Ocean Engineering 26 (1999) 507–518
Stochastic Doppler shift and encountered wave
period distributions in Gaussian waves
G. Lindgrena,*, I. Rychlika, M. Prevostob
a
Department of Mathematical Statistics, Lund University, Box 118, SE-221 00 Lund, Sweden
b
IFREMER, Technopole de Brest-Iroise, BR 70, FR-29280, Plouzané, France
Received 18 August 1997; accepted 13 November 1997
Abstract
We show how to calculate the encountered wave period distribution for a ship traveling
with constant speed on a Gaussian random sea with a directionally distributed frequency spectrum.  1998 Elsevier Science Ltd. All rights reserved.
Keywords: Directional spectrum; Encountered spectrum; Random waves; Wave length; Wave period
1. Introduction
Encountered wave period is the time elapsed between successive up- or downcrossings of a mean level for an object moving with constant speed on a random
sea. This is a random quantity. The Doppler effect changes the frequency of each
elementary deterministic periodic wave that makes up the random sea. These changes
result in a complicated change in the statistical distribution of the encountered
wave period.
We shall be concerned with the (half) zero crossing wave period, defined as the
time between a down-crossing of the mean level and the following up-crossing,
measured at a fixed point, and the (half) zero crossing encountered wave period, i.e.
the time between mean level down- and up-crossings for a moving object that moves
with constant horizontal speed on the surface. We shall also have use for the (half)
zero crossing wave length along a line, which is the distance between mean level
* Corresponding author.
0029-8018/99/$—see front matter  1998 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 9 - 8 0 1 8 ( 9 8 ) 0 0 0 1 5 - 8
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G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
down- and up-crossings for a section of the surface along a straight line, observed
at a fixed time instant.
Wave period and wave length can be considered as extreme cases of encountered
wave period, which are found when moving with speed ␯ across the sea surface.
Then, ␯ ⫽ 0 will give wave period, and ␯ ⫽ ⬁ will give the equivalent of wave
length after normalization. With finite ␯, the Doppler shift in frequency for elementary waves will result in a complicated change in the encountered wave period pattern
for a random sea.
For a genuinely random sea, wave periods, measured at a fixed position, and
encountered wave period when moving at constant speed, and wave length, are all
random quantities. Their statistical distributions are in general very difficult to obtain,
but for Gaussian sea states, where sea elevation is described by a Gaussian random
process, one can calculate these distributions numerically with high accuracy, based
on the spectral density function of the process.
This paper addresses the relation between the statistical distribution of wave period
measured at a fixed point, and the speed dependent encountered wave period on a
Gaussian random sea. We shall calculate and compare these distributions using the
very accurate method for zero crossing distributions in Gaussian processes described
in Rychlik and Lindgren (1993, 1995); Lindgren et al. (1997). This method uses as
input the spectral density function for the encountered speed dependent Gaussian
process. This method to calculate wave period distributions is the only known method
that gives correct answers valid for general spectra.
The encountered spectrum is found by transformation of variables and numerical
integration of the fixed point directional frequency spectrum. The transformation
agrees with the one given by St. Denis and Pierson in their pioneering paper (St.
Denis and Pierson, 1953) on ship motion in a random sea. We use however, slightly
different notation, resulting a more explicit formula for the encountered spectrum.
2. Basic definitions
2.1. Covariance and spectra
We consider a time dependent random wave ␰(t,x) with time-parameter t 苸 ᑬ
and space parameter x ⫽ (x1,x2) 苸 ᑬ2. Thus, {␰(t,x0), t 苸 ᑬ} describes the elevation
at a fixed location x0 as a function of time t, while {␰(t0,x),x 苸 ᑬ2} is the elevation
at time t0 as a function of the coordinate x. The mean level is set to zero, E(␰(t,x))
⫽ 0, and the covariance function is denoted, (with ␳ ⫽ (␳1,␳2)),
r(␶,␳) ⫽ Cov(␰(t,x),␰(t ⫹ ␶,x ⫹ ␳)).
To distinguish between time and space descriptions, we use superscripts ⍀ and
K, respectively. Thus, we denote the wave angular frequency spectral density by
R(⍀)(␻) and the wave number spectral density by R(K)(␬), with
G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
冕
exp(i␻␶)R(⍀)(␻)d␻
冕冕
exp(i␬␳)R(K)(␬)d␬.
