Abstract
The modeling of sub-daily extreme rainfall has long constituted a challenge for hydrologists in view of the limited availability of data, both in time and space. In this context, the scale invariance principle, which formally links precipitation intensities across a range of time scales under a rigorous theoretical underpin, comprises a useful and parsimonious tool for deriving probabilistic models for the referred stochastic variables. This paper aims at reviewing the history of application of the scale invariance principle to the marginal distributions of intense precipitation, as well as presenting recent developments in this field. We first address the basic concepts and the mathematical formalism of scale invariant models, discussing distinct scaling regimes across durations and estimation procedures. Next, we present a comprehensive review on at-site and regional stationary scaling models applied in several regions of the world, with focus on their underlying assumptions and main limitations. Finally, we address extensions of the rationale for nonstationary models as a means of accommodating potential effects of climate change in extreme short-duration rainfall. While discussing each of these frameworks, we indicate potential research gaps and modeling developments which could further advance the understanding of scaling behavior of extreme precipitation and improve statistical inference. Hence, this review may constitute a useful guide for practitioners and motivate future research in the modeling of short-duration extreme rainfall.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avanzi F, De Michele C, Gabriele S, Ghezzi A, Rosso R (2015) Orographic signature on extreme precipitation of short durations. J Hydrometeorol 16(1):278–294. https://doi.org/10.1175/JHM-D-14-0063.1
Bairwa AK, Khosa R, Maheswaran R (2016) Developing intensity duration frequency curves based on scaling theory using linear probability weighted moments: a case study from India. J Hydrol 542:850–859. https://doi.org/10.1016/j.jhydrol.2016.09.056
Blanchet J, Ceresetti D, Molinié G, Creutin J-D (2016) A regional GEV scale-invariant framework for intensity–duration–frequency analysis. J Hydrol 540:82–95. https://doi.org/10.1016/j.jhydrol.2016.06.007
Bonaccorso B, Brigandì G, Aronica GT (2020) Regional sub-hourly extreme rainfall estimates in Sicily under a scale invariance framework. Water Resour Manag 34(14):4363–4380. https://doi.org/10.1007/s11269-020-02667-5
Boni G, Parodi A, Rudari R (2006) Extreme rainfall events: learning from raingauge time series. J Hydrol 327(3–4):304–314. https://doi.org/10.1016/j.jhydrol.2005.11.050
Borga M, Vezzani C, Fontana GD (2005) Regional Rainfall depth–duration–frequency equations for an Alpine Region. Nat Hazards 36(1–2):221–235. https://doi.org/10.1007/s11069-004-4550-y
Bougadis J, Adamowski K (2006) Scaling model of a rainfall intensity-duration-frequency relationship. Hydrol Process 20(17):3747–3757. https://doi.org/10.1002/hyp.6386
Burlando P, Rosso R (1996) Scaling and multiscaling models of depth-duration-frequency curves for storm precipitation. J Hydrol 187(1–2):20
Casas-Castillo MC, Rodríguez-Solà R, Navarro X, Russo B, Lastra A, González P, Redaño A (2018) On the consideration of scaling properties of extreme rainfall in Madrid (Spain) for developing a generalized intensity-duration-frequency equation and assessing probable maximum precipitation estimates. Theor Appl Climatol 131(1–2):573–580. https://doi.org/10.1007/s00704-016-1998-0
Ceresetti D, Molinié G, Creutin J-D (2010) Scaling properties of heavy rainfall at short duration: a regional analysis. Water Resour Res. https://doi.org/10.1029/2009WR008603
Chevalier C, Martius O, Ginsbourger D (2021) Modeling nonstationary extreme dependence with stationary max-stable processes and multidimensional scaling. J Comput Graph Stat 30(3):745–755. https://doi.org/10.1080/10618600.2020.1844213
