CHAPTER 4.4: UNCERTAINTY ANALYSIS OF
NON-BREAKING WAVES
J.K. VRIJLING1), P.H.A.J.M. VAN GELDER1)
1)
Delft University of Technology, Hydraulic and Offshore Engineering Section,
Stevinweg 1, NL-2628 CN, Delft, The Netherlands
e-mail: a.ramdjan@ct.tudelft.nl
ABSTRACT
In this chapter the concept of uncertainty analysis in the probabilistic design of vertical
breakwaters is described. In particular the uncertainty analysis of the wave height of nonbreaking waves and its forces and moments on vertical structures is presented. The uncertainty
analysis of impact waves can be found in Chapter 5.1 of Volume IIa.
Keywords: Uncertainty analysis, pulsating waves, long-term, short-term, maximum
individual wave, return periods.
1. INTRODUCTION
In Vrijling, (1997), a first introduction on uncertainties is presented. He describes that in most
practical cases data is far too limited to give reliable estimates for the distribution and the
autocorrelation function of wave heights, water levels, etc. The statistical uncertainty, i.e. the
uncertainty of the parameter estimates due to the limited number of observations, has to be
assessed. In that case the inherent uncertainty has to be increased with the statistical
uncertainty.
During the life of the structure hydraulic info will be gained and the statistical uncertainty will
normally be reduced. However due to the discovery of slowly running stochastic processes
(e.g. sea level rise, climate change) this type of uncertainty may well increase too. The
uncertainty of the type of p.d.f. that describes the data is mostly included in the category of
statistical uncertainty too. In a Bayesian approach the likelihood of various types can be
estimated from the data and in that way the choice reduces to an estimation problem.
Many of the engineering models that describe the behaviour of structures are imperfect. They
may be imperfect because the physical phenomena are not known (regression models without
underlying theory). They can also be imperfect because some variables of lesser importance in
the theoretical model, are omitted in the engineering model for reasons of efficiency and the
resulting scatter is accepted as model uncertainty.
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In the first case we only know the uncertainty in the tested or observed domain. The
development of a model founded in physics will reduce the uncertainty in this case. With the
physical model, observed data and experience can be translated to other conditions or even
other sites. If for instance the phenomena of wave breaking and local wave generation are
understood better, predictions of the (conditional) p.d.f. of significant wave height can be
made then by reliance on scarce statistical data only.
In the second case the uncertainty can also be estimated outside the domain of experience by
using the more sophisticated model. Care has however to be taken that the model is verified
with specific prototype data.
In case of complete lack of info, the engineering judgement or degree of belief can be
expressed as a probability distribution. If data becomes available the distribution can be
formally updated.
In this chapter the principles of an uncertainty analysis will be described followed by two
sections:
- an uncertainty analysis of wave heights,
- an uncertainty analysis of moments and forces.
An uncertainty analysis of wave impacts is given in Chapter 5.1 of Volume IIa.
2. UNCERTAINTY ANALYSIS
When studying the probabilistic design of vertical breakwaters it is important to develop a
philosophy to discern between inherent uncertainty (randomness) and uncertainty related to
the lack of information (statistical and model uncertainty).
The inherent uncertainty is natural, fundamental and irreducible. For instance the outcome of
the flipping of a coin is fundamentally uncertain. The outcome will be a head or a tail but no
one can predict it. The outcome of the throw of a die is characterised by the same type of
uncertainty. The outcome will be one of the set 1-6 but the real value will be uncertain, until
the experiment has been performed. Many natural phenomena like wind velocity, river
discharge, wave height, grain size, etc. exhibit an inherent uncertainty too. Theoretically any
positive value could be attained with a certain probability. The probabilities of the various
possible outcomes are mostly described by probability density functions (p.d.f.'s), ranging in
the examples given above from two valued Bernoulli p.d.f. for the coin to a Weibull, Gumbel
or other continuous p.d.f..
