COPULA APPROACH FOR FLOOD PROBABILITY ANALYSIS
OF THE HUANGPU RIVER DURING BARRIER CLOSURE
J.Y. NAI
Delft University of Technology, Faculty of Civil Engineering and Geosciences,
Stevinweg 1, 2600 GA Delft, the Netherlands
P.H.A.J.M. VAN GELDER
Delft University of Technology, Faculty of Civil Engineering and Geosciences,
Stevinweg 1, 2600 GA, Delft, the Netherlands
P.J.M. KERSSENS
WL | Delft Hydraulics, Rotterdamseweg 185, 2629 HD Delft, the Netherlands
Z.B. WANG
Delft University of Technology, Faculty of Civil Engineering and Geosciences,
Stevinweg 1, 2600 GA Delft, the Netherlands
E. VAN BEEK
Delft University of Technology, Faculty of Civil Engineering and Geosciences,
Stevinweg 1, 2600 GA Delft, the Netherlands
This paper introduces the use of copulas for flood probability analysis applied to the Huangpu
River during barrier closure. The Huangpu River meanders through the downtown area of Shanghai
City and connects the westward-located Tai Lake with the Yangtze River estuary. Storm surges of a
typhoon passing the offshore region of Shanghai in combination with high tide is the main cause for
flooding of the Huangpu River with inundation of the downtown area as a result. By the year 2010,
Shanghai should be protected for floods with a frequency of 1:1000 years. As re-heightening of the
present floodwalls alongside the river to meet this demand is not a sustainable solution anymore, the
Shanghai Municipal Government is investigating the feasibility to protect Shanghai with a storm
surge barrier in the mouth of the river. However, typhoons not only bring storm surges but also
torrential rainfall to the area. These intense rains temporarily increase the runoff into the Huangpu
River substantially, and since the river is the main drainage route in the area, flooding of the river
during barrier closure may occur after all. The objective of this study is to analyze the flood
probability of the Huangpu River during barrier closure due to upstream discharges into the river.
The paper shows that copulas have proven to be useful in flood probability analyses when limited
data is available in which only the marginal distribution functions are known.
1
2
1.
Introduction
The Huangpu River is the main shipping and drainage route to port city Shanghai in
the PRC of China. The city is located in the Tai Lake basin which is surrounded by the
East China Sea to the East, the Yangtze River in the North and the Hangzhou Bay in the
south. The Huangpu River meanders through the downtown area of Shanghai City and
connects the westward-located Tai Lake with the Yangtze River estuary in the North
East.
Typhoons, a regional specific name for tropical cyclones, in the vicinity of Shanghai
are the main trigger for flooding of the Huangpu River. Typhoons bring along storm
surge, torrential rainfall and strong winds. These hazards move along with the typhoon
and affect the areas the typhoon passes. When a typhoon passes Shanghai, the storm
surges caused will be driven into the Yangtze River estuary causing the storm tide levels
to increase additionally because of the shallow waters and confined dimensions within
the estuary. When this conjuncts with astronomical high tide, the storm tide traveling into
the Huangpu River can easily cause the water levels in the river rise to unparalleled levels
with inundation of urban Shanghai as a result.
The historical highest water levels recorded in the Huangpu River are caused by the
11th typhoon in 1997 (also named Winnie) passing the region. At Huangpu Park
observation station in the city centre, the water level reached the historical height of
5.72m above Wusongkou datum (WD), being about 1.32m above the flood warning level
and 0.5m higher than the previous record in 1981. Moreover the water level was only
0.14m lower than the design 1:1000 year’s water level at this location.
To protect urban Shanghai against flooding of the Huangpu River, a system of
floodwalls and control gates has been erected along almost its entire course as well as
along the downtown sections of the river’s major tributaries. For decades however, these
flood protection works have been heightened and reinforced continuously to keep pace
with the rising flood frequency of the Huangpu River associated with rapid ground level
subsidence in the area and changes in the upstream water regime. Even though some
sections of the urban floodwall raises for more than three meters above street level
nowadays, the flood protection of Shanghai City is currently not reflecting the current
and expected social and economic importance of the area to China.
Since heightening and reinforcement of the existing floodwalls has proven to be not
a sustainable solution to floods, the Shanghai Municipal Government intends to raise the
city’s flood frequency to 1:1000 via a storm surge barrier in the mouth of the Huangpu
River.
