Navigation

Statistics of extremes and estimation of extreme rainfall, 1, Theoretical investigation

Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall, 1, Theoretical investigation, Hydrological Sciences Journal, 49 (4), 575–590, 2004.

[doc_id=610]

[English]

The Gumbel distribution has been the prevailing model for quantifying risk associated with extreme rainfall. Several arguments including theoretical reasoning and empirical evidence are supposed to support the appropriateness of the Gumbel distribution. These arguments are examined thoroughly in this work and are put into question. Specifically, theoretical analyses show that the Gumbel distribution is quite unlikely to apply to hydrological extremes and its application may misjudge the risk as it underestimates seriously the largest extreme rainfall amounts. Besides, it is shown that hydrological records of typical length (some decades) may display a distorted picture of the actual distribution suggesting that the Gumbel distribution is an appropriate model for rainfall extremes while it is not. In addition, it is shown that the extreme value distribution of type II (EV2) is a more consistent alternative. Based on the theoretical analysis, in the second part of this study an extensive empirical investigation is performed using a collection of 169 of the longest available rainfall records worldwide, each having 100-154 years of data. This verifies the inappropriateness of the Gumbel distribution and the appropriateness of EV2 distribution for rainfall extremes.

PDF Full text (266 KB)

PDF Additional material:

See also: http://dx.doi.org/10.1623/hysj.49.4.575.54430

Our works referenced by this work:

1. Koutsoyiannis, D., A probabilistic view of Hershfield's method for estimating probable maximum precipitation, Water Resources Research, 35 (4), 1313–1322, 1999.
2. Koutsoyiannis, D., and G. Baloutsos, Analysis of a long record of annual maximum rainfall in Athens, Greece, and design rainfall inferences, Natural Hazards, 22 (1), 29–48, 2000.
3. Koutsoyiannis, D., The Hurst phenomenon and fractional Gaussian noise made easy, Hydrological Sciences Journal, 47 (4), 573–595, 2002.
4. Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall, 2, Empirical investigation of long rainfall records, Hydrological Sciences Journal, 49 (4), 591–610, 2004.

Our works that reference this work:

1. Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall, 2, Empirical investigation of long rainfall records, Hydrological Sciences Journal, 49 (4), 591–610, 2004.
2. Koutsoyiannis, D., Uncertainty, entropy, scaling and hydrological stochastics, 1, Marginal distributional properties of hydrological processes and state scaling, Hydrological Sciences Journal, 50 (3), 381–404, 2005.
3. Koutsoyiannis, D., An entropic-stochastic representation of rainfall intermittency: The origin of clustering and persistence, Water Resources Research, 42 (1), W01401, doi:10.1029/2005WR004175, 2006.
4. Koutsoyiannis, D., A. Montanari, H. F. Lins, and T.A. Cohn, Climate, hydrology and freshwater: towards an interactive incorporation of hydrological experience into climate research—DISCUSSION of “The implications of projected climate change for freshwater resources and their management”, Hydrological Sciences Journal, 54 (2), 394–405, 2009.
5. Koutsoyiannis, D., and A. Langousis, Precipitation, Treatise on Water Science, edited by P. Wilderer and S. Uhlenbrook, 2, 27–78, Academic Press, Oxford, 2011.
6. Koutsoyiannis, D., Hurst-Kolmogorov dynamics and uncertainty, Journal of the American Water Resources Association, 47 (3), 481–495, 2011.
7. Papalexiou, S.M., D. Koutsoyiannis, and A. Montanari, Can a simple stochastic model generate rich patterns of rainfall events?, Journal of Hydrology, 411 (3-4), 279–289, 2011.
8. Papalexiou, S.M., D. Koutsoyiannis, and C. Makropoulos, How extreme is extreme? An assessment of daily rainfall distribution tails, Hydrology and Earth System Sciences, 17, 851–862, 2013.
9. Papalexiou, S.M., and D. Koutsoyiannis, Battle of extreme value distributions: A global survey on extreme daily rainfall, Water Resources Research, 49 (1), 187–201, doi:10.1029/2012WR012557, 2013.
10. Efstratiadis, A., A. D. Koussis, D. Koutsoyiannis, and N. Mamassis, Flood design recipes vs. reality: can predictions for ungauged basins be trusted?, Natural Hazards and Earth System Sciences, 14, 1417–1428, doi:10.5194/nhess-14-1417-2014, 2014.

