Abstract
Estimates of probable maximum precipitation (PMP) and corresponding probable maximum flood (PMF) are necessary for planning, design, and risk assessment of flood control structures whose failure could have catastrophic consequences. For PMP estimation, multifractal approach (MA) is deemed to be better than conventional approaches, which are based either on statistical concepts or physical aspects. The MA yields physically meaningful PMP estimates by attempting to capture scaleinvariant multiplicative cascade mechanism inherent in rainfall. This paper attempts to gain insights into the performance of MA by comparing PMP estimates obtained using the approach with those resulting from the use of two widely used empirical approaches (storm maximization approach (SMA) and Hershfield method (HM)) on two floodprone river basins (Mahanadi and Godavari) in India. The results indicate that rainfall data of the two river basins exhibit multifractal properties, and the use of MA has an advantage over HM and SMA in estimating PMP corresponding to longer durations (>3 days). PMP estimates obtained using HM are generally lower (higher) than those obtained using SMA for 1day (higher) duration. PMP maps are prepared for the two Indian river basins corresponding to 1day to 5day durations. Further, PMP estimates obtained based on the PMP maps are provided for 18 catchments in the Mahanadi basin and 53 catchments in the Godavari river basin.
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Data availability
Daily gridded precipitation data having a resolution of 0.5°×0.5° was procured from the India Meteorological Department (IMD). Data is available for research purposes only.
References
Arora M, Kumar R, Singh RD, Malhotra J, Kumar N (2016) Analysis of unusual meteorological conditions that led to recent floods in Bhagirathi Basin (Uttarakhand Himalayas). Hydrol Sci J 61(7):1238–1243
Bernardara P, Schertzer D, Sauquet E, Tchiguirinskaia I, Lang M (2008) The flood probability distribution tail: how heavy is it? Stoch Env Res Risk A 22(1):107–122
Central Water Commission (CWC) (2012) Integrated hydrological data book. Hydrological data directorate, information systems organization, water planning and projects wing, New Delhi, India
Chavan SR, Srinivas VV (2017) Regionalization based envelope curves for PMP estimation by Hershfield method. Int J Climatol 37(10):3767–3779
Chen LC, Bradley AA (2006) Adequacy of using surface humidity to estimate atmospheric moisture availability for probable maximum precipitation. Water Resour Res 42(9)
Chen CT, Knutson T (2008) On the verification and comparison of extreme rainfall indices from climate models. J Clim 21(7):1605–1621
Chow VT, David RM, Larry WM (1988) Applied hydrology. McGrawHill
Coles S, Bawa J, Trenner L, Dorazio P (2001) An introduction to statistical modeling of extreme values, vol 208. Springer, London
Collier CG, Hardaker PJ (1996) Estimating probable maximum precipitation using a storm model approach. J Hydrol 183:277–306
De Lima MIP, Grasman J (1999) Multifractal analysis of 15min and daily rainfall from a semiarid region in Portugal. J Hydrol 220(1):1–11
Deshpande NR, Kulkarni BD, Verma AK, Mandal BN (2008) Extreme rainfall analysis and estimation of Probable Maximum Precipitation (PMP) by statistical methods over the Indus river basin in India. J Spat Hydrol 8(1)
Dooge JC (1959) A general theory of the unit hydrograph. J Geophys Res 64(2):241–256
Dooge JCI (1986) Looking for hydrologic laws. Water Resour Res 22(9):46S–58S
Douglas EM, Barros AP (2003) Probable maximum precipitation estimation using multifractals: application in the Eastern United States. J Hydrometeorol 4:1012–1024
Gupta VK, Waymire E (1990) Multiscaling properties of spatial rainfall and river flow distributions. J Geophys Res 95(D3):1999–2009
Hershfield DM (1961) Estimating the probable maximum precipitation. J Hydraul Div Am Soc Civ Eng 87(HY5):99–106
Hershfield DM (1965) Method for estimating probable maximum rainfall. J Am Water Works Assoc 57:965–972
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
Jain SK, Agarwal PK, Singh VP (2007) Hydrology and water resources of India (Vol. 57). Springer Science & Business Media.
