Determination of Optimum Two-Dimensional Copula Functions in Analyzing Groundwater Changes Using Meta Heuristic Algorithms

Document Type : Research Paper


1 1- Ph.D Student of Water Resources Management, University of Birjand, Iran.

2 Associate Professor, Department of Water Engineering, University of Birjand, Iran.

3 3- Professor, Department of Civil and Environmental Engineering, Politecnico di Milano, Italy.

4 4- Associate Professor, Department of Water Engineering, Shahrekord University, Iran.


One of the important factors in recharging the groundwater aquifers in each region is rainfall. Reducing rainfall, and even extreme values of it, can have a huge impact on reducing groundwater levels, which can cause serious damage to aquifers and their construction. Hydrological frequency analysis methods mainly consist of two steps, including selection of an appropriate sample distribution and estimation of selected distribution parameters. Using distribution functions for univariate hydrological analysis has been discussed by many researchers. Different probability distribution models have been used in hydrological studies including the two-parameter distribution such as those by Gumbel, Weibull, Gamma, and Lognormal in several studies (Du et al., 2005; Giraldo Osorio & García Galiano, 2012; Jiang et al., 2015; Villarini, 2009), the distribution of three parameters, such as GEV in other studies (Cannon, 2010; El Adlouni et al., 2007), and the Pearson-type III distribution in the studies by  Chen et al., (2010 , 2015). Therefore, due to the importance of drought and its effects on groundwater level, there are two objectives in this paper. The first is to investigate the artificial intelligence methods and meta-heuristic algorithms to determine the parameters of the copula functions, and the second is the bivariate analysis of precipitation and groundwater shortage signatures in the Nazloochai sub-basin located in the west of Lake Urmia.