This storm surge barrier will become the first line of defense against typhoon
induced storm surges traveling into the river. However, the existing urban floodwall will
not lose its function during barrier closure since it is then the only defense against the
upstream discharge into the river. And considering the Huangpu River’s role as the main
3
drainage route to the area, flooding might even occur during barrier closure due to the
increased runoff into the river as a result of torrential rainfall accompanying typhoons
passing the area. As part of the ongoing feasibility study of protecting the city with a
storm surge barrier in the mouth of the river, the objective of this study is to analyze the
flood probability of the Huangpu River during barrier closure due to upstream discharges
into the river.
2.
Limit state condition during barrier closure
During barrier closure, flooding of the Huangpu River occurs either when the barrier
fails to keep out storm tides or when the river is not able to store its discharge during the
duration of barrier closure. This is visualized in Figure 1.
Figure 1. Fault tree for flooding of the downtown area of Shanghai City during barrier closure; the
focus of this paper is on flooding as a result of insufficient storage capacity of the Huangpu River.
In this study we only focus on flooding as a result of insufficient storage capacity of
the Huangpu River during barrier closure. This situation is captured in a limit state
function expressed as:
Z  RS
(1)
The storage capacity and the discharged water volume during barrier closure are
represented with R and S respectively. When Z=0 flooding just not occurs, this is defined
as the limit state condition. In contrast, when Z<0, the Huangpu River floods.
The storage capacity is considered deterministic and quantified by the dimensions of
the river, which are governed by the length of the Huangpu River with regard to the
barrier locations in the mouth of the river, the height of the floodwalls and the initial
water level in the river immediately after barrier closure. The discharged water volume is
determined by the closure duration, the base discharge and the torrential rainfall runoff
triggered by the typhoon. The latter two are considered stochastic variables.
The flood probability of the river during barrier closure is defined as the probability
that the limit state condition is exceeded, which is expressed as
Pflooding  P( Z  0)  P( S  s )
(2)
4
The flood probability is retrieved by backward evaluation of the limit state function.
First, the critical discharges are determined per closure duration for each of the potential
barrier locations with a one-dimensional flow model of the Huangpu River in SOBEK
River. The critical discharge is defined as the uniform discharge required per closure
duration to cause a limit state condition.
Hence, the flood probability is the most likely combination of base discharge and
torrential rainfall runoff that causes this critical discharge. Figure 2 presents the critical
discharges at Huangpu Park in the downtown area of Shanghai for three potential storm
surge barrier locations in the mouth of the river. When the barrier closes at the most
downstream location at Wusongkou, the limit state condition at Huangpu Park is reached
after 36 hours of barrier closure for a uniform river discharge of 750m3/s. For three main
locations along the river the current warning water level is used to determine the
corresponding critical discharges. These locations are Wusongkou in the river mouth,
Huangpu Park in the city centre and Mishidu, the most upstream location about 80km
from Wusongkou. At Huangpu Park in the city centre, the warning level amounts 4.55m
WD.
Critical discharge with reference to Huangpu Park
2000
Wusongkou
Zhanghua Bang
Fishery Yard
critical discharge [m 3/s]
1750
1500
1250
1000
750
500
250
0
0
12
24
36
48
closure duration [hrs]
60
72
Figure 2. Critical discharges of the Huangpu River for three barrier locations with reference to the
warning water level at Huangpu Park in the down town area of Shanghai City.
3.
Upstream discharges into the Huangpu River
The next step in the analyses is to determine the upstream discharge into the
Huangpu River during a typhoon storm. The absence of historical discharge data during
typhoon storms requires us to decompose the discharge into its main contributors and
analyze them accordingly.
First, there is the base discharge, which because of the tidal dominance in the
Huangpu River strongly depends on the hydrological year and month of interest. In 1954,
the rain season brought more rain than expected and flood water from the Tai Lake
needed to be diverted into the river increasing the annual average discharge to 755m3/s.
5
In contrast, the annual average discharge can be as low as 153m3/s in a dry year as in
1979.
Secondly, there is the additional drainage into the river due to typhoon induced
torrential rainfall in the Tai Lake basin. Torrential rainfall is defined as rainfall with
intensity over 100mm/24hrs. According to Yuan (1999) torrential rains with intensities
ranging from 150 to over 200mm/24hrs occurs most frequently in Shanghai. These
intensities typically cause water logging hazards in the urban area. Theoretically,
torrential rainfall can occur throughout the year, depending on the actual meteorological
conditions. However, the main trigger for this type of rain is the tropical cyclones in the
vicinity of Shanghai from June to October.
