1 Introduction

Extreme environmental events, such as high-intensity rains and floods, can have substantial impacts on society and the economy. The estimation of their return periods and the magnitude of extreme event are of great importance in hydrologic modeling, reservoir management, and also design of hydraulic structures such as bridges, dams, spillways, culverts, and irrigation ditches.

In a relatively young country like Malaysia, annual rainfall or river flow records are all too often inadequate or unavailable to allow for reliable estimation of extreme events at a location of interest. This poses a difficulty for hydrologist or engineers to derive reliable flood estimates. Therefore, estimation using regional frequency analysis is a popular and practical means of providing flood information at stations with little or no flow data available for the purposes of flood control and engineering economic (Jingyi and Hall 2004).

Hosking and Wallis (1997) have developed a new approach to regional frequency analysis based on L-moments. This technique has been used at all stages of regional analysis including the identification of homogeneous regions, identification and testing of regional frequency distributions, and estimations of flood quantiles at stations of interest. The approach has been applied successfully in modeling floods in a number of cases studied in Malaysia (Lim and Lye 2003), New Zealand (Madsen et al. 1997), Southern Africa (Kjeldsen et al. 2002), Turkey (Saf 2010), Iran (Rahnama and Rostami 2007), China (Chen et al. 2006), Italy (Noto and La Loggia 2009; Cannarozzo et al. 2009), Pakistan (Hussain and Pasha 2009), Tunisia (Abida and Ellouze 2008), Canada (Glaves and Waylen 1997; Yue and Wang 2004), UK (Fowler and Kilsby 2003), and India (Parida et al. 1998; Kumar et al. 1999, 2003a).

Elamir and Seheult (2003) introduced trimmed L-moments (TL-moments) which are a generalization of L-moments. TL-moments have certain advantages over L-moments and conventional moments. TL-moments which assign zero weight to extreme observations are practically easy to compute and are more robust compared with L-moments when used to estimate from a sample containing outliers. However, as observed from the literature, TL-moments have not been widely applied in frequency analysis. At present, TL-moments have only been derived for the normal, logistic, Cauchy, exponential, and generalized pareto (GPA) distributions (Elamir and Seheult 2003).

Many statistical distributions for regional frequency analysis have been investigated for extreme hydrologic variables. In this study, three probability distributions were considered: generalized extreme value (GEV), generalized logistic (GLO), and GPA. The short-listed distributions were chosen based on previous studies such as those by Zin et al. (2009) and Zalina et al. (2002) of which these distributions were more prominent for tropical regions and by Kyselý (2009) for modeling precipitation extremes. This paper aims to provide a comprehensive evaluation of the L-moments and TL-moments with regards to the three aforementioned probability distributions by first revisiting regional homogeneity establishment based on the L-moments by Hosking and Wallis (1997). Subsequently, corresponding relationships for regional homogeneity analysis and the regional parameters of the GEV and GLO distributions using TL-moments are developed. Annual maximum rainfall from 40 stations within Selangor, Malaysia, is utilized to perform the regional frequency analysis using both procedures.

2 TL-moments

The fundamental concepts of TL-moments are essentially the same as L-moments. Elamir and Seheult (2003) defined TL-moments as

$$ \lambda_r^{{(t)}} = {r^{{ - 1}}}\sum\limits_{{k = 0}}^{{r - 1}} {{{\left( { - 1} \right)}^k}\left( {\begin{array}{*{20}{c}} {r - 1} \\ k \\ \end{array} } \right)} E\left( {{X_{{r + t - k:r + 2t}}}} \right) $$

where \( E\left( {{X_{{i:r}}}} \right) = \frac{{r!}}{{\left( {i - 1} \right)!\left( {r - 1} \right)!}}\int\limits_0^1 {x(F){F^{{i - 1}}}{{\left( {1 - F} \right)}^{{r - i}}}dF} \)

