CN102063527B - Method for calculating combined return period of ocean extreme value considering typhoon influence - Google Patents

Abstract

The invention discloses a typhoon influence considered method for calculating a combined return period of an ocean extreme value, comprising the specific steps of: (1), determining four constraint conditions of a two-dimensional maximum entropy distribution function; (2), deducing a two-dimensional maximum entropy combined distribution mode; (3), determining an integer explicit expression of a mixed distribution moment of an extreme value wave height and an extreme value water level; (4), deducing an undetermined parameter equation set relevant to the mixed distribution moment; (5) constructing a nested composite maximum entropy distribution new mode for calculating the combined return period of the extreme value wave height and the extreme value water level in a typhoon influenced sea area; and (6), verifying the effectiveness of the new method. In the invention, the extreme value wave height and the extreme value water level are comprehensively considered, the influence on the extreme value wave height and the extreme value water level from the typhoon is considered and the whole mode is based on the maximum entropy principle so that the apriority and the artificial assumption of the mode are avoided. The integer explicit expression of the new mode is convenient for the engineering application; and through two groups of ten undetermined parameters, the new mode can be used for fitting the observation data of different sea areas more precisely and flexibly, thereby having wider range of application.

Description

A kind of ocean extreme value associating reoccurrence period projectional technique of considering typhoon influence

Technical field

The present invention relates to a kind of Extreme Wave of considering typhoon influence and extreme water level associating reoccurrence period projectional technique, specifically seek a kind ofly to consider that typhoon influence factor, probability distribution avoid Extreme Wave and the extreme water level associating reoccurrence period projectional technique of apriority and artificial hypothesis.

Background technology

Along with global warming, more serious and frequent by the Oceanic disasters that typhoon/hurricane causes.Typhoon can cause unusual billow, if astronomical spring tide at a time when typhoon passes by is added typhoon and just formed extreme Oceanic disasters condition if astronomical tidal level adds Storm Surge, can cause destructive disaster.This just Hurricane Katrina in 2005 make U.S.'s Crescent City be subject to one of important mechanisms of huge disaster.Although typhoon storm tide and typhoon are a pair of monodidymus of typhoon in general, with regard to the offshore waters, both sizes are different because of geographical conditions and the landform in typhoon track and waters, that is to say the two neither synchronous, neither be independently.So the associating reoccurrence period of rationally calculating Extreme Wave and extreme water level under the typhoon influence is to marine engineering design and prevent and fight natural adversities very important.

When Extreme Wave and extreme water level being carried out probabilistic eigenspace analysis in the past, usual way is artificially to set first a joint probability distribution for example two-dimentional Gumbel distributes, two-dimentional Logistic distributes, and then estimates parameter in this probability function by given data.Obviously such way is priori, the probability distribution function of generally choosing can both be by test of hypothesis, but the different probability pattern push away associating reoccurrence period result different, human factor is very large, is difficult to reasonably reflect the uncertainty of Marine Environmental Elements.When in addition Marine Environmental Elements being carried out combined probability analysis, few probability nature that the key factor typhoon that causes the risk generation is discussed simultaneously.

Wave height maximum entropy distribution pattern existing application in oceanographic engineering based on entropy principle, it can to a certain degree avoid artificial priori to choose reckoning Extreme Wave and the error of extreme water level associating during the reoccurrence period that conceptual schema causes, but this pattern is single factor (being single argument) pattern all the time in application process, be difficult to describe complicated marine environment, can't provide probabilistic information relevant between Extreme Wave and the extreme water level, and ignored equally the factor of typhoon when inquiring into Return period design wave height, the inducement that risk is not occured is taken into account.

How to objectively respond uncertainty and the combined action of multiple Marine Environment Factors, taking into full account this key factor that causes disaster to take place frequently of typhoon affects wave height and water level, avoid simultaneously artificial priori to set the reckon error that conceptual schema causes, improving Extreme Wave and the extreme water level projection accuracy of associating reoccurrence period, is that Oceanic disasters statistical forecast and oceanographic engineering are built urgent problem.

Summary of the invention

Purpose of the present invention provides a kind of typhoon influence marine site Extreme Wave based on entropy principle and extreme water level associating reoccurrence period projectional technique, the method is based upon entropy principle, on Multivariate Extreme Value theory and the random point process basis, by considering the two correlationship of typhoon frequency and Extreme Wave and extreme water level, derive the maximum entropy joint distribution new model of Extreme Wave and extreme water level, and with the nested typhoon number of times of new model maximum entropy distribution, provide one and take into full account the typhoon influence factor, avoid reckoning extreme water level and the new method of Extreme Wave associating reoccurrence period of apriority and artificial supposition.

