Natural Hazards
and Earth System
Sciences
Open Access
Atmospheric
Chemistry
and Physics
Open Access
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
doi:10.5194/nhess-13-1841-2013
© Author(s) 2013. CC Attribution 3.0 License.
cess
Geosciences
G
Measurement
Techniques
H. Zhong1 , P.-J. van Overloop2 , and P. H. A. J. M. van Gelder1
1 Section
Open Access
A joint probability approach using a 1-D hydrodynamic model for
Atmospheric
estimating high water level frequencies in the Lower
Rhine Delta
Open Access
of Hydraulic Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology,
Delft, the Netherlands
2 Section of Water Resources Management, Faculty of Civil Engineering and Geosciences, Delft University of Technology,
Delft, the Netherlands
Biogeosciences
Correspondence to: H. Zhong ([email protected])
1
Introduction
In the Lower Rhine Delta in the Netherlands, the Rhine and
Meuse rivers run from theEarth
east andSystem
the south into the North
Sea at Hook of Holland, to the Haringvliet in the west and
Dynamics
into the Lake IJsselmeer in the north. The area is a center
of high economic activity and maritime transportation with a
dense population, which is at risk of being flooded by river
Geoscientific
or by sea, or by both. The water
level in this transitional area
is influenced by the upstream
river flows, the downstream
Instrumentation
sea level as well as the operation of the existing controllable
Methods and
structures. At the upstream boundary, the Rhine flow comes
Data Systems
from rainfall-runoff and snowmelt
in the Alps; the Meuse
flow is mainly determined by rainfall in France and Belgium.
At the downstream boundaries, the extreme still water level
(excluding waves) arises Geoscientific
from a combination of the astronomical tides and the meteorologically induced storm surge
Development
components. InModel
this article
the extreme still water level is
the so-called “storm surge”. Astronomical tides are driven
by astronomical forces and are deterministic, while the wind
induced storm surges occur
stochastically,
driven by meteoHydrology
and
rological forcing. To protect the delta from sea flooding, the
Earth
System
estuary can be closed off from
the sea
by large dams and controllable gates and pumps. Also,Sciences
along the rivers controllable
structures have been constructed in order to regulate the flow
and water levels. The operation of these structures was involved in the operational water management system of the
Netherlands (van Overloop, 2009).
Ocean
It is important to evaluate
the Science
high water level frequencies; first to evaluate the flood risk, and second, to support
Open Access
Open Access
Open Access
Open Access
Abstract. The Lower Rhine Delta, a transitional area between the River Rhine and Meuse and the North Sea, is at
risk of flooding induced by infrequent events of a storm surge
or upstream flooding, or by more infrequent events of a combination of both. A joint probability analysis of the astronomical tide, the wind induced storm surge, the Rhine flow
and the Meuse flow at the boundaries is established in order
to produce the joint probability distribution of potential flood
events. Three individual joint probability distributions are established corresponding to three potential flooding causes:
storm surges and normal Rhine discharges, normal sea levels
and high Rhine discharges, and storm surges and high Rhine
discharges. For each category, its corresponding joint probability distribution is applied, in order to stochastically simulate a large number of scenarios. These scenarios can be used
as inputs to a deterministic 1-D hydrodynamic model in order
to estimate the high water level frequency curves at the transitional locations. The results present the exceedance probability of the present design water level for the economically
important cities of Rotterdam and Dordrecht. The calculated
exceedance probability is evaluated and compared to the governmental norm. Moreover, the impact of climate change on
the high water level frequency curves is quantified for the
year 2050 in order to assist in decisions regarding the adaptation of the operational water management system and the
flood defense system.
Climate
of the Past
Open Access
Received: 30 October 2012 – Published in Nat. Hazards Earth Syst. Sci. Discuss.: –
Revised: 15 May 2013 – Accepted: 21 May 2013 – Published: 25 July 2013
Open Access
Solid Earth
Open Acces
Published by Copernicus Publications on behalf of the European Geosciences Union.
M
1842
H. Zhong et al.: A joint probability approach using a 1-D hydrodynamic model
the future flood defense design. The resistance of the river
dikes against flooding is captured in the design water level;
for instance, for the economically important city of Rotterdam the design water level is regarded as the water level
with the exceedance frequency of 1/10 000. However, climate change and human interventions complicate the computation of these high water level frequencies. For example,
being a very significant human intervention, the change in the
operational water management system has resulted in nonhomogeneous high water level observations.
The non-homogeneous extreme observations derived from
climate change and human interventions cannot be used to
estimate the high water level frequencies. Instead, a joint
probability approach, using a 1-D hydrodynamic model, can
be used to estimate the high water level frequencies (Mantz
and Wakeling, 1979; Acreman, 1994; Gorji-Bandpy, 2001;
Samuels and Burt, 2002; Adib et al., 2010). As a given high
water level at a transition location may result from a number
of combinations of sea level and upstream fluvial flow and
from how the operational water management system reacts
to the situation at hand, the occurrence of all these combinations together determines the frequency of the given water
level.
The first application of a joint probability approach in the
Lower Rhine Delta dates back to 1969. Van der Made (1969)
divided three joint probability distributions for three individual categories: high sea levels and normal discharges, normal sea levels and high discharges, high discharges and high
sea levels. For each category the joint probability distribution
was estimated from the observations of the peak values of the
sea level and the Rhine flow in the same day.
However, the above joint probability approach only considered the peak values of the sea level and the Rhine flow,
and assumed the other associated variables, for example the
surge duration, to be pre-determined. These associated variables also play an important role in the operational water
management system, and therefore influence the high water
level frequencies in the delta. For example, after the closure
of the Maeslant barrier and the Haringvliet dam, the water
level within the delta depends on the fluvial flow, the storage capacity and the closure duration (Zhong et al., 2012).
Recent research (De Michele et al., 2007; Wahl et al., 2012)
has shown how to include more variables in the probabilistic analysis of the hydrodynamic boundary conditions. In this
paper, more associated variables will be taken into account in
the high water level frequency estimation.
1-D hydrodynamic numerical models can be applied to assess the changing water level frequencies caused by different
changes, for example changes in mean sea level rise, peak
river flows and the construction and operation of controllable
hydraulic structures. The detailed models can produce accurate water levels at transitional locations by examining the
interaction of sea level, fluvial flow and infrastructure operations, but their computational time restrains the application
of those models in a Monte Carlo simulation method. For
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
example, only a very limited number of sampling scenarios
were used by Mantz and Wakeling (1979) and Samuels and
Burt (2002). However, for accurate Monte Carlo simulations
a very large number of stochastic sampling scenarios are necessary. For that reason, in this research, a strongly simplified
1-D hydrodynamic model of the Lower Rhine Delta is applied.
