Trivariate flood frequencies analysis with regional dependence and Copula Functions
DOI:
https://doi.org/10.24850/j-tyca-2025-01-08Keywords:
Frank and Gumbel-Hougaard CF, symmetric multivariate CF, asymmetric trivariate CF, Kendall's tau ratio, upper tail and observed dependences, secondary return period, design eventsAbstract
Design floods (DF) give dimension for hydrological security to the hydraulic protection works. The most reliable estimate is obtained through the univariate frequency analysis (FA), which represents the maximum annual flows available, with an appropriate probability distribution function (PDF), to estimate the predictions sought. In this study, the FA is carried out with the trivariate approach, processing a base record of flows QX and two other auxiliaries, QY and QZ, which are correlated to the first and have the same amplitude. The verification of the simultaneous character of the QX, QY and QZ flows (that they belong to the same event analyzed) is described in detail. The joint trivariate PDF of flows was obtained using the Gumbel-Hougaard Copula function, which showed an excellent fit and reproduced the observed dependency on flows. A numerical application exposed here processed 43 annual flows and was carried out at the hydrometric stations, Tempoal as base, and El Cardón and Terrerillos as auxiliaries of the Tempoal river system of Hydrological Region No. 26 (Pánuco), Mexico. In order to obtain the ideal marginal PDFs, the Moment Ratios Diagram L was used and, in addition, the Kappa and Wakeby PDFs were applied to contrast predictions. Finally, conclusions are formulated, which highlight the importance of the trivariate approach, based on regional dependence, to validate the behavior in magnitudes of the DF estimated with the univariate approach.
References
AghaKouchak, A., Sellars, S., & Sorooshian, S. (2013). Chapter 6. Methods of tail dependence estimation. In: AghaKouchak, A., Easterling, D., Hsu, K., Schubert, S., & Sorooshian, S. (eds.). Extremes in a changing climate (pp. 163-179). Dordrecht, The Netherlands: Springer.
Aldama, A. A., Ramírez, A. I., Aparicio, J., Mejía-Zermeño, R., & Ortega-Gil, G. E. (2006). Seguridad hidrológica de las presas en México. Jiutepec, México: Instituto Mexicano de Tecnología del Agua.
Barbe, P., Genest, C., Ghoudi, K., & Rémillard, B. (1996). On Kendall’s Process. Journal of Multivariate Analysis, 58(2), 197-229.
Beale, E. M. L. & Little, R. J. A. (1975). Missing values in multivariate analysis. Journal of Royal Statistical Society Series B, 37(1), 129-145. DOI: 10.1111/j.2517–6161.1975.tb01037.x
Bender, J., Wahl, T., & Jensen, J. (2014). Multivariate design in the presence of non–stationarity. Journal of Hydrology, 514(June), 123-130. DOI: 10.1016/j.jhydrol.2014.04.017
Bobée, B. (1975). The Log-Pearson type 3 distribution and its application to Hydrology. Water Resources Research, 11(5), 681-689. DOI: 10.1029/WR011i005p00681
Bobée, B., & Ashkar, F. (1991). Chapter 1. Data requirements for hydrologic frequency analysis. In: The Gamma Family and derived distributions applied in hydrology (pp. 1-12). Littleton, USA: Water Resources Publications.
Box, M. J. (1965). A new method of constrained optimization and a comparison with other methods. Computer Journal, 8(1), 42-52.
Bunday, B. D. (1985). Theme 6.2. The complex method. In: Basic optimisation methods (pp. 98-106). London, England: Edward Arnold publishers, Ltd.
Campos-Aranda, D. F. (2003). Capítulo 7. Integración numérica y Capítulo 9. Optimización numérica. En: Introducción a los métodos numéricos: software en Basic y aplicaciones en hidrología superficial (pp. 137-153, 172-211). San Luis Potosí, México: Editorial Universitaria Potosina.
Campos-Aranda, D. F. (2014). Predicción de crecientes usando la distribución Pareto Generalizada ajustada con tres métodos simples. Tláloc, 65, octubre-diciembre, 7-26.
Campos-Aranda, D. F. (2015). Estimación simultánea de datos hidrológicos anuales faltantes en múltiples sitios. Ingeniería. Investigación y Tecnología, 16(2), 295-306.
Campos-Aranda, D. F. (2022). Análisis de frecuencias de crecientes bivariado con dependencia regional y funciones Cópula. Aqua-LAC, 14(2), 47-61. DOI: 10.29104/phi–aqualac/2022–v14–2–11
Campos-Aranda, D. F. (2023). Selección y aplicación de funciones Cópula con dependencia en su extremo derecho al análisis de frecuencias conjunto (Q,V) de crecientes anuales. Tecnología y ciencias del agua, 14(5), 120-188. DOI: 10.24850/j–tyca–14–05–03
Capéraà, P., Fougéres, A. L., & Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84(3), 567-577. DOI: 10.1093/biomet/84.3.567
Chai, T., & Draxler, R. R. (2014). Root mean square error (RMSE) or mean absolute error (MAE)? – Arguments against avoiding RMSE in the literature. Geoscientific Model Development, 7(3), 1247-1250. DOI: 10.5194/gmd–7–1247–2014
Chebana, F., & Ouarda, T. B. M. J. (2021). Multivariate non-stationary hydrological frequency analysis. Journal of Hydrology, 593(February), 125907. DOI: 10.1016/j.jhydrol.2020.125907
Chen, L., & Guo, S. (2019). Chapter 2. Copula theory and Chapter 3. Copula-based flood frequency analysis. In: Copulas and its application in hydrology and water resources (pp. 13-38, 39-71). Gateway East, Singapore: Springer.