To achieve the above object of the invention, the technical solution used in the present invention is as follows:

Typhoon influence marine site design wave height and design water level associating reoccurrence period are calculated new method, it is characterized in that it comprises step:

Step 1: four constraint conditions determining the Two-dimensional maximum-entropy distribution function:

${\∫}_0^{+\∞}{\∫}_0^{+\∞}f(x,y)\mathrm{dxdy}=1---\left(a\right)$

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}f(x,y)(\ln x+\ln y)\mathrm{dxdy}={\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HSA00000391246100031.png,HSA00000391246100042.png"\; state="\{\{state\}\}">c</figure-callout>}}_1<+\&infin;---\left(b\right)$

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{x}^{m_1}f(x,y)\mathrm{dxdy}={\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">c</figure-callout>}}_2<+\&infin;---\left(c\right)$

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{y}^{m_2}f(x,y)\mathrm{dxdy}={\mathrm{<figure-callout\; id="3"\; label="c"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">c</figure-callout>}}_3<+\&infin;---\left(d\right)$

M wherein ₁, m ₂Be positive integer or positive mark, c ₁, c ₂And c ₃Be constant; F (x, y) is the joint probability density function of Extreme Wave X and extreme water level Y.

Step 2: by four constraint conditions that step 1 is determined, use entropy principle, determine to contain 6 parameter m undetermined ₁, m ₂, α, b, c, the expression formula of the Two-dimensional maximum-entropy joint density function of d:

$f(x,y)=a{\left(\mathrm{xy}\right)}^b{e}^{-{\mathrm{cx}}^{m_1}-{\mathrm{<figure-callout\; id="2"\; label="dy\; m"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">dy</figure-callout>}}^{m_{}}};$

Step 3: determine that by step 2 mixed distribution of typhoon influence marine site Extreme Wave and extreme water level measured data is apart from T _{M, n}Integration and explicit expression:

$T_{m,n}={\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{x}^m{y}^n\mathrm{\&alpha;}{\left(\mathrm{xy}\right)}^b{e}^{-c{x}^{m_1}-{\mathrm{<figure-callout\; id="2"\; label="dy\; m"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">dy</figure-callout>}}^{m_{}}}\mathrm{dxdy}$

$T_{m,n}=\frac{\mathrm{\&alpha;}}{m_1{m}_2{c}^{\frac{m+b+1}{m_1}}{d}^{\frac{n+b+1}{{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}}}\mathrm{\&Gamma;}\left(\frac{m+b+1}{m_1}\right)\mathrm{\&Gamma;}\left(\frac{n+b+1}{m_2}\right)$

Here m, n gets respectively 0,1,2, wherein the Euler integral of the second kind of Γ () for knowing;

Step 4: determine that by step 3 typhoon influence marine site Extreme Wave and extreme water level measured data mixed distribution are apart from T _{M, n}With parameter m undetermined ₁, m ₂, α, b, c, the Simultaneous Equations of d:

$\frac{{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100042.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{0,1}}}^2}{T_{\mathrm{0,0}}{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">T</figure-callout>}}_{0,2}}=\frac{{\mathrm{\&Gamma;}}^2\left(\frac{b+2}{m_2}\right)}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_2}\right)\mathrm{\&Gamma;}\left(\frac{b+3}{m_2}\right)}$

$\frac{{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100042.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{1,0}}}^2}{T_{\mathrm{0,0}}{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{2,0}}}=\frac{{\mathrm{\&Gamma;}}^2\left(\frac{b+2}{m_1}\right)}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_1}\right)\mathrm{\&Gamma;}\left(\frac{b+3}{m_1}\right)}$

$\mathrm{\&alpha;}=\frac{m_1{m}_2{c}^{\frac{b+1}{m_1}}{d}^{\frac{b+1}{{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_1}\right)\mathrm{\&Gamma;}\left(\frac{b+1}{m_2}\right)}$

$c={\left[\frac{\mathrm{\&Gamma;}\left(\frac{b+2}{m_1}\right)T_{\mathrm{0,0}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_1}\right){\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100042.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{1,0}}}\right]}^{{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HSA00000391246100031.png,HSA00000391246100042.png"\; state="\{\{state\}\}">m</figure-callout>}}_1}$

$d={\left[\frac{\mathrm{\&Gamma;}\left(\frac{b+2}{m_2}\right)T_{\mathrm{0,0}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_2}\right)T_{0,1}}\right]}^{{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}$

Parameter m undetermined by following formula equation solution Two-dimensional maximum-entropy distribution function ₁, m ₂, α, b, c, d;

Step 5 makes up the typhoon influence marine site by step 1, step 2 and calculates Extreme Wave and the extreme water level nested compound maximum entropy joint distribution pattern of associating reoccurrence period:

