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Uncertainty Analysis of Stage-Discharge Curve Using Bayesian and Bootstrap Methods

Bayesian과 Bootstrap 방법을 이용한 수위-유량 관계곡선의 불확실성 분석

  • Received : 2019.03.19
  • Accepted : 2019.05.10
  • Published : 2019.05.31

Abstract

The objective of this study is to reduce the uncertainty of the river discharge estimation method using the stage-discharge relation curve. It is necessary to consider the quantitative and accurate estimation method because the river discharge data is essential data for hydrological interpretation and water resource management. For this purpose, the parameters estimated by Bayesian and Bootstrap methods are compared with the ones obtained by stage-discharge relation curve. In addition, the Bayesian and Bootstrap methods are applied to assess uncertainty and then those are compared with the confidence intervals of the results from standard error method which has t-distribution. From the results of this study, The estimated value of the regression analysis developed through this study is less than 1 ~ 5%. Also It is confirmed that there are some areas where the applicability is better than the existing one according to the water level at each point. Therefore, if we use more suitable method according to the river characteristics, we could obtain more reliable discharge with less uncertainty.

본 연구는 수위-유량 관계곡선을 이용한 하천 유량 산정방법의 불확실성을 감소시키는 것을 목적으로 하였다. 하천 유량 자료는 수문해석과 수자원 관리를 하는데 있어서 필수적으로 요구되는 자료이기 때문에 정량적으로 정확한 산정 방법을 고찰할 필요가 있다. 이를 위해 Bayesian 및 Bootstrap 방법을 이용한 수위-유량 관계식의 매개변수와 기존의 매개변수를 비교하였으며, 불확실성을 평가하기 위해서 표준오차법에 t-분포를 적용한 추정치 결과의 신뢰구간을 비교하였다. 그 결과, 본 연구를 통해 개발된 회귀분석에 의한 추정값은 약 1~5 %미만의 차이가 보이며, 각 지점에서 수위에 따라 기존보다 더 적용성이 우수한 결과를 보이는 부분도 존재함을 확인하였다. 따라서 본 연구에서 제시한 방법별로 하천의 특성 및 수위에 맞게 적용한다면 보다 더 신뢰성 있는 유량 자료를 확보할 수 있을 것으로 생각된다.

Keywords

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Fig. 1. The stage-discharge Curves for water level stations

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Table 1. The stage-discharge curve equations for water level stations

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Fig. 2. Confidence interval for uncertainty of stage-discharge curve obtained by Bayesian method

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Fig. 3. Confidence interval for uncertainty of stage-discharge curve obtained by Bootstrap method

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Fig. 4. Confidence interval for uncertainty of rating curve obtained by each method

Table 2. The stage-discharge curve equations obtained by Bayesian method

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Table 3. The stage-discharge curve equations obtained by Bootstrap method

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Table 4. Statistical error evaluation for discharges obtained from the stage-discharge curves (Gilan)

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Table 5. Statistical error evaluation for discharges obtained from the stage-discharge curves (Docheon)

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Table 6. Statistical error evaluation for discharges obtained from the stage-discharge curves (Sanyang)

