Joint occurrence probability analysis of typhoon-induced storm surges and rainstorms using trivariate Archimedean copulas

Ocean Engineering 171 (2019) 533–539

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Joint occurrence probability analysis of typhoon-induced storm surges and rainstorms using trivariate Archimedean copulas

T

Xing Yang*, Jun Qian Jiangsu Hydraulic Research Institute, 97# Nanhu road, Nanjing, China

ARTICLE INFO

ABSTRACT

Keywords: Joint probability Trivariate copulas Typhoon Storm surge Rainstorm

In this paper, the trivariate frequency analysis method for joint probability assessment of typhoon-induced storm surges and rainstorms is presented. The proposed method includes three major processes. The first process is to the find the best fitted marginal cumulative distributions for wind speeds (W), storm surges (S) and rainstorms (R), respectively. It is found that the Lognormal distribution is more suitable for storm surges and rainstorms, and Weibull distribution is more suitable for wind speeds. The second process is to estimate the joint probability distributions of W, S and R using four trivariate copulas, namely, Gumbel, Clayton, Frank and AMH. It is found that the joint probability can be best modeled by Gumbel copula. The last process is to calculate the conditional joint distributions of typhoon-induced storm surges and rainstorms which is useful for government agencies for improving disaster system management during typhoon events. An effective Particle Swarm Optimization (PSO) is proposed for to estimate the parameters of marginal cumulative distributions and copula functions. Analysis results based on the maximum likelihood method (MLM) and log-likelihood function (LLF) demonstrate the effectiveness of the PSO.

1. Introduction Not only strong winds and wind-generated waves, typhoons also bring rainstorms and storm surges. The combination of these typhooninduced phenomena can lead to extremely high water levels, thereby increasing the coastal flooding risk from rainstorms. Actually, flooding disasters induced by typhoons are one of the major natural hazards in many coastal regions including the eastern coastal region of China. So, hydrological engineering planning, design, and management in these coastal regions require joint probability distributions as well as conditional probability distributions of these typhoon-induced phenomena. Some meaningful works on the joint probability distributions with regard to typhoon-induced flooding events have been done by Hashino (1985), Yue (2001), Dong et al. (2004), Doong et al. (2008), Lian et al. (2013), etc. Hashino (1985) generalized the Freund bivariate exponential distribution to represent the joint probability distribution of rainfall intensity and maximum storm surge in Osaka Bay, Japan. Yue (2001) used bivariate Gumbel copula to analyze the joint distribution of annual maximum storm peaks and the corresponding storm amounts at the Tokushima meteorological observation station in Japan. Dong et al. (2004) proposed a Poisson bivariate Gumbel logistic distribution to calculate the joint probability of extreme wave height and concomitant wind speed during typhoon process in the South China Sea. Doong et al. *

(2008) utilized the Monte Carlo simulation approach (Hawkes et al., 2002) to represent the joint probability distribution of significant wave heights and water levels during typhoons at four in-situ stations, namly Longdong, Hualien, Dapenwan and Eluanbi that locate at northern, eastern, southern and western of Taiwan respectively. Lian et al. (2013) used bivariate Gumbel copula to describe the joint probability distribution of rainfall and tidal level in Fuzhou City, China, which suffers from major flooding damage from typhoons twice a year on average. All of the above-mentioned joint probability studies focused on bivariate statistical methods. However, a typhoon-induced flooding event is mainly characterized by wind-generated waves, rainstorms and storm surges. So, trivariate frequency analysis can provide a more reasonable assessment of this event. Especially in recent years, there has been a growing interest in applying trivariate copulas to stochastic hydrological process analysis, e.g., rainfall frequency analysis (Zhang et al., 2007; Kao and Rao, 2008; Balistrocchi and Bacchi, 2011), flood frequency analysis (Escalantesandoval and Raynalvillasenor, 1994; Grimaldi and Serinaldi, 2006; Sandoval and Raynal, 2008; Zhang et al., 2014) and drought frequency analysis (Wong et al., 2010; Ma et al., 2013). Further in 2013, Ganguli and Reddy (2013) adopted trivariate Student's t copula for multivariate analysis of flood risks, and applied this analysis method to a case study of flood flows of Delaware River basin at Port Jervis, NY, USA. In 2014, Bezak et al. (2014). used three-

Corresponding author. E-mail address: [email protected] (X. Yang).

https://doi.org/10.1016/j.oceaneng.2018.11.039 Received 10 November 2017; Received in revised form 16 October 2018; Accepted 25 November 2018 0029-8018/ © 2018 Published by Elsevier Ltd.

