ORIGINAL PAPER

Hyun-Han Kwon Æ Young-Il Moon

Improvement of overtopping risk evaluations using probabilisticconcepts for existing dams

Published online: 19 November 2005� Springer-Verlag 2005

Abstract Hydrologic risk analysis for dam safety relieson a series of probabilistic analyses of rainfall-runoffand flow routing models, and their associated inputs.This is a complex problem in that the probability dis-tributions of multiple independent and derived randomvariables need to be estimated in order to evaluate theprobability of dam overtopping. Typically, parametricdensity estimation methods have been applied in thissetting, and the exhaustive Monte Carlo simulation(MCS) of models is used to derive some of the distri-butions. Often, the distributions used to model some ofthe random variables are inappropriate relative to theexpected behaviour of these variables, and as a result,simulations of the system can lead to unrealistic valuesof extreme rainfall or water surface levels and hence ofthe probability of dam overtopping. In this paper, threemajor innovations are introduced to address this situa-tion. The first is the use of nonparametric probabilitydensity estimation methods for selected variables, thesecond is the use of Latin Hypercube sampling to im-prove the efficiency of MCS driven by the multiplerandom variables, and the third is the use of Bootstrapresampling to determine initial water surface level. Anapplication to the Soyang Dam in South Korea illus-trates how the traditional parametric approach can leadto potentially unrealistic estimates of dam safety, whilethe proposed approach provides rather reasonable esti-mates and an assessment of their sensitivity to keyparameters.

Keywords Dam safety Æ Overtopping probability ÆNonparametric Monte Carlo simulation Æ Latinhypercube sampling Æ Bootstrap

1 Introduction

In recent years, many countries have experienced fre-quent floods that may have overtop dams. Such defi-ciencies can cause dams to break and extreme floods tooccur downstream. This leads to various problems suchas loss of social capital, large scaled economical ex-penses, and the loss of life. For this reason, emergencyspillways or storage capacity must be designed rationallyenough to satisfy safety needs.

The design standard of dams is a very complex issuethat has not yet been definitely resolved. In manycountries, a large deterministic approach has been usedto account for hydraulic and hydrologic uncertainty.The height of the dam is determined by flood hydro-graph estimation and reservoir routing, while spillwaycapacity has been determined by using a design floodand the probable maximum flood (PMF).

Most countries currently accept the PMF as a stan-dard for dam safety. Generally, it is considered that adam spillway that has been designed using PMF repre-sents a fewer risks and security at least at the time ofconstruction. However, viewed in the existing dam, thePMF has two problems. First of all, the PMF haschanged over time. Secondly, the PMF standard doesnot take into account any reciprocal among risk, costs,and economic benefits.

In the case of modification of the structure, includingrepairs and reconstruction of the dam, economic feasi-bility and social safety goals need to be addressed. Whenevaluating the priority of the dam rehabilitation, a risk-based analysis is a potentially useful approach.

In recent years, numerous studies have attempted tofind and explore the possibility of a risk-based analysisin dam safety. Von Thun (1987) studied the risk analysismethod with USBR (U.S. Bureau of Reclamation) in

H.-H. KwonDepartment of Earth and Environmental Engineering,Columbia University, S.W. Mudd,Room 842A 500 West 120th Street, MC 4711,New York, NY 10027, USAE-mail: hk2273@columbia.edu

Y.-I. Moon (&)Department of Civil Engineering,University of Seoul, Seoul,130-743, KoreaE-mail: ymoon@uos.ac.kr

Stoch Environ Res Risk Assess (2006) 20: 223–237DOI 10.1007/s00477-005-0017-2

order to estimate dam risk and risk expense and twoconferences were held in 1985 (McCann 1985; Haimesand Stakhiv 1986; Moser and Stakhiv 1987). When thesafety of a dam is evaluated, the risk analysis is ordi-narily applied to estimate failure probability related torare events and to estimate the relative frequency ofextreme events associated with the PMF. Foufoula-Georgiou (1989) illustrated how the exceedance proba-bility of extreme precipitation could be derived andapplied to two watersheds in Iowa. Langseth and Per-kins (1983) proposed a procedure for dam safety anal-ysis. Stedinger and Grygier (1985), Von Thun (1987),Resendiz-Carillo and Lave (1987), Haimes et al.(1988)and Petrakian et al. (1989) investigated the developmentof dam safety standards.

Moreover, various studies have attempted to evaluatethe safety of dams (Bowles 1987; McCann et al. 1985;Von Thun 1987; Resendiz-Carrill and Lave 1987; Kar-lsson and Haimes 1988a, 1988b; Haimes et al. 1988;Karlsson and Haimes 1989; Petrakian et al. 1989). Thegeneral approach is to estimate a probability distribu-tion associated with extreme precipitation and runoff(NRC, 1988).

Hydrologic dam risk analysis depends on a series ofprobabilistic analyses of rainfall-runoff and their associ-ated input variables. This is a complex problem in that theprobability distributions of multiple independent andderived random variables need to be estimated to estimatethe probability of dam overtopping. Typically, paramet-ric density estimation (PDE) methods and the MonteCarlo simulation (MCS) of models are used to derivesome of the distributions. Often, the distributions used tomodel some of the random variables are inappropriaterelative to the expected behavior of these variables.

Major innovations are introduced in this paper toaddress this situation. Much focus is placed on ahydrologic dam risk analysis approach using nonpara-metric kernel density estimation (NKDE) applied toextreme events. In addition, the use of Latin hypercubesampling (LHS) and bootstrapping for uncertaintyanalysis of the estimated extreme event risk are alsoproposed. The paper is organized to first compare somealternate strategies for the dam safety hydrologic riskassessment. A hydrologic dam risk model is then pre-sented. The next section is intended to evaluate theexisting approach for PDE estimation. The fourth sec-tion provides the proposed approach. Finally, the pro-posed approach is applied to the analysis of the safety ofan existing dam in South Korea.

2 The hydrologic dam risk model

In exploring the hydrologic dam risk analysis, we willlimit hydrologic random variables to rainfall, initialreservoir water level, wind velocity, time of concentra-tion, storage coefficient and SCS Curve Number (CNNo.). If we expect some of them not to be independent,these are assumed to be independent.

