Poisson bivariate

8/12/2019 Poisson bivariate

1/17

O R I G I N A L P A P E R

Estimating storm surge intensity with Poisson bivariate

maximum entropy distributions based on copulas

Shanshan Tao Sheng Dong Nannan Wang C. Guedes Soares

Received: 10 January 2013 / Accepted: 18 March 2013 / Published online: 29 March 2013 Springer Science+Business Media Dordrecht 2013

Abstract This paper introduces four kinds of novel bivariate maximum entropy distri-

butions based on bivariate normal copula, GumbelHougaard copula, Clayton copula and

Frank copula. These joint distributions consist of two marginal univariate maximum

entropy distributions. Four types of Poisson bivariate compound maximum entropy dis-

tributions are developed, based on the occurrence frequency of typhoons, on these novel

bivariate maximum entropy distributions and on bivariate compound extreme value theory.

Groups of disaster-induced typhoon processes since 19492001 in Qingdao area areselected, and the joint distribution of extreme water level and corresponding significant

wave height in the same typhoon processes are established using the above Poisson

bivariate compound maximum entropy distributions. The results show that all these four

distributions are good enough to fit the original data. A novel grade of disaster-induced

typhoon surges intensity is established based on the joint return period of extreme water

level and corresponding significant wave height, and the disaster-induced typhoons in

Qingdao verify this grade criterion.

Keywords Poisson bivariate maximum entropy distribution

Typhoon-induced storm surge Disaster intensity Joint period Water levelSignificant wave height

Abbreviations

UMED Univariate maximum entropy distribution

BMED Bivariate maximum entropy distributions

NBMED Bivariate maximum entropy distributions with normal copula

GHBMED Bivariate maximum entropy distributions with GumbelHougaard copula

CBMED Bivariate maximum entropy distributions with Clayton copula

S. Tao S. Dong (&) N. WangCollege of Engineering, Ocean University of China, Qingdao 266100, China

e-mail: [email protected]

C. Guedes Soares

Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Tecnico,

Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

1 3

Nat Hazards (2013) 68:791807

DOI 10.1007/s11069-013-0654-6


2/17

FBMED Bivariate maximum entropy distributions with Frank copula

EBMED Equivalent bivariate maximum entropy distribution

PBCEVD Poisson bivariate compound extreme value distribution

PGMCD PoissonGumbel mixed compound distribution

PBCMED Poisson bivariate compound maximum entropy distributionMOM Method of moments

ECFM Empirical curve-fitting method

MLM Maximum likelihood method

PNBMED Poisson normal bivariate maximum entropy distribution

PGHBMED Poisson GumbelHougaard bivariate maximum entropy distribution

PCBMED Poisson Clayton bivariate maximum entropy distribution

PFBMED Poisson Frank bivariate maximum entropy distribution

1 Introduction

In the design of marine structures, many kinds of long-term and extreme distributions, such

as the Gumbel, Weibull, lognormal and Pearsons type-3 distributions, have been applied

to fit annual extreme data. Review papers have been produced (Isaacson and MacKenzie

1981; Muir and El-Shaarawi 1986; Guedes Soares and Scotto 2011), and comparisons

between the performance of various models have also been addressed (Van Vledder et al.

1993; Guedes Soares and Scotto 2001).

As alternative to these distributions, Zhang and Xu (2005) proposed a type of univariate

maximum entropy distribution (UMED) function that contains four parameters, so it hasmore flexibility and adaptability than other long-term distributions used in ocean engi-

neering. As almost all distributions generally used in frequency analysis of ocean data are

special cases of this distribution (Dong et al. 2012a,b), the UMED can be applied more

extensively to this type of data. An alternative application of maximum entropy to uni-

variate wave data can be found in Petrov et al. (2013).

Researches show that considering only one environmental variable in the design of

offshore structures is too limited. In fact, the correlation between environmental variables

is often important as for example, the astronomical tide and storm surge, which often

happen at the same time, or large wave heights and higher wind speed often come together

in the same typhoon process (Wahl et al. 2012). So the simultaneous influence of two or

more environmental elements should be taken into consideration as has been considered in

various approaches (Bitner-Gregersen and Guedes Soares 1997).

The earlier ones dealt with the joint distribution of significant wave heights and char-

acteristic periods and examples of different bivariate models are Haver ( 1985), Athanas-

soulis et al. (1994), Ferreira and Guedes Soares (2002), Repko et al. (2004) and Jonathan

et al. (2010), while a more recent approach dealing with bivariate maximum entropy

distributions was presented by Dong et al. (2013).

