The Determination of the Most Appropriate Probability Distribution Models for the Meteorological Variables

Abstract

Every component of the hydrological cycle is essential for controlling water supplies and assessing the potential catastrophic events like floods and droughts. The variables of hydrological system are unexpected and unique to each place. In this paper, the most crucial variables including precipitation, temperature, relative humidity, and evaporation are examined for Ankara province. For meteorological parameters, the Lognormal, Log-logistic, Gamma, Weibull, Normal, and Gumbel models are used to find the best suitable distributions. Kolmogorov-Smirnov, Cramers-von Mises, Akaike's Information Criterion, Bayesian Information Criterion, Anderson-Darling, and Maximum Loglikelihood methods are utilized to test these models. Results shows that there is a distinct distribution model for each parameter. In particular, it has been determined that the Gumbel distribution is a better model for annual total precipitation, whereas the Normal distribution is a better model for annual minimum temperature. At stations 17130 and 17664, the gamma distribution is observed to be the best fit distribution at annual total precipitation, but station 17128 is found to be the most appropriate Log-logistic and normal distribution. Stations 17128, 17130, and 17664 for annual maximum temperature series are fitted with the Normal, Log-logistic, and Lognormal, respectively. Gamma is found to be the best fit when analyzing annual mean temperature for stations 17128 and 17130, whereas Lognormal is selected for station 17664. It is expected that these results will contribute to the planning of water resources projects in the region.

Keywords

• Meteorological Variables
• Distribution
• Goodness of Fit
• Ankara

References

1. [1] M. A. Alam, K. Emura, C. Farnham, and J. Yuan, “Best-Fit Probability Distributions and Return Periods for Maximum Monthly Rainfall in Bangladesh,” Climate, vol. 6, no. 1, Art. no. 1, Mar. 2018, doi: 10.3390/cli6010009.
2. [2] M. J. Mamman, O. Y. Martins, J. Ibrahim, and M. I. Shaba, “Evaluation of Best-Fit Probability Distribution Models for the Prediction of Inflows of Kainji Reservoir, Niger State, Nigeria,” Air, Soil and Water Research, vol. 10, p. 1178622117691034, Jan. 2017, doi: 10.1177/1178622117691034.
3. [3] I. E. Ahaneku and M. Y. Otache, “Stochastic Characteristics and Modelling of Monthly Rainfall Time Series of Ilorin, Nigeria.,” NONE, 2014, Accessed: Aug. 27, 2022. [Online]. Available: http://repository.futminna.edu.ng:8080/jspui/handle/123456789/8342
4. [4] M. Esit, S. Kumar, A. Pandey, D. M. Lawrence, I. Rangwala, and S. Yeager, “Seasonal to multi-year soil moisture drought forecasting,” npj Climate and Atmospheric Science, vol. 4, no. 1, Art. no. 1, Mar. 2021, doi: 10.1038/s41612-021-00172-z.
5. [5] E. A. Njoku and D. E. Tenenbaum, “Quantitative assessment of the relationship between land use/land cover (LULC), topographic elevation and land surface temperature (LST) in Ilorin, Nigeria,” Remote Sensing Applications: Society and Environment, vol. 27, p. 100780, Aug. 2022, doi: 10.1016/j.rsase.2022.100780.
6. [6] P. Sharma, S. Singh, and S. D. Sharma, “Artificial Neural Network Approach for Hydrologic River Flow Time Series Forecasting,” Agric Res, Jun. 2021, doi: 10.1007/s40003-021-00585-5.
7. [7] M. M. Khudri, “Determination of the Best Fit Probability Distribution for Annual Extreme Precipitation in Bangladesh,” European Journal of Scientific Research, Jan. 2013, Accessed: Aug. 27, 2022. [Online]. Available: https://www.academia.edu/38182722/Determination_of_the_Best_Fit_Probability_Distribution_for_Annual_Extreme_Precipitation_in_Bangladesh
8. [8] J. Yuan, K. Emura, C. Farnham, and M. A. Alam, “Frequency analysis of annual maximum hourly precipitation and determination of best fit probability distribution for regions in Japan,” Urban Climate, vol. 24, pp. 276–286, Jun. 2018, doi: 10.1016/j.uclim.2017.07.008.

