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Uncertainty and variability in bivariate modeling of hydrological droughts

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Abstract

There are two kinds of uncertainty factors in modeling the bivariate distribution of hydrological droughts: the alteration of predefined critical ratios for pooling droughts and excluding minor droughts and the temporal variability of univariate and/or bivariate characteristics of droughts due to the impact of human activities. Daily flow data covering a period of 56 hydrological years from two gauging stations from a humid region in South China are used. The influences of alterations of threshold values of flow and critical ratios of pooling droughts and excluding minor droughts on drought properties are analyzed. Six conventional univariate models and three Archimedean copulas are employed to fit the marginal and joint distributions of drought properties, the Kolmogorov–Smirnov and Anderson–Darling methods are used for testing the goodness-of-fit of the univariate model, and the Cramer-von Mises method based on Rosenblatt’s transform is applied for the test of the bivariate model. The change point analysis of the copula parameter of bivariate distribution of droughts is first made. Results demonstrate that both the statistical characteristics of each drought property and their bivariate joint distributions are sensitive to the critical ratio of excluding minor droughts. A model can be selected to fit the marginal distribution for drought deficit volume or maximum deficit, but it is not determined for drought duration with the lower ratios of the pooling and excluding droughts. The statistical uncertainty of drought duration makes the modeling of bivariate joint distribution of drought duration and deficit volume or of drought duration and maximum deficit undermined. Change points significantly occurred in the period from the late 1970s to the middle 1980s for a single drought property and the copula parameter of their joint distribution due to the impact of human activities. The difference between two subseries separated by the change point is remarkable in the magnitudes of drought properties and the joint return periods. A copula function can be selected to optimally fit the bivariate distribution, provided that the critical ratios of pooling and excluding droughts are great enough such as the optimal value of 0.4 in the case study. It is valuable that the modeling and designing of the bivariate joint correlation and distribution of drought properties can be performed on the subseries separated by the change point of the copula parameter.

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References

  • Abdul Rauf UF, Zeephongsekul P (2014) Copula based analysis of rainfall severity and duration: a case study. Theor Appl Climatol 115:153–166

    Article  Google Scholar 

  • Burn DH, Wychreschuk J, Bonin DV (2004) An integrated approach to the estimation of stream flow drought quantiles. Hydrol Sci J 49(6):1011–1024

    Article  Google Scholar 

  • Byzedi M (2010) Analysis of hydrological drought based on daily flow series. Proc World Acad Sci Eng Technol 70:249–252

    Google Scholar 

  • Cancelliere A, Salas JD (2004) Drought length properties for periodic-stochastic hydrological data. Water Resour Res 40(2):W02503. doi:10.1029/2002WR001750

    Article  Google Scholar 

  • Chebana F, Ouarda TBMJ (2011) Depth-based multivariate descriptive statistics with hydrological applications. J Geophys Res 116:D10120. doi:10.1029/2010JD015338

    Article  Google Scholar 

  • Chen YD, Zhang Q, Xiao M, Singh VP (2013) Evaluation of risk of hydrological droughts by the trivariate Plackett copula in the East River basin (China). Nat Hazards 68(2):529–547

    Article  Google Scholar 

  • Chung CH, Salas JD (2000) Drought occurrences probabilities and risks of dependent hydrologic processes. J Hydrol Eng 5(3):259–268

    Article  Google Scholar 

  • De Michele C, Salvadori G, Vezzoli R, Pecora S (2013) Multivariate assessment of droughts: frequency analysis and dynamic return period. Water Resour Res 49(10):6985–6994

    Article  Google Scholar 

  • Dobric J, Schmid F (2007) A goodness of fit test for copulas based on Rosenblatt’s transformation. Compu Stat Data An 51(9):4633–4642

    Article  Google Scholar 

  • Dracup J, Lee K, Paulson EJ (1980) On the definition of droughts. Water Resour Res 16(2):297–302

    Article  Google Scholar 

  • Durante F, Salvadori G (2010) On the construction of multivariate extreme value models via copulas. Environmetrics 21(2):143–161

    Google Scholar 

  • Fernandez B, Salas JD (1999) Return period and risk of hydrologic events I: mathematical formulation. J Hydrol Eng 4(4):297–307

    Article  Google Scholar 

  • Fleig AK, Tallaksen LM, Hisdal H, Demuth S (2006) A global evaluation of streamflow drought characteristics. Hydrol Earth Syst Sci 10:535–552

