Abstract: In the present study L-moment approach was used for flood frequency analysis in Halil-River basin. To identify the homogeneous regions, the Ward hierarchical cluster method was used. For independent testing of the cluster in the station for homogeneity, site data information was used. The Halil-River basins were divided in to two regions (region A and B) and parameters of the regional frequency distribution were evaluated by L-moment ratios. For the selection of appropriate distributions, L-moment diagram, goodness of fit test and plotting position methods were used. The results showed that in Halil-River basin, Generalized Pareto distribution for region A, Generalized extreme values, Pearson type III, Lognormal, Generalized Logistic and Generalized Pareto for region B are appropriate distributions. The relative Root Mean Square Error (rRMSE) between observed and estimated data in all stations was calculated. The results showed a good agreement between observed and estimated data. Regional model evaluated for determination of mean flood discharge magnitude by liner and multiple regression method.
Introduction
Estimation of extreme events is an important practical application in hydrology, especially because the planning and design of water resource projects and flood-plain management, which depend on the frequency and magnitude of peak discharges. Information on flood magnitudes and their frequencies is needed for design of hydraulic structures such as dams, spillways, road and railway bridges, culverts, urban drainage systems, flood plain zoning, economic evaluation of flood protection projects etc (Kumar et al., 2003). Regional flood frequency analysis is usually applied when no local data at a site of interest are available, or the data are insufficient for a reliable estimation of flood quantiles for the required return period. Regional flood frequency analysis has three major components, namely, delineation of homogeneous region, determination of appropriate probability density function, (or frequency curves), of the observed data and the development of a regional flood frequency model, such as a relationship between flows of different return periods, basin characteristics and climatic data. This study includes identification of homogeneous regions based on cluster analysis of site characteristics, identification of suitable regional frequency distribution and development of a regional flood frequency models.
L-moments
Recently, Hosking (1990) has defined L-moments approaches, which are
analogous to conventional moments and can be expressed in terms of linear
combinations of order statistics. Basically, L-moments are linear functions
of Probability-Weighted Moments (PWMs) (Sankarasubramania and Srinivasan,
1999). Procedures based on PWM and L-moment, are equivalent but L-moment
is more convenient, because they are directly interpretable as measure
of the scale and of the shape of probability distributions (Mckerchur,
1994). L-moments are robust to outliers and virtitually unbiased for small
samples, making them suitable for flood frequency analysis (Adamowski,
2000; Lee and Meang, 2003). Similar to conventional moments, the purpose
of L-moments and probability-weighted moments is to summaries theoretical
distribution and observed samples. Greenwood summarizes the theory of
PWM and defined them as below (Schulze and Smithers, 2002):
βr = E{X[FX(x)r} |
(1) |
where βr is the rth order PWM and FX(x) is the cumulative distribution function (cdf) of X. Unbiased sample estimators (βj) of the first four PWMs are given as:
| (2) |
where X(j) represents the ranked Annual Maximum Series (AMS) with X(1) being the highest value and X(n) the lowest value, respectively. The first four L-moment are given as follow:
| (3) |
Unbiased sample estimators of the first four L-moments are obtained by substituting the PWM sample estimators from Eq. (2) into Eq. (3). The first L-moment λ0 is equal to the mean value of X. Finally, the L-moment ratios are calculated as:
| (4) |
Sample estimates of L-moment ratios are obtained by substituting the L-moments in Eq. (4) with sample L-moments (Hosking and Wallis, 1997).
Index Flood
The T-year event XT is defined as the event exceeded on
average once every T years (Schulze and Smithers, 2002). When the annual
maximum floods are distributed according to a specified frequency distribution
with cdf, the T-year event can be calculated as Cadman et al. (2003):
XT = F-1(1-1/T) |
(5) |
Regional frequency analysis methods, such as the index flood method, include information from nearly stations exhibiting similar statistical behavior as at the site under consideration in order to obtain more reliable estimates (Schulze and Smithers, 2002; Rakesh et al., 2003). Regional methods can also be used to obtain estimates at ungauged sites, which are important in region such as Halil-River basin, where the flow gaugaing network density is relatively low. Consider a homogeneous region with N sites, each site i having sample size ni and observed AMS xij, j = 1,…,ni. The AMS from a homogeneous region are identically distributed except for a site-specific scaling factor and the index flood. At each site the AMS is normalized using the index flood as:
| (6) |
Where μii is the Mean Annual Flood (MAF) at site i, which is often used as the index flood. The sample L-moment ratios are estimated at each site and the regional record length weighted average L-moment ratios are calculated as:
| (7) |
Where
| (8) |
Where
Identification of Homogeneous Regions
In Halil-River basin six hydrometric sites, which have sufficient
length record of data and are important for frequency analysis, were selected.
For identification of homogeneous regions, Hosking and Wallis recommended
using Ward’s method, which is a hierarchical clustering method based
on minimizing the Euclidean distance in site characteristics space within
each cluster. The site characteristics selected in this study for each
site included: Latitude (LAT) and longitude (LON) of the flow gauging
weir, Mean Annual Flood (MAF), station area (AREA), altitude (ALT) and
design storm intensity (ID). Table 1 shows the site
characters for six stations in Halil-River basin. Using this method Halil-River
basin divided to two regions (A and B). After identification of homogeneous
regions, using Hosking’s method discordancy measure (Di) of the sites
was determinate in each region. Table 2 shows the L-moment
ratios and discordances measure for region A and B stations. Figure
1 shows location of gauging sites and homogenous regions in Halil-River
basin.
