CN108229060B - Parameterized joint distribution model based on effective wave height, average wave period and wave direction - Google Patents

Parameterized joint distribution model based on effective wave height, average wave period and wave direction Download PDF

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CN108229060B
CN108229060B CN201810096107.9A CN201810096107A CN108229060B CN 108229060 B CN108229060 B CN 108229060B CN 201810096107 A CN201810096107 A CN 201810096107A CN 108229060 B CN108229060 B CN 108229060B
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董胜
林逸凡
巩艺杰
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Ocean University of China
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Abstract

The invention relates to the technical field of wave energy resource assessment, in particular to a parameterized joint distribution model based on effective wave height, average wave period and wave direction, which comprises the following steps of S1, constructing edge distribution f of linear variable effective wave height and average wave periodHs(hs) and fTm(tm); s2, constructing edge distribution f of circular variable wave directionΘ(θ); s3, constructing combined distribution c of effective wave height and wave direction and average wave period and wave directionHs‑Dir(u, v) and cTm‑Dir(v, w), conditional distribution h of effective wave height and direction and average wave period and directionHs|Dir(u | v) and hTm|Dir(v | w) and two-dimensional conditional distribution c of effective wave height, average wave period and wave directionHs‑Tm|Dir(hHs|Dir(u|v),hTm|Dir(v | w)); and S4, constructing a three-dimensional joint distribution model of the effective wave height, the average wave period and the wave direction. According to the invention, three main wave factors of effective wave height, average wave period and wave direction are considered, so that the wave conditions can be more accurately described.

Description

Parameterized joint distribution model based on effective wave height, average wave period and wave direction
Technical Field
The invention relates to the technical field of wave energy resource assessment, in particular to a parameterized joint distribution model based on effective wave height, average wave period and wave direction.
Background
Resource assessment of wave energy is closely related to wave spectrum elements, and the wave spectrum elements also affect the energy capture efficiency of equipment and the safety of structures. The establishment of the wave element probability model is very important for analyzing wave energy characteristics and resources, designing devices and evaluating and optimizing power generation efficiency.
In wave energy evaluation and subsequent device and array optimization design, effective wave height and characteristic wave period (such as spectral peak period, zero crossing period or average period) are two most important wave spectrum elements which are closely related, and the accurate construction of a combined probability model is always the key point of related research for many years. In the prior art, the construction of an effective wave height and characteristic wave period model can adopt, such as an original two-dimensional joint distribution model, but the model requires data to be normally distributed and is not suitable for extreme wave height values; for example, a two-dimensional conditional probability model is to study one-dimensional probability distribution of effective wave height under a specified characteristic wave period and then establish a mathematical relationship between the characteristic wave period and the effective wave height probability distribution model parameters, but the model has the disadvantages of complex construction and large data support, and the established function of the characteristic wave period and the effective wave height probability distribution model parameters has certain experience and is inaccurate in estimation of extreme wave conditions. In addition, with the Copula method being more and more widely applied in recent years, expert scholars propose various different probability models for analyzing the two-dimensional joint distribution of wave elements based on the two-dimensional Copula function, but relevant researches still have difficulty in establishing a proper statistical model through comparing a traditional two-dimensional lognormal model, a conditional model and several parameterized models based on the Copula function.
In practice, the direction of the waves is also an important parameter in marine environments. Considering the influence of environmental factors such as monsoon, the wave direction changes continuously in one year, and the long-term probability distribution of the wave direction is often expressed as a multimodal form, namely, a plurality of dominant wave directions exist, the wave form propagated from each dominant wave direction has obviously different influence, and the wave height-period combined probability distribution of the wave directions is also different. Therefore, the sea state characteristics are more accurately described by using the combined probability distribution model of the wave direction, the corresponding effective wave height and the characteristic period, and compared with the two-dimensional distribution of the effective wave height and the characteristic wave period, the research on the combined probability model is less.
Disclosure of Invention
In order to describe wave conditions more accurately, the invention provides a parameterized joint distribution model based on effective wave height, average wave period and wave direction.