In order to describe the dependence between wave height and wind speed, Prince-Wright

(1995), Morton and Bowers (1996), Zachary et al. (1998) and Nerzic and Prevosto (2000)have develop different approaches. Dong et al. (2005) simulated the joint return periods of

wind and wave with bivariate lognormal distribution (BLND); Leira (2010) compared the

Nataf model, the NKR model and the Plackett model of bivariate Weibull distribution.

Sklar (1959) proposed the concept of copula, and several other researchers constructed

many kinds of copulas to obtain joint probability distributions. A copula can combine the

792 Nat Hazards (2013) 68:791807

1 3


3/17

marginal distributions of different ocean environmental parameters with some correlations

among them, and eventually get a joint distribution (Nelsen2006). So the copula is a good

way to construct bivariate joint distributions. Favre et al. (2004) discussed the application

of copulas in the construction of multivariate joint models and applied these models to

analyze the joint distribution of flood peak and flood volume; Hanne et al. (2004) obtainedthe joint distribution of wave height and wave period based on bivariate normal copula,

and successfully applied it to data from the Japan Sea; De Waal and van Gelder (2005)

compared the joint distribution of extreme value wave height and wave period based on

BurrParetoLogistic copula with the result of a physical model; Muhaisen et al. (2010)

established the bivariate probability model of significant wave height and storm duration

based on a copula for the optimum design of gravel breakwaters, and Antao (2012) has

used different formulations of copulas to model the joint distribution of wave height and

steepness.

Here, four kinds of commonly used bivariate copulas are applied to obtain bivariate

maximum entropy distributions (BMED) based on the two margins of UMED, as fol-

lows: normal copula, GumbelHougaard copula, Clayton copula and Frank copula

(Nelsen 2006). The BMED given by these four copulas are abbreviated as NBMED,

GHBMED, CBMED and FBMED, respectively. Liu et al. (2010) proposed an equivalent

bivariate maximum entropy distribution (EBMED) to determine the joint return period of

wind speed and wave height considering the lifetime of platform structures, and this

model is consistent with bivariate maximum entropy distributions with normal copula

(NBMED).

The concept of the compound distribution was firstly proposed by Feller (1957). Ma and

Liu (1979) proposed the theory of the compound extreme value distribution by consideringthe occurrence frequency number of typhoons in different marine regions of China. Muir

and El-Shaarawi (1986) compared this model with other five commonly used distributions

in engineering design and found that it fitted the observations best, and the prediction effect

was also very well. Dong et al. (2009) constructed Poisson maximum entropy distribution

and utilized it to calculate the return typhoon wave heights. In order to estimate the

combined effects on the marine platforms of the wind speed and the wave height in every

typhoon process, Liu et al. (2002) generalized the Poisson univariate compound extreme

value distribution to one kind of Poisson bivariate compound extreme value distribution

(PBCEVD), and that is PoissonGumbel mixed compound distribution (PGMCD). Four

kinds of Poisson bivariate compound maximum entropy distributions (PBCMEDs) basedon NBMED, bivariate maximum entropy distributions with GumbelHougaard copula

(GHBMED), bivariate maximum entropy distributions with Clayton copula (CBMED) and

bivariate maximum entropy distributions with Frank copula (FBMED) are constructed in

Sect. 2, and they could be chosen for the engineering designs.

Typhoon is a kind of ocean dynamic phenomenon that often induces severe disasters.

For the past 40 years, coastal scientists applied the SarrirSimpson Scale to classify

tropical cyclones (Dolan and Davis 1994). This scale categorizes hurricanes into five

classes based on wind speed, and it provides the storm surges and central pressures

simultaneously. Halsey (1986) proposed a classification of Atlantic Coast extratropicalstorms based on damage potential index. Mendoza and Jimenez (2005) provided a storm

classification based on the beach erosion potential in Catalonian Coast. In this paper, based

on the data of annual extreme water level and corresponding significant wave height about

the typhoon surge from Qingdao area, the above four Poisson bivariate compound maxi-

mum entropy distributions (PBCMEDs) are applied to judge the intensity grade of storm

surge by its joint return period in the third section.

Nat Hazards (2013) 68:791807 793

1 3


4/17

2 Poisson bivariate compound maximum entropy distributions based on copulas

2.1 Univariate maximum entropy distribution

Jaynes (1968) introduced the maximum entropy principle as: the probability distributionwith smallest error is the distribution which makes the entropy the largest under the

additional constraints conditions with known information. Zhang and Xu (2005) proposed

one kind of UMED function given by:

Gx Zxa0

gtdtZxa0

at a0c expbt a0nh i

dt 1

in which X is a random variable, and b, c, n and a0 (the location parameter) are the

parameters of the UMED function. Here, a is a combination ofb,c,n and a0

, which can be

given by the following expression.

anbc1n C1 c 1n

2

whereC() represents the gamma function defined as:

Cs Z10

xs1exdx 3

The UMED has four parameters, so it has very strong flexibility and adaptability.