Coles S, Pericchi LR, Sisson S (2003) A fully probabilistic approach to extreme rainfall modeling. J Hydrol 273(1–4):35–50. https://doi.org/10.1016/S0022-1694(02)00353-0
Cooley D, Cisewski J, Erhardt RJ, Mannshardt E, Jeon S, Omolo BO, Sun Y (2012) A survey of spatial extremes: measuring spatial dependence and modeling spatial effects. Revstat-Stat J 10(1):135–165. https://doi.org/10.57805/REVSTAT.V10I1.114
de Aguilar MG, Costa VAF (2020) A regional similarity-based approach for sub-daily rainfall nonparametric generation. Rev Bras Recur Hídricos 25:e5. https://doi.org/10.1590/2318-0331.252020190054
Diez-Sierra J, del Jesus M (2019) Subdaily sainfall estimation through daily rainfall downscaling using random forests in Spain. Water 11(1):125. https://doi.org/10.3390/w11010125
Emmanouil S, Langousis A, Nikolopoulos EI, Anagnostou EN (2020) Quantitative assessment of annual maxima, peaks-over-threshold and multifractal parametric approaches in estimating intensity-duration-frequency curves from short rainfall records. J Hydrol 589:125151. https://doi.org/10.1016/j.jhydrol.2020.125151
Fischer EM, Knutti R (2016) Observed heavy precipitation increase confirms theory and early models. Nat Clim Change 6(11):986–991. https://doi.org/10.1038/nclimate3110
Forster C, Oesting M (2022) Non-stationary max-stable models with an application to heavy rainfall data
Fowler HJ, Lenderink G, Prein AF, Westra S, Allan RP, Ban N, Barbero R, Berg P, Blenkinsop S, Do HX, Guerreiro S, Haerter JO, Kendon EJ, Lewis E, Schaer C, Sharma A, Villarini G, Wasko C, Zhang X (2021) Anthropogenic intensification of short-duration rainfall extremes. Nat Rev Earth Environ 2(2):107–122. https://doi.org/10.1038/s43017-020-00128-6
García-Marín AP, Ayuso-Muñoz JL, Jiménez-Hornero FJ, Estévez J (2013) Selecting the best IDF model by using the multifractal approach. Hydrol Process 27(3):433–443. https://doi.org/10.1002/hyp.9272
Gerold LA, Watkins DW (2005) Short duration rainfall frequency analysis in Michigan using scale-invariance assumptions. J Hydrol Eng 10(6):450–457.
Ghanmi H, Bargaoui Z, Mallet C (2016) Estimation of intensity-duration-frequency relationships according to the property of scale invariance and regionalization analysis in a Mediterranean coastal area. J Hydrol 541:38–49. https://doi.org/10.1016/j.jhydrol.2016.07.002
Gómez-Gómez J, Carmona-Cabezas R, Sánchez-López E, Gutiérrez de Ravé E, Jiménez-Hornero FJ (2022) Multifractal fluctuations of the precipitation in Spain (1960–2019). Chaos Solitons Fractals 157:111909. https://doi.org/10.1016/j.chaos.2022.111909
Gupta RS (2017) Hydrology & hydraulic systems, 4th edn. Waveland Press, Inc, Long Grove
Gupta VK, Waymire E (1990) Multiscaling properties of spatial rainfall and river flow distributions. J Geophys Res 95(D3):1999. https://doi.org/10.1029/JD095iD03p01999
Horton RE (1945) Erosional Development of Streams and their drainage basins: Hydrophysical Approach to quantitative morphology. Geol Soc Am Bull 56(3):275. https://doi.org/10.1130/0016-7606(1945)56[275:EDOSAT]2.0.CO;2
Huang Q, Chen Y, Xu S, Liu Y, Li X (2010) Scaling models of a rainfall intensity-duration-frequency relationship. In: 2010 6th international conference on natural computation. IEEE, Yantai, China, pp 3415–3419
Hubert P, Tessier Y, Lovejoy S, Schertzer D, Schmitt F, Ladoy P, Carbonnel JP, Violette S, Desurosne I (1993) Multifractals and extreme rainfall events. Geophys Res Lett 20(10):931–934. https://doi.org/10.1029/93GL01245
Hurst HE (1951) Long-term storage capacity of reservoirs. Trans Am Soc Civ Eng 116(1):770–799. https://doi.org/10.1061/TACEAT.0006518
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
Innocenti S, Mailhot A, Frigon A (2017) Simple scaling of extreme precipitation in North America. Hydrol Earth Syst Sci 21(11):5823–5846. https://doi.org/10.5194/hess-21-5823-2017
