ISSH - Stochastic Hydraulics 2005 - 23 and 24 May 2005 - Nijmegen - The Netherlands
Implications of uncertainties on flood defence policy
J.K. Vrijling, P.H.A.J.M. van Gelder
Delft University of Technology, Faculty of Civil Engineering and Geosciences, P.O. Box 5048, NL-2600 GA
Delft, The Netherlands
1 ABSTRACT
This paper is concerned with the implications on
flood defence policy connected with the lack of information in hydraulic engineering models. The paper focusses on hydraulic engineering models which
result in a probability of failure. The different types
of uncertainty in such models will be discussed. Future research might give information and reduce
some of the uncertainties. It will be discussed how to
present the current and the uncertain future probability of failure. A simplified model which calculates
the probability of failure of a dike along the Dutch
lake IJssel (1500 km2) will be used as an example.
2 INTRODUCTION
The present safety standards of the Dutch dikes and
flood defences date to the report of the "Staatscommissie voor den Waterweg" (1920) and are expressed as an exceedance frequency of the design
water level (see Fig. 1). Much research on safetyand risk analysis has been carried out since 1920,
especially in the area of extreme value statistics. In
civil engineering practice many parameter estimation methods for probability distribution functions
are in circulation. Well known methods are for example:
the method of moments (Johan Bernoulli, 16671748),
- the method of maximum likelihood (Daniel Bernoulli, 1700-1782),
the method of least squares (on the original or on
the linearized data), (Gauss, 1777-1855),
the method of Bayesian estimation (Bayes, 1763),
the method of minimum cross entropy (Shannon,
1949),
- the method of probability weighted moments
(Greenwood et al., 1979),
- the method of L-moments (Hosking, 1990).
Many attempts (for instance, Goda and Kobune,
(1990), Burcharth and Liu, (1994), Yamaguchi,
(1997), and Van Gelder and Vrijling, (1997)), have
been made to find out which estimation method is
preferable for the parameter estimation of a particular probability distribution in order to obtain a reliable estimate of the p-quantiles. It is desirable that the
quantile estimate be unbiased, that is, its expected
value should be equal to the true value. It is also desirable that an unbiased estimate be efficient, i.e., its
variance should be as small as possible. The problem of unbiased and efficient estimation of extreme
quantiles from small samples is commonly encountered in the civil engineering practice. For example,
wave height data may be available for past 10 to 20
years and on that basis one may have to estimate a
design flood level corresponding to a 100 years return period. The engineer who has to design a
breakwater will hesitate a lot, or rather reject, the extrapolation of a wave height to a 10-2 quantile, if it
is lower than the highest observation in the collected
data (see Fig. 2). The engineer will prefer a method
which takes physical reasoning into consideration.
Physical reasoning to judge the homogeneity of a
data set are for instance:
1. Type of spectrum (swell, wind, single or double
peaked wave height spectrum);
2. Season, calendar period of the data set (sea level data in the winter or summer);
3. Physical characteristics of the phenomena
(breaking, non-breaking waves).
Combining statistical models with physical arguments leads to more homogeneous data and realistic
outcomes than applying statistical methods only.
The research of the design of the Eastern Scheldt
barrier (Vrijling and Bruinsma, 1980 and Mulder
and Vrijling, 1980) shows how wave height, sea level, and wave period data should be analysed with
help of physical models. Also the case study of wave
height and peak period data measured in the Bay of
Bengal near the city of Madras by Van Gelder and
Vrijling, 1998, demonstrates this importence.
Klemes (1993) proposes to find the statistical distributions of river discharges by using orographic and
Fig. 1: Frequency analysis 'avant-la-lettre'; 1920 publication with number of storm surges above water level at Hoek van Holland.
POT data, exponential fit, and quantile uncertainty (1 std)
0
10
-1
Frequency
10
-2
10
-3
10
-4
10
2
3
4
5
6
X
7
8
9
10
Fig. 2: Extrapolation of peaks-over-threshold data
with a ML method to a 10-2 quantile which is lower
than the highest observation (6.0 m).
ISSH - Stochastic Hydraulics 2005 - 23 and 24 May 2005 - Nijmegen - The Netherlands
synoptic considerations, rather than fitting the annual maxima or peaks over threshold values of discharges to distributions.