Although the UMED function has many advantages in practical applications, the estima-

tion of these four parameters is difficult. Zhang and Xu (2005) adopted the method of

moments (MOM) to fit the UMED (exclusive location parameter). The MOM needs to use

the third-order moment, so the sampling error is greater than that when estimating the

distribution with only two parameters. Dong et al. (2009) considered engineering practice

and proposed an empirical curve-fitting method (ECFM). Dong et al. (2012b) derived the

maximum likelihood method (MLM) for the UMED and compared it with MOM and

ECMF using data of annual extreme wave heights. The results show that the MLM and

ECFM are better parameter estimation methods.

2.2 Bivariate maximum entropy distributions based on copulas

The study of copulas and their applications in statistics is a rather modern development.

The reason is that copulas provide a way to construct families of the bivariate distributions

with univariate margins and their correlation (Nelsen 2006). Based on Sklars theorem, if

the two marginals are both UMEDs, then the joint distributions of the bivariate stochastic

variables can be obtained by copulas, and these joint distributions can be called BMED.

There are four kinds of commonly used bivariate copulas: the normal copula, the GumbelHougaard copula, the Clayton copula and the Frank copula (Nelsen 2006). A brief intro-

duction to these copulas is as follows.

The probability distribution function and density function of the bivariate normal copula

are as follows:

794 Nat Hazards (2013) 68:791807

1 3


5/17

Cu; v; h ZU1u

1

ZU1v1

1

2pffiffiffiffiffiffiffiffiffiffiffiffiffi

1 h2p exp s

2 2hst t221 h2

dsdt 4

cu; v; h 1ffiffiffiffiffiffiffiffiffiffiffiffiffi1 h2

p exp 2 h2U1u2 2hU1uU1v 2 h2U1v221 h2

( );

5respectively, where U() is the univariate standard normal distribution function and U-1()is the inverse function of U(); 1 B h B 1 is the linear dependent coefficient ofU-1(U) and U-1(V). U, V are independent when h = 0 and completely correlated if

|h| = 1.

The probability distribution function and density function of the bivariate Gumbel

Hougaard copula are as follows:

Cu; v; h exp ln uh ln vhh i1

h

6

cu; v; h ln u ln vh1 ln uh ln vh

h i1hh 1

uv ln uh ln vh21h

Cu; v; h; 7

respectively, where h C 1 is the correlated parameter and the relationship between h and

the Kendall ranks is s = 1 - 1/h.U, Vare independent whenh = 1, andU,Vtend to be

completely correlated ifh ? ??.The probability distribution function and density function of the bivariate Clayton

copula are as follows:

Cu; v; h uh vh 1 1h 8cu; v; h 1 h uvh1 uh vh 1 21h; 9

respectively, where 0\ h\?? is the correlated parameter and the relationship between

hand the Kendall ranks iss = h/(h ? 2).U, Vtend to be independent when h ? 0, and

U, Vtend to be completely correlated when h ? ??.The probability distribution function and density function of the bivariate Frank copula

are as follows:

Cu; v; h 1h

ln 1 ehu 1ehv 1eh 1

10

cu; v; h heh 1ehuveh 1 ehu 1ehv 1 2

; 11

respectively, whereh = 0 is the correlated parameter, and the relationship between h and

the Kendall rank is s1 4h

1h

Rh

0t

et1 dt 1h i

. h[ 0 denotes U, V have positive correla-

tion, while h\ 0 denotes U, Vhave negative correlation, and when h ? 0 the random

variables U, Vtend to be independent.

Let X and Y both follow UMEDs GX(x) and GY(y), respectively, and the distribution

functions of them are as follows:

Nat Hazards (2013) 68:791807 795

1 3


6/17

GXx Zxa1

a1s a1c1 expb1s a1n1h i

ds 12

GYy Zya2

a2t a2c2 expb2t a2n2h i

dt 13

in which a1n1bc11n1

1 C1c11

n1 and a2n2b

c21n2

2 C1c21

n2:

According to Sklars theorem and the above four bivariate copulas C(u, v), the joint

probability distribution (BMED) functions and density functions ofXand Ycan be written

as

G

x;y

C

GX

x

; GY

y

14

gx;y cGXx; GYy gXx gYy 15where c(u, v), gX(x) and gY(y) are the probability density functions ofC(u, v), GX(x) and

GY(y), respectively. Four kinds of BMEDs (NBMED, GHBMED, CBMED and FBMED)

based on according copulas have been constructed.