In order to achieve the purpose, the invention adopts the following technical scheme: the parameterized joint distribution model based on the effective wave height, the average wave period and the wave direction comprises the following steps,
s1, constructing edge distribution f of effective wave height and average wave period of linear variableHs(hs) and fTm(tm)；
S2, constructing edge distribution f of circular variable wave directionΘ(θ)；
S3, constructing combined distribution c of effective wave height and wave direction and average wave period and wave directionHs-Dir(u, v) and cTm-Dir(v, w), conditional distribution h of effective wave height and direction and average wave period and directionHs|Dir(u | v) and hTm|Dir(v | w) and two-dimensional conditional distribution c of effective wave height, average wave period and wave directionHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v|w))；
And S4, constructing a three-dimensional joint distribution model of the effective wave height, the average wave period and the wave direction.
Further, the improved maximum entropy distribution is adopted to fit the edge distribution of the linear variable in the step S1.
Further, in the step S2, a von Mises distribution is adopted to fit the edge distribution of the circular variable.
Further, the step S3 specifically includes,
s31, constructing combined distribution c of effective wave height and wave direction and average wave period and wave directionHs-Dir(u, v) and cTm-Dir(v, w), conditional distribution h of effective wave height and direction and average wave period and directionHs|Dir(u | v) and hTm|Dir(v | w) and a significant wave height;
s32, constructing two-dimensional conditional distribution c of effective wave height, average wave period and wave directionHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v|w))。
Further, the step S31 specifically includes,
s311, constructing a probability density function of linear-circular distribution;
s312, taking the linear-circular distribution as a Copula function in a special form, namely linear-circular Copula, and obtaining the combined accumulated frequency distribution;
s313, estimating a distribution function of circular correlation coefficients in the linear-circular Copula combined accumulated frequency distribution by utilizing von Mises distribution;
s314, obtaining the effective wave height, the wave direction and the average wave periodCombined distribution of phase and wave direction cHs-Dir(u, v) and cTm-Dir(v, w) obtaining a conditional probability distribution h of the effective wave height and the wave direction and the average wave period and the wave directionHs|Dir(u | v) and hTm|Dir(v|w)。
Further, in the step S32, an Archimedean Copula model is adopted to fit the two-dimensional conditional distribution c of the effective wave height, the average wave period and the wave directionHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v|w))。
Further, the step S4 specifically includes,
s41, constructing a three-dimensional joint cumulative probability distribution function according to a Copula function;
and S42, obtaining a three-dimensional joint distribution model of the effective wave height, the average wave period and the wave direction.
The parameterized joint distribution model based on the effective wave height, the average wave period and the wave direction has the following beneficial effects:
(1) the linear distribution and the circular distribution are organically combined.
In engineering applications, where wind fields or wave fields usually involve several different prevailing directions, the observed data cannot be modeled with unimodal distributions, to overcome this difficulty hybrid von Mises distributions have been proposed and proved to be a flexible model. In order to couple the linear-circular distribution with the pair-copula, the linear-circular distribution is taken as a copula function of a special form, called linear-circular copula, and a joint cumulative frequency distribution of the linear-circular copula function is obtained by using a periodic function with a period of 2 pi.
(2) The wave conditions can be more accurately described by considering three main wave elements, namely the effective wave height, the average wave period and the wave direction.
The invention provides a wave period, wave height and wave direction combined distribution parameter model. The model can consider the mutual relation among three variables to more vividly represent the wave condition; meanwhile, the method has enough precision to adapt to multivariate distribution, and can be applied to wave energy resource assessment and other calculations.