Future climate change will affect the high water level frequencies in the Lower Rhine Delta. For the Rhine flow, climate change is expected to increase winter precipitation with
earlier snowmelt (Middelkoop and Kwadijk, 2001), which
will lead to an increase in the frequency and magnitude of
extreme Rhine flows (Hooijer et al., 2004; Pinter et al., 2006;
Linde et al., 2010). For mean sea levels along the Dutch
coast, a range of 0.15 to 0.35 m rise until 2050 and a range of
0.35 to 0.85 m rise until 2100, corresponding to the reference
year of 1990, are commonly used extrapolation values (van
den Hurk et al., 2006; Second Delta Commission, 2008). In
fact, the relative mean sea level rise will be larger when taking mean land subsidence due to glacial isostasy and subsoil
compaction into consideration. The effects of climate change
on the characteristics of the wind induced surge along the
Dutch coastline was investigated and no evidence of significant changes was detected (Sterl et al., 2009), and, hence, we
can assume that the characteristics are not influenced by climate change. Although there are still inherent uncertainties
in the prediction of climate change on the hydraulic boundary conditions within climate change scenario studies, it can
be assumed that the future changes in high water level frequencies can be assessed by applying an appropriate climate
change scenario. In this paper estimates of mean sea level rise
and increases of peak Rhine discharge in the future scenario
of 2050 are included to assess the future high water level frequencies. The results can assist in adapting the operational
water management system to control the negative effects of
climate change in the Lower Rhine Delta.
This paper aims to assess the high water level frequencies
in the Lower Rhine Delta and to identify the exceedance frequencies of the design water level. The paper starts with the
introduction of the method and an illustration of the model,
followed by the joint probability analysis. The results and
discussion are presented, followed by the conclusions and
future research recommendations.
2
Method
Applying the joint probability approach using a 1-D hydrodynamic model to estimate the high water level frequencies
in the estuary delta is illustrated in Fig. 1. The importance
sampling Monte Carlo simulation method will help generate
a large number of stochastic scenarios as inputs for the above
model. The outline of the method is as follows:
– from historical observations, three categories are selected in order to separate different combinations of sea
www.nat-hazards-earth-syst-sci.net/13/1841/2013/


1-D hydrodynamic model including infrastructure operations is simulated with the
scenarios of the three categories and result in the same number of high water
levels (10,000) at transitional locations in the delta respectively;
The high water level frequency curves are computed. In addition, the exceedance
H. Zhong
et al.:
A joint
approach
a 1-D
model
probabilities
of the
design
waterprobability
levels at Rotterdam
and using
Dordrecht
arehydrodynamic
calculated
from the resulting peak water levels of the simulations.
Fig. 1. Flowchart of the methodology
Fig. 1. Flowchart of the methodology.
water levels and Rhine flows that may cause high water
levels in the Lower Rhine Delta;
– for each category a corresponding joint probability distribution is estimated;
– a large set of stochastic scenarios are generated by
means of the importance sampling Monte Carlo simulation for each category (10 000 per category);
– 1-D hydrodynamic model including infrastructure operations is simulated with the scenarios of the three categories and result in the same number of high water levels (10 000) at transitional locations in the delta; and
– the high water level frequency curves are computed.
In addition, the exceedance probabilities of the design
water levels at Rotterdam and Dordrecht are calculated
from the resulting peak water levels of the simulations.
3
The 1-D simplified hydrodynamic model of the
Lower Rhine Delta
The Rhine Delta is a system of inter-connected rivers, canals,
reservoirs, and adjustable structures as can be seen in Fig. 2
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
1843
(left). A strongly simplified 1-D hydrodynamic model was
used to simulate the complex flows (van Overloop, 2009).
The model used a large grid size of 20 km and constant
bed slopes. The large water bodies, such as the IJsselmeer,
were modeled as reservoirs with a level-area description. Figure 2 (right) presents the model consisting of 36 nodes and
40 reaches. The total number of water level nodes, including
the extra grid points on reaches longer than 20 km, is 56. The
hydraulic structures, as indicated in Fig. 2 (left), are under the
present operating rules of the national water board. The structures were modeled by flows derived from their discharge–
water level relation. Tortosa (2012) calibrated and validated
this simplified model using simulation results of an accurate
high-order numerical model of the Netherlands over the period 1970 to 2003. The model’s accuracy is sufficient for this
research (inaccuracies in the order of a few centimeters for
the water levels).
The hydrodynamic characteristics of the delta are mainly
governed by discharge of the rivers Rhine (Lobith: node 14
in Fig. 2), Meuse (Borgharen: node 1) and by the water level
at the sea boundaries (Hook of Holland: node 36 and Haringvliet: node 29). In the model calculations, the sea level at
Haringvliet is assumed to be the same as at Hook of Holland.
As the focus of the research is on the Lower Rhine Delta surrounding the economically important cities of Rotterdam and
Dordrecht, the other sea level boundaries in the North (node
30, 33, 35), which do not affect Rotterdam and Dordrecht,
are set to 0 m m.s.l. during the running of the model. At Rotterdam (node 24) and Dordrecht (node 22) the high water
level frequencies are estimated. In addition, the dike heights
along the rivers are assumed to be high enough so that no
4
overflowing or breaching can occur.
4
The joint probability analysis
This section starts with the description of the observation
data, followed by the estimation of the joint probability distributions for three categories. The data is shown in Table 1.
The behavior of the River Rhine, Meuse and the sea conditions may change considerably over the longer periods
due to artificial or natural causes. Commonly, three popular statistical tests are employed to check whether a trend
exists in an observation time series. The Mann–Kendall test
can be applied to assess the significance of trends in hydrometeorological time series such as stream flow, temperature
and precipitation (Mann, 1945; Kendall, 1975). The Spearman’s rho test can also be used to detect monotonic trends in
a time series (Lehmann, 1975; Sneyers, 1990). The Wilcoxon
rank sum test can test if abrupt shifts exist in a time series
(Wall, 1986). No significant trends or shifts have been detected with these three tests in the annual maximum series
of sea level and Rhine and Meuse discharges, except that
a least squares linear regression suggests a gradual increase
of 0.20 m mean sea level rise per century. This result is in
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
1844
H. Zhong et al.: A joint probability approach using a 1-D hydrodynamic model
Table 1. Data description.
Station
Data
time
data description
Hook of Holland
observed sea level (m m.s.l.)
1939–2009
Hook of Holland
predicted astronomical tidal level (m m.s.l.)
1939–2009
Lobith
Borgharen
Rhine discharge (m3 s−1 )
Meuse discharge (m3 s−1 )
1901–2009
1911–2009
1939–1970 water level per 1 h; 1971–
2009 water level per 10 min
time unit is the same as the above sea
level
daily-average discharge
daily-average discharge
Note: the source of these data is the Rijkswaterstaat website: http://www.rijkswaterstaat.nl/waterbase.
1
2
3 The 1-D simplified hydrodynamic model of the Lower Rhine Delta
3
4
Fig. 2. (Left) Description of the Rhine delta with existing operated structures and (Right)
Fig. 2. (left) Description
ofoverview
the Rhineofdelta
with existing
structures
and(van
(right)
overview
of the simplified 1-D hydrodynamic model
5
the simplified
1-D operated
hydrodynamic
model
Overloop,
2009).
(van Overloop,62009).
7
The Rhine Delta is a system of inter-connected rivers, canals, reservoirs, and adjustable
8
structures as can be seen in Fig. 2 (Left). A strongly simplified 1-D hydrodynamic model
9
was
used to(van
simulate
the complex
flows (van
The model
used a along
large the lower Rhine branch. A
line with previous
research
Gelder,
1996; Zhong
et al.,Overloop,
the 2009).
floodplains
inundated
10
grid size of 20 km and constant bed slopes. The large water bodies, such as the IJsselmeer,
2012).
discharge slightly exceeding 6000 m3 s−1 was more or less
11
were modeled as reservoirs with a level-area description. Fig. 2 (Right) presents the
The division
threeconsisting
categories
of The total
assumed
to of
be water
the critical
value which resulted in the high12 into
model
of is
36based
nodeson
andthresholds
40 reaches.
number
level nodes,
the peak surge
residual
and
the
peak
of
Rhine
flow
occurring
est
floodplains
inundated
(Kwadijk
and Middelkoop, 1994).