2.3 Poisson bivariate compound maximum entropy distribution

In order to estimate the combined effects on the marine platforms of wind speed and wave

height in each typhoon process, Liu et al. (2002) generalized the Poisson univariatecompound extreme value distribution to a Poisson bivariate compound extreme value

distribution.

Assume that the frequency n of a marine extreme event that happens in some marine

area is a discrete variable and its distribution probability is Pk. Let two marine extreme

elements when the extreme event happens be n andg, otherwise be f and c. Suppose the

joint distribution functions of (n,g) and (f,c) areG(x,y) andQ(x,y), respectively, the joint

probability density function of (n, g) is g(x, y) and the distribution function ofnis GX(x).

Let (ni,gi) be theith observation of (n,g).nis a random variable which follows the Poisson

distribution and is independent from (n, g). The distribution function ofn is denoted by

Pkekkk

k! ; k0; 1; 2;. . . 16

Define

X; Y f; c; n0nj; gj; nj max

1 i nni; n 1

( 17

Then, the distribution function of (X, Y) is as follows:

F0x;y ek X1k1

Zy1

Zx1

ekkkk!

kGXuk1gu; vdudv

ek 1 kZy

1

Zx1

ekGXugu; vdudv0@

1A

18

796 Nat Hazards (2013) 68:791807

1 3


7/17

If g(x, y) is the probability density function of one kind of the BMEDs obtained in

Sect. 2.2, F0(x, y) is called PBCMED. The four new PBCMEDs are named as Poisson

normal bivariate maximum entropy distribution (PNBMED), Poisson GumbelHougaard

bivariate maximum entropy distribution (PGHBMED), Poisson Clayton bivariate maxi-

mum entropy distribution (PCBMED) and Poisson Frank bivariate maximum entropydistribution (PFBMED).

2.4 Tests for PBCMED

Before using PBCMED models which are proposed in Sect. 2.3, a test of the model must be

conducted to ensure its goodness of fit of the original data. Here, three steps of test for

PBCMED are needed, they are: (1) Pearsonsv2 test for Poisson distribution, (2) KStest

for UMED margins and (3) v2 test for copulas (Hu2002).

2.4.1 Pearsons v2 test for Poisson distribution

Assume that the observations of the population of F(x) are x1, x2, , xn, and F0(x) is a

theoretical distribution. Let the null hypothesis be H0: F(x) = F0(x) and the alternative

hypothesis is H1: F(x) = F0(x). Divide the samplex1,x2, , xn intokgroups. Denote the

number of individuals in the group (xi1, xi) as vi. Evaluate the expected frequency nPi.

Suppose that the theoretical distribution function is F0x Pk

x0kxek

x! , then

Pi = F0(xi) - F0(xi1), i = 1, 2, , k, where 0\Pi\ 1 and

Pki1Pi1. Evaluate the

statistic: v2

Pk

i1v2

i

nPi n, then under the significance level a, ifv2[ v2k2, reject H0,

otherwise cannot reject H0.

2.4.2 KS test for UMED margins

Assume thatF(x) is the actual distribution function ofX,F0(x) is a known distribution, and

the sample size is n. Let the null hypothesis be H0: F(x) = F0(x), and the alternative

hypothesis is H1: F(x) = F0(x). Choose the statistic Dnsup1\k\1 jFnx F0xj;whereFn(x) is the empirical probability distribution function.

Letdk(1)= |Fn(xk) - F0(xk)| anddk

(2)= |Fn(xk?1) - F0(xk)|. Then, the observation value

of the statistic Dn is ^Dmax1 k n jd1k d2kj: If the significance level is a, then fordifferent sample size n, the KS critical value is Dn(a). If ^Dn\Dna; we admit H0,otherwise reject H0.

2.4.3 Pearsons v2 test for copulas

Hu (2002) introduces an Mstatistic which follows the v2 distribution, and it can estimate

the degree of fitting of the bivariate copulas. By using the Mstatistic, it is possible to know

whether the copula functions can describe the dependent structure of the variables or not,

and find how they fit the actual data.The detailed steps are as follows.

Assume that the observation data of X and Y are {xt} and {yt}, t= 1, 2, , n. Let

ut= GX(xt), vt= GY(yt). Construct a form R which contains k9 kgrid units. The unit in

theith row and the jth column is denoted by R(i, j), i, j = 1, 2, , k. For each {ut, vt}, if

(i - 1)/kB utB i/k and (j - 1)/kB vtB j/k both set up, then denote {ut, vt}2 R(i, j).