Chowdhary, H., & Singh, V. P. (2019). Chapter 11. Multivariate frequency distributions in hydrology. In: Teegavarapu, R. S. V., Salas, J. D., & Stedinger, J. R. (eds.). Statistical analysis of hydrologic variables (pp. 407-489). Reston, USA: American Society of Civil Engineers.
Davis, P. J., & Polonsky, I. (1972). Chapter 25. Numerical interpolation, differentiation and integration. In: Abramowitz, M., & Stegun, I. A. (eds.). Handbook of mathematical functions (9th print.) (pp. 875-926). New York, USA: Dover Publications.
Dupuis, D. J. (2007). Using Copulas in hydrology: Benefits, cautions, and issues. Journal of Hydrologic Engineering, 12(4), 381-393. DOI: 10.1061/(ASCE)1084–0699(2007)12:4(381)
Escalante-Sandoval, C. A., & Raynal-Villaseñor, J. A. (1994). A trivariate extreme value distribution applied to flood frequency analysis. Journal of Research of the National Institute of Standards and Technology, 99(4), 369-375.
Escalante-Sandoval, C., & Raynal-Villaseñor, J. (2008). Trivariate generalized extreme value distribution in flood frequency analysis. Hydrological Sciences Journal, 53(3), 550-567.
Frahm, G., Junker, M., & Schmidt, R. (2005). Estimating the tail–dependence coefficient: Properties and pitfalls. Insurance: Mathematics and Economics, 37(1), 80-100. DOI: 10.1016/j–insmatheco.2005.05.008
Genest, C., & Favre, A. C. (2007). Everything you always wanted to know about Copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12(4), 347-368. DOI: 10.1061/(ASCE)1084–0699(2007)12:4(347)
Genest, C., & Chebana, F. (2017). Copula modeling in hydrologic Frequency Analysis. In: Singh, V. P. (ed.). Handbook of Applied Hydrology (2nd ed.) (pp. 30.1-30.10). New York, USA: McGraw-Hill Education.
Goel, N. K., Seth, S. M., & Chandra, S. (1998). Multivariate modeling of flood flows. Journal of Hydraulic Engineering, 124(2), 146-155.
Gräler, B., van den Berg, M. J., Vandenberghe, S., Petroselli, A., Grimaldi, S., De Baets, B., & Verhoest, N. E. C. (2013). Multivariate return periods in hydrology: A critical and practical review focusing on synthetic design hydrograph estimation. Hydrology and Earth System Sciences, 17(4), 1281-1296. DOI: 10.5194/hess–17–1281–2013
Grimaldi, S., & Serinaldi, F. (2006a). Design hyetograph analysis with 3–copula function. Hydrological Sciences Journal, 51(2), 223-238. DOI: 10.1623/hysj.51.2.223
Grimaldi, S., & Serinaldi, F. (2006b). Asymmetric copula in multivariate flood frequency analysis. Advances in Water Resources, 29(8), 1155–1167. DOI: 10.1016/j.advwatres.2005.09.005
Hosking, J. R. M. (1994). The four-parameter Kappa distribution. IBM Journal of Research and Development, 38(3), 251-258.
Hosking, J. R., & Wallis, J. R. (1997). Appendix: L-moments for some specific distributions. In: Regional Frequency Analysis. An approach based on L–moments (pp. 191-209). Cambridge, England: Cambridge University Press.
IMTA, Instituto Mexicano de Tecnología del Agua. (2003). Banco Nacional de Datos de Aguas Superficiales (Bandas) (8 CD). Jiutepec, México: Comisión Nacional del Agua, Secretaría de Medio Ambiente y Recursos Naturales, Instituto Mexicano de Tecnología del Agua.
Kite, G. W. (1977). Chapter 12. Comparison of frequency distributions. In: Frequency and risk analyses in hydrology (pp. 156-168). Fort Collins, USA: Water Resources Publications.
Kjeldsen, T. R., Ahn, H., & Prosdocimi, L. (2017). On the use de a four–parameter kappa distribution in regional frequency analysis. Hydrological Sciences Journal, 62(9), 1354-1363. DOI: 10.1080/02626667.2017.1335400
Meylan, P., Favre, A. C., & Musy, A. (2012). Chapter 3. Selecting and checking data series and Theme 9.2. Multivariate Frequency Analysis using Copulas. In: Predictive hydrology. A frequency analysis approach (pp. 29-70, 164-176). Boca Raton, USA: CRC Press.
