Abstract
The bivariate hydrological quantile estimation may inevitably induce large sampling uncertainty due to short sample size. It is crucial to quantify such uncertainty and its impacts on reservoir routing. In this study, a copula-based parametric bootstrapping uncertainty (C-PBU) method is proposed to characterize the bivariate quantile estimation uncertainty and the impact of such uncertainty on the highest reservoir water level is also investigated. The Geheyan reservoir in China is selected as a case study. Four evaluation indexes, i.e. area of confidence region, mean horizontal deviation, mean vertical deviation and average Euclidean distance, are adopted to quantify the quantile estimation uncertainty. The results indicate that the uncertainty of quantile estimation and the highest reservoir water level increases with larger return period. The 90% confidence interval (CI) of highest reservoir water level reaches 1.56 m and 2.52 m under 20-year and 50-year JRP respectively for the sample size of 100. It is also indicated that the peak over threshold (POT) sampling method contribute to uncertainty reduction comparing with the annual maximum (AM) method. This study could provide not only the point estimator of design floods and corresponding design water level, but also the rich uncertainty information (e.g. 90% confidence interval) for the references of reservoir flood risk assessment, scheduling and management.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bezak N, Mikoš M, Šraj M (2014) Trivariate frequency analyses of peak discharge, hydrograph volume and suspended sediment concentration data using copulas. Water Resour Manag 28(8):2195–2212. https://doi.org/10.1007/s11269-014-0606-2
Bhunya PK, Berndtsson R, Jain SK, Kumar R (2013) Flood analysis using negative binomial and generalized Pareto models in partial duration series (PDS). J Hydrol 497(7):121–132. https://doi.org/10.1016/j.jhydrol.2013.05.047
Chebana F, Ouarda TBMJ (2011) Multivariate quantiles in hydrological frequency analysis. Environmetrics 22(1):63–78. https://doi.org/10.1002/env.1027
Chen L, Singh VP (2017a) Generalized beta distribution of the second kind for flood frequency analysis. Entropy 19(6):1–17
Chen L, Singh VP (2017b) Entropy based derivation of generalized distributions for hydrometeorological frequency analysis. J Hydrol 557:699–712. https://doi.org/10.1016/j.jhydrol.2017.12.066
Chen L, Singh VP, Xiong F (2017) An entropy-based generalized gamma distribution for flood frequency analysis. Entropy 19(6):1–15
De Michele C, Salvadori G (2003) A generalized Pareto intensity–duration model of storm rainfall exploiting 2-copulas. J Geophys Res 108(D2):4067. https://doi.org/10.1029/2002JD002534
De Michele C, Salvadori G, Canossi M, Petaccia A, Rosso R (2005) Bivariate statistical approach to check adequacy of dam spillway. J Hydrol Eng 10(1):50–57. https://doi.org/10.1061/(ASCE)1084-0699(2005)10:1(50)
Dung NV, Merz B, Bárdossy A, Apel H (2015) Handing uncertainty in bivariate quantile estimation-an application to flood hazard analysis in the Mekong Delta. J Hydrol 527:704–717. https://doi.org/10.1016/j.jhydrol.2015.05.033
Duong T (2007) Ks: Kernel density estimation and kernel discriminant analysis for multivariate data in R.J. Stat. Software 21 (October)
Favre AC, El Adlouni S, Perreault L, Thiémonge N, Bobée B (2004) Multivariate hydrological frequency analysis using copulas. Water Resour Res 40(1). https://doi.org/10.1029/2003WR002456
Fischer S, Schumann A (2015) Robust flood statistics: comparison of peak over threshold approaches based on monthly maxima and TL-moments. Hydrol Sci J 61(3):457–470
Goel NK, Kurothe RS, Mathur BS, Vogel RM (2000) A derived flood frequency distribution for correlated rainfall intensity and duration. J Hydrol 228(1-2):56–67. https://doi.org/10.1016/S0022-1694(00)00145-1
Gräler B, van den Berg M, Vandenberghe S, Petroselli A, Grimaldi S, De Baets B, Verhoest N (2013) Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation. Hydrol Earth Syst Sci 17(4):1281–1296. https://doi.org/10.5194/hess-17-1281-2013
Guo S, Chen J, Li Y, Liu P, Li T (2011) Joint operation of the multi-reservoir system of the three gorges and the Qingjiang cascade reservoirs. Energies 4(7):1036–1050. https://doi.org/10.3390/en4071036
Kwon HH, Lall U (2016) A copula-based nonstationary frequency analysis for the 2012-2015 drought in california. Water Resour Res 52(7):5662–5675. https://doi.org/10.1002/2016WR018959
Li T, Guo S, Liu Z, Xiong L, Yin J (2016) Bivariate design flood quantile selection using copulas. Hydrol Res 48(4):997–1013
