CN106326526A - A Method for Calculating Wave Height of Wave Design under Non - Narrow Constraints - Google Patents

Abstract

The present invention discloses a method for calculating wave height of wave design without narrow spectrum constraint. The method includes the joint distribution function of wave height and period and the probability density function of wave height and period. 1) The joint distribution function of wave height and period: as shown in the picture. The joint distribution function of the wave height period of the invention is better than the measured data, and the combined design value calculated by this distribution function is more reasonable and can be used to describe the general wave height and period, which can provide theoretical basis for ocean engineering.

Description

A kind of without the projectional technique of wave Design Wave under narrow spectrum constraints

Technical field

The present invention relates to a kind of without the projectional technique of wave Design Wave under narrow spectrum constraints.

Background technology

The one that wave is to occur in ocean is sufficiently complex and important fluctuations phenomenon, and ocean engineering is built by research wave If, ocean development, traffic shipping, marine fishing be significant with activities such as cultivation.And to wave since the forties Research achieved with bigger development.Also achieve numerous achievements, and these achievements have had become as dynamic oceanography A part, have substantial connection with numerous areas such as ocean remote sensing, ocean engineering and upper ocean kinetics.It addition, along with The development of economic technology and the change of international situation, wave achievement in research is to marine military activity and the using value of sea economy More and more important.

At present to the research contents of wave widely, including wave generation, grow up, disappear the process declined, and these During the mutual relation of factor.And set up ocean statistical models and drive marine by the principal element during these Learn model wave to be simulated and forecast.At present, Chinese scholar the most all achieves great successes at a lot of aspects.Including The aspects such as ocean wave spectrum, ocean wave factor statistical distribution, wave forecast method and Nearshore Wave.

Wave study general is divided into two kinds of method statistics and kinetics.Often by two kinds of method knots in practical study Close.Owing to wave has randomness, so being considered as stochastic process in practical study, both can be from the internal structure of wave Carry out studying namely ocean wave spectrum, it is also possible to carry out studying namely Wave parameters statistical distribution from the external performance characteristic of wave.And The key problem of random seaway research is the research of the statistical distribution to ocean wave factor.Research to the statistical distribution of ocean wave factor Being the random nature studying wave in terms of outward appearance, therefore it has very important meaning in terms of the engineer applied of wave, Also have been a great concern.

Ocean wave factor is used to describe some important amounts of the looking randomness matter of wave, such as wavelength, wave height, cycle Deng.Since the fifties, a lot of scholars begin one's study its statistical distribution, and its research method can be largely classified into three kinds, the first Method is to observe actual wave obtain ocean wave factor data and be analyzed measured data, therefrom finds out relevant statistics rule Rule.Second method is to utilize laboratory equlpment or electronic computer to be simulated wave.The third method is just based on sea Wave theory seeks the rule of ocean wave factor distribution.

Longuet-Higgins derived ocean wave factor distribution function first in 1975, and this result is in corrugated displacement Deriving under the conditions of supposing for the normal process of even and narrow spectrum, its conclusion is that ocean wave factor is distributed as Rayleigh.By Relatively backward in early stage ocean observation technology and observation method so that observational data is abundant not, and observation data are the most accurate.No Can accurately judge whether the ocean wave factor of reality better conforms to this distribution.And constantly sending out along with ocean observation technology By the observation of substantial amounts of off-lying sea and laboratory experiment, exhibition, proves that Rayleigh is under the supposition that corrugated displacement is normal process repeatedly It is good approximation to some problems, such as the description etc. for deep water wave.But owing to this derivation being distributed is to assume wave Under conditions of being normal process, therefore which limits its application in some respects, as to ocean microwave remote sensing and modern times sea The problems such as ocean military technology, this is accomplished by studying abnormal wave.

The wave distribution function derived due to Longuet-Higgins (1975) has two obvious shortcomings: one is when week When phase is 0, functional value is not 0, and two be function about the average in cycle is symmetrical.And actual ocean wave factor was distributed about the cycle Average be asymmetrical.In order to overcome disadvantages mentioned above Longuet-Higgins (1983) to give again one by Mathematical treatment Plant and be distributed about cycle asymmetrical ocean wave factor.Grandson inspires confidence in (1988) to derive according to the ray theory of wave linear model and fluctuation A kind of distribution of new ocean wave factor.Recently, wave theory of statistics is carried out by Stansell etc. (2004) and Zheng etc. (2004) Improve.The distribution function pushed away by the method that theory of statistics is made discretization correction overcome original theory some lack Fall into.

All the time, although ocean wave factor distribution function is constantly improved, but most of derivation result is with sea Wave corrugated displacement be normal process supposition under the conditions of derive, its result is all consistent, i.e. ocean wave factor is distributed as Rayleigh is distributed.Although make this to assume make theory analysis and derive greatly simplified, but the observation of substantial amounts of off-lying sea and experiment Room experiment proves the most repeatedly, and actual ocean wave factor the most all obeys Rayleigh distribution.Zhang, Xu (2004) are based on maximum Entropy principle, conciliates a variational problem by coordinate transform, derives the maximum entropy probability density letter of a kind of new wave corrugated displacement Number, this density function has four parameters undetermined with proposed in the past compared with, can be with finer matching observation data and wider Be suitable for generally various in the case of non-linear wave.And this functional form is simple, does not has the restriction of small nonlinearity, consequently facilitating Theory and actual application.

