Part Two - Applications
Published online by Cambridge University Press: 03 January 2019
Summary
In this chapter, we will illustrate the application of copulas in rainfall frequency analysis. This chapter is divided into two parts: (1) rainfall depth-duration frequency (DDF) analysis; and (2) multivariate rainfall frequency (i.e., four-dimensional) analysis. The rainfall data from the watersheds in the United States are collected and applied for analyses. The Archimedean, meta-elliptical, and vine copulas are applied to model the dependence among rainfall variables. Application shows that the DDF may be modeled by the Gumbel–Hougaard copula. Both vine and meta-elliptical copulas may be applied to model the spatial dependence of rainfall variables. Compared to the vine copula, modeling is easier to do when applying the meta-elliptical copula.
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- Copulas and their Applications in Water Resources Engineering , pp. 365 - 599Publisher: Cambridge University PressPrint publication year: 2019
References
References
Abdul Rauf, U. F. A. and Zeephongsekul, P. (2014). Copula based analysis of rainfall severity and duration: a case study. Theoretical and Applied Climatology, 115(1–2), 153–166.CrossRefGoogle Scholar
Bacchi, B., Becciu, G., and Kottegoda, N. T. (1994). Bivariate exponential model applied to intensities and durations of extreme rainfall. Journal of Hydrology, 155, 225–236.CrossRefGoogle Scholar
Cantet, P. and Arnaud, P. (2014). Extreme rainfall analysis by a stochastic model: impact of the copula choice on the sub-daily rainfall generation. Stochastic Environmental Research and Risk Assessment, 28, 1479–1492.CrossRefGoogle Scholar
Cong, R.-G. and Brady, M. (2012). The interdependence between rainfall and temperature: copula analysis. Scientific World Journal, 405675, doi:10.1100/2012/405675.Google Scholar
Cordova, J. R. and Rodriguez-Iturbe, I. (1985). On probabilistic structure of storm surface runoff. Water Resources Research, 21(5), 755–763.Google Scholar
Favre, A.-E., El Adlouni, S., Perreault, L, Thiemonge, N., and Bobee, B. (2004). Multivariate hydrological frequency analysis using copulas. Water Resources Research, 40, W01101.CrossRefGoogle Scholar
Goel, N. K., Kurothe, R. S., Mathur, B. S., and Vogel, R.M. (2000). A derived flood frequency distribution for correlated rainfall intensity and duration. Journal of Hydrology, 228, 56–67.CrossRefGoogle Scholar
Grimaldi, S., Serinaldi, F., Napolitano, F., and Ubertine, L. (2005). A 3-copula function application for design hyetograph analysis. IAHS Publication, 293, 1–9.Google Scholar
Hao, Z. and Singh, V. P. (2013). Entropy-based method for extreme rainfall analysis in Texas. Journal of Geophysical Research, 118, 263–273, doi:10.1029/2011JD017394.CrossRefGoogle Scholar
Hashino, M. (1985). Formulation of the joint return period of two hydrologic variates associated with a poisson process. Journal of Hydroscience and Hydraulic Engineering, 3(2), 73–84.Google Scholar
Joe, H. (1997). Multivariate Models and Multivariate Dependence Concepts. Chapman & Hall/CRC, New York.Google Scholar
Kao, S.-C. and Govindaraju, R. S. (2007). A bivariate frequency analysis of extreme rainfall with implications for design. Water Resources Research, 112, D13119.Google Scholar
Kao, S.-C. and Govindaraju, R. S. (2008). Tivariate statistical analysis of extreme rainfall events via Plackett family of copulas. Water Resources Research, 44, W02415.Google Scholar
