FLOOD RISK CALCULATED WITH DIFFERENT RISK MEASURES
S.N.Jonkman
1
, P.H.A.J.M. van Gelder
2
, J.K. Vrijling
3
Abstract: In this paper it is investigated whether flood risks of an existing area in the Netherlands can be determined with different risk measures. An overview is given of risk measures used in the field of quantitative risk analysis.
Furthermore, a case study is described, in which a flood risk analysis is performed. The results of these risk calculations are compared to existing standards and the risks of other activities in the Netherlands.
1 INTRODUCTION
Large parts of the Netherlands lie below sea level and are threatened by river floods. Without the protection of dunes, dikes and hydraulic structures more than half of the country would be almost permanently flooded, see figure 1.
The Netherlands therefore has a long history of floods. The first well-documented flood took place in the year 1287 and flooded large parts of the North and Southwest of the country. About
50.000 persons died in this tragic event. The last disastrous flood occurred in 1953. A storm surge from the North Sea inundated large parts of the
Southwest of the country. Apart from immense economic damage, 1800 persons drowned during this disaster.
Figure 1: The Netherlands without flood protection (the dark area would be flooded due to sea and river floods)
1 Hydraulic engineer, Road and Hydraulic Engineering division,, vd Burghweg 1, NL-2628 CS Delft, the Netherlands, s.n.jonkman@dww.rws.minvenw.nl, fax +31152518555, (corresponding author)
2 Assistant professor, Delft University of Technology, Faculty of Civil engineering, Stevinweg 1, NL-
2600 GA Delft, the Netherlands, fax +31152785124, p.vangelder@ct.tudelft.nl
3 Professor, Delft University of Technology, Faculty of Civil engineering, Stevinweg 1, NL-2600 GA
Delft, the Netherlands, fax +31152785124, j.k.vrijling@ct.tudelft.nl
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
1

Up until 1953 dikes were constructed to withhold the highest known water level.
After the 1953 flood the Delta Committee was installed to investigate the possibilities for a new safety approach. In an econometric analysis the optimal safety level was determined for the largest flood prone area, Central Holland. This work laid the foundations for the new safety approach, in which dikes are dimensioned based on a design water level with a certain probability of occurrence.
Since the work of the Delta Committee, performed in the late 1950’s, some developments have resulted in an increased awareness of flood risks. Firstly, the economic value of the country has increased rapidly. It can be questioned if the protection standards as derived by the Delta Committee still provide enough protection when the current economic value is considered. Moreover, in 1993 and
1995 extremely high river discharges occurred in the river system, which made large scale evacuations necessary. Also other recent disasters, such as the crash of the El
Al Boeing in Amsterdam in 1993 and the explosion of the fire works storage in
Enschede in 2000, have lead to increasing awareness of risks that face society and it’s citizens. In the future climate change, the accompanying sea level rise and higher river discharges will lead to an increase of flood hazards for the Netherlands.
This quick overview of developments illustrates the necessity of a new societal debate, in which the level of flood protection should be re-considered. In this debate several issues are of importance, of social, economic and technical nature. From a technical background a contribution can be given to the discussion by providing more insight in the current flood risks. For a balanced decision on flood protection it is needed to consider both the probability of a flood and it’s consequences in connection to each other.
For a clearer insight in flood protection it should firstly be determined what expressions are available to quantify risks. It should also be investigated how flood risks can be determined and how the concept of flood risk can be applied in decision making. Finally the outcomes of the flood risk calculations will be analysed by comparing the calculated risk levels with the risks and standards in other sectors.
In this paper the results of research conducted by the Road and Hydraulic
Engineering Division and Delft University of Technology are presented. In section 2 an overview is given of risk measures used in the field of quantitative risk analysis.
Section 3 gives a description of the method of flood risk analysis as is performed in a case study. With the described risk measures the flood risk is calculated for an existing polder in the Netherlands. The results of these calculations are compared to current standards and the risks of other activities in the Netherlands in section 4. The conclusions from this study are summarized in section 5.
2 AN OVERVIEW OF QUANTATIVE RISK MEASURES
In the study of flood risk in the Netherlands it was found that a comprehensive overview of methods to quantify and limit risks arising from different sources is still missing in literature. Therefore a study of risk literature was performed with the aim
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
2