$F(x,y)=P_0+\underset{i=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\mathrm{\&eta;}\&CenterDot;i^{\mathrm{\&gamma;}+1}\&CenterDot;\exp (-{\mathrm{\&beta;i}}^{\mathrm{\&xi;}}){\&Integral;}_{-\&infin;}^y{\&Integral;}_{-\&infin;}^x{G}_x^{i-1}\left(u\right)g(u,v)\mathrm{dudv}$

Wherein i represents the number of times that the annual typhoon in marine site occurs, P ₀The probability of typhoon does not appear in the expression marine site, and g (u, v) is the Two-dimensional maximum-entropy joint density function, G _x(u) be the marginal distribution function of (X, Y), parameter η undetermined, beta, gamma, ξ is by the simultaneous solution system of equations acquisition relevant with year Extreme Wave data moment of distribution;

Step 6: the associating reoccurrence period N (year) that is pressed following formula to calculating Extreme Wave X and extreme water level Y by step 5:

$N=\frac{1}{P(X>x,Y>y)}=\frac{1}{1-F_x\left(x\right)-F_y\left(y\right)+F(x,y)}$

Wherein F (x, y) for step 5 push away maximum entropy joint distribution pattern, F _x(x) and F _y(y) be respectively its corresponding marginal distribution function.

Step 7: the validity of verifying new model according to the measured data data of Extreme Wave and extreme water level and typhoon occurrence number.

In the specific implementation process, the oceanographic data data is known, can obtain by newspaper data behind different oceanic observations or the boats and ships, and the difference of the oceanographic data data of different geographical characteristics is to embody by the parameter in the new model.

Four constraint conditions that propose in the described step 1 are the essential probability natures that satisfy of new model, but not priori and artificial appointment.

M in the described step 1 ₁, m ₂When getting positive integer, guarantee that the arbitrary order mixed moment of Extreme Wave and extreme water level and the Order Moments of marginal distribution exist.

The expression formula of Two-dimensional maximum-entropy joint distribution function obtains by finding the solution a broad sense isoperimetric variational problem in the described step 2.

Moment of distribution T in the described step 4 _{M, n}In the actual computation process, press the row expression formula by observation data and calculate acquisition:

$T_{m,n}\&ap;\frac{\underset{i,j=1}{\overset{N}{\mathrm{\&Sigma;}}}{x_i}^m{y_{\mathrm{<figure-callout\; id="2"\; label="j\; n\; N"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">j</figure-callout>}}}^n}{N^{}}$

M wherein, n gets respectively 0,1,2, x _ix _jBe observation data, N is total year number.

P in the described step 5 _i=η i ^γExp (β i ^ξ), i=0,1,2 ... be the discrete maximum entropy distribution function of the typhoon number of times inquire into based on entropy principle, parameter η undetermined wherein, beta, gamma, ξ by simultaneous solution with specify the relevant system of equations acquisition of marine site year typhoon occurrence number moment of distribution.

Described step 7 further comprises the measured data data of utilizing, illustrate checking with Logistic Two dimensional Distribution and Gumbel Two dimensional Distribution commonly used in the oceanographic engineering, by judging the validity of new model with the fitting degree of measured data, in addition in conjunction with observation website marine site of living in characteristic, determine the rationality of associating reoccurrence period of inquiring into.

The present invention takes into full account typhoon to wave height and water level impact, maximum entropy joint distribution by the Extreme Wave that will derive and extreme water level is nested with typhoon number of times maximum entropy distribution, set up the new model of extreme value generation randomness and correlativity more complete probabilistic information between the reflection three, the process of setting up of pattern follows entropy principle all the time, avoided apriority in the classic method and the interference of human factor.2 groups of 10 parameters undetermined in the Parameter Relation, can reach flexibly accurately match observation data, the integration explicit expression can make new model be convenient to engineering and use, the method of the reckoning ocean extreme value associating reoccurrence period of consideration typhoon influence is also very limited at present, and advantage of the present invention is comparatively obvious in application.

Description of drawings

Fig. 1 is process flow diagram of the invention process;

Fig. 2 is that the Two-dimensional maximum-entropy that the present invention derives distributes and two-dimentional Logistic distribution probability density function isoline comparison diagram (the 36 ° of 03 ' N in wheat island, 120 ° 25 ' E);

Fig. 3 is that the Two-dimensional maximum-entropy that the present invention derives distributes with two-dimentional Logistic distribution probability density function isoline comparison diagram (towards connecting the 35 ° of 53 ' N in island, 120 ° 52 ' E);

Fig. 4 is that the Two-dimensional maximum-entropy that the present invention derives distributes and two-dimentional Gumbel distribution probability density function isoline comparison diagram (Mai Dao);

Fig. 5 is that the Two-dimensional maximum-entropy that the present invention derives distributes and two-dimentional Gumbel distribution probability density function isoline comparison diagram (Chao Liandao);

Fig. 6 is the reckoning associating reoccurrence period isogram (Mai Dao) that the present invention derives;

Fig. 7 is the reckoning associating reoccurrence period isogram (Chao Liandao) that the present invention derives.