IPCC (2022) Climate change 2022: impacts, adaptation and vulnerability. Cambridge University Press, Cambridge and New york
Kottegoda NT, Rosso R (2008) Applied statistics for civil and environmental engineers, 2nd edn. Blackwell Pub, Oxford
Koutsoyiannis D (2004a) Statistics of extremes and estimation of extreme rainfall: I. theoretical investigation. Hydrol Sci J 49(4):3. https://doi.org/10.1623/hysj.49.4.575.54430
Koutsoyiannis D (2004b) Statistics of extremes and estimation of extreme rainfall: II. Empirical investigation of long rainfall records. Hydrol Sci J 49(4):4. https://doi.org/10.1623/hysj.49.4.591.54424
Koutsoyiannis D (2005) Uncertainty, entropy, scaling and hydrological stochastics. 1. Marginal distributional properties of hydrological processes and state scaling. Hydrol Sci J 50(3):1. https://doi.org/10.1623/hysj.50.3.381.65031
Koutsoyiannis D (2022) Stochastics of hydroclimatic extremes - a cool look at risk, 2nd edn. Kallipos, Open Academic Editions, Athens
Koutsoyiannis D, Foufoula-Georgiou E (1993) A scaling model of a storm hyetograph. Water Resour Res 29(7):2345–2361. https://doi.org/10.1029/93WR00395
Koutsoyiannis D, Onof C (2001) Rainfall disaggregation using adjusting procedures on a poisson cluster model. J Hydrol 246(1–4):109–122. https://doi.org/10.1016/S0022-1694(01)00363-8
Kristvik E, Johannessen B, Muthanna T (2019) Temporal downscaling of IDF curves applied to future performance of local stormwater measures. Sustainability 11(5):1231. https://doi.org/10.3390/su11051231
Langousis A, Veneziano D (2007) Intensity-duration-frequency curves from scaling representations of rainfall. Water Resour Res. https://doi.org/10.1029/2006WR005245
Langousis A, Veneziano D, Furcolo P, Lepore C (2009) Multifractal rainfall extremes: theoretical analysis and practical estimation. Chaos Solitons Fractals 39(3):1182–1194. https://doi.org/10.1016/j.chaos.2007.06.004
Lee J, Kim U, Kim S, Kim J (2022) Development and application of a rainfall temporal disaggregation method to project design rainfalls. Water 14(9):1401. https://doi.org/10.3390/w14091401
Lima CHR, Lall U, Troy T, Devineni N (2016) A hierarchical bayesian GEV model for improving local and regional flood quantile estimates. J Hydrol 541:816–823. https://doi.org/10.1016/j.jhydrol.2016.07.042
Lima CHR, Kwon H-H, Kim Y-T (2018) A local-regional scaling-invariant bayesian GEV model for estimating rainfall IDF curves in a future climate. J Hydrol 566:73–88. https://doi.org/10.1016/j.jhydrol.2018.08.075
Lovejoy S, Mandelbrot BB (1985) Fractal properties of rain, and a fractal model. Tellus Dyn Meteorol Oceanogr 37(3):209–232. https://doi.org/10.3402/tellusa.v37i3.11668
Martel J-L, Brissette FP, Lucas-Picher P, Troin M, Arsenault R (2021) Climate change and rainfall intensity–duration–frequency curves: overview of science and guidelines for adaptation. J Hydrol Eng 26(10):03121001. https://doi.org/10.1061/(ASCE)HE.1943-5584.0002122
Mehrotra R, Westra S, Sharma A, Srikanthan R (2012) Continuous rainfall simulation: 1. A regionalized subdaily disaggregation approach. Water Resour Res. https://doi.org/10.1029/2011WR010489
Menabde M, Harris D, Seed A, Austin G, Stow D (1997) Multiscaling properties of rainfall and bounded random cascades. Water Resour Res 33(12):2823–2830. https://doi.org/10.1029/97WR02006
Menabde M, Seed A, Pegram G (1999) A simple scaling model for extreme rainfall. Water Resour Res 35(1):335–339. https://doi.org/10.1029/1998WR900012
Morbidelli R (ed) (2022) Rainfall: modeling, measurement and applications. Elsevier, Amsterdam
Morbidelli R, Saltalippi C, Flammini A, Cifrodelli M, Picciafuoco T, Corradini C, Casas-Castillo MC, Fowler HJ, Wilkinson SM (2017) Effect of temporal aggregation on the estimate of annual maximum rainfall depths for the design of hydraulic infrastructure systems. J Hydrol 554:710–720. https://doi.org/10.1016/j.jhydrol.2017.09.050