Summary of the invention

The technical problem to be solved in the present invention is to provide a kind of without the reckoning side of wave Design Wave under narrow spectrum constraints Method.

In order to solve above-mentioned technical problem, the technical solution used in the present invention is, a kind of without wave under narrow spectrum constraints The projectional technique of Design Wave, including the probability density function of the joint distribution function in wave height and cycle and wave height with cycle；

1) wave height and the joint distribution function in cycle:

$F(H,T)=e^{-{\[{(-ln{\∫}_0^H{\mathrm{\αh}}^{\mathrm{\γ}}{e}^{-{\mathrm{\βh}}^n}dh)}^{\mathrm{\θ}}+{(-ln{\∫}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\θ}}\]}^{1/\mathrm{\θ}}}---\left(1\right)$

2) wave height and the probability density function in cycle:

$\begin{array}{c}f(H,T)=\\ \frac{e^{-{\[{(-ln{\∫}_0^H{\mathrm{\αh}}^{\mathrm{\γ}}{e}^{-{\mathrm{\βh}}^n}dh)}^{\mathrm{\θ}}+{(-ln{\∫}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\θ}}\]}^{1/\mathrm{\θ}}}{\ln }^{\mathrm{\θ}-1}({\∫}_0^H{\mathrm{\αh}}^{\mathrm{\γ}}{e}^{-{\mathrm{\βh}}^n}dh+{\∫}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}{{\∫}_0^H{\mathrm{\αh}}^{\mathrm{\γ}}{e}^{-{\mathrm{\βh}}^n}dh\·{\∫}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt}\\ \·{\[{(-\ln {\∫}_0^H{\mathrm{\αh}}^{\mathrm{\γ}}{e}^{-{\mathrm{\βh}}^n}dh)}^{\mathrm{\θ}}+{(-\ln {\∫}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\θ}}\]}^{2/\mathrm{\θ}-2}\\ +{\[{(-\ln {\∫}_0^H{\mathrm{\αh}}^{\mathrm{\γ}}{e}^{-{\mathrm{\βh}}^n}dh)}^{\mathrm{\θ}}+{(-\ln {\∫}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\θ}}\]}^{1/\mathrm{\θ}-2}\\ \·{\mathrm{\αH}}^{\mathrm{\γ}}{e}^{-{\mathrm{\βH}}^n}\·{\mathrm{aT}}^b{e}^{-{\mathrm{cT}}^d}\end{array}---\left(2\right)$

In formula (1) and formula (2):

H: sea wave height

T: Periods

α: undetermined constant

β: undetermined constant

γ: undetermined constant

θ: 1≤θ ＜+∞

N: undetermined constant

A: undetermined constant

B: undetermined constant

C: undetermined constant

D: undetermined constant

H: sea wave height variable

T: Periods variable.

The invention has the beneficial effects as follows:

The joint distribution function in wave height cycle is a kind of new nonlinear, is not to lead with normal process and narrow spectrum for condition The distribution gone out, reflects the uncertainty of ocean wave factor under certain physical significance.And utilize measured data to Joint Distribution Verify, and be compared with cycle Joint Distribution with conventional wave height.Then new wave height cycle Joint Distribution is applied to push away Calculate co-design value and compare with traditional Joint Distribution.Result shows, new distribution function is with measured data matching relatively Good, and the co-design value applying this distribution function to calculate is safer, can wider application describe general sea wave height with In the cycle, provide theoretical foundation for ocean engineering.

Accompanying drawing explanation

The present invention is further detailed explanation with detailed description of the invention below in conjunction with the accompanying drawings.

Fig. 1 is the actual measurement wave height scatterplot with the cycle of the embodiment of the present invention.

Fig. 2 is the probability density function of the zero dimension wave height of the embodiment of the present invention.

Fig. 3 is the probability density function in the zero dimension cycle of the embodiment of the present invention.

Fig. 4 is the probability graph of the wave height Normal distribution test of the embodiment of the present invention.

Fig. 5 is the probability graph of the cycle Normal distribution test of the embodiment of the present invention.

Fig. 6 is wave height and the Joint Distribution figure in cycle of the embodiment of the present invention.

Fig. 7 is wave height and the isogram of the Joint Distribution in cycle of the embodiment of the present invention.

Fig. 8 be the embodiment of the present invention different cycles under the conditions of the condition scattergram of sea wave height.

Fig. 9 be the embodiment of the present invention different sea wave heights under the conditions of the condition scattergram in cycle.

Detailed description of the invention

One, ocean wave factor statistical distribution based on principle of maximum entropy

Owing to conventional method is all to derive under the conditions of the normal process with corrugated displacement as even and narrow spectrum suppose 's.Its conclusion is that ocean wave factor is distributed as Rayleigh.And substantial amounts of observation and experiment prove that actual ocean wave factor is not Rayleigh distribution.Therefore the present embodiment is on the basis of principle of maximum entropy, derives a kind of new period profile, and this distribution is put Wide restrictive condition, is no longer limited by normal process and narrow spectrum, and is met principle of maximum entropy, can be reflected to a certain extent The uncertainty in cycle, its application can be the most extensive.

