Comparison and evaluation of uncertainties in extreme flood estimations of the upper Yangtze River by the Delta and profile likelihood function methods

  • Jianhua Wang
  • Fan Lu
  • Kairong Lin
  • Weihua Xiao
  • Xinyi Song
  • Yanhu He
Original Paper


Frequency calculation for extreme flood and methods used for its uncertainty estimation are popular subjects in hydrology research. In this study, uncertainties in extreme flood estimations of the upper Yangtze River were investigated using the Delta and profile likelihood function (PLF) methods, which were used to calculate confidence intervals of key parameters of the generalized extreme value distribution and quantiles of extreme floods. Datasets of annual maximum daily flood discharge (AMDFD) from six hydrological stations located in the main stream and tributaries of the upper Yangtze River were selected in this study. The results showed that AMDFD data from the six stations followed the Weibull distribution, which has a short tail and is bounded above with an upper bound. Of the six stations, the narrowest confidence interval can be detected in the Yichang station, and the widest interval was found in the Cuntan station. Results also show that the record length and the return period are two key factors affecting the confidence interval. The width of confidence intervals decreased with the increase of record length because more information was available, while the width increased with the increase of return period. In addition, the confidence intervals of design floods were similar for both methods in a short return period. However, there was a comparatively large difference between the two methods in a long return period, because the asymmetry of the PLF curve increases with an increase in the return period. This asymmetry of the PLF method is more proficient in reflecting the uncertainty of design flood, suggesting that PLF method is more suitable for uncertainty analysis in extreme flood estimations of the upper Yangtze River Basin.


Uncertainty Extreme flood Yangtze River Delta method Profile likelihood function 



The research is financially supported by the National Basic Research Program of China (“973” Program) (Grant Nos.: 2013CB036406 and 2015CB452701) and the National Natural Science Foundation of China (Grant Nos.: 51679252 and 51379223).