Khedun, C. P., Mishra, A. K., Singh, V. P., and Giardino, J. R. (2014). A copula-based precipitation model: investigating the interdecadal modulation of ENSO’s impacts on monthly precipitation. Water Resources Research, 50, 1–20, doi:10.1002/2013WR013763.CrossRefGoogle Scholar
Kurothe, R. S., Goel, N. K., and Mathur, B. S. (1997). Derived flood frequency distribution of negatively correlated rainfall intensity and duration. Water Resources Research, 33(9), 2103–2107.Google Scholar
Moazami, S., Golian, S., Kavianpour, M. R., and Hong, Y. (2014). Uncertainty analysis of bias from satellite rainfall estimates using copula method. Atmospheric Research, 137, 145–166CrossRefGoogle Scholar
Singh, K. and Singh, V. P. (1991). Derivation of bivariate probability density functions with exponential marginals. Stochastic Hydrology and Hydraulics, 6(1), 47–54.Google Scholar
Vernieuwe, H., Vandenberghe, S., De Bates, B., and Verhoest, N. E. C. (2015). A continuous rainfall model based on vine copulas. Hydrology and Earth System Sciences, 19, 2685–2699. doi:10.5194/hess-19-2685-2015.Google Scholar
Yue, S. (2000a). Joint probability distribution of annual maximum storm peaks and amounts as represented by daily rainfalls. Hydroscience Journal, 45(2), 315–326.Google Scholar
Yue, S. (2000b). The Gumbel logistic model for representing a multivariate storm event. Advances in Water Resources, 24(2), 179–185.CrossRefGoogle Scholar
Yue, S. (2000c). The Gumbel mixed model applied to storm frequency analysis. Water Resources Management. 14(5), 377–389.Google Scholar
Zhang, L. and Singh, V. P. (2007a). Bivariate rainfall frequency analysis using Archimedean copulas. Journal of Hydrology, 332, 93–109.Google Scholar
Zhang, L. and Singh, V. P. (2007b). IDF curves using Frank Archimedean copula. Journal of Hydrologic Engineering, 12(6), 651–662.Google Scholar
Zhang, L. and Singh, V. P. (2007c). Gumbel–Houggard copula for trivariate rainfall frequency analysis. Journal of Hydrologic Engineering 12(4), 409–419.Google Scholar
Zhang, Q., Li, J., and Singh, V. P. (2012). Application of Archimedean copulas in the analysis of the precipitation extremes: effects of precipitation change. Theoretical and Applied Climatology, 107(1–2), 255–264.Google Scholar
Zhang, Q., Li, J., Singh, V. P., and Xu, C.-Y. (2013). Copula-based spatio-temporal patterns of precipitation extremes in China. International Journal of Climatology, 33(5), 1140–1152.CrossRefGoogle Scholar
References
Abberger, K. (2005). A simple graphical method to explore tail-dependence in stock return pairs. Applied Financial Economics, 15(1), 43–51.Google Scholar
Bezak, N., Mikos, M., and Sraj, M. (2014). Trivariate frequency analysis of peak discharge, hydrograph volume and suspended sediment concentration data using copulas. Water Resources Management, 28, 2195–2212. doi:10.1007/s11269-014-0606-2.CrossRefGoogle Scholar
Capéraà, P., Fougeres, A.-L., and Genest, C. (1997). A nonparametric estimation procedure for bivariate exteme value copulas. Biometrika, 84, 567–577.Google Scholar
Chen, L., Singh, V. P., and Guo, S. (2013). Measure of correlation between river flows using the copula-entropy theory. Journal of Hydrologic Engineering, 18(12), 1591–1608.Google Scholar
Chen, L., Singh, V. P., Guo, S., Hao, Z., and Li, T. (2012). Flood coincidence risk analysis using multivariate copula functions. Journal of Hydrologic Engineering, 17(6), 742–755.Google Scholar