Although the probability p of a head or the probability of say a 3 for the die can be derived by
using symmetry considerations, the true value (the coin might be forged) will be in some
doubt unless sufficient experiments are made. A considerable number of observations is
needed to estimate the value of the parameters of the p.d.f's of the mentioned natural
phenomena. The uncertainty that is associated with the parameter estimates based on limited
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CHAPTER 4.4
UNCERTAINTY ANALYSIS
data, is called the statistical uncertainty. An increasing amount of data tends to decrease this
uncertainty.
Related to the statistical uncertainty is the problem of the limited accuracy of physicalmathematical models. In many cases the mathematical model of a physical process has to be
calibrated by means of small scale experiments or prototype observations. Due to the limited
number of experiments or observations and to imperfections of the model, differences will
exist between the predicted and the observed outcomes. These differences give rise to the
model uncertainty. Model uncertainty is related to the statistical uncertainty in the sense that
more data tend to reduce the parameter uncertainty. In case of an imperfect model some
uncertainty will persist unless the model itself is improved.
In a modern Bayesian approach both uncertainties are incorporated in the analysis of the
reliability of the structure. This reflects the approach of the designer, who tends to opt for
larger margins in case of an increased uncertainty of any kind.
2.1 Inherent uncertainty in time
Stochastic processes running in time (individual wave heights, significant wave heights, water
levels, discharges, etc.) are examples of the class of inherent uncertainty. Unlimited data will
not reduce this uncertainty. The realisations of the process in the future stay uncertain. The
p.d.f or the cumulative probability distribution function (c.d.f.) and the auto-correlation
function describe the process. In case of a periodic stationary process like a wave field the
autocorrelation function will have a sinusoidal form and the spectrum as the Fourier-transform
of the autocorrelation function gives an adequate description of the process.
Attention should be paid to the fact that the well known wave energy spectra as PM, Jonswap
and TMA are not always able to represent the wave field at a site. In quite some practical
cases swell and wind wave form a wave field together. The presence of two energy sources
may be clearly reflected in the double peaked form of the wave energy spectrum. A practical
description of a general spectrum and some examples are given in Vrijling (1997).
An attractive aspect of the spectral approach is that the inherent uncertainty can be easily
transferred through linear systems by means of transfer functions. By means of the linear wave
theory the incoming wave spectrum can be transformed into the spectrum of wave loads on a
vertical caisson wall. The p.d.f. of wave loads can be derived from this wave load spectrum.
Of course it is assumed here that no wave breaking takes place in the vicinity of the caisson.
In case of non-stationary processes, that are governed by meteorological and atmospheric
cycles (sign. wave height, discharges) the p.d.f. and the autocorrelation function are needed.
Here the autocorrelation function gives an impression of the persistence of the phenomenon.
The persistence of rough and calm conditions is of utmost importance in workability and
serviceability analyses.
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If the interest is directed to the analysis of ultimate limit states e.g. sliding of the caisson the
autocorrelation is eliminated by selecting only independent maxima for the statistical analysis.
If this selection method does not guarantee a set of homogeneous and independent
observations, physical or meteorological insights may be used to homogenise the data-set. For
instance if the fetch in NW-direction is clearly maximal, the dataset of maximum significant
wave height could be limited to NW-storms. If such insight fails, one could take only the
observations exceeding a certain threshold (P.O.T.) into account hoping that this will lead to
the desired result. In case of a clear yearly seasonal cycle the statistical analysis can be limited
to the yearly maxima.
Special attention should be given to the joint occurrence of significant wave height Hs and
spectral peak period Tp. A general description of the joint p.d.f. of Hs and Tp is not known. A
practical solution for extreme conditions considers the significant wave height and the wave
steepness as independent stochastic variables to describe the dependence. This is a
conservative approach as extreme wave heights are more easily realised than extreme peak
periods. For the practical description of daily conditions (SLS) the independence of sp and Tp
seems sometimes a better approximation.
Also the dependence of water levels and significant wave height should be explored because
the depth limitation to waves can be reduced by wind setup. Here the statistical analysis
should be clearly supported by physical insight. Moreover it should not be forgotten that
shoals could be eroded or accreted due to changes in current or wave regime induced by the
construction of the breakwater.