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References

  1. Arnason, S.(2005), Estimating nonlinear hydrological rating curves and discharge using the Bayesian approach, Master's Thesis, University of Iceland, Reykjavík, Iceland.
  2. Coz, J.Le., Renard, B., Bonnifait, L., Branger, F. and Boursicaudr R.Le.(2014), Combining hydraulic knowledge and uncertain gaugings tn the estimation of hydrometric rating curve: A Bayesian approach, Journal of Hydrology, Vol.509, pp.573-587. https://doi.org/10.1016/j.jhydrol.2013.11.016
  3. Efron, B.(1979), Bootstrap Methods: Another Look at the Jackknife, The Annals of Statistics, Vol. 7, No. 1, pp.1-26. https://doi.org/10.1214/aos/1176344552
  4. Efron, B. and Tibshirani, R.(1986), Bootstrap Methods for Standard Errors, confidence Intervals, and Other Measures of Statistical Accuacy, Statistical Science, Vol.1, No.1, pp.54-75. https://www.jstor.org/stable/2245500 https://doi.org/10.1214/ss/1177013815
  5. Efron, B. and Tibshirani, R.J.(1994), An Introduction to the Bootstrap, Chapman and Hall/CRC, UK.
  6. Gelman, A., Carlin, J.B., Sternm H.S. and Rubin, D.B.(2004), Bayesian data analysis 2th, Chapman & Hall, CRC.
  7. Hwang, SJ, Kim, SE, Lee, KS, Lee, JW, Jung, SW(2009). Relative Analysis of Computed Discharge and Actual Survey Discharge in Nak Dong Basin, 2009 Korea Water Resources Association Annual Conference, pp.1072-1077.
  8. Ingimarsson, K.M., Hrafnkelesson, B., Gardarsson, S.M. and Snorrason, A.(2010), Bayesian discharge rating curves based on B-spline smoothing functions, Hydrology and Earth System Sciences Discussions, Vol.7, pp.2747-2780. https://doi.org/10.5194/hessd-7-2747-2010
  9. ISO 1100-2(1998), Measurement of liquid flow in open channels: Part2 Determination of the stage-discharge relationship.
  10. Jhun, MS(1990). A Computer Intensive Method for Modern Statistical Data Analysis I ; Bootstrap Method and Its Applications, The Korean journal of applied statistics, Vol.3, No.1, pp.121-141.
  11. Kim, HS(2010). Hydrology, DongHwa Technology. [Korean Literature]
  12. Kim, JS, Yoon, SK, Moon, YI(2013). Development of Rating Curve for High Water Level in an Urban Stream using Monte Carlo Simulation, Journal of the Korean Society of Civil Engineers, Vol.33, No.4, pp.1433-1446. https://dx.doi.org/10.12652/Ksce.2013.33.4.1433
  13. Kim, JY, Kim, JG, Lee, JC, Kwon, HH(2016). A development of rating-curve using Bayesian Multi-Segmented model, Journal of Korea Water Resources Association, Vol.49, No.3, pp.253-262. https://doi.org/10.3741/JKWRA.2016.49.3.253
  14. Kim, SU, Lee, KS(2008). Identification of Uncertainty in Fitting Rating Curve with Bayesian Regression, Journal of Korea Water Resources Association, Vol.41, No.9, pp.943-958. https://doi.org/10.3741/JKWRA.2008.41.9.943
  15. Kim, YS, Kim, TG, Kim, HS, Noh, HS, Jang, DW(2018). Frequency Analysis Using Bootstrap Method and SIR Algorithm for Prevention of Natural Disasters, Journal of Wetlands Research, Vol. 20, No. 2, pp. 105-115. https://doi.org/10.17663/JWR.2018.20.2.105
  16. Kwon, HH, Kim, JG, Lee, JS, Na, BK(2012). Uncertainty Assessment of Single Event Rainfall-Runoff Model Using Bayesian Model, Journal of Korea Water Resources Association, Vol.25, No.5, pp.505-516. https://doi.org/10.3741/JKWRA.2012.45.5.505
  17. Kwon, HH, Moon, YI(2004). Application of Bootstrap Confidence Limit Estimation Methodology of Hydrologic Time Series, Journal of the Korean Society of Civil Engineers, Vol.24, No.6, pp.567-576.
  18. Kwon, HH, Moon, YI, Choi, BK, Kim, SM(2008). Derivation and Uncertainty Analysis of Rating Curve Using Hierarchical Bayesian Model, Journal of Korea Water Resources Association, pp.1211-1214.
  19. Lee, KH, Lee, JK, Kim, SJ, Kim, HS(2011). Uncertainty Analysis of Flood Damage Estimation Using Bootstrap Method and SIR Algorithm, Journal of Wetlands Research, Vol. 13, No. 1, pp. 53-66. https://doi.org/10.17663/JWR.2011.13.1.053
  20. Lee, MW, Yi, CS, Kim, HS, Shim, MP(2005). Rainfall Frequency Determination by Bootstrap Method and SIR Algorithm and Risk Analysis, Journal of the Korean Society of Civil Engineers, Vol.25, No.5B, pp.365-373.
  21. Lee, SH(2012). Rating Method Using Improved Stage-Fall-Discharge Relationships in the Backwater Affected River, Ph.D. Dissertation, Konkuk university, Seoul, Korea. [Korea Literature]
  22. Livestock Research Institute(1999). Gibbs Sampling in Quantitative Genetics, 31225-51890-37-9902, Livestock Research Institute [Korean Literature]
  23. Moon, KH(2010). Rainfall Frequency Analysis Considering Likelihood of Extreme Rainfall in SIR Algorithm, Master's Thesis, Inha University, Incheon, Korea. [Korea Literature]
  24. Moyeed, R.A. and Clarke, R.T.(2005), The use of Bayesian methods for fitting rating curves with case studies, Advance in Water Resources, Vol.28, pp.807-818. https://doi.org/10.1016/j.advwatres.2005.02.005
  25. Noortwijk, J.M., Kalk, H.J. and Chbab, E.H.(2003), Bayesian Computation of Design Discharges, European Safety and Reliability Conference.
  26. Reitan, T. and Petersen-Overleir, A.(2008), Bayesian methods for estimating multi-segment discharge rating curves, Stochastic Environmental Research and Risk Assessment, Vol.23, No.5, pp.627-642. https://doi.org/10.1007/s00477-008-0248-0.
  27. Rustomji, P., & Wilkinson, S. N. (2008)., Applying bootstrap resampling to quantify uncertainty in fluvial suspended sediment loads estimated using rating curves, Water Resources Research, Vol. 44(9), https://doi.org/10.1029/2007WR006088.
  28. Vigiak, O., & Bende-Michl, U. (2013). Estimating bootstrap and Bayesian prediction intervals for constituent load rating curves. Water Resources Research, Vol. 49(12), pp.8565-8578. https://doi.org/10.1002/2013WR013559.
  29. Westerberg, I., Guerrero, J. L., Seibert, J., Beven, K. J., & Halldin, S. (2011). Stage-discharge uncertainty derived with a non-stationary rating curve in the Choluteca River, Honduras. Hydrological Processes, 25(4), 603-613. https://doi.org/10.1002/hyp.7848.
  30. Woo, KT(2012). Uncertainty Assessment of the Rating Curve, Master's Thesis, Gyeongsang National University, Gyeongsangnam-do, Jinju-si, Korea. [Korea Literature]
  31. Yoon, YN(2007). Basics and Applications of Hydrology, Cheongmon. [Korean Literature]