Stochastic Model of the Annual Maximum Rainfall Series Using Probability Distributions


  • Nurul Azizah Musakkir Department of Statistics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, 90245 Makassar, Sulawesi Selatan, Indonesia
  • Nurtiti Sunusi Department of Statistics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, 90245 Makassar, Sulawesi Selatan, Indonesia
  • Sri Astuti Thamrin Department of Statistics, Faculty of Mathematics and Natural Sciences, Hasanuddin University, 90245 Makassar, Sulawesi Selatan, Indonesia



Maximum rainfall, GEV distribution, Gumbel distribution, return period, rainy season


Rainfall is a natural process that is often characterized by significant variability and uncertainty. Stochastic models of rainfall typically involve the use of probability distributions to describe the likelihood of different outcomes occurring. This study aimed to model the annual maximum of daily rainfall in Makassar City, Indonesia for the period 1980–2022, specifically focusing on the rainy season (November to April) using probability distributions to estimate return periods. The study used the Generalized Extreme Value (GEVD) and Gumbel distributions. The Kolmogorov-Smirnov test was used to determine the suitability of each distribution, and the likelihood ratio test was employed to determine the best-fit model. The Mann-Kendall test was used to detect any trends in the data. The results indicated that the Gumbel distribution was the best-fit model for data in November, December, January, March, and April, while GEV was appropriate for February. No trends were observed in any of the months. The study then estimated the maximum rainfall for various return periods. January produced the highest maximum rainfall estimates for the 2, 3, and 5-year return periods, while February produced the highest maximum rainfall estimates for the 10 and 20-year return periods. Information about maximum rainfall can be valuable for the government and other stakeholders in developing flood prevention strategies and mitigating the effects of heavy rainfall, particularly during the peak months of the rainy season in Makassar City, which are December, January, and February.


De Luca, D. L., and Capparelli, G. (2022). Rainfall nowcasting model for early warning systems applied to a case over Central Italy. Nat. Hazards, 112, 501-520.

Harmel, R. D., Richardson, C. W., and King, K. W. (2000). Hydrologic response of a small watershed model to generated precipitation. Trans. ASAE, 43(6), 1483-1488.

Trenberth, K. (2011). Changes in precipitation with climate change. Clim. Res., 47(1), 123-138.

Chapman, T. (1998). Stochastic modelling of daily rainfall: the impact of adjoining wet days on the distribution of rainfall amounts. Environ. Model. Softw., 13(3-4), 317-324.

Du, H., Xia, J., Zeng, S., She, D., and Liu, J. (2014). Variations and statistical probability characteristic analysis of extreme precipitation events under climate change in Haihe River Basin, China. Hydrol. Process., 28(3), 913-925.

Li, Z., Brissette, F., and Chen, J. (2014). Assessing the applicability of six precipitation probability distribution models on the Loess Plateau of China. Int. J. Climatol., 34(2), 462-471.

WMO. (2009). Guide to hydrological practices, volume II: Management of water resources and application of hydrological practices. 6th edition. Geneva, Switzerland: World Meteorological Organization.

Ball, J., et al. (2016). Australian rainfall and runoff: A guide to flood estimation, Commonwealth of Australia.

Salinas, J. L., Castellarin, A., Viglione, A., Kohnová, S., and Kjeldsen, T. R. (2014). Regional parent flood frequency distributions in Europe – Part 1: Is the GEV model suitable as a pan-European parent? Hydrol. Earth Syst. Sci., 18(11), 4381-4389.

Sanusi, W., Mulbar, U., Jaya, H., Purnamawati, and Side, S. (2017). Modeling of rainfall characteristics for monitoring of the extreme rainfall event in Makassar City. Am. J. Appl. Sci., 14(4), 456-461.

Sanusi, W., Chaerunnisa, S., Annas, S., Side, S., and Abdy, M. (2022). Estimated parameters of rain flow distribution using L-Moment method in South Sulawesi, Indonesia. J. Appl. Math. Comput., 6(1), 30-40.

Susanti, W., Adnan, A., Yendra, R., and Muhaijir, M. N. (2018). The analysis of extreme rainfall events in Pekanbaru city using three-parameter generalized extreme value and generalized Pareto distribution. Appl. Math. Sci., 12(2), 69-80.

Fisher, R. A., and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math. Proc. Camb. Philos. Soc., 24(2), 180-190.

Jenkinson, A. F. (1955). The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Q. J. R. Meteorol. Soc., 81(348), 158-171.