13
including the extra grid points on reaches longer than 20 km, is 56. The hydraulic
14 1.00
structures,
as indicated
in Fig.
(Left),m3are
present
rules
of the
in the same day:
m in Hook
of Holland
and2 6000
s−1under the
Thirdly,
theoperating
threshold
value
of 6000 m3 s−1 was applied by
15 surge
national
water board.
structures were
modeled
flows derived
from
their
at Lobith. The
residual
is theThe
difference
of the
ob- by Chbab
(1995),
with
thedischargegeneralized Pareto distribution to es16
water
level
relation. Tortosa
(2012) calibrated
model
served sea level
and the
predicted
astronomical
tide level.and validated
timate this
the simplified
frequencies
of using
high Rhine flows. In this article
17
simulation results of an accurate high-order numerical model of the Netherlands over the
This threshold
value
for
the
peak
surge
residual
is
chosen
for
the
application
of
this
threshold,
as well as the fitted gen18
period 1970 to 2003. The model’s accuracy is sufficient for this research (inaccuracies in
two reasons:19first the
of all,
this
value
is
related
to
the
operation
eralized
Pareto
distribution
function,
leads to a Rhine deorder of a few centimeters for the water levels).
of the Maeslant
sign discharge (with a probability of 1/1250 per year) of
20 Barrier. The peak surge residual of 1.0 m,
21 the
Thehigh
hydrodynamic
characteristics
the adelta
governed
discharge
of the
coinciding with
astronomical
tide levelofand
highare mainly
15 250
m3 s−1by
, which
is comparable
to commonly used val22
rivers
Rhine
(Lobith:
node
14
in
Fig.
2),
Meuse
(Borgharen:
node 1) and by the water
Rhine flow, could make the Rotterdam water level exceed
ues.
23
level at the sea boundaries (Hook of Holland: node 36 and Haringvliet: node 29). In the
the critical value
of
3.0 m m.s.l. (the decision level of the cloThe selected events from 1939 to 2009 in Fig. 3 are used
24
model calculations, the sea level at Haringvliet is assumed to be the same as at Hook of
sure of the barrier).
Secondly,
the
threshold
of
1.0
m
was
apto
the joint
probability
25
Holland. As the focus of the research is on the Lowerestimate
Rhine Delta
surrounding
the distributions of three cate-
plied in the estimation of the frequency of the wind induced
surge peak level (Bijl, 1997). The threshold of 6000 m3 s−1
for Rhine discharge is determined by means of three reasons:
first of all, this value is related to the operation of the Maeslant Barrier (Bol, 2005). Secondly, this value is related to
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
gories. The biggest sea flooding in 1953 is missing from the
website database. Instead, the information of this surge resid5
ual comes from Gerritsen (2005), in which the peak surge
residual is 3.00 m and the duration is 50 h.
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
11
12
13
14
15
16
17
23 11in Category I of Fig. 3. The probability distribution of the storm surge
24 12Scheldt
by separating
thetheastronomical
tide and
Thesewas
surgeestimated
residual curves
are taken into
probability analysis
with the
two wind
param
The selected events from 1939 to 2009 in Fig. 3 are used to estimate the joint probability
25
surge
(Vrijling
and
Bruinsma,
1980;
Praagman
and
Roos,
1987).
Th
13
surge
residual
h
and
surge
duration
T
.
The
probability
distributions
of
smax
s
distributions
of three
categories.
Theprobability
biggest sea flooding
in 1953using
is missing
from
the
H. Zhong
et al.:
A joint
approach
a 1-D
hydrodynamic
model
1845
website database. Instead, the information
of
this
surge
residual
comes
from
Gerritsen
parameters
applied
to simulate
pseudo of
surge
residual
curves with an
26 14introduced
andare
further
validated
formany
the station
Hook
of Holland.
(2005), in which the peak surge residual is 3.00 m and the duration is 50 hours.
design discharge (with a probability of 1/1,250 per year) of 15250 m3/s, which is comparable
to commonly used values.
15
shape function. The astronomical tide curves can also be simulated by the
16
As a result, the simulated surge residual curves and the simulated tide curves can
II
III
17
combined into the simulated sea level curves.
18
19
300 extreme
I surge residuals in Category I are analyzed in Fig. 3 in order to estima
20
curve in Hook of Holland. The observed peak surge residuals and associated d
21
plotted in Fig 5. Their linear correlation coefficient is 0.0474, and therefore they
22
Peaksurgeresidual (m) to be linearly independent. For a surge residual curve at Hook of Holland, the
Fig. 3. Selected events: Category I, storm surge 23
and normal
Rhine flow; and
Category
II. high
residual
duration
are generated and constrained by complex physical fact
Rhine
and normalevents:
sea water
level; Category
III. surge
storm surge
and highRhine
Rhine flow
Fig.flow
3. Selected
Category
and normal
27 I, storm
24
offshore
surge,
the
shallow
water depth, the interaction between tide and surge, et
II,normal
high Rhine
flow
4.1 flow;
StormCategory
surge and
Rhine
flowand normal sea water level; Catfrom a statistical point of view, the assumption of independence between the
III,events
stormofsurge
high
Rhine25
flow.
Theegory
historical
stormand
surges
coinciding
with normal upstream flows are shownFig. 4. Variation with time of the extreme still water level.
26 of the
residual
andin the
duration is acceptable.
in Category I of Fig. 3. The probability distribution
storm surges
the Eastern
14000
10000
3
Rhineflow(m/s)
12000
8000
6000
4000
2000
0
-0.5
21
22
23
24
25
26
27
0
0.5
1
1.5
2
2.5
3
3.5
Scheldt was estimated by separating the astronomical tide and the wind induced storm
surge (Vrijling and Bruinsma, 1980; Praagman and Roos, 1987). This method was
introduced
and further
validated
fornormal
the stationRhine
of Hook flow
of Holland.
4.1 Storm
surge
and
90
80
Surge duration (hours)
18
19
20
70
60
The historical events of storm surges coinciding with normal upstream flows are shown in Category I of Fig. 3. The
50
probability distribution of the storm surges in the Eastern
40
Scheldt was estimated by separating the astronomical tide
30
and the wind induced storm surge (Vrijling and Bruinsma,
20
1980; Praagman and Roos, 1987). This method was intro10
duced and further validated for the station of Hook of Hol7
0
1
1.5
2
2.5
3
land.
Peak surge residual (m)
27
From a statistical point of view, the occurrence of the as28
5. The peak surge residuals and associated durations
tronomical tide component is independent
of the occurrence Fig.
Fig. 5. The peak surge residuals and associated durations.