Jarvis A, Reuter HI, Nelson A, Guevara E (2008) Holefilled SRTM for the globe Version 4, available from the CGIARCSI SRTM 90m Database (http://srtm.csi.cgiar.org)
Kalnay E, Kanamitsu M, Kistler R, Collins W, Deaven D, Gandin L, Iredell M, Saha S, White G, Woollen J, Zhu Y, Chelliah M, Ebisuzaki W, Higgins W, Janowiak J, Mo KC, Ropelewski C, Wang J, Leetmaa A, Reynolds R, Jenne R, Joseph D (1996) The NCEP/NCAR 40year reanalysis project. B Am Meteorol Soc 77(3):437–471
Koutsoyiannis D (1999) A probabilistic view of Hershfield’s method for estimating probable maximum precipitation. Water Resour Res 35:1313–1322
Kulkarni AK, Mandal BN, Sangam RB (1998) Estimation of probable maximum precipitation (PMP) of the Sina river catchment in Maharashtra for design purposes. Maharashtra Bhugol Shastra Sansthan Patrika 12:135–148
Lavallée D, Schertzer D, Lovejoy S (1991) On the determination of the codimension function. In: Nonlinear variability in geophysics. Springer, Netherlands, pp 99–109
London J (1962) A study of the atmospheric heat balance. Research Division, College of Engineering, Department of Meteorology and Oceanography, New York University
Lovejoy S, Schertzer D (1985) Generalized scale invariance in the atmosphere and fractal models of rain. Water Resour Res 21(8):1233–1250
Lovejoy S, Schertzer D (1990) Multifractals, universality classes and satellite and radar measurements of cloud and rain fields. J Geophys ResAtmos 95(D3):2021–2034
Lovejoy S, Schertzer D (2007) Multifractals and rain. In: Kundzewicz ZW (ed) New uncertainty concepts in hydrology and water resources. Cambridge University Press, Cambridge, pp 61–103
Lu F, Wang H, Yan D, Zhang D, Xiao W (2013) Application of profile likelihood function to the uncertainty analysis of hydrometeorological extreme inference. Sci China Technol Sci 56(12):3151–3160
Olsson J (1995) Limits and characteristics of the multifractal behavior of a highresolution rainfall time series. Nonlinear Process Geophys 2:23–29
Osborn TJ, Hulme M (1997) Development of a relationship between station and gridbox rainday frequencies for climate model evaluation. J Clim 10(8):1885–1908
Papalexiou SM, Koutsoyiannis D (2006) A probabilistic approach to the concept of Probable Maximum Precipitation. Adv Geosci 7:51–54
Papalexiou SM, Koutsoyiannis D (2013) Battle of extreme value distributions: a global survey on extreme daily rainfall. Water Resour Res 49(1):187–201
Parisi G, Frisch U (1985) A multifractal model of intermittency. In: Ghil M, Benzi R, Parisi G (eds) Turbulence and predictability in geophysical fluid dynamics and climate dynamics. Elsevier, NorthHolland, pp 84–88
Rajeevan M, Bhate J (2008) A high resolution daily gridded rainfall data set (19712005) for Mesoscale Meteorological Studies. NCC research report no.9
Rajeevan M, Bhate J (2009) A high resolution daily gridded rainfall dataset (19712005) for mesoscale meteorological studies. Curr Sci 96(4):558–562
Rakhecha PR, Clark C (2002) The probable maximum flood at the Ukai and Lakhwar dam sites in India. In: The extremes of lite extremes: Extraordinary Floods, July 2000, Reykjavik, Iceland
Rakhecha PR, Soman MK (1994) Estimation of probable maximum precipitation for a 2day duration: part 2—North Indian region. Theor Appl Climatol 49(2):77–84
Rakhecha PR, Deshpande NR, Soman MK (1992) Probable maximum precipitation for a 2day duration over the Indian peninsula. Theor Appl Climatol 45:277–283
Rakhecha PR, Mandal BN, Kulkarni AK, Deshpande NR (1995) Estimation of probable maximum precipitation for catchments in eastern India by a generalized method. Theor Appl Climatol 51(12):67–74