When the typhoon season starts in the Shanghai region, the water levels in the
Huangpu River are swollen already because of the plum rain season earlier. These plum
rains are monsoon induced wide spread prolonged rains and can cause severe flooding of
the inland areas as they can fall for a period of three months continuously and swell the
water levels in the lakes and rivers of the Tai Lake basin and raise the groundwater table
in this deltaic area. When the typhoon season then follows, the typhoon induced heavy
downpour will temporarily increase the already higher discharges into the Huangpu
River. Consequently, flooding of the Huangpu River due to insufficient storage capacity
for the upstream discharge can be reality even though a storm surge barrier successfully
stops storm tides from the East China Sea traveling into the Huangpu River.
4.
Copula for torrential rainfall and storm surges
The key driving mechanism of a tropical cyclone is the conversion of latent and
sensible heat from the ocean into wind. In fact, torrential rainfall, like storm surge and
wind gusts, is an inevitable exhaust product of the mechanism of tropical cyclones; a
positive correlation between storm surge and torrential rainfall seems likely. As a
consequence the torrential rainfall runoff into the Huangpu River depends on the storm
surge level that in combination with the tide in the mouth of the river enforces a barrier
closure.
When we assume the future barrier closure water level equals the current warning
water level in the river mouth at 4.80m WD, then the annually exceedance frequency of
this water level is 4.12 times according to observations of the Shanghai Water Authority.
We can now derive the mean surge level required (1.18m) given the occurrence barrier
closure by using the typhoon induced storm surge levels probability distribution derived
from historical typhoon storms from 1921 to 1997 at Wusongkou (Hohai, 1999).
The torrential rainfall corresponding to this surge level is hence derived by linking
the known probability distribution functions of typhoon induced storm surge levels at
Wusongkou and torrential rainfall in the Tai Lake Basin for different event durations (Tai
6
Lake Basin Authority, 2000) into their joint probability distribution function via the
Gumbel copula.
Copulas are functions that link arbitrary univariate distributions into their joint
multivariate distribution. The study of copulas and its applications is rather new but a
rapid growing field in the literature of statistics. Copulas originate from the study on
probabilistic metric spaces. At present, copulas are particularly used in the field of
financial mathematics and actuaries.
The relation between copulas and multivariate distribution functions is stated in
Sklar’s theorem (1959). The version of this theorem for a bivariate situation, which is of
interest in this study, is as follows:
Let H be joint distribution function with margins F and G, then there exists a copula
C that, for all x, y in [0,1]2,
H ( x, y)  C ( F ( x), G( y))
(3)
If F and G are continuous, then C is unique; otherwise C is uniquely determined on,
RanF × RanG, where Ran is the range of the margins. Conversely, if C is a copula and F
and G are distribution functions then H, as in (3), is a joint distribution function with
margins F and G.
From Sklar’s theorem, we see that for continuous multivariate distribution functions,
the univariate margins and the multivariate dependence structure can be separated, and
the dependence structure is represented by the copula. Clearly, this provides a large
freedom in choosing the margins once the desired dependence structure has been
specified. Because of the strong physical relationship between storm surge and torrential
rainfall of a typhoon we use the Gumbel copula to model their joint distribution. The
Gumbel copula belongs to the special class of Archimedean copulas and is characterized
by its stronger dependence in the lower probabilities.
The main characteristic of Archimedean copulas is that they allow us to reduce the
study of multivariate copulas to a single univariate functions, the so-called generator of
the copula. Different choices of generators yield several important families. A complete
listing of Archimedean copulas and their properties can be found in Nelson (1999).
Because of the relative simplicity of the Archimedean copula together with its coverage
of the common probability distributions, this class of copulas is used in a wide range of
applications. The general expression of Archimedean copulas for variables u, v  [0, 1]2
is
C (u, v)   1 ( (u )   (v))
(4)
in which  is the generator of the copula. The generator  is a strictly decreasing
function, which maps the interval (0,1] onto [0,). The generator of the Gumbel copula
is given by
7
 (t )  ( ln t ) , for   [1, )
(5)
hence, the Gumbel copula is expressed as

CGumbel  (u, v)  exp [( ln u)  ( ln v) ]1 

(6)
Much of the usefulness of copulas derives from the fact that copulas are invariant
under strictly increasing transformations of the random variables. Since Spearman’s 
and Kendall’s  share this characteristic as well, they can be expressed in terms of
copulas. Note that these are not the ordinary product moment correlation coefficients.
