Evaluation of generalized extreme value and Gumbel distributions for estimating maximum daily rainfall

Álvaro José Back; Fernanda Martins Bonfante

doi:10.5327/Z217694781015

Authors

Álvaro José Back Empresa de Pesquisa Agropecuária e Extensão Rural de Santa Catarina (Epagri) - Brazil https://orcid.org/0000-0002-0057-2186
Fernanda Martins Bonfante Universidade do Extremo Sul Catarinense (UNESC) - Brazil https://orcid.org/0000-0002-9773-4742

DOI:

https://doi.org/10.5327/Z217694781015

Keywords:

Heavy rain, drainage, probability, territorial management

Abstract

Extreme rain events can cause social and economic impacts in various sectors. Knowing the risk of occurrences of extreme events is fundamental for the establishment of mitigation measures and for risk management. The analysis of frequencies of historical series of observed rain through theoretical probability distributions is the most commonly used method. The generalized extreme value (GEV) and Gumbel probability distributions stand out among those applied to estimate the maximum daily rainfall. The indication of the best distribution depends on characteristics of the data series used to adjust parameters and criteria used for selection. This study compares GEV and Gumbel distributions and analyzes different criteria used to select the best distribution. We used 224 series of annual maximums of rainfall stations in Santa Catarina (Brazil), with sizes between 12 and 90 years and asymmetry coefficient ranging from -0.277 to 3.917. We used the Anderson–Darling, Kolmogorov-Smirnov (KS), and Filliben adhesion tests. For an indication of the best distribution, we used the standard error of estimate, Akaike’s criterion, and the ranking with adhesion tests. KS test proved to be less rigorous and only rejected 0.25% of distributions tested, while Anderson–Darling and Filliben tests rejected 9.06% and 8.8% of distributions, respectively. GEV distribution proved to be the most indicated for most stations. High agreement (73.7%) was only found in the indication of the best distribution between Filliben tests and the standard error of estimate.

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References

Abreu, M.C.; Cecílio, R.A.; Pruski, F.F.; Santos, G.R.; Almeida, L.T.; Zanettim, S.S., 2018. Critérios para escolha de distribuições de probabilidade em estudos de Eventos extemos de precipitação. Revista Brasileira de Meteorologia, v. 33, (4), 601-613. http://dx.doi.org/10.1590/0102-7786334004.

Affonso, V.; Faria, G.A.; Lopes, B.G.; Tsutsumoto, N.Y.; Fonseca, A.D.; Felizardo, L.M., 2020. Análise dos dados de precipitação máxima no noroeste paulista pela teoria dos valores extremos. Research, Society and Development, v. 9, (10), e9709109396. https://doi.org/10.33448/rsd-v9i10.9396.

Agência Nacional das Águas (ANA). Sistema de Informações Hidrológicas (Accessed October 8, 2020) at: http://hidroweb.ana.gov.br/.

Aiyelokun, O.; Ojelabi, A., Malomo, S.; Agbede, O., 2017. Efficient flood forecasting for the operation of hydraulic structures in a typical river basin. International Journal of Scientific & Engineering Research, v. 8, (11), 463-481.

Alam, M.A.; Emura, K.; Farnham, C.; Yuan, J., 2018. Best-Fit Probability Distributions and Return Periods for Maximum Monthly Rainfall in Bangladesh. Climate, v. 6, (1), 9. https://doi.org/10.3390/cli6010009.

Almeida, K.N., Reis, J.A.T.; Mendonça, A.S.F., 2015. Avaliação do desempenho dos métodos expeditos de determinação de equações de chuvas intensas. Brazilian Journal of Environmental Sciences, (35), 63-77 (Accessed December, 2020) at: http://www.rbciamb.com.br/index.php/Publicacoes_RBCIAMB/article/view/207

Al-Suhili, R.H.; Khanbilvardi, R., 2014. Frequency Analysis of the Monthly Rainfall Data at Sulaimania Region, Iraq. American Journal of Engineering Research, v. 3, (5), 212-222.