Then the present embodiment is by analyzing wave height and the dependency relation in cycle, and the probability distribution that wave height and cycle meet Dependency structure relation between pattern, utilizes Copula function, derives the joint distribution function in a kind of new wave height cycle.This Distribution function is a kind of new nonlinear, and its univariate distribution all meets principle of maximum entropy, in certain physical significance The uncertainty of ocean wave factor can be reflected down.Derivation be described below:

1.1 Periods based on principle of maximum entropy distributions

Cycle is one of main ocean wave factor, has important effect in ocean engineering.From the fifties, people Periods is carried out the semi-theoretical research of substantial amounts of semiempirical, but the theoretical of the period profile being generally used at present has been tied Fruit is the most few.The present embodiment utilizes entropy principle, derives a kind of new period profile.

Periods T regards the non-negative continuous random variables taking finite value as, i.e.

0 ＜ T ＜+∞

Its comentropy is

$H\left(T\right)=-{\∫}_0^{+\mathrm{\∞}}f\left(t\right)lnf\left(t\right)dt---\left(1.01\right)$

Wherein f (t) is the density function of T.Obviously constraints is met

${\∫}_0^{+\mathrm{\∞}}f\left(t\right)dt=1---\left(1.02\right)$

Further, f (t) also should be retrained as follows:

${\&Integral;}_0^{+\mathrm{\&infin;}}f\left(t\right)\ln tdt<+\mathrm{\&infin;}---\left(1.03\right)$

${\&Integral;}_0^{+\mathrm{\&infin;}}{t}^{\mathrm{\&xi;}}f\left(t\right)dt<+\mathrm{\&infin;}---\left(1.04\right)$

Wherein ξ is a constant.And retrain (1.03) and (1.04) not priori and specify.Formula (1.03) reflects about T's One general fact: when t → 0 or t →+∞, have f (t) → 0；T always takes positive finite value in practice, and formula (1.04) is also Meet objective fact.When ξ round numbers, formula (1.04) is represented by:

$\stackrel{\&OverBar;}{T}={\&Integral;}_0^{+\mathrm{\&infin;}}{t}^{\mathrm{\&xi;}}f\left(t\right)dt<+\mathrm{\&infin;},m=1,2,...---\left(1.05\right)$

The arbitrary order statistical moment of i.e. T all exists.

By principle of maximum entropy, to find in formula (1.02), under formula (1.03) and formula (1.04) constraint, make f maximum for H (T) (t).This has been transformed into a conditional problem of variation.

Formula (1.01) is regarded as a functional:

$H\left(T\right)=-{\&Integral;}_{-\mathrm{\&infin;}}^{+\mathrm{\&infin;}}F(t,y)dt---\left(1.06\right)$

Wherein

Y=f (t), F (t, y)=ylny

Theorem (Shen Yong vigorously waits 2001) according to variation generalized isoperimetric problem has following inference, in condition

(constant), i=1,2 ..., n (1.07)

Under constraint, the Eular equation that when H (T) takes extreme value, y (t) function meets is

Formula (1.02), formula (1.03), (1.04) three constraintss of formula are substituted into the Eular equation shown in formula (1.08) just Have

$\frac{\&part;}{\&part;f}\&lsqb;-fln\left(f\right)+\mathrm{\&lambda;}(f-1)+bfln(t-)-{\mathrm{ct}}^{d}f\&rsqb;=0---\left(1.09\right)$

Wherein f=f (t), and λ, b, c and d are undetermined constant.The cycle solved by formula (1.09) retrains bar in above three The form of the maximum entropy probability density function under part is

$f\left(t\right)={\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}---\left(1.10\right)$

Wherein a=e^λ-1For undetermined constant.

Formula (1.10) is the maximum entropy probability density function of cycle T.Use T moment of distributionCan be in the hope of in formula (1.10) Parameter.Its expression formula is

$\left\{\begin{array}{c}\frac{{\mathrm{\&Gamma;}}^2\left(\frac{b+2}{d}\right)}{\mathrm{\&Gamma;}\left(\frac{b+1}{d}\right)\mathrm{\&Gamma;}\left(\frac{b+3}{d}\right)}=\frac{A_1^2}{A_2}\\ \frac{\mathrm{\&Gamma;}\left(\frac{b+2}{d}\right)\mathrm{\&Gamma;}\left(\frac{b+4}{d}\right)}{{\mathrm{\&Gamma;}}^2\left(\frac{b+3}{d}\right)}=\frac{A_1{A}_3}{A_2^2}\\ c=\frac{{\mathrm{\&Gamma;}}^d\left(\frac{b+2}{d}\right)}{{\&lsqb;A_1\mathrm{\&Gamma;}\left(\frac{b+1}{d}\right)\&rsqb;}^d}\\ a=\frac{{\mathrm{dc}}^{\frac{b+1}{d}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{d}\right)}\end{array}\right.---\left(1.11\right)$

Wherein A_mM=1,2,3, in practice can be byEstimate.x_iFor X I observation.Represent A_mEstimated value.

The Joint Distribution in 1.2 wave height-cycles based on single argument maximum entropy distribution

1.2.1 Copula function

Joint distribution function comprises information of both variable, and one is the information of variable marginal distribution, and two is between variable The information of dependency structure.Compared with single argument distribution function, joint distribution function can preferably describe the character of random vector.

If random vector (X, Y) joint distribution function is that (x, y), then the marginal distribution of X and Y is respectively F to F₁(x)=F (x, + ∞) and F₂(y)=F (+∞, y).After removing the information of marginal distribution in Joint Distribution, it is left with copula i.e. The information of Copula function.Existence function C makes

F (x, y)=C (F₁(x), F₂(y)) (1.12)

Then title C is the Copula function of distribution function F.