Embodiment

As shown in Figure 1, below in conjunction with drawings and Examples the specific embodiment of the invention is done and is described in further detail:

Step 1: the constraint condition of determining the Two-dimensional maximum-entropy distribution function:

Extreme Wave X and the extreme water level Y integral body in typhoon influence marine site are made as two-dimensional random vector (X, Y), and wherein X is with Y both asynchronous also dependent but relevant to a certain extent.

If the joint probability density function of X and Y is f (x, y), then the entropy of (X, Y) is defined as

H(f)＝-∫ _R∫ _Rf(x，y)log f(x，y)dxdy (1)

Entropy principle: describe the information obtain but be to have maximum entropy person to the preferred probability of unavailable Information preservation maximum uncertainty.

According to entropy principle, seek to make the f (x, y) of the entropy maximum of following formula, for present problem, we propose following constraint

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}f(x,y)\mathrm{dxdy}=1---\left(a\right)$

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}f(x,y)(\ln x+\ln y)\mathrm{dxdy}={\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HSA00000391246100031.png,HSA00000391246100042.png"\; state="\{\{state\}\}">c</figure-callout>}}_1<+\&infin;---\left(b\right)$

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{x}^{m_1}f(x,y)\mathrm{dxdy}={\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">c</figure-callout>}}_2<+\&infin;---\left(c\right)$

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{y}^{m_2}f(x,y)\mathrm{dxdy}={\mathrm{<figure-callout\; id="3"\; label="c"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">c</figure-callout>}}_3<+\&infin;---\left(d\right)$

M wherein ₁, m ₂Be positive integer or positive mark, c ₁, c ₂, c ₃Be constant.In above condition, it is because Extreme Wave and extreme water level are non-negative that lower limit of integral is taken as 0.What be worth stressing is, above-mentioned four constraints are to meet axiom and generally know true and be not priori.Constraint (a) is to guarantee that the regularity of probability density function is necessary.Constraint (b) guarantee x →+∞ and y →+∞, f (x, y) → 0 in x → 0 and y → 0, x → 0 and four kinds of situations such as y=constant, y → 0 and x=constant, this meets the general known fact.Constraint (c) and constraint (d) guarantee on the one hand when x →+∞ and y →+f (x, y) → 0 during ∞, on the other hand, the arbitrary order mixed moment of assurance f (x, y) and the existence of the Order Moments of f (x, y) marginal distribution, this also meets axiom.

Step 2: the expression formula of derivation Two-dimensional maximum-entropy distribution function:

Set function

M wherein ₁, m ₂, t, b, c, d are parameter undetermined, will be determined by constraint condition and measured data.

In formula (a), (b), (c) and the broad sense isoperimetric variational problem under the constraint condition (d) be: find the solution f (x, y), make it satisfy Eulerian equation

$\frac{\&PartialD;L}{\&PartialD;f(x,y)}=0$

$-\ln f(x,y)-1+t+b\ln \mathrm{xy}-{\mathrm{cx}}^{m_1}-{\mathrm{<figure-callout\; id="2"\; label="dy\; m"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">dy</figure-callout>}}^{m_{}}=0---\left(3\right)$

By formula (3), thereby obtain

$f(x,y)=\mathrm{\&alpha;}{\left(\mathrm{xy}\right)}^b{e}^{-{\mathrm{cx}}^{m_1}-{\mathrm{dy}}^{m_2}}---\left(4\right)$

Here α=e ^T-1

Step 3: determine that by step 2 mixed distribution of typhoon influence marine site Extreme Wave and extreme water level measured data is apart from T _{M, n}Integration and explicit expression:

Stochastic variable (X, Y) mixed distribution is apart from T _{M, n}Be defined as

$T_{m,n}={\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{x}^m{y}^{n}f(x,y)\mathrm{dxdy}---\left(5\right)$

In formula (4) substitution following formula

$T_{m,n}=\mathrm{\&alpha;}{\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{x}^{m+b}{y}^{n+b}{e}^{-c{x}^{m_1}-{\mathrm{<figure-callout\; id="2"\; label="dy\; m"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">dy</figure-callout>}}^{m_{}}}\mathrm{dxdy}---\left(6\right)$

Finish integration and obtain its Explicit Expression formula

$T_{m,n}=\frac{\mathrm{\&alpha;}}{m_1{m}_2{c}^{\frac{m+b+1}{m_1}}{d}^{\frac{n+b+1}{{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}}}\mathrm{\&Gamma;}\left(\frac{m+b+1}{m_1}\right)\mathrm{\&Gamma;}\left(\frac{n+b+1}{m_2}\right)---\left(7\right)$

