Evaluation of three approaches to probable maximum precipitation estimation: a study on two Indian river basins


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 scale-invariant 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 flood-prone 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 1-day (higher) duration. PMP maps are prepared for the two Indian river basins corresponding to 1-day to 5-day 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.


  1. 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

    Google Scholar 

  2. 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

    Article  Google Scholar 

  3. Central Water Commission (CWC) (2012) Integrated hydrological data book. Hydrological data directorate, information systems organization, water planning and projects wing, New Delhi, India

  4. Chavan SR, Srinivas VV (2017) Regionalization based envelope curves for PMP estimation by Hershfield method. Int J Climatol 37(10):3767–3779

    Article  Google Scholar 

  5. Chen LC, Bradley AA (2006) Adequacy of using surface humidity to estimate atmospheric moisture availability for probable maximum precipitation. Water Resour Res 42(9)

  6. Chen CT, Knutson T (2008) On the verification and comparison of extreme rainfall indices from climate models. J Clim 21(7):1605–1621

    Article  Google Scholar 

  7. Chow VT, David RM, Larry WM (1988) Applied hydrology. McGraw-Hill

  8. Coles S, Bawa J, Trenner L, Dorazio P (2001) An introduction to statistical modeling of extreme values, vol 208. Springer, London

    Book  Google Scholar 

  9. Collier CG, Hardaker PJ (1996) Estimating probable maximum precipitation using a storm model approach. J Hydrol 183:277–306

    Article  Google Scholar 

  10. De Lima MIP, Grasman J (1999) Multifractal analysis of 15-min and daily rainfall from a semi-arid region in Portugal. J Hydrol 220(1):1–11

    Article  Google Scholar 

  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)

  12. Dooge JC (1959) A general theory of the unit hydrograph. J Geophys Res 64(2):241–256

    Article  Google Scholar 

  13. Dooge JCI (1986) Looking for hydrologic laws. Water Resour Res 22(9):46S–58S

    Article  Google Scholar 

  14. Douglas EM, Barros AP (2003) Probable maximum precipitation estimation using multifractals: application in the Eastern United States. J Hydrometeorol 4:1012–1024

    Article  Google Scholar 

  15. Gupta VK, Waymire E (1990) Multiscaling properties of spatial rainfall and river flow distributions. J Geophys Res 95(D3):1999–2009

    Article  Google Scholar 

  16. Hershfield DM (1961) Estimating the probable maximum precipitation. J Hydraul Div Am Soc Civ Eng 87(HY5):99–106

    Google Scholar 

  17. Hershfield DM (1965) Method for estimating probable maximum rainfall. J Am Water Works Assoc 57:965–972

    Article  Google Scholar 

  18. 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

    Article  Google Scholar 

  19. Jain SK, Agarwal PK, Singh VP (2007) Hydrology and water resources of India (Vol. 57). Springer Science & Business Media.

  20. Jarvis A, Reuter HI, Nelson A, Guevara E (2008) Hole-filled SRTM for the globe Version 4, available from the CGIAR-CSI SRTM 90m Database (http://srtm.csi.cgiar.org)

  21. 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 40-year reanalysis project. B Am Meteorol Soc 77(3):437–471

    Article  Google Scholar 

  22. Koutsoyiannis D (1999) A probabilistic view of Hershfield’s method for estimating probable maximum precipitation. Water Resour Res 35:1313–1322

    Article  Google Scholar 

  23. 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

    Google Scholar 

  24. Lavallée D, Schertzer D, Lovejoy S (1991) On the determination of the codimension function. In: Non-linear variability in geophysics. Springer, Netherlands, pp 99–109

    Google Scholar 

  25. London J (1962) A study of the atmospheric heat balance. Research Division, College of Engineering, Department of Meteorology and Oceanography, New York University

  26. Lovejoy S, Schertzer D (1985) Generalized scale invariance in the atmosphere and fractal models of rain. Water Resour Res 21(8):1233–1250

    Article  Google Scholar 

  27. Lovejoy S, Schertzer D (1990) Multifractals, universality classes and satellite and radar measurements of cloud and rain fields. J Geophys Res-Atmos 95(D3):2021–2034