These are so-called rank correlations. Given a set where the data values x and y are
organized in order of size. The rank correlation coefficients can then be computed for
the given numerical values, which are in the form of ranks.
Moreover, with Archimedean copulas, the expression for correlation can be
expressed directly in terms of the generator  Eq. (7) and (8), which simplifies
computations significantly.
 (t )
dt

'(t )
0
1
  1  4
(7)
For the Gumbel copula, the Kendall’s  is hence expressed as
  1   1 , for   [0,1]
(8)
Similarly, simulating just the generators of the copula performs simulations of the
Archimedean copula. Many algorithms exist for random variable generation with
copulas. A universal algorithm to generate random variables (u v) from bivariate
Archimedean copulas is proposed by Genest and Mackay (1986). By simply using the
inverted margins of the individual variables we obtain the values for storm surge levels
and torrential rainfall. Results are shown in Figures 3 and 4 respectively  = 0 and  =
0.5.
P(H  D), =0
storm surge level, h [cm]
1
P(H>h)
0.8
0.6
0.4
0.2
0
0
0.5
P(D>d)
1
Joint storm surge and torrential rainfall
250
200
150
100
50
0
0
200
400
1 day torrential rainfall, d [mm]
Figure 3. Simulation of the Gumbel 2-copula,  = 0, for storm surge and torrential rainfall with H:
storm surge level [cm] and D: rainfall [mm].
8
P(H  D), =0.5
storm surge level, h [cm]
1
P(H>h)
0.8
0.6
0.4
0.2
0
0
0.5
P(D>d)
Joint storm surge and torrential rainfall
250
200
150
100
1
50
0
0
100
200
300
1 day torrential rainfall, d [mm]
Figure 4. Simulation of the Gumbel 2-copula,  = 0.5, for storm surge and torrential rainfall with H:
storm surge level [cm] and D: rainfall [mm].
Subsequently, the conditional probabilities of 1 and 3 days of torrential rainfall in
the Tai Lake basin given a storm surge level of at least 1.18m are presented in Figure 5.
The correlation between storm surge levels and torrential rainfall is adjusted to 0.3 which
is representative according to studies of Hohai University (2000).
From Figure 5 we can see that the conditional probabilities are shifted to the right in
comparison with the margins as expected. Furthermore, the Gumbel copula gives stronger
upper tail dependence, which is visible in the conditional probabilities. The factor
between the joint probabilities and the conditional probabilities is the probability for a
storm surge level of at least 1.18m (P=10%). Indeed, a lower surge level will cause the
joint probability curve to lower since the probability of this surge level is higher.
Moreover, the conditional probability curve will move towards the marginal distribution
curve and it will become the marginal distribution curve when the given surge level
becomes nil indeed. A higher correlation between storm surge and torrential rainfall will
cause the conditional probability curve to shift more to the right from the marginal curve.
Rainfall distribution with Gumbel copula,  = 0.3, hs  118.46cm
Rainfall distribution with Gumbel copula,  = 0.3, hs  118.46cm
100
100
60
40
conditional
marginal
joint
60
40
20
20
0
80
probability [%]
probability [%]
80
conditional
marginal
joint
50
100
150
200
250
300
350
1 day torrential rainfall in Tai Lake Basin, d
[mm]
Basin
0
50
100
150
200
250
300
350
3 days torrential rainfall in Tai Lake Basin, d
[mm]
Basin
Figure 5. 1 and 3 days torrential rainfall distributions from Gumbel copula generated samples with
 = 0.3 and a given surge level of at least 118cm P=10%.
9
5.
Flood probability during barrier closure
The conditional probabilities of torrential rainfall given a barrier closure in the
mouth of the Huangpu River are found via the Gumbel copula. These results are
subsequently translated into actual torrential rainfall runoff into the river. In the final step
of the analyses we determine the flood probability of the Huangpu River during barrier
closure.
During barrier closure, the warning water level will be exceeded first at Mishidu
because of its upstream position in the river and its lower warning water level. The
critical discharges at Mishidu are hence dominant in the flood probability analysis. The
flood probability at Huangpu Park is eventually analyzed with the required torrential
rainfall runoff with respect to the critical discharges at Mishidu.