Morbidelli R, García-Marín AP, Mamun AA, Atiqur RM, Ayuso-Muñoz JL, Taouti MB, Baranowski P, Bellocchi G, Sangüesa-Pool C, Bennett B, Oyunmunkh B, Bonaccorso B, Brocca L, Caloiero T, Caporali E, Caracciolo D, Casas-Castillo MC, G.Catalini C, Chettih M, Kamal Chowdhury AFM, Chowdhury R, Corradini C, Custò J, Dari J, Diodato N, Doesken N, Dumitrescu A, Estévez J, Flammini A, Fowler HJ, Freni G, Fusto F, García-Barrón L, Manea A, Goenster-Jordan S, Hinson S, Kanecka-Geszke E, Kar KK, Kasperska-Wołowicz W, Krabbi M, Krzyszczak J, Llabrés-Brustenga A, Ledesma JLJ, Liu T, Lompi M, Marsico L, Mascaro G, Moramarco T, Newman N, Orzan A, Pampaloni M, Pizarro-Tapia R, Puentes Torres A, Rashid MM, Rodríguez-Solà R, Manzor MS, Siwek K, Sousa A, Timbadiya PV, Filippos T, Vilcea MG, Viterbo F, Yoo C, Zeri M, Zittis G, Saltalippi C (2020) The history of rainfall data time-resolution in a wide variety of geographical areas. J Hydrol 590:125258. https://doi.org/10.1016/j.jhydrol.2020.125258
Morbidelli R, Saltalippi C, Dari J, Flammini A (2021) A review on Rainfall data resolution and its role in the hydrological practice. Water 13(8):1012. https://doi.org/10.3390/w13081012
Mujumdar PP (2018) Dependence structure of urban precipitation extremes. Adv Water Resour 121:206–218. https://doi.org/10.1016/j.advwatres.2018.08.003
Muller A, Bacro J-N, Lang M (2008) Bayesian comparison of different rainfall depth–duration–frequency relationships. Stoch Environ Res Risk Assess 22(1):33–46. https://doi.org/10.1007/s00477-006-0095-9
Nadarajah S, Anderson CW, Tawn JA (1998) Ordered multivariate extremes. J R Stat Soc Ser B Stat Methodol 60(2):473–496. https://doi.org/10.1111/1467-9868.00136
Naghettini M (2000) A study of the properties of scale Invariance as applied to intensity-duration-frequency relationships of heavy storms. Building partnerships. American Society of Civil Engineers, Minneapolis, pp 1–8
Naghettini M (ed) (2017) Fundamentals of statistical hydrology. Springer, Cham
Nair J, Wierman A, Zwart B (2022) The Fundamentals of heavy tails: properties, emergence, and estimation. Cambridge University Press, Cambridge
Nguyen T-H, Nguyen V-T-V (2020) Linking climate change to urban storm drainage system design: an innovative approach to modeling of extreme rainfall processes over different spatial and temporal scales. J Hydro-Environ Res 29:80–95. https://doi.org/10.1016/j.jher.2020.01.006
Nguyen TH, Nguyen VTV (2022) Development of new extreme Rainfall maps for Urban infrastructure design in Canada using the scale-invariance generalized extreme value distribution. In: Walbridge S, Nik-Bakht M, Ng KTW, Shome M, Alam MS, El Damatty A, Lovegrove G (eds.), Proceedings of the Canadian society of civil engineering annual conference 2021. Springer, Singapore, pp 603–614
Nguyen VTV, Nguyen TD, Wang H (1998) Regional estimation of short duration rainfall extremes. Water Sci Technol. https://doi.org/10.1016/S0273-1223(98)00311-4
Nguyen VTV, Nguyen TD, Ashkar F (2002) Regional frequency analysis of extreme rainfalls. Water Sci Technol 45(2):75–81. https://doi.org/10.2166/wst.2002.0030
Nguyen TH, Nguyen VTV, Nguyen HL (2018) A spatio-temporal statistical downscaling approach to deriving extreme Rainfall IDF relations at ungauged sites in the context of climate change. In: Proceedings of 13th international conference on hydroinformatics. pp 1539–1530
Olsson J, Niemczynowicz J, Berndtsson R (1993) Fractal analysis of high-resolution rainfall time series. J Geophys Res 98(D12):23265. https://doi.org/10.1029/93JD02658
Papalexiou SM, Koutsoyiannis D (2013) Battle of extreme value distributions: a global survey on extreme daily rainfall: survey on extreme daily rainfall. Water Resour Res 49(1):187–201. https://doi.org/10.1029/2012WR012557
Park J, Cross D, Onof C, Chen Y, Kim D (2021) A simple scheme to adjust poisson cluster rectangular pulse rainfall models for improved performance at sub-hourly timescales. J Hydrol 598:126296. https://doi.org/10.1016/j.jhydrol.2021.126296
