Abstract
In nonstationary hydrological frequency analysis, the nonstationarity of hydrological series is often characterized by expressing the distribution parameters as functions of some covariates. The distribution parameters with the stochastic covariates such as climatic indices could result in time variations in forms of wide random fluctuations, which would fail to give a clear-cut description and explanation for the nonstationarity. In this paper, a hierarchical model is proposed to derive the nonstationary hydrological distribution with smooth changes by characterizing stochastic covariates as random processes. Under this model, the conditional distribution of the interested hydrological variable given its covariates is first constructed, and then the distribution of the interested hydrological variable is derived by combining its conditional distribution with the probability distribution of the stochastic covariates. The results of a simulation study indicate satisfactory performance of the proposed hierarchical model in fitting the generated series. The annual runoff series of the Weihe River is employed to perform a case study. The results reveal that the hierarchical model is able to give an intuitive description of the nonstationarity of the annual runoff. On the basis of the smoothly changing hydrological distribution derived by the hierarchical model, the nonstationarity of annual runoff series can be clearly described and explicitly associated with the changes of the covariates. It is found that the decline in the mean of annual runoff of the Weihe River is attributed to precipitation decline, climate warming, and agricultural irrigation. The hierarchical model with stochastic covariates is more applicable in engineering practice than the theoretical distribution with time-varying parameters.
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Data Availability Statement
The discharge data used in this study are available from the corresponding author upon reasonable request, and the meteorological data are available from the National Meteorological Information Center, China (http://data.cma.cn/).
Acknowledgments
This research was jointly financially supported by the National Natural Science Foundation of China (NSFC Grants U2240201, 52109002, and 41890822) and the Open Research Fund of Jiangxi Provincial Key Laboratory of Water Resources and Environment of Poyang Lake, Jiangxi Provincial Institute of Water Sciences (2020GPSYS06), all of which are greatly appreciated.
References
Aas, K., C. Czado, A. Frigessi, and H. Bakken. 2009. “Pair-copula constructions of multiple dependence.” Insur. Math. Econ. 44 (2): 182–198. https://doi.org/10.1016/j.insmatheco.2007.02.001.
Akaike, H. 1974. “A new look at the statistical model identification.” IEEE Trans. Automat. Contr. 19 (6): 716–723. https://doi.org/10.1109/TAC.1974.1100705.
Bevacqua, E., D. Maraun, I. Hobæk Haff, M. Widmann, and M. Vrac. 2017. “Multivariate statistical modeling of compound events via pair-copula constructions: Analysis of floods in Ravenna (Italy).” Hydrol. Earth Syst. Sci. 21 (6): 2701–2723. https://doi.org/10.5194/hess-21-2701-2017.
Blöschl, G., et al. 2017. “Changing climate shifts timing of European floods.” Science 357 (6351): 588–590. https://doi.org/10.1126/science.aan2506.
Bracken, C., K. D. Holman, B. Rajagopalan, and H. Moradkhani. 2018. “A Bayesian hierarchical approach to multivariate nonstationary hydrologic frequency analysis.” Water Resour. Res. 54 (1): 243–255. https://doi.org/10.1002/2017WR020403.
Breymann, W., A. Dias, and P. Embrechts. 2003. “Dependence structures for multivariate high frequency data in finance.” Quant. Finance 3 (1): 1–14. https://doi.org/10.1080/713666155.
Casella, G., and R. L. Berger. 2002. Statistical inference. London: Thomson Learning.
Chang, J., Y. Wang, E. Istanbulluoglu, T. Bai, Q. Huang, D. Yang, and S. Huang. 2015. “Impact of climate change and human activities on runoff in the Weihe River Basin, China.” Quat. Int. 380 (Aug): 169–179. https://doi.org/10.1016/j.quaint.2014.03.048.
Debele, S. E., W. G. Strupczewski, and E. Bogdanowicz. 2017. “A comparison of three approaches to non-stationary flood frequency analysis.” Acta Geophys. 65 (4): 863–883. https://doi.org/10.1007/s11600-017-0071-4.
Dingman, S. L. 2008. Physical hydrology. Englewood Cliff, NJ: Prentice Hall.
Du, T., L. Xiong, C.-Y. Xu, C. J. Gippel, S. Guo, and P. Liu. 2015. “Return period and risk analysis of nonstationary low-flow series under climate change.” J. Hydrol. 527 (Aug): 234–250. https://doi.org/10.1016/j.jhydrol.2015.04.041.
Frank, J., and J. R. Massey. 1951. “The Kolmogorov-Smirnov test for goodness of fit.” J. Am. Stat. Assoc. 46 (253): 68–78. https://doi.org/10.1080/01621459.1951.10500769.
Hu, Y., Z. Liang, V. P. Singh, X. Zhang, J. Wang, B. Li, and H. Wang. 2018. “Concept of equivalent reliability for estimating the design flood under non-stationary conditions.” Water Resour. Manage. 32 (3): 997–1011. https://doi.org/10.1007/s11269-017-1851-y.