The Dutch Ministry of Public Works wants to
change the policy of estimating failure probabilities
to bring it in line with the approach in areas like
planning and transport, where failure probabilities
are given in a framework of acceptable risk. The
transition from water level criteria towards flooding
probabilities -and finally to a flood risk approachrequires a model to calculate the probability of a
failure of a dike system with several boundary conditions:
1 The model should be widely accepted by the flood
defence community and in some sense by the general public.
2 The results of the model should be sufficiently robust. The answers should not vary substantially with
slight moderations of the input.
3 The results of the model should not actuate the decision to alter the dike systems. The dike systems are
now in full compliance with the Deltacommittee
standard and are generally considered to be sufficiently safe.
In the past few years models have been developed in
order to determine the probability of failure of a dike
system, including several failure mechanisms and
their statistical dependencies. The mentioned boundary conditions to the model were approached by carrying out case studies of typical, slightly schematised polders in the Netherlands. In the case studies a
comparison between the old and the new safety approach has been made, to show that the differences
are minor (condition 3). Subsequently these case
studies are a good first step on the road to acception
of the model (condition 1).
A problem that still has to be investigated in accordance with the second condition is whether a future
change in the uncertainties in the model could cause
a significant change in the resulting probability of
failure. To investigate this an uncertainty analysis of
the model has to be carried out. An uncertainty analysis can give insight into the contributions and the
effects that various types of uncertainty have on the
resulting probability of flooding.
The discussion of this paper will be the problems
connected with an uncertainty analysis of a model
with a probability of failure as a result. The paper is
organized as follows. First the different types of uncertainty will be discussed followed by the available
options to present the results of an uncertainty analysis. A model for the effects of the uncertainty due
to lack of information will be presented and applied
to a case study. Finally the conclusions are given.
3 TYPES OF UNCERTAINTY
Uncertainties in decision and risk analysis can primarily be divided into two categories: uncertainties
that stem from variability in known (or observable)
populations and therefore represent randomness in
samples (inherent uncertainty), and uncertainties that
come from basic lack of knowledge of fundamental
phenomena (epistemic uncertainty).
Inherent uncertainties represent randomness or the
variations in nature. For example, even with a long
history of data, one cannot predict the maximum water level that will occur in, for instance, the coming
year at the North Sea. It is not possible to reduce inherent uncertainties.
Epistemic uncertainties are caused by lack of
knowledge of all the causes and effects in physical
systems, or by lack of sufficient data. It might be
possible to obtain the type of the distribution, or the
exact model of a physical system, when enough research could and would be done. Epistemic uncertainties may change as knowledge increases.
3.1 Five types of uncertainty
The inherent uncertainty and epistemic uncertainty
can be subdivided in the following five types of uncertainty (Paté-Cornell 1996): inherent uncertainty
in time and in space, parameter uncertainty and distribution type uncertainty (together also known as
statistical uncertainty) and finally model uncertainty.
Fig. 3: Types of uncertainty.
i) Inherent uncertainty in time
Stochastic processes running in time such as the
occurrence of water levels and wave heights are examples of the class of inherent uncertainty in time.
Unlimited data will not reduce this uncertainty because the realizations of the process in the future
stay uncertain.
ii) Inherent uncertainty in space
Random variables that represent the fluctuation in
space, such as the dike height. Just as for inherent
uncertainty in time it holds that unlimited data (e.g.
if the height would be known every centimeter) will
not reduce this uncertainty. There will always still be
a fluctuation in space.
iii) Parameter uncertainty
This uncertainty occurs when the parameters of a
distribution are determined with a limited number of
data. The smaller the number of data, the larger the
parameter uncertainty.
iv) Distribution type uncertainty
This type represents the uncertainty of the distribution type of the variable. It is for example not
clear whether the occurrence of the water level of
the North Sea is exponentially or Gumbel distributed or whether it has a completely different distribution.
Remark: a choice was made to divide statistical
uncertainty into parameter- and distribution type uncertainty although it is not always possible to draw
the line; in case of unknown parameters (because of
lack of observations), the distribution type will be
uncertain as well.
Later in this paper we will try to distinguish what
type of uncertainty a stochastic parameter represents.
Since parameter uncertainty and distribution type
uncertainty can not be discerned, another practical less scientific- division has been chosen. The statistical uncertainty is divided in two parts: ‘statistical
uncertainty of variations in time’ and ‘statistical uncertainty of variations in space’.