Back, Á.J., 2018. Análise de frequência de vazões máximas para projetos de drenagem. Revista Técnico-Científica de Engenharia Civil, v. 1, (2), 1-14. http://dx.doi.org/10.18616/civiltec.v1i2.5065.

Back, Á.J.; Cadorin, S.B., 2020. Extreme rainfall and intensity-duration-frequency equations for the state of Acre, Brazil. Brazilian Journal of Environmental Sciences, v. 55, (2), 159-170. https://doi.org/10.5327/Z2176-947820200597.

Bella, N.; Dridi, H.; Kalla, M., 2020. Statistical modeling of annual maximum precipitation in Oued El Gourzi Watershed, Algeria. Applied Water Science, v. 10, 94. https://doi.org/10.1007/s13201-020-1175-6.

Beskow, S.; Caldeira, T.L.; Mello, C.R.; Faria L.C.; Guedes, H.A.S., 2015. Multiparameter probability distributions for heavy rainfall modeling in extreme southern Brazil. Journal of Hydrology: Regional Studies, 4, (part B), 123-133. https://doi.org/10.1016/j.ejrh.2015.06.007.

Bork, C.K.; Castro, A.S.; Leandro, D.; Corrêa, L.B.; Siqueira, T.M., 2017. Índices de precipitação extrema para os períodos atual (1961-1990) e futuro (2011-2100) na bacia do rio Taquari-antas, RS. Brazilian Journal of Environmental Sciences, (46), 29-45. https://doi.org/10.5327/Z2176-947820170233.

Caldeira, T.L.; Beskow, S.; Mello, C.R.; Faria, L.C.; Souza, M.R.; Guedes, H.A.S., 2015. Modelagem probabilística de eventos de precipitação extrema no estado do Rio Grande do Sul. Revista Brasileira de Engenharia Agrícola e ambiental, v. 19, (3), 197-203. http://dx.doi.org/10.1590/1807-1929/agriambi.v19n3p197-203.

Chow, V.T., 1964. Handbook of applied hydrology. McGraw-Hill Co., New York, 42 pp.

Coelho Filho, J.A.P.; Melo, D.C.R.; Araújo, M.L.M., 2017. Estudo de chuvas intensas para a cidade de Goiânia/GO por meio da modelação de eventos máximos anuais pela aplicação das distribuições de Gumbel e generalizada de valores extremos. Ambiência, v. 13, (1), p. 75-88. http://dx.doi.org/10.5935/ambiencia.2017.01.05.

Coronado-Hernández, O.C.; Merlano-Sabalza, E.; Díaz-Vergara, Z.; Coronado-Hernández, J.R., 2020. Selection of Hydrological Probability Distributions for Extreme Rainfall Events in the Regions of Colombia. Water, v. 12, (5), 1397. http://dx.doi.org/10.3390/w12051397.

Costa, J.N.; Silva Júnior, J.B.; Araújo, S.M.S., 2018. Riscos e desastres relacionados a eventos extremos (climáticos e meteorológicos) no estado da Paraíba. Revista de Geociências do Nordeste, v. 4, 110-125.

Coulson, C.H., 1991. Manual of operational hydrology in B.C., 2ndedn B.C. Water Management Division, Hydrology Section, Ministry of Environment, Landsand Parks, BC, Canada.

Cremoneze, I.Z.; Peralta, D.; Mazucheli, J.; Emanuelli, I.P., 2017. Análise de frequências das precipitações máximas mensais observadas nos estados do Paraná e Rio Grande do Sul. Enciclopédia Biosfera, v. 14, (25), 48-57.

Cunnane, C., 1973. A particular comparison of annual maxima and partial duration series methods of flood frequency prediction. Journal of Hydrology, v. 18, (3-4), 257-271. https://doi.org/10.1016/0022-1694(73)90051-6.

Das, N.M.S.; Simonovic, S.P., 2011. The comparison of GEV, Log-Pearson type 3 and Gumbel distributions in the Upper Thames River Watershed under global climate models. Water Resources Research Report, London, 54 pp.

De Paola, F.; Giugni, M.; Pugliese, F.; Annis, A; Nardi, F., 2018. GEV parameter estimation and stationary vs. non-stationary analysis of extreme rainfall in African test cities. Hydrology, v. 5, (2), 28. https://doi.org/10.3390/hydrology5020028.