Here m, n gets respectively 0,1,2;

The Euler integral of the second kind of Γ () for knowing;

Step 4: set up the two-dimensional random variable mixed distribution distance and parameter m undetermined determined by step 3 ₁, m ₂, α, b, c, the Simultaneous Equations that d is relevant

$\frac{{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100042.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{1,0}}}^2}{T_{\mathrm{0,0}}{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{2,0}}}=\frac{{\mathrm{\&Gamma;}}^2\left(\frac{b+2}{m_1}\right)}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_1}\right)\mathrm{\&Gamma;}\left(\frac{b+3}{m_1}\right)}---\left(8\right)$

$\frac{{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100042.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{0,1}}}^2}{T_{\mathrm{0,0}}{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">T</figure-callout>}}_{0,2}}=\frac{{\mathrm{\&Gamma;}}^2\left(\frac{b+2}{m_2}\right)}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_2}\right)\mathrm{\&Gamma;}\left(\frac{b+3}{m_2}\right)}---\left(9\right)$

$\mathrm{\&alpha;}=\frac{m_1{m}_2{c}^{\frac{b+1}{m_1}}{d}^{\frac{b+1}{{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_1}\right)\mathrm{\&Gamma;}\left(\frac{b+1}{m_2}\right)}---\left(10\right)$

$c={\left[\frac{\mathrm{\&Gamma;}\left(\frac{b+2}{m_1}\right)T_{\mathrm{0,0}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_1}\right){\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000391246100031.png,HSA00000391246100042.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{1,0}}}\right]}^{m_1}---\left(11\right)$

$d={\left[\frac{\mathrm{\&Gamma;}\left(\frac{b+2}{m_2}\right)T_{\mathrm{0,0}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_2}\right)T_{0,1}}\right]}^{m_2}---\left(12\right)$

As long as T _{M, n}Known, then through type (8)-Shi (12) obtains the parameter undetermined in the formula (4).

Mixed moment T _{M, n}In the actual computation process, press the row expression formula by observation data and calculate acquisition

$T_{m,n}\&ap;\frac{\underset{i,j=1}{\overset{N}{\mathrm{\&Sigma;}}}{x_i}^m{y_j}^n}{N^2}---\left(13\right)$

Here it is total year number that m, n get respectively 0,1,2, N;

Step 5: make up the typhoon influence marine site and calculate Extreme Wave and the extreme water level nested compound maximum entropy distribution pattern of associating reoccurrence period:

The discrete maximum entropy distribution function expression of typhoon number of times is

p _i＝ηi ^γ exp(-βi ^ξ) (14)

Parameter η undetermined, beta, gamma, ξ parametric equation

$\frac{A_1^2}{A_0{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}=\frac{\mathrm{\&Gamma;}{\left(\frac{\mathrm{\&gamma;}+2}{\mathrm{\&xi;}}\right)}^2}{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}\right)\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+3}{\mathrm{\&xi;}}\right)}---\left(15\right)$

$\frac{A_0{A}_3}{A_1{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HSA00000391246100031.png,HSA00000391246100032.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}=\frac{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+4}{\mathrm{\&xi;}}\right)\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}\right)}{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+2}{\mathrm{\&xi;}}\right)\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+3}{\text{\&xi;}}\right)}---\left(16\right)$

$\mathrm{\&beta;}={\left[\frac{A_1\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}\right)}{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+2}{\mathrm{\&xi;}}\right)}\right]}^{-\mathrm{\&xi;}}---\left(17\right)$

$\mathrm{\&eta;}=\ln \left[\frac{{\mathrm{\&beta;}}^{\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}}\&CenterDot;\mathrm{\&xi;}}{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}\right)}\right]+1---\left(18\right)$

A wherein _k(k=1,2,3) are k-rank squares, namely

$A_k=E\left(i^k\right)=\underset{i=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{i}^k{p}_i,k=\mathrm{0,1,2},\&CenterDot;\&CenterDot;\&CenterDot;---\left(19\right)$

In actual applications, A _kEstimated as follows by observation data

$\stackrel{\&OverBar;}{A_k}=\frac{1}{N}\underset{i=0}{\overset{N}{\mathrm{\&Sigma;}}}{i}^k,k=\mathrm{0,1,2},\&CenterDot;\&CenterDot;\&CenterDot;N---\left(20\right)$

With the A in the formula (20) _kReplace

Just from then on solve η, beta, gamma, ξ formula numerical value.

