Uncertainty assessment of estimation of hydrological design values

  • Yi-Ming Hu
  • Zhong-Min Liang
  • Yong-Wei Liu
  • Xiao-Fan Zeng
  • Dong Wang
Original Paper


Hydrological frequency analysis is the foundation for hydraulic engineering design and water resources management. However, the existence of uncertainty in sample representation usually causes the uncertainty in quantile or design value estimation. Standard deviation (SD) of the design value with a given non-exceedance probability, as an extremely valuable indicator, is suggested to quantify the uncertainties of hydrological frequency analysis. In order to assess the impact of the sampling uncertainty on design value, in China’s national standard for design flood calculation, an empirical formula for estimating SD of the design value was suggested; however, it is applicable only to the case that Pearson type three probability distribution function (PE3) and method of moment are used to analyze the hydrological samples. In principle, for other types of probability distribution functions and parameter estimation methods, the suggested empirical formula is not suitable. The aim of this article is to propose a general approach to estimate the SD of a design value by using the Bootstrap resampling technique. In order to testify the applicability of the approach, under the condition of PE3 distribution, three kinds of schemes were designed to calculate SD, i.e. the suggested empirical formula in China’s national standard for design flood calculation (SEF), Bootstrap technique with method of moment for parameter estimation and Bootstrap with linear moment for parameter estimation. The annual precipitation observations from 50 gauges around China were analyzed using these three approaches. Results show that the SD values of quantile estimates are significantly different among these three approaches, which means that when parameter estimation methods rather than method of moment are employed, the SD provided by SEF is inapplicable. It also indicates that, in terms of modified design value, i.e., original design value adding security correction value, the proposed method using the Bootstrap method, as a general method for SD estimation, is effective and feasible.


Hydrological frequency analysis Hydrological design value or quantile Standard deviation Security correction value Bootstrap method Uncertainty analysis 



This study was supported by the Major Program of National Natural Science Foundation of China (Grant no. 51190095), the National Natural Science Foundation of China (Grant nos. 51079039, 51309105), and Graduate Research and Innovation Program for Ordinary University of Jiangsu Province (1043-B1305324). We wish to thank the reviewers and editors, whose suggestions have helped us to improve the quality of the paper.