Drawings
FIG. 1 is a seasonal distribution scatter plot of data for different seasons of the sea area under study; wherein the shape of the data represents different average wave periods Tm ("o": 0-2, "+": 2-4, ": 4-6," x ": 6-8," □ ": 8-10,": 10-12, ". DELTA": 12-14);
FIG. 2 shows the effective wave height HsA fitted graph of the edge distribution of (a);
FIG. 3 shows the period T of the average wavemA fitted graph of the edge distribution of (a);
FIG. 4 is a fitting graph of the edge distribution of the wave direction θ;
FIG. 5 shows the effective wave height and direction Hs-a joint distribution fit of θ;
FIG. 6 is a plot of a joint distribution fit of mean wave period and wave direction Tm-theta;
FIG. 7 is a graph fitting the distribution of the effective wave height and the mean period CHs-Tm (u, w);
fig. 8 is a PPi and PFi image of the model.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention relates to a parameterized joint distribution model based on effective wave height, average wave period and wave direction, which comprises the following steps,
s1, constructing edge distribution f of effective wave height and average wave period of linear variableHs(hs) and fTm(tm)；
In step S1, the improved maximum entropy distribution is used to simulate the edge distribution of the linear variable, and the specific steps are as follows:
linear variables include the effective wave height Hs and the average period Tm, assuming that the annual average frequency of occurrence of the variables is n (0,1,2, …, k, …), with a corresponding probability of P (n-k) -Pk(Pk＝P0,P1,P2,…,Pk…) and a composite extremum distribution function of
Figure BDA0001565044550000041
Corresponding Poisson distribution of
Figure BDA0001565044550000042
Substituting the Poisson distribution function into the composite extremum distribution to obtain:
Figure BDA0001565044550000043
in order to obtain the maximum entropy distribution, the following conditions are proposed:
Figure BDA0001565044550000044
let y be f (x), then its Euler equation is
Figure BDA0001565044550000045
Order to
Figure BDA0001565044550000046
Then lny ═ α' + γ lnx- β xξTherefore, under the constraint of the above three conditions, the probability density function of the maximum entropy distribution is
Figure BDA0001565044550000047
Maximum entropy distribution function of
Figure BDA0001565044550000048
If mathematical expectation of random variables
Figure BDA0001565044550000049
The second central moment V and the third central moment B exist in any statistical sequence, and then the expression is carried outThe formula is as follows:
Figure BDA00015650445500000410
substituting g (x) into F (x) to obtain
Figure BDA0001565044550000051
The probability density function that improves the maximum entropy distribution is then:
Figure BDA0001565044550000052
in the formula, beta, gamma, xi and a0The (location parameters) are parameters of the function, and the parameter estimation adopts a Maximum Likelihood Estimate (MLE) method.
S2, constructing edge distribution f of circular variable wave directionΘ(θ)；
In probability theory, the von Mises distribution is used as a normal distribution for the circular variables because it is close to a normal distribution. Given an ideal value z ═ eiθThe von Mises distribution is the maximum entropy density of z. The order of the probability density function and Bessel function of von Mises is:
Figure BDA0001565044550000053
in which theta is belonged to [0,2 pi) and is radian form of random circular variable, 0 is more than or equal to mu<2 π is the mean, reflecting the center of symmetry of the von Mises distribution; k>0 is the center parameter, reflecting how dense the circular variables are from the center. I isj(K) is jth(j ═ 0,1,2, …) a first class of modified Bessel functions, the expression:
Figure BDA0001565044550000054
based on this, the probability density function of the hybrid von Mises distribution is:
Figure BDA0001565044550000055
wherein wh >0 is a non-negative parameter of the mixture distribution, and the sum is 1. H is the number of von Mises distributions required. The cumulative frequency function of the mixed von Mises distribution is:
Figure BDA0001565044550000056
based on the maximum entropy principle, the parameter estimation can use EM algorithm to estimate the parameters according to Akaike's Information Criterion (AIC), see the following equation.
Figure BDA0001565044550000057
S3, constructing combined distribution c of effective wave height and wave direction and average wave period and wave directionHs-Dir(u, v) and cTm-Dir(v, w), conditional distribution h of effective wave height and direction and average wave period and directionHs|Dir(u | v) and hTm|Dir(v | w) and two-dimensional conditional distribution c of effective wave height, average wave period and wave directionHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v|w))；
S31, constructing combined distribution c of effective wave height and wave direction and average wave period and wave directionHs-Dir(u, v) and cTm-Dir(v, w), conditional distribution h of effective wave height and direction and average wave period and directionHs|Dir(u | v) and hTm|Dir(v | w) and a significant wave height;
