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

Main Article Content

Álvaro José Back
http://orcid.org/0000-0002-0057-2186
Fernanda Martins Bonfante
https://orcid.org/0000-0002-9773-4742

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.

Article Details

How to Cite
Back, Álvaro, & Bonfante, F. (2021). Evaluation of generalized extreme value and Gumbel distributions for estimating maximum daily rainfall. Brazilian Journal of Environmental Sciences (Online), 56(4), 654-664. https://doi.org/10.5327/Z217694781015
Section
Articles

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.