Heterogeneity Test
Hosking and Wallis (1997) proposed a statistical test based on L-moment
ratios for testing the heterogeneity of the proposed regions. The test
compares the between-site variation in sample L-CV with the expected variation
for a homogeneous region. The method fits a four parameters kappa distribution
to the regional average L-moment ratios. The estimated kappa distribution
is used to generate 500 homogeneous regions with population parameters
equal to the regional average sample L-moment ratios. The properties of
the simulated homogeneous region are compared to the sample L-moment ratios
as
| (9) |
| Table 1: | Site characteristic for Halil-River stations |
| Table 2: | L-moment ratios and discordances measure for region A and B stations |
| Fig. 1: | Homogeneous regions (A and B) and Halil-River stations (Konarueyeh, Henjan, Maydan, Cheshm Aroos, Pol Baft and Soltani) |
Where μV is the mean of simulated V values and σV is the standard deviation of simulated V values. For the sample and simulated regions, respectively, V is calculated as:
| (10) |
Where N is the number of sites, ni is the record length at site i, t(i) is the sample L-CV at site I and tR is the regional average sample L-Cv. If H<1, the region can be regarded as ‘acceptable homogeneous’, 1≤H<2 is ‘possible homogeneous’ and H≥2 is ‘definitely heterogeneous’ (Hosking and Wallis, 1997).
Goodness-of-fit Test
The goodness-of-fit test described by Hosking and Wallis (1997) is
based on a comparison between sample L-kurtosis and population L-kurtosis
for different distributions. The test statistic is termed ZDIST
and given as follows:
| (11) |
Where DIST refer to a candidate distribution. τ4DIST is the population L-kurtosis of selected distribution, t4R is the regional average sample L-kurtosis and σ4 is the standard deviation of regional average sample L-kurtosis. A four-parameter kappa distribution is fitted to the regional average sample L-moment ratios. The kappa distribution was used to simulate 500 regions similar to the observed regions. From these simulated regions B4 and σ4 are estimated. Declare the fit to be adequate if ZDIST is sufficiently close to zero, a reasonable reiteration for selection of suitable being |ZDIST| (Hosking and Wallis, 1997). The test described above applied to the four homogeneous regions. For each region the data were tested against the General Logistic (GLO), General Pareto (GPA), General Extreme Value (GEV), General Normal (GNO) and Pearson Type 3 (PE3) distribution. Table 3 shows the results.
Regional Flood Frequency Distribution
Several methods are available for selecting appropriate regional distributions.
In this study the regional frequency distributions were selected based
on the results of L-moment ratios as described by Hosking and Wallis (1997).
Additionally probability plots (plotting position) were used to verify
that the selected distributions provided a satisfactory description of
the observed AMS.
L-moment Ratio Diagrams
An L-moment ratio diagram of L-kurtosis versus L-skewness compares
sample estimates of the dimension less ratios with their population counterparts
for ranges of statistical distributions include GLO, GEV, GNO, PE3 and
GPA. L-moment diagrams are useful for discerning grouping of sites with
similar flood frequency behavior and identifying the statistical distribution
likely to adequately describe this behavior. Figure 2
shows the L-moment ratio diagram for homogeneous regions in Halil-River
basin (A and B). As the sample L-moments, are unbiased, the sample points
should be distributed above and below the theoretical line of a suitable
distribution (Hosking and Wallis, 1997). From the above L-moment diagrams,
it appears that the GPA distribution for region A and the GPA, GLO, GEV,
PE3 and GNO for region B are appropriate.
Plotting Position
As pointed out by Hosking et al. (1985), comparison of different
regional frequency distributions against observed data cannot be used
to discriminate between different distributions, as the observed data
represents only one of an infinite number of realizations of the ‘true’
underlying population (Schulze and Smithers, 2002). However, the probability
plots may reveal tendencies such as systematic regional bias in the estimation
of the extreme events. To assess how well the proposed regional frequency
distribution fit to the observed AMS, the calculated XT-T relationships
for Konarueyeh station in region B are shown in Fig. 3.
The empirical existence probability for the ordered observations x(i)
were calculated using the median probability plotting position as described
by Hosking and show below:
| Fig. 2: | L-moment ratio diagram for homogeneous regions (A and B) in Halil-River basin |
| Fig. 3: | Probability plot for Konarueyeh station in region B |
| Fig. 4: | rRMSE between computed and observed data for Konarueyeh station |
| Table 3: | Regional parameters for the various distributions for the region B |
| (12) |
From above three methods, goodness-of-fit test, L-moment ratio diagram and plotting position, the GPA distribution for region A and the GNO, GEV, GLO, GPA and PE3 distributions for region B were selected as regional frequency distributions.
Quantizes Estimation
After the regional distributions were selected, using these distributions
the quantiles with different nonexceedance probability estimated for regions
A and B in Halil-River basin. Table 3 shows the estimated
value using GPA distribution in regions A and B.
The accuracy of estimated values (regional and at-site estimations) was determinate using relative Root Mean Squire Error (rRMSE). Figure 4 shows the rRMSE in Konarueyeh station. From these charts the rRMSE values in high return period are low. This indicates that both at-site and regional estimation procedure in high return period give accurate results.
Conclusions
In this study using site characteristic and Ward’s method, hierarchical clustering method based on minimizing the Euclidean distance in site characteristics space within each cluster, the Halil-River basin divided into two acceptably homogeneous regions. The heterogeneity measures based on H1 were -1.67 and 1.88 for regions A and B, respectively. The identification of suitable regional distribution for each of two regions was based on the L-moment diagram, a goodness-of-fit test and evaluated using probability plots. The GPA distribution for region A and PE3, GNO, GLO, GPA and GEV distributions for region B were suitable and selected. The rRMSE values between computed and observed data were obtained. These values in high return period were low and indicate that both at-site and regional estimation procedure in high return period give accurate results. Regional models for homogeneous regions was obtained using the multiple regression and step-wise method and with catchment and hydrologic characteristics.