29
of the wind induced storm surge component at the mouth of
The
surge residual
curve can be approximated by a squared cosine function. I
the Lower Rhine Delta. However, 30
these two
components
can
However,
from residual
a statisticalcurves
point ofand
view,the
the simulated
asinteract with each other when they
into the delta
31propagate
comparison
betweensurge,
the etc.
observed
surge
cu
sumption
of
independence
between
the
peak
surge
residual
(de Ronde, 1985). Their nonlinear
generallycosine
in32interaction
symmetric
function of six extreme surge events agrees with this
duration is acceptable.
creases the surge height at a rising33
astronomical
tide andIn
de-Fig. 7and
assumption.
thethesurge
residual curve from the big sea flooding in 1953
The
surge
residual curve can be approximated by a squared
creases the surge height at a high astronomical tide (Bijlsma,
34
a symmetric curve (Gerritsen,
2005).
cosine function. In Fig. 6 the comparison between the ob1989). Quantifying the nonlinear effect is beyond the scope
35
served surge residual curves and the simulated curves by the
of this study. For the sake of convenience, it can be assumed
36
The
design
surge
residual
curve
can
be derived
from surge
the observed
surge residu
symmetric
cosine
function
of six extreme
events agrees
that the wind induced storm surge is independent of the aswith
this
reasonable
assumption.
In
Fig.
7
the
surge
residual
tronomical tide as indicated in Fig. 4.
curve from the big sea flooding in 1953 also showed a symThese surge residual curves are taken into the probability
metric curve (Gerritsen, 2005).
analysis with two parameters: peak surge residual hsmax and
The design surge residual curve can be derived from the
surge duration Ts . The probability distributions of these two
observed surge residual curves.
parameters are applied to simulate many pseudo surge residual curves with an appropriate shape function. The astronomπ ·t
hs (t) = hsmax · cos2 (
),
(1)
ical tide curves can also be simulated by the same logic. As a
Ts
result, the simulated surge residual curves and the simulated
where hsmax stands for the peak value of the surge residual,
tide curves can be linearly combined into the simulated sea
level curves.
and its unit is m; Ts is the duration of the surge residual, and
In order to estimate the surge curve in Hook of Holland,
its unit is hours. Here we assume that the surge peak occurs
300 extreme surge residuals in Category I are analyzed in
when t is 0.
Fig. 3. The observed peak surge residuals and associated duThe generalized Pareto distribution and the Weibull Disrations are plotted in Fig. 5. Their linear correlation coeffitribution fit the distributions of peaks and durations of these
cient is 0.0474, and therefore they are assumed to be linearly
selected surge residuals, respectively. In this research, all paindependent. For a surge residual curve at Hook of Holland,
rameters of distributions are estimated by the maximum likethe peak surge residual and duration are generated and conlihood method.
strained by complex physical factors like the offshore surge,
The semi-diurnal astronomical tide in Hook of Holland, is
the shallow water depth, the interaction between tide and
almost symmetric, and can therefore be approximated by a
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
Ts
1
2
3
where hsmax stands for the peak value of the surge residual, and its unit is m; Ts is the
Zhong
al.: Aisjoint
probability
using athat
1-D hydrodynamic
model
of the surge residual,H.and
itsetunit
hours.
Hereapproach
we assume
the surge peak
occurs when t is 0.
1846
duration
(t )  hs max  cos 2 (
 t
Ts
)
(1)
ere hsmax stands for the peak value of the surge residual, and its unit is m; Ts is the
ration of the surge residual, and its unit is hours. Here we assume that the surge peak
curs when t is 0.
1
2
3
4
5
of the water level ha with a period of 12.4 hours and with amplitude of hHW -hLW. Where
hHW is the high tide level; hLW is the low tide level; their unit is m MSL; u is the time shift
between peaks of tide and surge. Fig. 8 shows that the simulated tide level from the
sinusoidal function represents the tide well.
4
h h
h h
2
5 Fig.
Fig.6. 6.
The surge
historical
surge
and (right)
design curves;
(t )  curves;
sin( simulated
(t correlation
hdesign
the
u )) 
(left)(Left)
The historical
residual curves
andresidual
the simulatedcurves
the
coefficient
squared R 2 .(Right)
2
2
g. 6. (Left) The historical surge residual curves and the simulated design curves;2 (Right)12.4
6
the correlation 6coefficient squared R
the correlation coefficient squared R2
ig. 7. (Left)
HW
LW
HW
LW
a
31.5
3
Observed surge curve
Surge residual simulated (m)
Surge residual (m)
1
0.5
0
Observed data
1.5
1
2
1.5
1
0.5
0
0.5
-0.5
31/01
2.5
Simulated surge curve
2
1.5
2
R =0.77
-0.5
Astronomical tide level (m MSL)
2.5
2
Surge residual (m)
2.5
Simulated surge curve
Tide data
Tide level simulated
Surge residual simulated (m)
2.5
1.5
Tide level
Observed surge curve
Observed data
1
2
R =0.77
20.5
1.5
0
1-0.5
-1
0.5
01/01/1971
02/01/1971
Time
Astronomical tide level simulated (m MSL)
3
3
(2)
03/01/1971
1
2
R =0.73
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Astronomical tide level (m MSL)
1.5
7 3
1953
Observed surge residual (m) 8
Fig. 8. The stochastic astronomical tide function
0
0
9 designFig.
8. The stochastic astronomical tide function.
The surge residual curve of the largest flood in 1953 and the
simulated
As a consequence of the assumed independency of the tide and the wind induced storm
Fig. 7. (left) The surge residual curve of the largest flood in10
1953
2
curve; (Right) the correlation coefficient squared R
11 surge, -0.5
the time shift between peaks u fits a uniform probability density function. Time
and the design simulated -0.5
curve;
(right)
the
correlation
coefficient
31/01
01/02
02/02
03/02shifts u larger 0
1
2
12
than 12.4 hours
are irrelevant,
so3 considering a symmetrical shape, the
2
squared R .
1953
Observed
surge
Time
shifts
u larger
than (m)
12.4 h are irrelevant, so considerdensity
function
of uresidual
becomes:
Pareto Distribution and the Weibull Distribution
fit13the probability
distributions
of
01/02
02/02
03/02
0
1
2
e General
7
ing a symmetrical
shape, the probability density function of
1
aks and
selected
surge residuals,
In this
8 durations
Fig. of7.these
(Left)
The surge
residualrespectively.
curve of the
1953
and the design simulated
plargest
(u ) research,
0
uflood
 all
12.4in
hours
u
becomes
2
rameters of distributions
are estimated
by the
Maximum
Likelihood
Method.
sinusoidal
curve
and
modeled
as
a
periodical
fluctuation
of
(3)
9
curve; (Right) the correlation coefficient
squared R2
1
1
the water level ha with a period of 12.4 h and with ampli- p(u ) 
1
u  12.4 hours
12.4and
2|u| > · 12.4 h
10 tudeastronomical
p(u)
= 0can
(3)
e semi-diurnal
in Hook
is almost
of hHW − hLWtide
. Where
hHWofis Holland,
the high tide
level; hsymmetric,
LW
2
refore
be
approximated
by
a
sinusoidal-curve
and
modeled
as
a
periodical
fluctuation
14
the low
tide level;Pareto
their unitDistribution
is m m.s.l.; u is the
timethe
shiftWeibull Distribution
11 isThe
General
and
fit the distributions of
1 surge water
1 level
15 In conclusion,
is:
p(u) =the storm|u|
h.
< · 12.4
between
peaks
of
tide
and
surge.
Figure
8
shows
that
the
sim12 peaks and durations of these selected surgeh(t )residuals,
respectively.