Other works that reference this work:

1. Mora, R.D., C. Bouvier, L. Neppel and H. Niel, Regional approach for the estimation of low-frequency distribution of daily rainfall in the Languedoc-Roussillon region, France, Hydrological Sciences Journal, 50(1), 17-29, 2005.
2. Gaume, E., On the asymptotic behavior of flood peak distributions, Hydrology and Earth System Sciences, 10(2), 233-243, 2006.
3. Bacro, J.N., and A. Chaouche, Uncertainty in the estimation of extreme rainfalls around the Mediterranean Sea: an illustration using data from Marseille, Hydrological Sciences Journal, 51(3), 389-405, 2006.
4. Lombardo, F., F. Napolitano and F. Russo, On the use of radar reflectivity for estimation of the areal reduction factor, Natural Hazards and Earth System Sciences, 6(3), 377-386, 2006.
5. Di Baldassarre, G., A. Brath and A. Montanari, Reliability of different depth-duration-frequency equations for estimating short-duration design storms, Water Resources Research, 42(12), W12501, 2006.
6. #Re, M., R. Saurral and V.R. Barros, Extreme precipitations in Argentina, Proceedings of the 8th ICSHMO, Foz do Iguacu, Brazil, INPE, 1575-1583, 2006.
7. Endreny, T.A., and S. Pashiardis, The error and bias of supplementing a short, arid climate, rainfall record with regional vs. global frequency analysis, Journal of Hydrology, 334 (1-2), 174-182, 2007.
8. Tie, A.G.B., B. Konan, Y.T. Brou, S. Issiaka, V. Fadika & B. Srohourou, Estimation of daily extreme rainfall in a tropical zone: Case study of the Ivory Coast by comparison of Gumbel and lognormal distributions, Hydrol. Sci. J., 52(1), 49-67, 2007.
9. Langousis, A., and D. Veneziano, Intensity-duration-frequency curves from scaling representations of rainfall, Water Resources Research, 43(2), W02422, 2007.
10. Hultman, N.E., J.G. Koomey and D.M. Kammen, What history can teach us about the future costs of U.S. nuclear power, Environmental Science & Technology, 41(7), 2087 - 2094, 2007.
11. Arnaud, P., J.A. Fine and J. Lavabre, An hourly rainfall generation model applicable to all types of climate, Atmospheric Research, 85(2), 230-242, 2007.
12. Pujol, N., L. Neppel and R. Sabatier, Regional tests for trend detection in maximum precipitation series in the French Mediterranean region, Hydrological Sciences Journal, 52(5), 956-973, 2007.
13. Norbiato, D., M. Borga, M. Sangati and F. Zanon, Regional frequency analysis of extreme precipitation in the eastern Italian Alps and the August 29, 2003 flash flood, Journal of Hydrology, 345(3-4), 149-166, 2007.
14. Buishand, T.A., Estimation of a large quantile of the distribution of multi-day seasonal maximum rainfall: the value of stochastic simulation of long-duration sequences, Climate Research, 34(3), 185-194, 2007.
15. Muller, A., J.N. Bacro and M. Lang, Bayesian comparison of different rainfall depth-duration-frequency relationships, Stochastic Environmental Research and Risk Assessment, 22(1), 33-46, 2008.
16. Kochanek, K., W.G. Strupczewski, V.P. Singh and S. Weglarczyk, The PWM large quantile estimates of heavy tailed distributions from samples deprived of their largest element, Hydrological Sciences Journal, 53(2), 367-386, 2008.
17. Soro, G.E., B.T.A. Goula, F.W. Kouassi, K. Koffi, B. Kamagate, I. Doumouya, I. Savane & B. Srohorou, Courbes intensité durée fréquence des précipitations enclimat tropical humide…, European J. Sci. Res., 21(3), 394-405, 2008.
18. Sutcliffe, J.V., F.A.K. Farquharson, E.L. Tate and S.S. Folwell, Flood regimes in the Southern Caucasus: the influence of precipitation on mean annual floods and frequency curves, Hydrology Research, 39 (5-6), 385-401, 2008.
19. #Kohnová, S., J. Parajka, J. Szolgay and K. Hlavčová, Mapping of Gumbel Extreme Value Distribution Parameters for Estimation of Design Precipitation Totals at Ungauged Sites, Bioclimatology and Natural Hazards, 129-136, Springer, Netherlands, 2008.