The maximum entropy joint distribution of Extreme Wave and extreme water level is nested with typhoon number of times maximum entropy distribution, obtain considering that the Extreme Wave of typhoon influence and extreme water level associating reoccurrence period calculate new model

$F(x,y)=P_0+\underset{i=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\mathrm{\&eta;}\&CenterDot;i^{\mathrm{\&gamma;}+1}\&CenterDot;\exp (-{\mathrm{\&beta;i}}^{\mathrm{\&xi;}}){\&Integral;}_{-\&infin;}^y{\&Integral;}_{-\&infin;}^x{G}_x^{i-1}\left(u\right)g(u,v)\mathrm{dudv}---\left(21\right)$

Wherein i represents the number of times that the annual typhoon in marine site occurs, P ₀The probability of typhoon does not appear in the expression marine site, and g (u, v) is the Two-dimensional maximum-entropy joint density function, G _x(u) be the marginal distribution function of (X, Y), parameter η undetermined, beta, gamma, ξ is by the simultaneous solution system of equations acquisition relevant with year Extreme Wave data moment of distribution.

Step 6: calculate by the Extreme Wave X of following formula definition and the associating reoccurrence period N (year) of extreme water level Y:

$N=\frac{1}{P(X>x,Y>y)}=\frac{1}{1-F_x\left(x\right)-F_y\left(y\right)+F(x,y)}---\left(22\right)$

Wherein F (x, y) is maximum entropy joint distribution pattern formula (21), F _x(x) and F _y(y) be respectively its corresponding marginal distribution function.

Step 7: with Mai Dao hydrometric station (36 ° of 03 ' N, 120 ° of 25 ' E, data are from 1984 to calendar year 2001) and towards the Lian Dao hydrometric station (35 ° of 53 ' N, 120 ° of 52 ' E, data are from nineteen sixty-five to 1989 year) field data, provide specific implementation process of the present invention, and Logistic Two dimensional Distribution and Gumbel Two dimensional Distribution commonly used in result of calculation and the oceanographic engineering illustrated checking, by judging the validity of new model with the fitting degree of measured data, in addition in conjunction with towards marine site of living in, Lian Dao hydrometric station characteristic, determine the rationality of associating reoccurrence period of inquiring into.

Calculate wave height and water level mixed moment T by formula (13) _{M, n}, calculate (4) formula parameter m undetermined by formula (8)-(12) ₁, m ₂, α, b, c, d the results are shown in table 1:

Table one Mai Dao and Chao Lian island Two-dimensional maximum-entropy distribution function value of consult volume undetermined [formula (9)]

Calculate typhoon frequency k-rank square A by formula (20) _k, calculate (14) formula parameter η undetermined by formula (15)-(18), beta, gamma, ξ the results are shown in table 2:

Table 2 Mai Dao and Chao Lian island be typhoon number of times maximum entropy distribution function value of consult volume undetermined separately

With the parameter undetermined in the table one, in the substitution formula (4), get the expression of density function, draw the density function isogram in conjunction with Logistic Two dimensional Distribution and Gumbel Two dimensional Distribution, can find out that from Fig. 2, Fig. 3 the maximum entropy Two dimensional Distribution will be better than the Logistic Two dimensional Distribution significantly to the match of data, especially in the concentrated zone of data point, Fig. 4, Fig. 5 show that maximum entropy Two dimensional Distribution and Gumbel Two dimensional Distribution fitting effect approach, and the former slightly is better than the latter.

By formula (21) and formula (22), can draw the Extreme Wave of Mai Dao marine site and marine site, Chao Lian island consideration typhoon factor and the isogram of extreme water level associating reoccurrence period N (year) is Fig. 6, Fig. 7.

Because pattern is a Two dimensional Distribution pattern shown in the formula (21), the various combination of Extreme Wave and extreme water level can be corresponding to same associating reoccurrence period N.Usually take following mode to calculate design wave height and water level in the marine engineering design: wave height at first to be used with the corresponding maximum entropy one dimension distribution of pattern as preferential variable calculated that N one meets Extreme Wave, then calculate (perhaps being checked in by Fig. 6, Fig. 7) corresponding extreme water level by formula (21) and formula (22), the Mai Dao marine site of so calculating and marine site, Chao Lian island the results are shown in table 3, for ease of the validity of check new model, this table is also shown the result that Poisson-Mixed-Gumbel two dimension composite mode is calculated.

Two kinds of methods of table 3 are Mai Dao and Chao Lian island associating reoccurrence period design wave height and design water level value relatively

The wave height value of listing in this table is considered as preferential variable with Extreme Wave and calculates, is that N one meets Extreme Wave.Water level value in this table be specify in the situation of these Extreme Waves with formula (21) push away corresponding extreme water level.For example, listed first wave height value 7.38m is that Extreme Wave was met in the Mai Dao marine site in 10 years one in the table, and its corresponding water level value 5.51m is the extreme water level of meeting in simultaneous 10 years with it.

As seen from Table 3, for the Mai Dao marine site, the pattern that the present invention proposes calculates that design water level and the result who is calculated by the Poisson-Mixed-Gumbel pattern approach.

