Abstract
Reservoir storage plays an important role in water supply during the dry season when precipitation is insufficient. In a watershed where the streams are controlled by reservoirs, drought occurrences depend on not only precipitation variations but also reservoir regulation. In this study, the joint dependence structure of the reservoir storage and its relevant variables of precipitation and/or upstream outflow were analyzed for two cascade reservoirs in a headwater basin of the Huaihe River, China. Correlation analysis indicates that the reservoir storage in October (the end of the wet season) depends highly on the regional precipitation at time scales of several months, e.g., 7 months for the upstream and 9 months for the downstream. Additionally, the downstream storage is correlated with outflow from the upstream reservoir at the 5-month timescale significantly. For estimation of the joint probability of pairs of the storage and its relevant variables, univariate marginal distributions and bivariate copula were appropriately selected in terms of statistical tests. The bivariate return period of \(T(X < x,Y < y)\) and \(T(X \le x,Y \ge y)\) and the conditional probability of \(P(Y \ge y|X \le x)\) were estimated by using the selected Clayton copula. The results from contour lines of the bivariate return period demonstrate that the probability of drought occurrences affected by both reservoir storage and precipitation/outflow is smaller than that by either of the variables. Meanwhile, the concurrent drought probability between precipitation and reservoir storage in the upstream is higher than that in the downstream. The estimated conditional probability offers useful information on how much the regular storage could be remained under some specified drought levels of precipitation/upstream outflow. Therefore, the results are helpful for improving the operation strategies of the cascade reservoirs for the adaptive management of drought under different climate variations.
Similar content being viewed by others
References
Adler J, Parmryd I (2010) Quantifying colocalization by correlation: the Pearson correlation coefficient is superior to the Mander’s overlap coefficient. Cytom Part A 77A:733–742. doi:10.1002/cyto.a.20896
Aghakouchak A, Ciach G, Habib E (2010) Estimation of tail dependence coefficient in rainfall accumulation fields. Adv Water Resour 33:1142–1149. doi:10.1016/j.advwatres.2010.07.003
Agresti A (2010) Analysis of ordinal categorical data, 2nd edn. Wiley, New York
Aissia MAB, Chebana F, Ouarda TBMJ, Roy L, Desrochers G, Chartier I, Robichaud É (2012) Multivariate analysis of flood characteristics in a climate change context of the watershed of the Baskatong reservoir, Province of Québec, Canada. Hydrol Process 26(1):130–142. doi:10.1002/hyp.8117
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723. doi:10.1109/TAC.1974.1100705
Anderson TW, Darling DA (1952) Asymptotic theory of certain “goodness-of-fit” criteria based on stochastic processes. Ann Math Stat 23:193–212. doi:10.1214/aoms/1177729437
Ane T, Kharoubi C (2003) Dependence structure and risk measure. J Bus 76(3):411–438
Borroni CG (2013) A new rank correlation measure. Stat Pap 54(2):255–270. doi:10.1007/s00362-011-0423-0
Bozdogan H (2000) Akaike’s information criterion and recent developments in information complexity. J Math Psychol 44(1):62–91. doi:10.1006/jmps.1999.1277
Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65:141–151
Cook DR, Johnson ME (1981) A family of distributions for modeling non-elliptically symmetric multivariate data, Journal of the Royal Statistical Society. Ser B Methodol 43:210–218
Deheuvels P (1979) La fonction de dépendance empirique et ses propriétés: Un test non paramétrique d’indépendance. Acad Roy Belgique Bull Cl Sci 65(6):274–292