2.2 Inherent uncertainty in space
Soil properties can be described as stochastic processes in space. From a number of field tests
the p.d.f. of the soil property and the (three-dimensional) autocorrelation function can be fixed
for each homogeneous soil layer. Here the theory is further developed than the practical
knowledge. Numerous mathematical expressions are proposed in the literature to describe the
autocorrelation. No clear preference has however emerged yet as to which functions describe
the fluctuation pattern of the soil properties best . Moreover the correlation length (distance
where correlation becomes zero) seems to be of the order of 30 to 100m while the spacing of
traditional soil mechanical investigations for breakwaters is of the order of 500m. So it seems
that the intensity of the soil mechanical investigations has to be increased considerably if
reliable estimates have to be made of the autocorrelation function.
The acquisition of more data has a different effect in case of stochastic processes in space than
in time. As breakwater structures are immobile, there is only one single realisation of the field
of soil properties. Therefore the soil properties at the location could be exactly known if
sufficient soil investigations were done. Consequently the actual soil properties are fixed after
construction, although not completely known to man. The uncertainty can be described by the
distribution and the autocorrelation function, but it is in fact a case of lack of info.
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UNCERTAINTY ANALYSIS
An important aspect of this type of uncertainty is that without further investigations the
knowledge of the foundation increases with time. During the life of the structure information
will be gained as each storm exceeding the previous that is survived by the structure, pushes
the lower limit of the strength upward (Bayesian updating of the strength).
2.3 The effect of the various types of uncertainty on reliability
From an engineering point of view the Bayesian approach that takes all uncertainties into
account as p.d.f's reflects the designer's intuition very well. Keeping the physical structure
equal an increase in uncertainty of any variable increases the formal probability of failure too.
From this point of view there is no difference between inherent, statistical and model
uncertainty all have to be incorporated in the probabilistic calculations.
In the probabilistic calculations however a difference occurs between uncertainties that have
many (e..g. yearly) realisations during the lifetime of the structure and those that have only
one connected to the specific structure and the site. Every storm season shows a independent
maximum Hs every year. The properties of the foundation and the strength of the caisson have
only one realisation per structure.
Consequently that the probability of failure is not solely a property of the structure but also a
result of our lack of knowledge. This can be illustrated by calculating the probability
distribution of the life time to failure of a structure with resistance R with p.d.f. fR(x)
subjected to a yearly maximal load S with c.d.f. FS(x):
N
F L (N) =  F S (x ) . f R (x) . dx
(1)
From this distribution the conditional failure rate h(N) can be derived according to:
h(N) =
f L (N)
1 F L (N)
(2)
The formal relation between the probabilistic calculation of the probability of failure in a year
and the conditional failure rate is illustrated in Annex I, where two examples are given. If the
standard deviation of the yearly maximal wave load is large in relation to the standard
deviation of the resistance the dependence between failure in two consequent years is low. If
the probability of failure is say p per year then the failure probability is approximately N.p
during the lifetime of N years.
If however, keeping the standard deviation of the reliability function equal, the opposite is
true, the failure in consequent years are dependent. In that case the probability of failure in the
first year is p and the probability of failure in the life time too. In the first case the conditional
failure rate is constant over time and equal to p. In the second case however the conditional
failure rate equals p in the first year and falls to zero afterwards.
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The second case was already mentioned above, where it was stated that every storm that was
survived by the structure improved the knowledge of the p.d.f. of the resistance (pushing the
lower tail to the right). Thus every year the knowledge of the owner grows and the probability
of failure of the structure falls.
In exactly the same way the failure probability of the structure can be improved by
investigating e.g. the quality of the foundation assuming that this leaves the average
unchanged and reduces the uncertainty.
2.4 Uncertainty modelling
Suppose that the true state of nature is X. Prediction of X may be modelled by X*. As X* is a
model of the real world, imperfections may be expected; the resulting predictions will
therefore contain error and a correction N may be applied. Consequently, the true state of
nature may be represented by Ang, (1973).
X = NX*
(3)
If the state of nature is random, the model X* naturally is also a random variable. The inherent
variability is described by the coefficient of variation of X*, given by (x*)/(x*).The
necessary correction N may also be considered a random variable, whose mean value (N)
represents the mean correction for systematic error in the predicted mean value, whereas the
c.o.v. of N, given by (N)/(N), represents the random error in the predicted mean value.