Many studies used event models for rainfall-runoffand then considered certain probabilities for rainfallevents – in this case watershed antecedent conditions arean issue, and one needs a probability density function(PDF) for them; but also the initial reservoir water levelis an issue. Antecedent conditions and the water level arelikely to be highly correlated. Since this was difficult tomodel previous most studies took these two to beindependent. Others, in the name of being conservative,have assumed that the water level comes from the dis-tribution of annual maximum water level. If we keeptrack of all days and then calculate annual probabilities,then both approaches could give the same result. How-ever, simulating from the daily water levels would re-quire many more samples to get the right representation,hence one prefer to use the annual maximum water leveland take it to be independent from the antecedent con-ditions.

2.1 Modeling concept

To evaluate the hydrologic risk associated with damfailure, we need to establish a method to transform thewater surface level into overtopping probability. Withregard to this point, this study is focused on the methodsproposed by Cheng et al. (1982) and Pohl (1999). Fig-ure 1 indicates the associated PDF of each variablewhen hydrologic dam risk analysis is demonstrated byMCS approach (Pohl, 1999).

The formulation associated with MCS for hydrologicdam risk analysis can be represented as follows.

WS1 þ WS2 þ WS3 > Dam Crest Level ð1Þ

where WS1 is initial water surface level and WS2, WS3 isthe increasing water surface level by the flood, windrespectively. Assume that WS1, WS2, WS3 are inde-pendent variables. Then:

Fig. 1 Design concept for probabilistic dam risk approach usingMonte Carlo simulation

224

PT ¼ P WS1 þ WS2 þ WS3 > Dam Crest Levelð Þ

¼Z 1

BT

Z 1BT

Z 1BT

f ðWS1 þ WS2 þ WS3Þ � dWS1 � dWS2

� dWS3

ð2Þ

PT ¼Z 1

BT

Z 1BT

Z 1BT

fWS1ðWS1Þ � fWS2ðWS2Þ � fWS3ðWS3Þ

� dWS1 � dWS2 � dWS3

ð3Þ

where, P is probability, PT is dam overtopping proba-bility induced by the three variables, f(.) is the PDF ofeach variable, and BT is dam height.

Figure 2 indicates the overall procedure of modelingfor the overtopping probability. After calibrating rain-fall-runoff model, hydrologic variables are generated byNKDE method. The simulated variables are applied torainfall-runoff model and the estimated discharges aretransformed to water surface level using reservoir rout-ing. Overtopping probability is estimated by risk mod-eling function. In the proposed approach, level-poolreservoir routing is used to estimate water surface level.Reservoir characteristics, e.g., storage–discharge–stagerelationships, are used with a model such as HEC-1.

2.2 Risk modeling for flood-induced overtopping

For modeling flood-induced overtopping, rainfall, initialwater level of reservoir, SCS CN No., time of concen-tration and storage constant are considered as uncertainhydrologic variables. HEC-1 is used as an event rainfall-runoff model. Equation (4) provides an indicator used toestimate overtopping probability induced by flood.

gF ¼ HC � H0 � hF ð4Þ

where, HC, H0 and hF indicate the dam height, undis-turbed reservoir water surface level and increasing levelinduced by flood, respectively.

2.3 Risk modeling for wind-induced overtopping

In risk modeling of wind-induced overtopping, uncer-tainty variables are the initial water level of reservoir andthe wind velocity. Equation (5) provides an indicatorvariable used to estimate overtopping probability in-duced by wind (Cheng et al. 1982).

gW ¼ HC � H0 � ðhT þ hrÞ¼ HC � H0

� V 2w F

63200Dþ azw exp �b 0:031V 0:18

w F �0:09e

� �� �� �

ð5Þ

where, HC represents the dam height, H0 is undisturbedreservoir water surface level, HT is increasing level in-duced by wind,Hr is wave run-up, VW is wind velocity, Fis fetch length, D is the average depth of the reservoir, Fe

is the effective fetch length, and a, b are coefficientscorrespond to embankment slopes.

2.4 Risk modeling induced by concurrent floodand wind

The combined load of simultaneous occurrences of dif-ferent natural forces may be more significant than theload induced by a single natural force. For this reason,

Fig. 2 Proposed hydrologicdam risk analysis using MonteCarlo simulation

225

Wen (1977) proposed formulation of combined load riskmodel. Equation (6) provides an expression for theovertopping probability induced by concurrence of floodand wind (Cheng et al., 1982).

PFW ¼lw

lF þ lwP�

hF þ hmz > HC � H0

�

þ lF

lF þ lwMAX

"P

hF þ

hmz

2> HC � H0

!;

P

hF þ hT þ hr þ hmz

2> HC � H0

!#ð6Þ

Where, PFW is total dam overtopping probability in-duced by concurrence of flood and wind, hmz is theweighted increasing level induced by wind consideringthe present reservoir level, l f is duration of flood, l w isduration of wind.

3 Present hydrologic dam risk analysis

Before developing hydrologic dam risk method, it isworth identifying some of the major issues about presentdam risk approach. Figure 3 indicates problems ofpresent hydrologic dam risk analysis

Most of variables in the hydrologic dam risk analysisare random, but this has not been appropriately ad-dressed in the present approach to dam safety analysis.Typically the uncertainty associated with an individualvariable is modeled using a particular univariate prob-ability distribution like Gaussian distribution in thehydrologic dam risk analysis. Though Gaussian distri-bution cannot effectively reflect the statistical propertyin the dam risk analysis, it has been used until now. It isbecause it is easy to use (Lombardi, 2002).

In the hydrologic dam risk analysis, overtoppingprobability frequently indicates unrealistic values be-cause of applying the unbounded probability distribu-tion to a potentially bounded (e.g., the PMF concept

implies a bound) variable like dam water surface level.Further, triangular, rectangular and beta distributionthat are bounded often do not represent the nature ofhydrologic variables. The traditional PDE methodmakes it difficult to represent multimodal probabilitydensity functions. Suppose that the Figure 4 indicatesPDF of real data. If it is estimated by normal distribu-tion following the Figure 5, it cannot reflect the statis-tical property of data.

