussian
random process and histogram constructed
from data