It is reasonable to assume that N and X* are statistically independent. Therefore we can write
the mean value of X as
(X)=(N)(x*)
(4)
The total uncertainty in the prediction of X becomes:
Cov(X) = sqrt(Cov2(N) + Cov2(x*))
(5)
Instead of using a multiplicative model error, it is also possible but not very frequently used,
to apply an additive model error M in which:
X = X* + M
(6)
3. UNCERTAINTY ANALYSIS OF WAVE HEIGHTS
In this section an overview will be given of the uncertainty analysis of wave heights. A
comparison is made between the statistical behaviour of short term and long term wave
heights. The uncertainty of the maximum wave height is studied and results are compared
with the design wave height formula of Goda Hdesign = 1.8Hs.
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UNCERTAINTY ANALYSIS
3.1 Short term and long term wave heights
Battjes, (1975), investigated the probability distributions of the significant wave heights on a
long term time scale (e.g. one year). He noticed that symmetric distributions, such as the
normal distribution, were not suitable to describe the long term distribution for the wave
heights. Skewed distributions, such as the Gumbel and Weibull distribution, fitted much
better.
Longuet-Higgins, 1962, investigated the probability distributions of the wave heights on a
short term time scale (e.g. three hours). With theoretical arguments, he could derive that the
short term distributions is given by a Rayleigh distribution, given a few easily-satisfied,
boundary conditions (such as stationarity conditions). The Rayleigh distribution is a oneparameter distribution. The free parameter is given by the significant wave height:
F(H|Hs) = 1 - exp(-2 (H/Hs)2)
(7)
The short term distribution of the wave height is conditionalized on Hs.
3.2 Maximum individual wave height
In this section the distribution function of the maximum wave height from a data set of N
individual wave heights (from one sea state) is studied. This distribution function is called the
extreme value distribution of Hmax and is given by:
F(Hmax|Hs) = F(H|Hs)N = (1 - exp(-2 (H/Hs)2)N
(8)
The long term distribution of Hs is usually given by one if the following distributions:
- Exponential
- Gumbel
- Weibull
- Generalized extreme value
- etc.
For instance, consider a Gumbel and Rayleigh distribution for the long term and short term
distributions respectively:
FL(Hs) = EXP(-EXP(-(Hs-A)/B))
FS(H|Hs) = 1 - EXP(-2 (H/Hs)2)
(9)
(10)
From the long term and short term distributions, the joint probability distribution (jpd) can be
studied. In figure 1 the jpd of H and Hs and the jpd of Hmax and Hs are presented for certain
values of the distribution parameters.
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The jpd of Hmax and Hs is shown for N=1 and N=3000 individual waves.
Contour plot of joint probability of Hmax and Hs
10
9
8
7
Hs [m]
6
5
4
N=1
3
N=3000
2
1
0
0
2
4
6
8
10
H [m]
Fig.1: Contour plots
The distribution of Hmax is given by:
Conditionalized:
F(Hmax|Hs) = F(H|Hs)N = (1 - EXP(-2 (H/Hs)2))N
(11)
Unconditionalized:
F(Hmax) = F(Hmax | Hs) FL(Hs) d Hs
(12)
In Figure 2 the unconditionalized distribution of Hmax (N=3000) is shown (right line) in
comparison with the significant wave height Gumbel-distribution (left line) and the Goda line
1.8Hs (middle line).
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CHAPTER 4.4
10
Exceedance probability
10
10
10
10
10
UNCERTAINTY ANALYSIS
Exceedance probability of Hmax
0
-1
-2
Gumbel
-3
1.8Hs
N=3000
-4
-5
5
10
15
20
H [m]
Fig. 2: Exceedance probabilities
Note that the exceedance probabilities for the maximum wave height (out of 3000 individual
waves) are much higher than on basis of the Goda line 1.8Hs would be predicted.
In the next table the multiplication factor are given for the 1/100 years wave heights.