If one focuses on the extremes of precipitation, andwind velocity, the parametric density function, and beestimated using one of three methods —the method ofmoments(MOM), maximum likelihood method (MLM)and probability weighted moments (PWM). Since eachof them can often lead to different choices of the bestdistribution and its parameters, parameter and modeluncertainty make it difficult to select a probability dis-tribution objectively. The bounded triangular and uni-form distribution also can not reflect the property ofdata (Figure 6).

When the bounded probability distribution like tri-angular one is used, it is important to determine thebounds reasonably. However, the maximum, minimumand average value of the annual maximum data are used

Fig. 3 Problems of present hydrologic dam risk analysis

Fig. 4 PDF of real data

226

as bounds in the current analysis method. In this mind,it is not sufficient to give representativeness into thebounds for initial reservoir level. The parametric ap-proach to determine the bound for triangular distribu-tion is shown in Figure 7.

In hydrologic dam risk analysis, Monte Carlo simu-lation (MCS) is a key technique for solving technicalproblems of the uncertainty. In MCS, it is important togenerate random number corresponding to the proba-bility distribution of each random variable. Generally,SRS (Simple Random Sampling) is used to generateuniform random number between 0 and 1. In manycases of dam risk analysis, the overtopping probabilityrepresents rare chance like 10,000-year return periods.Thus one needs to simulate many times, and besides, thegreater the number of input variables the longer thesimulation time (Sacks et al., 1989).

4 Improvement of hydrologic dam risk analysis

4.1 Latin Hypercube Sampling

In MCS, it is important to generate random numbercorresponding to the probability distribution of each

random variable. Generally, the present risk analysismethod uses simple random sampling (SRS) to generateuniform random number but it is required a lot ofrealizations to reflect rare events. In this point of view,we introduced Latin hypercube sampling (LHS) to thehydrologic dam risk model.

In order to complement SRS, LHS method was pre-sented by McKay et al. (1979). This method for uncer-tainty analysis is very easy to construct and can besimply applied to the MCS. At the same time, it ispossible to uniformly distribute the generated values oncorresponding distribution. Therefore, LHS is mainlyused for simulating large-scale computation. Stein(1987) investigated that LHS decreases dispersion com-pared with SRS, and Owen (1992) proposed a centrallimit theorem for LHS.

After dividing the grids of input variables into thenumber of n, LHS can extract the random number fromeach grid in the space of input variable without repeti-tion. LHS can be defined as the following Equation (7).

xi;j ¼ F �1j1

nPi;j � ri;j� �

ð7Þ

where ri,j means the random number that independentlyfollows the uniform distribution [0, 1] and Pi,j representsrandom permutation. Pi,j, ... , Pn,d decide the gridscontaining xi,j and the correct position in the grids isdetermined by ri,j, ... , rn,d. The LHS having ri,j=1/2 forall i,j is called Center LHS (CLHS). In case ri,j usuallyfollows U[ 0,1 ], it is called Random LHS (RLHS).Table 1 represents the procedure of Latin hypercubesampling. To implement realization, 1) random numberis generated (1st column) and then 2) random permu-tation is carried out (3rd column). LHS can be obtainedas follow: random permutation�1+random number)/total number of random number, that is,(6�1+0.254887)/10=0.525489(1st RLHS).

Figures 8–10 simply compare the results using SRSand CLHS. Figure 8 represents random number gener-ation using SRS. The only 10 random numbers were

Fig. 5 PDF using Normal distribution

Fig. 6 Bounded/Unbounded PDF of W.S.L., The bounds need tobe estimated a priori

Fig. 7 Parametric approach to determine bounds in triangulardistribution

227

sampled, and SRS is not uniformly sampled between 0and 1 because the number of random numbers is notsufficient (Figure 8). Therefore it is a disadvantage inthat the number of simulations will greatly increase inthe uncertainty analysis or risk analysis that deal withthe small chance. However, LHS decreases the numberof simulations, that is, it is possible to sample only onerandom number in 10 positions between 0 and 1. Fig-ure 9 shows ten random numbers using CLHS. CLHS isa random number with the probability of 1/2 in eachgrid (total ten grids) and the random number is simul-taneously sampled one by one at 10 positions between 0and 1.

Figure 10 shows the process of sampling fromGaussian distribution. LHS has a merit in that morecorrectly expresses probability distribution becauserandom number can be uniformly sampled corre-sponding to cumulative probability.

4.2 Monte Carlo Simulation (MCS) using NKDE

The most critical disadvantage in the dam risk analysisusing parametric MCS that it can not define proper

probability distribution for uncertainty variables; i.e.hydrologic variable such as precipitation, wind velocity,coefficient of discharge, initial water surface level, heightof spillway and water quantity are applied to the trian-gular distribution, uniform distribution and the normaldistribution (Cheng et al., 1982). Thus, we employedNKDE method to improve the PDE.

Applying NKDE to hydrologic random variables, wewill only deal with rainfall, wind velocity and initialreservoir water level due to insufficient data. Other threevariables (concentration of time, storage coefficient andSCS CN number) will be applied to parametric boundeddistribution like triangular one.

In MCS, the expected risk value was estimated byexamining large number of repetitive simulation runs.The theoretical failure probability (pf) can be estimatedas follows:

Table 1 Procedure of Latin Hypercube sampling for 10 realizations

RandomNumber

Order ofRandomNumber

RandomPermutation

SRS RLHS CLHS

0.2548878 1 6 0.254887 0.525489 0.550.8998391 2 10 0.899839 0.989984 0.950.5953296 3 8 0.595329 0.759533 0.750.7055610 4 1 0.705561 0.070556 0.050.3636529 5 4 0.363652 0.336365 0.350.9152143 6 7 0.915214 0.691521 0.650.0071959 7 2 0.007195 0.100720 0.150.9428642 8 9 0.942864 0.894286 0.850.7192367 9 5 0.719236 0.471924 0.450.2120264 10 3 0.212026 0.221203 0.25

Fig. 8 Random number generation using SRS, random numbersare not uniformly sampled and two numbers are in the same grid

Fig. 9 Random number generation using CLHS, random numbersare uniformly sampled and only one number is in each grid

Fig. 10 An example of LHS–MCS from Gaussian distribution

228

pf ¼Z� � �Z

D

f ðx1; x2; . . .;xnÞdx1x2; . . . ;xn¼Z

D

f ðxÞdx ð8Þ

where xi represents the basic random variable vector,f(x) represents the joint probability density function ofrandom vector (xi), and D means failure areas.