Nelsen, R. B. (2006). Chapter 4. Archimedean Copulas. In: An introduction to Copulas (2nd ed.) (pp. 109-155). New York, USA: Springer Series in Statistics.
Nieves, A., & Domínguez, F. C. (1998). Secciones 6.2 y 6.3. Cuadratura de Gauss e integrales múltiples. Métodos numéricos. Aplicados a la ingeniería (pp. 416-434). México, DF, México: Compañía Editorial Continental.
Poulin, A., Huard, D., Favre, A. C., & Pugin, S. (2007). Importance of tail dependence in bivariate frequency analysis. Journal of Hydrologic Engineering, 12(4), 394-403. DOI: 10.1061/(ASCE)1084–0699(2007)12:4(394)
Rao, A. R., & Hamed, K. H. (2000). Theme 1.8. Tests on hydrologic data and Theme 8.3: The Generalized Pareto distribution. Flood frequency analysis (pp. 12-21, 271-290). Boca Raton, USA: CRC Press.
Requena, A. I., Mediero, L., & Garrote, L. (2013). A bivariate return period based on copulas for hydrologic dam design: Accounting for reservoir routing in risk estimation. Hydrology and Earth System Sciences, 17(8), 3023-3038. DOI: 10.5194/hess-17-3023-2013
Salvadori, G., & De Michele, C. (2004). Frequency analysis via copulas: Theoretical aspects and applications to hydrological events. Water Resources Research, 40(W12511), 1-17. DOI: 10.1029/2004WR003133
Salvadori, G., & De Michele, C. (2007). On the use of Copulas in Hydrology: Theory and Practice. Journal of Hydrologic Engineering, 12(4), 369-380. DOI: 10.1061/(ASCE)1084–0699(2007)12:4(369)
Salvadori, G., De Michele, C., Kottegoda, N. T., & Rosso, R. (2007). Chapter 3. Bivariate analysis via Copulas; Appendix B: Dependence, and Appendix C: Families of Copulas. Extremes in nature. An approach using Copulas (pp. 131-175, 219-232, 233-269). Dordrecht, The Netherlands: Springer.
Salvadori, G., De Michele, C., & Durante, F. (2011). On the return period and design in a multivariate framework. Hydrology and Earth System Sciences, 15(11), 3293-3305. DOI: 10.5194/hess–15–3293–2011
Stedinger, J. R. (2017). Flood frequency analysis. In: Singh, V. P. (ed.). Handbook of applied hydrology (2nd ed.) (pp. 76.1-76.8). New York, USA: McGraw-Hill Education.
Stegun, I. A. (1972). Chapter 27. Miscellaneous functions. In: Abramowitz, M., & Stegun, I. A. (eds.). Handbook of mathematical functions (9th print.) (pp. 997-1010). New York, USA: Dover Publications.
WRC, Water Resources Council. (1977). Guidelines for determining flood flow frequency (revised edition). Bulletin #17A of the Hydrology Committee. Washington, DC, USA: Water Resources Council.
Willmott, C. J., & Matsuura, K. (2005). Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance. Climate Research, 30(1), 79-82. DOI: 10.3354/cr030079
Xu, C., Yin, J., Guo, S., Liu, Z., & Hong, X. (2016). Deriving design flood hydrograph based on conditional distribution: A case study of Danjiangkou reservoir in Hanjiang basin. Mathematical Problems in Engineering, 2016(4319646), 1-16. DOI: 10.1155/2016/4319646
Yue, S., Ouarda, T. B. M. J., Bobée, B., Legendre, P., & Bruneau, P. (1999). The Gumbel mixed model for flood frequency analysis. Journal of Hydrology, 226(1-2), 88-100. DOI: 10.1016/S0022–1694(99)00168–7
Yue, S. (2000). Joint probability distribution of annual maximum storm peaks and amounts as represented by daily rainfalls. Hydrological Sciences Journal, 45(2), 315-326. DOI: 10.1080/02626660009492327
Yue, S., & Rasmussen, P. (2002). Bivariate frequency analysis: Discussion of some useful concepts in hydrological application. Hydrological Processes, 16(14), 2881-2898. DOI: 10.1002/hyp.1185
Zhang, L., & Singh, V. P. (2006). Bivariate flood frequency analysis using the Copula method. Journal of Hydrologic Engineering, 11(2), 150-164. DOI: 10.1061/(ASCE)1084–0699(2006)11:2(150)
Zhang, L., & Singh, V. P. (2007). Trivariate flood frequency analysis using the Gumbel-Hougaard Copula. Journal of Hydrologic Engineering, 12(4), 431-439. DOI: 10.1061/(ASCE)1084–0699(2007)12:4(431)
Zhang, L., & Singh, V. P. (2019). Chapter 3. Copulas and their properties and Chapter 4. Symmetric Archimedean Copulas. Copulas and their applications in water resources engineering. pp. (pp. 62-122, 123-171). Cambridge, United Kingdom: Cambridge University Press.
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