Empresa de Pesquisa Agropecuária e Extensão Rural de Santa Catarina (EPAGRI). 2020. Banco de dados de variáveis ambientais de Santa Catarina. Epagri, Florianópolis, 20 p. (Epagri, Documentos, 310).

Esteves, L., 2013. Consequences to flood management of using different probability distributions to estimate extreme rainfall. Journal of Environmental Management, v. 115, 98-105. https://doi.org/10.1016/j.jenvman.2012.11.013.

Fernandes, RA.; Valverde, M.C., 2017. Análise da resiliência aos extremos climáticos de chuva: estudo preliminar na região de Mauá no ABC Paulista – São Paulo. Brazilian Journal of Environmental Sciences, (44), 1-17. https://doi.org/10.5327/z2176-947820170183.

Feyissa, T.A.; Tukura, N.G., 2019. Evaluation of the best-fit probability of distribution and return periods of river discharge peaks. Case study: Awetu River, Jimma, Ethiopia. Journal of Sedimentary Environments, v. 4, (4), 360-368. https://doi.org/10.12957/jse.2019.46128.

Filliben, J.J., 1975. The probability plot correlation coefficient test for normality. Technometrics, v. 17, (1), 11-117. https://doi.org/10.2307/1268008.

Fischer, T.; Su, B.; Luo, Y.; Scholten, T., 2012. Probability Distribution of Precipitation Extremes for Weather Index–Based Insurance in the Zhujiang River Basin, South China. Journal of Hydrometeorology, v. 13, (3), 1023-1037. https://doi.org/10.1175/JHM-D-11-041.1.

González-Álvarez, Á.; Viloria-Marimón, O.M.; Coronado-Hernández, Ó.E.; Vélez-Pereira, A.M.; Tesfagiorgis, K.; Coronado-Hernández, J.R., 2019. Isohyetal maps of daily maximum rainfall for different return periods for the Colombian Caribbean Region. Water, v. 11, (2), 358. https://doi.org/10.3390/w11020358.

Heo, J.H.; Kho, Y.W.; Shin, H.; Kim, S.; Kim, T., 2008. Regression equations of probability plot correlation coefficient test statistics from several probability distributions. Journal of Hydrology, v. 355, (1-4), 1-15. https://doi.org/10.1016/j.jhydrol.2008.01.027.

Hosking, J.R.M., 1990. L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society. Series B. Statistical Methodological, v. 52, (1), 105-124 (Accessed December, 2020) at: https://www.jstor.org/stable/2345653.

Hosking, J.R.M., 1994. The four-parameter Kappa distribution. IBM Journal of Research and Development, v. 38, (3), 251-258. https://doi.org/10.1147/rd.383.0251.

Hosking, J.R.M., 2005. FORTRAN routines for use with the method of L-moments. Version 3.04, Rep. No. RC 20525 (90933). IBM Research Division, T.J. Watson Research Center, Yorktown Heights, NY.

Kite, G.W., 1977. Frequency and risk analyses in hydrology. Water Resources Publications, Fort Collins, Colorado, 224 pp.

Leite, M.L.; Virgens Filho, J.S., 2011. Ajuste de modelos de distribuição de probabilidade a séries horárias de velocidade do vento para o município de Ponta Grossa, Estado do Paraná. Acta Scientiarum. Technology, v. 33, (4), 447-455. https://doi.org/10.4025/actascitechnol.v33i4.7072.

Lima, A.O.; Lyra, G.B.; Abreu, M.C.; Oliveira-Júnior, J.F.; Zeri, M.; Cunha-Zeri, G., 2021. Extreme rainfall events over Rio de Janeiro State, Brazil: Characterization using probability distribution functions and clustering analysis. Atmospheric Research, v. 247, 1052212. https://doi.org/10.1016/j.atmosres.2020.105221.

Mandal, S.; Choudhury, B.U., 2015. Estimation and prediction of maximum daily rainfall at Sagar Island using best fit probability models. Theoretical Appied Climatolology, v. 121, (1-2), 87-97. https://doi.org/10.1007/s00704-014-1212-1.