The present hydrologic dam risk analysis is used ofthe concept of parametric statistical inferences. Someproblems associated with PDE method arise from (1) theobjective selection of a distribution, (2) the reliability ofparameters (especially for skewed data with a short re-cord length), (3) the inability to analyze multimodaldistributions, and (4) the treatment of outliers. There-fore, the classical PDE may not be suitable for modelingthe hydrologic dam risk. However, the NKDE approachdoes not require assumptions of the underlying popu-lations from which the data are obtained and multi-modal distributions can be applied better. Thus,recently, several nonparametric approaches that ispromising for estimating the PDF of hydrologic variablehave been introduced by Moon and Lall (1994), Moonet al. (1993), and many others.

Rosenblatt (1956) introduced the kernel estimator,defined for all real x where x1, x2, ... ,xn are independentidentically distributed real observations, K(.) is a kernelfunction, and h is a bandwidth assumed to tend to zeroas n tends to infinity. Nonparametric kernel densityfunction is defined as Equation (9).

f̂ ðxÞ ¼ 1

n

Xn

i¼ 1

1

hk

x� Xi

h

�ð9Þ

quantile according to cumulative distribution functionF(.) can be estimated by direct integration of kerneldensity function and this equation is as below.

F̂ ðxÞ ¼ 1

n

Xn

i¼ 1

1

hK

x� Xi

h

�ð10Þ

KðtÞ ¼Z t

�1kðuÞdu ð11Þ

If p represents probability which is according to cumu-lative distribution function F(.), and F� 1 is inversefunction of F(.), then the quantile about any real numberx is defined as Equation (12).

x ¼ F �1ðpÞ ¼ infðx 2 < : F ðxÞ>pÞ; p 2 ð0; 1Þ ð12Þ

To implement LHS–MCS using NKDE, randomnumber is sampled by CLHS and NKDE uses annualmaximum data. In the NKDE, the kernel functiondepends on the characteristics of data, that is, theCauchy kernel function is suitable to the extreme pre-cipitation and wind velocity while the Epanechnikovbounded kernel function is a desirable one like theinitial water surface level. The quantile of each variableis generated numerically from cumulative kernel den-sity function corresponding to random number. Thequantile estimation method applied in this study is the

kernel quantile function estimator proposed by Moonand Lall (1994).

4.3 Simulation of water surface level using Bootstrapapproach

In the hydrologic dam risk analysis, the initial watersurface level may be considered as the most sensitiveuncertainty variable. In particular, it is important toconsider a bounded probability distribution within damheight and reflect the properties of data. However, thepresent PDE is not easy to solve these problems. Inaddition, the triangular distribution, uniform distribu-tion and beta distribution having the boundary do notreflect the property of statistics. Thus three approacheshave been suggested to simulate initial water surfacelevel. Figure 11 shows three approaches.

First, for better estimation of probability density ofwater level, we focused on the NKDE using the boundedkernel function like Epanechnikov. In this case, annualmaximum water surface level data is used for simulation.Second, the confidence interval based on bootstrap re-sampling was employed to determine the bounds oftriangular distribution. Third, if the annual maximumwater surface level is used as an initial reservoir level inthe hydrologic risk analysis, the overtopping probabilitywould be overestimated because previous floods alreadyhave been reflected in the water surface level. Thus,bootstrap resampling was considered to simulate theinitial water surface level on purpose to compare withother cases. In advance, the water surface level was di-vided into flood season (June–September) and non-floodseason (October–May), and the only flood season dailywater surface level was used.

To estimate the confidence intervals, it is desirable tobe approximated with Fourier series using equation (13).Figure 12 represents bootstrap resampling of watersurface level using Fourier series. The water surface levelis fitted using Fourier series as follows (Sutherland et al.,1988).

f ðtÞ ¼ lþXK

k¼1ak cos

2pktTÞ þ bk sinð

2pktT

�� ð13Þ

To transform the time series data into the Fourierseries curve, the least square method is used to estimateparameters. The confidence interval is estimated throughthe following Equation (14) and Equation (15). Fig-ure 13 shows the procedure for estimating confidenceintervals using bootstrap.

1

B�XB

b¼1max

f̂ bðtÞ � f̂ ðtÞ

r̂bf̂ bðtÞ

!6CC

" #¼ 1� a ð14Þ

f̂ ðtÞ � CC � r̂f̂ ðtÞ ð15Þ

where, B means the numbers of bootstrap and a meansthe confidence limit.

229

5 Hydrologic dam risk analysis of existing dam

5.1 Study catchment and data

The proposed hydrologic dam risk model is applied theSoyang Dam in South Korea. The Soyang multipurposeDam was constructed in 1973 and the dam height andlength are 123 m (from 150 El.m to 203 El.m) and530 m respectively. The spillway of Soyang Dam wasdesigned as a 1,000-year flood. The effective reservoirstorage of dam is 1,900 106 m3. The normal water sur-face is 193.5 El.m and the restricted water surface in

flood season is 190.3 El.m. The primary spillway is lo-cated on the left embankment and the maximum flowrate is 5,500m3/sec.

The Soyang Dam was chosen to target region becausethere are comparatively abundant hydrologic data, andbesides, that is main dam in South Korea. The SoyangDam includes three catchments, Inbuk, Naerin andSoyang. Soyang River totally has 166 km length and thelargest tributary of north Han River. The region ismostly enclosed with mountains. The mean slope of thearea is approximately 1/800–1/1,000. The catchment’scharacteristics are summarized in Table 2.

