Abstract
Exceptionally large realizations of hydrometeorological variables summarize our knowledge on the underlying processes under extreme conditions. However, the formal probabilistic modeling of record-breaking precipitation and flooding events has received limited attention from researchers and practitioners, particularly with respect to estimation of the future behavior of the variable of interest once a large magnitude event is observed. This paper discusses the use of the two-parameter Rayleigh distribution for describing the dynamics of record-breaking precipitation events, as well as predicting the distributions of future realizations of the process, on the basis of the theory of records. The main motivation for this study was an unprecedented precipitation event which accumulated 186 mm in 3 h—almost twice the previous record—and caused severe damage in the city of Belo Horizonte, in the Brazilian state of Minas Gerais. Our results suggested that, before observing this large magnitude event, the evolution of record-breaking precipitation amounts is properly captured by the model. However, this exceptional event was severely underestimated during prediction—the model estimate was more than 80 mm smaller than the observed rainfall amount. On the other hand, the inclusion of this event in inference entailed a lack of fit of the model and strongly disrupted the dynamics of previous record events, which further highlighted the difficulties of dealing with such unexpected large observations of hydrometeorological variables even under theoretically sound mathematical frameworks.
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References
Ahsanullah M (1995) Record statistics. Nova Science Publishers, Hauppauge
Ahsanullah M, Nevzorov VB (2015) Records via probability theory. Atlantis Press Paris, Paris
Alvares CA, Stape JL, Sentelhas PC, Gonçalves JLM, Sparovek G (2013) Köppen’s climate classification map for Brazil. Meteorol Z 22(6):711–728. https://doi.org/10.1127/0941-2948/2013/0507
Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Willey, New York
Arnold BC, Balakrishnan N, Nagaraja HN (2008) Record values. In: First course in order statistics, classics in applied mathematics. SIAM
Castellarin A (2007) Application of probabilistic envelope curves for design-flood estimation at ungauged sites. Water Resour Res 43(4):W04406. https://doi.org/10.1029/2005WR004384
Castellarin A, Merz R, Blöschl G (2009) Probabilistic envelope curves for extreme rainfall events. J Hydrol 378(3–4):263–271. https://doi.org/10.1016/j.jhydrol.2009.09.030
Cebrián AC, Castillo-Mateo J, Asín J (2022) Record tests to detect non-stationarity in the tails with an application to climate change. Stoch Environ Res Risk Assess 36:313–330. https://doi.org/10.1007/s00477-021-02122-w
Chandler KN (1952) The distribution and frequency of record values. J R Stat Soc Ser B (statistical Methodology) 14(2):220–228
Coles S (2001) An introduction to statistical modeling of extreme values. Springer-Verlag. https://doi.org/10.1007/978-1-4471-3675-0
Costa V, Sampaio J (2021) Bayesian approach for estimating the distribution of annual maximum floods with a mixture model. J Hydrol Eng 26(6):04021017. https://doi.org/10.1061/(ASCE)HE.1943-5584.0002091
Douglas EM, Vogel RM (2006) The probabilistic behavior of floods of record in the United States. J Hydrol Eng 11(5):482–488. https://doi.org/10.1061/(ASCE)1084-0699(2006)11:5(482)
Empacher C, Kamps U, Volovskiy G (2023) Statistical prediction of future sports records based on record values. Stats 6(3):131–147. https://doi.org/10.3390/stats6010008
Falkensteiner M, Schellander H, Ehrensperger G, Hell T (2023) Accounting for seasonality in the metastatistical extreme value distribution. Weather Clim Extremes 42:100601. https://doi.org/10.1016/j.wace.2023.100601
Iliopoulou T, Malamos N, Koutsoyiannis D (2022) Regional ombrian curves: design rainfall estimation for a spatially diverse rainfall regime. Hydrology 9(5):67. https://doi.org/10.3390/hydrology9050067
INMET - Instituto Nacional de Meteorologia (2022) Normais Climatológicas 1991–2020 (Climatological Standards of Brazil 1991–2020). https://portal.inmet.gov.br/normais. Accessed 27 June 2023
Kjeldsen TR, Prosdocimi I (2018) Assessing the element of surprise of record-breaking flood events. J Flood Risk Manag 11:S541–S553. https://doi.org/10.1111/jfr3.12260
Koutsoyiannis D, Kozonis D, Manetas A (1998) A mathematical framework for studying rainfall intensity-duration-frequency relationships. J Hydrol 206(1–2):118–135. https://doi.org/10.1016/S0022-1694(98)00097-3
Kron W (2015) Flood disasters–a global perspective. Water Policy 17(1):6–24. https://doi.org/10.2166/wp.2015.001
Lima CHR, Lall U, Troy T, Devineni N (2016) A hierarchical Bayesian GEV model for improving local and regional flood quantile estimates. J Hydrol 541(B):816–823. https://doi.org/10.1016/j.jhydrol.2016.07.042
Marani M, Ignaccolo M (2015) A metastatistical approach to rainfall extremes. Adv Water Resour 79:121–126. https://doi.org/10.1016/j.advwatres.2015.03.001
