Journal of Korea Water Resources Association (한국수자원학회논문집)

Korea Water Resources Association (한국수자원학회)
권호 표지
DOI QR코드

DOI QR Code

Identification of Uncertainty in Fitting Rating Curve with Bayesian Regression

베이지안 회귀분석을 이용한 수위-유량 관계곡선의 불확실성 분석

  • Kim, Sang-Ug (SNU BK21 SIR Group, Seoul National University) ;
  • Lee, Kil-Seong (Dept. of Civil and Environmental Engineering, Seoul National University)
  • 김상욱 (서울대학교 BK21 안전하고 지속가능한 사회기반건설 사업단) ;
  • 이길성 (서울대학교 공과대학 건설.환경공학부)
  • Published : 2008.09.02
Download PDF
⟨ Previous Next ⟩

Abstract

This study employs Bayesian regression analysis for fitting discharge rating curves. The parameter estimates using the Bayesian regression analysis were compared to ordinary least square method using the t-distribution. In these comparisons, the mean values from the t-distribution and the Bayesian regression are not significantly different. However, the difference between upper and lower limits are remarkably reduced with the Bayesian regression. Therefore, from the point of view of uncertainty analysis, the Bayesian regression is more attractive than the conventional method based on a t-distribution because the data size at the site of interest is typically insufficient to estimate the parameters in rating curve. The merits and demerits of the two types of estimation methods are analyzed through the statistical simulation considering heteroscedasticity. The validation of the Bayesian regression is also performed using real stage-discharge data which were observed at 5 gauges on the Anyangcheon basin. Because the true parameters at 5 gauges are unknown, the quantitative accuracy of the Bayesian regression can not be assessed. However, it can be suggested that the uncertainty in rating curves at 5 gauges be reduced by Bayesian regression.

본 연구는 수위-유량 관계곡선식의 매개변수 추정을 수행하기 위하여 Bayesian 회귀분석을 적용하였다. 또한 불확실성측면에서의 효과를 탐색하기 위하여 Bayesian 회귀분석에 의한 추정치와 t 분포를 이용하여 산정한 일반 최소자승법(ordinary least square, OLS)에 의한 회귀분석의 추정치를 각각 산정하여 산정결과의 신뢰구간을 비교분석 하였다. 등분산케이스의 통계적 실험결과 t 분포를 이용하여 산정된 평균 추정치와 Bayesian 회귀분석에 의한 평균 추정치는 크게 다르지 않았으나, 비등분산 케이스의 경우에는 Bayesian 회귀분석이 참값에 가까운 추정치를 산정함을 알 수 있었다. 또한 불확실성 측면에서 평가해 볼 때 신뢰구간의 상한추정치와 하한추정치의 차이는 Bayesian 회귀분석을 사용한 경우가 기존 방법을 사용한 경우보다 작은 것으로 나타났으며, 이로부터 수위-유량 관계곡선식의 매개변수를 추정하는 경우 Bayesian 회귀분석이 일반 회귀분석보다 불확실성을 표현하는데 있어서 우수하다는 결과를 얻을 수 있었다. 적용된 두 가지의 추정방법은 비등분산성을 고려한 통계적 실험을 통하여 장점과 단점이 비교되었으며, 안양천 유역의 5개 지점으로부터 얻어진 유량측정성과를 이용하여 적용성을 알아보았다. 현장 적용결과는 참값을 알지 못하므로 정량적 우수성은 평가할 수 없었으나, 기존에 사용되는 불확실성 산정방법보다 Bayesian 회귀 분석 불확실성은 감소시켜 나타냄을 알 수 있었다.