to give an overview of quantitative risk measures used in quantitative risk analysis
(QRA) to express the risks of various activities. A risk measure is a mathematical function of the probability of an event and the consequences of that event. Measures that deal with the risk qualitatively were not considered. This paper focuses mainly on risk measures that consider loss of life and economic damage as a consequence.
For a more complete overview of the study of risk measures reference is made to
(Jonkman, 2003). For every risk measure first the characteristics and mathematical expression are described. Consequently the field of application and the standards used are given.
Individual risk measures
The first measure is the individual risk (IR) as used by the Dutch Ministry of
Housing, Spatial Planning and Environment (VROM). It is defined as the probability that an average unprotected person permanently present at that point location, would get killed due to an accident at the hazardous activity (Bottelberghs, 2000).
IR
 
(1)
Where:
P f
P d|f probability of failure (yr
-1
) probability of dying of the individual in the case of failure, assuming the permanent unprotected presence of the individual.
A slightly different definition, in which the actual presence of the individual is considered, is used by the Dutch Technical Advisory Committee on Water Defences
(TAW, 1985) and by Bohenblust (1998) to describe the actual personal risk. An overview of other measurements to express the individual risk is given by Bedford and Cooke (2001), examples are the “activity specific hourly mortality rate” and the
“death per unit activity”.
Individual risk measures: fields of application and standards
The IR is a property of the place and as such useful in spatial planning. Locations with equal individual risk levels can be shown on a map with so-called risk contours.
The measure of individual risk is used in the Netherlands to limit the risks nearby hazardous installations and transport routes. The Dutch Ministry of Housing, Spatial planning and Environment (VROM) has set the following standard for populated areas (Bottelberghs, 2000).
IR

10
-6
(2)
Risks lower than 10
-6
per year should always be reduced to a level as low as is reasonably achievable (ALARA). This standard is set for more or less involuntary imposed risks related to the siting of hazardous activities. The method of TAW gives the opportunity to limit a broader set of risks ranging from voluntary activities, such as mountaineering, to more involuntary risks, such as those of hazardous installations. The following standard is proposed by TAW:
IR 10
-4
(3)
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
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In this expression the value of the policy factor

varies according to the degree of voluntariness of the activity and with the benefit perceived, see table 1.
Table 1: Policy factor

as function of voluntariness and benefit of the activity
(Vrijling, 1998).
 Voluntariness Benefit Example
100 Completely voluntary
Direct benefit Mountaineering
10
1
0.1
0.01 voluntary
Neutral
Involuntary
Involuntary
Direct benefit
Direct benefit
Some benefit
No benefit
Motorbiking
Car driving
Factory
Living nearby an LPG station
Societal risk measures
Societal risk is defined by Ichem (1985) as “the relationship between frequency and the number of people suffering from a specified level of harm in a given population from the realisation of specified hazards”. Where individual risk gives the probability of dying on a certain location, the societal risk gives a number for a whole area no matter where precisely within that area the harm will occur. The difference is shown in figure 2.
A Persons B
Risk source
IR
IR’
Risk source
IR
IR’
IR
A
= IR
B
SR
A
< SR
B
Figure 2: The difference between individual and societal risk. Both situations have equal individual risk levels (shown by IR’ and IR). Because of the larger population density of situation B, B has a larger societal risk (based on (Stallen, 1996))
The basis of the calculation of societal risk is formed by the probability density function (pdf) of the yearly number of fatalities. From the pdf an FN curve can be derived, which shows the probability of exceedance as a function of the number of fatalities, on a double logarithmic scale.
1

F
N
( x )

P ( N
 x )


 x f
N
( x )
 dx ( 4 )
Where: f n
(x)
F
N
(x) the probability density function (pdf) of the number of fatalities per year probability distribution function of the number of fatalities per year, signifying the probability of fewer
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
4

than x fatalities per year
4
A simple measure for societal risk is the expected value of the number of fatalities per year, E(N), in literature often referred to as the Potential Loss of Life (PLL):
E(N)



