Frequency analysis and rainfall-runoff simulation based on the tree sequence of the Vine copula

Document Type : Research Paper


Department of Water Engineering, Shahrekord University, Shahrekord, Iran.



Rainfall-runoff simulation is one of the most important challenges in the management of water resources in each basin, considering the climatic changes and the increase of extreme values in recent years, especially in different regions of Iran. In this study, the rainfall and runoff values in the statistical period of 1993-2019 were used regarding the rainfall-runoff simulation in the Qale Shahrokh sub-basin in the Zayandeh-Rood Dam basin. In this study, the Vine copula-based simulation approach was used to simulate and joint frequency analysis the flow discharge in Qale Shahrokh station given by the rainfall in Chelgerd, Meyheh and Marghmalek stations. By analyzing the tree sequence of vine copulas and using internal rotated copulas, D-vine copula was selected as the best copula based on the studied statistics. By using the best copula and the conditional relationship c(u4|u1,u2,u3), the flow discharge simulation was done given by rainfall of the upstream stations. The simulation results showed an error rate equal to 18.57 (m3/s) based on the RMSE statistic and the efficiency of the model was 86% based on the NSE statistic. The results of the simulation of flow discharge values given by rainfall in the upstream stations showed that the proposed approach has simulated the average of observed values with high accuracy. By considering the condition of rainfall occurrence in the frequency analysis of flow discharge in Qale Shahrokh station, the joint return period curve and the conditional probability of occurrence in this sub-basin were obtained. Using this curve, it is possible to estimate the flow discharge values of the studied station with high accuracy along with different return periods and different probability of occurrence. Considering the fact that the proposed method is based on the condition of rainfall in the region as well as the marginal distribution according to the data, it has no implementation limitations and is somehow specific to the studied region.