Keywords

References

  1. 김상욱과 이길성 (2008a). "Bayesian MCMC를 이용한 저수량 점 빈도분석: Ⅰ. 이론적 배경과 사전분포의 구축." 한국수자원학회 논문집, 한국수자원학회, 제41권, 제1호, pp. 35-47 https://doi.org/10.3741/JKWRA.2008.41.1.035
  2. 김상욱과 이길성 (2008b). "Bayesian MCMC를 이용한 저수량 점 빈도분석: II. 적용과 비교분석." 한국수자원학회 논문집, 한국수자원학회, 제41권, 제1호, pp. 49-63 https://doi.org/10.3741/JKWRA.2008.41.1.049
  3. 김원, 김상호, 차준호, 김동구 (2001). "수위-유량관계의 정확도 분석." 대한토목학회 학술발표회 논문집, 대한토목학회, pp. 1-4
  4. 김치영, 차준호, 김동구, 김원 (2004). "하천 유량측정에서 제한된 측선수에 따른 불확실도." 한국수자원학회 학술발표회 논문집, 한국수자원학회, pp. 1086-1091
  5. 서규우, 김수현, 김대곤 (2005). "유량측정방법에 따른 형산강유량의 불확실도 분석." 한국수자원학회 학술발표회 논문집, 한국수자원학회, pp. 1538-1542
  6. 이길성 (1996). 낙동강 수계 실시간 최적 저수관리 시스템 개발 (분석모델 부문) 보고서. 한국수자원공사
  7. 이길성 (2001). 수자원 분석시스템 구축기법에 관한 연구보고서. 한국수자원공사
  8. 이길성, 이경호 (2004). "유량측정시 불확실성의 검토와 수위-유량곡선식의 개발." 한국수자원학회 학술발표회 논문집, 한국수자원학회, pp. 1242-1246
  9. 이길성 (2007). 안양천 유역의 물순환 건전화 기술개발. 서울대학교, 과학기술부
  10. 정성원, 김동희, 문장원, 김동필 (2003) "유량자료 품질개선을 위한 정확도 제고방안(수위-유량 관계곡선)." 한국수자원학회 학술발표회 논문집(1), 한국수자원학회, pp. 121-124
  11. 차준호, 김상호, 김원, 김동구 (2001). "유량측정자료의 불확실도." 대한토목학회 학술발표회 논문집, 대한토목학회, pp. 5-8
  12. 차준호, 김원, 윤광석, 김동구 (2002). “유량측정자료의 불확실도 분석." 한국수자원학회 학술발표회 논문집(2), 한국수자원학회, pp. 989-994
  13. 한국수자원공사 (2001). HYMOS. pp. 8.8-8.12
  14. 한국수자원공사 (2000, 2001, 2002, 2003, 2004, 2005, 2006). 수문자료집
  15. Bickel, P.J., and Doksum, K.A. (1977). Mathematical statistics: basic ideas and selected topics. Holden-Day, Inc., San Francisco, CA
  16. DeGagne, M.P.J., Douglas, G.G., Hudson, H.R., and Simonovic, S.P. (1996). "A decision support system for the analysis and use of stagedischarge rating curves." Journal of Hydrology, Vol. 184, pp. 225-241 https://doi.org/10.1016/0022-1694(95)02973-7
  17. Herschy, R.W. (1980). Manual on stream gauging: Computation of discharge, Vol. 2, Report No. 13, WMO
  18. Herschy, R.W. (1985). Streamflow measurement, Elsevier Applied Science Publishers. M.O
  19. ISO (1998). "Determination of the stage-discharge relationship." Measurement of liquid flow in open channels-Part 2, ISO Standard 1100-2, International Organization of Standards, pp. 133-153
  20. Kuczera, G. (1999). "Comprehensive at-site flood frequency analysis using Monte Carlo Bayesian inference." Water Resources Research, Vol. 35, No. 5, pp. 1551-1557 https://doi.org/10.1029/1999WR900012
  21. Kuczera, G., and Parent E. (1998). "Monte Carlo assessment of parameter uncertainty in conceptual catchment models: The Metropolis algorithm." Journal of Hydrology, Vol. 211, pp. 69-85 https://doi.org/10.1016/S0022-1694(98)00198-X
  22. Lambie, J.C. (1978). "Measurement of flowvelocity-area methods." In: Hershcy RW, (Ed.), Hydrometry: Principles and Practices, Wiley, Chichester, Chapter 1
  23. Lee, K.S., and Kim, S.U. (2007). "Identification of uncertainty in low flow frequency analysis using Bayesian MCMC method." Hydrological Processes, In press(on-line published)
  24. Madsen, H., and Rojsberg, H.D. (1997). "Generalized least squares and empirical Bayes estimation in regional partial duration series index flood modeling." Water Resources Research, Vol. 33, No. 4, pp. 771-781 https://doi.org/10.1029/96WR03850
  25. Mosley, M.P., and McKerchar, A.I. (1993). "Streamflow." Handbook of Hydrology, Chap. 8, McGraw-Hill. N.Y
  26. Moyeed R.A., and Clarke, R.T. (2005). "The use of Bayesian methods for fitting rating curves, with case studies." Advances in Water Resources, Vol. 28, pp. 807-818 https://doi.org/10.1016/j.advwatres.2005.02.005
  27. O'Connell, D.R.H., Ostenaa, D.A., Levish, D.R., and Klinger, R.E. (2002). "Bayesian flood frequency analysis with paleohydrologic bound data." Water Resources Research, Vol. 38, No. 5, pp. 1-14 https://doi.org/10.1029/2000WR000028
  28. Petersen-Overleir, A. (2004). "Accounting for heteroscedasticity in rating curve estimates." Journal of Hydrology, Vol. 292. pp. 173-181 https://doi.org/10.1016/j.jhydrol.2003.12.024
  29. Petersen-Overleir, A., and Reitan, T. (2005). "Objective segmentation in compound rating curves." Journal of Hydrology Vol. 311, pp. 188-201 https://doi.org/10.1016/j.jhydrol.2005.01.016
  30. Rantz, S.E. (1982). "Measurement and computation of streamflow." Vol. II. Computation of Discharge. US Geological Survey Water Supply Paper 2175, Washington
  31. Reis, D.S. Jr, and Stedinger, J.R. (2005). "Bayesian MCMC flood frequency analysis with historical information." Journal of Hydrology, Vol. 313, pp. 97-116 https://doi.org/10.1016/j.jhydrol.2005.02.028
  32. Reis, D.S. Jr, Stedinger, J.R., and Martins, E.S. (2005). "Bayesian generalized least squares regression with application to log Pearson type III regional skew estimation." Water Resources Research, Vol. 41, W10419 https://doi.org/10.1029/2004WR003445
  33. Sauer, V.B. and Meyer, R.W. (1992). Determination of error in individual discharge measurements. US Geological Survey Open-file Report, pp. 92-144
  34. Seber, G.A.F. and Wild, C.J. (1989). Nonlinear regression, John Wiley & Sons, Inc., N.Y
  35. Seidou, O., Ouarda, T.B.M.J., Barbet, M., Bruneau, P., and Bobee, B. (2006). "A parametric Bayesian combination of local and regional information in flood frequency analysis." Water Resources Research, Vol. 42, W11408, DOI: 10.1029/2005 WR004397
  36. Sorensen, D. and Gianola, D. (2002). Likelihood, Bayesian, and MCMC methods in Quantitative Genetics. Springer-Verlag, New York
  37. Thiemann, M., Trosset, M., Gupta, H.V., and Sorooshian, S. (2001). "Bayesian recursive parameter estimation for hydrologic models." Water Resources Research, Vol. 37, No. 10, pp. 2521-2535 https://doi.org/10.1029/2000WR900405
  38. Venables, W.N., and Smith, D.M. (2008). An introduction to R, R Development Core Team
  39. Vicens, G.J., Rodriguez-Iturbe, I., and Schaake, J.C. Jr. (1975). "A Bayesian framework for the use of regional information in hydrology." Water Resources Research, Vol. 11, No. 3, pp. 405-414 https://doi.org/10.1029/WR011i003p00405
  40. Vrugt, J.A., Gupta, H.V., Bouten, W., and Sorooshian, S. (2003). "Shuffled complex evolution Metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters." Water Resources Research, Vol. 39, No. 8, SWC 1-16
  41. Wood, E.F., and Rodriguez-Iturbe, I. (1975a). "Bayesian inference and decision making for extreme hydrologic events." Water Resources Research, Vol. 11, No. 4, pp. 533-542 https://doi.org/10.1029/WR011i004p00533
  42. Wood, E.F., and Rodriguez-Iturbe, I. (1975b). A Bayesian approach to analyze uncertainty among flood frequency models. Water Resources Research, Vol. 11, No. 6, pp. 839-843 https://doi.org/10.1029/WR011i006p00839
  43. WMO (1994). Guide to Hydrological Practices, 5th Ed., pp. 170-172