Marra F, Nikolopoulos EI, Anagnostou EN, Morin E (2018) Metastatistical extreme value analysis of hourly rainfall from short records: estimation of high quantiles and impact of measurement errors. Adv Water Resour 117:27–39. https://doi.org/10.1016/j.advwatres.2018.05.001
Marra F, Zoccatelli D, Armon M, Morin E (2019) A simplified MEV formulation to model extremes emerging from multiple nonstationary underlying processes. Adv Water Resour 127:280–290. https://doi.org/10.1016/j.advwatres.2019.04.002
Mendes S, Scotti A (2021) The rayleigh-haring-tayfun distribution of wave heights in deep water. Appl Ocean Res 113:102739. https://doi.org/10.1016/j.apor.2021.102739
Miniussi A, Marani M, Villarini G (2020) Metastatistical extreme value distribution applied to floods across the continental United States. Adv Water Resour 136:103498. https://doi.org/10.1016/j.advwatres.2019.103498
Nevzorov VB (2001) Records: mathematical theory. American Mathematical Society, Providence
Pishgar-Komleh SH, Keyhani A, Sefeedpari P (2015) Wind speed and power density analysis based on Weibull and Rayleigh distributions (a case study: Firouzkooh county of Iran). Renew Sustain Energy Rev 42:313–322. https://doi.org/10.1016/j.rser.2014.10.028
Seo JI, Kim Y (2017) Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information. J Appl Stat 44(12):2222–2237. https://doi.org/10.1080/02664763.2016.1251886
Serinaldi F, Kilsby CG (2018) Unsurprising surprises: the frequency of record-breaking and over threshold hydrological extremes under spatial and temporal dependence. Water Resour Res 54(9):6460–6487. https://doi.org/10.1029/2018WR023055
Shcherbakov R (2023) Statistics of weibull record-breaking events. Mathematics 11(3):635. https://doi.org/10.3390/math11030635
SIMGE – Sistema de Meteorologia e Recursos Hídricos de Minas Gerais (2020) Previsão do Tempo e Avisos Meteorológicos. http://simge.mg.gov.br. Accessed 27 June 2023
Stan Development Team (2023) Stan Modeling Language Users Guide and Reference Manual, version 2.32. https://mc-stan.org
Stephenson AG, Lehmann EL, Phatak A (2016) A max-stable process model for rainfall extremes at different accumulation durations. Weather Clim Extremes 13:44–53. https://doi.org/10.1016/j.wace.2016.07.002
Vidrio-Sahagún CH, He J (2022) Hydrological frequency analysis under nonstationarity using the metastatistical approach and its simplified version. Adv Water Resour 166:104244. https://doi.org/10.1016/j.advwatres.2022.104244
Villa C (2017) Bayesian estimation of the threshold of a generalized Pareto distribution for heavy-tailed observations. TEST 26(1):95–118. https://doi.org/10.1007/s11749-016-0501-7
Vogel RM, Castellarin A, Matalas NC, England JF Jr, Zafirakou A (2019) Hydrologic record events. In: Salas JD, Stedinger JR (eds) Teegavarapu RSV. Statistical Analysis of Hydrologic Variables. Methods and Applications. American Sociaty of Civil Engineers, Reston
Vogel RM, Zafirakou-Koulouris A, Matalas NC (2001) Frequency of record-breaking floods in the United States. Water Resour Res 37(6):1723-1731. https://doi.org/10.1029/2001WR900019
Volovskiy G, Kamps U (2020) Maximum observed likelihood prediction of future record values. TEST 29:1072–1097. https://doi.org/10.1007/s11749-020-00701-7
Wu S, Chang W (2023) Power comparison of the testing on the lifetime performance index for Rayleigh lifetime products under progressive type I interval censoring. Commun Stat Simul Comput 52(4):1435–1448. https://doi.org/10.1080/03610918.2021.1884716
Zambrano-Bigiarini M (2020) hydroGOF: Goodness-of-fit functions for comparison of simulated and observed hydrological time series. R package version 0.4–0. https://github.com/hzambran/hydroGOF
Acknowledgements
The authors acknowledge the support to this research from CAPES (“Coordenação de Aperfeiçoamento de Pessoal de Nível Superior”) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico). The authors also wish to acknowledge the editors, an anonymous reviewer and Dr. Theano Iliopoulou for the valuable comments and suggestions, which greatly helped improving the paper.
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This work was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior–Brazil (CAPES) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico).
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All authors contributed to the study conception and design. The first draft of the manuscript was written by VAFC and WSF and all authors commented on it. All authors read and approved the final manuscript.
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Costa, V., Sampaio, J., Fernandes, W. et al. Assessing the unexpectedness of a very large observed rainfall event in the metropolitan region of Belo Horizonte, Brazil. Nat Hazards 120, 3979–3994 (2024). https://doi.org/10.1007/s11069-023-06369-0
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DOI: https://doi.org/10.1007/s11069-023-06369-0