7th International Conference on Hydroinformatics
HIC 2006, Nice, FRANCE
CALCULATION OF EXPECTED ANNUAL DAMAGE
CONSIDERING OF DISCHARGE-PROBABILITIES FUNCTION
WOOSUNG NAM
School of Civil and Environmental Engineering, Yonsei University,
134 Sinchon-Dong, Seodaemun-Gu, Seoul, 120-749, Korea
JUN-HAENG HEO
School of Civil and Environmental Engineering, Yonsei University,
134 Sinchon-Dong, Seodaemun-Gu, Seoul, 120-749, Korea
In this study, flood frequency analysis and Monte Carlo simulation are used to
determine the expected annual damage (EAD) including uncertainties of dischargeprobability function. USACE (U.S. Army Corps of Engineers) recommends the logPearson type III distribution as a flood frequency model. However, it is not the case of
Korean flood data. Therefore, frequency analysis is applied to flood data of two stations,
Dongchon and Seongseo in the Kumho River basin in Korea. Appropriate probability
distributions for those sites are selected based on the method of probability weighted
moments and several goodness-of-fit tests such as χ2-test, Kolmogorov-Smirnov test,
Cramer von Mises test, PPCC (Probability Plot Correlation Coefficient) test. Sample data
for each return period of an appropriate underlying probability distribution are generated
by Monte Carlo simulation and then an appropriate sampling distribution is found based
on the same frequency analysis procedure. Then EAD, including uncertainties, is
estimated by applying sampling distributions to discharge-probability functions. As the
results, gamma, Gumbel, GEV and Weibull distributions are found to be appropriate to
flood data of these two sites. EAD considering uncertainties shows the larger value than
EAD without considering uncertainties. The accuracy of EAD depends on the sample
size, that is, the accuracy of EAD is decreased as sample size is decreased. EAD
calculated by the lognormal sampling distribution is similar to that evaluated by the
Weibull sampling distribution. EAD from the Gumbel sampling distribution leads to
smaller value than others and the results are reversed in case of 2- parameter Weibull
sampling distribution.
INTRODUCTION
Uncertainty in hydrologic, hydraulic, and economic functions has been considered by
safety factors or freeboard due to lack of uncertainty analysis techniques. Recent
development of statistics and advances in efficient computer applications enable
uncertainty to be explained. Therefore, the concept of risk analysis has come to be
included in design of hydraulic structures.
EAD (Expected Annual Damage) is the prevailing index of flood damage. EAD is
defined as the average flood damage and is calculated by integrating damage-probability
1
2
function. A damage-exceedance function is obtained by combining discharge-exceedance
function, stage-discharge function (rating curve), and stage-damage function [4].
Hydraulic, hydrologic, and economic uncertainties are included in the above three
contributing functions and lead to uncertainty in damage-exceedance probability
function. That results in uncertainty in EAD.
The log-Pearson type III (LP III) distribution has been used for estimating flood
quantile and non-central t distribution used for describing uncertainty in dischargeexceedance function [1, 3, 4, 5]. However, it is sometimes inappropriate to apply LP Ⅲ
model to the regions which have different regional hydrologic characteristics. Therefore,
the appropriate probability distribution should be selected by frequency analysis of flood
data from observation or conversion of stage data. Uncertainty should be considered by
selecting and adding a sampling distribution according to the probability distribution for
flood. In this paper, EAD considering uncertainties in discharge-exceedance function is
calculated and the effects of uncertainties on the calculation of EAD is investigated by
considering probability distribution of flood data and its sampling distribution.
METHODOLOGY
Damage-exceedance probability function is necessary to estimate EAD. However, it is
not easy to express damage-exceedance probability function in the form of numerical
formula. For this purpose, another approach such as a Monte Carlo simulation is used to
calculate EAD in this study. Sampling distribution is needed to explain uncertainty of
discharge-exceedance probability function. And it is selected from frequency analysis of
quantiles for each return period from Monte Carlo simulation. The random numbers for
each selected probability distribution function are generated and then a goodness of fit
test, such as a chi-square test, is carried out for checking the homogeneity of generated
data. Then, EAD based on each probability distribution is computed and the effect of
sample size on EAD is analyzed.
Monte Carlo simulation is performed by the following procedures:
(1) Generate flood data for a given parameter estimates of the selected probability
distribution,
(2) Apply a goodness of fit test for each generated data set to check whether the
data set satisfies the selected probability model.
(3) Select appropriate sampling distribution for the quantile values of specific
return period T (nonexceedance probability q).
(4) Generate flood data for selected sampling distribution.
(5) Estimate damages for generated flood data from sampling distribution.
(6) Estimate EADs by using damages-exceedance probability relation from step
(5).
(7) Average EADs.
Theoretically damage-exceedance function is obtained by collecting damage data
and fitting to a suitable statistical model. However, it is hard to gather reliable data in the
3
field. Therefore, damage-exceedance probability function is calculated by using
hydraulic, hydrologic, and economic information. In other words, it is obtained by three
contributing distributions such as discharge-exceedance probability function, rating
curve, and stage-damage function. Discharge-exceedance probability function and rating
curve are combined into stage-exceedance probability function. It means that the
probability exceeding a target stage is equal to that exceeding a target discharge.
Damage-exceedance probability function is obtained by combining stage-damage and
stage-exceedance probability functions. Finally, EAD is calculated by integrating
damage-exceedance probability function [5].
Another EAD is calculated by using generated data from each selected probability
distribution, that is, Monte Carlo simulation. Monte Carlo simulation continues until the
average EAD comes to be in the limit of error. EAD is generally estimated by Eq. (1)
(Bao, 1987).