Main Subjects

1-    Ahmadi, A., Radmanesh, F., Parham, G.A. and Mirabbasi, R., 2017. Comparison of conventional and intelligent methods in estimating copula function parameters for multivariate frequency analysis of low flows (Case study: Dez river basin), ECO Hydrology, 4(2), pp. 315-325 (In Persian).
2-    Brunner, M.I., Seibert, J. and Favre, A.C., 2016. Bivariate return periods and their importance for flood peak and volume estimation, Wiley Interdisciplinary Reviews: Water, 3(6), pp. 819-33.
3-    Cannon, A.J., 2010. A flexible nonlinear modelling framework for nonstationary generalized extreme value analysis in hydroclimatology. Hydrological Processes, 24, pp. 673–685.
4-    Chen, L., Guo, S., Yan, B., Liu, P. and Fang, B., 2010. A new seasonal design flood method based on bivariate joint distribution of flood magnitude and date of occurrence, Hydrological Sciences Journal, 55(8), pp. 1264–1280.
5-    Chen, L., Singh, V.P., Guo, S. and Zhou, j., 2015. Copula-based method for multisite monthly and daily streamflow simulation, Journal of Hydrology, 528, pp. 369–384
6-    De Michele, C. and Salvadori, G., 2003. A generalized Pareto intensity duration model of storm rainfall exploiting 2-copulas, Journal of Geophysical Research, 108(2), pp. 1–11
7-    De Michele, C., Salvadori, G., Passni, G. and Vezzoli, R., 2007. A multivariate model of sea storms using copulas, Coastal Engineering, 54(10), pp. 734–751
8-    Du, T., Xiong, L., Xu, C.Y., Gippel, C.J., Guo, S. and Liu, P., 2015. Return period and risk analysis of nonstationary low-flow series under climate change, Journal of Hydrology, 527, pp. 234–250
9-    El Adlouni, S., Ouarda, T., Zhang, X., Roy, R. and Bobée, B., 2007. Generalized maximum likelihood estimators for the nonstationary generalized extreme value model, Water Resources Research, 43(3), pp. 1-13.
10- Favre, A.C., Adlouni, S., Perreault, L., Thiémonge, N. and Bobée, B., 2004. Multivariate hydrological frequency analysis using copulas, Water Resources Research, 40(1), pp. 1-11.
11- Giraldo Osorio, J.D. and García Galiano, S.G., 2012. Non-stationary analysis of dry spells in monsoon season of Senegal River Basin using data from regional climate models (RCMs), Journal of Hydrology, 45, pp. 82–92.
12- Grimaldi, S. and Serinaldi, F., 2006a. Asymmetric copula in multivariate flood frequency analysis. Advances in Water Resources, 29(8), pp. 1155–1167
13- Grimaldi, S. and Serinaldi, F., 2006b. Design hyetographs analysis with 3-copula function, Hydrological Sciences Journal, 51(2), pp. 223–238
14- Hui-Mean, F., Yusof, F., Yusop Z. and Suhaila, J., 2019. Trivariate copula in drought analysis: a case study in peninsular Malaysia, Theoretical and Applied Climatology, 138(1), pp. 657-671.
15- Jiang, C., Xiong, L., Xu, C.Y. and Guo, S., 2015. Bivariate frequency analysis of nonstationary low-flow series based on the time-varying copula, Hydrological Processes, 29(6), pp. 1521–1534.
16- Joe, H., 1997. Multivariate models and multivariate dependence concepts. CRC Press.
17- Kao, S.C. and Govindaraju, R. S. 2010. A copula-based joint deficit index for droughts. Journal of Hydrology (Amsterdam), 380(1–2), pp. 121–134
18- Kao, S.C. and Govindaraju, R.S., 2007. A bivariate frequency analysis of extreme rainfall with implications for design. Journal of Geophysical Research, 112(13), 131-159
19- Keef, C., Svensson, C. and Tawn, J.A., 2009. Spatial dependence in extreme river flows and precipitation for Great Britain. Journal of Hydrology (Amsterdam), 378(3–4), pp. 240–252
20- Khalili, K., Tahoudi, M. N., Mirabbasi, R. and Ahmadi, F., 2016. Investigation of spatial and temporal variability of precipitation in Iran over the last half century. Stochastic Environmental Research and Risk Assessment, 30(4), pp. 1205-21.
21- Kuhn, G., Khan, S., Ganguly, A.R., and Branstetter, M.L., 2007. Geospatial temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America, Advances in Water Resources, 30(12), pp. 2401–2423
22- Mirabbasi, R., Anagnostou, E.N., Fakheri-Fard, A., Dinpashoh, Y. and Eslamian, S., 2013. Analysis of meteorological drought in northwest Iran using the Joint Deficit Index, Journal of Hydrology, 492, pp. 35-48.
23- Mirjalili, S. and Lewis, A., 2016. The whale optimization algorithm, Advances in Engineering Software, 95, pp. 51-67.
24- Nazeri Tahroudi, M., Pourreza-Bilondi, M. and Ramezani, Y., 2019. Toward coupling hydrological and meteorological drought characteristics in Lake Urmia Basin, Iran, Theoretical and Applied Climatology.
25- Nelsen, R.B., 2007. An introduction to copulas. Springer Science & Business Media.
26- Renard, B. and Lang, M., 2007. Use of a Gaussian copula for multivariate extreme value analysis: some case studies in hydrology, Advances in Water Resources, 30(4), pp. 897–912
27- Salvadori, G. and De Michele, C., 2007. On the use of copulas in hydrology: theory and practice. Journal of Hydrologic Engineering, 12(4), pp. 369-80.
28- Salvadori, G. and De Michele, C., 2010. Multivariate multiparameter extreme value models and return periods: a copula approach, Water Resources Research, 46(10), pp. 1-11.
29- Salvadori, G., De Michele, C., Kottegoda, N.T., Rosso, R., 2007. Extremes in nature: an approach using copulas. Springer, New York
30- Saremi, S., Mirjalili, S. and Lewis, A., 2017. Grasshopper optimisation algorithm: Theory and application, Advances in Engineering Software, 105, pp. 30-47.
31- Serinaldi, F., Bonaccorso, B., Cancelliere, A. and Grimaldi, S., 2009. Probabilistic characterization of drought properties through copulas, Physics and Chemistry of the Earth, 34(10–12), pp. 596–605.
32- Shiau, J.T., 2006. Fitting drought duration and severity with two-dimensional copulas, Water Resour Manage, 20(5), pp. 795–815.
33- Shiau, J.T., Wang, H. Y. and Tsai, C.T., 2006. Bivariate frequency analysis of floods using copulas, Journal of the American Water Resources Association, 42(6), pp. 1549–1564.
34- Singh, V.P., 2010. Meta-elliptical copulas for drought frequency analysis of periodic hydrologic data, Stochastic Environmental Research and Risk Assessment, 24(3), pp. 425–444.
35- Sklar, M., 1959. Fonctions de répartition à n dimensions et leurs marges. Université Paris.
36- Tahroudi, M.N., Khalili, K., Ahmadi, F., Mirabbasi, R. and Jhajharia, D., 2019. Development and application of a new index for analyzing temperature concentration for Iran’s climate, International Journal of Environmental Science and Technology, 16(6), pp. 2693-2706.
37- Vaziri, H., Karami, H., Mousavi, S.F. and Hadiani, M., 2018. Analysis of hydrological drought characteristics using copula function approach, Paddy and water environment, 16(1), pp. 153-161.
38- Villarini, G., Serinaldi, F., Smith, J.A. and Krajewski, W.F., 2009. On the stationarity of annual flood peaks in the continental United States during the 20th century, Water Resources Research, 45, pp. 1-17.
39- Xiao, Y., Guo, S.L., Liu, P., Yan, B.W. and Chen, L., 2009. Design flood hydrograph based on multi-characteristic synthesis index method, Journal of Hydrologic Engineering, 14(12), pp. 1359–1364.
40- Yue, S. and Rasmussen, P., 2002. Bivariate frequency analysis: discussion of some useful concepts in hydrological application, Hydrological Processes, 16(14), pp. 2881-98.
41- Zeynali, M.J. and Pourreza Bilondi, M., 2018. Matlab and its application in water resources, Publication of University of Birjand, 345 Pp (In Persian).
42- Zhang, L. and Singh, V.P., 2006. Bivariate flood frequency analysis using the copula method, Journal of Hydrologic Engineering, 11(2):150–164.
43- Zhang, L. and Singh, V.P., 2007a. Gumbel Hougaard copula for trivariate rainfall frequency analysis. Journal of Hydrologic Engineering, 12(4), pp. 409–419.
44-  Zhang, L. and Singh, V.P., 2007b. Trivariate flood frequency analysis using the Gumbel-Hougaard copula, Journal of Hydrologic Engineering, 12(4), pp. 431–439.
Volume 44, Issue 1
June 2021
Pages 93-109
  • Receive Date: 17 October 2019
  • Revise Date: 31 December 2019
  • Accept Date: 04 March 2020
  • First Publish Date: 21 March 2021