Nat Hazards (2013) 68:791807 797

1 3


8/17

Let Aij denote the number of the actual observation points which fall into the unit R(i, j),

and Bij denote the predicted frequency of the predicted points produced by different

copulas which fall into the unit R(i, j). Then,

MXki1

Xkj1

AijBij2Bij

v2k 12 19

If the significance level is a, the rejection region isfM[v21ak 12g, wherev21ak 12 is the downside 1 - a quantile of the v2 distribution with the free degree

4906 4908 5116 5622 8114 8406 8509 9005 9015 9216 9414 9415 9711

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

Typhoon Number

ExtremeWaterlevel(m)

4906 4908 5116 5622 8114 8406 8509 9005 9015 9216 9414 9415 9711

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Typhoon Number

SignificantWaveHeight(m)

a

b

Fig. 1 a Extreme water levels of typhoons. b Significant wave heights of typhoons

798 Nat Hazards (2013) 68:791807

1 3


9/17

(k- 1)2

. So if M[ v21ak 12, the copula is not suitable for this data; ifM\v21ak 12, the copula could be used to construct the bivariate model.

Choose the bivariate copula which makes Mthe smallest as the best bivariate copula to

construct new trivariate models by integral.

3 Intensity grade judgment of storm surge

There are 77 typhoons affecting Qingdao in 53 years since 19492001 (which means its

center enters the area north of 35N). It means that typhoons occur three times every

2 years on average in this area. Although there are 1.5 typhoons every year that affect

Qingdao, disaster-induced typhoon surge does not occur every year. The storm surge

Table 1 Estimations of parameters for different distributions

Environmental element Distributions A1 A2 A3 A4

Extreme water level UMED 7.87 9 10-3 9.99 5.30 2.26

Gumbel 4.86 0.27

Weibull 3.36 1.79 5.61

Lognormal 1.61 6.80 9 10-2

Significant wave height UMED 11.86 9.99 0.50 0

Gumbel 3.02 0.99

Weibull 0.29 3.71 2.81

Lognormal 1.22 0.34

For UMED A1 = b, A2 = c, A3 = n, A4 = a0; for Gumbel distribution A1 = l, A2 = r; for Weibull dis-tribution A1 = l, A2 = r, A3 = c; for lognormal distribution A1 = l, A2 = r

0 1 20

5

10

15

20

25

30

35

40

45

50

Typhoon Occurence Times Each Year

TyphoonFrequency

observed data

fitting curve

Fig. 2 Poisson distribution of typhoon frequency

Nat Hazards (2013) 68:791807 799

1 3


10/17

disaster intensity in Qingdao depends on various influencing factors such as the intensity,

duration and route of the passing typhoon. In particular, when strong storm surge happens,

the joint tidal level can be high enough and the concomitant huge wave height toward

shoreline can be large enough.Select 13 disaster-induced typhoon processes in Qingdao since 19492001, and con-

sider the extreme water level (L) and corresponding significant wave height (H) in the same

typhoon processes, the data are shown in Fig. 1a, b. The four PBCEVDs proposed in Sect.

2.3can be applied to judge the intensity grade of storm surge by its joint return period of

(L, H).

0.1 0.2 0.5 1 2 5 10 20 30 40 50 60 70 80 90 95 98 994

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8a

b

P /%

ExtremeWaterLevel(m)

Observed data

OMED fitting

Gumbel fitting

Weibull fitting

Lognormal fitting

0.10.2 0.5 1 2 5 10 20 30 40 50 60 70 80 90 95 98 990.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

10.5

11.5

P /%


Observed data

OMED fitting

Gumbel fitting

Weibull fitting

Lognormal fitting

Fig. 3 a Fittings of extreme water level. b Fittings of significant wave height

800 Nat Hazards (2013) 68:791807

1 3


11/17


12/17

Applyv2 test in Sect.2.4.3to judge the BMED models (NBMED, GHBMED, CBMED

and FBMED) fit the observations of (L,H) well or not. Because the sample size of (L,H) is

13, choose k= 3 or k= 4 in order to make the number of grid units 9 or 16 close to the

sample size. Then, two forms R1 and R2 which contains 3 9 3 and 4 9 4 grid units,respectively, could be obtained. Let significance level a = 0.05. The v2 statistics M of

different copulas are calculated and compared with v20:954 9:49 and v20:959 16:92.The calculation results are presented in the Table4.

All thev2 statistics in Table4are smaller thanv2 test values, so all the BMED models

are good enough to fit the data pairs of (L, H). Table4also shows that GHBMED fits the

observations best, and CBMED is the worst.

Above all, the three kinds of tests for occurrence frequency of disaster-induced

typhoons, two UMED margins of (L,H) and the four BMED models verify the adaptability

of the four PBMEDs of the data.