  1. Becker V, Schilling M, Bachmann J, Baumann U, Raue A, Maiwald T, Timmer J, Klinqmuller U (2010) Covering a broad dynamic range: information processing at the erythropoietin receptor. Science 328:1404–1408CrossRefGoogle Scholar
  2. Bernardara P, Schertzer D, Sauquet E, Tchiguirisnkaia I, Lang M (2008) The flood probability distribution tail: how heavy is it? Stoch Environ Res Risk Assess 22:107–122. doi: 10.1007/s00477-006-0101-2 CrossRefGoogle Scholar
  3. Bhunya PK, Jain SK, Ojha CSP, Agarwal A (2007) Simple parameter estimation technique for three-parameter generalized extreme value distribution. J Hydrol Eng 12(6):682–689CrossRefGoogle Scholar
  4. Bobee B (1973) Sample error of T-year events computed by fitting a Pearson type 3 distribution. Water Resour Res 9(5):1264–1270CrossRefGoogle Scholar
  5. Chen L, Singh VP, Guo S, Zhou J, Zhang J (2015) Copula-based method for multisite monthly and daily streamflow simulation. J Hydrol 528:369–384CrossRefGoogle Scholar
  6. Coles S (2001) An introduction to statistical modeling of extreme values. Springer, LondonCrossRefGoogle Scholar
  7. Coles SG, Dixon MJ (1999) Likelihood-based inference for extreme value models. Extremes 2(1):5–23CrossRefGoogle Scholar
  8. Gilli M, Kellezi E (2006) An application of extreme value theory for measuring financial risk. Comput Econ 27:207–228CrossRefGoogle Scholar
  9. Gu H, Yu Z, Wang G, Wang J, Ju Q, Yang C, Fan C (2015) Impact of climate change on hydrological extremes in the Yangtze River Basin, China. Stoch Env Res Risk A. 29:693–707CrossRefGoogle Scholar
  10. Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc B 52(1):105–124Google Scholar
  11. Hosking JRM, Wallis JR (1993) Some statistics useful in regional frequency analysis. Water Resour Res 29(2):271–281CrossRefGoogle Scholar
  12. Hosking JRM, Wallis JR (1997) Regional frequency analysis. Cambridge Unviersity Press, CambridgeCrossRefGoogle Scholar
  13. Hosking JRM, Wallis JR, Wood EF (1985) Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27(3):251–261CrossRefGoogle Scholar
  14. Hromadka TV II, Whitley RJ, Smith MJ (2011) Stability of confidence levels for flood frequencies using additional data. J Water Res Prot 3:228–234CrossRefGoogle Scholar
  15. Hu YM, Liang ZM, Liu YW, Zeng XF, Wang D (2015) Uncertainty assessment of estimation of hydrological design values. Stoch Environ Res Risk Assess 29(2):501–511CrossRefGoogle Scholar
  16. Huang W, Xu S, Nnaji S (2008) Evaluation of GEV model for frequency analysis of annual maximum water levels in the coast of United States. Ocean Eng 35(11–12):1132–1147CrossRefGoogle Scholar
  17. Huard D, Mailhot A, Duchesne S (2010) Bayesian estimation of intensity-duration-frequency curves and of the return period associated to a given rainfall event. Stoch Environ Res Risk Assess 24(3):337–347CrossRefGoogle Scholar
  18. Javelle P, Ouarda T, Lang M, Bobee B (2002) Development of regional flood-duration frequency curves based on the index-flood method. J Hydrol 258(1–4):249–259CrossRefGoogle Scholar
  19. Jenkinson AF (1955) The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Q J R Meteorol Soc 81:158–171CrossRefGoogle Scholar
  20. Kjeldsen TR, Jones DA (2006) Prediction uncertainty in a median-based index flood method using L moments. Water Resour Res 42:W07414. doi: 10.1029/2005WR004069 CrossRefGoogle Scholar
  21. Kreutz C, Raue A, Kaschek D, Timmer J (2013) Profile likelihood in systems biology. FEBS J280:2564–2571. doi: 10.1111/febs.12276 Google Scholar
  22. Kuczera G (1999) Comprehensive at-site flood frequency analysis using Monte Carlo Bayesian inference. Water Resour Res 35(5):1551–1557CrossRefGoogle Scholar
  23. Kumar R, Chatterjee C (2005) regional flood frequency analysis using L-moments for north Brahmaputra region of India. J Hydrol Eng 10(1):1–7CrossRefGoogle Scholar
  24. Lee KS, Kim SU (2008) Identification of uncertainty in low flow frequency analysis using Bayesian MCMC method. Hydrol Process 22(12):1949–1964CrossRefGoogle Scholar
  25. Liang ZM, Chang WJ, Li BQ (2012) Bayesian flood frequency analysis in the light of model and parameter uncertainties. Stoch Environ Res Risk Assess 26:721–730CrossRefGoogle Scholar
  26. Liu P, Lin KR, Wei XJ (2015) A two-stage method of quantitative flood risk analysis for reservoir real-time operation using ensemble-based hydrologic forecasts. Stoch Env Res Risk A. 29:803–813CrossRefGoogle Scholar
  27. Loucks E, Oriel K, Heineman M (2005) Frequency characteristics of long-duration rainfall events. Impacts Glob Clim Change. doi: 10.1061/40792(173)125 Google Scholar