It is F, F that theorem 1.1 (Sklar theorem) sets the joint distribution function of random vector (X, Y)₁And F₂It is respectively edge to divide , then there is a copula C in cloth function so that

F (x, y)=C (F₁(x), F₂(y)) ,-∞≤x, y≤+ ∞ (1.13)

Set up.If F₁And F₂Be continuous distribution function, then C is unique；Whereas if C is a dependency structure letter Number, two unitary distribution functions are respectively F₁And F₂, then the function F that defined by formula (1.13) (x, y) be one with F₁And F₂For limit The binary combination distribution function of fate cloth.

By Sklar theorem, Joint Distribution can be splitted into the long-pending of unitary marginal distribution and Copula.

$f(x,y)=\frac{\&part;F(x,y)}{\&part;x\&part;y}=\frac{\&part;C\left(F_1\right(x),F_2\left(y\right))}{\&part;x\&part;y}=\frac{\&part;C(u,v)}{\&part;u\&part;v}=c(u,v)f_1\left(x\right)f_2\left(y\right)---\left(1.14\right)$

Wherein, f₁(x), f₂Y () is marginal distribution, (u v) is Copula function C (u, density function v) to c.

Several conventional dimensional Co pula functions and the suitability thereof are as follows:

(1) Gumbel-Hougaard (GH) Copula:

C (u, v)=exp{-[(-lnu)^θ+(-lnv)^θ]^1/θ, wherein θ ∈ [1 ,+∞]；

GH Copula function is only applicable to variable and there is positively related situation, mainly portrays the upper tail phase between stochastic variable Guan Xing.

(2) Clayton Copula:

C (u, v)=(u^-θ+v^-θ-1)^-1/θ, wherein θ ∈ [0 ,+∞]；

With GH function, Clayton Copula is applicable to describe positively related stochastic variable, is mainly used to description and combines point Lower tail dependency between stochastic variable in cloth.

(3) Ali-Mikhail-Haq (AMH) Copula:

C (u, v)=uv/ [1-θ (1-u) (1-v)], wherein θ ∈ [-1,1)；

AMH function can describe the stochastic variable that plus or minus is relevant, but is not suitable for positive correlation or negative correlation is the highest Variable, AMH Copula structure symmetrically property.

(4) Frank Copula:

Wherein θ ∈ R；

Similar with AMH Copula function, but degree of relevancy is not limited.Frank Copula structure has symmetry Property, i.e. at its upper tail being distributed and lower tail, the dependency between variable symmetrically increases.

1.2.2 sea wave height based on single argument maximum entropy distribution and the Joint Distribution in cycle

Sea wave height probability density function is

$f\left(H\right)={\mathrm{\&alpha;H}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;H}}^n}---\left(1.15\right)$

Owing to the stochastic variable that sea wave height is non-negative is so its distribution function can be expressed as:

$F_1\left(H\right)={\&Integral;}_0^{H}f\left(h\right)dh={\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh---\left(1.16\right)$

The distribution function in corresponding cycle corresponding to formula (1.10) is:

$F_2\left(T\right)={\&Integral;}_0^{T}f\left(t\right)dt={\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt---\left(1.17\right)$

The dependency in sea wave height and cycle can be obtained according to experimental data, and can be seen that wave ripple according to scatterplot The high tail-dependence coefficient with the cycle, selects different copulas in different situations, obtains its Joint Distribution letter Number.

If measured data checking sea wave height and cycle are positively related stochastic variables, and its scatterplot Joint Distribution medium wave High and the cycle has lower tail dependency.Meet feature and the range of Clayton copula, with F₁And F (H)₂(T) Single argument for wave height and cycle is distributed, and takes the connectivity function that Clayton copula is two marginal distribution, Clayton copula form is:

C (u, v)=(u^-θ+v^-θ-1)^-1/θ (1.18)

Above formula derivation be can get C (u, density function v), i.e.

$C(u,v)=\frac{\&part;C(u,v)}{\&part;u\&part;v}=(1+\mathrm{\&theta;})u^{-\mathrm{\&theta;}-1}{v}^{-\mathrm{\&theta;}-1}{(u^{-\mathrm{\&theta;}}+v^{-\mathrm{\&theta;}}-1)}^{-\frac{1}{\mathrm{\&theta;}}-2}---\left(1.19\right)$

Order

$u=F_1\left(H\right)={\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh---\left(1.20\right)$

$v=F_2\left(T\right)={\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt---\left(1.21\right)$

Formula (1.20) and formula (1.21) are substituted into the joint distribution function of the available wave height of formula (1.18) and cycle, i.e.

$F(H,T)={\&lsqb;{\left({\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh\right)}^{-\mathrm{\&theta;}}+\left({\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt\right)-1\&rsqb;}^{-1/\mathrm{\&theta;}}---\left(1.22\right)$

Formula (1.20) and formula (1.21) first can be substituted into formula by the probability density function obtaining wave height and cycle wanted (1.19) (u, density function v) is to obtain C

$\begin{array}{c}c(u,v)=(1+\mathrm{\&theta;}){\left({\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh\right)}^{-\mathrm{\&theta;}-1}{\left({\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt\right)}^{-\mathrm{\&theta;}-1}\\ \&CenterDot;{\&lsqb;{\left({\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh\right)}^{-\mathrm{\&theta;}}+{\left({\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt\right)}^{-\mathrm{\&theta;}}-1\&rsqb;}^{-\frac{1}{\mathrm{\&theta;}}-2}\end{array}---\left(1.23\right)$