Fig. 4: Statistical uncertainty.
v) Statistical uncertainty of variations in time
When determining the probability distribution of
random variable that represents the variation in time
of a process (like the occurrence of a water level),
there essentially is a problem of information scarcity. Records are usually too short to ensure reliable
estimates of low-exceedance probability quantiles in
many practical problems. The uncertainty caused by
this shortage of information is the statistical uncertainty of variations in time. This uncertainty can theoretically be reduced by keeping record of the process for the coming centuries.
vi) Statistical uncertainty of variations in space
When determining the probability distribution of
random variable that represents the variation in
space of a process (like the fluctuation in the height
of a dike), there essentially is a problem of shortage
of measurements. It is usually too expensive to
measure the height or width of a dike in great detail.
This statistical uncertainty of variations in space can
be reduced by taking more measurements (see also
Vrijling and Van Gelder, 1998).
vii) Model uncertainty
Many of the engineering models that describe the
natural phenomena like wind and waves are imperfect. They may be imperfect because the physical
phenomena are not known (for example when regression models without underlying theory are
used), or they can be imperfect because some variables of lesser importance are omitted in the engineering model for reasons of efficiency.
3.2 Reduction of uncertainty
In Sec. 3.1, it was mentioned that inherent uncertainties represent randomness or the variations in nature.
Inherent uncertainties cannot be reduced.
Epistemic uncertainties, on the other hand, are
caused by lack of knowledge. Epistemic uncertainties may change as knowledge increases.
In general there are three ways to increase
knowledge:
- Gathering data
- Research
- Expert-judgment
Data can be gathered by taking measurements or by
keeping record of a process in time. Research can
for instance be done into the physical model of a
phenomenon or into the better use of existing data.
By using expert opinions it is possible to acquire the
probability distributions of variables that are too expensive or practically impossible to measure.
The goal of all this research obviously is to reduce
the uncertainty in the model. Nevertheless it is also
thinkable that uncertainty will increase. Research
might show that an originally flawless model actually contains a lot of uncertainties. Or after taking
ISSH - Stochastic Hydraulics 2005 - 23 and 24 May 2005 - Nijmegen - The Netherlands
some measurements the variations of the dike height
can be a lot larger. It is also thinkable that the average value of the variable will change because of the
research that has been done.
The consequence is that the calculated probability of
failure will be influenced by future research. In order
to guarantee a stable and convincing flood defence
policy after the transition, it is important to understand the extent of this effect.
In the next section will be discussed what influence
the future reduction of uncertainty can have on the
probability of failure and how to present this influence. It is assumed that in practice research will
rarely increase uncertainty or change the average
value of a variable.
4 THE EFFECT OF UNCERTAINTY DUE TO
LACK OF INFORMATION
The problem of lack of information in hydraulic engineering models is studied in detail in this section.
Data records are usually too short to ensure reliable
estimates of low-exceedance probability quantiles in
many practical problems .
The information about a random variable can be
updated by the help of expert judgements. Cooke
(1991) describes various methods to do so. Expert
judgements can also be used to reduce the uncertainty of the quantiles. In order to include expert
judgements in the quantile estimation of a certain
quantity, the following approach is proposed in this
paper. The approach will be applied to a case study
of quantile estimation of water levels at Lake IJssel.
Consider S the random variable describing the
loads on the structure.
Consider R the random variable describing the resistance of the structure.
The effect of the value of information on the random variables S and R may be modelled by correcting its original mean value by subtracting U in
which U is the distribution which represents the uncertainty. Furthermore the standard deviation of U
will be added in (S2+R2) to take account for the
uncertainty. Summarized in a table:
The implications of the changes in  and the related flooding frequency of the case study Lake IJssel will be discussed in Sec. 5.
As discussed in Sec. 3.2 it is possible that some
uncertainties might be reduced in the future. One
could say that those uncertainties add to the ‘uncertainty’ of the probability of failure. The obvious
problem is that it can not be predicted which uncertainties will be reduced in the future. There are several philosophies thinkable about what uncertainty
will remain part of the future probability of failure.
It is important to realise that, for every philosophy, the difference between the ‘true’ probability of
failure and the ‘uncertainty’ of the probability of
failure, is an artificial difference. If all uncertainties
are integrated this will result in the same, current
probability of failure.