For towards the Lian Dao marine site, to such an extent as to too high 20 years one chances of the water level value associating water level of being calculated by the Poisson-Mixed-Gumbel pattern reaches 8.34 (m), surpass 9 (m) and met the associating water level in 50 years one.It is an island that is positioned at the off-lying sea that tide connects the island, and the water level that marine site occurs around it should be lower than river mouth and marine site, bay, and extrapolate so high extreme water level not too tallies with the actual situation.In fact, from towards the year extreme water level observed reading that connects island nineteen sixty-five to 1989 year, the soprano only is 6.90 (m).

By Fig. 2 to Fig. 5, in conjunction with damp Lian Dao research station and Mai Dao research station marine site of living in characteristic, and table 3 result of calculation as seen: calculate that with the method that the present invention provides the water level value that obtains is relatively reasonable, verified with this to take into full account the typhoon influence factor, to avoid in the situation of apriority and artificial supposition that the reckoning extreme water level that the present invention provides is rational with the method that Extreme Wave is united the reoccurrence period.

Claims (4)

1. an ocean extreme value of considering typhoon influence is united the reoccurrence period projectional technique, it is characterized in that it comprises step:

Step 1: four constraint conditions determining the Two-dimensional maximum-entropy distribution function:

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}f(x,y)\mathrm{dxdy}=1---\left(a\right)$

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}f(x,y)(\ln x+\ln y)\mathrm{dxdy}=c_1<+\&infin;---\left(b\right)$

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{x}^{m_1}f(x,y)\mathrm{dxdy}=c_2<+\&infin;---\left(c\right)$

${\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{y}^{m_2}f(x,y)\mathrm{dxdy}=c_3<+\&infin;---\left(d\right)$

M wherein ₁, m ₂Be positive integer or positive mark, c ₁, c ₂And c ₃Be constant; F (x, y) is the joint probability density function of Extreme Wave X and extreme water level Y;

Step 2: by four constraint conditions that step 1 is determined, use entropy principle, determine to contain 6 parameters undetermined

m ₁, m ₂, α, b, c, the expression formula of the Extreme Wave of d and extreme water level Two-dimensional maximum-entropy joint density function:

$f(x,y)=a{\left(\mathrm{xy}\right)}^b{e}^{-{\mathrm{cx}}^{m_1}-{\mathrm{dy}}^{m_2}};$

Step 3: determine that by step 2 mixed distribution of typhoon influence marine site Extreme Wave and extreme water level measured data is apart from T _{M, n}Integration and explicit expression:

$T_{m,n}=\mathrm{\&alpha;}{\&Integral;}_0^{+\&infin;}{\&Integral;}_0^{+\&infin;}{x}^{m+b}{y}^{n+b}{e}^{-{\mathrm{cx}}^{m_1}-d{y}^{m_2}}\mathrm{dxdy}$

$T_{m,n}=\frac{\mathrm{\&alpha;}}{m_1{m}_2{c}^{\frac{m+b+1}{m_1}}{d}^{\frac{n+b+1}{m_2}}}\mathrm{\&Gamma;}\left(\frac{m+b+1}{m_1}\right)\mathrm{\&Gamma;}\left(\frac{n+b+1}{m_2}\right)$

Here m, n gets respectively 0,1,2, wherein the Euler integral of the second kind of Γ () for knowing;

Step 4: set up by step 3 and determine that typhoon influence marine site Extreme Wave and extreme water level measured data mixed distribution are apart from T _{M, n}With parameter m undetermined ₁, m ₂, α, b, c, the Simultaneous Equations of d:

$\frac{{T_{\mathrm{1,0}}}^2}{T_{\mathrm{0,0}}{T}_{\mathrm{2,0}}}=\frac{{\mathrm{\&Gamma;}}^2\left(\frac{b+2}{m_1}\right)}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_1}\right)\mathrm{\&Gamma;}\left(\frac{b+3}{m_1}\right)}$

$\frac{{T_{\mathrm{0,1}}}^2}{T_{\mathrm{0,0}}{T}_{0,2}}=\frac{{\mathrm{\&Gamma;}}^2\left(\frac{b+2}{m_2}\right)}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_2}\right)\mathrm{\&Gamma;}\left(\frac{b+3}{m_2}\right)}$

$\mathrm{\&alpha;}=\frac{m_1{m}_2{c}^{\frac{b+1}{m_1}}{d}^{\frac{b+1}{m_2}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_1}\right)\mathrm{\&Gamma;}\left(\frac{b+1}{m_2}\right)}$

$c={\left[\frac{\mathrm{\&Gamma;}\left(\frac{b+2}{m_1}\right)T_{\mathrm{0,0}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_1}\right)T_{\mathrm{1,0}}}\right]}^{m_1}$