In this research, all (4)
12.4
2
ulated tide level from the sinusoidal function represents the  hs (t )  ha (t )  h0
13 tide
parameters
of distributions are estimated16by here
theh0 Maximum
In conclusion,
the storm surgeMethod.
water level is
9 Likelihood
is mean
sea level.
well.
17
14
2π
hHW + hLW 18 The characteristics of the high tide level (hHW) at Hook of Holland can be captured in a
hHW − hLW
= hs (t) + ha (t) + h0 ,
(4)
· sin(
(t + u)) +
(2)
a (t) =
19 Normal
Distribution
which is
is estimated
by one
year data of high astronomical
levels
15 hThe
semi-diurnal
astronomical
tide
ofh(t)
Holland,
almost
symmetric,
and cantide
2
12.4
2 in Hook
16 therefore
be approximated
a sinusoidal-curve
andh0modeled
a periodical fluctuation
As a consequence
of the assumedby
independency
of the
where
is mean sea as
level.
The characteristics of the high tide level (hHW ) at Hook
tide and the wind induced storm surge, the time shift between peaks u fits a uniform probability density function.
of Holland can be captured in a normal distribution which 10
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
9
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
H. Zhong
et al.: Aanalysis
joint probability
using a 1-D hydrodynamic
derived from
the harmonic
of waterapproach
level observations
(see Fig 9). Inmodel
Fig.
ow tide level (hLW) is approximately a linear function of hHW.
derived from the
harmonic analysis of water level observations (see Fig3000
9). In Fig.
1
low tide level (hLW) is approximately a linear function of hHW.
Gaussian Copula simulation
2500
0.9
0.8
0.6
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.2
0.3
0.1
0.2
0
0.1
Empirical probability
Normal distribution fitted
Empirical probability
Normal distribution fitted
3
0.9
0.7
Observations
Meuse flow (m /s)
Cumulative
probability
Cumulative
probability
0.81
1847
2000
1500
1000
500
0
0
1000
2000
3000
4000
5000
6000
Rhine flow (m3/s)
Low astronomical
tide(m
level
(m MSL)
Low astronomical
tide level
MSL)
1
2
Fig. 11. Results from a graphical based goodness fit of the Gaussian Copula simul
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Fig. 11. Results from a graphical based goodness fit of the Gaussian
3
High astronomical tide level in Hook of Holland (m MSL)
copula simulation.
0
0.7 tide
0.8 level
0.9
1
1.1by 1.2
1.3
1.5
1.6(5) computes the exceedance probability of a certain Rotterdam water
4 1.4Equation
Fig. 9. High
fitted
the
Normal
Distribution
High level
astronomical
in Hook ofdistribution.
Holland (m MSL)
Fig. 9. High tide
fitted tide
by level
the normal
5 in one year for Category I. The Peak over Threshold (POT) method (1.0 m for
Fig. 09. High tide level fitted by the Normal Distribution
Equation
(5) computes
the exceedance
of a cer-surge event
6 surge residual in Fig.3)
is applied
to detect
the windprobability
induced storm
-0.1
∗ in one year for Category I.
0
tain
Rotterdam
water
level
h
R
7 average the annual-average number of events
of 4.38 is chosen. The number o
-0.2
hLW=-0.18*hHW-0.48
-0.1
The
peak
over
threshold
(POT) method
(1.0parameter
m for the peak
8
occurring
in
one
year
fits
a
Poisson
distribution
and the
 is 4.38. The
-0.3
-0.2
surge
residual
in
Fig.
3)
is
applied
to
detect
the wind inhLW=-0.18*hHW-0.48
9
GPD
process
can
be
transformed
to
the
GEV
distribution
for
annual
maxima (show
-0.4
-0.3
duced storm surge events and on average the annual-average
10 appendix). The detailed information is available from Smith (2004). The probabil
-0.5
-0.4
number of events of 4.38 is chosen.
The number of events
-0.6
11
refers to the exceedance
probability
offits
a specific
in one year.and the pahR* distribution
-0.5
occurring
in one year
a Poisson
-0.7
-0.6
-0.8
-0.7
-0.9
-0.8
rameter
The Poisson-GPD process can be transP ( hR* )    I (*)
p ( hs maxλ) pis(T4.38.
s ) p ( hHW ) p (u ) p (Qr , Qm ) dhs max dTs dhHW dudQr dQm
formed
to
the
GEV
distribution for annual maxima (shown
I  1: hR*  hR (hin
, Tappendix).
)
s max
s , hHW , u , QThe
r , Qm
the
detailed
information is available from
0.4
0.6
0.8
1
1.2
1.4
1.6
∗ ) refers to the exceedance
*
-0.9
(2004).
The
probability
p(h
High astronomical tide level (m MSL) I  0 : h  h ( h Smith
R
R 1.6 R
s max , Ts , hHW , u , Qr , Qm )
0.4
0.6
0.8
1
1.2
1.4
∗
probability of a specific hR in one year.
Fig. 10. LinearHigh
relationship
between
astronomical tide level
(m MSL) hHW and hLW
12 where I is an indicator function and hR is calculated from the specific input par
Fig. 10. Linear relationship between
and hthe
LW hydrodynamic model. In a event in Category I, the upstream Rhine
13 hHW
using
Z Z Z Z Z Z
Fig. 10. Linearof
relationship
between h
HW and h
LW .
obability distribution
the associated
normal
upstream
flow
be storm
estimated
by
of the
14 independentcan
al., 1960;)p(T
Jorigny
et al., 2002).
∗
P (hR ) = surge (DantzigIet
(∗)p(h
smax
s )p(hHW )
obability distribution
of theRhine
associated
normalflows.
upstream
can beCopula
estimated
by
ompanying
daily-average
and Meuse
Theflow
Gaussian
function
companying
daily-average
Rhine
and Meuse
flows.
The
Gaussian
Copula
function
nts the dependence
structure
between
Rhine
discharge
Meuse
discharge
15 astronomical
4.2 Q
High
Rhine
flow
and normal
seadT
water
r and
m,m )dhsmax
p(u)p(Q
,Q
dudQr dQm
(5)
rQ
s dhHWlevel
is estimated
by the one
year data
of high
tide
ents
the
dependence
structure
between
Rhine
discharge
Q
and
Meuse
discharge
Q
,
∗
r
m
the marginal
distributions
fit
Lognormal
Distribution
for
Q
and
the
Gamma
r
levels that is derived from the harmonic
water of high
16analysis
The of
events
flows
with
normal
I = 1Rhine
: hR <=
hR (hcoinciding
, u, Q
smax , Ts , hHW
r , Qm ) sea water levels are s
the marginal
fit Fig.
Lognormal
Distribution
for
Q
the This
Gamma
ution
for level
Qm. distributions
Figure 11 (see
indicates
the
Copula
presents
well hthe
observations
9).that
In Fig.
10
the low
tide
level
17Gaussian
Category
IIr ofand
Fig.
focuses
on
this
kind
of combinations which
I =3.
0 : h∗R >category
(h
,
T
,
h
,
u,
Q
,
Q
s HW
r
m)
bution
for (h
QLW
Figure
11 indicates
that
theMeuse
Gaussian
presents
theR smax
m. )daily-average
ence between
Rhine
and
flows.
The
Gaussian
Coupla
is approximately
a linear
function
of hHW
. Copula
18
extreme
water
level inwell
Rotterdam.