the step S31 specifically includes the steps of,
s311, constructing a probability density function of linear-circular distribution;
the probability density function of the linear-circular distribution is:
fX,Θ＝2πg(ξ)fX(x)fΘ(θ)
in the formula (f)X(x) Probability of being a linear variableDensity function, fΘ(θ) is the probability density function of the angle variable, and g (ξ) is the probability density function of the circular correlation coefficient ξ. Let the cumulative frequencies of the linear and circular variables be U and V, respectively, and ξ can be defined as:
Figure BDA0001565044550000061
s312, taking the linear-circular distribution as a Copula function in a special form, namely linear-circular Copula, and obtaining the combined accumulated frequency distribution;
in order to couple the linear-circular distribution with the Copula function of a common linear variable and construct a three-dimensional joint distribution model based on the Pair-Copula theory, the invention takes the linear-circular distribution as the Copula function of a special form, namely the linear-circular Copula, and records the function as CL-C(u, v) the probability density function of which is:
cL-C＝2πg(ξ)
but C isL-CThe analytical function of (u, v) cannot be directly obtained from the double integration of the joint density function. Thus, let
Figure BDA0001565044550000062
Figure BDA0001565044550000063
g (xi) is a circular distribution on a unit circle, i.e. g (xi) is a periodic function with a period of 2 pi, for any v 'and theta',
Figure BDA0001565044550000064
thus jointly accumulating the probabilities CL-C(u, v) can be expressed as:
Figure BDA0001565044550000065
Figure BDA0001565044550000066
is connected withAnd (5) continuing. Let the first derivative of g (-) be g1(. o) second derivative is g2(. 2) then
Figure BDA0001565044550000071
After simplification, the joint cumulative frequency distribution of the linear-circular copula function is:
CL-C(u,v)＝{g2(2πu)-g2[2π(u-v)]+g2(-2πv)}/2π
to obtain CL-CAfter the analytic function of (u, v), the conditional probability hL|C(u | v) can be obtained by the following formula:
Figure BDA0001565044550000072
cHs-Dir(u,v)、cTm-Dir(v, w) and cHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v | w)) may be
S313, estimating a distribution function of circular correlation coefficients in the linear-circular Copula combined accumulated frequency distribution by utilizing von Mises distribution;
distribution function g (-) using generalized von Mises distribution (GvM2) to estimate circular correlation coefficient xi
Figure BDA0001565044550000073
To obtain the first and second order analytical functions g1 (-) and g2 (-) of gGvM2 (-) the above equation is Fourier transformed:
Figure BDA0001565044550000074
wherein R (n mu)1) Is a rotation matrix, defined as:
Figure BDA0001565044550000075
an, Bn are equal to
Figure BDA0001565044550000076
Figure BDA0001565044550000077
In the formula, HnIs defined as:
Figure BDA0001565044550000078
let Sn be 1, if n/2 is a non-negative integer and Sn is 0, Gn, and Hn may be calculated by numerical methods:
Figure BDA0001565044550000081
Figure BDA0001565044550000082
therefore, there are:
Figure BDA0001565044550000083
Figure BDA0001565044550000084
s314, obtaining the joint distribution c of the effective wave height and the wave direction and the average wave period and the wave directionHs-Dir(u, v) and CTm-Dir(v, w) obtaining a conditional probability distribution h of the effective wave height and the wave direction and the average wave period and the wave directionHs|Dir(u | v) and hTm|Dir(v|w)；
The joint distribution of the effective wave height and the wave direction, and the joint distribution of the mean period and the wave direction can be fitted by a linear-circular Copula function. The maximum likelihood estimate of the generalized von Mises distribution can be obtained approximately from the following equation
Figure BDA0001565044550000085
Figure BDA0001565044550000086
Figure BDA0001565044550000087
Figure BDA0001565044550000088
In the formula, A-1(. is an inverse function of A (K), and A (K) is I1(К)/I0(K) is a monotonically increasing function, A-1(. cndot.) needs to be calculated by numerical methods.
S32, constructing two-dimensional conditional distribution c of effective wave height, average wave period and wave directionHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v|w))。
Wherein, an Archimedean Copula model is adopted to fit two-dimensional conditional distribution c of effective wave height, average wave period and wave directionHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v|w))。
The last step is to establish a two-dimensional conditional distribution model cHs-Tm|DirFitting c with an Archimedean Copula model (see table above) commonly used for fitting the joint distribution of the effective wave height-characteristic periodHs-Tm|DirThe Copula parameter alpha fitting is estimated according to the following formula by the maximum likelihood method
Figure BDA0001565044550000091
Archimedean Copula model
Figure BDA0001565044550000092
And S4, constructing a three-dimensional joint distribution model of the effective wave height, the average wave period and the wave direction.