Fig. 11 Procedure ofsimulation for initial reservoirlevel

Fig. 12 Bootstrap resampling of water surface level using Fourier series

230

To evaluate the overtopping probability, this studyfocuses on the annual maximum data except water sur-face level for bootstrap technique. The available data inthe Soyang Dam is represented in Table 3. The 12rainfall stations are available in the region.

5.2 Uncertainty analysis of rainfall-runoff modeling

The rainfall-runoff model is a core one in the hydrologicdam risk analysis. In this study, HEC-1, a representativerainfall-runoff event model developed by U.S. ArmyCorps of Engineers, was used to transform rainfall intopeak discharge. In the rainfall-runoff modeling, thecalibration is always required but one often may expe-rience that each event indicates different parameters.These uncertainties may be explained by antecedentmoisture condition (AMC), the differences of temporaland spatial distribution of rainfall. Thus the range of theparameters will be evaluated by uncertainty analysis,and the estimated range is applied to generate peakdischarge in the proposed dam risk model.

To calculate the discharge, the Clark instantaneousunit hydrograph, SCS CN No. and Muskingum channelrouting methods are used in the HEC-1. The procedureof rainfall-runoff is summarized in Fig. 14.

In three catchments, the uncertainty bands of time ofconcentration, storage coefficient and SCS CN No. wereevaluated. Based on the estimated values, a lot of peakdischarge data was generated by using LHS–MCS.

The major flood events in 1984, 1990, 1995 and 1998were used to calibrate HEC-1. The parameters werecalibrated with the optimization method of HEC-1applying the following Equation (16). The Equation (16)shows the error in percentage of peak discharge, qs (t)and q0 (t) indicates predictive discharge and observeddischarge, respectively.

Z ¼ 100qsðpeakÞ � q0ðpeakÞ

q0ðpeak

ð16Þ

Thus the minimum value of time of concentration andstorage coefficient, their maximum value and meanvalues were used for estimating the range of uncertain-ties. In addition, the uncertainty band of the SCS CNNo. is evaluated by using AMC-I, AMC-II and AMC-III that is the minimum, mean and maximum value,respectively. The estimated range in the uncertainty ofmajor parameters is shown in Table 4.

In order to verify rainfall-runoff model, 1991 floodevent was used and generated 200 times to determineuncertainty bands. The time of concentration and stor-age coefficient was simulated as a uniform distribution,and SCS CN No. was generated as a triangular distri-bution. Figure 15 shows the uncertainty bands of esti-mated hydrograph. The uncertainty band was estimatedas follows: mean value±2·standard deviation.

Fig. 13 Procedure ofconfidence intervals usingbootstrap method

Table 2 Catchments characteristics

Catchment Area (km2) Watershed Slope (m/m)

Inbuk 923.8 0.01268Naerin 1069.3 0.01168Soyang 709.9 0.03529

Stream Stream Length (m) Stream Slope (m/m)Soyang River 55,800 0.00155

Table 3 Data used in the studyHydrologic Data No. of

StationPeriodofrecord

Max. Average standarddeviation

Remark

Rainfall(mm) 12 1964–2003 (40) 287.18 129.09 46.73 Annual Maximum(duration:24 hr)

Water surface level(El.m) 1 1974–2003 (30) 197.99 186.51 6.29 Annual MaximumWater surface level(El.m) 1 1974–2003 (30) 197.93 174.47 8.67 Daily AverageWind velocity(km/hr) 1 1964–2003 (40) 82.08 45.35 10.57 Annual Maximum

231

5.3 Rainfall duration time

To investigate overtopping probability of dam, rainfallduration time to cause the largest-scale flood is neces-sarily estimated. For this objectives, rainfall durationtime has been estimated by each frequency and dura-tion time. After estimating design rainfall according toeach duration time from Intensity–Duration–Fre-quency (IDF) curve, the estimated design rainfall istemporally distributed by Huff distribution. The effec-tive rainfall is transformed into direct discharge byClark instantaneous unit hydrograph, SCS CN No.and Muskingum channel routing like the previousuncertainty analysis.

They show somewhat different values with HuffQuartile and the rainfall duration time causing thelargest-scale flood in Soyang Dam was approximatelyestimated to fourth Huff quartile 24 h. The estimatedpeak discharge is summarized in Table 5.

Fig. 14 Procedure of rainfall-runoff modeling using HEC-1

Table 4 Range of rainfall-runoff parameters Sub Basin Time of Concentration

(hr)Storage Coefficient (hr) SCS CN Number

Min. Mean Max. Min. Mean Max. Min. Mean Max.

Inbuk 1.34 3.53 4.84 1.80 6.31 9.68 39.8 61.1 78.3Naerin 1.43 2.46 3.30 1.84 2.56 2.81 49.6 70.1 84.4Soyang Dam 0.99 1.26 1.52 1.30 2.26 3.16 56.6 75.6 87.7

Table 5 Rainfall durationtime in study area Frequency

(year)1st Huff Quartile 2nd Huff Quartile 3rd Huff Quartile 4th Huff Quartile

RainfallDurationTime (min)

PeakFlood(CMS)

CriticalDurationTime (min)

PeakFlood(CMS)

CriticalDurationTime (min)

PeakFlood(CMS)

CriticalDurationTime (min)

PeakFlood(CMS)

50 1980 11,685 1260 11,803 1680 11,583 1440 14,294100 1980 13,280 1380 13,367 1680 13,041 1440 16,073200 1980 14,871 1440 14,913 1680 14,458 1440 17,824300 2040 15,817 1260 15,829 1680 15,314 1440 18,861500 2040 17,004 1260 16,989 1680 16,376 1440 20,154

Fig. 15 Uncertainty analysis of rainfall-runoff model (CLHS)

232

5.4 Outline of simulation for uncertainty variables

Before initiating the proposed approach, we shouldevaluate the probability density function for eachuncertainty variables such as precipitation, wind veloc-ity, initial water level, time of concentration, storagecoefficient, and SCS CN No..

The PDF of rainfall and wind velocity were simulatedusing NKDE method. In estimating quantile function,Cauchy kernel function was selected among variouskernel functions because it is suitable to extreme event,and Plug-In method (Sheather and Jones 1991) was usedto estimate density function. We also applied regionalfrequency analysis to rainfall and point frequencyanalysis to wind velocity on purpose to compare withother methods. Two approaches take advantage of an-nual maximum data. The cases for hydrologic dam riskanalysis are summarized in Table 6.