Then wushu (1.15) and formula (1.23) substitute into formula (1.14) and can draw the probability density function of wave height and cycle For

$\begin{array}{c}f(H,T)=(1+\mathrm{\&theta;}){\left({\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh\right)}^{-\mathrm{\&theta;}-1}{\left({\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt\right)}^{-\mathrm{\&theta;}-1}\\ \&CenterDot;{\&lsqb;{\left({\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh\right)}^{-\mathrm{\&theta;}}+{\left({\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt\right)}^{-\mathrm{\&theta;}}-1\&rsqb;}^{-\frac{1}{\mathrm{\&theta;}}-2}\end{array}$

$\&CenterDot;{\mathrm{\&alpha;H}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;H}}^n}\&CenterDot;{\mathrm{aT}}^b{e}^{-{\mathrm{cT}}^d}---\left(1.24\right)$

If measured data checking sea wave height and cycle are positively related stochastic variables, and its scatterplot Joint Distribution medium wave High and the cycle has upper tail dependency.Meet the feature of Gumbel-Hougaard Copula copula and use model Enclose, equally with F₁And F (H)₂(T) be wave height and cycle single argument distribution, take Gumbel-Hougaard Copula dependency structure Function is the connectivity function of two marginal distribution, and Gumbel-Hougaard Copula copula form is:

C (u, v)=exp{-[(-lnu)^θ+(-lnv)^θ]^1/θ} (1.25)

Above formula derivation be can get C (u, density function v), i.e.

$\begin{array}{c}c(u,v)=\frac{e^{-{\&lsqb;{(-\ln u)}^{\mathrm{\&theta;}}+{(-lnv)}^{\mathrm{\&theta;}}\&rsqb;}^{1/\mathrm{\&theta;}}}{\ln }^{\mathrm{\&theta;}-1}(u+v)}{uv}\{{\&lsqb;{(-\ln u)}^{\mathrm{\&theta;}}+{(-lnv)}^{\mathrm{\&theta;}}\&rsqb;}^{-2/\mathrm{\&theta;}-2}+\\ {\&lsqb;{(-\ln u)}^{\mathrm{\&theta;}}+{(-lnv)}^{\mathrm{\&theta;}}\&rsqb;}^{1/\mathrm{\&theta;}-2}\}\end{array}---\left(1.26\right)$

The single argument making sea wave height and cycle is distributed as:

$u=F_1\left(H\right)={\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh---\left(1.27\right)$ $v=F_2\left(T\right)={\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt---\left(1.28\right)$

Formula (1.24) and formula (1.25) are substituted into respectively the joint distribution function of the available wave height of formula (1.22) and cycle, i.e.

$F(H,T)=e^{-{\&lsqb;{(-ln{\&Integral;}_0^H{\mathrm{ah}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh)}^{\mathrm{\&theta;}}+{(-ln{\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\&theta;}}\&rsqb;}^{1/\mathrm{\&theta;}}}---\left(1.29\right)$

Then wushu (1.27), formula (1.28) and formula (1.26) substitute into formula (1.14) can draw the probability of wave height and cycle Density function is:

$\begin{array}{c}f(H,T)=\\ \frac{e^{-{\&lsqb;{(-ln{\&Integral;}_0^H{\mathrm{ah}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh)}^{\mathrm{\&theta;}}+{(-ln{\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\&theta;}}\&rsqb;}^{1/\mathrm{\&theta;}}}{\ln }^{\mathrm{\&theta;}-1}({\&Integral;}_0^H{\mathrm{ah}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh+{\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}{{\&Integral;}_0^H{\mathrm{ah}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh\&CenterDot;{\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt}\end{array}$

$\begin{array}{c}\&CenterDot;{\&lsqb;{(-\ln {\&Integral;}_0^H{\mathrm{ah}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh)}^{\mathrm{\&theta;}}+{(-\ln {\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\&theta;}}\&rsqb;}^{2/\mathrm{\&theta;}-2}\\ +{\&lsqb;{(-\ln {\&Integral;}_0^H{\mathrm{ah}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh)}^{\mathrm{\&theta;}}+{(-\ln {\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\&theta;}}\&rsqb;}^{1/\mathrm{\&theta;}-2}\\ \&CenterDot;{\mathrm{\&alpha;H}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;H}}^n}\&CenterDot;{\mathrm{aT}}^b{e}^{-{\mathrm{cT}}^d}\end{array}---\left(1.30\right)$

Thus method is by analyzing wave height and the dependency relation in cycle, and maximum entropy wave height and maximum entropy cycle single argument Dependency structure between distribution obtains the distribution of new sea wave height and cycle, the method not by even normal process and The restriction of narrow spectrum supposition condition, and marginal distribution all derivations under entropy principle, therefore energy under certain physical significance Preferably reflect the uncertainty of wave.

Two, the inspection of the Joint Distribution in sea wave height and cycle and application

Through above to the derivation of model and theoretical introduce, we can be come new wave height and week by measured data The Joint Distribution of phase is tested.Verify its reasonability, and compare with the Joint Distribution of conventional wave height and cycle.This reality Execute marine site, Li Yichaolian island actual measurement sea wave height and mean wave height data (1963-1989) and cycle and data average period (1963-1989) new Joint Distribution it is analyzed and applies.