    Article  Google Scholar 

  28. 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

    Google Scholar 

  29. 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

    Article  Google Scholar 

  30. Olsson J (1995) Limits and characteristics of the multifractal behavior of a high-resolution rainfall time series. Nonlinear Process Geophys 2:23–29

    Article  Google Scholar 

  31. Osborn TJ, Hulme M (1997) Development of a relationship between station and grid-box rainday frequencies for climate model evaluation. J Clim 10(8):1885–1908

    Article  Google Scholar 

  32. Papalexiou SM, Koutsoyiannis D (2006) A probabilistic approach to the concept of Probable Maximum Precipitation. Adv Geosci 7:51–54

    Article  Google Scholar 

  33. Papalexiou SM, Koutsoyiannis D (2013) Battle of extreme value distributions: a global survey on extreme daily rainfall. Water Resour Res 49(1):187–201

    Article  Google Scholar 

  34. 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, North-Holland, pp 84–88

    Google Scholar 

  35. Rajeevan M, Bhate J (2008) A high resolution daily gridded rainfall data set (1971-2005) for Mesoscale Meteorological Studies. NCC research report no.9

  36. Rajeevan M, Bhate J (2009) A high resolution daily gridded rainfall dataset (1971-2005) for mesoscale meteorological studies. Curr Sci 96(4):558–562

    Google Scholar 

  37. 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

  38. Rakhecha PR, Soman MK (1994) Estimation of probable maximum precipitation for a 2-day duration: part 2—North Indian region. Theor Appl Climatol 49(2):77–84

    Article  Google Scholar 

  39. Rakhecha PR, Deshpande NR, Soman MK (1992) Probable maximum precipitation for a 2-day duration over the Indian peninsula. Theor Appl Climatol 45:277–283

    Article  Google Scholar 

  40. 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(1-2):67–74

    Article  Google Scholar 

  41. Rakhecha PR, Kulkarni AK, Mandal BN, Sangam RB, Deshpande NR (1998) Probable maximum precipitation for the catchment of Koyna dam. Mausam 49:211–216

  42. Rao AR, Hamed KH (2000) Flood Frequency Analysis. CRC Press, Boca Raton

    Google Scholar 

  43. 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

    Article  Google Scholar 

  44. Schertzer D, Lovejoy S (1992) Hard and soft multifractal processes. Phys A 185(1):187–194

    Article  Google Scholar 

  45. Smith WL (1966) Note on the relationship between total precipitable water and surface dew point. J Appl Meteorol 5(5):726–727

    Article  Google Scholar 

  46. Srivastava AK, Rajeevan M, Kshirsagar SR (2009) Development of a high resolution daily gridded temperature data set (1969-2005) for the Indian region. Atmos Sci Lett. https://doi.org/10.1002/asl.232

  47. Swain RE, England JF, Bullard KL, Raff DA (2004) Hydrologic hazard curve estimating procedures. Research report DSO-04-08, U.S. Department of Interior, Bureau of Reclamation

  48. Tessier Y, Lovejoy S, Schertzer D (1993) Universal multifractals: theory and observations for rain and clouds. J Appl Meteorol 32:223–250

    Article  Google Scholar 

  49. 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

    Article  Google Scholar 

  50. 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

    Article  Google Scholar 

  51. 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

  52. World Meteorological Organization (WMO) (2009) Manual on estimation of Probable Maximum Precipitation (PMP), World Meteorological Organization, WMO-No. 1045, Geneva, Switzerland

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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, writing-original draft. V. V. Srinivas: investigation, writing- review and editing.

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Correspondence to Sagar Rohidas Chavan.

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Generalized extreme value distribution

The GEV distribution is a continuous probability distribution which is useful to model annual maximum values x1, …, xm extracted from the observed precipitation data. The cumulative distribution function F of the GEV distribution is of the form:

$$ F(x)=\left\{\begin{array}{l}\exp \left\{-{\left[1+k\frac{\left(x-\xi \right)}{\beta}\right]}^{-1/k}\right\}\kern0.5em \mathrm{for}\ k\ne 0\\ {}\exp \left\{-\exp \left[\frac{\left(x-\xi \right)}{\beta}\right]\right\}\kern2em \mathrm{for}\ k=0\end{array}\right. $$

where ξ, β, and k are the location, scale, and shape parameters respectively (β > 0, ξ, k ∈ ) 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 non-exceedance probability F (where 0 < F < 1).