Dobric J, Schmid F (2005) Nonparametric estimation of the lower tail dependence lambda(L) in bivariate copulas. J Appl Stat 32(4):387–407. doi:10.1080/02664760500079217
Favre AC, El Adlouni S, Perreault L, Thiémonge N, Bobée B (2004) Multivariate hydrological frequency analysis using copulas. Water Resour Res 40:W01101
Frahm G, Junker M, Schmidt R (2005) Estimating the tail-dependence coefficient: properties and pitfalls. Insur Math Econ 37(1):80–100
Frank MJ (1979) On the simultaneous associativity of f(x, y) and x + y−f(x, y). Aequationes Math 19:194–226
Frank J, Masse J (1951) The Kolmogorov–Smirnov test for goodness of fit. J Am Stat Assoc 46(253):68–78
Gayen AK (1951) The frequency distribution of the product-moment correlation coefficient in random samples of any size drawn from non-normal universes. Biometrika 38:219–247. doi:10.2307/2332329
Genest C (1987) Frank’s family of bivariate distributions. Biometrika 74:549–555
Genest C, Favre AC (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368
Genest C, Rivest LP (1993) Statistical inference procedures for bivariate Archimedean copulas. J Am Stat Assoc 88(423):1034–1043
Genest C, Ghoudi K, Rivest LP (1995) A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3):543–552
Gumbel EJ (1960) Bivariate exponential distributions. J Am Stat Assoc 55:698–707
Hougaard P (1986) A class of multivariate failure time distributions. Biometrika 73:671–678
Hu WW, Wang GX, Deng W, Li SN (2008) The influence of dams on ecohydrological conditions in the Huaihe River basin, China. Ecol Eng 33(3):233–241
Janga Reddy M, Ganguli P (2012) Application of copulas for derivation of drought severity–duration–frequency curves. Hydrol Process 26:1672–1685. doi:10.1002/hyp.8287
Jaranilla-Sanchez PA, Wang L, Koike T (2011) Modeling the hydrologic responses of the Pampanga River basin, Philippines: a quantitative approach for identifying droughts. Water Resour Res. doi:10.1029/2010wr009702
Jenkinson AF (1955) The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Q J R Meteorol Soc 87:158–171
Joe H (1997) Multivariate models and dependence concepts. Chapman and Hall, London
Johnson NL (1994) Continuous univariate distributions, vol 1. Wiley, New York
Kendall MG (1938) A new measure of rank correlation. Biometrika 30:81–93. doi:10.1093/biomet/30.1-2.81
Kendall MG (1955) Rank correlation methods. Griffin, London
Kendall M, Gibbons JD (1990) Rank correlation methods. Oxford University Press, New York
Lee T, Modarres R, Ouarda T (2013) Data-based analysis of bivariate copula tail dependence for drought duration and severity. Hydrol Process 27(10):1454–1463
López-Moreno JI, Begueria S, Garcia-Ruiz JM (2004) The management of a large Mediterranean reservoir: storage regimens of the Yesa Reservoir, Upper Aragon River Basin, central Spanish Pyrenees. Environ Manag 34(4):508–515. doi:10.1007/s00267-003-0249-1
López-Moreno JI, Vicente-Serrano SM, Zabalza J, Beguería S, Lorenzo-Lacruz J, Azorin-Molina C, Morán-Tejeda E (2013) Hydrological response to climate variability at different time scales: a study in the Ebro basin. J Hydrol 477:175–188. doi:10.1016/j.jhydrol.2012.11.028
Mann HB (1945) Nonparametric tests against trend. Econometrica 13:245–259
McKee TB, Doesken NJ, Kleist J (1993) The relationship of drought frequency and duration to time scales. In: Proceedings of the 8th conference on applied climatology, vol 22. American Meteorological Society Boston, pp 179–183
Nelsen RB (1986) Properties of a one-parameter family of bivariate distribution with specified marginals. Commun Stat Theory Methods 15:3277–3285
Nelsen RB (1999) An introduction to copulas. Springer, New York
Oakes D (1982) A model fo association in bivariate survival data. J R Stat Soc Ser B 44:414–422
Otieno H, Yang J, Liu W, Han D (2014) Influence of rain gauge density on interpolation method selection. J Hydrol Eng 19(11):04014024. doi:10.1061/(asce)he.1943-5584.0000964