The distribution function of 2.1 single arguments (wave height, cycle)

First wave zero dimension wave height and the scatter plot of data of zero dimension periodic sequence (lower abbreviation wave height and cycle) are given (Fig. 1).

The marginal distribution in wave height and cycle selects the distribution shown in formula (1.13) and formula (1.14) respectively, and distribution function is respectively For:

$F_1\left(H\right)={\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh---\left(2.01\right)$

$F_2\left(T\right)={\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt---\left(2.02\right)$

1 can be shown in Table in the hope of its corresponding parameter by formula (1.23) and formula (1.8).

The parameter value of table 1 distribution of wave height and period function

The parameter of table 1 is substituted into formula (1.12) and formula (1.07) can obtain the maximum entropy probability density letter in wave height and cycle Number (such as Fig. 2 and Fig. 3) is respectively

$f\left(H\right)=65.4215H^{-10.2738}{e}^{-0.1338H^{5.6174}}---\left(2.03\right)$

$f\left(T\right)=0.9651T^{11.2852}{e}^{-0.3649T^{6.4199}}---\left(2.04\right)$

By Fig. 2 and Fig. 3 it can be seen that wave height based on principle of maximum entropy derivation, the probability density function in cycle are with actual Data fit is preferable.Owing to conventional probability density function is all to release, by wave height and cycle under the hypothesis of normal process The probability such as Fig. 4 and Fig. 5 of Normal distribution test can be seen that wave height and cycle are in very large range not meet normal distribution.

2.2 sea wave heights and the joint probability distribution in cycle

For in the research of wave, owing to there is certain dependency relation between different ocean wave factors, if examining the most merely Consider wave height or the cycle studies its characteristic, be some statistical properties of the ocean wave factor that can not comprehensively understand complexity.Therefore, Consider that wave height and cycle i.e. its Joint Distribution could preferably grasp some characteristics of wave simultaneously, thus design for coastal engineering Rational foundation is provided.Wave height that the present embodiment is studied and the joint probability density function in cycle do not rely on normal process and The restrictive condition of narrow spectrum, and meet principle of maximum entropy, say the uncertainty that can preferably describe ocean wave factor in theory.

2.2.1 sea wave height and the dependency in cycle and tolerance thereof

For ocean wave factor, its characteristic variable wave height and there is certain dependency between the cycle.Generally use Kendall Rank correlation coefficient τ carries out the tolerance of dependency.The calculating formula of Kendall rank correlation coefficient τ is:

$\mathrm{\&tau;}=\frac{2}{n(n-1)}{\mathrm{\&Sigma;}}_{1\&le;i\&le;j\&le;n}sign\&lsqb;(x_i-x_j)(y_i-y_j)\&rsqb;---\left(2.05\right)$

In formula: (x_i, y_i) it is eyeball data, sign () is sign function, as (x_i-x_j)(y_i-y_j) ＞ 0 time, sign =1；As (x_i-x_j)(y_i-y_j) ＜ 0 time, sign=-1；As (x_i-x_j)(y_i-y_jDuring)=0, sign=0.N is series length.

Observation data according to Huanghai Sea hydrometric station, can be obtained the Kendall order phase in wave height and cycle by formula (2.05) Close coefficient τ=0.0102.

2.2.2Copula function parameter is estimated and wave height and the Joint Distribution in cycle

The method for parameter estimation of conventional Copula function have correlation metric method, the IFM estimation technique, maximum likelihood method and Suitable collimation method etc..The present embodiment uses correlation metric method to estimate the parameter of Copula function.Correlation metric method is profit Parameter θ is calculated by the relation between parameter θ and the Kendall rank correlation coefficient τ of Copula function.Permissible according to formula (2.06) Draw the physical relationship formula of parameter θ and τ.

$\mathrm{\&tau;}=4{\&Integral;}_0^1{\&Integral;}_0^{1}C(u,v)dC(u,v)-1---\left(2.06\right)$

τ=0.0102 is substituted into formula (2.06), the parameter of four kinds of Copula functions can be calculated.Can be calculated by formula (2.10) Obtain sum of deviation square, be shown in Table 2.

The parameter of table 2 Copula function and Fitness Test OLS (sum of deviation square) result of calculation

Function name	Gumbel	Clayton	AMH	Frank
					Parameter θ	1.0103	2.2056	0.8595	6.7852
Sum of deviation square OLS	0.415	0.326	0.528	0.476

In order to select fitting effect preferable Copula function, sum of deviation square is used to be fitted inspection, permissible by table 2 Finding out, the sum of deviation square of Clayton function is minimum, i.e. Clayton function is imitated for the matching between sea wave height and cycle The most best.And it is positively related stochastic variable by measured data checking sea wave height and cycle, and its scatterplot (Fig. 1) is permissible Find out that in Joint Distribution, wave height and cycle have lower tail dependency, meet the feature of Clayton copula and use model Enclose.The most theoretically with from empirically saying, Clayton function should be chosen and the Joint Distribution in sea wave height and cycle is carried out point Analysis calculates, i.e. wave height shown in formula (1.22) and the joint distribution function in cycle.