Tab.1: Multiplification factors (Short-term Rayleigh, Long-term Gumbel)
1/100
YEARS
Hs [m]
Hmax
[m]
FACTOR
N=250
6.50
11.90
1.83
N=1000
6.50
12.99
1.99
N=3000
6.50
13.80
2.12
3.3 Uncertainties in determining return periods of wave heights
In this section the wave height which has an exceedance frequency of 1% will be studied with
help of Monte Carlo simulations. Given a dataset of 100 individual wave heights from one sea
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state, a first guess for the "1% exceedance wave height" would be the maximum value of this
dataset. However, in the next it will be shown that this guess is too conservative.
The short term wave height distribution conditionalized on Hs is Rayleigh distributed. Given a
dataset {h1, h2,..., h100} of 100 individual wave heights. On the basis of this dataset it is
determined what the wave height will be which is exceeded with probability 1/100. I.e. the
wave height h1/100 which satisfies the equation P(h>h1/100) = 1/100.
It follows that h1/100 is a random variable. Its distribution function can be approximated by a
normal distribution with mean 1.51Hs and standard deviation 0.79N-2Hs in which N is the
length of the available dataset (100 in the above example) and Hs is estimated by hi /0.63N.
A Monte Carlo simulation is performed. A set of 100 wave heights is generated from a
Rayleigh distribution with significant wave height of 5m. The empirical distribution function
is plotted in figure 3. The probability distribution function of h1/100 is included and given by
the normal distribution with a mean value of 7.55m and a standard deviation of 0.39m. Also
the distribution of the maximum wave height hmax = max{h1, h2,..., h100} can be determined. In
the above Monte Carlo simulation with Hs=5m, it is given by a normal distribution with a
mean value of 8.18m and a standard deviation of 1.16m. Remark that the answer is more
conservative than the former prediction.
In the former it is shown that P(Hind>1.51Hs) = 1/100. Goda already showed that
P(Hind>1.8Hs) = 2/1000. However, after 3000 individual waves, it appears that P(Hi such that
Hi>1.8Hs) = 0.9975! If we would require P(Hi such that Hi>Hs) = 1/100 after 3000
individual waves, then  = 2.51 (=sqrt(-2ln(-(1-1/100)1/3000+1)).
Probability of failures over the lifetime of the structures is analysed by Vrijling and Van
Gelder, (1997). Correlation in failures in subsequent years of the structure is taken into
account in this paper.
10
-1
Probability
10
Monte Carlo (Hs=5m)
0
10
10
-2
-3
0
2
4
6
Wave height [m]
Fig.3: Return period uncertainty
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CHAPTER 4.4
UNCERTAINTY ANALYSIS
4. UNCERTAINTY ANALYSIS OF MOMENTS AND FORCES
In Van der Meer et.al., (1994), the model uncertainty of Goda's formula has been studied. For
the horizontal force they found that the model bias of Goda was 90% and the model
uncertainty was 25%. The model bias and -uncertainty were based on the calculated and
measured values of the average of the highest of 250 waves. However, the average of the
highest of 250 waves has inherent uncertainty, as was described earlier in this paper. Goda's
model uncertainty should be corrected for the highest wave uncertainty in the following way:

Total = (Goda2 + Wave2)1/2
(13)
In case of the horizontal forces it follows by
Total = 0.25 and Wave = 0.15 that Goda = 0.20; a reduction of 5% in the model uncertainty.
The uncertainty in the horizontal moment was originally estimated by Van der Meer et.al. as
40%. However, taking the wave height uncertainty into account, this uncertainty reduces to
37%.
The new calculated Goda model uncertainties are presented in the following table:
Tab.2: Goda model uncertainties
Model
Uncertainty
Horizontal Force
20%
Horizontal Moment
37%
Vertical Force
20%
Vertical Moment
34%
5. CONCLUSIONS
In this chapter an overview is given on the concept of uncertainty analyses. In particular,
attention has been paid on:
- an uncertainty analysis of non-breaking wave heights (Fig.1-3, Table 1),
- an uncertainty analysis of moments and forces of non-breaking waves (Table 2).
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