It is assumed that the wind induced surge com
dence
between
Rhine and
The
The
probability
distribution
ofdistributions
theMeuse
associated
up- Gaussian
ence structure
asdaily-average
well as the
marginal
isnormal
well
applied
to
simulate
thelevelfunction
where
ICoupla
is anpeak
indicator
and hRresidual
is calculated
from than
the 1.0 m. T
19 flows.
can
be discarded
when
the
of the surge
is lower
dence
structure
as
well
as
the
marginal
distributions
is
well
applied
to
simulate
the
stream
flow
can
be
estimated
by
the
accompanying
dailyspecificexceeding
input the
parameters
using the
hydrodynamic
model. In to be co
m flows for Category I where the very few occurrences
of
Rhine
flows
20
in
this
kind
of
combinations
astronomical
tide
is
the
only
component
3m flows for
3
Category
I and
where
very few
occurrences
of funcRhine flows
exceeding
average
Rhine
The Gaussian
a event
in Category I, the upstream Rhine flow is independent
/s are maximized
at 6000
mMeuse
/s. the flows.
21 incopula
the sea
water level.
m3/s are maximized
at 6000the
m3dependence
/s.
tion represents
structure22
between Rhine disof the storm surge (Dantzig et al., 1960; Jorigny et al., 2002).
charge
discharge
Qmcan
, where
marginal
r and Meuse
companying
low Q
Rhine
and Meuse
flows
be the
assumed
todisbe constant
during
the large scale storm depressions which probably al
23
The
high
Rhine
flows
comeRhine
from
4.2 during
High
flow and normal sea water level
companying
low
Rhine
and
Meuse
flows
can
be
assumed
to
be
constant
the
tributions
fit
log-normal
distribution
for
Q
and
the
gamma
r
urges’ period, which is supposed not to influence
water
in Rotterdam
in flows. The Gumbel copula is applied to desc
24 the
about
thelevels
associated
high Meuse
distribution
Qm . Figure
that
Gaussian
surges’ period,
which for
is supposed
not11toindicates
influence
thethewater
levels in Rotterdam in
calculation.
25 dependency, as The
it exhibits
stronger
in with
the positive
tail. The as
events ofahigh
Rhine dependency
flows coinciding
normal sea
calculation.copula presents well the dependence between daily-average
26
Meuse
flows
are
selected
at
the
same
day
when
the
Rhine
peaks
occur.
A Genera
water
levels
are
shown
in
Category
II
of
Fig.
3.
This
cateRhine and Meuse flows. The Gaussian copula dependence
gory
focuses
on
this
kind
of
combinations
which
result
in
ex27 isDistribution
structure as well as the marginal distributions
well applied and a Lognormal Distribution fit the selected Rhine and Meus
treme water levels in Rotterdam. It is assumed that the wind
28 I where
respectively.
to simulate the upstream flows for Category
the few
occurrences of Rhine flows exceeding 6000 m3 s−1 are maximized at 6000 m3 s−1 .
The accompanying low Rhine and Meuse flows can be assumed to be constant during the storm surge period, which
is not supposed to influence the water levels in Rotterdam in
the model calculation.
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
induced surge component can be discarded when the peak
level of the surge residual is lower than 1.0 m. Therefore, in
this kind of combination the astronomical tide is the only
component to be considered in the sea water level.
The high Rhine flows come from large scale storm de1111probably also bring about the associated
pressions which
high Meuse flows. The Gumbel copula is applied to describe
this dependency, as it exhibits a stronger dependency in the
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
6 The Chi-square (  2 ) test is used to determine the goodness of fit between observ
7 with expected values derived from the Gumbel copula. The calculated value of 
27.8
is far
below
the critical
value ofusing
61 for
47hydrodynamic
degrees of freedom
H. 8Zhong
et al.:
A joint
probability
approach
a 1-D
model at the sign
9 level of 0.5%. In addition, the good fit is shown in Fig. 12.
1848
positive tail. The associated Meuse flows are selected from
the same day when the Rhine peaks occur. A generalized
Pareto distribution and a log-normal distribution fit the selected Rhine and Meuse flows, respectively.
FQr, Qm (Qr, Qm ) = Cα (u, v)
1
= exp{−[(− ln u)α ] + (− ln v)α ] α }
u = Fr (Qr )
v = Fm (Qm )
1
α=
,
1−τ
(6)
10
11
Fig. 12. Results from a graphical based goodness fit of the Gumbel Copula simula
Fig. 12. Results from a graphical based goodness fit of the Gumbel
where the relationship between the Gumbel
12 copula paramcopula simulation.
eter α and Kendall’s tau τ is also shown. 13
α is estimated
as
High Rhine and Meuse flow curves can be generated by the design hydrographs
1.7158; Fr , is the marginal distribution of 14
high Rhine
flow;
et al., 2002a; Parmet et al., 2002b) multiplied by the ratio between the generated
Fm , is the marginal distribution of the associated Meuse
15 and the design peak
values.
Equation
(8) shows the exceedance probability of a certain
flow; FQr, Qm (Qr, Qm ) is the joint cumulative probability.
16
Rotterdam
water
level h∗R in one year for Category III.
2
The Chi-square (χ ) test is used to determine the goodness
Equation
of a certain Rotterdam water lev
17 values
Z Z Z Z Zprobability
Z
of fit between observed data with expected
derived(7) shows the exceedance
9
2
∗
18
one
year
for
Category
II.
from the Gumbel copula. The calculated value of χ being
P (hR ) =
·
I (∗)p(hsmax )p(hHW )p(Ts )
70
27.8 is far below the critical value of 61 for 47 degrees
of
P ( h* )   I (*) p ( hHWp(u)p(Q
) p (Qr , Qm, )Q
dhHW
dQr dQdh
m HW dTs dudQr dQm
(8)
r
m )dhsmax
freedom at the significance level of 0.5 %. In addition, the
∗
good fit is shown in Fig. 12.
1 : hR <= hR (hsmax , hHW , Ts , u, Qr , Qm )
I  1: h*  hR I(h=
HW , Qr , Qm )
High Rhine and Meuse flow curves can be generated Rby
I = 0 : h∗ > hR (hsmax , hHW , Ts , u, Qr , Qm )
 0 : hR*  hR (hHW , Qr , RQm )
the design hydrographs (Parmet et al., 2002a, b) Imultiplied
by the ratio between the generated values and the design peak
values.
5 Monte Carlo simulation
Equation (7) shows the exceedance probability of a certain
Rotterdam water level h∗R in one year for Category II.