509
⬁
r(⍀)(␶) ⫽ Cov(␰(t,x0),␰(t ⫹ ␶,x0)) ⫽
␻⫽⫺⬁
and, for ␬ ⫽ (␬1,␬2), ␬␳ ⫽ ␬1␳1 ⫹ ␬2␳2,
r(K)(␳) ⫽ Cov(␰(t0,x),␰(t0,x ⫹ ␳)) ⫽
␬ 苸 ᑬ2
For unidirectional waves, with one-dimensional space parameter x and one-dimensional wave number ␬, there is a simple relation between the spectral densities
R(⍀)(␻) and R(K)(␬), (␬ one-dimensional), see e.g. (Price and Bishop, 1974; Sect.
9.2). We express the relation between the one-sided forms of the spectra, R(⍀)
+ (␻)
and R(K)
(
␬
)
defined
for
␻
>
0
and
␬
>
0,
respectively.
The
relation
follows
from
the
+
dispersion relation,
␻2 ⫽ g␬ tanh(␬h), ␻ > 0, ␬ > 0,
(1)
where h is the water depth, and g denotes the constant of gravity. With ␯␬ ⫽ ∂␻/∂␬,
the group velocity, the relation becomes
R(⍀)
+ (␻) ⫽
1 (K)
R (␬),
兩␯␬兩 +
␻ > 0, ␬ > 0.
(2)
This follows from linear wave theory and superposition of the simple harmonic
waves ␰(t,x) ⫽ A cos(t␻ ⫺ x␬ ⫹ ␾), whose frequency ␻ and period ␬, are connected
via Eq. (1).
For deep water, where ␬hÀ1 and tanh(␬h) ⬇ 1, the relations simplify to ␻2 ⫽
g␬ and
R(⍀)
+ (␻) ⫽
2␻ (K) 2
R (␻ /g),
g +
0 ⬍ ␻ ⬍ ⬁.
(3)
The two-sided frequency density is R(⍀)(␻) ⫽ R(⍀)
+ (兩␻兩)/2, ⫺ ⬁ ⬍ ␻ ⬍ ⬁.
2.2. Directional spectrum
To find the distribution of the encountered wave frequency, we have to take into
account the direction of the elementary waves, relative to the direction of the ship
movement. We define a directional wave frequency density, G(␪,␻), relative to the
direction ␪0 ⫽ 0, as a function of frequency ␻ and direction ␪. For computational
simplicity, we define it for ⫺ ⬁ ⬍ ␻ ⬍ ⬁, and ⫺ ␲/2 ⬍ ␪ ⱕ ␲/2. This means that
waves with ␻ > 0 are trailing waves which travel in the direction ␪ and waves with
␻ ⬍ 0 are meeting and travel in the direction ⫺␪.
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G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
With independence between frequency and direction one has
G(␪,␻) ⫽ R(⍀)(␻)D(␪),
where
冕
␲/2
D(␪)d␪ ⫽ 1.
⫺␲/2
In the examples we shall use a truncated Pierson–Moskowitz frequency spectrum,
with
R(⍀)
+ (␻) ⫽
g2␣ −␤(␻ /␻)4
·e p
for 0 ⱕ ␻ ⱕ ␻max,
␻5
(4)
where ␻max is a truncation limit, in the examples taken as a multiple of the peak
frequency ␻p.
2.2.1. Remark 1
In our notation, wave propagation direction is defined by the combination of ␪ in
(⫺␲/2,␲/2] and ␻ in (⫺⬁,⬁), with the sign of ␻ determining the propagation in
direction ␪ or ⫺␪. This does not agree with conventional notation, where ␪ in [0,2␲)
determines the direction and ␻ is always positive. Our reason for choosing this
notation is that it gives more transparent formulas for the calculation of encountered
spectrum. The difference between meeting and trailing waves is also more easily
illustrated; see for example Fig. 1.