Main Subjects

  1. Aas, K., Czado, C., Frigessi, A., & Bakken, H. (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and economics44(2), 182-198.
  2. Adamson, P.T., Metcalfe, A.V., & Parmentier, B. (1999). Bivariate extreme value distributions: an application of the Gibbs sampler to the analysis of floods. Water Resources Research35(9), 2825-2832.
  3. Bedford, T., & Cooke, R. (2001). Probabilistic risk analysis: foundations and methods. Cambridge University Press.
  4. Czado, C. (2019). Analyzing dependent data with vine copulas. Lecture Notes in Statistics, Springer222.
  5. Dastourani, M., & Nazeri Tahroudi, M. (2022). Toward coupling of groundwater drawdown and pumping time in a constant discharge. Applied Water Science12(4), 1-13.
  6. Favre, A. C., El Adlouni, S., Perreault, L., Thiémonge, N., & Bobée, B. (2004). Multivariate hydrological frequency analysis using copulas. Water Resources Research40(1).
  7. Favre, A. C., Musy, A., & Morgenthaler, S. (2002). Two‐site modeling of rainfall based on the Neyman‐Scott process. Water Resources Research38(12), 43-1.
  8. Joe, H. (1997). Multivariate models and multivariate dependence concepts: Chapman and Hall/CRC.
  9. Kao, S. C., & Govindaraju, R. S. (2007). A bivariate frequency analysis of extreme rainfall with implications for design. Journal of Geophysical Research: Atmospheres112(D13).
  10. Khalili, K., Tahoudi, M. N., Mirabbasi, R., & Ahmadi, F. (2016). Investigation of spatial and temporal variability of precipitation in Iran over the last half century. Stochastic Environmental Research and Risk Assessment30(4), 1205-1221.
  11. Khashei, A., Shahidi, A., Nazeri-Tahroudi, M., & Ramezani, Y. (2022). Bivariate simulation and joint analysis of reference evapotranspiration using copula functions. Iranian Journal of Irrigation & Drainage16(3), 639-656.
  12. Khozeymehnezhad, H., & Nazeri-Tahroudi, M. (2020). Analyzing the frequency of non-stationary hydrological series based on a modified reservoir index. Arabian Journal of Geosciences13(5), 1-13.
  13. Kurowicka, D., & Cooke, R. M. (2007). Sampling algorithms for generating joint uniform distributions using the vine-copula method. Computational statistics & data analysis51(6), 2889-2906.
  14. Li, F., & Zheng, Q. (2016). Probabilistic modelling of flood events using the entropy copula. Advances in Water Resources97, 233-240.
  15. Nash, J. E., & Sutcliffe, J. V. (1970). River flow forecasting through conceptual models part I-A discussion of principles. Journal of Hydrology10(3), 282-290.
  16. Nazeri Tahroudi, M., Ramezani, Y., De Michele, C., & Mirabbasi, R. (2022a). Application of Copula Functions for Bivariate Analysis of Rainfall and River Flow Deficiencies in the Siminehrood River Basin, Iran. Journal of Hydrologic Engineering27(11), 05022015.
  17. Nazeri Tahroudi, M., Ramezani, Y., De Michele, C., & Mirabbasi, R. (2022b). Application of copula‐based approach as a new data‐driven model for downscaling the mean daily temperature. International Journal of Climatology. DOI: 10.1002/joc.7752
  18. Nazeri Tahroudi, M., Ramezani, Y., De Michele, C., & Mirabbasi, R. (2022c). Trivariate joint frequency analysis of water resources deficiency signatures using vine copulas. Applied Water Science12(4), 1-15.
  19. Nazeri Tahroudi, M., Ramezani, Y., De Michele, C., & Mirabbasi, R. (2022d). Multivariate analysis of rainfall and its deficiency signatures using vine copulas. International Journal of Climatology42(4), 2005-2018.
  20. Nazeri Tahroudi, M., Ramezani, Y., De Michele, C., & Mirabbasi, R. (2021). Flood routing via a copula-based approach. Hydrology Research52(6), 1294-1308.
  21. Nelsen, R. B. (2006). An introduction to copulas, ser. Lecture Notes in Statistics. New York: Springer.
  22. Pham, M. T., Vernieuwe, H., De Baets, B., & Verhoest, N. (2018). A coupled stochastic rainfall-evapotranspiration model for hydrological impact analysis. Hydrology and Earth System Sciences22(2), 1263-1283.
  23. Pronoos Sedighi, M., Ramezani, Y., Nazeri Tahroudi, M., & Taghian, M. (2022). Joint frequency analysis of river flow rate and suspended sediment load using conditional density of copula functions. Acta Geophysica, 1-13.
  24. Salvadori, G., & De Michele, C. (2007). On the use of copulas in hydrology: theory and practice. Journal of Hydrologic Engineering, 12(4), 369-380.
  25. Sklar, M. (1959). Fonctions de repartition an dimensions et leurs marges. inst. statist. univ. Paris, 8, 229-231.
  26. Tabatabaei, S. M., Dastourani, M., Eslamian, S., & Nazeri Tahroudi, M. (2022). Ranking and optimizing the rain-gauge networks using the entropy–copula approach (Case study of the Siminehrood Basin, Iran). Applied Water Science12(9), 1-13.
  27. Nazeri Tahroudi, M., Pourreza-Bilondi, M., & Ramezani, Y. (2019). Toward coupling hydrological and meteorological drought characteristics in Lake Urmia Basin, Iran. Theoretical and Applied Climatology138(3), 1511-1523.
  28. Wang, R., Zhao, C., Zhang, J., Guo, E., Li, D., Alu, S., & Dong, Z. (2019). Bivariate copula function-based spatial–temporal characteristics analysis of drought in Anhui Province, China. Meteorology and Atmospheric Physics131(5), 1341-1355.
  29. Xiao, Y., Guo, S., Liu, P., & Fang, B. (2008). A new design flood hydrograph method based on bivariate joint distribution. IAHS Publications-Series of Proceedings and Reports319, 75-82.
  30. Yue, S., Ouarda, T. B. M. J., & Bobée, B. (2001). A review of bivariate gamma distributions for hydrological application. Journal of Hydrology246(1-4), 1-18.
  31. Zhang, D., Yan, M., & Tsopanakis, A. (2018). Financial stress relationships among Euro area countries: an R-vine copula approach. The European Journal of Finance24(17), 1587-1608.
  32. Zhang, L., & Singh, V. (2006). Bivariate flood frequency analysis using the copula method. Journal of Hydrologic Engineering, 11(2), 150-164.