For t = 0, TL-moments yields the original L-moments defined by Hosking (1990). For t = 1, the first four TL-moments are expressed as

$$ \lambda_1^{{(1)}} = E\left( {{X_{{2:3}}}} \right) = 6{\beta_1} - 6{\beta_2} $$
$$ \lambda_2^{{(1)}} = \tfrac{1}{2}E\left( {{X_{{3:4}}} - {X_{{2:4}}}} \right) = 6\left( { - 2{\beta_3} + 3{\beta_2} - {\beta_1}} \right) $$
$$ \lambda_3^{{(1)}} = \tfrac{1}{3}E\left( {{X_{{4:5}}} - 2{X_{{3:5}}} + {X_{{2:5}}}} \right) = \frac{{20}}{3}\left( { - 5{\beta_4} + 10{\beta_3} - 6{\beta_2} + {\beta_1}} \right) $$
$$ \lambda_4^{{(1)}} = \tfrac{1}{4}E\left( {{X_{{5:6}}} - 3{X_{{4:6}}} + 3{X_{{3:6}}} - {X_{{2:6}}}} \right) = \frac{{15}}{2}\left( { - 14{\beta_5} + 35{\beta_4} - 30{\beta_3} + 10{\beta_2} - {\beta_1}} \right) $$

The TL-moment ratios: TL-coefficient of variation (TL-Cv, \( \tau_2^{{(1)}} \)), TL-coefficient of skewness (L-Cs, \( \tau_3^{{(1)}} \)) and TL-coefficient of kurtosis (TL-Ck, \( \tau_4^{{(1)}} \)) are defined as \( \tau_2^{{(1)}} = \frac{{\lambda_2^{{(1)}}}}{{\lambda_1^{{(1)}}}} \), \( \tau_3^{{(1)}} = \frac{{\lambda_3^{{(1)}}}}{{\lambda_2^{{(1)}}}} \) and \( \tau_4^{{(1)}} = \frac{{\lambda_4^{{(1)}}}}{{\lambda_2^{{(1)}}}} \)

Detailed discussions on L-moments and TL-moments can be found in numerous literatures such as Hosking (1990) and Elamir and Seheult (2003).

3 Regional frequency analysis based on L-moments

Hosking and Wallis (1993, 1997) organized regional frequency analysis into four stages: (a) screening of the data, (b) identifying homogeneous regions, (c) choosing a regional frequency distribution, and (d) estimating the regional frequency distribution. A discussion of the first three stages is given next.

3.1 Discordance test

Discordancy test, D i , is used to screen out data from stations whose point sample L-moments are markedly different from other stations. The objective is to check the suitability of the data for carrying out the regional analysis. The discordancy test, D i , for station i is defined as

$$ {D_i} = \frac{1}{3}N{\left( {{u_i} - \overline u } \right)^T}{S^{{ - 1}}}\left( {{u_i} - \overline u } \right) $$
$$ S = \sum\limits_{{i = 1}}^N {\left( {{u_i} - \overline u } \right){{\left( {{u_i} - \overline u } \right)}^T}} $$

where \( {u_i} = {\left[ {t_2^i\;t_3^i\;t_4^i} \right]^T} \) for station i, N is the number of stations, S is covariance matrix of u i , and \( \overline u \) is the mean of vector, u i . Critical values for discordancy statistic are tabulated by Hosking and Wallis (1993); for N ≥ 15 stations, the critical value is 3. If the D-statistic of a station exceeds 3, its data is considered to be discordant from the rest of the regional data.

3.2 Heterogeneity test

An essential task in regional frequency analysis is the determination of homogeneous regions. Hosking and Wallis (1993) suggested the heterogeneity test, H, where L-moments are used to assess whether a group of stations may reasonably be treated as belonging to a homogeneous region. The proposed heterogeneity tests are based on: the L-coefficient of variation (L-Cv), the L-Cv and L-skewness (L-Cs); and L-Cs and L-kurtosis (L-Ck). These tests are defined respectively as

$$ {V_1} = \sqrt {{\sum\nolimits_{{i = 1}}^N {{n_i}{{\left( {t_2^{{(i)}} - t_2^R} \right)}^2}/\sum\nolimits_{{i = 1}}^N {{n_i}} } }} $$
$$ {V_2} = \sum\nolimits_{{i = 1}}^N {\left\{ {{n_i}{{\left[ {{{\left( {t_2^{{(i)}} - t_2^{{\left. R \right)}}} \right)}^2} + {{\left( {t_3^{{(i)}} - t_3^R} \right)}^2}} \right]}^{{1/2}}}} \right\}/\sum\nolimits_{{i = 1}}^N {{n_i}} } $$
$$ {V_3} = \sum\nolimits_{{i = 1}}^N {\left\{ {{n_i}{{\left[ {{{\left( {t_3^{{(i)}} - t_3^R} \right)}^2} + {{\left( {t_4^{{(i)}} - t_4^R} \right)}^2}} \right]}^{{1/2}}}} \right\}/\sum\nolimits_{{i = 1}}^N {{n_i}} } $$