    Article  Google Scholar 

  • Ganguli P, Reddy MJ (2012) Risk assessment of droughts in Gujarat using bivariate copulas. Water Resour Manage 26(11):3301–3327

    Article  Google Scholar 

  • Ganguli P, Reddy MJ (2014) Evaluation of trends and multivariate frequency analysis of droughts in three meteorological subdivisions of western India. Int J Climatol 34(3):911–928

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Genest C, Rivest LP (1993) Statistical inference procedures for bivariate Archimedean copulas. J Am Stat Assoc 88(423):1034–1043

    Article  Google Scholar 

  • Genest C, Remillard B, Beaudoin D (2009) Goodness-of-fit tests for copulas: a review and a power study. Insur Math Econ 44(2):199–213

    Article  Google Scholar 

  • Graler B, Van Den Berg MJ, Vandenberghe S, Petroselli A, Grimaldi S, De Baets B, Verhoest NEC (2013) Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph. Hydrol Earth Syst Sci 17:1281–1296

    Article  Google Scholar 

  • Gringorten II (1963) A plotting rule for extreme probability paper. J Geophys Res 68(3):813–814

    Article  Google Scholar 

  • Hong XJ, Guo SL, Zhou YL, Xiong LH (2015) Uncertainties in assessing hydrological drought using streamflow drought index for the upper Yangtze River basin. Stoch Environ Res Risk Assess 29(4):1235–1247

    Article  Google Scholar 

  • Jiang C, Xiong LH, Xu CY, Guo SL (2015) Bivariate frequency analysis of nonstationary low-flow series based on the time-varying copula. Hydrol Process 29(6):1521–1534

    Article  Google Scholar 

  • Kao SC, Govindaraju RS (2010) A copula-based joint deficit index for droughts. J Hydrol 380(1–2):121–134

    Article  Google Scholar 

  • Khedun CP, Mishra AK, Singh VP, Giardino JR (2014) A copula-based precipitation forecasting model: investigating the interdecadal modulation of ENSO’s impacts on monthly precipitation. Water Resour Res 50(1):580–600

    Article  Google Scholar 

  • Kroll CN, Vogel RM (2002) Probability distribution of low streamflow series in the United States. J Hydrol Eng 7(2):137–146

    Article  Google Scholar 

  • Kuchment LS, Demidov VN (2013) Probabilistic characterization of hydrological droughts. Russ Meteorol Hydrol 38(10):694–700

    Article  Google Scholar 

  • Lavielle M, Teyssiere G (2006) Detection of multiple change-points in multivariate time series. Lith Math J 46(3):287–306

    Article  Google Scholar 

  • Lee T, Salas JD (2011) Copula-based stochastic simulation of hydrological data applied to Nile River flows. Hydrol Res 42(4):318–330

    Article  Google Scholar 

  • Lee T, Modarres R, Ouarda TBMJ (2013) Data-based analysis of bivariate copula tail dependence for drought duration and severity. Hydrol Process 27(10):1454–1463

    Article  Google Scholar 

  • Liu DD, Chen XH, Lian YQ, Lou ZH (2010) Impacts of climate change and human activities on surface runoff in the Dongjiang River Basin of China. Hydrol Process 24(21):1487–1495

    Article  Google Scholar 

  • Ma MW, Song SB, Ren LL, Jiang SH, Song JL (2013) Multivariate drought characteristics using trivariate Gaussian and Student t copulas. Hydrol Process 27(8):1175–1190

    Article  Google Scholar 

  • Madsen H, Rosbjerg D (1995) On the modelling of extreme droughts. In: Modelling and management of sustainable basin-scale water resource systems (Proceedings of a boulder Symposium). IAHS Publ NO. 231

  • Milly PCD, Betancourt J, Falkenmark M, Hirsch RM, Kundzewicz ZW, Lettenmaier DP, Stouffer RJ (2008) Stationarity is dead: whither water management? Science 319:573–574

    Article  CAS  Google Scholar 

  • Mishra AK, Singh VP (2010) A review of drought concepts. J Hydrol 391(1–2):202–216

    Article  Google Scholar 

  • Mishra AK, Singh VP (2011) Drought modeling—a review. J Hydrol 403(1–2):157–175

    Article  Google Scholar 

  • Modarres R, Sarhadi A (2010) Frequency distribution of extreme hydrologic drought of southeastern semiarid region. Iran. J Hydrol Eng 15(4):255–264