Marques, R.F.P.V.; Mello, C.R.; Silva, A.M.; Franco, C.S.; Oliveira, A.S., 2014. Desempenho de distribuições de probabilidades aplicadas a eventos extremos de precipitação diária. Ciência Agrotecnologia, v. 38, (4), 335-342. https://doi.org/10.1590/S1413-70542014000400003.

Marra, F.; Morin, E.; Peleg, N.; Mei, Y.; Anagnostou, E.N., 2017. Intensity–duration–frequency curves from remote sensing rainfall estimates: comparing satellite and weather radar over the eastern Mediterranean. Hydrology and Earth System Sciences, v. 21, (5), 2389-2404. https://doi.org/10.5194/hess-21-2389-2017.

Mello, C.R.; Silva, A.M., 2005. Métodos estimadores dos parâmetros da distribuição de Gumbel e sua influência em estudos hidrológicos de projeto. Irriga, v. 10, (4), 318-334. https://doi.org/10.15809/irriga.2005v10n4p334-350.

Mistry, P.B.; Suryanarayana, M.V., 2019. Estimation of annual one day maximum rainfall using probability distributions for Waghodia Taluka, Vadodara. Global Research and Development Journal for Engineering, 296-300 (Accessed November, 2020) at: https://www.grdjournals.com/uploads/conference/GRDCF/012/059/GRDCF012059.pdf

Molina-Aguilar, J.P.; Gutierrez-Lopez, A.; Raynal-Villaseñor, J.A.; Garcia-Valenzuela, L.G., 2019. Optimization of Parameters in the Generalized Extreme-Value Distribution Type 1 for Three Populations Using Harmonic Search. Atmosphere, v. 10, (5), 257. http://dx.doi.org/10.3390/atmos10050257.

Monteiro, J.B.; Zanella, M.E., 2017. A metodologia dos máximos de precipitação aplicada ao estudo de eventos extremos diários nos municípios de Crato, Fortaleza e Sobral-CE. GeoTextos, v. 13, (2), 135-159. http://dx.doi.org/10,977/194-5537geo.v13i2.24011.

Moretti, A.R.; Mendes, B.V.M., 2003. Sobre a precisão das estimativas de máxima verossimilhança nas distribuições bivariadas de valores extremos. Pesquisa Operacional, v. 23, (2), 301-324. https://doi.org/10.1590/S0101-74382003000200004.

Mouri, G.; Minoshima, D.; Golosov, V.; Chalov, S.; Seto, S.; Yoshimura, K.; Nakamura, S.; Oki, T., 2013. Probability assessment of flood and sediment disasters in Japan using the total runoff-integrating pathways model. International Journal of Disaster Risk Reduction, v. 3, 31-43. https://doi.org/10.1016/J.IJDRR.2012.11.003.

Naghettini, M.; Pinto, E.J.A., 2007. Hidrologia estatística. CPRM, Belo Horizonte (Accessed Mês xx, 20xx) at: http://rigeo.cprm.gov.br/jspui/handle/doc/454.

Namitha, M.R.; Vinothkumar, V., 2019. Development of empirical models from rainfall-intensity-duration-frequency curves for consecutive Days maximum rainfall using GEV distribution. Journal of Pharmacognosy and Phytochemistry, v. 8, (1), 2705-2709.

Nanvapisheh, A.A., 2021. The comparison between Gumbel and exponentiated Gumbel distributions and their applications in hydrological process. American Journal of Computer Science and Information Technology, v. 9, (3), 80.

Ottero, C.R.; Chargel, L.T.; Hora, M.A.G.M., 2018. Análise de frequência dos dados pluviométricos observados em 2011 a 2013 na região Serrana do Rio de Janeiro. Revista Brasileira de Meteorologia, v. 33, (1), 131-139. http://dx.doi.org/10.1590/0102-7786331007.

Pereira, D.C.; Duarte, L.R.; Sarmento, A.P., 2017. Intensity-duration-frequency curves determination of Ipameri – Goiás. Revista Eletrônica de Engenharia Civil, v. 13, (2), 233-246. http://dx.doi.org/10.5216/reec.V13i2.43330.