Ocean Engineering 171 (2019) 533–539

X. Yang, J. Qian

dimensional Gumbel-Hougaard copula to analyze the joint distributions of peak discharge, hydrograph volume and suspended sediment concentration data form one hydrological station in Slovenia and five stations in USA. A substantial advantage of the copula-based methodology is that it does not assume marginal distributions to be normal or independent. However, the joint occurrence as well as conditional occurrence of storm surges, rainstorms and wind speeds during typhoon events which may lead to failure of flooding protection systems, has not been fully investigated. Therefore, in this study, Shenzhen, a coastal megacity in subtropical southern China, was selected for a joint probability assessment of these variables. The objectives of the study are (1) to review the historical record of damage caused by typhoon-induced flooding events, (2) to find the most appropriate copula representing the trivariate dependence structure of these variables, and (3) based on the above, to derive trivariate joint distributions and conditional distributions of these variables using copula on the basis of the marginal distributions of these variables. With this aim, first the study area and data used in this study are described, then the proposed trivariate copula analysis method is presented, and, finally, results are summarized.

Table 1 Basic statistic characteristics of the analyzed typhoon data during the period of 1970–2012. 24-h rainfall (mm) Mean Minimum Maximum

117.3 26.2 338.6

Wind Speed (m/s)

Storm surges (m)

14.3 9.10 27.0

1.14 0.53 2.23

Typhoon No. 9318, which made landfall in Shenzhen on 26 September 1993. The highest 24-h precipitation recorded at the Shenzhen National Climate Observatory amounted to 338.6 mm. This typhoon took 14 lives and caused 764 million RMB (about 111. 38 million U.S. dollars) in direct economic losses. Table 2 provides a summary of death tolls, casualties, and direct economic losses caused by selected typhoons (shown in Fig. 2) in Shenzhen since 1970. It can be observed that, though the excessive rainfalls were not always occurring together with excessive storm surges for the zone, high storm surges and strong rainfalls were easily superposed during typhoon periods. Thus, these typhoons made serious natural disasters. Shenzhen National Climate Observatory typically reports winds over a 10-min average. The typhoon moves very fast, and the typhoon-induced storm surge and continuous heavy rainfall durations are generally within 24 h in Shenzhen. The maximum 10-min average wind speeds with the simultaneous 24 h rainfalls and storm surges during ∼60 typhoons were used in this paper. A vertical datum is needed in deriving height or elevation of points on the surface of the earth. The Zhujiang vertical datum is a datum based on the Pearl River water level, which is widely used in Guangdong province, China. This paper adopted Zhujiang vertical datum as a datum for storm surges.

2. Study area and typhoon data Shenzhen is located in South China and on the eastern bank of the Pearl River. Shenzhen neighbors Hong Kong. It is the first Special Economic Zone in China. As a megacity, Shenzhen occupies 1991.64 square kilometers and there are 11.38 million permanent residents by the end of 2015. The city, being located at the subtropical zone in western Pacific Ocean, has a subtropical marine climate with a prominent monsoonal influence. The wet southwest monsoon in the summer and autumn is associated with typhoons, and the northeast monsoon in winter and spring is appreciably drier (Yim, 1996). Shenzhen suffers from an average of about four or five typhoons per year. The high winds, heavy rain and storm surges associated with typhoons in Shenzhen had caused substantial economic damage and heavy loss of lives. This paper is aimed at assessing the joint occurrence probability of typhoon-induced storm surges and rainstorms in Shenzhen. For the Chiwan Tide Gage station (Fig. 1), the tide data is available since 1964. For the Shenzhen National Climate Observatory (Fig. 1), the 10-min average wind speed data and 24 h rainfall records are available since 1970. Some basic statistical characteristics of these variables are presented in Table 1. There were ∼60 typhoons within the range of 500 km of Shenzhen from 1970 to 2012, leading to severe damage to agriculture and industry, and some loss of human life. An outstanding example was

