Bivariate drought frequency curves and confidence intervals: a case study using monthly rainfall generation

  • Jiyoung Yoo
  • Ungtae Kim
  • Tae-Woong KimEmail author
Original Paper


Although water resources management practices recently use bivariate distribution functions to assess drought severity and its frequency, the lack of systematic measurements is the major hindrance in achieving quantitative results. This study aims to suggest a statistical scheme for the bivariate drought frequency analysis to provide comprehensive and consistent drought severities using observed rainfalls and their uncertainty using synthesized rainfalls. First, this study developed a multi-variate regression model to generate synthetic monthly rainfalls using climate variables as causative variables. The causative variables were generated to preserve their correlations using copula functions. This study then focused on constructing bivariate drought frequency curves using bivariate kernel functions and estimating their confidence intervals from 1,000 likely replica sets of drought frequency curves. The confidence intervals achieved in this study may be useful for making a comprehensive drought management plan through providing feasible ranges of drought severity.


Drought Frequency Confidence interval Copulas 



This work was supported by Grants from Korean National Research Foundation (No. 2010-0016717) and Korean National Emergency Management Agency (NEMA-11-NH-40). The authors also thank the anonymous reviewers for their constructive comments and corrections.


  1. Adamowski K (1985) Nonparametric kernel estimation of flood frequencies. Water Resour Res 21(11):1585–1590CrossRefGoogle Scholar
  2. Bonaccorso B, Cancelliere A, Rossi G (2003) An analytical formulation of return period of drought severity. Stoch Env Res Risk Assess 17(3):157–174CrossRefGoogle Scholar
  3. Cancelliere A, Salas JD (2004) Drought length properties for periodic-stochastic hydrologic data. Water Resour Res 40(2):W02503CrossRefGoogle Scholar
  4. Chen L, Singh VP, Gui S (2011) Drought analysis based on copulas. 2011 Symposium on data-driven approaches to droughts. Purdue University, West LafayetteGoogle Scholar
  5. Chung C, Salas JD (2000) Drought occurrence probabilities and risks of dependent hydrologic processes. J Hydrol Eng 5(3):259–268CrossRefGoogle Scholar
  6. Delleur JW, Kavva ML (1978) Stochastic models for monthly rainfall forecasting and synthetic generation. J Appl Meteorol 17(10):1528–1536CrossRefGoogle Scholar
  7. Dracup JA, Lee KS, Paulson EG Jr (1980) On the definition of droughts. Water Resour Res 16(2):297–302CrossRefGoogle Scholar
  8. Fernandez B, Salas JD (1999) Return period and risk of hydrologic events. I: mathematical formulation. J Hydrol Eng 4(4):297–307CrossRefGoogle Scholar
  9. Genest C, Favre AC (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368CrossRefGoogle Scholar
  10. Gonzalez J, Valdes JB (2003) Bivariate drought recurrence analysis using tree ring reconstructions. J Hydrol Eng 8(5):247–258CrossRefGoogle Scholar
  11. Kao SC, Govindaraju RS (2010) A copula-based joint deficit index for droughts. J Hydrol 380(1–2):121–134CrossRefGoogle Scholar
  12. Kim T-W, Valdes JB, Yoo C (2003) Nonparametric approach for estimating return periods of droughts in arid regions. J Hydrol Eng 8(5):237–246CrossRefGoogle Scholar
  13. Kim T-W, Valdes JB, Yoo C (2006) Nonparametric approach for bivariate drought characterization using Palmer drought index. J Hydrol Eng 11(2):134–143CrossRefGoogle Scholar
  14. Loaiciga H, Leipnik R (1996) Stochastic renewal model of low-flow streamflow sequences. Stochastic Hydrology and Hydraulics 10(1):65–85CrossRefGoogle Scholar
  15. Mirakbari M, Ganji A, Fallah S (2010) Regional bivariate frequency analysis of meteorological droughts. J Hydrol Eng 15(12):985–1000CrossRefGoogle Scholar
  16. Moon YI, Lall U (1994) Kernel quantile function estimator for flood frequency analysis. Water Resource Research 30(11):3095–3103CrossRefGoogle Scholar
  17. O’Brien RM (2007) A caution regarding rules of thumb for variance inflation factors. Qual Quant 41:673–690CrossRefGoogle Scholar
  18. Oliveria JDT (1975) Bivariate extremes: extensions. Bull Int Stat Inst 46(2):241–251Google Scholar
  19. Salas JD, Fu C, Cancelliere A, Dustin D, Bode D, Pineda A, Vincent E (2005) Characterizing the severity and risk of drought in the Poudre River, Colorado. J Water Resour Plan Manag 131(5):383–393CrossRefGoogle Scholar
  20. Scott DW (1992) Multivariate density estimation: theory, practice and visualization. Wiley, New YorkCrossRefGoogle Scholar
  21. Sharma A, O’Neill R (2002) A nonparametric approach for representing interannual dependence in monthly streamflow sequences. Water Resour Res 38(7): 5–1:5-10Google Scholar
  22. Shiau JT (2006) Fitting drought duration and severity with two-dimensional copulas. Water Resour Manage 20(5):795–815CrossRefGoogle Scholar
  23. Shiau JT, Shen HW (2001) Recurrence analysis of hydrologic droughts of differing severity. J Water Resour Plan Manag 127(1):30–40CrossRefGoogle Scholar
  24. Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall/CRC, LondonGoogle Scholar
  25. Sklar A (1959) Fonctions de repartition a n dimensions et leurs marges. Publ Inst Statist Univ Paris 8(1):11Google Scholar
  26. Smakhtin VU (2001) Low flow hydrology: a review. J Hydrol 240(3–4):147–186CrossRefGoogle Scholar
  27. Ünal N, Aksoy H, Akar T (2004) Annual and monthly rainfall data generation schemes. Stoch Env Res Risk Assess 18(4):245–257CrossRefGoogle Scholar
  28. Wilhite DA (2000) Drought as a natural hazard: concepts and definitions. Drought A Global Assess 1:3–18Google Scholar
  29. Yevjevich V (1967) Objective approach to definitions and investigations of continental hydrologic droughts. Hydrology Paper 23, Colorado State U, Fort CollinsGoogle Scholar
  30. Yue S, Ouarda TBMJ, Bobee B, Legendre P, Bruneau P (1999) The Gumbel mixed model for flood frequency analysis. J Hydrol 226(1–2):88–100CrossRefGoogle Scholar
  31. Zhang L, Singh VP (2006) Bivariate flood frequency analysis using the copula method. J Hydrol Eng 11(2):150–164CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringHanyang UniversitySeoulKorea
  2. 2.Department of Civil and Environmental EngineeringThe University of TennesseeKnoxvilleUSA
  3. 3.Department of Civil and Environmental EngineeringHanyang UniversityAnsanKorea

Personalised recommendations