0 x
 f
N
(x)
 dx ( 5 )
The British Health and Safety Executive (HSE) has defined a risk integral as a measure for societal risk (Carter, 1995):
RI



0 x

( 1

F
N
(x))
 dx ( 6 )
By Vrijling and van Gelder (1997) it is mathematically proven that the RI can be expressed in two characteristics of the pdf of the number of fatalities, the expected value E(N) and the standard deviation

(N):
RI

1
2
( E 2 ( N )
 
2 ( N )) ( 7 )
From the pdf also the expected value of the number of deaths, E(N) (or potential loss of life (PLL)) and the standard deviation,

(N), can be derived. Combining these two elements the total risk (TR) has been proposed by Vrijling et al. (1998) as a measure for societal risk.
TR

E ( N )
 k
 
( N ) ( 8 )
This measure takes into account the risk aversion towards rare accidents with large numbers of fatalities by multiplying a risk aversion index k with the standard deviation. Although the formulae may look different, the societal risk measures described here are all based on the pdf of the number of fatalities.
Societal risk measures: fields of application and standards
In several countries the risks of various hazardous activities are limited by a FN criterion line. These standards can be described with the following general formula:
1F x

C
(9) x n
Where: n
C steepness of the limit line constant that determines the position of the limit line.
A standard with a steepness of n=1 is called risk neutral. If the steepness n=2, the standard is called risk averse. In this case larger accidents are weighted more heavily and accepted with a relatively lower probability. In the Netherlands the VROM standard is applied for hazardous installations, with constants C = 10
-3
and n = 2.
The expected number of fatalities is used in the regulation of the risks of dams.
Standards have been proposed by British Columbia Hydro (Bowles, 1999) and the
United States Bureau of Reclamation (1997).
4 Note that a different convention can be found in other published works, in which the symbol F
(or F(x)) signifies the probability of “x or more” fatalities per year.
N
(x)
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
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BC Hydro : E ( N )

10

3 ( fat / yr ) USBR : E ( N )

10

2 fat / yr ( 10 )
The risk integral is used in the United Kingdom as a part of HSE’s examination of proposals for siting new hazardous installations. However, no limiting value is yet attached to it.
As a standard a limiting value for the Total risk of TR <
 .
100 has been proposed
(Vrijling, 1998), taking into account the already described policy factor

.
Economic risk measures
The FD curve displays the probability of exceedance as a function of the economic damage. The FD curve and the expected value of the economic damage can be derived from the pdf of the economic damage ( f
D
(x ))
1

F
D
( x )

P ( D
 x )


 x f
D
( x )
 dx ( 11 )



D
( )

(12)
0

Where:
F
D
(x) the probability distribution function of the economic damage
E(D) expected value of the economic damage
The problem of the acceptable level of risk can also be formulated as an economic decision problem (van Danzig, 1956). In the method of economic optimisation the total costs in a system ( C tot
) are determined by summing up the investments ( I ) for a safer system and the expected value of the economic damage. In the optimal economic situation the total costs in the system are minimised: min( C tot
)
 min( I

( )) (13)
With this criterion the optimal probability of failure in a system can be determined, when investments ( I ) and the expected economic damage ( E(D) ) are a function of the probability of failure.
Economic risk measures: fields of application and standards
The expected value of the economic damage is used as a part of cost benefit analysis of flood prevention measures in the UK (Parker, 1987). The benefits of a measure are determined by calculating the expected value of the economic damage before and after the measure has been taken. The difference between these two values results in the benefits, which can be weighed against the costs of the measures. A limit for the expected economic damage per year for dams has been proposed by BC Hydro (Hydropower & Dams, 1998), in which the financial risks for one dam should not exceed:
( )