Here, the regional average L-moment ratios are calculated using the following formula

$$ t_2^R = \sum\nolimits_{{i = 1}}^N {{n_i}t_2^{{(i)}}/} \sum\nolimits_{{i = 1}}^N {{n_i}}, t_3^R = \sum\nolimits_{{i = 1}}^N {{n_i}t_3^{{(i)}}/} \sum\nolimits_{{i = 1}}^N {{n_i}} $$
$$ t_4^R = \sum\nolimits_{{i = 1}}^N {{n_i}t_4^{{(i)}}/} \sum\nolimits_{{i = 1}}^N {{n_i}} $$

where N is the number of stations and n i is the record length at station i. The heterogeneity test is then defined as

$$ {H_j} = \left( {{V_j} - {\mu_{{{V_j}}}}} \right)/{\sigma_{{{V_j}}}}\quad j = 1,2,3 $$

where \( {\mu_{{{V_j}}}} \) and \( {\sigma_{{{V_j}}}} \) are the mean and standard deviation of simulated V j values, respectively. The estimated Kappa distribution is used to generate homogeneous regions with population parameters equal to the regional average sample L-moment ratios. Hosking and Wallis (1993, 1997) proposed the four parameters Kappa distribution in the simulation. The cumulative probability density function and quantile function for the Kappa distribution are

$$ F(x) = {\left\{ {1 - h{{\left[ {1 - \frac{k}{\alpha }\left( {x - \xi } \right)} \right]}^{{1/k}}}} \right\}^{{1/h}}} $$
$$ Q(F) = \xi + \frac{\alpha }{k}\left[ {1 - {{\left( {\frac{{1 - {F^h}}}{h}} \right)}^k}} \right] $$

The value of the H-statistic indicates that the region is acceptably homogeneous with a corresponding order of L-moments if H < 1, possibly homogeneous when 1 ≤ H < 2 and definitely heterogeneous when H ≥ 2.

3.3 Selection of a regional frequency distribution

After confirming the homogeneity of the studied region, an appropriate distribution needs to be selected for the regional frequency analysis. Selection of the distribution is very crucial especially for relatively large return periods because the distribution type can affect, to a great extent, the magnitude of the estimated floods. The L-moment ratio diagram and Z-test are employed for this purpose.

The L-moment ratio diagram is a graph of the L-Cs and L-Ck which compares the fit of several distributions on the same graph. However, direct visual inspection of the L-moment ratio diagram is somewhat subjective as more than one distribution could pose as a possible candidate.

The Z-test judges how well the simulated L-Cs and L-Ck of a fitted distribution matches the regional average L-Cs and L-Ck values. For each selected distribution, the Z-test is calculated as follows

$$ {Z^{{DIS}}} = \left( {\tau_4^{{Dis}} - t_4^R} \right)/{\sigma_4} $$

where DIS refers to a particular distribution, \( \tau_4^{{Dis}} \) is the L-kurtosis of the fitted distribution while the standard deviation of \( t_4^R \) is given by

$$ {\sigma_4} = {\left[ {{{\left( {{N_{{sim}}}} \right)}^{{ - 1}}}\sum\limits_{{m = 1}}^{{{N_{{sim}}}}} {{{\left( {t_4^{{(m)}} - t_4^R} \right)}^2}} } \right]^{{1/2}}} $$

\( t_4^{{(m)}} \) is the average regional L-kurtosis and has to be calculated for the m th simulated region. This is obtained by simulating a large number of kappa using Monte Carlo simulations. The value of the Z-statistics is considered to be acceptable at the 90% confidence level if \( \left| {{Z^{{DIS}}}} \right| \leqslant 1.64 \). If more than one candidate distribution is acceptable, the one with the lowest \( \left| {{Z^{{DIS}}}} \right| \) is regarded as the best-fit distribution.

Details on the method of L-moments in estimating the parameters of several distributions can be found in Sankarasubramaniam and Srinivasan (1999).