Paschalis A, Molnar P, Fatichi S, Burlando P (2014) On temporal stochastic modeling of precipitation, nesting models across scales. Adv Water Resour 63:152–166. https://doi.org/10.1016/j.advwatres.2013.11.006
Renard B (2011) A bayesian hierarchical approach to regional frequency analysis. Water Resour Res 47(11):10089. https://doi.org/10.1029/2010WR010089
Rodriguez-Iturbe I, Cox DR, Isham V (1987) Some models for rainfall based on stochastic point processes. Proc R Soc Lond Math Phys Sci 410(1839):269–288. https://doi.org/10.1098/rspa.1987.0039
Rodriguez-Iturbe I, Cox DR, Isham V (1988) A point process model for Rainfall: further developments. Proc R Soc Lond Ser Math Phys Sci 417:283–298
Rodríguez-Solà R, Casas-Castillo MC, Navarro X, Redaño Á (2017) A study of the scaling properties of rainfall in spain and its appropriateness to generate intensity-duration-frequency curves from daily records. Int J Climatol 37(2):770–780. https://doi.org/10.1002/joc.4738
Rosso R, Burlando P (1990) Scale invariance in temporal and spatial rainfall. In: Proceedings XV general assembly European geophysical society. Annales Geophysicae, Copenhagen, pp 23–27
Salvadori G, De Michele C (2001) From generalized pareto to extreme values law: scaling properties and derived features. J Geophys Res Atmospheres 106(D20):24063–24070. https://doi.org/10.1029/2001JD900091
Sampaio J, Costa V (2021) Bayesian regional flood frequency analysis with GEV hierarchical models under spatial dependency structures. Hydrol Sci J 66(3):422–433. https://doi.org/10.1080/02626667.2021.1873997
Sane Y, Panthou G, Bodian A, Vischel T, Lebel T, Dacosta H, Quantin G, Wilcox C, Ndiaye O, Diongue-Niang A, Diop Kane M (2018) Intensity–duration–frequency (IDF) rainfall curves in Senegal. Nat Hazards Earth Syst Sci 18(7):1849–1866. https://doi.org/10.5194/nhess-18-1849-2018
Schlef KE, Kunkel KE, Brown C, Demissie Y, Lettenmaier DP, Wagner A, Wigmosta MS, Karl TR, Easterling DR, Wang KJ, François B, Yan E (2023) Incorporating non-stationarity from climate change into rainfall frequency and intensity-duration-frequency (IDF) curves. J Hydrol 616:128757. https://doi.org/10.1016/j.jhydrol.2022.128757
Sebille Q, Fougères A-L, Mercadier C (2017) Modeling extreme rainfall a comparative study of spatial extreme value models. Spat Stat 21:187–208. https://doi.org/10.1016/j.spasta.2017.06.009
Serinaldi F (2010) Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models. Nonlinear Process Geophys 17(6):697–714. https://doi.org/10.5194/npg-17-697-2010
Serinaldi F, Kilsby CG (2014) Rainfall extremes: toward reconciliation after the battle of distributions. Water Resour Res 50(1):336–352. https://doi.org/10.1002/2013WR014211
Serinaldi F, Kilsby CG, Lombardo F (2018) Untenable nonstationarity: an assessment of the fitness for purpose of trend tests in hydrology. Adv Water Resour 111:132–155. https://doi.org/10.1016/j.advwatres.2017.10.015
Silva AT, Portela MM, Naghettini M, Fernandes W (2017) A bayesian peaks-over-threshold analysis of floods in the Itajaí-açu River under stationarity and nonstationarity. Stoch Environ Res Risk Assess 31(1):185–204. https://doi.org/10.1007/s00477-015-1184-4
Silva DF, Simonovic S, Schardong A, Goldenfum JA (2021) Introducing non-stationarity into the development of intensity-duration-frequency curves under a changing climate. Water 13(8):1008. https://doi.org/10.3390/w13081008
Sivakumar B (2000) Fractal analysis of rainfall observed in two different climatic regions. Hydrol Sci J 45(5):727–738. https://doi.org/10.1080/02626660009492373
Stephenson AG, Lehmann EA, Phatak A (2016) A max-stable process model for rainfall extremes at different accumulation durations. Weather Clim Extrem 13:44–53. https://doi.org/10.1016/j.wace.2016.07.002