2.3. Wave characteristics
For a time dependent wave process ␰(t,x) with time parameter t and one-dimensional space parameter x we define the following wave characteristics, describing a
half wave:
Fig. 1. Speed dependent encountered frequency spectrum for isotropic (left), meeting (middle) and trailing (right) waves with Pierson–Moskowitz spectrum. Curves from right to left in each figure correspond
to speed ␯ ⫽ c␯m, when c ⫽ 0.125(0.125)2.5.
G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
T
511
⫽ zero crossing wave period
⫽ time from down ⫺ to up ⫺ crossing,
L
⫽ zero crossing wave length
⫽ distance from down ⫺ to up ⫺ crossing,
For a Gaussian wave the corresponding mean values can be expressed by the
spectral moments, mk ⫽ 兰␻kR⍀(␻)d␻ and Mk ⫽ 兰␬kR(K)(␬)d␬. The explicit
expressions are
Tz ⫽ E[T] ⫽ ␲√m0/m2,
Lz ⫽ E[L] ⫽ ␲√M0/M2.
These expectations are to be taken in the ergodic sense, i.e. they are equal to the
averages of observed wave periods or wave lengths, for a random wave observed
over infinite time or space horizon.
2.4. Encountered wave frequency spectrum
Wave length and wave period are two extremes of encountered wave period, which
is what is observed on a ship traveling at a constant speed across the random surface.
With speed ␯, the elevation at time t is ␰␯(t) ⫽ ␰(t,(␯t,0)), assuming the movement
is in the direction ␪0 ⫽ 0. Wave length and wave period are the two cases found
by letting ␯ →⬁ (with t →0), and ␯ →0, respectively. With finite ␯, the Doppler
change in frequency for the elementary waves results in a complicated change in
the distribution of the observed wave period.
The distribution of the encountered wave period can be found via the encountered
frequency spectrum, by the numerical algorithms presented in Section 5. These algorithms use the spectral density of the encountered wave process, or alternatively the
covariance function, and we therefore need an explicit expression for the encountered, speed dependent, wave frequency spectrum R␯(K)(␻˜ ). (We use the notation ␻˜
for the encountered frequencies to distinguish from the fixed location frequencies
␻.) The derivation is found in Section 4 and it gives the same result as the one given
by St. Denis and Pierson (1953), with slightly different notation.
3. Examples
3.1. Specifications
We shall now give three examples of encountered wave frequency spectrum, and
of the corresponding distributions of encountered wave period, calculated by the
tools described in Section 5.
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We use as fixed point frequency spectrum a truncated Pierson–Moskowitz spectrum, defined in one-sided form as
R(⍀)
+ (␻) ⫽
g2␣ −␤(␻ /␻)4
·e p ,
␻5
for 0 ⱕ ␻ ⱕ ␻max,
with peak frequency ␻p ⫽ 1, ␤ ⫽ 1.25, and truncated at ␻max ⫽ 2␻p. (The truncation
is necessary in order to avoid too many waves with very small period; for more
details on the effect of truncation, see Lindgren et al. (1997)). This gives an average
(half) zero crossing period at a fixed location (i.e. ␯ ⫽ 0), of
Tz ⫽ ␲√m0/m2 ⫽ 3.6596,
where
␻max
mk ⫽
冕
␻kR(⍀)
+ (␻)d␻.
0
We use three different types of directional dependence:
Example 1: isotropic waves
In an isotropic case the average energy of waves is the same in all directions, and
the directional spectrum is
Gisotropic(␪,␻) ⫽ R(⍀)
+ (兩␻兩)·
1
,
2␲
(5)
for ⫺␲/2 ⬍ ␪ ⬍ ␲/2 and ⫺ ⬁ ⬍ ␻ ⬍ ⬁.
Example 2: meeting non-isotropic waves
For a non-isotropic spectrum with meeting waves, all waves meet the ship in an
acute angle, and we take in this example the directional spectrum
Gmeeting(␪,␻) ⫽ R(⍀)
+ (⫺␻)·
2
cos2(␪),
␲
(6)
for ⫺␲/2 ⬍ ␪ ⬍ ␲/2 and ␻ ⬍ 0.