$10.000

(14)
The method of economic optimisation has originally been applied by van Danzig
(1956) to determine the optimal level of flood protection (i.e. dike height) for Central
Holland (this polder forms the economic centre of the Netherlands). The total
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
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investments in dike heightening ( I tot
) are determined by the initial costs ( I
0
) and the variable costs (
I’
). The dike is heightened X metres, the difference between the new dike height ( h ) and the current dike height ( h
0
).
I tot
  
' AND X

-
0
(15)
The expected value of the economic damage can be calculated from the probability of flooding ( P b
), the damage caused by the flood ( D ), and the discount rate ( r’
). The flood level h is modelled as exponentially distributed with parameters
A and B.
P b
 e
 h

A
B AND E ( D )

P b

D
( 16 ) r '
Now the total costs are formulated as the sum of investments and the expected value of the economic damage. The economic optimum is found by minimising the total costs. By taking the derivative of the total costs and the dike height, the optimal flooding probability ( P b,opt
) and the optimal dike height ( h opt
) are found.
P

 
D
'
AND h opt

-
 ln( P ) (17)
Overview of risk measures: fields of application and standards
In the table 2 the most important characteristics of the described risk measures have been shown: the mathematical expression, the field of application, the standards and a reference to literature.
Table 2: Overview of risk measures
Risk measure Mathematical
Individual risk
VROM expression
IR

P f
.
P d | f
Individual risk
TAW
FN curve
IR

P f
( )

.
P d | f

 x f
N
( )
 dx
Field of application
VROM: hazardous installations/ transport routes
Case studies, floods
Hazardous installations
Standard
<10 -6
<

.10
-4
Reference
Bottelberghs,
2000
TAW, 1985
VROM:
1-F
N
(x)<10 -3 /x 2
Stallen et al.,
1996
Expected value of the
( ) number of fatalities/year
Risk Integral
Total risk
RI

TR




N
( )
 dx
0



0
( 1

F
N
( x ))
 x
 dx
E ( N )
 k .

( N )
FD Curve
Dams (USA,
( 14 )
HSE (UK): land use planning
Different case studies
-
Expected economic damage
Economic
Optimisation
1

F
D
( x )


 x f
D
( x )
 dx
E ( D )



0 x
 f
D
( x )
 dx
Dam safety min( C tot
)
 min( I

E ( D )) Flood protection
<10 -2 USBR USBR, 1997
<10 -3 BC Hydro Bowles, 1999
-
<

100
-
E(D) <
$ 10.000
Economic optimum
Carter, 1995
Vrijling et al.,1998
-
Parker, 1987 van Danzig,
1956
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
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3 PILOT CASE FLOOD RISK
The project Pilot Case Flood Risk (PICASO) has been performed to investigate whether current knowledge and the accompanying models are suitable for a flood risk analysis in the Netherlands. A full description of the project and it’s results is given in (van Manen, 2001). A case study has been performed for the "Betuwe,
Tieler- en Culemborger Waarden" (BTCW). This area is situated in the eastern part of the Netherlands and measures about 80 by 25 kilometres (figure 3). The polder is inhabited by approximately 360.000 persons and has an estimated economic value of about 40 billion Euro. The polder is threatened by river floods, in the north by the
Lek river and in the south by the Waal river. Dikes that surround the whole polder
(or dike-ring as it is called in the Netherlands) achieve protection against high water.
Figure 3: The Betuwe, Tieler- en Culemborger Waarden area and a picture of the flood protection with dikes
Within the Pilot Case Flood Risk the following steps have been taken:
Selection of breach locations
The number of fatalities and the damage caused by a flood is mainly determined by the location of the initial breach. Therefore the total dike ring is divided in fourteen distinct sections, each will lead to a distinctly different flood pattern. Also multiple breaches were considered.
Probability of flooding
For every section the probability of flooding has been determined with a model that takes into account the different failure modes of the dike (for example overtopping or instability of the dike). In this model both the load on the dike, the hydraulic boundary conditions, and the strength, determined by geometry and geotechnical profile, are considered. This results in an overall probability of flooding of the area of once in a thousand years. An important uncertainty in the assessment of the probability of failure is the remaining strength of the dike after occurrence of the failure mechanism.
Modelling of breach
After that the breach growth has been modelled, based on the geotechnical and geometric characteristics of the dike. The fraction of the river discharge that flows trough the breach varies between 40% and 80%. It has to be noted that breach discharges are thought to be over estimated.
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
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Flood simulation
The flood process was simulated with a two dimensional hydraulic model, which results in water depths and water velocities throughout the flooded area, as a function of time.
Damage assessment
Finally the economic damage and the number of fatalities for every flood scenario are determined with the available damage model (Vrisou van Eck, 2000). The economic damage is mainly determined by the water depth in the area. The total number of fatalities is a function of water depth and rate of rising of the water. It has to be noted that the fatality modelling involves a lot of uncertainties, evacuation has for instance not been modelled in it. The outcomes of the case study are the basis of the calculation of the flood risk for the area with some of the described risk measures. The results of these calculations are presented in the next section.
4 RESULTS: FLOOD RISK CALCULATED WITH DIFFERENT RISK
MEASURES
In this section the results of the risk calculations for the “Betuwe, Tieler- en
Culemborger Waarden”, based on the outcomes of the Pilot Case Flood Risk, will be presented. Also the possible application of the risk measures in decision making about flood risks is briefly discussed.
Individual Risk
The individual risk is calculated with the data from the different flood simulations. For all the flood scenarios the probability of drowning at every location in the polder given a certain flood ( P d|i
) is determined. By multiplication with the probability of the flood ( P i
) and summing up the values for all n defined scenarios, the individual risk for every location ( IR(x,y) ) can be calculated. The calculated individual risk levels for the area are shown in figure 4.
 i n 