S41, constructing a three-dimensional joint cumulative probability distribution function according to a Copula function;
assuming H is a joint cumulative frequency function with an edge distribution of two random variables x and y, there is a copula function C that satisfies for any (x, y):
H(x,y)＝C(FX(x),FY(y))
c is unique if Fx and Fy are consecutive; conversely, C can uniquely identify Fx and Fy. If u is FX(x)，v＝FY(y), the copula function C (u, v) can be defined as:
C(u,v)＝FXY(x,y)＝FXY[FX -1(u),FY -1(v)]
let fXY(x, y) represents a joint probability density function, cUV(u, v) denotes copula density function, according to the definition of copula function, fXY(x, y) can be obtained by the following formula:
Figure BDA0001565044550000093
assuming that the cumulative frequency distribution function of the three variables (X, Y, Z) is defined as F (·), let the edge cumulative distribution u ═ F (X), v ═ F (Y), w ═ F (Z), according to pair-copula theory, the joint cumulative probability distribution function F for (X, Y, Z) is given as F (·), andXYZis defined by the formula:
Figure BDA0001565044550000094
in the formula, FX|ZAnd FY|ZFor the conditional distribution, it is calculated by the following formula:
Figure BDA0001565044550000101
Figure BDA0001565044550000102
in the above equation, F is calculatedXYZIn the process of (2), three copula functions are mentioned: cXZ(u,w),CYZ(v, w) and one-dimensional conditional distribution function are denoted CXY|ZThe Z-timing (X, Y) joint distribution is shown. Z is a key variable because when Z changes, the combined distribution of X and Y also changes. The wave direction is Z-matched because the state of waves in different directions is significantly different, especially in areas where the wave direction is dominant. Assuming that the other two linear variables X and Y are the effective wave height Hs and the average period Tm, the combined probability density of Hs, Tm and theta can be estimated by the following formula:
fHs-Tm-Dir(hs,tm,θ)＝fHs(hs)·fTm(tm)·fΘ(θ)×cHs-Dir(u,w)·cTm-Dir(v,w)·cHs-Tm|Dir(hHs|Dir(u|w),hTm|Dir(v|w))
wherein u is FHs(hs),v＝FTm(tm),w＝FDir(θ), f represents the density function of the edge distribution, h is the abbreviation of copula conditional probability function, and f is determinedHs-Tm-Dir(hs, tm, θ) by obtaining fHs(hs)、fTm(tm)、fΘ(θ)、cHs-Dir(u,v)、cTm-Dir(v, w) and cHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v | w)).
Given array [ hs ]n,tmn,θn]n＝1,…,TThen the log-likelihood function is
Figure BDA0001565044550000103
S42, obtaining a three-dimensional joint distribution model of effective wave height, average wave period and wave direction;
will f isHs(hs)、fTm(tm)、fΘ(θ)、cHs-Dir(u,v)、cTm-Dir(v,w)、hHs|Dir(u|v)、hTm|Dir(v | w) and cHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v | w)) into the formula:
fHs-Tm-Dir(hs,tm,θ)＝fHs(hs)·fTm(tm)·fΘ(θ)×
cHs-Dir(u,w)·cTm-Dir(v,w)·cHs-Tm|Dir(hHs|Dir(u|w),hTm|Dir(v|w))
and obtaining a three-dimensional joint distribution model of the effective wave height, the average wave period and the wave direction.
To verify the effectiveness of the method of the invention, verification is made by the following example.
In order to ensure that the data is sufficient and representative, effective wave height Hs, average wave period Tm and wave direction theta data observed every three hours for 5 continuous years at certain observation stations 1997 to 2001 in the Atlantic are applied to the method. Due to seasonal changes of wave climate, the winter (december to february) in most regions of the northern hemisphere is the most abundant season of wave energy, the total number of data in the winter is 3608, and further analysis is sufficient, so that the wave data in the winter is taken as an example. FIG. 1 shows a scatter plot of the distribution of the data used. The radius in the figure represents the effective wave height Hs.
One, one factor analysis
Fig. 2 and 3 show fitted images of linear edge distributions of the effective wave height Hs and the mean wave period Tm. The result shows that the improved maximum entropy distribution is well fitted with Hs and Tm data.
Fig. 4 shows a fitting image of the distribution of wave direction edges, and the degree of fitting is relatively decreased when H is 5 and 6 in winter.
Two, two dimensional joint distribution
FIGS. 5 and 6 show graphs of the combined empirical cumulative frequency of Hs-theta and Tm-theta and the cumulative frequency of the corresponding theoretical distributions. As shown, the regression coefficient R2 ranged from 0.9137-0.9729, indicating that the model was built very close to the original data.
FIG. 7 is a plot of a joint distribution fit of mean wave period and wave direction Tm-theta.