The parameter value of table 1 distribution of wave height and period function and the parameter θ of Copula function are substituted into formula (1.21) and formula (1.19) obtaining the joint probability density function in sea wave height and cycle is:

$\begin{array}{c}f(H,T)=3.2056{\left({\&Integral;}_0^{H}65.4215h^{-10.2738}{e}^{-0.1338H^{5.6174}}dh\right)}^{-3.2056}\\ \&CenterDot;{\left({\&Integral;}_0^{T}0.9651t^{11.2852}{e}^{-0.3649t^{6.4199}}dt\right)}^{-3.2056}\\ \&CenterDot;\&lsqb;{\left({\&Integral;}_0^{H}65.4215h^{-10.2738}{e}^{-0.1338H^{5.6174}}dh\right)}^{-2.2056}+\\ {\left({\&Integral;}_0^{T}0.9651t^{11.2852}{e}^{-0.3649t^{6.4199}}dt\right)}^{-2.2056}-1{\&rsqb;}^{-2.4534}\\ \&CenterDot;65.4215H^{-10.2738}{e}^{-0.1338H^{5.6174}}\&CenterDot;0.9651T^{11.2852}{e}^{-0.3649T^{6.4199}}\end{array}---\left(2.07\right)$

And distribution function is accordingly:

$\begin{array}{c}F(H,T)=\&lsqb;{\left({\&Integral;}_0^{H}65.4215h^{-10.2738}{e}^{-0.1338H^{5.6174}}dh\right)}^{-2.2056}\\ +{\left({\&Integral;}_0^{T}0.9651t^{11.2852}{e}^{-0.3649t^{6.4199}}dt\right)}^{-2.2056}-1{\&rsqb;}^{-0.4534}\end{array}---\left(2.08\right)$

Wave height and the Joint Distribution in cycle and contour are as shown in Fig. 6 and Fig. 7

Sum of deviation square criterion is used to carry out with the wave height derived under normal process and narrow spectral condition and cycle joint density Function i.e. formula

$f(\mathrm{\&alpha;},\mathrm{\&tau;})=\frac{{\mathrm{\&pi;\&alpha;}}^2}{4{\mathrm{v\&tau;}}^2}(1+e^{\frac{{\mathrm{\&pi;\&alpha;}}^2}{v^2\mathrm{\&tau;}}})\exp \{-\frac{{\mathrm{\&pi;\&alpha;}}^2}{4}\&lsqb;1+\frac{1}{v^2}{(\frac{1}{\mathrm{\&tau;}}-1)}^2\&rsqb;\}---\left(2.09\right)$

The wave height based on single argument maximum entropy derived with the present embodiment and the goodness of fit of cycle joint density function formula Relatively.The calculating formula of sum of deviation square OLS is as follows

$OLS=\sqrt{\frac{1}{n}{\mathrm{\&Sigma;}}_{i=1}^n{({\mathrm{pe}}_i-p_i)}^2}---\left(2.10\right)$

Wherein, pe_iFor empirical Frequency；p_iFor theoretic frequency.

Calculated the OLS value of two kinds of Joint Distribution by formula (2.10) and parameter is shown in Table 3 and table 4

The parameter of table 3 wave height based on single argument maximum entropy and cycle Joint Distribution and OLS value

The parameter of the wave height shown in table 4 formula (2.09) and cycle Joint Distribution and OLS value

V	OLS
		0.4	0.672

Be can be seen that wave height based on single argument maximum entropy and cycle joint density function are to measured data by Fig. 6 and Fig. 7 Matching good.And enter with cycle Joint Distribution with the wave height to derive under normal process and narrow spectral condition according to sum of deviation square Row compares, by table 3 and table 4 can be seen that wave height based on single argument maximum entropy that the present embodiment derives and cycle Joint Distribution from Difference quadratic sum is less.Show its superiority to a certain extent.

2.2.3 new joint distribution function is applied to calculate wave height and the associating return period in cycle

The single argument distribution function of sea wave height H and cycle T is respectively formula (2.01) and formula (2.02), is designated as F (H) respectively With F (T), equal to or more than wave height and the list in cycle of certain set-point according to the definition of Copula function, sea wave height and cycle The variable return period is:

$N_H=\frac{1}{1-F\left(H\right)}---\left(2.11\right)$

$N_T=\frac{1}{1-F\left(T\right)}---\left(2.12\right)$

Wherein, N_H, N_TIt is respectively wave height and the single argument return period in cycle.

If during specified criteria cycle T >=t, the conditional probability distribution of sea wave height H is:

$\begin{array}{c}{F}_{H/T}=P(H\&le;h|T\&GreaterEqual;t)\\ =\frac{P(T\&GreaterEqual;t|H\&le;h)P(H\&le;h)}{P(T\&GreaterEqual;t)}\\ =\frac{(1-P(T\&GreaterEqual;t|H\&le;h)\left)P\right(H\&le;h)}{P(T\&GreaterEqual;t)}\\ =\frac{P(H\&le;h)-P(T\&GreaterEqual;t|H\&le;h))P(H\&le;h)}{P(T\&GreaterEqual;t)}\\ =\frac{F\left(H\right)-F(H,T)}{1-F\left(T\right)})\end{array}---\left(2.13\right)$

During specified criteria sea wave height H >=h, the conditional probability distribution of cycle T is:

$\begin{array}{c}{F}_{T/H}=P(T\&le;t|H\&GreaterEqual;h)\\ =\frac{P(H\&GreaterEqual;h|T\&le;t)P(T\&le;t)}{P(H\&GreaterEqual;h)}\\ =\frac{(1-P(H\&GreaterEqual;h|T\&le;t)\left)P\right(T\&le;t)}{P(H\&GreaterEqual;h)}\\ =\frac{P(T\&le;t)-P(H\&GreaterEqual;h|T\&le;t))P(T\&le;t)}{P(H\&GreaterEqual;h)}\\ =\frac{F\left(T\right)-F(H,T)}{1-F\left(H\right)})\end{array}---\left(2.14\right)$

Conditional probability figure in the case of sea wave height and cycle various combination can be provided by formula (2.13) and formula (2.14), as Fig. 8 and Fig. 9.As can be seen from the figure under the probit of the sea wave height under different cycles value and different sea wave height value Period probability value (being shown in Table 5).

