Unit III Art Rev (MOA)

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, Gumbel–Hougaard 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 1949–2001 in Qingdao area are

selected, 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 level � Significant 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 Gumbel–Hougaard copula

CBMED Bivariate maximum entropy distributions with Clayton copula

S. Tao � S. Dong (&) � N. Wang College 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 Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

123

Nat Hazards (2013) 68:791–807 DOI 10.1007/s11069-013-0654-6

FBMED Bivariate maximum entropy distributions with Frank copula

EBMED Equivalent bivariate maximum entropy distribution

PBCEVD Poisson bivariate compound extreme value distribution

PGMCD Poisson–Gumbel mixed compound distribution

PBCMED Poisson bivariate compound maximum entropy distribution

MOM Method of moments

ECFM Empirical curve-fitting method

MLM Maximum likelihood method

PNBMED Poisson normal bivariate maximum entropy distribution

PGHBMED Poisson Gumbel–Hougaard 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 Pearson’s 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 has

more 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) 68:791–807

123

marginal distributions of different ocean environmental parameters with some correlations

among them, and eventually get a joint distribution (Nelsen 2006). 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) obtained

the 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

Burr–Pareto–Logistic 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 Antão (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, Gumbel–Hougaard 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 considering

the 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 Poisson–Gumbel mixed compound distribution (PGMCD). Four

kinds of Poisson bivariate compound maximum entropy distributions (PBCMEDs) based

on NBMED, bivariate maximum entropy distributions with Gumbel–Hougaard 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 Sarrir–Simpson 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 extratropical

storms based on damage potential index. Mendoza and Jiménez (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) 68:791–807 793

123

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 distribution

with 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:

GðxÞ¼ Zx

a0

gðtÞdt ¼ Zx

a0

aðt � a0Þc exp �bðt � a0Þn h i

dt ð1Þ

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 of b, c, n and a0, which can be given by the following expression.

a ¼ nb cþ1 n C�1

c þ 1 n

� � ð2Þ

where C(�) represents the gamma function defined as:

CðsÞ¼ Z1

0

xs�1e�xdx ð3Þ

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 Sklar’s 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 Gumbel–

Hougaard 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) 68:791–807

123

Cðu; v; hÞ¼ ZU�1ðuÞ

�1

ZU�1ðvÞ

�1

1

2p ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � h2 p exp �

s2 � 2hst þ t2

2ð1 � h2Þ

� � dsdt ð4Þ

cðu; v; hÞ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 � h2 p exp �

ð2 � h2Þ½U�1ðuÞ�2 � 2hU�1ðuÞU�1ðvÞþð2 � h2Þ½U�1ðvÞ�2

2ð1 � h2Þ

( ) ;

ð5Þ

respectively, 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 of U-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:

Cðu; v; hÞ¼ exp � ð� ln uÞh þð� ln vÞh h i1

h

� � ð6Þ

cðu; v; hÞ¼ ðln u � ln vÞh�1 ð� ln uÞh þð� ln vÞh

h i1 h

þh � 1 � �

uv½ð� ln uÞh þð� ln vÞh�2� 1 h

� Cðu; v; hÞ; ð7Þ

respectively, where h C 1 is the correlated parameter and the relationship between h and the Kendall rank s is s = 1 - 1/h. U, V are independent when h = 1, and U, V tend to be completely correlated if h ? ??.

The probability distribution function and density function of the bivariate Clayton

copula are as follows:

Cðu; v; hÞ¼ u�h þ v�h � 1 � ��1h ð8Þ

cðu; v; hÞ¼ ð1 þ hÞ � ðuvÞ�h�1 � u�h þ v�h � 1 � ��2�1h

; ð9Þ

respectively, where 0 \ h \ ?? is the correlated parameter and the relationship between h and the Kendall rank s is s = h/(h ? 2). U, V tend to be independent when h ? 0, and U, V tend to be completely correlated when h ? ??.