1
P P ( , ) i

(18)
Figure 4 : Individual risk per year for the “Betuwe, Tieler- en Culemborger Waarden” polder
The IR of the polder can be compared to standards of VROM (accepting an individual risk of 10
-6
/ year) and TAW. By TAW a

value of 0,1 for flood prone areas has been proposed (TAW, 1985) which results in an acceptable IR of 10
-5
per year. In figure 5 it has been shown in which areas the calculated IR for the polder would not be acceptable by the VROM and the TAW standards (the dark areas exceed the limit)
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
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TAW standard
IR < 10
-5
VROM standard
IR < 10
-6
Figure 5: Individual risk of the Betuwe polder compared to VROM standard and the TAW standard, dark areas exceed the limit
Societal Risk
For every flood scenario the probability of occurrence and the number of fatalities are determined. With this data a probability density function of the number of fatalities has been formed, from which an FN curve can be derived. The FN curve for the flooding of the polder is compared to some other risks in the Netherlands in figure 6. The figure shows that the risks of a flood of the polder are larger than the sum of the risks for airports, hazardous installations and railway yards in the
Netherlands. Apart from the uncertainties in loss of life modelling, it has to be noted that this comparison concerns risks from various sources with different natures.
1.00E-01
1.00E-02
1
1.00E-03
1.00E-04
1.00E-05
1.00E-06
1.00E-07
1.00E-08
1.00E-09 flood
10 100 fatalities (N)
1000 10000 flood risk
100000 installations railway yards airports total
Figure 6: FN curve for the polde r “Betuwe, Tieler- en Culemborger Waarden” compared tot the FN curves of other activities in the Netherlands
From the pdf the expected value of the number of deaths per year and the standard deviation are derived:
E ( N )

2 , 4 ( fat / yr )

( N )

104 , 4 ( fat / yr ) ( 19 )
Also the total risk (TR) is determined. For a value of the risk aversion index k of 3 this leads to a TR value of 315,6 fatalities per year. The TAW criterion limits the total risk to a value of
 .
100, resulting in an acceptable total risk of 10 fatalities per year for a proposed

value of 0,1 (TAW, 1985).
It has to be noted that the fatality modelling involves a lot of uncertainties, evacuation has for instance not been modelled in it. It is found that these uncertainties have a major impact on the magnitude of the calculated individual and societal risks.
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
10