Rakhecha PR, Kulkarni AK, Mandal BN, Sangam RB, Deshpande NR (1998) Probable maximum precipitation for the catchment of Koyna dam. Mausam 49:211–216
Rao AR, Hamed KH (2000) Flood Frequency Analysis. CRC Press, Boca Raton
Schertzer D, Lovejoy S (1987) Physical modeling and analysis of rain and clouds by anisotropic scaling of multiplicative processes. J Geophys Res 92D:9693–9714
Schertzer D, Lovejoy S (1992) Hard and soft multifractal processes. Phys A 185(1):187–194
Smith WL (1966) Note on the relationship between total precipitable water and surface dew point. J Appl Meteorol 5(5):726–727
Srivastava AK, Rajeevan M, Kshirsagar SR (2009) Development of a high resolution daily gridded temperature data set (19692005) for the Indian region. Atmos Sci Lett. https://doi.org/10.1002/asl.232
Swain RE, England JF, Bullard KL, Raff DA (2004) Hydrologic hazard curve estimating procedures. Research report DSO0408, U.S. Department of Interior, Bureau of Reclamation
Tessier Y, Lovejoy S, Schertzer D (1993) Universal multifractals: theory and observations for rain and clouds. J Appl Meteorol 32:223–250
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 ResAtmos 101(D21):26427–26440
Volosciuk C, Maraun D, Semenov VA, Park W (2015) Extreme precipitation in an atmosphere general circulation model: impact of horizontal and vertical model resolutions. J Clim 28(3):1184–1205
World Meteorological Organization (WMO) (1986) Manual for estimation of probable maximum precipitation, Operational Hydrology Report No. 1, 2nd edition, WMO No. 332, Secretariat of the World Meteorological Organization, Geneva, Switzerland
World Meteorological Organization (WMO) (2009) Manual on estimation of Probable Maximum Precipitation (PMP), World Meteorological Organization, WMONo. 1045, Geneva, Switzerland
Acknowledgements
The authors express their gratitude to the Editor for the constructive reviews that resulted in improving quality of the work. The authors are also thankful to the India Meteorological Department, which provided the gridded rainfall data for the study.
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Sagar Rohidas Chavan: formal analysis, investigation, writingoriginal draft. V. V. Srinivas: investigation, writing review and editing.
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Appendixes
Appendixes
Generalized extreme value distribution
The GEV distribution is a continuous probability distribution which is useful to model annual maximum values x_{1}, …, x_{m} extracted from the observed precipitation data. The cumulative distribution function F of the GEV distribution is of the form:
where ξ, β, and k are the location, scale, and shape parameters respectively (β > 0, ξ, k ∈ ℜ) and 1 + k(x − ξ)/β > 0. Estimates of the parameters are obtained in this study using the maximum likelihood method. Equation (A1) can be rearranged to get \( {\hat{x}}_F \) (i.e., the quantile estimate of x) for any specified value of nonexceedance probability F (where 0 < F < 1).
The shape parameter k is important, as it determines the tail behavior of the distribution. If k > 0, it will correspond to Fréchet distribution, which has a long tail. If k = 0, it is referred to as Gumbel distribution whose tail is exponential. If k < 0, it will correspond to Weibull distribution with an upper limit for x. The sampling error caused by the limited sample data (observations on the extreme precipitation) is inevitable, which increases the uncertainty in \( \hat{k} \) and consequently x_{F} that depends on \( \hat{k} \). The uncertainty can be assessed by constructing the profile likelihood function to determine the confidence intervals of the shape parameter k and the quantile estimate x_{F} (e.g., Coles et al. 2001; Lu et al. 2013).