Let us consider four different philosophies (options) about the uncertainties that might be reduced
and form the ‘uncertainty’ of the probability of
flooding, and the uncertainties that cause the ‘true’
probability of flooding, as given in table 2:
Inherent uncertainty (in time)
Inherent uncertainty (in space)
Statistical uncertainty (of variations in time)
Statistical uncertainty (of variations in space)
Model uncertainty
Opt
1
Pf
Opt
2
Pf
Opt
3
Pf
Opt
4
Pf
Pf
Pf
Pf
-
Pf
Pf
-
-
Pf
-
-
-
Pf
-
-
-
Table 2: Four different options
Option 1
No incertainties will be reduced. All uncertainties
in the model will be integrated to determine the
probability of failure. This probability of failure also
represents the current, actual probability of failure.
Option 2
A practical division is made between the uncertainties that might be influenced by taking measurements or doing research and the remaining uncertainties. Statistical uncertainty of variations in time
will be a part of the probability of failure. This uncertainty can only theoretically be reduced by keeping record of the underlying process for the coming
centuries.
Table 1: The effect of information U on the random
variables R and S.
Option 3
A theoretical division is made between inherent
uncertainty and epistemic uncertainty. The assumption is that through research all epistemic uncertainty
disappears. This would result in a probability of failure that is solely caused by inherent uncertainties.
Option 4
Assumed is that only inherent uncertainty in time
causes the probability of failure. This is also the
philosohy used in the present safety standards.
wind surge, wave height, wave steepness, wave runup, and lake oscillations were taken into account.
Two locations along the Lake IJssel were considered
in detail: Enkhuizen and Rotterdamsche Hoek
(Fig.1).
A short summary of the uncertainty analysis
study of Vrijling et al. (1999) will be given in Sec.
5.1. The goal of Sec. 5.2 is to analyze the influences
of the uncertainties in the water levels on the reliability-based optimal design of the dikes at the locations of Enkhuizen and Rott.Hoek.
1.6
Option 1
1.4
Option 2
1.2
Option 3
1
Option 4
0.8
0.6
0.4
0.2
0
2
2.5
3
3.5
4
4.5
5
5.5
6
Reliability index
Fig. 5: Probability Distributions of 
In Fig. 5 the probability of failure of a hypothetical example is shown in four different ways. Option
1 gives the probability with no uncertainty. With
every next option, more uncertainty is removed from
the ‘true’ probability of failure and added to the ‘uncertainty’ of the probability of failure.
It is clear that the more uncertainty is removed
from the resulting probability of failure; the higher
the mean and the larger the standard deviation of the
distribution of the reliability index will be.
5 CASE STUDY
Lake IJssel is situated in the northern part of the
Netherlands (Fig. 6). It has an area of approximately
1500km2. The lake is surrounded by dikes in order
to protect the low-lying polders from flooding. The
required safety against inundation of the polders is
1/4000 yr-1. In Westphal et al. (1997), a physical
model has been developed for the water levels of
Lake IJssel.It is based on WAQUA (a twodimensional water flow model) and HISWA (a wave
model). The model of Westphal has been analyzed
in an uncertainty study by Vrijling et al. (1999). Uncertainties with respect to water level, wind speed,
Fig. 6: Lake IJssel and the two locations of inteest
(Enkhuizen and Rott.Hoek).
5.1 Uncertainty analysis of the physical model
The probability of Z < 0 (overtopping of the Lake
IJssel dikes) was calculated by a first order reliability method (FORM) in Vrijling et al. (1999). Given
the uncertainties in water level, wind speed, wind
surge, wave height, wave steepness, wave run-up,
and lake oscillations, the uncertainties in the probability of overtopping at the two locations were determined.
The results of the uncertainty calculations are
summarized graphically in Figs. 7 and 7 for Rotterdamsche Hoek and Enkhuizen respectively. Notice
the differences between the required crest heights for
the three cases: intrinsic uncertainty, instrinsic + statistical uncertainty, and intrinsic + statistical + model uncertainty. These differences can be up to 1 metre. There has also been made a distinction between
additive and multiplicative uncertainty modelling
(Vrijling et al., 1999), but this distinction is not important for the next part of this paper.