  28. Lu F, Wand H, Yan D, Zhang DD, Xiao WH (2013) Application of profile likelihood function to the uncertainty analysis of hydro meteorological extreme inference. Sci China Technol Sci 56(12):3151–3160CrossRefGoogle Scholar
  29. Madsen H, Rasmussen PF, Rosbjerg D (1997) Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events 1. At-site modeling. Water Resour Res 33(4):746–757Google Scholar
  30. Martins ES, Stedinger JR (2000) Generalized maximum-likelihood generalizedextreme-value quantile estimators for hydrologic data. Water Resour Res 36(3):737–744CrossRefGoogle Scholar
  31. Merz B, Thieken AH (2005) Separating natural and epistemic uncertainty in flood frequency analysis. J Hydrol 309(1–4):114–132CrossRefGoogle Scholar
  32. Morrison JE, Smith JA (2002) Stochastic modeling of flood peaks using the generalized extreme value distribution. Water Resour Res 38(12):1305. doi: 10.1029/2001WR000502 CrossRefGoogle Scholar
  33. Murphy SA, Van der Vaart AW (2000) On profile likelihood. J Am Stat Assoc 95(450):449–465CrossRefGoogle Scholar
  34. Nguyen V, Tao D, Bourque A (2002) On selection of probability distributions for representing annual extreme rainfall series. Glob Solut Urban Drain. doi: 10.1061/40644(2002)250 Google Scholar
  35. Park JS (2005) A simulation-based hyper parameter selection for quantile estimation of the generalized extreme value distribution. Math Comput Simul 70(4):227–234CrossRefGoogle Scholar
  36. Prescott P, Walden AT (1980) Maximum likelihood estimation of the parameters of the generalized extreme-value distribution. Biometrika 67(3):723–724CrossRefGoogle Scholar
  37. Raue A, Kreutz C, Maiwald T (2009) Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25:1923–1929CrossRefGoogle Scholar
  38. Saf B (2009) Regional flood frequency analysis using L-moments for the west mediterranean region of turkey. Water Resour Manag 23:531–551CrossRefGoogle Scholar
  39. Schaber J (2012) Easy parameter identifiability analysis with COPASI. Biosystems 110:183–185CrossRefGoogle Scholar
  40. Schelker M, Raue A, Timmer J, Kreutz C (2012) Comprehensive estimation of input signals and dynamics in biochemical reaction networks. Bioinformatics 28:529–534CrossRefGoogle Scholar
  41. Shamir E, Georgakakos KP, Murphy M (2012) Frequency analysis of the 7-8 December 2010 extreme precipitation in the Panama Canal watershed. J Hydrol 480:136–148CrossRefGoogle Scholar
  42. Shao Q, Leratb J, Podger G, Dutta D (2014) Uncertainty estimation with bias-correction for flow series based on rating curve. J Hydrol 510:137–152CrossRefGoogle Scholar
  43. Siliverstovs B, Otsch R, Kemfert C, Jaeger CC, Haas A, Kremers H (2009) Climate change and modelling of extreme temperatures in Switzerland. Stoch Environ Res Risk Assess 24:311–326. doi: 10.1007/s00477-009-0321-3 CrossRefGoogle Scholar
  44. Smith JA (1987) Estimating the upper tail of flood frequency distributions. Water Resour Res 23(8):1657–1666CrossRefGoogle Scholar
  45. Virtanen A, Uusipaikka E (2008) Computation of profile likelihood-based confidence intervals for reference limits with covariates. Stat Med 27(7):1121–1132CrossRefGoogle Scholar
  46. Walshaw D (2000) Modelling extreme wind speeds in regions proneto hurricanes. J R Stat Soc Ser C 49(1):51–62CrossRefGoogle Scholar
  47. Yoon S, ChoW Heo JH, Kim CE (2010) A full Bayesian approach to generalized maximum likelihood estimation of generalized extreme value distribution. Stoch Environ Res Risk Assess 24(5):761–770CrossRefGoogle Scholar
  48. Zhang Q, Gu X, Singh VP, Xiao M, Chen X (2015) Evaluation of flood frequency under non-stationarity resulting from climatechange and human activities in the East River basin, China. J Hydro 527:565–575CrossRefGoogle Scholar
  49. Zong YQ, Chen XQ (2000) The 1998 Flood on the Yangtze, China. Nat Hazards 22(2):165–184CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Jianhua Wang
    • 1
  • Fan Lu
    • 1
  • Kairong Lin
    • 2
    • 3
  • Weihua Xiao
    • 1
  • Xinyi Song
    • 1
  • Yanhu He
    • 2
    • 3
  1. 1.State Key Laboratory of Simulation and Regulation of Water Cycle in River BasinChina Institute of Water Resources and Hydropower ResearchBeijingChina
  2. 2.Department of Water Resources and Environment, School of Geography and PlanningSun Yat-sen UniversityGuangzhouChina
  3. 3.Key Laboratory of Water Cycle and Water Security in Southern China of Guangdong High Education InstituteGuangzhouChina

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