Jiang, C., L. Xiong, S. Guo, J. Xia, and C.-Y. Xu. 2017. “A process-based insight into nonstationarity of the probability distribution of annual runoff.” Water Resour. Res. 53 (5): 4214–4235. https://doi.org/10.1002/2016WR019863.
Jiang, C., L. Xiong, C.-Y. Xu, and S. Guo. 2015. “Bivariate frequency analysis of nonstationary low-flow series based on the time-varying copula.” Hydrol. Process. 29 (6): 1521–1534. https://doi.org/10.1002/hyp.10288.
Khaliq, M. N., T. B. M. J. Ouarda, J. C. Ondo, P. Gachon, and B. Bobée. 2006. “Frequency analysis of a sequence of dependent and/or nonstationary hydrometeorological observations: A review.” J. Hydrol. 329 (3–4): 534–552. https://doi.org/10.1016/j.jhydrol.2006.03.004.
Koutsoyiannis, D. 2006. “Nonstationarity versus scaling in hydrology.” J. Hydrol. 324 (Feb): 239–254. https://doi.org/10.1016/j.jhydrol.2005.09.022.
Koutsoyiannis, D., and A. Montanari. 2015. “Negligent killing of scientific concepts: The stationarity case.” Hydrol. Sci. J. 60 (7–8): 1174–1183. https://doi.org/10.1080/02626667.2014.959959.
Kumar, D. N., and R. Maity. 2008. “Bayesian dynamic modelling for nonstationary hydroclimatic time series forecasting along with uncertainty quantification.” Hydrol. Process. 22 (17): 3488–3499. https://doi.org/10.1002/hyp.6951.
Kwon, H.-H., C. Brown, and U. Lall. 2008. “Climate informed flood frequency analysis and prediction in Montana using hierarchical Bayesian modeling.” Geophys. Res. Lett. 35 (5): L05404. https://doi.org/10.1029/2007GL032220.
Li, S., Y. Qin, X. Song, S. Bai, and Y. Liu. 2021. “Nonstationary frequency analysis of the Weihe River annual runoff series using de-nonstationarity method.” J. Hydrol. Eng. 26 (11): 04021034. https://doi.org/10.1061/(ASCE)HE.1943-5584.0002125.
Lima, C. H. R., and U. Lall. 2009. “Hierarchical Bayesian modeling of multisite daily rainfall occurrence: Rainy season onset, peak, and end.” Water Resour. Res. 45 (7): W07422. https://doi.org/10.1029/2008WR007485.
Lima, C. H. R., and U. Lall. 2010. “Spatial scaling in a changing climate: A hierarchical Bayesian model for non-stationary multi-site annual maximum and monthly streamflow.” J. Hydrol. 383 (3–4): 307–318. https://doi.org/10.1016/j.jhydrol.2009.12.045.
Milly, P. C. D., J. Betancourt, M. Falkenmark, R. M. Hirsch, Z. W. Kundzewicz, D. P. Lettenmaier, and R. J. Stouffer. 2008. “Stationarity is dead: Whither water management?” Science 319 (5863): 573–574. https://doi.org/10.1126/science.1151915.
Montanari, A., et al. 2013. “‘Panta Rhei–Everything flows’, Change in hydrology and society–The IAHS scientific decade 2013–2022.” Hydrol. Sci. J. 58 (6): 1256–1275. https://doi.org/10.1080/02626667.2013.809088.
Montanari, A., and D. Koutsoyiannis. 2014. “Modeling and mitigating natural hazards: Stationarity is immortal!” Water Resour. Res. 50 (12): 9748–9756. https://doi.org/10.1002/2014WR016092.
Najafi, M., and H. Moradkhani. 2013. “Analysis of runoff extremes using spatial hierarchical Bayesian modeling.” Water Resour. Res. 49 (10): 6656–6670. https://doi.org/10.1002/wrcr.20381.
Nelsen, R. B. 2006. An introduction to copulas. New York: Springer.
Niederreiter, H. 1978. “Quasi-Monte Carlo methods and pseudo-random numbers.” Bull. Am. Math. Soc. 84 (6): 957–1041. https://doi.org/10.1090/S0002-9904-1978-14532-7.
Nogaj, M., S. Parey, and D. Dacunha-Castelle. 2007. “Non-stationary extreme models and a climatic application.” Nonlinear Process Geophys. 14 (3): 305–316. https://doi.org/10.5194/npg-14-305-2007.
Obeysekera, J., and J. Salas. 2014. “Quantifying the uncertainty of design floods under nonstationary conditions.” J. Hydrol. Eng. 19 (7): 1438–1446. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000931.
Obeysekera, J., and J. Salas. 2016. “Frequency of recurrent extremes under nonstationarity.” J. Hydrol. Eng. 21 (5): 04016005. https://doi.org/10.1061/(ASCE)HE.1943-5584.0001339.
Read, L. K., and R. M. Vogel. 2015. “Reliability, return periods, and risk under nonstationarity.” Water Resour. Res. 51 (8): 6381–6398. https://doi.org/10.1002/2015WR017089.