Chowdhary, H., Escobar, L. A., and Singh, V. P. (2011). Identification of suitable copulas for bivariate frequency analysis of flood peak and flood volume data. Hydrology Research, 42(2–3), 193–216.CrossRefGoogle Scholar
Coles, S. G., Heffernan, J. E., and Tawn, J. A. (1999). Dependence measures for extreme value analyses. Extremes, 2, 339–365.Google Scholar
Durocher, M., Chebana, F., and Ouarda, T. B. M. J. (2016). On the prediction of extreme flood quantiles at ungauged locations with spatial copula. Journal of Hydrology, 533, 523–532. doi:10.1016/j.jhydrol.2015.12.029.CrossRefGoogle Scholar
Frahm, G., Junker, M., and Schmidt, R. (2005). Estimating the tail dependence coefficient: properties and pitfalls, Insurance Mathematics & Economics, 37, 80–100.Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, New York.Google Scholar
Poulin, A., Huard, D., Favre, A.-C., and 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).CrossRefGoogle Scholar
Requena, A. I., Chebana, F., and Mediero, L. (2016). A complete procedure for multivariate index-flood model application. Journal of Hydrology, 535, 559–580. doi:10.1016/j.jhydrol.2016.02.004.Google Scholar
Schmidt, R. and Stadtmuller, U. (2006). Non-parametric estimation of tail dependence. Scandinavian Journal of Statistics: Theory and Applications, 33(2), 307–335.Google Scholar
Serinaldi, F. (2015). Dismissing return periods. Stochastic Environmental Research and Risk Assessment, 29, 1179–1189, doi:10.1007/s00477–014–0916–1.Google Scholar
Sraj, M., Bezak, N., and Brilly, M. (2015). Bivariate flood frequency analysis using the copula function: a case study of the Litija station on the Sava River. Hydrological Processes, 29, 225–238. doi:10.1002/hyp.10145.CrossRefGoogle Scholar
Yue, S., Ouarda, T. B. M. J., Bobee, B., Legendre, P., and Bruneau, P. (1999). The Gumbel mixed model for flood frequency analysis. Journal of Hydrology, 226, 88–100.CrossRefGoogle Scholar
References
Box, G. E. P., Jenkis, G. M., and Reinsel, G. C. (2007) Time Series Analysis: Forecasting and Control, 5th edition. John Wiley and Sons, Inc., Hoboken.Google Scholar
Kotz, S., and Nadarajah, S. (2004) Multivariate t Distributions and Their Applications. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Additional Reading
References
Aas, K., Czado, C., Frigessi, A., and Bakken, H. (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, 44, 182–198, doi:10.1016/j.insmatheco.2007.02.001.Google Scholar
AghaKouchak, A. (2015). A multivariate approach for persistence-based drought prediction: application to the 2010–2011 East Africa drought. Journal of Hydrology, 526, 127–135. doi:10.1016/j.jhydrol.2014.09.063.CrossRefGoogle Scholar
Chen, L., Singh, V. P., Guo, S., Mishra, A. K., and Guo, J. (2013) Drought analysis using copulas. Journal of Hydrologic Engineering, 18(7), 797–808. doi:10.1061/(ASCE)HE.1943-5584.0000697.CrossRefGoogle Scholar
Chen, Y. D., Zhang, Q., Xiao, M., and Singh, V. P. (2013). Evaluation of risk of hydrological droughts by the trivariate Plackett copula in the East River basin (China). Natural Hazards, 68, 529–547.Google Scholar
De Michele, C., Salvadori, G., Vezzoli, R., and Pecora, S. (2013). Multivariate assessment of droughts: frequency analysis and dynamic return period. Water Resources Research, 49, 6985–6994. doi:10.1002/wrcr.20551.Google Scholar
Genest, C., Rémillard, B., and Beaudoin, D. (2007). Goodness-of-fit tests for copulas: a review and a power study. Insurance: Mathematics and Economics. doi:10.1016/j.insmatheco.2007.10.1005.Google Scholar