$d={\left[\frac{\mathrm{\&Gamma;}\left(\frac{b+2}{m_2}\right)T_{\mathrm{0,0}}}{\mathrm{\&Gamma;}\left(\frac{b+1}{m_2}\right)T_{0,1}}\right]}^{m_2}$

Mixed distribution is apart from T _{M, n}In the actual computation process, press the row expression formula by observation data and calculate acquisition:

$T_{m,n}\&ap;\frac{\underset{s,t=1}{\overset{M}{\mathrm{\&Sigma;}}}{x_s}^m{y_t}^n}{M^2}$

M wherein, n gets respectively 0,1,2, x _s, y _tBe the observation data of Extreme Wave and extreme water level year by year, M is total year number;

Step 5: make up the typhoon influence marine site by step 1, step 2 and calculate Extreme Wave and the extreme water level nested compound maximum entropy joint distribution pattern of associating reoccurrence period:

$F(x,y)=p_0+\underset{i=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{p}_i{\&Integral;}_{-\&infin;}^y{\&Integral;}_{-\&infin;}^x{G}_x^{i-1}\left(u\right)g(u,v)\mathrm{dudv}$

Wherein i represents the number of times that the annual typhoon in marine site occurs, p ₀The probability of typhoon does not appear in the expression marine site, and g (u, v) is the Two-dimensional maximum-entropy joint density function, G _x(u) be the marginal distribution function of (X, Y), p _i=η ^{I γ}Exp (β i ^ξ) be the discrete maximum entropy distribution function of the typhoon number of times of inquiring into based on entropy principle, parameter η undetermined, beta, gamma, ξ specifies the relevant system of equations of marine site year typhoon occurrence number moment of distribution to obtain by finding the solution:

$\frac{A_1^2}{A_0{A}_2}=\frac{\mathrm{\&Gamma;}{\left(\frac{\mathrm{\&gamma;}+2}{\mathrm{\&xi;}}\right)}^2}{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}\right)\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+3}{\mathrm{\&xi;}}\right)}$

$\frac{A_0{A}_3}{A_1{A}_2}=\frac{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+4}{\mathrm{\&xi;}}\right)\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}\right)}{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+2}{\mathrm{\&xi;}}\right)\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+3}{\text{\&xi;}}\right)}$

$\mathrm{\&beta;}={\left[\frac{A_1\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}\right)}{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+2}{\mathrm{\&xi;}}\right)}\right]}^{-\mathrm{\&xi;}}$

$\mathrm{\&eta;}=\ln \left[\frac{{\mathrm{\&beta;}}^{\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}}\&CenterDot;\mathrm{\&xi;}}{\mathrm{\&Gamma;}\left(\frac{\mathrm{\&gamma;}+1}{\mathrm{\&xi;}}\right)}\right]+1$

A _kBe k-rank square, can press the row expression formula according to observation data and estimate:

$\stackrel{\&OverBar;}{A_k}=\frac{1}{M}\underset{k=0}{\overset{M}{\mathrm{\&Sigma;}}}{i}^k$

In actual computation, A _kValue be exactly

Wherein k gets the number of times that 0,1,2,3, i represents that the annual typhoon in marine site occurs, and M is total year number;

Step 6: the associating reoccurrence period N (year) that is pressed following formula to calculating Extreme Wave X and extreme water level Y by step 5:

$N=\frac{1}{P(X>x,Y>y)}=\frac{1}{1-F_x\left(x\right)-F_y\left(y\right)+F(x,y)}$

Wherein F (x, y) for step 5 push away maximum entropy joint distribution pattern, F _x(x) and F _y(y) be respectively its corresponding marginal distribution function;

Step 7: the validity of verifying new model according to the measured data data of Extreme Wave and extreme water level and typhoon occurrence number.

2. a kind of ocean extreme value associating reoccurrence period projectional technique of considering typhoon influence according to claim 1 is characterized in that:

M in the described step 1 ₁, m ₂When getting positive integer, guarantee that the arbitrary order mixed moment of Extreme Wave and extreme water level and the Order Moments of marginal distribution exist.

3. according to claim 1 a kind of considers the ocean extreme value associating reoccurrence period projectional technique of typhoon influence, it is characterized in that:

The expression formula of Two-dimensional maximum-entropy joint distribution function obtains by finding the solution a broad sense isoperimetric variational problem in the described step 2.

4. a kind of ocean extreme value associating reoccurrence period projectional technique of considering typhoon influence according to claim 1 is characterized in that:

Described step 7 further comprises the measured data data of utilizing, illustrate checking with Logistic Two dimensional Distribution and Gumbel Two dimensional Distribution commonly used in the oceanographic engineering, by judging the validity of new model with the fitting degree of measured data, in addition in conjunction with observation website marine site of living in characteristic, determine the rationality of associating reoccurrence period of inquiring into.

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