The joint probability density contours of (L, H) for PNBMED, PGHBMED, PCBMEDand PFBMED are as Fig. 4ad; the joint co-occurrence probability contours of (L, H) for

PNBMED, PGHBMED, PCBMED and PFBMED are as Fig. 5ad. Based on the charac-

teristics of these four bivariate copulas (normal, GumbelHougaard, Clayton and Frank

copula), PNBMED and PFBMED are less sensitive to tail dependence, PGHBMED has

higher top tail, and PCBMED has lower top tail. So in Fig. 5b, the tidal level and wave

0.001

0.005

0.01

0.05

Extreme Water Level (m)

SignificantWaveH

eight(m)

3 3.5 4 4.5 5 5.5 6 6.50

1

2

3

4

5

6

7

8

9

10

0.001

0.005

0.01

0.05


SignificantWaveH

eight(m)

3 3.5 4 4.5 5 5.5 6 6.50

1

2

3

4

5

6

7

8

9

10

0.001

0.005

0.01

0.05



3 3.5 4 4.5 5 5.5 6 6.5

0

1

2

3

4

5

6

7

8

9

10

0.001

0.005

0.01

0.05



3 3.5 4 4.5 5 5.5 6 6.5

0

1

2

3

4

5

6

7

8

9

10

a b

cd

Fig. 4 aJoint probability density contoursof (L,H) for PNBMED.b Joint probability density contoursof

(L, H) for PGLBMED. c Joint probability density contours of (L, H) for PCBMED. d Joint probability

density contours of (L, H) for PFBMED

802 Nat Hazards (2013) 68:791807

1 3


13/17

height corresponding to 5- or 100-year joint return period are both larger; in Fig. 5c, these

two elements are both smaller; in Fig.5a, d, they are all modest.

According to the joint return period of (L,H), a novel grade of disaster-induced typhoon

surge intensity is presented in this study (see Table 5).

It should be pointed out here that, since there is only complete data about tidal level,

wave height and disaster degree in Qingdao area, the typhoon surge intensity classificationonly applies to Qingdao area at present. If the disaster-induced factors are both tidal level

and wave height in other areas too, this bivariate model can be also applied, but the

observations of tidal level, wave height and disaster loss should be considered in the

typhoon surge intensity classification.

Table 5 Grade of typhoon surge intensity

Grade 1 2 3 4

Disaster-inducing intensity grade Mild (MI) Moderate (M) Severe (SE) Destructive (D)

Joint return period of typhoon surge 010 1025 2550 50200

0.01

0.02

0.05

0.10.2



4 4.5 5 5.5 61

2

3

4

5

6

7

0.01

0.02

0.05

0.10.2



4 4.5 5 5.5 6

1

2

3

4

5

6

7

0.01

0.02

0.05

0.10.2



4 4.5 5 5.5 6

1

2

3

4

5

6

7

0.01

0.02

0.05

0.10.2


SignificantWaveH

eight(m)

4 4.5 5 5.5 61

2

3

4

5

6

7a b

dc

Fig. 5 a Joint probability contours of (L, H) for PNBMED. b Joint probability contours of (L, H) for

PGLBMED. c Joint probability contours of (L, H) for PCBMED.d Joint probability contours of (L,H) for

PFBMED

Nat Hazards (2013) 68:791807 803

1 3


14/17

The joint return periods of typhoon surge disasters in Qingdao since 19492001 cal-

culated by PNBMED, PGHBMED, PCMED and PFMED are listed in Table6 separately.

The corresponding grades of disaster-induced typhoon surge intensity are presented

simultaneously.

The practical typhoon surge disaster grades (based on disaster range and economicloss) in references (Guo et al. 1998; Li 1998) are also listed for comparison in Table6.

They pointed out that the warning water level (5.25 m in Qingdao) method cannot

reflect the actual disaster, and suggested that we need to consider tidal level and wave

height simultaneously. For example, based on Fig.1 and Table6, we know that for

typhoon No. 4906, its tidal level is 4.75 m and not larger than warning water level

(5.25 m), but it introduced severe disaster. The reason is that its corresponding wave

height is very high (5.0 m). The intensity levels of disaster-induced typhoons calculated

by all four PBMEDs are in excellent agreement with the given disaster grades except

typhoon No. 9415. The reason is that typhoon No. 9414 happened just before typhoon

No. 9415 (the return period is 17 or 18 years), and the defense facilities were just

reinforced and the peoples consciousness of typhoon prevention extremely increased

before this typhoon happened, at the same time timely prediction reduced the disaster

loss.