Example 3: trailing non-isotropic waves
For a non-isotropic spectrum with trailing waves, all waves travel more or less in
the same direction as the ship. We take as the directional spectrum
Gtrailing(␪,␻) ⫽ R+(⍀)(␻)·
2
cos2(␪),
␲
for ⫺␲/2 ⬍ ␪ ⬍ ␲/2 and ␻ > 0.
(7)
G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
513
3.2. Encountered frequency spectrum
The speed dependent encountered frequency spectrum R␯(⍀)(␻˜ ) has to be found by
numerical integration according to Eq. (14) in Section 4. We shall first study how
the encountered wave spectrum depends on the speed ␯. In order to calibrate the
models we shall compare ␯ with the velocity ␯p of the elementary wave with frequency ␻p ⫽ 1, equal to the peak frequency in the studied Pierson–Moskowitz spectrum. From the deep water dispersion relation, ␬p ⫽ ␻2p/g, it follows that it has velocity
␯p ⫽
␻p
⫽ g/␻p, since ␻p ⫽ 1,
␬p
(8)
and we therefore normalize speed by dividing by g.
We have calculated the encountered spectrum for the three cases, ␯ ⫽ c␯p, c ⫽
0.125(0.125)2. Fig. 1 shows the spectrum for isotropic, meeting, and trailing waves.
As it will be noted in Remark 2, the spectrum may be singular for ␻˜ ⫽ g/(4␯), which
is equal to half the frequency of the encountered standing wave, coming from right
behind the ship. The singularity is clearly seen for the isotropic and trailing cases.
The figures show the spectral densities in two-sided, unsymmetric, form, defined
for both positive and negative ␻˜ . Frequencies ␻˜ > 0 correspond to encountered trailing waves, ␻˜ ⬍ 0 to meeting waves. Thus, one can see how trailing and meeting
encountered waves contribute to the total encountered wave energy. To get a onesided frequency spectrum, one just has to add the two sides.
3.3. Encountered wave period
With the encountered spectra obtained by integration according to Eq. (14) we
can investigate the statistical properties of the encountered wave period.
The mean encountered period gives the average dependence on the speed. The
average Tz ⫽ ␲√m0/m2 is shown in Fig. 2 for the three cases. For isotropic and
meeting waves the mean encountered period decreases with inceasing speed, while
for trailing waves, the mean period first increases to a maximum, when the ship
speed is less than the speed of the dominant waves, after which it decreases with
further increased speed. (In fact, the maximum Tz occurs with ␯ slightly larger
than ␯p.)
If one wishes to obtain more detailed information about the encountered period
distribution one can compute the entire probability density function. This can be
done by means of the routine wave t in the WAT toolbox as described in Section
5; see Rychlik and Lindgren (1995). The results are shown in Fig. 3 for the three
examples. Note in particular the general shift towards shorter periods in the case of
meeting and isotropic waves, but note also that, for trailing waves, with speed ␯ ⫽
␯p/2, equal to half the speed of the peak frequency wave, the encountered wave
period is almost doubled. Even for higher speed, ␯ ⫽ 2␯p, we can see that although
the average Tz is small there is a large probability that very long wave periods occur.
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G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
Fig. 2. Encountered mean wave period as a function of speed, ␯ ⫽ c␯p, for c ⫽ 0.125 (0.125)2.5.
Isotropic (solid), meeting (dash-dotted), and trailing (dashed) waves, with Pierson–Moskowitz spectrum.
The three speed values illustrated in Fig. 3 are marked with circles.
Fig. 3. Encountered wave period density functions for Gaussian waves with isotropic (left), meeting
(middle) and trailing (right) Pierson–Moskowitz waves; ␯ ⫽ 0 (solid), ␯p/2 (dashed), ␯p (dash-dotted),
2␯p (dotted).
4. Derivation of encountered spectrum
We use the symbol ␻ to denote frequencies in the fixed location spectrum and
the symbol ␻˜ for encountered angular frequencies. If the directional fixed location
spectral density is G(␪,␻), the covariance between ␰(0,0) and ␰(t,x) becomes, with
x ⫽ (x1,x2),
冕 冕
⬁
Cov(␰(0,0),␰(␶,x)) ⫽
␲/2
cos(␻␶ ⫺ x1␬␻cos␪ ⫺ x2␬␻sin␪)G(␪,␻)d␪ d␻,
⫺ ⬁ ⫺␲/2
where ␬␻ ⫽ ␻ /g > 0 at infinite depth; cf. Eq. (3). Note that we allow ␻ between
⫺⬁ and ⬁ and restrict ␬␻ to positive values.