Economic Risk
For every flood scenario the economic damage is determined. With information from the different scenarios a pdf is formed. From this pdf the FD curve (figure 7) and the expected value of the economic damage can be derived. This results in an expected value of the economic damage of 13,2 million Euro per year.
1.00E-02
0
1.00E-03
5 economic damage (Euro)
10 15 20 25
1.00E-04
1.00E-05
1.00E-06
Figure 7: FD curve for the economic damage for the polder (x-as on a non-logarithmic scale)
Also the method of economic optimisation has been applied for area. In figure 8 the investments in dike improvement, the expected value of the economic damage and their sum, the total costs, are shown as a function of the reliability index. This index can be converted to a probability of flooding. The economic optimum is found when the total costs are minimal. The situation occurs for a reliability index of 4, which equals a probability of flooding of 3,16·10 -5
per year (or once in 32.000 year).
The current probability of flooding is estimated at a failure probability of about 10 -3 or a beta of 3,1.
2.50E+09
2.00E+09
1.50E+09
1.00E+09
5.00E+08
0.00E+00
3.0
investments risk total costs
3.2
3.4
3.6
3.8
reliabilty index
4.0
4.2
optimum
4.4
Figure 8: Economic optimisation of the "Betuwe, Tieler-, Culemborger Waarden"
Application of risk measures in decision making
In this paragraph a short analysis of the potential application of some of the risk measures in the decision making about flood risks is given. The measure of individual risk gives the risk for every location within the flood prone area. IR is therefore suitable as instrument in spatial planning. Risks of new housing developments can be assessed with the IR. By determining an acceptable IR level for flood risk a basic level of flood protection can be guaranteed for every citizen.
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
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The effect of risk reducing measures can clearly be shown in an FN curve. It is also a very suitable measure for comparing the risks of various activities. Also the hazards of different flood prone areas can be compared in an FN curve. The method of economic optimisation is a suitable instrument for the determination of the optimal risk reduction strategy. The economic effectiveness of alternative measures can be analysed with the economic optimisation.
It is recommended to apply these three measures in decision making, because they all fill in important aspects of the safety considerations. With the three measures risks are judged on various levels: local (by the individual risk), and for the whole area or country (with the FN curve). Also the most important types of damage, loss of life and economic damage, are considered. Furthermore, the measures provide insight in the effect of risk reducing actions and their cost effectiveness.
5 CONCLUSIONS AND RECOMMENDATIONS
In this paper it is investigated whether flood risks of an existing area in the
Netherlands can be determined with some of the risk measures used in the field of quantitative risk analysis.
Firstly an overview of quantitative risk measures has been made based on a study of literature. From this analysis it can be concluded that societal risk measures can be derived from the probability density function of the number of fatalities.
In the Pilot Case Flood Risk it has been shown that it is possible to conduct a full flood risk analysis with the available knowledge and methodology. The flood risk calculations contain the following steps: selection of breach locations, determination of probability of flooding, modelling of breach, flood simulation, damage assessment. Most important uncertainties in this calculations include the actual strength of the dikes, breach modelling, and loss of life and evacuation modelling.
In a case study it has been shown that flood risks can be determined with different risk measures for an existing area based on the outcomes of Pilot Case Flood Risk. It is expected that flood risks for other areas can be assessed similarly. The calculated individual and societal flood risks are high compared to the hazards of other activities in the Netherlands. However, it has to be noted that loss of life and evacuation modelling involves many uncertainties and that the comparison in this paper concerns risks from different sources and nature. Also economic risks have been assessed and economic optimisation has been performed.
Based on this study it is recommended to apply the risk measures: individual risk,
FN curve and economic optimisation for future decision making on flood risks. Also more research on loss of life and evacuation modelling is needed to come to a better calculation of the existing risks. In the case study only one flood prone area has been assessed. To come to a better overview of the risks of the Netherlands it is recommended to determine risks of other areas in the country. Finally the desired level of flood risk acceptance has to be determined in a societal debate.
S.N. Jonkman, P.H.A.J.M. van Gelder, J.K. Vrijling
12

Disclaimer
Any opinions expressed in this paper are those of the authors and do not necessarily reflect the position of the Ministry of Transport, Public Works and Water
Management.
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