Hao, Z. and AghaKouchak, A. (2014). A nonparametric multivariate multi-index drought monitoring framework. Journal of Hydrometeorology, 15, 89–101. doi:10.1175/JHM-D-12-0160.1.Google Scholar
Hao, Z., Hao, F., Singh, V. P., Sun, A. Y. and Xia, Y. (2016). Probabilistic prediction of hydrologic drought using a conditional probability approach based on the meta-Gaussian model. Journal of Hydrology, 542, 772–780. doi:10.1016/j.jhydrol.2016.09.048.Google Scholar
Kao, S-C. and Govindaraju, R. S. (2010). A copula-based joint deficit index for droughts. Journal of Hydrology, 380, 121–134. doi:10.1016/j.jhydrol.2009.10.029.Google Scholar
Janga Reddy, M. and Singh, V. P. (2014). Multivariate modeling of droughts using copulas and meta-heuristic methods. Stochastic Environmental Research an Risk Assessment, 28, 475–489.CrossRefGoogle Scholar
Kwak, J., Kim, S., Kim, G., Singh, V. P., Park, J., and Kim, H. S. (2016). Bivariate drought analysis using tree ring streamflow reconstruction in the Sacramento Basin, California, USA: a case study. Water, 8(122), 1–16. doi:10.3390/w8040122.Google Scholar
Madadgar, S. and Moradkhani, H. (2013). Drought analysis under climate change using copula. Journal of Hydrologic Engineering, 18(7), 746–759. doi:10.1061/(ASCE)HE.1943-5584.0000532.Google Scholar
McKee, T. B., Doesken, N. J., and Kleist, J. (1993). The relationship of drought frequency and duration to time scales. 8th Conference on Applied Climatology, American Meteorological Society, Anaheim. www.droughtmanagement.info/literature/AMS_Relationship_Drought_Frequency_Duration_Time_Scales_1993.pdf.Google Scholar
Mishra, A. K. and Singh, V. P. (2010). A review of drought concepts. Journal of Hydrology, 391, 202–216. doi:10.1016/j.jhydrol.2010.07.012.Google Scholar
Palmer, W. C. (1965). Meteorologic drought. US Department of Commerce, Weather Bureau, Research paper No. 45.Google Scholar
Palmer, W.C. (1968). Keeping track of crop moisture conditions, nationwide: the new crop moisture index. Weatherwise, 21, 156–161.Google Scholar
Rao, A. R. and Padamanabhan, G. (1984). Analysis and modeling of Palmer’s drought index series. Journal of Hydrology, 68, 211–229.Google Scholar
Salvadori, G. and De Michele, C. (2015). Multivariate real-time assessment of droughts via copula-based multi-site hazard trajectories and fans. Journal of Hydrology, 526, 101–115. doi:10.1016/j.jhydrol.2014.11.056.CrossRefGoogle Scholar
Salvadori, G., Durante, F., and De Michele, C. (2013). Multivariate return period calculation via survival functions. Water Resources research, 49, 2308–2311. doi:10.1002/wrcr.20204.Google Scholar
Santos, M. A. (1983). Regional droughts: a stochastic characterization. Journal of Hydrology, 66, 183–211.Google Scholar
Shukla, S. and Wood, A. W. (2008). Use of a standardized runoff index for characterizing hydrologic drought. Geophysical Research Letters, 35, L02405. doi:10.1029/2007GL032487.Google Scholar
Song, S. and Singh, V. P. (2010a). Frequency analysis of droughts using the Plackett copula and parameter estimation by genetic algorithm. Stochastic Environmental Research and Risk Assessment, 24(5), 783–805. doi:10.1007/s00477-010-0364-5.Google Scholar
Song, S. and Singh, V. P. (2010b). Meta-elliptical copulas for drought frequency analysis of periodic hydrologic data. Stochastic Environmental Research and Risk Assessment, 24(3), 425–444.Google Scholar
Texas State Historical Association. (n.d.). Nueces River, Handbook of Texas, www.tshaonline.org/handbook/online/articles/rnn15.Google Scholar