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

are as follows:

Cðu; v; hÞ¼� 1

h ln 1 þðe

�hu � 1Þðe�hv � 1Þ ðe�h � 1Þ

ð10Þ

cðu; v; hÞ¼ �hðe �h � 1Þe�hðuþvÞ

ðe�h � 1Þþðe�hu � 1Þðe�hv � 1Þ½ �2 ; ð11Þ

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

the Kendall rank is s ¼ 1 þ 4h 1h Rh

0 t

et�1 dt � 1 h i

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

tion, while h \ 0 denotes U, V have negative correlation, and when h ? 0 the random variables U, V tend 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) 68:791–807 795

123

GXðxÞ¼ Zx

a1

a1ðs � a1Þc1 exp �b1ðs � a1Þ n1

h i ds ð12Þ

GYðyÞ¼ Zy

a2

a2ðt � a2Þc2 exp �b2ðt � a2Þ n2

h i dt ð13Þ

in which a1 ¼ n1b c1þ1 n1

1 C �1ðc1þ1n1 Þ and a2 ¼ n2b

c2þ1 n2

2 C �1ðc2þ1n2 Þ:

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

probability distribution (BMED) functions and density functions of X and Y can be written

as

Gðx; yÞ¼ CðGXðxÞ; GYðyÞÞ ð14Þ

gðx; yÞ¼ cðGXðxÞ; GYðyÞÞ � gXðxÞ � gYðyÞ ð15Þ

where c(u, v), gX(x) and gY(y) are the probability density functions of C(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 univariate

compound 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 and g, otherwise be f and c. Suppose the joint distribution functions of (n, g) and (f, c) are G(x, y) and Q(x, y), respectively, the joint probability density function of (n, g) is g(x, y) and the distribution function of n is GX(x). Let (ni, gi) be the ith observation of (n, g). n is a random variable which follows the Poisson distribution and is independent from (n, g). The distribution function of n is denoted by

Pk ¼ e�kkk

k! ; k ¼ 0; 1; 2; . . . ð16Þ

Define

ðX; YÞ¼ ðf; cÞ; n ¼ 0 ðnj; gjÞ; nj ¼ max

1� i�n ni; n�1

( ð17Þ

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

F0ðx; yÞ¼ e�k þ X1 k¼1

Zy

�1

Zx

�1

e�kkk

k! k½GXðuÞ�k�1gðu; vÞdudv

¼ e�k 1 þ k Zy

�1

Zx

�1

ekGXðuÞgðu; vÞdudv

0 @

1 A

ð18Þ

796 Nat Hazards (2013) 68:791–807

123

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 Gumbel–Hougaard

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

mum entropy distribution (PCBMED) and Poisson Frank bivariate maximum entropy

distribution (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) Pearson’s v2 test for Poisson distribution, (2) K–S test for UMED margins and (3) v2 test for copulas (Hu 2002).

2.4.1 Pearson’s 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 sample x1, x2, …, xn into k groups. Denote the number of individuals in the group (xi–1, xi) as vi. Evaluate the expected frequency nPi.

Suppose that the theoretical distribution function is F0ðxÞ¼ Pk

x¼0 kx e�k

x! , then

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

i¼1 Pi ¼ 1. Evaluate the statistic: v̂2 ¼

Pk i¼1

v2 i

nPi

� � � n, then under the significance level a, if v̂2 [ v2k�2, reject H0,

otherwise cannot reject H0.

2.4.2 K–S test for UMED margins

Assume that F(x) is the actual distribution function of X, 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 Dn ¼ sup�1\k\1jFnðxÞ� F0ðxÞj; where Fn(x) is the empirical probability distribution function.

Let dk (1)

= |Fn(xk) - F0(xk)| and dk (2)

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

of the statistic Dn is D̂ ¼ max1�k�n jd ð1Þ k � d

2 kj: If the significance level is a, then for

different sample size n, the K–S critical value is Dn(a). If D̂n\DnðaÞ; we admit H0, otherwise reject H0.

2.4.3 Pearson’s v2 test for copulas

Hu (2002) introduces an M statistic which follows the v2 distribution, and it can estimate the degree of fitting of the bivariate copulas. By using the M statistic, 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 k 9 k grid units. The unit in

the ith row and the jth column is denoted by R(i, j), i, j = 1, 2, …, k. For each {ut, vt}, if (i - 1)/k B ut B i/k and (j - 1)/k B vt B j/k both set up, then denote {ut, vt} 2 R(i, j).