3. Analysis method 3.1. Conditional joint distribution function In this study, the trivariate approach using the copula functions for estimating multivariate joint occurrence distribution has been chosen. For typhoon-induced storm surges and rainstorms, conditional joint distributions may be of greater interest. Let variables W, R and S denote 10-min average wind speed, rainfall, storm surge, respectively. The following two general forms of the conditional joint distribution (CJD) function will be used based on the generalization obtained by Zhang et al. (2007). (1) The CJD-I of R and S, given W ≤ w, can be expressed as

FR, S|W (r , s|W

w) =

F (r , s , w ) F (w )

(1)

where F (r , s, w ) = P (R r , S s, W w ) with P =& #x202F;non-exceedance probability; r, s, w =& #x202F;values of variables R, S and W, respectively; F (w ) = P (W w )= marginal distribution of random variable W. (2) The CJD-II of R and S, given W = w, can be expressed as

FR, S|W (r , s|W = w ) =

F (r , s, w )/ w F (r , s , w ) = F (w )/ w F (w )

(2)

The Archimedean copulas are very popular in hydrologic application (Favre et al., 2004; Mirabbasi et al., 2012; Lee et al., 2013). F (r , s, w ) and F (r , s, w )/ F (w ) in Eq. (1) and Eq. (2) can be solved by four one-parameter Archimedean copulas (Gumbel, Clayton, Frank and AMH) based on almost any choice of marginal distributions (Exponential, Lognormal, Gamma and Weibull). Let Fr = P (R r ) ,

Fig. 1. Map of Shenzhen showing the Shenzhen National Climate Observatory and Chiwan Tide Gage station. 534

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X. Yang, J. Qian

Table 2 Summary of damage of selected typhoons in Shenzhen since 1970. Typhoon

Time

Peak intensity (mm/h)

Highest 24-h rainfall (mm)

Beaufort scale

Highest tidal level (m)

Death toll

Persons Injured or Missing

Economic losses (million RMB)

7908 8309 9318 9908 0313 0814

1979-8-2 1983-9-9 1993-9-26 1999-8-22 2003-9-2 2008-9-24

18.7 15.7 68.9 61.4 43.1 23.3

109.8 137.5 338.6 329.1 101.5 85.6

12 12 12 11 12 12

0.96 1.77 0.93 0.91 1.18 2.21

Unavailable 7 14 7 22 0

Unavailable 63 Unavailable 10 100 3

20 50 764 150 250 35

3.2. Evaluation of optimal parameters Parameters in Tables 3 and 4 can be estimated using a Particle Swarm Optimization (PSO) method proposed firstly by Kennedy and Eberhart (1995). PSO can be described by bird foraging behaviors. A group of birds are randomly searching the only one piece of food in a region. All birds (particles) do not know where the food is, but they know how far their current positions are from the food. To find the food, each bird will have to study its own previous best position (personal best), and bird group's previous best position (global best), and then adjusts its position and velocity accordingly. Repeating this process, the optimal position will be found by the bird which is nearer to the food (Geetha, 2013). Taking marginal cumulative distribution functions in Table 4 as an example, PSO is outlined as follows: Step 1. Initializing the velocity and position (parameter value) of each particle at time t = 0. Let L =& #x202F;the number of particles and M = the number of parameters. Then the position and the velocity of the kth particle in the M-dimensional parameter search space can be represented as k0, i = [ k0,1, k0,2, , k0, M ] and V k0, i = [V k0,1, V k0,2, ,V k0, M ], respectively. (1) The initial position of the kth particle can be expressed by

Fig. 2. Paths of some typhoons in Shenzhen. Table 3 List of the selected trivariate copulas. Copula Gumbel Clayton

F (r, s, w, θ)

]1/

exp{ [( ln Fr ) + ( ln Fs ) + ( ln Fw )

(Fr

Frank AMH

θ space

+ Fs

1

[1

ln[1 + (1

+ Fw

2)