4 Regional frequency analysis based on TL-moments

The procedures discussed in Section 3 are similarly employed for the TL-moments. The L-Cv, L-Cs, and L-Ck are equally replaced by the TL-Cv, TL-Cs, and TL-Ck for the discordancy and the homogeneity test. Selection of an adequately fitted distribution is carried out based on the TL-ratio diagram and Z-test using the regional TL-Cs and TL-Ck. As discussed before, distribution whose absolute Z-test value is less than 1.64 qualify as the possible candidate, the best distribution being the one with the lowest value. Formulas for the TL-Cv, TL-Cs, and TL-Ck are as discussed in Section 2.

Estimation of the design floods for specific return periods are generally needed in flood studies. Presently, of the three distributions under consideration, TL-moment parameter estimates are only available for the GPA distribution. In this research, TL-moment parameter estimates of GEV and GLO are developed and used to test for robustness of the distributions. Table 1 shows the TL-moments, TL-moment ratios, and the associated parameters derived for the GEV, GPA, and GLO distributions. Table 2 shows the coefficients for the newly developed relationships of TL-Cs and TL-Ck of the GPA, GEV, and GLO distributions based on TL-moments for the range \( - 1 \leqslant \tau_3^{{(1)}} \leqslant 1 \).

Table 1 Parameter development for the GLO, GEV, and GPA distributions
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Table 2 Equation coefficient of TL-moment ratio for the GEV, GPA, and GLO distributions
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5 Case study

Records of daily rainfall from 40 stations in Selangor with record lengths of 22 to 38 years were acquired from the Department of Irrigation and Drainage, Malaysia. The statistics and basic information of the data are listed in Table 3. All the stations, numbered 1 to 40, are located throughout Selangor with latitudes ranging from 26o up to 38o and longitudes from 8o to 18o, as shown in Fig. 1.

Table 3 Station name and statistics of maximum daily rainfall for all the stations in Selangor
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Fig. 1
figure 1

Location of rain gauge stations used in study

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6 Results and discussions

Initially, the whole of Selangor was assumed as one homogeneous region, and the discordancy test was used for data verification and quality control. Results of the discordancy test, D i , are given in Table 3. It is observed that, for the L-moments method, D critical = 3.0, is exceeded at four locations: stations 9, 20, 30, and 37, with D-statistic values of 3.167, 3.187, 3.577, and 3.949, respectively. Therefore, these four stations are excluded from the regional frequency analysis.

However, for the TL-moments method, it is observed that the D-statistic values for the 40 stations vary from 0.120 to 2.860. The largest D-statistic value is 2.860 for station 15, hence none of the stations have a D-statistic exceeding the critical value. Thus, for the TL-moments method, data from all stations are used for the development of regional frequency analysis.

The regional average L-moment ratios and average TL-moment ratios of the respective study regions are calculated, and the corresponding parameter values of the fitted Kappa distribution are found as presented in Table 4.

Table 4 Moment ratios and parameter values of the fitted Kappa distribution
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The H-statistics are computed by carrying out 500 simulations using the Kappa distribution based on the data from 36 stations for the L-moments and 40 stations for the TL-moments. Results of the H-tests for the L-moments and TL-moments, as given in Table 5, indicate that the study region demonstrates acceptable homogeneity (H < 1), therefore a subdivision of the stations into more regions was not necessary.

Table 5 Results of the homogeneity test and goodness-of-fit test
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After confirming the homogeneity of the study region, an appropriate distribution needs to be selected for the regional frequency analysis. Figures 2 and 3 show a comparison of the observed and theoretical relations between the L-moments and the TL-moments, respectively. In the L-moment ratio diagram of Fig. 2, the point defined by the sample average values of \( t_3^R = 0.2388 \) and \( t_4^R = 0.2055 \), lies closest to the L-moments of the GLO distribution followed by GEV and GPA distributions.

Fig. 2
figure 2

L-moment ratio diagram of L-kurtosis versus L-skewness for annual maximum rainfall of the Selangor region

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Fig. 3
figure 3

TL-moment ratio diagram of L-kurtosis versus L-skewness for annual maximum rainfall of the Selangor region

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Analysis of the TL-moment ratio diagram reveals that the sample average values of \( t_3^{{R(1)}} = 0.1657 \)and \( t_4^{{R(1)}} = 0.1065 \) in the diagram are better described by the theoretical TL-moments of the GLO and GEV rather than the GPA, hence the GLO and GEV distributions should be preferred.

Results of the Z-test for the three distributions are given in Table 5. For both the L-moments and TL-moments methods, the GPA distribution failed the test with a Z-test exceeding the critical value of 1.64. However, the GLO should be considered as the best-fit distribution, as this distribution gives the minimum Z-test for both the L-moments and TL-moments methods.