Liu Q, Hu D, Yan Q (2010) Decision tree algorithm based on average Euclidean distance. International Conference on Future Computer and Communication. IEEE, pp 507–511
Massoudieh A, Dentz M, Alikhani J (2017) A spatial markov model for the evolution of the joint distribution of groundwater age, arrival time, and velocity in heterogeneous media. Water Resour Res 53(7):5495–5515. https://doi.org/10.1002/2017WR020578
Mediero L, Jiménez-Álvarez A, Garrote L (2010) Design flood hydrographs from the relationship between flood peak and volume. Hydrol Earth Syst Sci 14(12):2495–2505. https://doi.org/10.5194/hess-14-2495-2010
MWR (Ministry of Water Resources) (2006) Regulations for calculating design flood of water resources and hydropower projects. Water Resources and Hydropower Press, Beijing (in Chinese)
Nelsen R (2006) An introduction to copulas, 2nd edn. Springer-Verlag, New York
Özban A (2004) Some new variants of Newton's method. Appl Math Lett 17(6):677–682. https://doi.org/10.1016/S0893-9659(04)90104-8
Pham-Gia T, Hung TL (2001) The mean and median absolute deviations. Math Comput Model 34(7):921–936. https://doi.org/10.1016/S0895-7177(01)00109-1
Radi NFA, Zakaria R, Piantadosi J, Boland J, Wan ZWZ, Azman AZ (2017) Generating synthetic rainfall total using multivariate skew- t, and checkerboard copula of maximum entropy. Water Resour Manag 31(6):1–16
Requena AI, Mediero Orduña L, Garrote de Marcos L (2013) A bivariate return period based on copulas for hydrologic dam design: accounting for reservoir routing in risk estimation. Hydrol Earth Syst Sci 17(8):3023–3038. https://doi.org/10.5194/hess-17-3023-2013
Salvadori G, De Michele C (2011) Estimating strategies for multiparameter multivariate extreme value copulas. Hydrol Earth Syst Sci 15(1):141–150. https://doi.org/10.5194/hess-15-141-2011
Salvadori G, De Michele C (2015) Multivariate real-time assessment of droughts via copula-based multi-site hazard trajectories and fans. J Hydrol 526:101–115. https://doi.org/10.1016/j.jhydrol.2014.11.056
Salvadori G, De Michele C, Durante F (2011) On the return period and design in a multivariate framework. Hydrol Earth Syst Sci 15(11):3293–3305. https://doi.org/10.5194/hess-15-3293-2011
Serinaldi F (2013) An uncertain journey around the tails of multivariate hydrological distributions. Water Resour Res 49(10):6527–6547. https://doi.org/10.1002/wrcr.20531
Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de L’UniversitéParis 8:229–231
Sraj M, Bezak N, Brilly M (2015) Bivariate flood frequency analysis using the copula function: a case study of the Litija station on the Sava River. Hydrol Process 29(2):225–238. https://doi.org/10.1002/hyp.10145
Volpi E, Fiori A (2012) Design event selection in bivariate hydrological frequency analysis. Hydrol Sci J 57(8):1506–1515. https://doi.org/10.1080/02626667.2012.726357
Xiao Y, Guo S, Xiong L, Fang B (2007) Joint analysis of peak and volume based on bivariate distribution. J Yangtze River Sci Res Inst 24(2):13–16 (in Chinese)
Xiao Y, Guo S, Liu P, Yan B, Chen L (2009) Design flood hydrograph based on multicharacteristic synthesis index method. J Hydrol Eng 14(12):1359–1364. https://doi.org/10.1061/(ASCE)1084-0699(2009)14:12(1359)
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. Math Probl Eng 2016:1–16. https://doi.org/10.1155/2016/4319646
Yin J, Guo S, Liu Z, Chen K, Chang FJ, Xiong F (2017) Bivariate seasonal design flood estimation based on copulas. J Hydrol Eng 22(12):05017028. https://doi.org/10.1061/(ASCE)HE.1943-5584.0001594
Zhang Q, Xiao M, Singh VP (2015a) Uncertainty evaluation of copula analysis of hydrological droughts in the East River basin, China. Glob Planet Chang 129:1–9. https://doi.org/10.1016/j.gloplacha.2015.03.001
Zhang Q, Qi T, Singh VP, Chen YD, Xiao M (2015b) Regional frequency analysis of droughts in China: a multivariate perspective. Water Resour Manag 29(6):1767–1787. https://doi.org/10.1007/s11269-014-0910-x
Acknowledgements
This study was financially supported by the National Key Research and Development Plan of China (2016YFC0402206) and the National Natural Science Foundation of China (51539009; 51579183). We are very grateful to the editor and three anonymous reviewers for their valuable comments and constructive suggestions that helped us to greatly improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yin, J., Guo, S., Liu, Z. et al. Uncertainty Analysis of Bivariate Design Flood Estimation and its Impacts on Reservoir Routing. Water Resour Manage 32, 1795–1809 (2018). https://doi.org/10.1007/s11269-018-1904-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11269-018-1904-x