Fr

1) (e

(e

Fr Fs Fw Fr )(1 Fs )(1

}

Fs

1) (e

Fw

1)

1)2

]

Fw )]

Exponential (exp) Lognormal (log)

F (x, δ)

1

V k0, i = rand () ×

2 1

0.5 + 0.5 [

(x ) =

2

ln(x

x

)

2

2) 3 2

x, 1

1

],

δ1: δ2: δ1: δ2: δ3:

>0 2

< x where

exp( t 2) dt

0

Gamma (gam)

Weibull (wei)

x

(x ) = 2

1

x 1 2 ( 2) 0 1

t

2 1

exp( t / 1) dt ,

>0

exp[

(x

2) 3 ], 2 1

x

1

>0

(

min,i )

max,i

(4)

2 ( max,i

min,i)

with Vmax, i = Vmin, i = To control excessive flying of parti2 cles outside the search space, V k0, i = Vmax, i if V k0, i > Vmax, i and V k0, i = Vmin, i if V k0, i < Vmin, i .

δ

exp(

(3)

min,i )

max,i

(2) The initial velocity of the kth particle can be expressed by

Table 4 List of the selected marginal cumulative distribution functions. Distribution

= rand () × (

where max,i and min,i are the upper and lower bounds of i position value, respectively, and rand () is the random number distributed in the range of [0, 1]. At time t = 0, the kth particle has its own best position denoted by pbk, i = k0, i .

[-1, ∞) θ≠0 (-∞, ∞) θ≠0 [-1, 1)

1/

(e

0 k,i

[1, ∞)

scale location scale location shape

Step 2. Comparing with all pbk, i to find the global best particle position gbi = [gb1, gb2, ..., gbM ] so far at time t. Let the current position of the kth particle is kt , i = [ kt,1, kt,2, , kt , M ]. Let F (x , pbk, i ) & #x202F;= the kth particle theoretical cumulative probability shown in Table 2 and F (x )  = the kth particle empirical cumulative probability defined by Gringorten (1963):

δ1: scale δ2: shape δ1: scale δ2: location δ3: shape

F (x j ) = P (X

xj ) =

N N

0.44 0.12

(5)

where N  = the jth smallest observation in the data set arranged in ascending order and N  = sample number of observations. The Nash-Sutcliffe efficiency coefficient (Nash and Sutcliffe, 1970), which is the one criterion most widely used for calibration and evaluation of hydrological models with observed data

Fs = P (S s ) and Fw = P (W w ) are marginal cumulative distribution functions of variables R, S and W, respectively. Table 3 shows the selected trivariate copula functions, where θ is the copula parameter. Table 4 shows the selected marginal cumulative distribution functions.

535

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(Gupta et al., 2009), was used for evaluate each particle's best position. Compared to many other criteria, the Nash–Sutcliffe model efficiency coefficient is more sensitive to differences between model predictions and observations, and thus is a better indicator of goodness of fit (Legates and McCabe, 1999). Let BEk  = each particle's Nash-Sutcliffe efficiency coefficient associated with its best position. Then the global best particle position gbi = pbk, i max{BEk} . N j=1

BEk = 1

[F (xj ) N j=1

#x202F;0.8. Four copulas (Gumbel, Clayton, Frank and AMH) were examined and compared to find the best fitted copula. Table 6 lists the estimated parameter θ and the results of PSO. According to CE, Gumbel copula is found to be more suitable to represent the joint distribution of rainfall, storm surge and wind speed. The probability–probability plots (Fig. 4) show that the estimated theoretical cumulative probabilities by Gumbel copula are quite close to the empirical ones. Eq. (2) can be modified by Gumbel copula as follows:

F (x j , pbk, i )]2

[F (xj )

FX ]2

F (r , s , w ) F (w )

(6)

=

where FX = N j = 1 F (x j )=the mean of the empirical cumulative probability data; An efficiency of 1 (BEk = 1) corresponds to a perfect match between theoretical and empirical cumulative probabilities. An efficiency of 0 indicates that the theoretical probabilities are as accurate as the mean of the empirical data. The better the goodness of fit is, the closer the value of BE is to 1. N

1

=

t k, i )

+ c2 × rand () × (gbi

=

t k, i

1 + Vkt + ,i

t k, i )

(8)

Updating the personal best position pbk, i of the kth particle according to (6) and (9) at time t+1, where CEk  =& #x202F;each particle's Nash-Sutcliffe model efficiency coefficient associated with its current position. For the kth particle, compare CEk with BEk . If current value is more closer to 1, then pbk, i is updated with the current position kt +, i 1 .