9. [9] E. Eris et al., “Frequency analysis of low flows in intermittent and non-intermittent rivers from hydrological basins in Turkey,” Water Supply, vol. 19, no. 1, pp. 30–39, Feb. 2019, doi: 10.2166/ws.2018.051.
10. [10] J. Liu, C. D. Doan, S.-Y. Liong, R. Sanders, A. T. Dao, and T. Fewtrell, “Regional frequency analysis of extreme rainfall events in Jakarta,” Nat Hazards, vol. 75, no. 2, pp. 1075–1104, Jan. 2015, doi: 10.1007/s11069-014-1363-5.
11. [11] M. I. Yuce and M. Esit, “Drought monitoring in Ceyhan Basin, Turkey,” Journal of Applied Water Engineering and Research, vol. 0, no. 0, pp. 1–22, Jun. 2021, doi: 10.1080/23249676.2021.1932616.
12. [12] R. W. Katz and B. G. Brown, “Extreme events in a changing climate: Variability is more important than averages,” Climatic Change, vol. 21, no. 3, pp. 289–302, Jul. 1992, doi: 10.1007/BF00139728.
13. [13] H. B. Unal, S. Asik, M. Avci, S. Yasar, and E. Akkuzu, “Performance of water delivery system at tertiary canal level: a case study of the Menemen Left Bank Irrigation System, Gediz Basin, Turkey,” Agricultural Water Management, vol. 65, no. 3, pp. 155–171, Mar. 2004, doi: 10.1016/j.agwat.2003.10.002.
14. [14] A. S. Anli and A. S. Anli, “Giresun Aksu Havzası Maksimum Akımlarının Frekans Analizi,” Akdeniz Üniversitesi Ziraat Fakültesi Dergisi, vol. 19, no. 1, Art. no. 1, Mar. 2006.
15. [15] H. Yavuz and S. Erdoğan, “Spatial Analysis of Monthly and Annual Precipitation Trends in Turkey,” Water Resour Manage, vol. 26, no. 3, pp. 609–621, Feb. 2012, doi: 10.1007/s11269-011-9935-6.
16. [16] M. Sandalci, “Flood Frequency Analysis of Akçay Stream,” Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 22, no. 5, Art. no. 5, 2018, doi: 10.16984/saufenbilder.402190.
17. [17] A. W. Salami, “Best-fit Probability Distribution model for peak daily rainfall of selected Cities in Nigeria,” New York Science Journal, Jan. 2009, Accessed: Aug. 27, 2022. [Online]. Available: https://www.academia.edu/1593242/Best_fit_Probability_Distribution_model_for_peak_daily_rainfall_of_selected_Cities_in_Nigeria
18. [18] M. T. Amin, M. Rizwan, and A. A. Alazba, “A best-fit probability distribution for the estimation of rainfall in northern regions of Pakistan,” Open Life Sciences, vol. 11, no. 1, pp. 432–440, Jan. 2016, doi: 10.1515/biol-2016-0057.
19. [19] H. Sun, G. Wang, X. Li, J. Chen, B. Su, and T. Jiang, “Regional frequency analysis of observed sub-daily rainfall maxima over eastern China,” Adv. Atmos. Sci., vol. 34, no. 2, pp. 209–225, Feb. 2017, doi: 10.1007/s00376-016-6086-y.
20. [20] G. Chen, J. Norris, J. D. Neelin, J. Lu, L. R. Leung, and K. Sakaguchi, “Thermodynamic and Dynamic Mechanisms for Hydrological Cycle Intensification over the Full Probability Distribution of Precipitation Events,” Journal of the Atmospheric Sciences, vol. 76, no. 2, pp. 497–516, Feb. 2019, doi: 10.1175/JAS-D-18-0067.1.
21. [21] N. Boudrissa, H. Cheraitia, and L. Halimi, “Modelling maximum daily yearly rainfall in northern Algeria using generalized extreme value distributions from 1936 to 2009,” Meteorological Applications, vol. 24, no. 1, pp. 114–119, 2017, doi: 10.1002/met.1610.
22. [22] M. Douka, T. S. Karacostas, E. Katragkou, and C. Anagnostolpoulou, “Annual and Seasonal Extreme Precipitation Probability Distributions at Thessaloniki Based Upon Hourly Values,” in Perspectives on Atmospheric Sciences, Cham, 2017, pp. 521–527. doi: 10.1007/978-3-319-35095-0_75.