Pérez-Sánchez, J.; Senent-Aparicio, J., 2017. Intensity-duration-frequency curves of short-duration storms in the Segura River Basin, Spain. Agrociência, v. 51, (6), 607-616.

Ramos, P.L.; Moala, F.A., 2014. A aplicação da distribuição exponencial geométrica estendida para modelagem de dados pluviométricos. Revista Brasileira de Meteorologia, v. 29, (4), 613-620. https://doi.org/10.1590/0102-778620130612.

Rizwan, M.; Guo, S.; Xiong, F.; Yin, J., 2018. Evaluation of various probability distributions for deriving design flood featuring right-tail events in Pakistan. Water, v. 10, (11), 1603. https://doi.org/10.3390/w10111603.

Salinas, J.L.; Castellarin, A.; Kohnová, S.; Kjeldsen, T.R., 2014. Regional parent flood frequency distributions in Europe-Part 2: Climate and scale controls. Hydrology and Earth System Sciences, v. 18, (11), 4391-4401. https://doi.org/10.5194/hess-18-4391-2014.

Santos, G.G.; Griebeler, N.P.; Oliveira, L.F.C., 2010. Chuvas intensas relacionadas à erosão hídrica. Revista Brasileira de Engenharia Agrícola e Ambiental, v. 14, (2), 115-123. https://doi.org/10.1590/S1415-43662010000200001.

Selge, F.; Hagel, H.; Gunkel, G.; Doluschitz, R., 2015. Annual rainfall variability and economical dependency of smallholder agriculture in the Semi-Arid Northeastern region of Brazil. Brazilian Journal of Environmental Sciences, (36), 155-167. https://doi.org/10.5327/Z2176-947820151009.

Silva Neto, V.L.; Viola, M.R.; Silva, D.D.; Mello, C.R.; Pereira, S.B.; Giongo, M., 2017. Daily rainfall disaggregation for Tocantins State, Brazil. Ambiente & Água, v. 12, (4), 605-617. https://doi.org/10.4136/ambi-agua.2077.

Skees, J.R., 2010. State of Knowledge Report — data requirements for the design of weather index insurance. GlobalAgRisk, Inc., Broadway, 153 pp.

Souza, V.A.S.; Nunes, M.L.A.; Francener, S.F.; Rosa, A.L., 2014. Eventos de precipitações extremas na Amazônia Ocidental: Rondônia – Brasil. Revista Brasileira de Climatologia, v. 14, 295-315. http://dx.doi.org/10.5380/abclima.v14i1.36816.

U.S. Water Resources Council (USWRC). 1981. Guidelines for determining flood flow frequency. Bulletin 17A. Washington, D.C., U.S Geological Survey, 148 pp. https://doi.org/10.3133/tm4B5.

Vivekanandan, N., 2015a. Comparison of L-moments of probability distributions for extreme value analysis of rainfall for estimation of peak flood discharge for ungauged catchments. International Journal of Scientific Research in Science and Technologic, v. 1, (5), 2395-6011.

Vivekanandan, N., 2015b. Quantitative assessment on fitting of Gumbel and Frechet distributions for extreme value analysis of rainfall. International Journal of Scientific Research in Science and Technology, v. 1, (2), 68-73. https://doi.org/10.32628/IJSRST151224.

Vivekanandan, V.; Shukla, S, 2015. Flood frequency analysis using method of moments and L-moments of probability distributions. Cogent Engineering, v. 2, (1), 1018704. https://doi.org/10.1080/23311916.2015.1018704.

Yuan, J.; Emura, K.; Farnham, C.; Alam, M.A., 2018. Frequency analysis of annual maximum hourly precipitation and determination of best fit probability distribution for regions in Japan. Urban Climate, v. 24, 276-286. https://doi.org/10.1016/j.uclim.2017.07.008.

Zhang, W.B.; Xie, Y.; Liu, B.Y., 2002. Rainfall erosivity estimation using daily rainfall amounts. Scientia Geographica Sinica, v. 22, (6), 705-711. https://doi.org/10.13249/j.cnki.sgs.2002.06.705.