A large number of boundary stochastic scenarios need to be
generated based on the joint probability distribution for each
Z Z Z
category. Then the 1-D model can run using these scenarios
P (h∗R ) =
I (∗)p(hHW )p(Qr , Qm )dhHW dQr dQm
and the outputs are the same number of peak water levels at
locations of interest in the Lower Rhine Delta. The resulting
I = 1 : h∗R <= hR (hHW , Qr , Qm )
(7)
series of peak water levels as well as the accompanying ocI = 0 : h∗R > hR (hH W, Qr , Qm )
currence probabilities can be transformed into the high water
level frequency curves in the delta. Due to the important eco4.3 Storm surge and high Rhine flow
nomic role only the results at Rotterdam and Dordrecht are
The very limited number of observations of the joint high
presented.
surge residual and high Rhine flow events in Category III is
Importance sampling is applied to reduce the number of
not appropriate for estimating the joint probability distribusamples in the Monte Carlo simulation but still get suftion. A rather simple method is introduced. In the historical
ficiently accurate estimations (Glynn and Iglehart, 1989;
observations, 9 events occurred in Category III and therefore
Roscoe and Diermanse, 2011). In the Monte Carlo simulait is assumed that the occurrence probability of a combinations, the exceedance probability Pf of a specific h∗R is simtion event in Category III is 9/70 per year. The marginal
ply taken to be nf /n, where nf is the number of samples
distributions of the peak surge residual, the surge duration,
which lead to hR ≥ h∗R , and n is the total number of genthe astronomical tide and the high Rhine flow are the same
erated samples. In importance sampling, the number of samas Category I and II, respectively. In a combination event,
ples which lead to hR ≥ h∗R increases largely because boundthe high peak surge residual is assumed to be independent to
ary inputs are not generated from their original probability
high Rhine flow (Dantzig et al., 1960; Jorigny et al., 2002).
distributions, but from alternative distributions which focus
on exceedance of the critical water level at Rotterdam h∗R . In
this study, we have used normal distributions for the most important input variables hsmax , Ts , Qr (high Rhine flow) and
Qm (high Meuse flow), centered around the values that lead
to h∗R . Note that for different h∗R , the corresponding normal
R
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
H. Zhong et al.: A joint probability approach using a 1-D hydrodynamic model
distributions are different in order to locate around the area
leading to hR ≥ h∗R . The changes in distributions need to be
compensated for.
Pf (h∗R ) =
n
1X
f
I (∗)
n i=1
g
(9)
I = 1 : h∗R <= hR
I = 0 : h∗R > hR ,
where Pf (h∗R ) is the exceedance probability of a specific
Rotterdam water level h∗R ; n is the total number of samples;
I is an indication function inside which the input parameters
are generated from the distributions g, f stands for the original probability density distributions of related variables and
g is the corresponding normal density distributions.
In the importance sampling method only input parameters
referring to hsmax , Ts , Qr (high Rhine flow) and Qm (high
Meuse flow) applied the new normal distributions instead of
the original distributions. The other input parameters were
sampled from their original probability distributions. Generally there are no upper bounds for these normal distributions.
To estimate the high water level frequencies and
exceedance probabilities of the design water levels,
10 000 boundary events were generated with the importance
sampling method and the model output were 10 000 maximum water levels at Rotterdam and Dordrecht for each of the
three categories. In order to test whether 10 000 simulations
was enough to get consistent results, another two groups of
10 000 simulations were generated to compare the difference.
These differences were found negligible.
To estimate the effect of climate change on the high water level frequencies, an increase of the peak discharge of the
Rhine as well as an increase of the mean sea level in the scenario of 2050 is studied. The mean sea level rise is assumed
to be 0.35 m (van den Hurk et al., 2006) and the peak Rhine
discharge increases by 10 % in reference to the year 2000 (Jacobs et al., 2000). In a second set of the Monte Carlo simulations, the input boundary conditions valid for scenario 2050
are generated by simply re-scaling the present boundary variables.
6
6.1
Results and discussion
Exceedance probability of the present design water
level
The design water level is crucial for the design, construction and maintenance of the flood defense system. According to the Dutch law, the design water level in Rotterdam is
regarded as the water level with the exceedance frequency
of 1/10 000; the design water level in Dordrecht is with the
exceedance frequency of 1/2000. The present design water
level is 3.6 m m.s.l. for Rotterdam and 3.0 m m.s.l. for Dordrecht (Ministerie van Verkeer and Waterstaat, 2007).
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
1849
The exceedance probabilities of the design water levels estimated in our method are a little lower than the official values based on Hydra-B (Ministerie van Verkeer and Waterstaat, 2007). The results are listed in Table 2.
From Table 2 the exceedance probability of the design
water level is 0 for Rotterdam in Category I. The result indicates that in current conditions the delta area can be protected from storm surges until the year of 2050. This agrees
with the design standard of the Maeslant Barrier, one key hydraulic structure at the mouth of the delta. Note that the result
depends on the assumption that the operation of the storm
surge barriers and dams at the mouth of the delta does not
fail during the storm surge events. It is believed that taking
the failure of the operation into consideration could result in
a higher exceedance probability in Rotterdam and Dordrecht
for Category I. Detailed research on the failure probabilities
of the hydraulic structures at the mouth is urgently required.
The exceedance probability in Category II is also 0 in
Rotterdam for both the present and the year 2050. The exceedance probability in Dordrecht is also very low. The results indicate that high fluvial flow (Rhine and Meuse flow)
has very limited influence on the extreme water levels of the
downstream of the delta. Instead, the extreme upstream fluvial flow could easily result in the breaching or overflowing
in the Dutch Upper Rhine Delta, which agrees with the nearcatastrophic floods of 1993 and 1995 (Engel, 1997).
The exceedance probability in Category III is still far
lower than the official standard 10−4 in Rotterdam for the
present and the year 2050, while the exceedance probability
in Dordrecht is higher than the official standard 1/2000 for
the year of 2050. Moreover, the sum of the exceedance probability in three categories shows that the Dutch Lower Rhine
Delta complies with the required norm for flood safety, except for the Dordrecht in future climate scenario of 2050.
The results show that the exceedance probability of the design water level is much higher in Category III than in Category I and Category II. It indicates that the combinated events
of the storm surge and the high Rhine flow become the main
source of floods in the Lower Rhine Delta. The high water
level frequency curves derived from these combinated events
can be drawn in Rotterdam and Dordrecht.
6.2
High water level frequency curves
The high water level frequency curves derived from the combination events of storm surges coinciding with high Rhine
flows are shown in Fig. 13.
The future high water level frequency curves (the dash
lines in Fig. 13) are about 0.2 to 0.4 m higher than the present
curves (the solid lines in Fig. 13) in Rotterdam and Dordrecht. It indicates that climate change will lead to more extreme events which increase the high water level frequency
in the future. The differences between the present and future high water level frequency curves are quantified in order to provide an indication for the further adaptation of the
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
1850
H. Zhong et al.: A joint probability approach using a 1-D hydrodynamic model
Table 2. Exceedance frequency of the design water level.
Annual exceedance frequency
Year
Category II
Category III
Rotterdam (3.6 m m.s.l.)
2010
2050
0
0
0
0
2.1 × 10−8
2.0 × 10−6
Dordrecht (3.0 m m.s.l.)
2010
2050
10−8
2.1 × 10−6
0
10−8
1.7 × 10−5
7.2 × 10−4
b
a
3.5
4
Present
2050
3.4
Water level in Dordrecht (m MSL)
Water level in Rotterdam (m MSL)
Category I
3.8
3.6
3.4
3.2
3
Present
2050
3.3
3.2
3.1
3
2.9
2.8
2.7
2.8
0
10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
-2
10
-4
10
-6
10
-8
10
-10
10
2.6
0
10
-2
10
-4
10
-6
10
-8
10
Exceedance probaiblity
Exceedance probability
Fig. 13. The high water level frequency curves due to the combination events of storm
surges
highwater
Rhine flows
III) incurves
a. Rotterdam.
Dordrecht
Fig. 13.