20. Kysely, J., A cautionary note on the use of nonparametric bootstrap for estimating uncertainties in extreme-value models, Journal of Applied Meteorology and Climatology, 47 (12), 3236-3251, 2008.
21. Hernandez, A., L. Guenni and B. Sanso, Extreme limit distribution of truncated models for daily rainfall, Environmetrics, 20 (8), 962-980, 2009.
22. Clarke, R. T., R. D. de Paiva, and C. B. Uvo, Comparison of methods for analysis of extremes when records are fragmented: A case study using Amazon basin rainfall data, Journal of Hydrology, 368 (1-4), 26-29, 2009.
23. Suhaila, J., and A. A. Jemain, Investigating the impacts of adjoining wet days on the distribution of daily rainfall amounts in Peninsular Malaysia, Journal of Hydrology, 368 (1-4), 7-25, 2009.
24. Twardosz, R., Probabilistic model of maximum precipitation depths for Kraków (southern Poland, 1886-2002), Theoretical and Applied Climatology, 98 (1-2), 37-45, 2009.
25. Endreny, T. A., and N. Imbeah, Generating robust rainfall intensity-duration-frequency estimates with short-record satellite data, Journal of Hydrology, 371(1-4), 182-191, 2009.
26. Twardosz, R., Analysis of hourly precipitation characteristics in Krakow, southern Poland, using a classification of circulation types, Hydrology Research, 40 (6), 553-563, 2009.
27. Huard, D., A. Mailhot, and S. Duchesne, Bayesian estimation of intensity–duration–frequency curves and of the return period associated to a given rainfall event, Stochastic Environmental Research and Risk Assessment, 24 (3), 337-347, 2010.
28. Saf, B., Assessment of the effects of discordant sites on regional flood frequency analysis, Journal of Hydrology, 380 (3-4), 362-375, 2010.
29. Abolverdi, J., and D. Khalili, Development of regional rainfall annual maxima for Southwestern Iran by L-moments, Water Resources Management, 24 (11), 2501-2526, 2010.
30. Bodini, A., and Q. A. Cossu, Vulnerability assessment of Central-East Sardinia (Italy) to extreme rainfall events, Natural Hazards and Earth System Sciences, 10, 61–72, 2010.
31. Zheng, X., J. Renwick, and A. Clark, Simulation of multisite precipitation using an extended chain-dependent process, Water Resour. Res., 46, W01504, doi:10.1029/2008WR007526, 2010
32. Gaume, E., L. Gaál, A. Viglione, J. Szolgay, S. Kohnová and G. Blöschl, Bayesian MCMC approach to regional flood frequency analyses involving extraordinary flood events at ungauged sites, Journal of Hydrology, 394 (1-2), 101-117, doi:10.1016/j.jhydrol.2010.01.008, 2010.
33. Hossain, F., I. Jeyachandran and R. Pielke Sr., Dam safety effects due to human alteration of extreme precipitation, Water Resources Research, 46, W03301, doi:10.1029/2009WR007704, 2010.
34. AghaKouchak, A., and N. Nasrollahi, Semi-parametric and parametric inference of extreme value models for rainfall data, Water Resources Management, 24 (6), 1229-1249, 2010.
35. Baek, C., and V. Pipiras, Estimation of parameters in heavy-tailed distribution when its second order tail parameter is known, Journal of Statistical Planning and Inference, 140 (7), 1957-1967, 2010.
36. Soro, G.E., T.A. Goula Bi, F.W. Kouassi and B. Srohourou, Update of Intensity-Duration-Frequency Curves for Precipitation of Short Durations in Tropical Area of West Africa (Cote D'ivoire), Journal of Applied Sciences, 10 (9), 704-715, 2010.
37. Blumenfeld, K. A., The frequency of high-impact convective weather events in the Twin Cities Metropolitan Area, Minnesota, Journal of Applied Meteorology and Climatology, 49, 619-631, 2010.
38. Van de Vyver, H., and G. R. Demarée, Construction of Intensity–Duration–Frequency (IDF) curves for precipitation at Lubumbashi, Congo, under the hypothesis of inadequate data, Hydrol. Sci. J., 55(4), 555–564, 2010.