For an object moving with speed ␯ along the x1-axis the covariance function of
the encountered waves is
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G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
515
r(⍀)
␯ (␶) ⫽ Cov兵␰(0,0),␰(␶,(␯␶,0))其
冕 冕 再冉
⬁
⫽
␲/2
cos ␶ ␻ ⫺ ␯
(9)
冊冎
␻2
cos␪ G(␪,␻)d␪ d␻.
g
⫺ ⬁ ⫺␲/2
To find the encountered frequency spectrum, we make a change of variables,
␻˜ ⫽ ␻ ⫺ ␯
冉
冊
␻2
␻
,
cos␪ ⫽ ␻ 1 ⫺
g
2␻0(␪)
where
␻0(␪) ⫽
g
.
2␯cos␪
The ␪-dependent constant ␻0(␪) is important in the future. In fact, 2␻0(␪) is equal
to the ␪-dependent frequency of an encountered standing wave, seen from the ship.
Letting ␻˜ 0(␪) ⫽ ␻0(␪)/2, we further define
T(␻˜ ) ⫽ 兵␪ 苸 (⫺␲/2,␲/2]; ␻˜ 0(␪) > ␻˜ 其,
(10)
␻ (␻˜ ) ⫽ ␻0(␪) ⫾ √␻0(␪) ⫺ 2␻0(␪)␻˜ , for ␻˜ ⬍ ␻˜ 0(␪).
(11)
±
2
From Eq. (9) we get
冕
冉冕
⬁
r(⍀)
␯ (␶ ) ⫽
cos␶␻˜
␻
˜ ⫽⫺⬁
冊
兵G̃␯(␪,␻+(␻˜ )) ⫹ G̃␯(␪,␻−(␻˜ ))其d␪ d␻˜ ,
␪ 苸 T(␻
˜)
(12)
where
G̃␯(␪,␻) ⫽
G(␪,␻)
⫽
兩∂␻˜ /∂␻兩
|
|
G(␪,␻)
.
2␯
1⫺
␻cos␪
g
(13)
Inserting ␻±(␻˜ ) into the denominator of Eq. (13), we get
1⫺
2␯ ±
␻±
␻ cos␪ ⫽ 1 ⫺
⫽⫾
g
␻0(␪)
␻˜
冪1 ⫺ ␻˜ (␪),
0
(note that ␻0(␪) > 0).
The sum of integrals within large parentheses in Eq. (12) is the encountered spectrum. We summarize these results in a theorem.
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G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
Theorem 1
The spectral density for the encountered waves, when moving with speed ␯ along
the x1-axis on a homogeneous random sea, is
R(⍀)
˜) ⫽
␯ (␻
冕
G̃␯(␪,␻+(␻˜ ))d␪ ⫹
␪ 苸 T(␻
˜)
冕
G̃␯(␪,␻−(␻˜ ))d␪,
(14)
␪ 苸 T(␻
˜)
where
G̃␯(␪,␻±(␻˜ )) ⫽
G(␪,␻±(␻˜ ))
␻˜
1⫺
␻˜ 0(␪)
冪
,
(15)
with ␻˜ 0(␪) ⫽ g/(4␯cos␪), and where T(␻˜ ), ␻±(␻˜ ), are defined by Eq. (10) and Eq.
(11), respectively. Here ␻˜ is used as argument for encountered wave frequency, to
distinguish it from ␻, the frequency of the individual wave components at a fixed
point.
In particular, as ␯↓0, ␻+(␻˜ )→␻˜ , while ␻−(␻˜ )→ ⫺ ⬁, and ␻0(␪)→⬁. Hence,
冕
␲/2
lim R(⍀)
˜) ⫽
␯ (␻
␯↓0
G(␪,␻˜ )d␪ ⫽ R(⍀)
˜ ).