Recall that flooding of the Huangpu River is defined as the probability that the
upstream discharge exceeds the storage capacity of the river or critical discharge.
P( Z  0)  P(Qupstream  Qcritical )
(10)
Since the only the upstream discharge is considered stochastic, flooding solely
depends on the variables that build up this upstream discharge, namely the base discharge
and the torrential rainfall runoff given barrier closure.
When we now define Qr as the required torrential rainfall runoff, which is just the
difference between the critical and base discharge, then the flood probability can be
denoted as
P(Z  0)  P(Qtorrential  Qr )
(11)
The base discharge data used is however discrete for a given month and type of
hydrological year. The base discharge depends on the prevailing type of hydrological
year. In a wet hydrological year, the base discharges are significant higher, but their
probability of occurrence is lower. As a consequence of the discrete data the flood
probability is first computed for given the barrier closure and the month and hydrological
year of interest. Next, the flood probability given the barrier closure for a given month
regardless of the hydrological year is obtained by the sum of the products of the monthly
flood probabilities and the normalized probabilities of the corresponding hydrological
years. The expression for the flood probability then reads
P( Z  0 | month) 
3

P(type _ year ) P(Qtorrential  Qr )
(12)
type _ year
The main goal is to determine the annual flood probability during barrier closure.
For this purpose the probability of a barrier closure is required, i.e. the assumed barrier
closure water level of 4.80m WD. This probability can be computed via random
combination of tide and surge levels. Results, however, show a probability of exceedance
10
of only 16%. This seems to be underestimated in comparison with the historical
exceedance frequency of 4.12 annually. Only for extreme events the difference between
frequency of exceedance and probability of exceedance can be neglected as an extreme
event is not likely to happen twice a year. Apparently, random combination of tide and
storm surge at Wusongkou only gives representative results for the higher probabilities as
these return periods are more reasonable compared with the results from the Shanghai
Water Authority. Probably, the linear interaction between storm surge of a tropical
cyclone and astronomical tide can not be neglected. As a consequence we continue the
analysis with use of the historical frequency of exceedance.
The frequency of exceedance is defined as the annual exceedance of a certain
threshold. Hence, the monthly distribution of the exceedance frequency of the barrier
closure water level can be found with the monthly probability of occurrence of tropical
cyclones in the Shanghai region, see table below.
Table 1. Normalized probability of occurrence of tropical cyclones in the Shanghai
area in the regional typhoon season
P [%]
June
8
July
20
August
28
September
25
October
9
Naturally, not every tropical cyclone in the Northwest Pacific will cause barrier
closure; this is however already taken into account in the frequency of exceedance of the
barrier closure water level. When we combine the monthly distribution of tropical
cyclones in the region with the barrier closure frequency we obtain the monthly
distribution of the barrier closure frequency. Hence the monthly frequency of exceedance
of the storage capacity of the Huangpu River during barrier closure reads:
f month (Z  0)  fclosure P(Z  0 | month)
(13)
Finally, we derive the annual flood frequency of the Huangpu River during barrier
closure for the three proposed barrier location with reference to the locations Mishidu
and Huangpu Park along the river. This flood frequency derived by summation of the
monthly flood frequencies, which can be expressed as
f (Z  0) 
5

f month (Z  0)
(14)
month
The results are shown in Figure 6 below. The effects of the discrete base discharges
are still clearly visible in the frequency curves in Figure 6. The flood frequencies for a
representative 24 hours barrier closure for each barrier location for a 1 day torrential
rainfall are presented in Table 2.
11
10
10
10
10
Flood frequency Huangpu River, Huangpu Park
frequency of exceedance [per year]
frequency of exceedance [per year]
Flood frequency Huangpu River, Mishidu
0
-1
-2
Wusongkou
Zhanghua Bang
Fishery Yard
-3
0
12
24
36
closure duration [hrs]
48
60
10
10
10
10
0
-1
-2
Wusongkou
Zhanghua Bang
Fishery Yard
-3
12
24
36
48
60
72
closure duration [hrs]
84
96
Figure 6. Flood frequency of the Huangpu River with reference to warning water levels at Huangpu
Park and Mishidu, 4.55 m WD and 3.50 respectively, during barrier closure for three proposed
barrier locations with 1 day of torrential rainfall.