 
E D qc* 

qc*
Dq  f q dq
(1)
qc* is the capacity of a hydraulic structure for certain flood, f q  is a probability
distribution of flood data and Dq  is damage function for flood ( q ). It is, however,
where
difficult to find an analytical solution due to complexity of calculation. So, such a
discrete formula in Eq. (2) is generally used

  D F
E D qc* 
j
j
(2)
j
where
D j and F j are the average flood damage and a probability deviate of the j th
interval.
Sampling distribution is employed to consider hydrologic uncertainty. If a sampling
distribution is included in Eq. (1), then EAD becomes
 

E D qc*   D(q T )h(qT )dqT  f (q)dq
*

q *c 
q c


 
(3)
qT is a flood of return period T and hqT  is a corresponding sampling
distribution for qT [3].
where
ESTIMATION OF EAD CONSIDERING UNCERTAINTIES
4
Flood data are constructed by converting 63 annual stage peak data points at Dongchon
site (site number 2 in Fig. 1) and 33 annual maximum stage data points at the Seongseo
site (site number 1 in Fig. 1) into flood data by using rating curves [2]. Appropriate
probability distributions are selected from frequency analysis of flood data. In this study,
Monte Carlo simulation technique is performed to estimate EAD. First of all, the random
numbers are generated for given parameters of the selected probability models by using
inverse function. In this case, the random numbers such as 10, 20, 30, 50, 80 and 100 are
generated to figure out the effect of finite sample size. Appropriate sampling distributions
are selected to include uncertainty of parameters by frequency analysis of flood data for
each return period from Monte Carlo simulation. Incorporating the sampling distribution,
EAD considering uncertainties is estimated.
Figure 1 Stations in Kumho river basin
FLOOD FREQUENCY ANALYSIS
Flood frequency analysis was performed for given flood data of the two sites considered.
The applied probability distributions for frequency analysis are the 13 models: 2- and 3parameter gamma, 2- and 3-parameter lognormal, log-Pearson type III, GEV (general
extreme value), Gumbel, 2- and 3-parameter log-Gumbel, 2- and 3-parameter Weibull, 4and 5-parameter Wakeby. The parameters were estimated based on the method of
probability weighted moments (PWM) which is known to be good for frequency analysis
5
of relatively short records. And then the goodness of fit tests such as chi-square test,
Kolmogorov-Smirnov test, probability plot correlation coefficient test, and Cramer von
Mises test were performed to find an appropriate probability model for a given flood
data. Finally, appropriate probability distributions for these sites are displayed in Table 1.
As shown in Table 1, these two sites have the same appropriate probability models as
those that are located close to each other. Note that as shown in Table 1, log-Pearson type
III model is not appropriate to flood data in Kumho River basin even though five models
are selected as appropriate models.
Table 1. Appropriate probability distribution based on PWM
Site
Probability distributions
Dongchon
Gamma-2, GEV, Gumbel, Weibull-2, Weibull-3
Seongseo
Gamma-2, GEV, Gumbel, Weibull-2, Weibull-3
CALCULATION OF EAD CONSIDERING UNCERTAINTIES
Appropriate sampling distributions are determined for given return period considered and
underlying probability distribution. As the results of simulations, the sampling
distributions for each site and underlying model are obtained as shown in Table 2. Even
though two sites have the same probability models for flood data, their sampling