CEk = 1

N j =1

[F (xj ) N j=1

F (x j ,

[F (x j )

t+1 2 k , i )]

FX ]2

F (r , s, w )( ln Fw )

1 [( ln F ) + ( ln F ) + ( ln F ) ](1/ r s w

1)

Fw F (r , s, w )( ln Fw )0.383

Fw [( ln Fr )1.383 + ( ln Fs )1.383 + ( ln Fw )1.383 ]0.383/1.383

(10)

The preceding analysis shows that it is possible to construct joint distributions of typhoon-induced storm surges and rainstorms using copulas. The parameters of marginal cumulative distributions and joint cumulative distributions via copula functions were estimated using the PSO. By comparisons of CE, we selected the Lognormal distribution for the rainfall and storm surge, Weibull distribution for wind speed, and Gumbel copula for the joint distribution of rainfall, storm surge and wind speed. Scale parameter (δ1), location parameter (δ2) and shape parameter (δ3) for rainfall and storm surge 4.93, −40.94, 0.52 and 0.34, −0.33, 0.29, respectively. Scale parameter (δ1), location parameter (δ2) and shape parameter (δ3) for wind speed are 14.33, 7.69, 1.33, respectively. Copula parameter (θ) is 1.383. Combining these considerations, a set of contour plots were then created. Fig. 5 shows the contour plots of the conditional joint distribution function of R and S, given that the wind speed does not exceed various thresholds (Eq. (1)). Fig. 6 shows the contour plots given that the wind speed equals various thresholds (Eq. (2)). Both figures imply that the CJD-I and CJD-II increase with rainfall and storm surge, respectively, and decrease with wind speed. To see this more clearly, the conditional joint probabilities of pairs (R = 200, S& #x202F;= 2.0) and (R = 300, S& #x202F;= 2.5) with a wind speed of 10, 15, 20, and 25& #x202F;m/s were computed as an example. As can be seen in Fig. 7, we can then conclude that (1) CJD has a significant reduction with increasing W, (2) compared with CDJ-I, CDJ-II has a more faster downward trend in the corresponding R and S, (3) a larger R or a higher S may lead to greater CJD, and (4) CJD is lower when W does not exceeds, rather than equals a threshold.

(7) t+1 k,i

exp{ [( ln Fr ) + ( ln Fs ) + ( ln Fw ) ]1/ } F (w )

4.3. Conditional joint distributions of the typhoon events

1 t+1 Step 3. Updating the velocity Vkt + , i and position k , i of the kth particle according to (7) and (8) at time t+1, where c1 andc2 are the 1 acceleration coefficients; w is the inertia factor. if Vkt + , i > Vmax, i or t+1 t+1 t Vk, i < Vmin, i , then Vk, i = Vk, i . 1 t Vkt + , i = w × V k, i + c1 × rand () × (pbk , i

=

(9)

Step 4. If a stopping criterion max{CEk } 0.9999 is met, then output gbi = [gb1, gb2, ..., gbM ] otherwise go to Step 2. 4. Results and discussion 4.1. Selected marginal cumulative distributions When compiling the calculation program according to PSO mentioned above, it is necessary to pay attention to the data overflow error or floating point division by zero. Let c1 = c2& #x202F;= 2, and w = 0.8. Let MNI& #x202F;= maximum number of calculation iterations. Four marginal distributions (Exponential, Lognormal, Gamma and Weibull) were examined and compared to find the best fitted distribution. Table 5 lists the estimated parameters and the results of PSO. According to CE, Lognormal distribution is found to be more suitable for 24& #x202F;h rainfall and storm surge, and Weibull distribution is found to be more suitable for wind speed. The probability–probability plots (Fig. 3) show that the estimated theoretical cumulative probabilities are quite close to the empirical ones associated with the best fitted distribution.

