poisson bivariate
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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
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DOI 10.1007/s11069-013-0654-6
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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
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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.
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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:
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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:
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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
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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).
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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
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(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
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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 /%
SignificantWaveHeight(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
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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
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
Extreme Water Level (m)
SignificantWaveHeight(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)
SignificantWaveHeight(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
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
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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
Extreme Water Level (m)
SignificantWaveHeight(m)
4 4.5 5 5.5 61
2
3
4
5
6
7
0.01
0.02
0.05
0.10.2
Extreme Water Level (m)
SignificantWaveHeight(m)
4 4.5 5 5.5 6
1
2
3
4
5
6
7
0.01
0.02
0.05
0.10.2
Extreme Water Level (m)
SignificantWaveHeight(m)
4 4.5 5 5.5 6
1
2
3
4
5
6
7
0.01
0.02
0.05
0.10.2
Extreme Water Level (m)
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
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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
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) 68:791807
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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.
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