    Article  Google Scholar 

  • Nadarajah S (2009a) A bivariate distribution with gamma and beta marginals with application to drought data. J Appl Stat 36(3):277–301

    Article  Google Scholar 

  • Nadarajah S (2009b) A bivariate pareto model for drought. Stoch Environ Res Risk A 23(6):811–822

    Article  Google Scholar 

  • Nelsen RB (2006) An introduction to copulas. Springer, New York

    Google Scholar 

  • Panchenko V (2005) Goodness-of-fit test for copulas. Phys A 355(1):176–182

    Article  Google Scholar 

  • Pettitt AN (1979) A non-parametric approach to the change-point problem. J Appl Stat 28:126–135

    Article  Google Scholar 

  • Pettitt AN (1980) A simple cumulative sum type statistic for the change-point problem with zero-one observations. Bio-metrika 67(1):79–84

    Google Scholar 

  • Poulin A, Huard D, Favre AC, Pugin S (2007) Importance of tail dependence in bivariate frequency analysis. J Hydrol Eng 12(4):394–403

    Article  Google Scholar 

  • Reddy MJ, Ganguli P (2012) Application of copulas for derivation of drought severity-duration-frequency curves. Hydrol Process 26(11):1672–1685

    Article  Google Scholar 

  • Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Statist 23(3):470–472

    Article  Google Scholar 

  • Ryu JH, Lee JH, Jeong S, Park SK, Han K (2011) The impacts of climate change on local hydrology and low flow frequency in the Geum River Basin. Korea. Hydrol Process 25(22):3437–3447

    Article  Google Scholar 

  • Salvadori G, De Michele C (2004) Frequency analysis via copulas: theoretical aspects and applications to hydrological events. Water Resour Res 40(12):W12511. doi:10.1029/2004WR003133

    Article  Google Scholar 

  • Salvadori G, De Michele C (2010) Multivariate multiparameter extreme value models and return periods: a copula approach. Water Resour Res 46(10):W10501. doi:10.1029/2009WR009040

    Article  Google Scholar 

  • Salvadori G, De Michele C, Durante F (2011) On the return period and design in a multivariate framework. Hydrol Earth Syst Sci 15:3293–3305

    Article  Google Scholar 

  • Salvadori G, Durante F, De Michele C (2013) Multivariate return period calculation via survival functions. Water Resour Res 49(4):2308–2311

    Article  Google Scholar 

  • Serinaldi F, Bonaccorso B, Cancelliere A, Grimaldi S (2009) Probabilistic characterization of drought properties through copulas. Phys Chem Earth 34(10–12):596–605

    Article  Google Scholar 

  • Shiau JT (2006) Fitting drought duration and severity with two dimensional copulas. Water Resour Manag 20(5):795–815

    Article  Google Scholar 

  • Shiau JT, Modarres R (2009) Copula-based drought severity-duration-frequency analysis in Iran. Meteorol Appl 16(4):481–489

    Article  Google Scholar 

  • Shiau JT, Feng S, Nadarajah S (2007) Assessment of hydrological droughts for the Yellow River China, using copulas. Hydrol Process 21(16):2157–2163

    Article  Google Scholar 

  • Singh VP, Zhang L (2007) IDF curves using the Frank Archimedean copula. J Hydrol Eng 12(6):651–662

    Article  Google Scholar 

  • Sklar A (1959) Fonctions de repartition an dimensions et leurs marges. Publ Inst Stat Univ Paris, pp 229–231

  • Smakhtin VY (2001) Low flow hydrology: a review. J Hydrol 240(3–4):147–186

    Article  Google Scholar 

  • Song SB, Singh VP (2010a) Meta-elliptical copulas for drought frequency analysis of periodic hydraulic data. Stoch Env Res Risk A 24(3):425–444

    Article  Google Scholar 

  • Song SB, Singh VP (2010b) Frequency analysis of droughts using the Plackett copula and parameter estimation by genetic algorithm. Stoch Env Res Risk A 24(5):783–805

    Article  Google Scholar 

  • Tallaksen LM, Madsen H, Clausen B (1997) On the definition and modelling of streamflow drought duration and deficit volume. Hydrol Sci J 42(1):15–33

    Article  Google Scholar 

  • Tu XJ, Zhang Q, Singh VP, Chen XH, Liu CL, Wang SB (2012) Space-time changes in hydrological processes in response to human activities and climatic change in the south China. Stoch Env Res Risk A 26(6):823–834