4.4. Reliability of the PSO In this study, the PSO was used to estimate the parameters of marginal cumulative distributions and copula functions. Eq. (9) was used for the goodness-of-fit-test for these univariate and trivariate distributions. Although comparison results between the empirical distributions and the theoretical distributions show good agreements, the reliability of PSO for estimating the parameters of these probability distributions is still worth discussing. For this, the maximum likelihood method (MLM) and log-likelihood function (LLF) were used. Mitková and Halmová (2014) used MLM to calculate location, shape and scale parameters of some marginal distributions for observed flood discharge, volume and duration. Genest et al. (1995) developed a semiparametric estimation method for copula parameter θ using LLF. Zhang et al. (2007) used the semiparametric estimation method to analyze Gumbel copula function of rainfall intensity, depth, and duration using LLF. The root mean square error (RMSE) (Willmott and Matsuura, 2005) and Nash-Sutcliffe efficiency coefficient calulated were applied and given in Table 7. The results show RMSE values and CE values by PSO is very

4.2. Selected copula for joint distributions The joint distributions could be obtained by combining the best fitted marginal distributions of rainfall (Lognormal), storm surge (Lognormal) and wind speed (Weibull), respectively, using copulas. Let c1 = c2 = 2, and w =& 536

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X. Yang, J. Qian

Table 5 The suitability of the marginal cumulative distribution functions. Marginal distribution

R

PSO

exp log gam wei

S

exp log gam wei

W

exp log gam wei

[δmin, δmax]

L

MNI

δ value

CE

δ1:[0,500] δ2: [-500,500] δ1: [-500,500] δ2: [-500, 500] δ3: [-500, 500] δ1: [0,100] δ2: [0, 100] δ1: [0,1000] δ2: [-500,500] δ3: [-500, 500] δ1:[0,50] δ2: [-50,50] δ1: [-50,50] δ2: [-50, 50] δ3: [-50, 50] δ1: [0,50] δ2: [0, 50] δ1: [0,50] δ2: [-50,50] δ3: [-50, 50] δ1:[0,500] δ2: [-500,500] δ1: [-500,500] δ2: [-500, 500] δ3: [-500, 500] δ1: [0,100] δ2: [0, 100] δ1: [0,500] δ2: [-500,500] δ3: [-500, 500]

100

10000

0.9705

100

10000

100

10000

1000

10000

100

10000

100

10000

100

10000

1000

10000

100

10000

100

10000

100

10000

100

10000

δ1 = 115.56; δ2 = 14.2 δ1 = 4.93; δ2 = -40.94 δ3 = 0.52 δ1 = 58.51; δ2 = 2.0 δ1 = 893.47; δ2 = 4.76; δ3 = 1.42 δ1 = 0.71; δ2 = 0.53 δ1 = 0.34; δ2 = -0.33; δ3 = 0.29 δ1 = 0.20; δ2 = 5.81; δ1 = 2.66; δ2 = -0.11; δ3 = 3.49 δ1 = 7.50; δ2 = 7.8 δ1 = 2.10; δ2 = 5.06 δ3 = 0.57 δ1 = 1.65; δ2 = 8.52 δ1 = 14.33; δ2 = 7.69; δ3 = 1.33

0.9897 0.9883 0.9873 0.9504 0.9891 0.9844 0.9851 0.970 0.9938 0.9876 0.9955

Fig. 3. Probability–probability plots of marginal distributions. Table 6 The suitability of the copula functions. Copula

Gumbel Clayton Frank AMH

PSO [δmin, δmax]

L

MNI

θ value

CE

[1,10] [-1,10] [-10,10] [-1,1]

100 100 100 100

10000 10000 10000 10000

1.383 0.703 2.549 0.599

0.9898 0.9859 0.9884 0.9418

Fig. 4. Probability–probability plots of joint distributions.

close to that by MLM and LLF. PSO is a reliable and intelligent algorithm proposed to estimate parameters of the marginal cumulative distributions and copula functions simply and effectively.