Nat Hazards (2013) 68:791–807 797

123

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,

M ¼ Xk i¼1

Xk j¼1

ðAij � BijÞ2

Bij �v2ððk � 1Þ2Þ ð19Þ

If the significance level is a, the rejection region is fM [ v21�aððk � 1Þ 2Þg, where

v21�aððk � 1Þ 2Þ 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

E xt

re m

e W

a te

r le

ve l (

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

S ig

n ifi

ca n t W

a ve

H e ig

h t (m

)

a

b

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

798 Nat Hazards (2013) 68:791–807

123

(k - 1) 2 . So if M [ v21�aððk � 1Þ

2Þ, the copula is not suitable for this data; if M\v21�aððk � 1Þ

2Þ, the copula could be used to construct the bivariate model. Choose the bivariate copula which makes M the 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 1949–2001 (which means its

center enters the area north of 35�N). 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 2 0

5

10

15

20

25

30

35

40

45

50

Typhoon Occurence Times Each Year

T yp

h o

o n

F re

q u

e n

cy

observed data fitting curve

Fig. 2 Poisson distribution of typhoon frequency

Nat Hazards (2013) 68:791–807 799

123

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 1949–2001, 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.

2.3 can 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 99 4

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 /%

E xt

re m

e W

a te

r L e ve

l ( 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 99 0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

10.5

11.5

P /%

S ig

n ifi

ca n t W

a ve

H e ig

h t (m

)

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) 68:791–807

123

The occurrence frequency number n of typhoon agrees with the Poisson distribution

with parameter k = 13/49 at the 0.05 significance level by Pearson’s v2 test, see Fig. 2. Fit the extreme water level (L) and corresponding significant wave height (H) in the

same disaster-induced typhoon processes with UMEDs separately, and the results are as

Table 1 and Fig. 3a, b. In these fittings, the parameter estimation method of ECFM is used.

The calculated return values of L and H are as in Table 2.

K–S tests for the two UMED margins are conducted, and average sum of squares of

deviations (Q) between fitting UMEDs and empirical distribution functions are calculated.

The results are as Table 3. For comparison, Gumbel, Weibull and lognormal distributions

are also used to fit the data of L and H, see Tables 1, 2, 3 and Fig. 3a, b, and their

distributions can be seen in ‘‘Appendix.’’

In Table 3, K–S tests show that both the UMED margins are the best fitting for the

observations of L and H; Q values are both the smallest. Figure 3a, b also display that

UMEDs fit the data best.

Table 2 Return values (m)

Marine environmental element Distributions Return periods (a)

5 10 20 25 50 100 200 500

Extreme water level UMED 5.34 5.50 5.62 5.65 5.75 5.84 5.91 6.00

Gumbel 5.26 5.46 5.65 5.71 5.90 6.08 6.27 6.51

Weibull 5.31 5.43 5.53 5.56 5.64 5.71 5.77 5.84

Lognormal 5.29 5.46 5.59 5.63 5.75 5.86 5.96 6.08

Significant wave height UMED 4.73 5.64 6.49 6.76 7.58 8.38 9.18 10.22

Gumbel 4.51 5.25 5.97 6.19 6.89 7.58 8.27 9.18

Weibull 4.68 5.28 5.77 5.91 6.31 6.67 7.00 7.39

Lognormal 4.52 5.26 5.96 6.18 6.86 7.53 8.21 9.11

Table 3 Results of K–S test and Q

Marine environmental element Distributions Dn (0.05) D̂n Q

Extreme water level UMED 0.36 0.14 1.28 9 10 -4

Gumbel 0.36 0.20 5.33 9 10 -4

Weibull 0.36 0.14 2.02 9 10 -4

Lognormal 0.36 0.15 2.77 9 10 -4

Significant wave height UMED 0.36 0.14 6.11 9 10 -3

Gumbel 0.36 0.18 9.80 9 10 -3

Weibull 0.36 0.20 7.37 9 10 -3

Lognormal 0.36 0.18 9.35 9 10 -3

Table 4 v2 statistics M for BMED models

BMED models NBMED GHBMED CBMED FBMED

3 9 3 7.3477 7.2239 9.2238 8.2046

4 9 4 13.4054 12.4279 14.7742 12.8542

Nat Hazards (2013) 68:791–807 801

123

Apply v2 test in Sect. 2.4.3 to 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 v20:95ð4Þ¼ 9:49 and v20:95ð9Þ¼ 16:92. The calculation results are presented in the Table 4.