  1. Changjiang Water Resources Commission (CWRC), Nanjing Institute of Hydrology and Water Resources (1995) Handbook for calculating design flood of water resources and hydropower projects. China Water Power Press, Beijing, pp 34–86Google Scholar
  2. Ding J, Yang R (1988) The determination of probability weighted moments with the incorporation of extraordinary values into sample data and their application to estimating parameters for the pearson type three distribution. J Hydrol 101(1):63–81Google Scholar
  3. Efron B (1979a) Bootstrap methods: another look at the Jackknife. Ann Stat 7:1–26CrossRefGoogle Scholar
  4. Efron B (1979b) Computers and the theory of statistics: thinking the unthinkable. SIAM Rev 21:460–480CrossRefGoogle Scholar
  5. Greenwood JA, Landwehr JM, Matals NC, Wallis JR (1979) Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form. Water Resour Res 15:1049–1054CrossRefGoogle Scholar
  6. Hosking JRM (1990) L-moments: analysis and estimation of distribution using linear combinations of order statistics. J R Stat Soc B 52:105–124Google Scholar
  7. Hu Yiming, Liang Zhongmin, Li Binquan, Yu Zhongbo (2013) Uncertainty assessment of hydrological frequency analysis using Bootstrap method. Math Probl Eng. doi: 10.1155/2013/724632
  8. Kendall MG (1975) Rank Correlation Methods, 4th edn. Charles Griffin, LondonGoogle Scholar
  9. Kirby W (1974) Algebraic boundedness of sample statistics. Water Resour Res 10:220–222CrossRefGoogle Scholar
  10. Kuczera G (1982) Combining site-specific and regional information: an empirical Bayes approach. Water Resour Res 18(2):306–314CrossRefGoogle Scholar
  11. Kuczera G (1999) Comprehensive at-site flood frequency analysis using Monte Carlo Bayesian inference. Water Resour Res 35(5):1551–1557CrossRefGoogle Scholar
  12. Lee KS, Kim SU (2008) Identification of uncertainty in low flow frequency analysis using Bayesian MCMC method. Hydrol Process 22(12):1949–1964. doi: 10.1002/hyp.6778 CrossRefGoogle Scholar
  13. Mann HB (1945) Non-parametric tests against trend. Econometrica 13:163–171Google Scholar
  14. Nguyen C, Gaume E, Payrastre O (2014) Regional flood frequency analyses involving extraordinary flood events at ungauged sites: further developments and validations. J Hydrol 508:385–396CrossRefGoogle Scholar
  15. Overeem Aart, Buishand Adri, Holleman Iwan (2008) Rainfall depth-duration-frequency curves and their uncertainties. J Hydrol 348:124–134CrossRefGoogle Scholar
  16. Payrastre O, Gaume E, Andrieu H (2011) Usefulness of historical information for flood frequency analyses: developments based on a case study. Water Resour Res 47:W08511CrossRefGoogle Scholar
  17. Reis DS Jr, Stedinger JR (2005) Bayesian MCMC flood frequency analysis with historical information. J Hydrol 313(1–2):97–116CrossRefGoogle Scholar
  18. Ribatet M, Sauquet E, Gresillon J-M, Ouarda TBMJ (2007) A regional Bayesian POT model for flood frequency analysis. Stoch Environ Res Risk Assess 21:327–339. doi: 10.1007/s00477-006-0068-z CrossRefGoogle Scholar
  19. Tang WH (1980) Bayesian frequency analysis. J Hydraul Div 106(7):1203–1218Google Scholar
  20. Tiwari Mukesh K, Chatterjee Chandranath (2010) Development of an accurate and reliable hourly flood forecasting model using wavelet–bootstrap–ANN (WBANN) hybrid approach. J Hydrol 394:458–470CrossRefGoogle Scholar
  21. Wallis JR, Matalas NC, Matas NC, Slack JR (1974) Just a moment! Water Resour Res 10:211–219CrossRefGoogle Scholar
  22. Wood EF, Rodriguez-Iturbe I (1975a) Bayesian inference and decision making for extreme hydrologic events. Water Resour Res 11(4):533–554CrossRefGoogle Scholar
  23. Wood EF, Rodriguez-Iturbe I (1975b) A Bayesian approach to analyzing uncertainty among flood frequency models. Water Resour Res 11(6):839–843CrossRefGoogle Scholar
  24. Yeou-koung Tung, Chi-leung Wong (2014) Assessment of design rainfall uncertainty for hydrologic engineering applications in Hong Kong. Stoch Environ Res Risk Assess 28:583–589CrossRefGoogle Scholar
  25. Zhongmin Liang, Li Binquan Yu, Zhongbo Wenjun Chang (2011) Application of Bayesian approach to hydrologic frequency analysis. Sci China Technol Sci 54(5):1183–1192CrossRefGoogle Scholar
  26. Zhongmin Liang, Wenjun Chang, Binquan Li (2012) Bayesian flood frequency analysis in the light of model and parameter uncertainties. Stoch Environ Res Risk Assess 26:721–730CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Yi-Ming Hu
    • 1
  • Zhong-Min Liang
    • 1
    • 2
  • Yong-Wei Liu
    • 1
  • Xiao-Fan Zeng
    • 3
  • Dong Wang
    • 1
  1. 1.College of Hydrology and Water ResourcesHohai UniversityNanjingChina
  2. 2.State Key Laboratory of Hydrology-Water Resources and Hydraulic EngineeringHohai UniversityNanjingChina
  3. 3.School of Hydropower & Information EngineeringHuaZhong University of Science and TechnologyWuhanChina

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