The joint periods of disaster-induced typhoons calculated by PCBMED are the largest,

then PNBMED, PFBMED and PGHMED. Table6 shows that the intensity levels of

typhoon Nos. 9711, 9216 and 8509 are the most destructive of all; the typhoon disaster

intensity is destructive when higher extreme water level and huge wave height appear

simultaneously (such as Nos. 9711, 9216 and 8509), and the disaster intensity is mild when

extreme water level is higher and concomitant wave height is relatively small (such as No.4908), in the process of typhoon surge. Furthermore, this verifies the hypothesis that

serious typhoon surge disasters in Qingdao area are caused by the higher tidal level and

concomitant huge wave heights toward shoreline.

Table 6 Disaster-induced intensity of typhoon surge in Qingdao area

Typhoon number 4906 4908 5116 5622 8114 8406 8509 9005 9015 9216 9414 9415 9711

Disaster grade SE M M M SE M D MI MI D MI MI D

PNBMED model

Return period (a) 29 17 11 10 27 11 67 7 8 106 8 18 132

Intensity grade SE M M M SE M D MI MI D MI M D

PGLBMED model

Return period (a) 28 16 11 10 25 10 53 7 8 73 8 17 86


PCBMED model

Return period (a) 28 16 11 10 28 10 74 7 8 128 8 17 166


PFBMED model

Return period (a) 28 16 11 10 25 10 61 7 8 97 8 17 125


804 Nat Hazards (2013) 68:791807

1 3


15/17

4 Conclusion

In this paper, four kinds of novel bivariate maximum entropy distributions based on

bivariate normal copula, GumbelHougaard copula, Clayton copula and Frank copula are

introduced. These joint distributions all consist of two margins with UMED. Consideringthe occurrence frequency of typhoons, Poisson bivariate compound maximum entropy

distributions based on these novel bivariate maximum entropy distributions and bivariate

compound extreme value theory are constructed. Based on the data of disaster-induced

typhoon processes since 19492001 in the Qingdao area, the joint distribution of extreme

water level and corresponding significant wave height in the same typhoon processes by

the above Poisson bivariate compound maximum entropy distributions are established. The

results show that all these four distributions are good enough to fit the original data. The

novel grade of disaster-induced typhoon surges intensity proposed based on the joint return

period of extreme water level and corresponding significant wave height is proposed, and

the disaster-induced typhoons in Qingdao verify this grade criterion.

Acknowledgments The study was partially supported by the National Natural Science Foundation of

China (51279186), the National Program on Key Basic Research Project (2011CB013704) and the Program

for New Century Excellent Talents in University (NCET-07-0778).

Appendix

The distribution function of Gumbel distribution is

Fx exp expx

l

r h i 20

in which l and r are the location parameter and scale parameter, respectively.

The distribution function of Weibull distribution is

F x 1 exp xl c

r

h i; x l

0; x\l

( 21

in whichl[ 0 is the location parameter, r[ 0 is the shape parameter, c[ 0 is the scale

parameter.

The distribution function of lognormal distribution is

Fx Zx

1

1

trffiffiffiffiffiffi

2pp exp 1

2r2 ln t l 2

dt; x a 22

in which l and r are the location parameter and scale parameter, respectively.

References

Antao E (2012) Probabilistic models of sea wave steepness. PhD thesis on naval architecture and marine

engineering, Instituto Superior Tecnico, Technical University of Lisbon

Athanassoulis GA, Skarsoulis EK, Belibassakis KA (1994) Bi-variate distributions with given marginals

with an application to wave climate description. Appl Ocean Res 16(1):117

Bitner-Gregersen E, Guedes Soares C (1997) Overview of probabilistic models of the wave environment

for reliability assessment of offshore structures. In: Guedes Soares C (ed) Advances in safety and

reliability. Pergamon, Oxford, pp 14451456

Nat Hazards (2013) 68:791807 805

1 3


16/17

De Waal DJ, van Gelder PHAJM (2005) Modelling of extreme wave heights and periods through copulas.