23. [23] K. Haddad, “Selection of the best fit probability distributions for temperature data and the use of L-moment ratio diagram method: a case study for NSW in Australia,” Theor Appl Climatol, vol. 143, no. 3, pp. 1261–1284, Feb. 2021, doi: 10.1007/s00704-020-03455-2.
24. [24] N. Vivekanandan, “Comparison of probability distributions in extreme value analysis of rainfall and temperature data,” Environ Earth Sci, vol. 77, no. 5, p. 201, Mar. 2018, doi: 10.1007/s12665-018-7356-z.
25. [25] B. C. Trewin, “Extreme temperature events in Australia,” 2001. Accessed: Aug. 27, 2022. [Online]. Available: https://scholar.google.com/scholar_lookup?title=Extreme+temperature+events+in+Australia&author=Trewin%2C+Blair+C.&publication_year=2001
26. [26] S. Sensoy and M. Demircan, “Climate of Turkey,” Mar. 2016.
27. [27] A. Danandeh Mehr, A. U. Sorman, E. Kahya, and M. Hesami Afshar, “Climate change impacts on meteorological drought using SPI and SPEI: case study of Ankara, Turkey,” Hydrological Sciences Journal, vol. 65, no. 2, pp. 254–268, Jan. 2020, doi: 10.1080/02626667.2019.1691218.
28. [28] A. Lyon, “Why are Normal Distributions Normal?,” Br J Philos Sci, vol. 65, no. 3, pp. 621–649, Sep. 2014, doi: 10.1093/bjps/axs046.
29. [29] R. D. Markovic, “Probability functions of best fit to distributions of annual precipitation and runoff,” 1965. Accessed: Aug. 27, 2022. [Online]. Available: https://scholar.google.com/scholar_lookup?title=Probability+functions+of+best+fit+to+distributions+of+annual+precipitation+and+runoff&author=Markovic%2C+Radmilo+D.&publication_year=1965
30. [30] C.-D. Lai, D. N. Murthy, and M. Xie, “Weibull Distributions and Their Applications,” in Springer Handbook of Engineering Statistics, H. Pham, Ed. London: Springer, 2006, pp. 63–78. doi: 10.1007/978-1-84628-288-1_3.
31. [31] K. J, O. Ngesa, and G. Orwa, “On Generalized Gamma Distribution and Its Application to Survival Data,” International Journal of Statistics and Probability, vol. 8, no. 5, pp. 85–102, 2019.
32. [32] R. Kissell and J. Poserina, Optimal Sports Math, Statistics, and Fantasy. Academic Press, 2017.
33. [33] K.-H. Chang, “Chapter 3 – Solid Modeling,” 2015. doi: 10.1016/B978-0-12-382038-9.00003-X.
34. [34] E. Castillo, Extreme Value Theory in Engineering. Elsevier, 2012.
35. [35] H. Akaike, “‘An information criterion (AIC).,’” Math Sci, vol. 14 (153), pp. 5–7, 1976.
36. [36] M. Stone, “Comments on Model Selection Criteria of Akaike and Schwarz,” Journal of the Royal Statistical Society. Series B (Methodological), vol. 41, no. 2, pp. 276–278, 1979.
37. [37] N. Smirnov, “Table for Estimating the Goodness of Fit of Empirical Distributions,” The Annals of Mathematical Statistics, vol. 19, no. 2, pp. 279–281, 1948.
38. [38] M. A. Stephens, “EDF Statistics for Goodness of Fit and Some Comparisons,” null, vol. 69, no. 347, pp. 730–737, Sep. 1974, doi: 10.1080/01621459.1974.10480196.
39. [39] N. Smirnov, “Estimate of deviation between empirical distribution functions in two independent samples,” Bulletin Moscow University, vol. 2 (2), no. 3–16, 1939.
40. [40] F. Laio, “Cramer–von Mises and Anderson-Darling goodness of fit tests for extreme value distributions with unknown parameters,” Water Resources Research, vol. 40, no. 9, 2004, doi: 10.1029/2004WR003204.
41. [41] C. Schwarz, “Sampling, Regression, Experimental Design and Analysis for Environmental Scientists, Biologists, and Resource Managers,” Mar. 2011.
42. [42] A. C. Cullen, H. C. Frey, and C. H. Frey, Probabilistic Techniques in Exposure Assessment: A Handbook for Dealing with Variability and Uncertainty in Models and Inputs. Springer Science & Business, 1999.