All the v2 statistics in Table 4 are smaller than v2 test values, so all the BMED models are good enough to fit the data pairs of (L, H). Table 4 also 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, PCBMED

and PFBMED are as Fig. 4a–d; the joint co-occurrence probability contours of (L, H) for

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

teristics of these four bivariate copulas (normal, Gumbel–Hougaard, 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)

S ig

n ifi

ca n t

W a ve

H e ig

h t

(m )

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

Extreme Water Level (m)

S ig

n ifi

ca n t

W a ve

H e ig

h t

(m )

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

Extreme Water Level (m)

S ig

n ifi

ca n

t W

a ve

H e

ig h

t (m

)

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

Extreme Water Level (m)

S ig

n ifi

ca n

t W

a ve

H e

ig h

t (m

)

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

c d

Fig. 4 a Joint probability density contours of (L, H) for PNBMED. b Joint probability density contours of (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) 68:791–807

123

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 classification

only 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 0–10 10–25 25–50 50–200

0.01

0.02

0.05

0.1 0.2

Extreme Water Level (m)

S ig

n ifi

ca n t W

a ve

H e ig

h t (m

)

4 4.5 5 5.5 6 1

2

3

4

5

6

7

0.01

0.02

0.05

0.1 0.2

Extreme Water Level (m)

S ig

n ifi

ca n t W

a ve

H e ig

h t (m

)

4 4.5 5 5.5 6 1

2

3

4

5

6

7 0.01

0.02

0.05

0.1 0.2

Extreme Water Level (m)

S ig

n ifi

ca n t W

a ve

H e ig

h t (m

)

4 4.5 5 5.5 6 1

2

3

4

5

6

7

0.01

0.02

0.05

0.1 0.2

Extreme Water Level (m)

S ig

n ifi

ca n t W

a ve

H e ig

h t (m

)

4 4.5 5 5.5 6 1

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) 68:791–807 803

123

The joint return periods of typhoon surge disasters in Qingdao since 1949–2001 cal-

culated by PNBMED, PGHBMED, PCMED and PFMED are listed in Table 6 separately.

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

simultaneously.

The practical typhoon surge disaster grades (based on disaster range and economic

loss) in references (Guo et al. 1998; Li 1998) are also listed for comparison in Table 6.

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 Table 6, 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 people’s 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. Table 6 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

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

PCBMED model

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

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

PFBMED model

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

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

804 Nat Hazards (2013) 68:791–807

123

4 Conclusion

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

bivariate normal copula, Gumbel–Hougaard copula, Clayton copula and Frank copula are

introduced. These joint distributions all consist of two margins with UMED. Considering

the 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 1949–2001 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

FðxÞ¼ exp �exp � x � l

r

� �h i ð20Þ

in which l and r are the location parameter and scale parameter, respectively. The distribution function of Weibull distribution is

F xð Þ¼ 1 � exp � x�lð Þc

r

h i ; x�l

0; x\l

( ð21Þ

in which l [ 0 is the location parameter, r [ 0 is the shape parameter, c [ 0 is the scale parameter.