ISSH - Stochastic Hydraulics 2005 - 23 and 24 May 2005 - Nijmegen - The Netherlands
Table 3. Distribution parameters and required
crest heights.
Notice that the probabilities of overtopping are
given by straight lines in the above semi-logarithmic
figures. This means that they can be described by
exponential distribution functions:
P(K<k) = 1-exp(-(k-A)/B).
When the following notation is adopted:
P = probability of overtopping [1/yr]
i.u. = intrinsic uncertainty
s.u. = statistical uncertainty
add. = additive model
mult. = multiplicative model
m.u. = model uncertainty
h = required crest height
table 3 can be derived.
The results of this table will serve as input for the
analysis of the economic optimal dike height in the
next section.
5.2 Reliability-based optimal design
In the reliability-based design of hydraulic structures, the idea is to determine the total costs function. By assuming the exponential distribution (with
parameters A and B) for the probabilities of overtopping, we can write:
The optimal dike height follows from the minimization of the total costs function and can be expressed by the formula:
and the optimal probability of failure is given by:
It is interesting to notice that the optimal probability of failure is independent of the A parameter of
the exponential distribution. The A values from table 1 are therefore neglected in the determination of
the optimal failure probability. An increase in the
slope of the exponential distribution (i.e. B>0), results in an increase in popt. From table 1, it was seen
that more uncertainty results in a higher B-value.
Consequently this leads to a higher optimal probability of failure.
The change in the optimal probability of failure
(from popt to popt’) caused by the increase in uncertainty (from B to B+B) can also be expressed as
follows:
Given an optimal probability of failure of 1/4000
yr-1 for Rott.Hoek, the inclusion of all uncertainties
(B=0.0852) leads to a new optimal probability of
failure of
Instead of an economic optimal dike height of
4.70 m, a height of 4.60 m is the result. For the location of Enkhuizen, the proposed approach leads to a
decrease in the dike height from 3.89 m to 3.77 m.
6 CONCLUSIONS
It can be concluded that the method described in
this paper is a very practical and simple method to
get insight in the problems concerned with the lack
of information in hydraulic engineering models.
It shows that the more uncertainty is expected to
be reduced in the future, the higher the mean and the
larger the standard deviation of the distribution of
the reliability index will be.
The method is demonstrated for a simple reliability function, but the advantage is that the method also works for a very complex reliability function with
a large number of variables.
The difference is that the result of the model is
equal to the mean of the reliability index. This creates a seeming difference between the probability of
flooding computed with the ‘old’ and the ‘new’
model.
This causes a problem for the transition from the
water level criteria towards flooding probabilities. It
is still not decided how to deal with this seeming difference. There are three possible reactions:
1 Accept the difference and do nothing.
2 Heighten the dikes in order to lower the ‘new’
probability of flooding to the ‘old’ value.
3 Do research before the transition takes place in
order to reduce some uncertainties and close the gap
between the ‘old‘ and the ‘new’ probability of flooding.
The reliability-based decision-making procedure
which has been applied in this paper can succesfully
be used in the analysis of the optimal failure probabilities for the dikes along the Lake IJssel. The influence of the uncertainties lead to an increase in the
probability of exceedance lines. When the hydraulic
boundary conditions are modelled in an exponential
way, analytical considerations can be given for the
optimal probabilities of failure and the optimal dike
heights.
Summarised, the following conclusions can be
drawn:
ISSH - Stochastic Hydraulics 2005 - 23 and 24 May 2005 - Nijmegen - The Netherlands
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Extreme value theory in the most refined
form is less fruitful
The limited amount of data and the various
sources of uncertainty have to be seen in the
context of the design decision
All uncertainties have to be taken into account in the design decision
A method to get insight in the effect of uncertainty in hydraulic engineering problems
is described.
The most influential random variables are
generally the ones with inherent uncertainty
(this uncertainty cannot be reduced).
In case of exponential distribution + normal
uncertainties, a simple expression for the
economic optimal probability of failure can
be derived
Larger location parameter leads to higher
optimal design, but has no influence on the
optimal probability of failure
Larger scale parameter leads to smaller optimal design and higher probability of failure
More options than structural
reduce uncertainty by data collection or research
decrease loads (μ down or σ down)
increase resistance (μ up or σ down)
reduce damage in case of failure
Economic optimal decisions should be proposed for the height as well as the timing of
the improvement
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