S.M. Papalexiou, and D. Koutsoyiannis, Entropy based derivation of probability distributions: A case study to daily rainfall, *Advances in Water Resources*, 45, 51–57, doi:10.1016/j.advwatres.2011.11.007, 2012.

[doc_id=1182]

[English]

The principle of maximum entropy, along with empirical considerations, can provide consistent basis for constructing a consistent probability distribution model for highly varying geophysical processes. Here we examine the potential of using this principle with the Boltzmann-Gibbs-Shannon entropy definition in the probabilistic modelling of rainfall in different areas worldwide. We define and theoretically justify specific simple and general entropy maximization constraints which lead to two flexible distributions, i.e., the three-parameter Generalized Gamma (GG) and the four-parameter Generalized Beta of the second kind (GB2), with the former being a particular limiting case of the latter. We test the theoretical results in 11 519 daily rainfall records across the globe. The GB2 distribution seems to be able to describe all empirical records while two of its specific three-parameter cases, the GG and the Burr Type XII distributions perform very well by describing the 97.6% and 87.7% of the empirical records, respectively.

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**See also:**
http://dx.doi.org/10.1016/j.advwatres.2011.11.007

**Remarks:**

Correction: As indicated in the attached pdfs, a caret '^' is missing above μ_G in eqn. (7).

**Our works referenced by this work:**

1. | D. Koutsoyiannis, Uncertainty, entropy, scaling and hydrological stochastics, 1, Marginal distributional properties of hydrological processes and state scaling, Hydrological Sciences Journal, 50 (3), 381–404, doi:10.1623/hysj.50.3.381.65031, 2005. |

2. | S.M. Papalexiou, and D. Koutsoyiannis, Ombrian curves in a maximum entropy framework, European Geosciences Union General Assembly 2008, Geophysical Research Abstracts, Vol. 10, Vienna, 00702, doi:10.13140/RG.2.2.23447.98720, European Geosciences Union, 2008. |

3. | S.M. Papalexiou, and D. Koutsoyiannis, Probabilistic description of rainfall intensity at multiple time scales, IHP 2008 Capri Symposium: “The Role of Hydrology in Water Resources Management”, Capri, Italy, doi:10.13140/RG.2.2.17575.96169, UNESCO, International Association of Hydrological Sciences, 2008. |

**Our works that reference this work:**

1. | F. Lombardo, E. Volpi, and D. Koutsoyiannis, Rainfall downscaling in time: Theoretical and empirical comparison between multifractal and Hurst-Kolmogorov discrete random cascades, Hydrological Sciences Journal, 57 (6), 1052–1066, 2012. |

2. | S.M. Papalexiou, D. Koutsoyiannis, and C. Makropoulos, How extreme is extreme? An assessment of daily rainfall distribution tails, Hydrology and Earth System Sciences, 17, 851–862, doi:10.5194/hess-17-851-2013, 2013. |

3. | S.M. Papalexiou, 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. |

4. | A. Efstratiadis, 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. |

5. | D. Koutsoyiannis, Entropy: from thermodynamics to hydrology, Entropy, 16 (3), 1287–1314, doi:10.3390/e16031287, 2014. |

6. | S.M. Papalexiou, and D. Koutsoyiannis, A global survey on the seasonal variation of the marginal distribution of daily precipitation, Advances in Water Resources, 94, 131–145, doi:10.1016/j.advwatres.2016.05.005, 2016. |

7. | D. Koutsoyiannis, and S.M. Papalexiou, Extreme rainfall: Global perspective, Handbook of Applied Hydrology, Second Edition, edited by V.P. Singh, 74.1–74.16, McGraw-Hill, New York, 2017. |

8. | T. Iliopoulou, S.M. Papalexiou, Y. Markonis, and D. Koutsoyiannis, Revisiting long-range dependence in annual precipitation, Journal of Hydrology, 556, 891–900, doi:10.1016/j.jhydrol.2016.04.015, 2018. |

9. | I. Tsoukalas, A. Efstratiadis, and C. Makropoulos, Stochastic periodic autoregressive to anything (SPARTA): Modelling and simulation of cyclostationary processes with arbitrary marginal distributions, Water Resources Research, 54 (1), 161–185, WRCR23047, doi:10.1002/2017WR021394, 2018. |

10. | I. Tsoukalas, S.M. Papalexiou, A. Efstratiadis, and C. Makropoulos, A cautionary note on the reproduction of dependencies through linear stochastic models with non-Gaussian white noise, Water, 10 (6), 771, doi:10.3390/w10060771, 2018. |

11. | P. Dimitriadis, D. Koutsoyiannis, T. Iliopoulou, and P. Papanicolaou, A global-scale investigation of stochastic similarities in marginal distribution and dependence structure of key hydrological-cycle processes, Hydrology, 8 (2), 59, doi:10.3390/hydrology8020059, 2021. |

12. | D. Koutsoyiannis, Stochastics of Hydroclimatic Extremes - A Cool Look at Risk, Edition 2, ISBN: 978-618-85370-0-2, 346 pages, doi:10.57713/kallipos-1, Kallipos Open Academic Editions, Athens, 2022. |

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1. | Zhang, L., and V. P. Singh, Bivariate rainfall and runoff analysis using entropy and copula theories, Entropy, 14, 1784-1812, 2012. |

2. | Kumphon, B., Maximum entropy and maximum likelihood estimation for the three-parameter kappa distribution, Open Journal of Statistics, 2 (4), 415-419, doi: 10.4236/ojs.2012.24050, 2012. |

3. | #Singh, V. P., Entropy Theory and its Application in Environmental and Water Engineering, Wiley, 2013. |

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6. | Serinaldi, F., and C. G. Kilsby, Rainfall extremes: Toward reconciliation after the battle of distributions, Water Resources Research, 50 (1), 336-352, 2014. |

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8. | Ridolfi, E., L. Alfonso, G. Di Baldassarre, F. Dottori, F. Russo, and F. Napolitano, An entropy approach for the optimization of cross-section spacing for river modelling, Hydrological Sciences Journal, 59 (1), 126-137, 2014. |

9. | Hosking, J. R. M., and N. Balakrishnan, A uniqueness result for L-estimators, with applications to L-moments, Statistical Methodology, 10.1016/j.stamet.2014.08.002, 2014. |

10. | Brouers, F., Statistical foundation of empirical isotherms, Open Journal of Statistics, 4, 687-701, 2014. |

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12. | Fan, Y.R., W.W. Huang, G.H. Huang, K. Huang, Y.P. Li, and X.M. Kong, Bivariate hydrologic risk analysis based on a coupled entropy-copula method for the Xiangxi River in the Three Gorges Reservoir area, China, 10.1007/s00704-015-1505-z, 2015. |

**Tagged under:**
Entropy,
Rainfall models