Section 4.1. The third process is to estimate the joint distribution of these variables using copula function, which are described in detail in Section 4.2. Finally, the conditional joint distributions of wind speed (W), rainfall (R) and storm surges (S) are calculated, which are described in detail in Section 4.3. The following conclusions are drawn from this study:

5. Conclusions The study is focused on the analysis of the joint probability of the occurrence of typhoon-induced storm surges and rainstorms. The study includes four major processes. The first process introduces the trivariate typhoon frequency analysis algorithm based on copula functions and PSO, which are described in detail in Section 3. The second process deals with the estimation of the marginal distributions of wind speeds (W), rainfalls (R) and storm surges (S), which are described in detail in

(1) Nash-Sutcliffe efficiency coefficient analysis showed the Lognormal distribution was the most appropriate for marginal cumulative distributions of R and S, and Weibull distribution was found to be more suitable for marginal cumulative distribution of W in 537

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Fig. 7. CJD of pairs (R = 200, S = 2.0) and (R = 300, S = 2.5) with various wind speed thresholds.

Fig. 5. Contour plots of CJD-I of R and S, given W ≤ w& #x202F;= [10, 15, 20, 25 m/s].

Table 7 Goodness-of-fit statistics. Method

R

Goodness-of-fit

log

MLM PSO

S

log

MLM PSO

W

wei

MLM PSO

(r, s, w)

Fig. 6. Contour plots of CJD-II of R and S, given W = w& #x202F;= [10, 15, 20, 25 m/s].

Shenzhen. The PeP plots showed good association between the theoritical distribution and the empirical distributions. Scale parameter (δ1), location parameter (δ2) and shape parameter (δ3) for W, R and S are (14.33, 7.69, 1.33), (4.93, −40.94, 0.52) and (0.34, −0.33, 0.29), respectively. (2) An effective Particle Swarm Optimization (PSO) was proposed to estimate the parameters of marginal cumulative distributions and copula functions. The maximum likelihood method (MLM) and loglikelihood function (LLF) were used to test the reliability of the proposed PSO. The root mean square error (RMSE) and NashSutcliffe efficiency values by the PSO is very close to that by MLM and LLF. Analysis results based on MLM and LLF demonstrate the effectiveness of the PSO. (3) Preliminary analysis showed that the Gumbel copula was the most appropriate for trivariate joint distribution analysis of W, R and S in Shenzhen. The joint distributions could be obtained by combining the Lognormal marginal distributions of R and S, and Weibull marginal distribution of W, respectively, using Gumbel copula.

Gumbel

LLF PSO

δ1 = 4.84; δ2 = -28.61 δ3 = 0.57 δ1 = 4.93; δ2 = -40.94 δ3 = 0.52 δ1 = 0.32; δ2 = -0.31; δ3 = 0.30 δ1 = 0.34; δ2 = -0.33; δ3 = 0.29 δ1 = 13.93; δ2 = 7.72; δ3 = 1.32 δ1 = 14.33; δ2 = 7.69; δ3 = 1.33 θ = 1.345 θ = 1.383

RMSE

CE

2.985

0.9894

2.945

0.9897

3.038

0.9893

3.057

0.9892

1.857

0.9954

1.854

0.9954

2.826 2.819

0.9894 0.9898

Parameter θ using PSO and LLF is 1.383 and 1.345, respectively. The corresponding (RMSE, CE) are (2.819, 0.9898) and (2.826, 0.9894), respectively. The results of PSO and LLF are quite similar. But PSO is a more simple and intelligent algorithm. (4) For government agencies for improving disaster system management during typhoon events, the conditional joint distributions (CJD) of typhoon-induced storm surges (S) and rainstorms (R) are important. This paper analyzed the CJD of R and S given that the wind speed (W) did not exceed a certain threshold or the wind speed (W) was equal to a certain threshold. Analysis results shows CJD has a significant reduction with increasing W, and a larger R or a higher S may lead to greater CJD. Acknowledgements This work was supported by Jiangsu Province Science and Technology (Grant No. BM2018028). 538

Ocean Engineering 171 (2019) 533–539

X. Yang, J. Qian

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