Lana, X., Martínez, M. D., Burgueño, A., Serra, C., Martín-Vide, J., and Gómez, L. (2006). Distributions of long dry spells in the iberian peninsula, years 1951–1990. Int. J. Climatol., 26(14), 1999-2021.

Park, J.-S., Kang, H.-S., Lee, Y. S., and Kim, M.-K. (2011). Changes in the extreme daily rainfall in South Korea. Int. J. Climatol., 31(15), 2290-2299.

Nadarajah, S., and Choi, D. (2007). Maximum daily rainfall in South Korea. J. Earth Syst. Sci., 116(4), 311-320.

Nashwan, M. S., Ismail, T., and Ahmed, K. (2019). Non-stationary analysis of extreme rainfall in Peninsular Malaysia. J. Sustain. Sci. Manag., 14(3), 17-34.

Kumar, V., Shanu, and Jahangeer. (2017). Statistical distribution of rainfall in Uttarakhand, India. Appl. Water Sci., 7(8), 4765-4776.

Babar, S., and Ramesh, H. (2014). Analysis of extreme rainfall events over Nethravathi basin. ISH J. Hydraul. Eng., 20(2), 212-221.

Abbas, K., Alamgir, Khan, S. A., Khan, D. M., Ali, A., and Khalil, U. (2012). Modeling the distribution of annual maximum rainfall in Pakistan. Eur. J. Sci. Res., 79(3), 418-429.

Aurangzeb, A., Abbas, K., Iqbal, A., Altaf, M., Nadeem, M. T., and Shahzad, F. (2022). Statistical analysis of rainfall trends in Balochistan and Sindh. Tech. J. Univ. Eng. Technol. UET Taxila, 27(1), 8-21.

Boudrissa, N., Cheraitia, H., and Halimi, L. (2017). Modelling maximum daily yearly rainfall in northern Algeria using generalized extreme value distributions from 1936 to 2009. Meteorol. Appl., 24(1), 114-119.

Min, J. L. J., and Halim, S. A. (2020). Rainfall modelling using generalized extreme value distribution with cyclic covariate. Math. Stat., 8(6), 762-772.

Malino, C. R., Arsyad, M., and Palloan, P. (2021). Analysis of rainfall and air temperature parameters in Makassar City related to climate change phenomena. J. Sains Dan Pendidik. Fis., 17(2), 139-145.

Coles, S. (2001). An introduction to statistical modeling of extreme values. London, New York: Springer.

Coles, S. G., and Dixon, M. J. (1999). Likelihood-based inference for extreme value models. Extremes, 2(1), 5-23.

Gilli, M., Maringer, D., and Schumann, E. (2019). Chapter 11-Basic Methods in Numerical Methods and Optimization in Finance. 2nd edition. Elsevier Inc, 229–271. [Online]. Available:

Berger, V. W., and Zhou, Y. (2014). Kolmogorov–Smirnov Test: Overview, in Wiley StatsRef: Statistics Reference Online. 1st ed. Balakrishnan, N., Colton, T., Everitt, B., Piegorsch, W., Ruggeri, F., and Teugels, J. L. Eds. Wiley.

Reiss, R.-D., and Thomas, M. (2007). Statistical Analysis of Extreme Values: with applications to insurance, finance, hydrology and other fields. 3rd edition. Birkhäuser.

Wu, H., Soh, L.-K., Samal, A., and Chen, X.-H. (2008). Trend analysis of streamflow drought events in Nebraska. Water Resour. Manag., 22(2), 145-164.

Cooley, D. (2013). Return periods and return levels under climate change in Extremes in a Changing Climate. A. AghaKouchak, D. Easterling, K. Hsu, S. Schubert, and S. Sorooshian. Eds. Dordrecht: Springer Netherlands. pp. 97–114.

Caires, S. (2016). A comparative simulation study of the annual maxima and the peaks-over-threshold methods. J. Offshore Mech. Arct. Eng., 138(5).

Thoban, M. I., and Hizbaron, D. R. (2020). Urban esilience to floods in parts of Makassar, Indonesia. E3S Web Conf., 200.

VOI. (2023). BMKG: The rob flood in Makassar City was triggered by sea waves rising to a height of 4 meters. Retrieved from

Alam, M., Emura, K., Farnham, C., and Yuan, J. (2018). Best-Fit probability distributions and return periods for maximum monthly rainfall in Bangladesh. Climate, 6(1), 1-16.

Sunusi, N., and Giarno. (2022). Comparison of some schemes for determining the optimal number of rain gauges in a specific area: A case study in an urban area of South Sulawesi, Indonesia. AIMS Environ. Sci, 9(3), 260-276.