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Regional Frequency Analysis of Extreme Rainfalls in the West Coast of Peninsular Malaysia using Partial L-Moments

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Abstract

This study was to reinstate the development of regional frequency analysis using L-moments approach. The Partial L-moments (PL-moments) method was employed and a new relationship for homogeneity analysis is developed. For this study, the PL-moments for generalized logistic (GLO), generalized pareto (GPA) and generalized value (GEV) distributions were derived based on the formula defined by Wang (Water Resour Res 32:1767–1771, 1996). The three distributions are used to develop the regional frequency analysis procedures. As a case of study, the Selangor catchment that consists of 30 sites which located on the west coast of Peninsular Malaysia has chosen as sample. Based on L-moment and PL-moment ratio diagrams as well as Z-test statistics, the GEV and GLO were identified as the best distributions to represent the statistical properties of extreme rainfalls in Selangor. Monte Carlo simulation shows that the method of PL-moments would outperform L-moments method for estimation of large returns period event.

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Acknowledgements

The authors thankfully acknowledged the financial support provided by Ministry of Higher Education, Malaysia and Universiti Sultan Zainal Abidin, Malaysia. The authors also would like to thank the Department of Irrigation and Drainage, Ministry of Natural Resources and Environment, Malaysia for providing the floods data and Universiti Teknologi Malaysia.

Author information

Authors and Affiliations

  1. Faculty of Informatics, Universiti Sultan Zainal Abidin Malaysia, Terengganu, Malaysia

    Zahrahtul Amani Zakaria

  2. Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, Johor, Malaysia

    Zahrahtul Amani Zakaria, Ani Shabri & Ummi Nadiah Ahmad

Authors
  1. Zahrahtul Amani Zakaria
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  2. Ani Shabri
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  3. Ummi Nadiah Ahmad
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Corresponding author

Correspondence to Zahrahtul Amani Zakaria.

Appendix

Appendix

1.1 PL-moments for the GEV Distribution (Wang 1996)

The CDF and quantile function of the GEV are given by

$$ F(x) = \exp \left\{ { - {{\left[ {1 - \frac{k}{\alpha}\left( {x - \xi } \right)} \right]}^{{1/k}}}} \right\}\quad k \ne 0 $$
(7)

and quantile function

$$ x(F) = \xi + \frac{\alpha }{k}\left[ {1 - {{\left( { - \ln F} \right)}^k}} \right]\quad k \ne 0 $$
(8)

Wang (1996) developed the partial PWMs of the GEV as

$$ (r + 1)\beta_r^{\prime} = \xi + \alpha H(r,{F_0},k) $$
(9)

where

$$ H(r,{F_0},k) = \frac{1}{k}\left\{ {1 - \frac{{\mathrm{P}\left[ {1 + k, - (r + 1)\ln {F_0}} \right]}}{{(1 - F_0^{{r + 1}}){{(r + 1)}^k}}}} \right\} $$

The first four PL-moments of the GEV are defined as

$$ \matrix{ {\lambda_1^{\prime} = \xi + \alpha H(0,{F_0},k)} \\ {\lambda_2^{\prime} = \alpha \left[ {H(1,{F_0},k) - H(0,{F_0},k)} \right]} \\ {\lambda_3^{\prime} = \alpha \left[ {2H(2,{F_0},k) - 3H(1,{F_0},k) + H(0,{F_0},k)} \right]} \\ {\lambda_4^{\prime} = \alpha \left[ {5H(3,{F_0},k) - 10H(2,{F_0},k) + 6H(1,{F_0},k) - H(0,{F_0},k)} \right]} \\ }<!end array> $$
(10)

In previous equation, P(.,.) is an Incomplete Gamma function

$$ \mathrm{P}(1 + k, - (r + 1)\ln {F_0}) = \int_0^{{ - (r + 1)\ln {F_0}}} {{\theta^k}} {e^{{ - \theta }}}d\theta $$
(11)

Then the first four PL-moments are computed to develop the PL-moment ratios (PL-Cv, PL-Cs and PL-Ck) for the GEV distribution.

1.2 Development of PL-moments for the GLO Distribution (This Research)

The CDF and quantile function of the GLO are given by

$$ F(x) = {\left\{ {1 + {{\left[ {1 - \frac{k}{\alpha}\left( {x - \xi } \right)} \right]}^{{1/k}}}} \right\}^{{ - 1}}}\quad k \ne 0 $$
(12)

and quantile function

$$ x(F) = \xi + \frac{\alpha }{k}\left[ {1 - {{\left( {\frac{{1 - F}}{F}} \right)}^k}} \right]\quad k \ne 0 $$
(13)