For initial water surface level, three approaches wereconsidered. First, the NKDE method was considered forthe annual maximum data. Plug-In method (Sheatherand Jones, 1991) and Epanechnikov kernel function wasused to estimate density function. Second, the confi-dence interval based on bootstrap resampling was em-ployed for determining the bounds of triangulardistribution. Third, bootstrap resampling was used formodeling the initial water surface level. Finally, theparameters of rainfall-runoff model were generated byuniform and triangular distribution.

5.4.1 Establishing PDF for Precipitation

For the PDE, the probability distributions are assumedto be underlying distribution for the rainfall. We adop-ted regional frequency analysis using L-moments (Hos-king, 1990). L-moments are linear combinations of orderstatistics which are robust to outliers and virtuallyunbiased for small samples (Hosking, 1990; Hosking

and Wallis, 1993). The results of the L-moment tests forthe entire study area, 12 stations, are homogeneous sincethe values of H are smaller than 1. Provided homoge-nous within a region, regional frequency analysis be-longs to a point frequency analysis of a singledistribution.

A goodness-of-fit test based on L-moments can beused to select an appropriate one of various distributions(Log Pearson type III, GEV, General Pareto, GeneralLogistic, etc.) and to estimate its parameters. Thegoodness-of-fit criterion for selecting the distribution isdefined as termed the Z-statistic (Hosking and Wallis,1993). The Z-statistic shown in Table 7 suggests that theGEV distribution can be selected for the rainfall in theregional frequency approach.

The NKDE method was applied to the rainfall.Figure 16 represents the PDF and shows the advantagesof NKDE, that is, NKDE effectively describes themultimodal.

5.4.2 Establishing PDF of wind velocity

To simulate wind velocity, the PDE and the NKDE areapplied to the annual maximum data. In case of windvelocity, point frequency analysis is available becausestudy area has one station. The parameters of each

Table 6 The cases of hydrologic dam risk analysis

Case Rainfall-RunoffModel Parameter

Rainfall WindVelocity

Initial ReservoirLevel

I Triangular and UniformDistribution

Regional FrequencyAnalysis (GEV Distribution)

Point FrequencyAnalysis (GEV Distribution)

TriangularDistribution

II Triangular and UniformDistribution

NonparametricApproach

NonparametricApproach

NonparametricApproach

III Triangular and UniformDistribution

NonparametricApproach

Nonparametric Approach Triangular Distribution(Confidence Interval)

IV Triangular and UniformDistribution

NonparametricApproach

NonparametricApproach

BootstrapResampling

Table 7 The results of goodness-of-fit tests

Probability Distribution L-Kurtosis Z Value

General Logistic 0.233 1.17General Extreme Value 0.204 �0.05General Normal 0.185 �0.86Pearson Type III 0.152 �2.26General Pareto 0.129 �3.25

Fig. 16 PDF of precipitation using NKDE

233

probability distribution are estimated by PWM andthen the goodness of fit tests of parameters such asKolmogorov-Smirnov, chi square and Cramer vonMises are applied at significance level 0.05. The good-ness of fit tests is summarized in Table 8. The GEV,Log Normal and Log Gumbel etc. distributions can beconsidered for hydrologic dam risk analysis. For thePDE, the GEV distribution was selected for point fre-quency approach.

The NKDE method was applied to the wind velocityand Figure 17 represents the PDF using the NKDEmethod. It effectively describes the multimodal and iscompared to the empirical density function using thenumber of class intervals presented by Sturges (1926).

5.4.3 Establishing a PDF for Initial Water Surface Level

The NKDE enabled us to estimate quantile within therange that does not exceed the height of the dam byusing bounded kernel function. Figure 18 represents thePDF using the NKDE method and it effectively de-scribes the multimodal. The bootstrap approach is usedfor two purposes. One is the confidence interval todetermine the bounds of triangular distribution, andanother is the bootstrap resampling as itself.

When parametric approaches are employed for initialreservoir level, it is important to set the bounds rea-sonably in bounded probability distribution. In thissense, we estimated the confidence intervals using thebootstrap technique. Figure 19 shows confidence inter-vals at significance level 0.1. The confidence interval canbe used as a bound for triangular distribution. Deter-mining bound for triangular distribution, we limitedbound to the highest band.

5.5 Estimation of hydrologic overtopping probability

To estimate the hydrologic overtopping probability, theproposed procedures are applied to the Soyang Dam

Table 8 The results of goodness-of-fit tests

ProbabilityDistribution

Chi Square Kolmogorov-Smirnov Cramer von Mises

Compute Table Remark Compute Table Remark Compute Table Remark

Gamma-2 4.742 5.990 O.K 0.203 0.238 O.K 0.225 0.461 O.KGamma-3 0.412 7.380 O.K 0.106 0.273 O.K 0.054 0.743 O.KGEV 0.912 3.840 O.K 0.145 0.238 O.K 0.177 0.461 O.KGumbel 6.258 5.990 N.G 0.165 0.238 O.K 0.158 0.461 O.KLog-Gumbel 2 3.355 5.990 O.K 0.128 0.238 O.K 0.147 0.461 O.KLog-Gumbel 3 3.355 3.840 O.K 0.136 0.238 O.K 0.145 0.461 O.KLog-Normal 3 1.294 7.380 O.K 0.099 0.273 O.K 0.045 0.743 O.KLog-Pearson type III 7.839 3.840 N.G 0.150 0.238 O.K 0.146 0.461 O.KWeibull-2 4.000 5.990 N.G 0.252 0.238 N.G 0.390 0.461 O.KWeibull-3 – – N.G – – N.G – – N.GWakeby-4 11.39 3.840 N.G 0.183 0.238 O.K 0.187 0.461 O.KWakeby-5 3.548 3.840 O.K 0.096 0.238 O.K 0.058 0.461 O.K