$$ {\hat{x}}_F=\left\{\begin{array}{l}\hat{\xi}-\frac{\hat{\beta}}{\hat{k}}\left(1-{\left(-\log F\right)}^{-\hat{k}}\right),\kern0.75em \hat{k}\ne 0\\ {}\hat{\xi}-\hat{\beta}\log \left(-\log F\right)\kern2.5em \hat{k}=0\end{array}\right. $$

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 xF 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 xF (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,

$$ T=\frac{1}{1-\exp \left\{-{\left[1+\frac{k}{\beta}\left(\mathrm{PMP}-\xi \right)\right]}^{-1/k}\right\}} $$

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 xF using profile likelihood function (Coles et al. 2001; Lu et al. 2013) involves the following steps.

  1. (1)

    Determine the maximum likelihood estimates of parameters \( \left(\hat{\xi},\hat{\beta},\hat{k}\right) \) and quantile xF of the GEV distribution, where \( \hat{\xi} \) and \( \hat{\beta} \) denote location and scale parameters, respectively.

Let x1, …, xm denote the data points in the annual maximum precipitation record. Construct the log-likelihood function to estimate parameters of GEV distribution. The function is given by Equation (A4) when k ≠ 0, and by Equation (A5) if k = 0.

$$ \mathrm{\ell}\left(\xi, \beta, k\right)=-m\log \beta -\left(1+1/k\right)\times \sum \limits_{i=1}^m\log \left[1+k\left(\frac{x_i-\xi }{\beta}\right)\right]-\sum \limits_{i=1}^m{\left[1+k\left(\frac{x_i-\xi }{\beta}\right)\right]}^{-1/k} $$
$$ \mathrm{\ell}\left(\mu, \beta \right)=-m\log \beta -\sum \limits_{i=1}^m\left(\frac{x_i-\xi }{\beta}\right)-\sum \limits_{i=1}^m\exp \left\{-\left(\frac{x_i-\xi }{\beta}\right)\right\} $$

Maximize the log-likelihood 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 xF corresponding to non-exceedance probability F (where 0 < F < 1).

  1. (2)

    Determine profile likelihood estimate of the shape parameter k of GEV distribution.

Suppose the shape parameter k = k0 ≠ 0 is a constant. Maximize the log-likelihood function given by Equation (A4) about ξ and β, repetitively in a certain range of k0. For each value assumed for k0, 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 c1 − α is 1 − α quantile of chi-square distribution with one degree of freedom.

  1. (3)

    Determine profile likelihood estimate of quantile xF of GEV distribution

To obtain a profile likelihood estimate of annual maximum precipitation quantile xF, redefine the parameters of GEV distribution by introducing xF 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 log-likelihood function of GEV as a function of (xF, β, k).

$$ \xi =\left\{\begin{array}{l}{x}_F+\frac{\beta }{k}\left(1-{\left(-\log F\right)}^{-k}\right),\kern0.75em k\ne 0\\ {}{x}_F+\beta \log \left(-\log F\right)\kern2.5em k=0\end{array}\right. $$

Finally, the profile likelihood estimate and confidence interval of quantile xF 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. (1)

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

  2. (2)

    Area weighted PMP corresponding to 1-day to 5-day durations was estimated for the catchment based on PMP map prepared for the basin as,

$$ {\mathrm{PMP}}^{(c)}=\sum \limits_{i=1}^{N_c}{w}_i\times {\mathrm{PMP}}_i=\sum \limits_{i=1}^{N_c}\left(\frac{A_i^{(c)}}{CA}\right)\times {\mathrm{PMP}}_i $$

where PMP(c) denotes area-weighted PMP for a catchment c having area CA and comprising Nc grids, \( {A}_i^{(c)} \) denotes the area of i-th grid (i = 1, …, Nc) contained in the catchment, and wi is the weight assigned to PMPi (i.e., PMP corresponding to i-th 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/s00704-021-03557-5

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  • Probable maximum precipitation
  • Storm maximization approach
  • Hershfield method
  • Multifractal approach
  • India