The distribution function of lognormal distribution is

FðxÞ¼ Zx

�1

1

tr ffiffiffiffiffiffi 2p p exp �

1

2r2 ln t � lð Þ2

dt; x�a ð22Þ

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

References

Antão 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):1–17

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 1445–1456

Nat Hazards (2013) 68:791–807 805

123

De Waal DJ, van Gelder PHAJM (2005) Modelling of extreme wave heights and periods through copulas. Extremes 8:345–356

Dolan R, Davis RE (1994) Coastal storm hazards. J Coast Res Spec Issue 12:103–114 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 435–439

Dong S, Xu PJ, Liu W (2009) Long-term prediction of return extreme storm surge elevation in Jiaozhou Bay. J Ocean Univ China 39(5):1119–1124

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 59:86–99

Favre AC, Adlouni SE, Perreault L, Thiémonge N, Bobée B (2004) Multivariate hydrological frequency analysis using Copulas. Water Resour Res 40(1):1–12

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 mean

wave period. Appl Ocean Res 24(1):31–45 Guedes Soares C, Scotto MG (2001) Modelling uncertainty in long-term predictions of significant wave

height. Ocean Eng 28(3):329–342 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 97–108

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):47–51 (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(3–4):114–136

Haver S (1985) Wave climate off northern Norway. Appl Ocean Res 7(2):85–92 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):93–109 Jaynes ET (1968) Prior probability. IEEE Trans Syst Sci Cybern 4:227–241 Jonathan P, Flynn J, Ewans KC (2010) Joint modelling of wave spectral parameters for extreme sea states.

Ocean Eng 37:1070–1080 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 6–11, 2010, Shanghai, China

Li PS (1998) Study on typhoon storm surge disaster forecasting in Qingdao area. Marine Forecast 15(3):72–78 (in Chinese)

Liu DF, Wen SQ, Wang LP (2002) Poisson–Gumbel Mixed compound distribution and its application. Chin Sci Bull 47(22):1901–1906

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 245–250

Ma FS, Liu DF (1979) Compound extreme distribution theory and its applications. Acta Math Appl Sin 2(4):366–375

Mendoza ET, Jiménez JA (2005) A storm classification based on the beach erosion potential in the Cata- lonian Coast. Coast Dyn 2005, 1–11. doi:10.1061/40855(214)98

Morton ID, Bowers J (1996) Extreme value analysis in a multivariate offshore environment. Appl Ocean Res 18:303–317

Muhaisen OSH, Elramlawee NJE, Garcı́a PA (2010) Copula-EVT-based simulation for optimal rubble- mound breakwater design. Civil Eng Environ Syst 27(4):315–328

Muir LR, El-Shaarawi AH (1986) On the calculation of extreme wave heights: a review. Ocean Eng 13(1):93–118

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

806 Nat Hazards (2013) 68:791–807

123

Nerzic R, Prevosto M (2000) Modeling of wind and wave joint occurrence probability and persistence duration from satellite observation data. In: Proceedings of the tenth international offshore and polar engineering conference, vol 3, May 28–June 2, 2000, Seattle, USA, pp 154–158

Petrov V, Guedes Soares C, Gotovac H (2013) Prediction of extreme significant wave heights using maximum entropy. Coast Eng 74:1–10

Prince-Wright R (1995) Maximum likelihood models of joint environmental data for TLP design. In: Proceedings of the 14th international conference on offshore mechanics and arctic engineering (OMAE 1995), vol 2. ASME, NY

Repko A, Van Gelder PHAJM, Voortman HG, Vrijling JK (2004) Bivariate description of offshore wave conditions with physics-based extreme value statistics. Appl Ocean Res 26:162–170

Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publ Inst State Univ Paris 8:229–231

Van Vledder G, Goda Y, Hawkes P, Mansard E, Martin MJ, Mathiesen M, Peltier E, Thompson E (1993) Case studies of extreme wave analysis: a comparative analysis. In: Proceedings of the second inter- national symposium on ocean wave measurement and analysis, pp 978–992

Wahl T, Mudersbach C, Jensen J (2012) Assessing the hydrodynamic boundary conditions for risk analyses in coastal areas: a multivariate statistical approach based on copula functions. Nat Hazards Earth Syst Sci 12:495–510

Zachary S, Feld G, Ward G, Wolfram J (1998) Multivariate extrapolation in the offshore environment. Appl Ocean Res 20(5):273–295

Zhang LZ, Xu DL (2005) A new maximum entropy probability function for the surface elevation of nonlinear sea waves. China Ocean Eng 19(4):637–646

Nat Hazards (2013) 68:791–807 807

123

Copyright of Natural Hazards is the property of Springer Science & Business Media B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.