39. Ben-Zvi, A., and B. Azmon, Direct relationships of discharges to the mean and standard deviation of the intervals between their exceedences, Hydrol. Sci. J., 55(4), 565–577, 2010.
40. Leviandier, T., A model capable of changing the heaviness of the tail of a probability distribution function, Hydrol. Sci. J., 55 (6), 992–1001, 2010.
41. Solaiman, T. A., L. M. King and S. P. Simonovic, Extreme precipitation vulnerability in the Upper Thames River basin: uncertainty in climate model projections, International Journal of Climatology, 31 (15), 2350-2364, 2011.
42. Blumenfeld, K. A., and R. H. Skaggs, Using a high-density rain gauge network to estimate extreme rainfall frequencies in Minnesota, Applied Geography, 31 (1), 5-11, DOI: 10.1016/j.apgeog.2010.03.013, 2011.
43. van den Brink, H. W., and G. P. Können, Estimating 10000-year return values from short time series, International Journal of Climatology, 31 (1), 115–126, 2011.
44. Pizarro-Tapia, R., C. Cabrera-Jofre, C. Morales-Calderon, and J. P. Flores-Villanelo, Temporal variations of rainfall and flows in the Maipo River Basin, central Chile, and the influence of glacier melting on water yield (1963-2006), Tecnologia y Ciencias del Agua, 2 (3), 5-19, 2011.
45. #Boukhelifa, M., B. Touaibia and P. Hubert, Storm water risk prevention through the establishment of intensity-duration-frequency curves (IDF): Application to the city of Tipasa in northwestern Algeria, IAHS-AISH Publication, 347, 36-42, 2011.
46. #Grimaldi, S., S.-C. Kao, A. Castellarin, S. M. Papalexiou, A. Viglione, F. Laio, H. Aksoy and A. Gedikli, Statistical Hydrology, In: Treatise on Water Science (ed. by P. Wilderer), 2, 479–517, Academic Press, Oxford, 2011.
47. Naumann, G., M. P. Llano and W. M. Vargas, Climatology of the annual maximum daily precipitation in the La Plata Basin, International Journal of Climatology, 32 (2), 247-260, 2012.
48. Li, C., V. P. Singh, and A. K. Mishra, Simulation of the entire range of daily precipitation using a hybrid probability distribution, Water Resour. Res., 48, W03521, doi: 10.1029/2011WR011446, 2012.
49. Endreny, T. A., and S. Diemont, Methods for assessing stormwater management at archaeological sites: Copan Ruins case study, Journal of Archaeological Science, 39 (8), 2637-2642, 2012.
50. Lee, T., T. B. M. J. Ouarda and C. S. Jeong, Nonparametric Multivariate weather generator and an extreme value theory for bandwidth selection, Journal of Hydrology, 452-453, 161-171, 2012.
51. Solaiman, T. A., S. P. Simonovic and D. H. Burn, Quantifying Uncertainties in the Modelled Estimates of Extreme Precipitation Events at Upper Thames River Basin, British Journal of Environment & Climate Change, 2 (2), 180-215, 2012.
52. Sørup, H. J. D., H. Madsen and K. Arnbjerg-Nielsen, Descriptive and predictive evaluation of high resolution Markov chain precipitation models, Environmetrics, 23 (7), 623-635, 2012.
53. #Benhattab, K., C. Christophe and M. Meddi, Extreme rainfall estimates using the station-year method in Cheliff basin, Algeria, 3rd STAHY International Workshop on Statistical Methods for Hydrology and Water Resources Management, Tunis, Tunisia, 2012.
54. #Liew, S. C., S. V. Raghavan, S.-Y. Liong and R. Sanders, Development of intensity-duration-frequency curves: incorporating climate change projection, Proc. 10th International Conference on Hydroinformatics, 2012.
55. #Ignaccolo, M., and M. Marani, Metastatistics of extreme values and its application in hydrology, arXiv: 1211.3087, 2012.
56. #Dyrrdal, A. V., Estimation of extreme precipitation in Norway and a summary of the state-of-the-art, Report no. 08/2012, Norwegian Meteorological Institute, 2012.
57. #Liew, S. L., S.-Y. Liong and S. V. Raghavan, A novel approach, using regional climate model, to derive present and future IDF curves for data scarce sites, Willis Research Network, 2012.