Poulin A, Huard D, Favre AC, Pugin S (2007) Importance of tail dependence in bivariate frequency analysis. J Hydrol Eng 12:394–403. doi:10.1061/(Asce)1084-0699(2007)12:4(394)
Salvadori G, De Michele C (2004) Frequency analysis via copulas: theoretical aspects and applications to hydrological events. Water Resour Res. doi:10.1029/2004wr003133
Shiau JT (2006) Fitting drought duration and severity with two-dimensional copulas. Water Resour Manag 20(5):795–815
Shiau JT, Modarres R (2009) Copula-based drought severity-duration-frequency analysis in Iran. Meteorol Appl 16(4):481–489
Shiau JT, Feng S, Nadarajah S (2007) Assessment of hydrological droughts for the Yellow River, China, using copulas. Hydrol Process 21(16):2157–2163. doi:10.1002/hyp.6400
Shiau JT, Modarres R, Nadarajah S (2012) Assessing multi-site drought connections in iran using empirical copula. Environ Model Assess 17(5):469–482. doi:10.1007/s10666-012-9318-2
Sklar A (1959) Fonctions de répartition á n dimensions et leurs marges, vol 8. Publications de l’Institut de Statistique de l’Université de Paris, Paris, pp 229–231
Thiessen AH (1911) Precipitation averages for large areas. Mon Weather Rev 39(7):1082–1084
Vicente-Serrano SM, López-Moreno JI (2005) Hydrological response to different time scales of climatological drought: an evaluation of the standardized precipitation index in a mountainous Mediterranean basin. Hydrol Earth Syst Sci Discuss 9(5):523–533
Wang X (2000) Characteristics analysis of rainfall in the upper reaches of the Shaying River Basin (in Chinese with English abstract). Hydrology 20(1):53–55
Weibull W (1939) A statistical theory of the strength of material. Ingeniors Vetenskaps Akademiens, Stockholm 151
Yevjevich V (1972) Probability and statistics in hydrology. Water Resources Publications, Fort Collins
Zhang Y, Xia J, Liang T, Shao Q (2010) Impact of water projects on river flow regimes and water quality in Huai River Basin. Water Resour Manag 24(5):889–908
Zhang Q, Singh VP, Sun P, Chen X, Zhang Z, Li J (2011a) Precipitation and streamflow changes in China: changing patterns, causes and implications. J Hydrol 410(3–4):204–216
Zhang Y, Shao Q, Xia J, Bunn SE, Zuo Q (2011b) Changes of flow regimes and precipitation in Huai River Basin in the last half century. Hydrol Process 25(2):246–257
Zhang Q, Li J, Singh VP, Xu C-Y (2013) Copula-based spatio-temporal patterns of precipitation extremes in China. Int J Climatol 33(5):1140–1152. doi:10.1002/joc.3499
Zhang R, Chen X, Zhang Z, Shi P (2015) Evolution of hydrological drought under the regulation of two reservoirs in the headwater basin of the Huaihe River, China. Stoch Environ Res Risk Assess 29(2):487–499. doi:10.1007/s00477-014-0987-z
Zhou Y, Ma Z, Wang L (2002) Chaotic dynamics of the flood series in the Huaihe River Basin for the last 500 years. J Hydrol 258:100–110. doi:10.1016/S0022-1694(01)00561-3
Acknowledgments
The research is financially supported by the National Natural Science Foundation of China (Grant No. 51190090) and Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 1501061B). We thank the editor and two anonymous reviewers for their constructive comments on the earlier manuscript, which lead to an improvement of the paper. The manuscript was edited by Jiayi Chen from University of California, Los Angeles.
Funding
This study was funded by the National Natural Science Foundation of China (Grant No. 51190090) and Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 1501061B).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Zhang, R., Chen, X., Cheng, Q. et al. Joint probability of precipitation and reservoir storage for drought estimation in the headwater basin of the Huaihe River, China. Stoch Environ Res Risk Assess 30, 1641–1657 (2016). https://doi.org/10.1007/s00477-016-1249-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-016-1249-z