O.K accept, N.G reject

Table 9 Statistics of observed and simulated variables

Variable Average StandardDeviation

Maximum

Rainfall Case Obs. (mm) Sim. (mm) Obs. Sim. Obs. (mm) Sim. (mm)I 129.09 135.07 46.73 71.77 287.18 2058.09II-IV 129.25 50.35 497.2

Wind Velocity Case Obs. (km/hr) Sim. (km/hr) Obs. Sim. Obs. (km/hr) Sim. (km/hr)I 45.35 45.35 10.57 9.19 82.08 145.49II-IV 45.37 10.97 116.15

Initial WaterSurface Level

Case Obs. (El.m) Sim. (El.m) Obs. Sim. Obs. (El.m) Sim. (El.m)I 186.51 185.89 6.29 5.07 197.99 197.98II 186.52 6.26 200.38III 174.47 183.13 8.67 7.11 197.93 200.52IV 175.17 9.44 197.93

Table 10 The overtopping probability of each case

Cause of Failure Overtopping Probability (p/year)

Flood Wind Flood andWind

ParametricApproach

I 1.02E-03 0.00E+00 1.22E-03

NonparametricApproach

II 6.45E-04 0.00E+00 1.16E-03III 2.15E-04 0.00E+00 3.73E-04IV 2.00E-06 0.00E+00 5.11E-06

234

and the four cases have been initiated. The four casesdepend on methods to evaluate probability densityfunction, and are summarized in Table 6. In this study,

200,000 simulations were implemented. The statistics ofsimulated variable is shown in Table 9. Table 10 showsthe overtopping probability of each case.

In the first case, the regional frequency analysis isapplied to the rainfall, and the overtopping probabilityindicates 1.22E-03 that is relatively high. With regard to1,000-year design flood, this seems reasonable but thesimulated rainfall indicates 2,058 mm which is unreal-istic exceeding the PMP (573 mm). Therefore, we draw aconclusion that the rainfall is the main cause for over-topping dam and the parametric approach is not enoughone for the hydrologic dam risk analysis.

Second, we focus on the NKDE approach for threemain variables, and the overtopping probability is esti-mated at 1.16E-03 that is a high value as much as re-gional frequency analysis of rainfall. In contrast to theregional frequency analysis of rainfall, the simulatedrainfall is reasonable with regard to PMP in study area.In addition to, the simulated maximum water surfacelevel is estimated at 200.38 El.m.

Third, the confidence limit based on bootstrap isapplied to the daily water surface level. The highest bandamong confidence limits is used as a bound in the tri-angular distribution. The overtopping probability indi-cates 3.73E-04 that is similar to that of case II. Thesesresults come from the simulated water surface level asmuch as NKDE approach. Therefore, it might be

Fig. 17 PDF of wind velocity using NKDE

Fig. 18 Probability density function for dam water surface level(NKDE)

Table 11 Results of the sensitivity analysis for the major variables

Rainfall(mm)

OvertoppingProbability (p/year)

Wind Velocity(km/hr)

OvertoppingProbability (p/year)

Water SurfaceLevel (El.m)

OvertoppingProbability (p/year)

100 0.00E+00 30 9.94E-04 185.0 0.00E+00200 1.25E-03 50 1.20E-03 187.5 0.00E+00300 3.38E-02 70 1.63E-03 190.0 0.00E+00400 8.03E-02 90 2.10E-03 192.5 0.00E+00500 1.47E-01 110 2.68E-03 193.0 0.00E+00600 2.50E-01 130 3.48E-03 193.5 5.00E-06

150 4.41E-03 194.0 5.11E-06195.0 5.11E-06197.5 3.03E-03200.0 8.58E-02

Fig. 19 Determination of bound using the bootstrap confidencelimit

235

interpreted as a suggestion of the possibility of bootstrapapproach to determine the bound of distribution inhydrologic dam risk analysis.

Finally, we employ the bootstrap resampling togenerate initial water surface level. The overtoppingprobability turned out to be relatively small valuecompared with other cases. In this study the bootstrapresampling approach extracts daily water surface levelduring the flood season. Otherwise other approaches useannual maximum data of variables. Thus, this resultmight be reasonable when considered above discussion.

5.6 Sensitivity analysis of model

We investigated the sensitivity of the proposed model.The sensitive analysis has the intention of understandingthe physical relation with models that are modeled withit (McCuen, 1973). The sensitive analysis is to estimatethe overtopping probability having a fixed value, andthen other variables are generated simultaneously. Ta-ble 11 shows the sensitive analysis results of each vari-able. The results from the sensitive analysis show thatthe initial water surface level shows the most sensitivevariable.

The overtopping probability increases remarkablyfrom 195E1.m. At this point, one can identify that thedam needs to maintain the initial water level below195E1.m in flood season. For the rainfall, when the 24-hour rainfall exceeds 300 mm during flood period, itleads the dam in a dangerous situation and we need damoperation. The wind velocity has little effect on theovertopping probability compared with the rainfall orinitial water level.

6 Conclusion

This study has introduced three major innovations. Firstof all, this article has attempted to use nonparametricprobability density estimation methods for selectedvariables. Secondly, much effort has been put on in or-der to improve the efficiency of Monte Carlo simulationsdriven by the multiple random variables by using LatinHypercube sampling. Thirdly, the Bootstrap approachhas employed to determine the initial water surface level.Based on the established risk analysis method, anapplication to the Soyang Dam in Korea has beenillustrated on how the traditional parametric approachcan lead to potentially unrealistic estimates of damsafety, while the proposed approach provided ratherreasonable estimates and an assessment of its sensitivityto key parameters. Overall, this study adds to theexisting literature on possible alternatives. To sum up,the following conclusions can be deduced.

All this considered, the parametric approach can leadto unreasonable results for the hydrologic dam riskanalysis. In contrast to the parametric approach, thenonparametric kernel density estimation approach

showed that the simulated variable is reasonable withregard to the PMP in the study area. The confidencelimit based on a bootstrap is applied to the daily watersurface level and might be interpreted as a suggestion ofa possibility of a bootstrap approach to determine thebound of distribution in hydrologic dam risk analysis. Inlight of the above results, the Soyang Dam needsstructural and nonstructural countermeasures, whichlessens the risks. Especially, the hydrologic risk takeslarge parts. Hence it is important to make structural riskreduction measures, such as the expansion of spillwaysand the construction of an outlet.