distributions are not exactly the same as each other.
Table 2. Sampling distributions for each appropriate probability distribution
Site
Probability
Dongchon
Seongseo
Distribution
GEV
Lognormal-2, -3
Gamma-2
Gumbel
Weibull-3
GEV
Gumbel
GEV
lognormal-2
lognormal-2, -3
Weibull-3
Gumbel
Gumbel
Gumbel
lognormal-2, -3
lognormal-2, -3
Weibull-2, -3
Weibull-3
Gumbel
Gumbel
Weibull-2
lognormal-2
lognormal-2, -3
Weibull-2
Weibull-3
Weibull-3
Lognormal-2
Weibull-2, -3
Weibull-3
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EAD considering uncertainties is estimated by using sampling distributions. Figures
2 and 3 show EAD without uncertainties and EAD considering uncertainties. Figure 2
shows EAD when the underlying model is GEV and five possible sampling models are
GEV, Gumbel, lognormal-2, lognormal-3, and Weibull-3 as shown in Table 2 for
Dongchon site. As expected, EAD is the smallest when considering no uncertainty, and
the second smallest when the underlying and sampling models are the same. Then the
other four sampling models have the largest EAD. Figure 3 shows EADs when Weibull
probability distribution is assumed as the appropriate model for Seongseo site. In this
case, EADs considering uncertainty are well above EAD without considering uncertainty.
It is also found that EAD generally converges into a constant value as the number of data
increases.
10
GEV
NO UNCERTAINTY
GEV
GUM
9
EAD (MILLION WON)
LN2
LN3
WBU3
8
7
6
0
10
20
30
40
50
60
70
80
90
100
110
SAMPLE SIZES
Figure 2 EAD without and with consideration of uncertainties (Dongchon, GEV)
7
21
WBU2
20
NO UNCERTAINTY
GUM
19
LN2
EAD (MILLION WON)
LN3
18
WBU3
17
16
15
14
13
12
0
10
20
30
40
50
60
70
80
90
100
110
SAMPLE SIZES
Figure 3. EAD without and with uncertainties (Seongseo, Weibull-2)
CONCLUSIONS
In this study, EAD incorporating uncertainty is estimated by using sampling distribution
for flood data at Dongchon and Seongseo sites in Korea. Several conclusions obtained
from this research are as follows:
(1) Several probability distributions are selected for the appropriate models for
flood data and several sampling distributions are also selected for each
underlying probability distribution.
(2) As the number of flood data points increase an estimate of EAD converges into
a constant value. That is, the more number of data points there are, the more
accurate an estimate of EAD becomes.
(3) EAD with uncertainties is bigger than that without uncertainties.
(4) EAD is estimated closely in the cases of lognormal and Weibull distributions as
sampling distribution of flood data at Dongchon and Seongseo while
underestimated in the case of the Gumbel model as sampling distribution and
overestimated in the case of the Weibull distribution as sampling distribution.
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REFERENCES
[1] Arnell, N. W., “Expected annual damage and uncertainties in flood frequency
estimation”, Journal of Water Resources Planning and Management, Vol. 115, No. 1,
(1989), pp. 94-107.
[2] Busan National Territory Management Office, “Basic Plan for River Maintenance at
Kumho River”, (1997).
[3] Bao, Y., Tung, Y., K. and Hasfurther, V. R., “Evaluation of Uncertainty in Flood
Magnitude Estimator on Annual Expected Damage Costs of Hydraulic Structures”,
Water Resources Research, Vol. 23, No. 11, (1987), pp. 2023-2029.
[4] Stedinger, J. R., “Confidence intervals for design events”, Journal of Hydraulic
Engineering, Vol. 109, (1983), pp. 13-27.
[5] USACE, “Engineering and Design Risk-based Analysis for Flood Damage
Reductions Studies”, EM 1110-2-1619, (1996).