Tu, X, Singh, V. P., Chen, X., Ma, M., Zhang, Q., and Zhao, Y. (2016) Uncertainty and variability in bivariate modeling of hydrological droughts. Stochastic Environmental Research and Risk Assessment, 30, 1317–1334.CrossRefGoogle Scholar
Van Rooy, M. P. (1965). A rainfall anomaly index independent of time and space. Notos, 14, 43–48.Google Scholar
Voss, R., May, W., and Roeckner, E. (2002). Enhanced resoluation modeling study on anthropogenic climate change: changes in extremes of the hydrological cycle. International Journal of Climatology, 22, 755–777.Google Scholar
Xu, K., Yang, D., Xu, X., and Lei, H. (2015). Copula based drought frequency analysis considering the spatio-temporal variability in Southwest China. Journal of Hydrology, 527, 630–640. doi:10.1016/j.jhydrol.2015.05.030.CrossRefGoogle Scholar
Yevjevich, V. (1967). An objective approach to definitions and investigations of continental hydrologic droughts. Hydrology Papers, Colorado State University, Fort Collins.Google Scholar
Yoo, J. Y., Shin, J. Y., Kim, D. K., and Kim, T.-W. (2013). Drought risk analysis using stochastic rainfall generation model and copula functions. Journal of Korea Water Resource Association, 46(4), 425–437.Google Scholar
Zelenhasic, E. and Salvai, A. (1987). A method of streamflow drought analysis. Water Resources Research, 23(1), 156–168.Google Scholar
Zhang, Q., Xiao, M., and Singh, V. P. (2015) Uncertainty evaluation of copula analysis of hydrological droughts in the East River Basin, China. Global and Planetary Change, 129, 1–9.Google Scholar
References
Hao, Z., AghaKouchak, A., and Phillips, T. J. (2013). Changes in concurrent monthly precipitation and temperature extremes. Environmental Research Letters, 8, 034014.Google Scholar
Miao, C., Sun, Q., Duan, Q., and Wang, Y. (2016). Joint analysis of changes in temperature and precipitation on the Loess Plateau during the period 1961–2011. Climate Dynamics. doi:10.1007/s00382–016–3022-x.Google Scholar
Pettitt, A. N. (1979). A non-parametric approach to the change-point problem. Journal of Applied Statistics, 18, 126–135.Google Scholar
Sedlmeier, K., Mieruch, S., Shädler, G., and Kottmeier, C. (2016). Compound extremes in a changing climate – a Markov chain approach. Nonlinear Processes in Geophysics, 23, 375–390.Google Scholar
Svensson, C. and Jones, D. A. (2002). Dependence between extreme sea surge, river flow and precipitation in eastern Britain. International Journal of Climatology, 22, 1149–1168.Google Scholar
Wahl, T., Jain, S., Bender, J., Meyers, S. D., and Luther, M. E. (2015) Increasing risk of compound flooding from storm surge and rainfall or major US cities. Nature Climate Change, 5, 1093–1097. doi:10.1038/NCLIMATE2736.Google Scholar
References
Al-Zahrani, M. and Husain, T. (1998). An algorithm for designing a precipitation network in south-western region of Saudi Arabia. Journal of Hydrology, 205, 205–216.CrossRefGoogle Scholar
Alfonso, L., Lobbrecht, A., and Rice, R. (2010). Information theory-based approach for location of monitoring water level gauges in polders. Water Resources Research, 46, W03528, doi:10.1029/2009WR008101.Google Scholar
Beirlant, J., Dudewicz, E. J., Gyorfi, L., and Van deMeulen, E. C. (2001). Nonparametric entropy estimation: an overview. http://jimbeck.caltech.edu/summerlectures/references/Entropy%20estimation.pdf.Google Scholar
Chow, C. K. and Liu, C. N. (1968). Approximating discrete probability distributions with dependence tree. IEEE Transactions on Information Theory, IT-14(3), 462–467.Google Scholar
