An improved typhoon simulation method based on the Latin hypercube sampling method

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The steps of the improved typhoon simulation method are summarized as follows.

Generation of simulated typhoon key parameters with Latin hypercube sampling method

Obtaining Independent Historical Parameters with the Cholesky Decomposition Method

Key typhoon parameters including central pressure difference ∆p0travel speed vsdirection of travel θthe minimum of the nearest distance Dmin and the radius at maximum winds rm obtained from historical typhoon wind data are expressed by a vector {X}, as following:

$$ { x}^{T} = left{ {lnleft( {Delta p_{0} } right) , lnleft( {r_{max} } right) , c , theta , d_{min} } right} $$

(1)

Then matrix [S] and [R] can be calculated according to the equations. (2) and (3). They represent respectively the correlation matrix and the covariance matrix of the key parameters of the typhoon, then the upper triangular matrix [T] can be obtained from [R] by Cholesky decomposition as shown in Equation. (4).

$$ left[ {S – lambda_{k} E} right]left{ {varphi_{k} } right} = 0{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} k = 1 ldots 5 $$

(2)

$$ left[ R right] = Eleft{ {left[ {x_{i} – Eleft( {x_{i} } right)} right]{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} left[ {x_{j} – Eleft( {x_{j} } right)} right]} right}{kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} left( {i,j = 1 ldots 5} right ) $$

(3)

$$ left[ R right] = left[ T right]left[ T right]^{^{first}} $$

(4)

The independent historical key parameter vector {Z} can be calculated from the lower triangular matrix [T]−1 (the inverse of the upper triangular matrix [T]) and the typhoon key parameter vector {X}, as shown in the equation. (5).

$$ left{ z right} = left[ T right]^{ – 1} left{ x right} $$

(5)

Generation of independent parameters simulated with the Latin hypercube sampling method

In vector {z}, the typhoon-independent key parameters represented by each data column correspond to vector {x}. The central pressure difference ∆p0 and t the radius at maximum winds rm can be fitted by a log-normal distribution, and the translational speed vsdirection of travel θ and the minimum of the nearest distance Dmin can be fitted by normal distribution. Based on the theory of the Latin hypercube sampling method, each probability function curve corresponding to the typhoon key parameters is divided into n equal intervals (not is the typhoon simulation numbers) and not random values ​​for not equal intervals can be generated14. Then the vector (left{ {z^{prime}} right}) for the simulated independent parameter can be generated with an inverse transformation of all probability functions for the historical independent parameters.

Generation of simulated typhoon key parameters

In order to obtain the key parameter vector of the simulated typhoon (left{ {x^{prime}} right})the upper triangular matrix [T] can be multiplied by the above vector (left{ {z^{prime}} right})expressed as follows:

$$ left{ {x^{prime}} right} = left[ T right]left{ {z^{prime}} right} $$

(6)

Unlike the Monte Carlo simulation methodsevenBy using the Latin hypercube sampling method, correlations between key typhoon parameters can be satisfactorily maintained when generating simulated typhoon key parameters.

Generation of typhoons with the typhoon model

In order to obtain an initial position of the typhoon, we bring the key parameters of the simulated typhoon into the typhoon wind field model. Next, it is assumed that the typhoon is moving in a straight line and the speed of movement vs is always maintained until it disappears. The model proposed by Vickery and Twisdaleten was used to calculate the central differential pressure ∆p0and the central differential pressure ∆p0 remains unchanged. This study adopts the empirical typhoon model proposed by Ishihara et al.15which is described in detail in Huang et al.16but I will not dwell on it here.

Perform extreme wind speed analysis

This article uses the extreme wind speed theory of the typhoon17 to calculate the extreme wind speed of the typhoon.

Assuming the probability that the wind speed of a typhoon is less than a specific wind speed v is Fvif the maximum score of a typhoon is youthe probability that the you in not typhoons is less than v is recorded as

$$ Fleft( {left. {U

(seven)

To calculate the probability that you v in τyears, eq. (7) can be transformed into:

$$ Fleft( {U

(8)

It is generally considered thatp( not, τ ) satisfies the Poisson distribution, and Eq. (8) is transformed into:

$$ Fleft( {U

(9)

where λ is the annual incidence of typhoons within a radius of 250 km around the research site. Whether τ = 1, Eq. (9) becomes:

$$ Fleft( {U

(ten)

It expresses the probability that you v in a year.

Record the maximum wind speed of m typhoons obtained by simulation from small to large asv1, v2,… vm. So the probability of youvI is

$$ F_{{v_{i} }} = frac{i}{m + 1};;;;left( {i = 1, cdots ,m} right) $$

(11)

Combine Eq. (10) to obtain the probability of youvI in a year

$$ Fleft( {U

(12)

Extreme wind speeds in typhoons are usually fitted to one of the generalized extreme value distributions. The type I (Gumbel) extreme value distribution is used in this study and can be written as:

$$ F(v) = exp left[ { – exp left( { – alpha left( {v – mu } right)} right)} right] $$

(13)

wherevis the maximum annual wind speed;αandµrepresent dispersion and mode respectively, which can be calculated by Eq. (12).

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