The probit of the sea wave height under table 5 different cycles value

The probability of Wave Height Value under the conditions of table 5 can find out different cycles intuitively, such as: specified criteria cycle T >=2 Time, P (H≤1, T >=2)=0.5613, P (H≤2, T >=2)=0.8449, can be that ocean engineering provides important theory to depend on According to.

The joint distribution function in wave height and cycle is formula (2.08), is designated as F (H, T).Its sea wave height and the associating in cycle Return period calculating formula is:

$N_{H,T}=\frac{1}{1-C\left(F\right(H),F(H,T)}=\frac{1}{1-F(H,T)}---\left(2.15\right)$

By formula (2.11) and formula (2.12) try to achieve single argument wave height and the single argument cycle is corresponding 5,10,20,50,100, 200, wave height and the value in cycle during 500 year return period, and associating return period of its correspondence can be obtained according to formula (2.15) (be shown in Table 6)。

The return period of table 6 single argument distribution and the return period of the Joint Distribution of correspondence thereof

As can be seen from Table 6, when wave height and cycle are respectively 4.74 and 2.17, the single argument weight in sea wave height and cycle Current it is 100 years, and combining the return period is 50.79.That is the associating return period in sea wave height and cycle is less than it The return period of single argument distribution.Describing from the angle of design load, the identical return period, single argument Wave Height Distribution and single argument cycle divide The design load that the design load that cloth calculates calculates less than Distribution of wave height and period.

In formula (2.15), joint distribution function is respectively joint distribution function formula (2.08) and the biography that the present embodiment is released The joint distribution function formula (2.09) of system, wave height and cycle when can try to achieve 5,10,20,50,100,200,500 years associating return periods Value (being shown in Table 7).

The wave height of 7 two kinds of distribution functions of table and cycle combine reproduction level

As can be seen from Table 7, under the conditions of the identical return period, the wave height cycle that classical joint distribution function formula (2.09) pushes away The reproduction level that pushes away of maximum entropy joint distribution function released less than the present embodiment of reproduction level.I.e. from actual ocean engineering From the point of view of design, wave height that what traditional method was too low have estimated and cycle combine reproduction level, thus reduce its peace Quan Xing.

The above results shows, design load that either single argument wave height or cycle calculate or traditional Joint Distribution pushes away Design load less than normal, therefore consider with security standpoint, push away with the joint distribution function of new sea wave height and cycle Design load is the safest, can be that the design of coastal engineering provides theoretical foundation.

Invention described above embodiment, is not intended that limiting the scope of the present invention.Any in the present invention Spirit and principle within amendment, equivalent and the improvement etc. made, should be included in the claim protection model of the present invention Within enclosing.

Claims (1)

1. one kind without the projectional technique of wave Design Wave under narrow spectrum constraints, it is characterised in that include wave height and cycle The probability density function in joint distribution function and wave height and cycle；

1) wave height and the joint distribution function in cycle:

$F(H,T)=e^{-{\&lsqb;{(-ln{\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh)}^{\mathrm{\&theta;}}+{(-ln{\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\&theta;}}\&rsqb;}^{1/\mathrm{\&theta;}}}---\left(1\right)$

2) wave height and the probability density function in cycle:

$\begin{array}{c}f(H,T)=\\ \frac{e^{-{\&lsqb;{(-ln{\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh)}^{\mathrm{\&theta;}}+{(-ln{\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\&theta;}}\&rsqb;}^{1/\mathrm{\&theta;}}}{\ln }^{\mathrm{\&theta;}-1}({\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh+{\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}{{\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh\&CenterDot;{\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt}\\ \&CenterDot;{\&lsqb;{(-\ln {\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh)}^{\mathrm{\&theta;}}+{(-\ln {\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\&theta;}}\&rsqb;}^{2/\mathrm{\&theta;}-2}\\ +{\&lsqb;{(-\ln {\&Integral;}_0^H{\mathrm{\&alpha;h}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;h}}^n}dh)}^{\mathrm{\&theta;}}+{(-\ln {\&Integral;}_0^T{\mathrm{at}}^b{e}^{-{\mathrm{ct}}^d}dt)}^{\mathrm{\&theta;}}\&rsqb;}^{1/\mathrm{\&theta;}-2}\\ \&CenterDot;{\mathrm{\&alpha;H}}^{\mathrm{\&gamma;}}{e}^{-{\mathrm{\&beta;H}}^n}\&CenterDot;{\mathrm{aT}}^b{e}^{-{\mathrm{cT}}^d}\end{array}---\left(2\right)$

In formula (1) and formula (2):

H: sea wave height；

T: Periods；

α: undetermined constant；

β: undetermined constant；

γ: undetermined constant；

θ: 1≤θ ＜+∞；

N: undetermined constant；

A: undetermined constant；

B: undetermined constant；

C: undetermined constant；

D: undetermined constant；

H: sea wave height variable；

T: Periods variable.