Extremes 8:345356

Dolan R, Davis RE (1994) Coastal storm hazards. J Coast Res Spec Issue 12:103114

Dong S, Liu YK, Wei Y (2005) Combined return value estimation of wind speed and wave height with

Poisson bi-variate lognormal distribution. In: Proceedings of the 15th international offshore and polar

engineering conference, vol 3. ISOPE, Seoul, pp 435439Dong S, Xu PJ, Liu W (2009) Long-term prediction of return extreme storm surge elevation in Jiaozhou

Bay. J Ocean Univ China 39(5):11191124

Dong S, Liu W, Zhang L, Guedes Soares C (2012a) Return value estimation of significant wave heights with

maximum entropy distribution. J Offshore Mech Arct Eng. doi:10.1115/1.4023248

Dong S, Tao SS, Lei SH, Guedes Soares C (2012b) Parameter estimation of the maximum entropy distri-

bution of significant wave height. J Coast Res. doi:10.2112/JCOASTRES-D-11-00185.1

Dong S, Wang N, Liu W, Guedes Soares C (2013) Bivariate maximum entropy distribution of significant

wave height and peak period. Ocean Eng 59:8699

Favre AC, Adlouni SE, Perreault L, Thiemonge N, Bobee B (2004) Multivariate hydrological frequency

analysis using Copulas. Water Resour Res 40(1):112

Feller W (1957) An introduction to probability theory and its applications, 2nd edn. Wiley, New York

Ferreira JA, Guedes Soares C (2002) Modeling bivariate distributions of significant wave height and meanwave period. Appl Ocean Res 24(1):3145

Guedes Soares C, Scotto MG (2001) Modelling uncertainty in long-term predictions of significant wave

height. Ocean Eng 28(3):329342

Guedes Soares C, Scotto MG (2011) Long term and extreme value models of wave data. In: Guedes Soares

C, Garbatov Y, Fonseca N, Teixeira AP (eds) Marine technology and engineering. Taylor & Francis,

UK, pp 97108

Guo KC, Guo MK, Jiang CB et al (1998) Preliminary analysis of disaster caused by typhoon no. 9711 in

Shandong Peninsula of China. Marine Forecast 15(2):4751 (in Chinese)

Halsey SD (1986) Proposed classification scale for major Northeast storms: East Coast USA, based on

extent of damage. Geol Soc Am Abstr Program (Northeast Sect) 18:21

Hanne TW, Dag M, Havard R (2004) Statistical properties of successive wave heights and successive wave

periods. Appl Ocean Res 26(34):114136Haver S (1985) Wave climate off northern Norway. Appl Ocean Res 7(2):8592

Hu L (2002) Essays in econometrics with applications in macroeconomic and financial modeling. Yale

University, New Haven

Isaacson M, MacKenzie NG (1981) Long-term distributions of ocean waves: a review. J Waterw Port Coast

Ocean Div 107(2):93109

Jaynes ET (1968) Prior probability. IEEE Trans Syst Sci Cybern 4:227241

Jonathan P, Flynn J, Ewans KC (2010) Joint modelling of wave spectral parameters for extreme sea states.

Ocean Eng 37:10701080

Leira BJ (2010) A comparison of some multivariate Weibull distributions. In: Proceedings of the ASME

2010 29th international conference on ocean, offshore and arctic engineering, OMAE2010-20678,

June 611, 2010, Shanghai, China

Li PS (1998) Study on typhoon storm surge disaster forecasting in Qingdao area. Marine Forecast15(3):7278 (in Chinese)

Liu DF, Wen SQ, Wang LP (2002) PoissonGumbel Mixed compound distribution and its application. Chin

Sci Bull 47(22):19011906

Liu W, Dong S, Chu XJ (2010) Study on joint return period of wind speed and wave height considering

lifetime of platform structure. In: Proceedings of the 29th international conference on offshore

mechanics and polar engineering, Shanghai, China, OMAE20247, vol 2, pp 245250

Ma FS, Liu DF (1979) Compound extreme distribution theory and its applications. Acta Math Appl Sin

2(4):366375

Mendoza ET, Jimenez JA (2005) A storm classification based on the beach erosion potential in the Cata-

lonian Coast. Coast Dyn 2005, 111. doi:10.1061/40855(214)98

Morton ID, Bowers J (1996) Extreme value analysis in a multivariate offshore environment. Appl Ocean

Res 18:303317Muhaisen OSH, Elramlawee NJE, Garca PA (2010) Copula-EVT-based simulation for optimal rubble-

mound breakwater design. Civil Eng Environ Syst 27(4):315328

Muir LR, El-Shaarawi AH (1986) On the calculation of extreme wave heights: a review. Ocean Eng

13(1):93118

Nelsen RB (2006) An introduction to copulas. Springer, New York

806 Nat Hazards (2013) 68:791807

1 3
http://dx.doi.org/10.1115/1.4023248http://dx.doi.org/10.2112/JCOASTRES-D-11-00185.1http://dx.doi.org/10.1061/40855(214)98http://dx.doi.org/10.1061/40855(214)98http://dx.doi.org/10.2112/JCOASTRES-D-11-00185.1http://dx.doi.org/10.1115/1.4023248


17/17

Poisson bivariate

Documents

Transcript of Poisson bivariate