Dong, X., Dohmen-Janssen, C. M., and Booij, M. J. (2005). Approximate spatial sampling of rainfall for flow simulation. Hydrological Sciences Journal, 50(2), 279–298.Google Scholar
Krstanovic, R. F. and Singh, V. P. (1992a). Evaluation of rainfall networks using entropy: I. Theoretical development. Water Resources Management, 6, 279–293.Google Scholar
Krstanovic, R.F. and Singh, V.P. (1992b). Evaluation of rainfall networks using entropy: II. Application. Water Resources Management, 6, 295–314.Google Scholar
Li, C., Singh, V. P., and Mishra, A. K. (2012). Entropy theory-based criterion for hydrometric network evaluation and design: maximum information minimum redundancy. Water Resources Research, 48, W05521, doi:10.1029/2011WR011251.Google Scholar
Markus, M., Knapp, H. V., and Tasker, G. D. (2003). Entropy and generalized least square methods in assessment of the regional value of streamgages. Journal of Hydrology, 283, 107–121, doi: 10.1016/S0022–1694(03)00224–0.Google Scholar
Mishra, A. K. and Coulibaly, P. (2009). Developments in hydrometric network design: a review, Reviews of Geophysics, 47, RG2001, doi:10.1029/2007RG000243.Google Scholar
Xu, H., Xu, C.-Y., Sælthun, N. R., Xu, Y., Zhou, B., and Chen, H. (2015). Entropy theory based multi-criteria resampling of rain gauge networks for hydrological modelling: a case study of humid area in southern China, Journal of Hydrology, 525, 138–151.Google Scholar
Xu, P. C., Wang, D., Singh, V. P., et al. (2017). A two-phase copula entropy-based multiobjective optimization approach to hydrometeorological gauge network design. Journal of Hydrology, 555, 328–341.Google Scholar
Yang, Y. and Burn, D. H. (1994). An entropy approach to data collection network design, Journal of Hydrology, 157, 307–324.Google Scholar
Yeh, M. S., Lin, Y. P., and Chang, L. C. (2006) Designing an optimal multivariate geostatistical groundwater quality monitoring network using factorial kriging and genetic algorithms, Environmental Geology, 50, 101–121, doi:10.1007/s00254–006–0190–8.Google Scholar
References
Li, B. Y. and Li, J. Z. (1994). Geomorphological Maps of China (1:4,000,000). Beijing Science Press, Beijing.Google Scholar
Ni, H. B., Zhang, L. P., Wu, X. Y. and Fu, X. T. (2008). Weathering of the Pisha-Sandstones in the wind-water erosion crisscross region on the Loess Plateau. Journal of Mountain Science, 5, 340–349.Google Scholar
USDA, United States Department of Agriculture, Agricultural Research Service. www.tucson.ars.ag.gov/dap/.Google Scholar
Wang, Y. C., Wu, Y. H., Kou, Q., Min, D. A., Chang, Y. Z., and Zhang, R. J. (2007). Definition of arsenic rock zone borderline and its classification. Science of Soil and Water Conservation, 5(1): 14–18 (in Chinese).Google Scholar
References
Arya, F. K. and Zhang, L. (2004). Time series analysis of water quality parameters at Stillaguamish River using order series method. Stochastic Environmental Research and Risk Assessment. doi:10.1007/s00477–014–0907–2. Climate of Texas, https://commons.wikimedia.org/wiki/File:Texas_K%C3%B6ppen.svg.Google Scholar
DuMouchel, W. H. (1973). On the asymptotic normality of the maximum-likelihood estimate when sampling from a stable distribution. Annals of Statistics, 1(5), 948–957.Google Scholar
Genest, C., Remillard, B., and Beaudoin, C. (2007). Goodness-of-fit tests for copulas: a review and a power study. Insurance: Mathematics and Economics. doi:10.1016/j.insmatheco.2007.10.1005.Google Scholar