For a given value of precipitation (PMP), the corresponding return period T can be estimated in the case of GEV distribution as,
Concept of profile likelihood function
In analysis with GEV distribution, the procedure to estimate confidence intervals of the shape parameter k and the quantile estimate x_{F} using profile likelihood function (Coles et al. 2001; Lu et al. 2013) involves the following steps.

(1)
Determine the maximum likelihood estimates of parameters \( \left(\hat{\xi},\hat{\beta},\hat{k}\right) \) and quantile x_{F} of the GEV distribution, where \( \hat{\xi} \) and \( \hat{\beta} \) denote location and scale parameters, respectively.
Let x_{1}, …, x_{m} denote the data points in the annual maximum precipitation record. Construct the loglikelihood function to estimate parameters of GEV distribution. The function is given by Equation (A4) when k ≠ 0, and by Equation (A5) if k = 0.
Maximize the loglikelihood function to determine the maximum likelihood estimates of the parameters \( \left(\hat{\xi},\hat{\beta},\hat{k}\right) \). Use the estimated values of the parameters in Equation (A2) to arrive at a quantile estimate x_{F} corresponding to nonexceedance probability F (where 0 < F < 1).

(2)
Determine profile likelihood estimate of the shape parameter k of GEV distribution.
Suppose the shape parameter k = k_{0} ≠ 0 is a constant. Maximize the loglikelihood function given by Equation (A4) about ξ and β, repetitively in a certain range of k_{0}. For each value assumed for k_{0}, the maximum value of the likelihood function can be obtained. The profile likelihood function of k is constructed using the resulting information. Finally, an approximation of 1 − α confidence interval of k is determined by using \( {C}_{\alpha }=\left\{k:2\left(\mathrm{\ell}\left(\hat{\xi},\hat{\beta},\hat{k}\right)\underset{\xi, \alpha }{\max}\mathrm{\ell}\left(\xi, \beta, k\right)\right)\le {c}_{1\alpha}\right\} \), where c_{1 − α} is 1 − α quantile of chisquare distribution with one degree of freedom.

(3)
Determine profile likelihood estimate of quantile x_{F} of GEV distribution
To obtain a profile likelihood estimate of annual maximum precipitation quantile x_{F}, redefine the parameters of GEV distribution by introducing x_{F} into the likelihood function. It can be achieved by rearranging Equation (A2) to get Equation (A6) and substituting Equation (A6) into Equation (A4) to derive the loglikelihood function of GEV as a function of (x_{F}, β, k).
Finally, the profile likelihood estimate and confidence interval of quantile x_{F} of GEV distribution can be obtained by following the procedure described (for shape parameter k) in step (2).
Area weighted PMP for a catchment
To estimate PMP for the catchment of a stream gauge in a river basin, the following steps were executed.

(1)
Catchment of the gauge was delineated by processing SRTM DEM data using ArcHYDRO tools in the ArcGIS framework.

(2)
Area weighted PMP corresponding to 1day to 5day durations was estimated for the catchment based on PMP map prepared for the basin as,
where PMP^{(c)} denotes areaweighted PMP for a catchment c having area CA and comprising N_{c} grids, \( {A}_i^{(c)} \) denotes the area of ith grid (i = 1, …, N_{c}) contained in the catchment, and w_{i} is the weight assigned to PMP_{i} (i.e., PMP corresponding to ith grid) as the ratio of \( {A}_i^{(c)} \) to CA.
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Chavan, S.R., Srinivas, V.V. Evaluation of three approaches to probable maximum precipitation estimation: a study on two Indian river basins. Theor Appl Climatol 144, 731–749 (2021). https://doi.org/10.1007/s00704021035575
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Keywords
 Probable maximum precipitation
 Storm maximization approach
 Hershfield method
 Multifractal approach
 India