Three, three dimensional analysis
To estimate the model and observed data X ═ Hs obs,Tm obs,θobs)N×3Similarity (N is the number of data) between the two, and the validity of the model is checked at the same time, so that a great number of combined pseudo-random numbers Y (H) need to be establisheds sim,Tm sim,θsim)，Hs,TmThe combined empirical cumulative frequency of θ may be defined as:
Figure BDA0001565044550000111
where I is an indicator function, which has a value of 1 when the condition is satisfied, and 0 otherwise. Using simulated data, PFiThe theoretical value can be estimated by
Figure BDA0001565044550000112
FIG. 8 shows PP in winter for the modeliAnd PFiAnd (4) an image. PP in winteriAnd PFiIs 0.9954, which means that the model and fitting algorithm can model the sea state of the sea area under study with a high level of confidence.
The invention provides a three-dimensional joint probability distribution model of wave direction, synchronous effective wave height and characteristic wave period. The circular distribution is effectively converted into linear distribution by utilizing the mixed von Mises distribution, and the joint accumulation frequency distribution of the linear-circular copula function is obtained by utilizing a periodic function taking 2 pi as a period. The invention can consider the interrelationship between the three variables to more vividly represent the wave condition with sufficient precision to adapt to multivariate distribution. Meanwhile, through example verification, the representation model and the fitting algorithm can simulate the sea state of the sea area under study at a high confidence level.
It will be understood that modifications and variations can be made by persons skilled in the art in light of the above teachings and all such modifications and variations are intended to be included within the scope of the invention as defined in the appended claims.

Claims (1)

1. The construction method of the parameterized three-dimensional joint distribution model based on the effective wave height, the average wave period and the wave direction is characterized by comprising the following steps,
s1, constructing edge distribution f of effective wave height and average wave period of linear variableHs(hs) and fTm(tm); the probability density function is respectively:
Figure FDA0003031118510000011
in the formula, beta, gamma, xi and a0Are both location parameters of the function; e is the base of the natural logarithm; f is a f function, x represents a corresponding linear variable, x is replaced by the effective wave height hs by calculating the edge distribution of the effective wave height, and x is replaced by the average period tm by calculating the edge distribution of the average period;
s2, adopting mixed von Mises distribution to construct edge distribution f of circular variable wave directionΘ(θ); the probability density function of the mixed von Mises distribution is:
Figure FDA0003031118510000012
where θ is the corresponding circular variable, here the wave direction; h is the number of main directions of wave directions; w is ah>0 is a non-negative parameter of the mixed distribution, and the sum is 1; kappah,μhRespectively as a center parameter and a position parameter; f. ofvMIs the probability density function of von Mises, which is formulated as follows:
Figure FDA0003031118510000013
in which theta is belonged to [0,2 pi) and is radian form of random circular variable, 0 is more than or equal to mu<2 π is the mean, reflecting the center of symmetry of the von Mises distribution; k>0 is a center parameter, reflecting the degree of density of the circular variable from the center; i isj(K) is jth(j ═ 0,1,2, …) modified Bessel functions of the first type, tabulatedThe expression is as follows:
Figure FDA0003031118510000014
s3, respectively constructing binary copula combined distribution c of effective wave height and wave direction and average wave period and wave directionHs-Dir(u, v) and cTm-Dir(v,w)：
CL-C(u,v)＝{g2(2πu)-g2[2π(u-v)]+g2(-2πv)}/2π
Wherein g (. cndot.) represents the density function of the mixed von Mises, g1(. and g)2(. cndot.) represents the integral of g (·) by one and two, respectively; u, v and w respectively represent three variables of joint distribution, u and v are respectively replaced by the effective wave height and the wave direction when the joint distribution of the effective wave height and the wave direction is calculated, and v and w are respectively replaced by the effective wave height and the wave direction when the joint distribution of the average period and the wave direction is calculated;
then constructing a conditional distribution h of the effective wave height and the wave direction and the average wave period and the wave directionHs|Dir(u | v) and hTm|Dir(v|w)：
Figure FDA0003031118510000015
Constructing two-dimensional conditional distribution of effective wave height, average wave period and wave direction
cHs-Tm|Dir(hHs|Dir(u|v),hTm|Dir(v|w))；
S4, constructing a three-dimensional joint distribution model of effective wave height, average wave period and wave direction:
Figure FDA0003031118510000021
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