Cited by

  1. Prediction of Stage Discharge Curve and Lateral Distribution of Unit Discharge in an Arbitrary Cross Section Channel with Floodplain Vegetation vol.44, pp.2, 2011, https://doi.org/10.3741/JKWRA.2011.44.2.157
  2. A Probabilistic Estimation of Changing Points of Seoul Rainfall Using BH Bayesian Analysis vol.43, pp.7, 2010, https://doi.org/10.3741/JKWRA.2010.43.7.645
  3. Analysis of Uncertainty of Rainfall Frequency Analysis Including Extreme Rainfall Events vol.43, pp.4, 2010, https://doi.org/10.3741/JKWRA.2010.43.4.337
  4. A development of rating-curve using Bayesian Multi-Segmented model vol.49, pp.3, 2016, https://doi.org/10.3741/JKWRA.2016.49.3.253
  5. Improvement of the Method using the Coefficient of Variation for Automatic Multi-segmentation Method of a Rating Curve vol.48, pp.10, 2015, https://doi.org/10.3741/JKWRA.2015.48.10.807
  6. Composite Loss Rate Model Combining Four Losses of Precipitation in a Watershed for Engineering Hydrology vol.17, pp.3, 2012, https://doi.org/10.1061/(ASCE)HE.1943-5584.0000439
  7. Development of Rating Curve for High Water Level in an Urban Stream using Monte Carlo Simulation vol.33, pp.4, 2013, https://doi.org/10.12652/Ksce.2013.33.4.1433