Table below shows that the return periods of the flood frequency for all barrier
locations for both Mishidu and Huangpu Park are significant. A low return period
indicates a high frequency of exceedance. Note that this does not mean that the threshold
will be exceeded every year.
Table 2. Return periods of the flood frequency of the Huangpu River during a 24 hours closure for
the three barrier locations with 1 day of torrential rainfall.
Wusongkou
Mishidu
Zhanghua Bang
Fishery Yard
f [years]
1.88
1.14
0.93
Huangpu Park f [years]
135
14.30
7.70
As expected, the most upstream barrier location at FisheryYard has the highest flood
frequency. However, even with a barrier at Wusongkou in the mouth of the river, the
flood frequency is higher than the aimed 1:1000 years for Shanghai City. To meet this
return period, the barrier may not close for longer than 8 hours and 16 hours with regard
to the warning water levels at Mishidu and Huangpu Park respectively. This closure
duration seems to be short to withstand any typhoon storm in the region. To prevent
flooding of the Huangpu River during barrier closure the upstream discharge volume has
to be either reduced or released. The first measure can be implemented simply with the
existing control gate at the Taipu River. During barrier closure, the Huangpu River can
be relieved from the stored water volume by facilitating hatches in the barrier doors. So
that, during low tide, the river can discharge the stored water volume into the Yangtze
River estuary through the hatches. The barrier itself remains closed, ready to protect
Shanghai City against the next storm.
12
6.
Conclusions and recommendations
This paper introduces the use of copulas to analyze the flood probability. In this
study we linked the known distribution functions of typhoon induced storm surge levels
in the mouth of the Huangpu River with torrential rainfall in the region to their joint
distribution function. From the results obtained we conclude that flooding of the
Huangpu River during barrier closure is a real threat with regard to the limited and
sometimes conflicting data.
The results are a strong indication that accurate data and precise knowledge on the
study area is vital for accurate analyses on the exact flood probability, since flooding of
the Huangpu River concerns the loss of property and even human lives. The methodology
developed in this study, including the concept of copulas, is suitable for future analyses
of the flood probability when accurate data becomes available. Copulas have proven to
be useful in flood probability analyses when limited data are available in which only the
individual distribution functions are known. Copulas allow us to study each individual
contribution to multivariate events like typhoon induced floods separately from each
other.
Acknowledgments
This paper results from the first author's MSc-graduation thesis at Delft
University of Technology in the Netherlands. The presentation at the
International Conference on Coastal Engineering 2003 has been made possible
with sponsorship from Delft University of Technology and the thesis sponsor
WL|Delft Hydraulics. Much gratitude is owed to prof. dr. ir. Vrijling and prof.
dr. ir. Stive both from Delft University of Technology for their efforts in this.
The author also acknowledges the support from the Shanghai Water Authority.
References
Chen, X. and Y. Zong, 1999. Typhoon hazards in the Shanghai area, Disasters,
1999, 23(1):66-88, Blackwell Publishers, Oxford.
Hohai University, 1999. Storm tide, flood, urban torrential rainfall joint
occurrence analysis, Huangpu River storm surge barrier study volume 1.3
(in Chinese), Shanghai Water Authority, Shanghai.
Hohai University, 2000. Final Report: Huangpu River storm tide levels analysis,
Huangpu River storm surge barrier study, Shanghai Water Authority (in
Chinese), Shanghai.
13
Genest, C. and J. Mackay, 1986. Copules archimédiennes et familles de lois
bidimensionnelles dont les marges sont données, Canadian Journal of
Statistics. 14, 145-159.
Nelson, Roger B. 1999. An introduction to copulas, Lecture notes in statistics,
Springer-Verlag, New York.
Sklar, A., 1959. Fonctions de répartition à n dimensions et leurs marges, Publ.
Inst. Statist. Univ. Paris 8, 229-231.
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14
KEYWORDS – ICCE 2004
COPULA APPROACH FOR FLOOD PROBABILITY ANALYSIS OF
THE HUANGPU RIVER DURING BARRIER CLOSURE
J.Y. Nai, P.H.A.J.M. van Gelder, P.J.M. Kerssens, Z.B. Wang, E. van Beek
Paper No:276
Flood probability
Storm surges
Tropical cyclones
Torrential rainfall
Joint probability distribution
Copula
Correlation
Dependence
Limit state
Floods
Fault tree