Read, L. K., and R. M. Vogel. 2016. “Hazard function analysis for flood planning under nonstationarity.” Water Resour. Res. 52 (5): 4116–4131. https://doi.org/10.1002/2015WR018370.
Renard, B. 2011. “A Bayesian hierarchical approach to regional frequency analysis.” Water Resour. Res. 47 (11): W11513. https://doi.org/10.1029/2010WR010089.
Salvadori, G., and C. De Michele. 2010. “Multivariate multiparameter extreme value models and return periods: A copula approach.” Water Resour. Res. 46 (10): W10501. https://doi.org/10.1029/2009WR009040.
Sankarasubramanian, A., and U. Lall. 2003. “Flood quantiles in a changing climate: Seasonal forecasts and causal relations.” Water Resour. Res. 39 (5): 1134. https://doi.org/10.1029/2002WR001593.
Santos, V., T. Wahl, R. Jane, S. Misra, and K. White. 2021. “Assessing compound flooding potential with multivariate statistical models in a complex estuarine system under data constraints.” J. Flood Risk Manage. 14 (4): e12749. https://doi.org/10.1111/jfr3.12749.
Schwarz, G. 1978. “Estimating the dimension of a model.” Ann. Statist. 6 (2): 461–464. https://doi.org/10.1214/aos/1176344136.
Steinschneider, S., and U. Lall. 2015. “A hierarchical Bayesian regional model for nonstationary precipitation extremes in Northern California conditioned on tropical moisture exports.” Water Resour. Res. 51 (3): 1472–1492. https://doi.org/10.1002/2014WR016664.
Strupczewski, W. G., V. P. Singh, and W. Feluch. 2001. “Non-stationary approach to at-site flood frequency modeling I. Maximum likelihood estimation.” J. Hydrol. 248 (1–4): 123–142. https://doi.org/10.1016/S0022-1694(01)00397-3.
Sun, X. 2005. “The development of the well-irrigation in Guanzhong region.” [In Chinese.] Ground Water 27 (1): 10–18. https://doi.org/10.3969/j.issn.1004-1184.2005.01.004.
Sun, X., U. Lall, B. Merz, and N. V. Dung. 2015. “Bayesian clustering for nonstationary flood frequency analysis: Application to trends of annual maximum flow in Germany.” Water Resour. Res. 51 (8): 6586–6601. https://doi.org/10.1002/2015WR017117.
Villarini, G., F. Serinaldi, J. A. Smith, and W. F. Krajewski. 2009a. “On the stationarity of annual flood peaks in the Continental United States during the 20th Century.” Water Resour. Res. 45 (8): W08417. https://doi.org/10.1029/2008WR007645.
Villarini, G., J. A. Smith, and F. Napolitano. 2010. “Nonstationary modeling of a long record of rainfall and temperature over Rome.” Adv. Water Resour. 33 (10): 1256–1267. https://doi.org/10.1016/j.advwatres.2010.03.013.
Villarini, G., J. A. Smith, F. Serinaldi, J. Bales, P. D. Bates, and W. F. Krajewski. 2009b. “Flood frequency analysis for nonstationary annual peak records in an urban drainage basin.” Adv. Water Resour. 32 (8): 1255–1266. https://doi.org/10.1016/j.advwatres.2009.05.003.
Vogel, R. M., C. Yaindl, and M. Walter. 2011. “Nonstationarity: Flood magnification and recurrence reduction factors in the United States.” J. Am. Water Resour. Assoc. 47 (3): 464–474. https://doi.org/10.1111/j.1752-1688.2011.00541.x.
Xiong, L., C. Jiang, and T. Du. 2014. “Statistical attribution analysis of the nonstationarity of the annual runoff series of the Weihe River.” Water Sci. Technol. 70 (5): 939–946. https://doi.org/10.2166/wst.2014.322.
Xiong, L., C. Jiang, C.-Y. Xu, K.-X. Yu, and S. Guo. 2015. “A framework of change-point detection for multivariate hydrological series.” Water Resour. Res. 51 (10): 8198–8217. https://doi.org/10.1002/2015WR017677.
Yan, L., L. Xiong, S. Guo, C.-Y. Xu, J. Xia, and T. Du. 2017. “Comparison of four nonstationary hydrologic design methods for changing environment.” J. Hydrol. 551 (Aug): 132–150. https://doi.org/10.1016/j.jhydrol.2017.06.001.
Zhang, W., G. Villarini, G. A. Vecchi, and J. A. Smith. 2018. “Urbanization exacerbated the rainfall and flooding caused by hurricane Harvey in Houston.” Nature 563 (7731): 384–388. https://doi.org/10.1038/s41586-018-0676-z.
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Journal of Hydrologic Engineering
Volume 28 • Issue 4 • April 2023
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© 2023 American Society of Civil Engineers.
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Received: Apr 24, 2022
Accepted: Nov 29, 2022
Published online: Feb 13, 2023
Published in print: Apr 1, 2023
Discussion open until: Jul 13, 2023
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