0 (␻
␪ ⫽ ⫺ ␲/2
Remark 2
Since the denominator in Eq. (15) is 0 for ␻˜ ⫽ ␻˜ 0(0) i.e. when cos␪ ⫽ y+: ⫽
g/(4␯␻˜ ), the integral in Eq. (14) may be generalized when ␻˜ ⱕ ␻˜ 0(0). However, a
substitution of variables,
cos␪ ⫽ y+(1 ⫺ u2),
removes the singularity for ␻˜ > ␻˜ 0(0). For ␻˜ ⫽ ␻˜ 0(0) ⫽ g/(4␯) the singularity can
not be removed, and the encountered spectral density R(⍀)
˜ ) can in fact be infinite
␯ (␻
for that ␻˜ -value.
5. Crossing distributions
5.1. Regression approximation
The statistical distribution of wave length or wave period in a random wave, is
in general impossible to express in closed analytic form. In general, the probability
density function for the distribution of the time T from 0 to the first up-crossing
zero of a random process {␰(␶),␶ ⱖ 0} is given by Durbin’s formula,
f T(t) ⫽ f␰(t)(0)·E(␰⬘(t)+I(t)兩␰(t) ⫽ 0),
(16)
see Durbin (1985), Rychlik (1987a), Lindgren and Rychlik (1991). Here, x ⫽
max(0,x) and I(t) is the indicator function of the event
+
G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
517
兵␰(s) ⬍ 0 for all s 苸 (0,t)其.
Thus, the density of the first passage time T involves the expectation of the variable
␰⬘(t)+I(t), which in turn depends on an infinite number of random variables. This
expectation is therefore not readily obtainable.
For a Gaussian process, one can approximate the crossing distance distribution
with high and controllable accuracy by replacing the infinite dimensional expectation
of ␰⬘(t)+ I(t) in Eq. (16) by a finite dimensional one. This is done by replacing
the infinite dimensional indicator I(t) by the finite dimensional indicator, IN(t), of
the event
兵␰(s) ⬍ 0 for s ⫽ sk, k ⫽ 1,…N其,
where s1,…,sN are suitably chosen time points between 0 and t. The theoretical justification for this procedure, called a regression approximation, can be found in Rychlik (1987b), Lindgren and Rychlik (1991). The choice of s1,…, sN can be made in
a recursive fashion, and by taking a moderately large N, one obtains high numerical
accuracy. In most cases, N ⫽ 5 suffices.
5.2. Numerical tools
To calculate the zero crossing density fT(t) one needs a dedicated algorithm that
can calculate finite dimensional integrals like
E(␰⬘(t)+IN(t)),
for any Gaussian process ␰(t). The authors have developed a package of such algorithms for use in crossings and extreme value theory. The general technique is
described in Rychlik and Lindgren (1993). Special adaptions of the algorithms for
use in wave analysis and for calculations of specific wave distributions are collected
in WAT, a toolbox of MATLAB algorithms, described in Rychlik and Lindgren
(1995).1
The WAT functions calculate, with desired high accuracy, the probability density
for mean level crossing distances, joint density of wave length and amplitude variables of different types, for example the time and height differences between a local
minimum and the next maximum, and wave characteristics based on the minimum
and maximum between successive mean level crossings. Processes which can be
handled are Gaussian processes and memoryless transformations of Gaussian processes.
The algorithms need as input the covariance function of the Gaussian process.
This can be of any form, analytical or as a tabulated function. The process can also
be specified by its spectral density function, in which case the covariances are calculated by means of the Fast Fourier Transform.
It should be emphasized that the density functions calculated by WAT give
1
See also http://www.maths.lth./se/matstat/staff/georg/watinfo.html.
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G. Lindgren et al. / Ocean Engineering 26 (1999) 507–518
numerically accurate values for the exact theoretical density functions in a Gaussian
process. To the authors’ knowledge, all other methods, suggested in the literature,
are approximations based on a fitting of a standard statistical distribution, e.g. by
means of moment fitting, or are based on too simplified approaches which do not
account for the full spectral content of the process. For typical examples, see
Gran (1992).
Acknowledgements
Part of this work was supported by the Office of Naval Research under Grant
N00014-93-1-0841 and carried out during a visit by George Lindgren to the Center
for Stochastic Processes, Chapel Hill.
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