58. Irizarry-Ortiz, M. M., J. Obeysekera, J. Park, P. Trimble, J. Barnes, W. Park-Said and E. Gadzinski, Historical trends in Florida temperature and precipitation, Hydrological Processes, 27 (16), 2225-2246, 2013.
59. Shahzadi, A., A. S. Akhter and B. Saf, Regional frequency analysis of annual maximum rainfall in monsoon region of Pakistan using L-moments, Pakistan Journal of Statistics and Operation Research, 9 (1), 111-136, 2013.
60. De Michele, C., and M. Ignaccolo, New perspectives on rainfall from a discrete view, Hydrological Processes, 10.1002/hyp.9782, 2013.
61. Habibi, B., M. Meddi and A. Boucefiane, Analyse fréquentielle des pluies journalières maximales Cas du Bassin Chott-Chergui, Nature & Technologie – C: Sciences de l'Environnement, 08, 41-48, 2013.
62. Kyselý, J., L. Gaál, J. Picek and M. Schindler,Return periods of the August 2010 heavy precipitation in northern Bohemia (Czech Republic) in the present climate and under climate change, Journal of Water and Climate Change, 4 (3), 265-286, 2013.
63. Lan, H. X., L. P. Li, Y. S. Zhang, X. Gao and H. J. Liu, Risk assessment of debris flow in Yushu seismic area in China: a perspective for the reconstruction, Nat. Hazards Earth Syst. Sci., 13, 2957-2968, 2013.
64. Vavrus, S. J., and R. J. Behnke, A comparison of projected future precipitation in Wisconsin using global and downscaled climate model simulations: implications for public health, International Journal of Climatology, 34 (10), 3106-3124, 2014.
65. Nastos, P. T., A. G. Paliatsos, K. V. Koukouletsos, I. K. Larissi and K. P. Moustris, Artificial neural networks modeling for forecasting the maximum daily total precipitation at Athens, Greece, Atmospheric Research, 144, 141-150, 2014.
66. Ryu, J., H. Lee, S. Yu and K. Park, Statistical evaluation on storm sewer design criteria under climate change in Seoul, South Korea, Urban Water Journal, 11 (5), 370-378, 2014.
67. Panagoulia, D., P. Economou and C. Caroni, Stationary and nonstationary generalized extreme value modelling of extreme precipitation over a mountainous area under climate change, Environmetrics, 25 (1), 29-43, 2014.
68. Serinaldi, F., and C. G. Kilsby, Rainfall extremes: Toward reconciliation after the battle of distributions, Water Resources Research, 50 (1), 336-352, 2014.
69. Asikoglu, O. L., and E. Benzeden, Simple generalization approach for intensity–duration–frequency relationships, Hydrol. Process., 28 (3), 1114-1123, 2014.
70. Bolívar-Cimé, A. M., E. Díaz-Francés and J. Ortega, Optimality of profile likelihood intervals for quantiles of extreme value distributions: applications to environmental disasters, Hydrological Sciences Journal, 10.1080/02626667.2014.897405, 2014.
71. Serinaldi, F., and C. G. Kilsby, Simulating daily rainfall fields over large areas for collective risk estimation, Journal of Hydrology, 10.1016/j.jhydrol.2014.02.043, 2014.
72. Kamali, A., Review of Progress in Analysis of Extreme Precipitation Using Satellite Data, Open Transactions on Geosciences, 2014.
73. Dyrrdal, A. V., T. Skaugen, F. Stordal and E. J. Førland, Estimating extreme areal precipitation in Norway from a gridded dataset, Hydrological Sciences Journal, 10.1080/02626667.2014.947289, 2014.
74. Manzano-Agugliaro, F., A. Zapata-Sierra, A. Fernández-Castañeda, A. García-Cruz and Q. Hernández-Escobedo, Extreme rainfall relationship in Mexico, Journal of Maps, 10.1080/17445647.2014.945105, 2014.
75. Weinerowska-Bords, K., Development of local IDF-formula using controlled random search method for global optimization, Acta Geophysica, 10.2478/s11600-014-0242-5, 2014.
76. Porto de Carvalho, J. R., E. Delgado Assad, A. Fortes de Oliveira and H. da Silveira Pinto, Annual maximum daily rainfall trends in the Midwest, southeast and southern Brazil in the last 71 years, Weather and Climate Extremes, 10.1016/j.wace.2014.10.001, 2014.

Tagged under: Extremes, Rainfall models