Svanidze GG (1980) Mathematical modeling of hydrologic series for hydroelectric and water resources computations. Water Resources Publications, Fort Collins
Tawn JA (1990) Modelling multivariate extreme value distributions. Biometrika 77(2):245–253. https://doi.org/10.2307/2336802
Tessier Y, Lovejoy S, Hubert P, Schertzer D, Pecknold S (1996) Multifractal analysis and modeling of rainfall and river flows and scaling, causal transfer functions. J Geophys Res Atmos 101(D21):26427–26440. https://doi.org/10.1029/96JD01799
Tyralis H, Langousis A (2019) Estimation of intensity–duration–frequency curves using max-stable processes. Stoch Environ Res Risk Assess 33(1):239–252. https://doi.org/10.1007/s00477-018-1577-2
Van de Vyver H (2018) A multiscaling-based intensity–duration–frequency model for extreme precipitation. Hydrol Process 32(11):1635–1647. https://doi.org/10.1002/hyp.11516
Van de Vyver H, Demarée GR (2010) 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. https://doi.org/10.1080/02626661003747390
Veneziano D, Furcolo P (2002) Multifractality of rainfall and scaling of intensity-duration-frequency curves. Water Resour Res 38(12):42. https://doi.org/10.1029/2001WR000372. 1-42–12
Veneziano D, Yoon S (2013) Rainfall extremes, excesses, and intensity-duration-frequency curves: a unified asymptotic framework and new nonasymptotic results based on multifractal measures. Water Resour Res 49(7):4320–4334. https://doi.org/10.1002/wrcr.20352
Verrier S, de Montera L, Barthès L, Mallet C (2010) Multifractal analysis of african monsoon rain fields, taking into account the zero rain-rate problem. J Hydrol 389(1–2):111–120. https://doi.org/10.1016/j.jhydrol.2010.05.035
Wilby RL, Dawson CW, Yu D, Herring Z, Baruch A, Ascott MJ, Finney DL, Macdonald DMJ, Marsham JH, Matthews T, Murphy C (2022) Spatial and temporal scaling of sub-daily extreme rainfall for data sparse places. Clim Dyn. https://doi.org/10.1007/s00382-022-06528-2
Willems P (2000) Compound intensity/duration/frequency-relationships of extreme precipitation for two seasons and two storm types. J Hydrol 233(1–4):189–205. https://doi.org/10.1016/S0022-1694(00)00233-X
Willems P (2012) Impacts of climate change on rainfall extremes and urban drainage systems. IWA Publ, London
Wójcik R, Buishand TA (2003) Simulation of 6-hourly rainfall and temperature by two resampling schemes. J Hydrol 273(1–4):69–80. https://doi.org/10.1016/S0022-1694(02)00355-4
Yan L, Xiong L, Jiang C, Zhang M, Wang D, Xu C (2021) Updating intensity–duration–frequency curves for urban infrastructure design under a changing environment. WIREs Water. https://doi.org/10.1002/wat2.1519
Yeo M, Nguyen V, Kpodonu TA (2021) Characterizing extreme rainfalls and constructing confidence intervals for IDF curves using scaling-GEV distribution model. Int J Climatol 41(1):456–468. https://doi.org/10.1002/joc.6631
Yu P-S, Yang T-C, Lin C-S (2004) Regional rainfall intensity formulas based on scaling property of rainfall. J Hydrol 295(1–4):108–123. https://doi.org/10.1016/j.jhydrol.2004.03.003
Acknowledgements
The authors acknowledge the support to this research from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG). The authors also wish to acknowledge the anonymous reviewer and editors for the valuable comments and suggestions, which greatly helped improving the paper.
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. Literature search was performed by AGB and improvements and review was performed by VAFC. The first draft of the manuscript was written by AGB and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Barbosa, A.G., Costa, V. A review on the scaling properties in maximum rainfall marginal distributions: theoretical background, probabilistic modeling, and recent developments. Stoch Environ Res Risk Assess 37, 4541–4553 (2023). https://doi.org/10.1007/s00477-023-02546-6
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-023-02546-6