These tentative conclusions await further refinementand improvement by conducting further research for thecontinuous rainfall-runoff modeling, and the extrapola-tion of variables and the fully statistical integration of arisk model with structure and geotechnology. All factorscontributing to the hydrologic risks (wind, flood, waveetc) have assumed to be independent but can be a prob-lem in situations where these factors maybe coupled.Under these circumstances, the multivariate kernel den-sity estimationmethod or the Bayesian approach that canconvey the dependence between variables will be useful.

References

Bowles DS (1987) A comparison of methods for integrated riskassessment of dams. In: Duckstein L, Plate E (eds) Engineeringreliability and risk in water resources. M. Nijhoff, Dordrecht

Cheng ST (1982) Overtopping risk evaluation for an existing dam.PhD Thesis, University of Illinois, Urbana

Falk M (1986) On the estimation of the quantile density function.Stat& Prob Lett 4:69–73

Foufoula-Georgiou E (1989) A probabilistic storm transpositionapproach for estimating exceedance probabilities of extremeprecipitation depths. Water Resour Res 25:799–815

Haimes YY, Stakhiv EZ (eds) (1986) Risk-based decision makingin water resources. (Proceedings of an engineering foundationconference, Santa Barbara, CA, November 3–5, 1985). ASCE,New York, NY

Haimes YY, Petrakian R, Karlsson PO, Mitsiopoulos J (1988) Riskanalysis of dam failure and extreme floods: application of thepartitioned multiobjective risk method. IWR Report 88-R-1,Environmental Systems Management, Inc, Charlottesville, VA

Hosking JRM (1990) L-moments: analysis and estimation of dis-tributions using linear combinations of ordered statistics. J RStatis Soc Ser B 52(1):105–124

Hosking JRM, Wallis JR (1993) Some statistics useful in regionalfrequency analysis. Water Resour Res 29(2):271–281

Hosking JRM, Wallis JR (1997) Regional Frequency Analysis.Cambridge University Press, Cambridge

Karlsson PO, Haimes YY (1988a) Risk-based analysis of extremeevents. Water Resour Res 24(1):9–20

Karlsson PO, Haimes YY (1988b) Probability distributions andtheir partitioning. Water Resour Res 24(1):21–29

Karlsson PO, Haimes YY (1989) Risk assessment of extremeevents: Application. J Water Resour Plan and Manag115(3):299–320

Langseth DE, Perkins FE (1983) The influence of dam failureprobabilities on spillway analysis. Proceedings of the Confer-ence on Frontiers in Hydraulic Engineering, ASCE: 459–464

Larrabee RD, Cornell CA (1979) A combination procedure for awide class of loading process. Proceedings of the SpecialtyConference on Probabilistic Mechanics and Structural Reli-ability, ASCE: 76–80

236

Lombardi G (2002) Preparatory works and regularmaintenance.Dam Maintenance and Rehabilitation: 7–15

McCann MW Jr (1985) Proceedings of a Workshop on CurrentDevelopments in Dam Safety Management

McCuen RH (1973) The role of sensitivity analysis in hydrologicmodeling. J Hydrol 18:37–53

McKay MD, Beckman RJ, Conover WJ (1979) A comparison ofthree methods for selecting values of input variables in theanalysis of output from a computer code. Technometrics21:239–245

Moon Y-I, Lall U (1994) Kernel Quantile Function Estimator forFlood Frequency Analysis. Water Resour Res 30(11):3095–3103

Moon Y-I, Lall U, Bosworth K (1993) A comparison of tailprobability estimators. Journal of Hydrology 151:343–363

Moser DA, Stakhiv EZ (1987) Risk analysis considerations fordam safety. In: Duckstein L, Plate E (eds) Engineering Reli-ability and Risk in Water Resources. M Nijhoff, Dordrecht

National Research Council (1988) Committee on techniques forestimating probabilities of extreme floods, estimating proba-bilities of extreme floods, methods and recommended research.National Academy Press, Washington DC

Owen AB (1992) A central limit theorem for latin hypercubesampling. Journal of the Royal Statistical Society 54:541–551

Petrakian R, Haimes YY, Stakhiv E, Moser D (1989)Risk analysisof dam failure and extreme floods. risk analysis and manage-ment of natural and man-made hazards, Haimes YY, StakhivEZ (eds) ASCE: 81–121

Pohl R (1999) Estimation of the probability of hydraulic-hydro-logical failure of dams. In: Proceedings of the InternationalWorkshop on risk analysis in dam safety assessment, Taipei,Taiwan, R.o.C: 143–157

Resendiz-Carrillo D, Lave LB (1987) Optimizing spillway capacitywith an estimated distribution of floods. Water Resour Res23(11):2043–2049

Rosenblatt M (1956) Remarks on some nonparametric estimates ofa density fuction. Ann Math Stat 27:832–837

Sacks J, Welch W, Mitchell T, Wynn H (1989) Designs and analysisof computer experiments (with discussion). Statistical Science4:409–435

Sheather SJ, Jones MC (1991) A reliable data-based bandwidthselection method for kernel density estimation. J Roy Stat SocB 53:683–690

Stedinger JR, Grygier J (1985) Risk-cost analysis and spillwaydesign. In: Torno HC (ed) Computer applications in water re-sources. ASCE: 1208–1217

Stein M (1987) Large sample properties of simulation using Latinhypercube sampling. Technometrics 29:143–151

Sturges HA (1926) The choice of a class interval. J Am StatAssn21:65–66

Sutherland DH, Olshen RA, Biden EN, Wyatt MP (1988) Thedevelopment of mature walking. MacKeith Press, London

Von Thun JL (1987) Use of risk-based analysis in making decisionson dam safety. In: Duckstein L, Plate E (eds) Engineeringreliability and risk in water resources. M Nijhoff, Dordrecht

Wen YK (1977) Statistical combination of extreme loads. J StructDiv 103(5):1079–1103

237