The partial PWMs of the GLO are developed as follows

$$ (r + 1)\beta_r^{\prime} = \xi + \frac{\alpha }{k} - \frac{{\alpha (r + 1)}}{{k(1 - F_0^{{r + 1}})}}{\mathrm{B}_{{1 - {F_0}}}}(1 + k,r - k + 1) $$
(14)

where \( {\mathrm{B}_{{1 - {F_0}}}}(.,.) \) is an Incomplete Beta function

$$ {\mathrm{B}_{{1 - {F_0}}}}(1 + k,r - k + 1) = \int_0^{{1 - {F_0}}} {{\theta^k}{{\left( {1 - \theta } \right)}^{{r - k}}}d} \theta $$

The first four PL-moments of the GLO are defined as

$$ \matrix{ {\lambda_1^{'} = \xi + \frac{\alpha }{k}\left[ {1 - \frac{{{\mathrm{B}_{{1 - {F_0}}}}(1 + k,1 - k)}}{{1 - {F_0}}}} \right]} \\ {\lambda_2^{'} = - \frac{\alpha }{k}\left[ {\frac{{2{\mathrm{B}_{{1 - {F_0}}}}(1 + k,2 - k)}}{{1 - F_0^2}} - \frac{{{\mathrm{B}_{{1 - {F_0}}}}(1 + k,1 - k)}}{{1 - {F_0}}}} \right]} \\ {\lambda_3^{'} = - \frac{\alpha }{k}\left[ {\frac{{6{\mathrm{B}_{{1 - {F_0}}}}(1 + k,3 - k)}}{{1 - F_0^3}} - \frac{{6{\mathrm{B}_{{1 - {F_0}}}}(1 + k,2 - k)}}{{1 - F_0^2}} + \frac{{{\mathrm{B}_{{1 - {F_0}}}}(1 + k,1 - k)}}{{1 - {F_0}}}} \right]} \\ {\lambda_4^{'} = - \frac{\alpha }{k}\left[ {\frac{{20{\mathrm{B}_{{1 - {F_0}}}}(1 + k,4 - k)}}{{1 - F_0^4}} - \frac{{30{\mathrm{B}_{{1 - {F_0}}}}(1 + k,3 - k)}}{{1 - F_0^3}} + \frac{{12{\mathrm{B}_{{1 - {F_0}}}}(1 + k,2 - k)}}{{1 - F_0^2}} - \frac{{{\mathrm{B}_{{1 - {F_0}}}}(1 + k,1 - k)}}{{1 - {F_0}}}} \right]} \\ }<!end array> $$
(15)

Then the first four PL-moments are computed to develop the PL-moment ratios (PL-Cv, PL-Cs and PL-Ck) for the GLO distribution.

1.3 Development of PL-moments for the GPA Distribution (This Research)

The CDF and quantile function of the GPA are given by

$$ F(x) = 1 - {\left\{ {1 - \frac{k}{\alpha}\left( {x - \xi } \right)} \right\}^{{1/k}}}\quad k \ne 0 $$
(16)

and quantile function

$$ x(F) = \xi + \frac{\alpha }{k}\left[ {1 - {{\left( {1 - F} \right)}^k}} \right]\quad k \ne 0 $$
(17)

The partial PWMs of the GPA are developed as follows

$$ (r + 1)\beta_r^{\prime} = \xi + \frac{\alpha }{k} - \frac{{\alpha (r + 1)}}{{k(1 - F_0^{{r + 1}})}}\int_{{{F_0}}}^1 {{{(1 - F)}^k}{F^r}dF} $$
(18)

The first four PL-moments of the GPA are defined as

$$ \matrix{ {\lambda_1^{'} = \xi + \frac{\alpha }{k}\left[ {1 - g{1_1}} \right]} \\ {\lambda_2^{'} = - \frac{\alpha }{k}\left[ {2g{2_1} - 2g{2_2} - g{1_1}} \right]} \\ {\lambda_3^{'} = - \frac{\alpha }{k}\left[ {6g{3_1} - 12g{3_2} + 6g{3_3} - 6g{2_1} + 6g{2_2} + g{1_1}} \right]} \\ {\lambda_4^{'} = - \frac{\alpha }{k}\left[ {20g{4_1} - 60g{4_2} + 60g{4_3} - 60g{4_4} - 30g{3_1} + 60g{3_2} - 30g{3_3} + 12g{2_1} - 12g{2_2} - g{1_1}} \right]} \\ }<!end array> $$
(19)

where

$$ g{s_r} = \frac{{{{\left( {1 - {F_0}} \right)}^{{k + r}}}}}{{\left( {k + r} \right)\left( {1 - F_0^s} \right)}} $$

Then the first four PL-moments are computed to develop the PL-moment ratios (PL-Cv, PL-Cs and PL-Ck) for the GPA distribution.

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Zakaria, Z.A., Shabri, A. & Ahmad, U.N. Regional Frequency Analysis of Extreme Rainfalls in the West Coast of Peninsular Malaysia using Partial L-Moments. Water Resour Manage 26, 4417–4433 (2012). https://doi.org/10.1007/s11269-012-0152-8

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