Theandhigh
level(category
frequency
due tob. the
combina-
tion
events
storm
high
flows
(category
III) 0.2
in to
The
future
high of
water
level surges
frequencyand
curves
(theRhine
dash lines
in Fig.
13) are about
0.4
higher than the
curves (the solid lines in Fig. 13) in Rotterdam and
(a)mRotterdam.
(b)present
Dordrecht.
Dordrecht. It indicates that climate change will lead to more extreme events which
increase the high water level frequency in the future. The differences between the present
and future high water level frequency curves are quantified in order to provide an
indication for the further adaptation of the operational water management system and the
operational water management system and the flood defense
flood defense system.
system.
The present high water level frequency curve in Rotterdam shows that the exceedance
The present high water level frequency curve in Rotterdam
probability of 3.0 m MSL is below 10-4. It is attributed by the controlled structures as
indicated
Fig.the
2 and
the operational
water management
behind
then.
shows inthat
exceedance
probability
of 3.0 m
m.s.l.
is During
belowthe
combination
of the storm surge and high Rhine flow, the Maeslant Storm Surge
−4 . It events
10
is
attributed
by
the
controlled
structures
as
indicated
Barrier and the Haringvliet sluices are closed in order to prevent the sea water from
in Fig. 2into
andthethedelta,
operational
water
management
then.and
propagating
and after the
closure
the water level behind
in Rotterdam
Dordrecht will increase due to the Rhine and Meuse flows coming into the delta. The
During
the
combination
events
of
the
storm
surge
and
high
mouth of the delta is open again to discharge fluvial water into the sea after the storm.
Rhine
the
Maeslant
Storm
Surge
and
the HarThe
closingflow,
decision
level
is 3.0 m MSL
in Rotterdam
andBarrier
2.7 m MSL
in Dordrecht.
present design water level in Rotterdam and in Dordrecht.
The high water level frequency complies with the required
norm for safety in the present. However, the threats of high
water levels exceeding the design water level are still exposed
to both cities at a low probability mainly due to the combination events of storm surges and high Rhine flows.
The new method enables assessment of high water level
frequencies in a changing environment with associated effects from climate change and operation of the infrastructures. In the future climate change will lead to more extreme events and increase the high water level frequency in
the Lower Rhine Delta. Moreover, the future development
of local economy and urbanization will increase the flood
induced damage when floods occur (te Linde et al., 2011).
Therefore the adaption measures are urged. The adaptation of
the present operational water management system was proposed by van Overloop et al. (2010). The method in the article, based on the Model Predictive Control method (in brief
MPC), can be applied to investigate the effect of the adaption measure on reducing the high water level frequency in
the delta.
In future research, the failure probability of the operation
of these controllable hydraulic structures should be further
incorporated. In addition, the statistical uncertainty in the
joint probability approach needs to be investigated.
ingvliet sluices are closed in order to prevent the sea water
To avoid extreme high water levels from the events in Category III, the construction of
from
propagating
into the
andoperational
after the
closure
the wanew
structures
needs exploration
and delta,
the present
water
management
system
requires
adaptation
in the future. and Dordrecht will increase due to the
ter level
in Rotterdam
7.Rhine
Conclusion
and future
research
and Meuse
flows
coming into the delta. The mouth of
This
the application
the joint probability
approach
thepaper
deltapresents
is open
again to ofdischarge
fluvial sampling
water into
thecoupled
sea
with a simplified 1-D hydrodynamic model to assess the exceedance probability of the
after
the
storm.
The
closing
decision
level
is
3.0
m
m.s.l.
in
present design water level in Rotterdam and in Dordrecht. The high water level frequency
Rotterdam and 2.7 m m.s.l. in Dordrecht.
To avoid extreme high water levels from the events in Cat-17
egory III, the construction of new structures needs exploration and the present operational water management system
requires adaptation in the future.
Appendix A
Probability distributions
1. hHW fits the normal distribution.
2
(h
−u)
1
− HW 2
2σ
f (hHW ; u, σ 2 ) = √
e
(A1)
2π σ 2
Here the mean u is 1.0861 m and the standard deviation
σ is 0.1790 m.
2. hsmax fits the generalized Pareto distribution (GPD).
7
Conclusion and future research
This paper presents the application of the joint probability
sampling approach coupled with a simplified 1-D hydrodynamic model to assess the exceedance probability of the
Nat. Hazards Earth Syst. Sci., 13, 1841–1852, 2013
hsmax − µ −( ξ1 +1)
1
(1 + ξ
)
(A2)
σ
σ
In this equation the shape parameter ξ is −0.0677; the
scale parameter σ is 0.3140; the location parameter u is
1.0 m.
f (hsmax ) =
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
H. Zhong et al.: A joint probability approach using a 1-D hydrodynamic model
3. Ts fits the Weibull distribution (two parameters).
k Ts k−1 −( Ts )k
( ) e λ
(A3)
λ λ
In the equation, Ts > 0, k is the shape parameter,
2.5237; λ is the scale parameter, 38.0887 h.
f (Ts ) =
4. u fits the Uniform distribution.
1
f (u) = 0 |u| > · 12.4 h
2
1
1
|u| < · 12.4 h
f (u) =
12.4
2
1
√ e
σ · Qr · 2π
− (ln Qr −µ)
2
(A5)
2·σ
6. Daily Meuse flow Qm fits the gamma distribution
1
− Qθm
· Qκ−1
m ·e
k
0(κ)θ
0(k) = (k − 1)
(A6)
In the equation, Qm is Meuse flow; κ is the shape parameter; θis the scale parameter; and their values are
1.2924, 329.14 m3 s−1 , respectively.
7. High Rhine flow Qr fits the generalized Pareto distribution.
1
Qr − µ (− ξ1 )−1
f (Qr ) = (1 + ξ
(A7)
)
σ
σ
In the equation, Qr is Rhine flow; ξ is the shape parameter; σ is the scale parameter; u is the location parameter;
and the parameters’ values are −0.0667, 1629.7 m3 s−1
and 6000 m3 s−1 , respectively.
8. The associated Meuse flow Qm fits the log-normal distribution.
1
− (ln Qm2−µ)
2·σ
f (Qm ) =
(A8)
√ e
σ · Qm · 2π
In the equation, Qm is Meuse flow; µ is the mean value
6.8667; σ is the stand deviation value, 0.3752.
9. hsmax the Generalized Extreme Value distribution transformed from the Poisson-GPD process.
(A9)
f (hsmax ) =
hsmax − µ
1
(1 + ξ(
))
σ
σ
−( ξ1 +1)
e
1
−µ − ξ
(−[1+ξ( hsmax
)]
σ
)
In this equation the shape parameterξ is −0.0677; the
scale parameter σ is 0.2841; the location parameter u is
1.4417 m. hsmax is larger than 1 m.
www.nat-hazards-earth-syst-sci.net/13/1841/2013/
Edited by: F. Castelli
Reviewed by: J. Beckers and one anonymous referee
References
In the equation, Qr is the associated normal Rhine flow,
µ is the mean value, 7.6808; σ is the stand deviation
value, 0.4782.
f (Qm ) =
Acknowledgements. We appreciate the professional comments
from Joost Beckers and an anonymous reviewer on the draft paper.
In addition, we would like to thank our colleague Mariette van
Tilburg for improving the draft writing.
(A4)
5. Daily Rhine flow Qr fits the log-normal distribution.
f (Qr ) =
1851
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