In regional frequency analysis, the final and important objective is to determine the robustness of the distribution in producing reasonably reliable estimates at all stations in the homogeneous region. Regional flood frequency relationships are developed using the chosen GLO and GEV distributions. The form of the regional frequency relationship or the growth factor for GLO and GEV distributions are respectively

$$ {Q_T} = \xi + \frac{\alpha }{k}\left\{ {1 - {{\left( {\frac{1}{{T - 1}}} \right)}^k}} \right\} $$

and

$$ {Q_T} = \xi + \frac{\alpha }{k}\left\{ {1 - {{\left[ { - \log \left( {1 - 1/T} \right)} \right]}^k}} \right\} $$

where Q T is quantile estimation at T-years return period. The regional parameters and the quantiles estimated for the GLO and GEV distributions for T = 2, 5, 10, 20, 50, and 100 years, using L-moments and TL-moments, are presented in Table 6.

Table 6 Regional parameter and quantile estimates of the GEV and GLO distributions for L-moments and TL-moments
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The robustness of the designated regional frequency distributions is further investigated for estimation of design flood quantiles. For this purpose, Meshgi and Khalili (2009) proposed Monte Carlo simulation to evaluate the errors between the simulated and calculated design flood quantiles. Two measures of performance that were used are the relative bias (RBIAS) and the relative root mean square error (RRMSE)

$$ RBIAS = {M^{{ - 1}}}\sum\limits_{{m = 1}}^M {\frac{{Q_T^{{(m)}} - Q_T^C}}{{Q_T^C}}} $$
$$ RRMSE = \sqrt {{{M^{{ - 1}}}\sum\limits_{{m = 1}}^M {{{\left( {\frac{{Q_T^{{\left[ m \right]}} - Q_T^C}}{{Q_T^C}}} \right)}^2}} }} $$

where M is the sample size, \( Q_T^{{(m)}} \) and \( Q_T^C \) is the simulated and true quantile of design flood at T-years, respectively.

Tables 7 and 8 present the RBIAS and RRMSE values of quantiles computed using the L-moments and TL-moments for the GEV and GLO distributions. Figures 4 and 5 provide the results of the RBIAS and the RRMSE for sample sizes of 20 and 80, respectively.

Table 7 RBIAS values for different quantiles of the GEV and GLO distributions for L-moments and TL-moments
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Table 8 RRMSE values for different quantiles of the GEV and GLO distributions for L-moments and TL-moments
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Fig. 4
figure 4

Results of the RBIAS for sample sizes 20 and 80 computed for quantile T = 2, 5, 10, 20, 50, and 100 years

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Fig. 5
figure 5

Results of the RRMSE for sample sizes 20 and 80 computed for quantiles T = 2, 5, 10, 20, 50, and 100 years

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The results show that the RBIAS and RRMSE values generally increase with a reduction in the sample size and for prediction of larger quantiles. The GEV distribution performs better than GLO distribution for the L-moments. However, the performances of GEV and GLO distributions are approximately similar for the TL-moments. Interestingly, the TL-moments method is significantly more efficient than the L-moments method for return periods, T, of less than 10 years. However, the performance of the L-moments method supersedes that of the TL-moments for higher return periods.

7 Conclusion

The study provides a comprehensive evaluation of the L-moments and TL-moments, by first revisiting regional frequency analysis based on the L-moments by Hosking and Wallis (1993; 1997). Regional homogeneity was investigated by first assuming the entire study area as one homogeneous regional cluster. The corresponding relationships for regional homogeneity analysis by the TL-moments are developed. TL-moments for the GPA, GLO, and GEV distributions are also developed and used to provide the corresponding TL-moment ratio diagrams and the goodness-of-fit test.

The results of this study have shown that, from 40 stations in the study region, 36 stations based on L-moments and all stations based on TL-moments are accepted statistically to be homogeneous using the discordancy test and heterogeneity test. The Z-test has also shown that the L-moments and TL-moments method produced the same results, where GLO and GEV were identified as the best distributions for modeling daily annual maximum rainfall in Selangor, Malaysia. Finally, Monte Carlo simulations which are used for performance evaluation showed that the TL-moments method is more efficient for lower quantile estimation but the L-moments method does outperform for higher quantile estimations.