    Article  Google Scholar 

  • Tu XJ, Singh VP, Chen XH, Ma MW, Zhang Q, Zhao Y (2015) Intra-annual distribution of dtreamflow and individual impacts of climate change and human activities in the Dongijang River basin. China. Water Resour Manage 29(8):2677–2695

    Article  Google Scholar 

  • Van Loon AF, Van Lanen HAJ (2013) Making the distinction between water scarcity and drought using an observation-modeling framework. Water Resour Res 49(3):1483–1502

    Article  Google Scholar 

  • Van Loon AF, Tijdeman E, Wanders N, Van Lanen HAJ, Teuling AJ, Uijlenhoet R (2014) How climate seasonality modifies drought duration and deficit. J Geophys Res Atmos 119(8):4640–4656

    Article  Google Scholar 

  • Wong G, Lambert MF, Leonard M, Metcalfe AV (2010) Drought analysis using trivariate copulas conditional on climate states. J Hydrol Eng 15(2):129–141

    Article  Google Scholar 

  • Yevjevich V (1967) An objective approach to definitions and investigations of continental hydrologic droughts. Hydrol Pap 23, Colorado State University, Fort Collins, Colorado

  • Zaidman MD, Keller V, Young AR, Cadman D (2003) Flow-duration-frequency behaviour of British rivers based on annual minima data. J Hydrol 277(3–4):195–213

    Article  Google Scholar 

  • Zelenhasic E, Salvai A (1987) A method of streamflow drought analysis. Water Resour Res 23(1):156–168

    Article  Google Scholar 

  • Zhang L, Singh VP (2007a) Bivariate rainfall frequency distributions using Archimedean copulas. J Hydrol 332(1–2):93–109

    Article  Google Scholar 

  • Zhang L, Singh VP (2007b) Gumbel-Hougaard copula for trivariate rainfall frequency analysis. J Hydrol Eng 12(4):409–419

    Article  Google Scholar 

  • Zhang L, Singh VP (2007c) Trivariate flood frequency analysis using the Gumbel–Hougaard copula. J Hydrol Eng 12(4):431–439

    Article  Google Scholar 

  • Zhang Q, Chen YD, Chen X, Li J (2011) Copula-based analysis of hydrological extremes and implications of hydrological behaviors in the Pearl River basin. China. J Hydrol Eng 16(7):598–607

    Article  Google Scholar 

  • Zhang Q, Li JF, Singh VP, Xu CY (2013) Copula-based spatio-temporal patterns of precipitation extremes in China. Int J Climatol 33:1140–1152

    Article  Google Scholar 

  • Zhang Q, Singh VP, Li K, Li JF (2014) Trend, periodicity and abrupt change in streamflow of the East River, the Pearl River basin. Hydrol Process 28(2):305–314

    Article  Google Scholar 

Download references

Acknowledgments

Supported by the National Natural Science Foundation of China (Grant NO: 51479217, 51479216), the Open Research Fund of State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin (China Institute of Water Resources and Hydropower Research, Grant NO: IWHR-SKL-201314), and the State Scholarship Fund of the China Scholarship Council (Grant No: 201308440054) are gratefully acknowledged.

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  1. Xinjun Tu

    Present address: Department of Water Resources and Environment, Sun Yat-sen University, Guangzhou, 510275, China

Authors and Affiliations

  1. Department of Water Resources and Environment, Sun Yat-sen University, Guangzhou, 510275, China

    Xiaohong Chen & Qiang Zhang

  2. State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing, 100038, China

    Xinjun Tu & Yong Zhao

  3. Center of Engineering and Technology of Water Security in Southern China of Guangdong, Sun Yat-sen University, Guangzhou, 510275, China

    Xinjun Tu, Xiaohong Chen & Qiang Zhang

  4. Department of Biological and Agricultural Engineering, Texas A&M University, Tamu 2117, College Station, TX, 77843, USA

    Xinjun Tu, Vijay P. Singh & Mingwei Ma

  5. Zachry Department of Civil Engineering, Texas A&M University, Tamu 2117, College Station, TX, 77843, USA

    Vijay P. Singh

  6. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, 210098, China

    Mingwei Ma

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Tu, X., Singh, V.P., Chen, X. et al. Uncertainty and variability in bivariate modeling of hydrological droughts. Stoch Environ Res Risk Assess 30, 1317–1334 (2016). https://doi.org/10.1007/s00477-015-1185-3

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