Sang-Ug Kim

Identification of Uncertainty in Fitting Rating Curve with 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. 【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.】

Regional Low Flow Frequency Analysis Using Bayesian Multiple Regression

This study employs Bayesian multiple regression analysis using the ordinary least squares method for regional low flow frequency analysis. The parameter estimates using the Bayesian multiple regression analysis were compared to conventional analysis using the t-distribution. In these comparisons, the mean values from the t-distribution and the Bayesian analysis at each return period are not significantly different. However, the difference between upper and lower limits is remarkably reduced using the Bayesian multiple regression. Therefore, from the point of view of uncertainty analysis, Bayesian multiple regression analysis is more attractive than the conventional method based on a t-distribution because the low flow sample size at the site of interest is typically insufficient to perform low flow frequency analysis. Also, we performed low flow prediction, including confidence interval, at two ungauged catchments in the Nakdong River basin using the developed Bayesian multiple regression model. The Bayesian prediction proves effective to infer the low flow characteristic at the ungauged catchment.

At-site Low Flow Frequency Analysis Using Bayesian MCMC: I. Theoretical Background and Construction of Prior Distribution

The low flow analysis is an important part in water resources engineering. Also, the results of low flow frequency analysis can be used for design of reservoir storage, water supply planning and design, waste-load allocation, and maintenance of quantity and quality of water for irrigation and wild life conservation. Especially, for identification of the uncertainty in frequency analysis, the Bayesian approach is applied and compared with conventional methodologies in at-site low flow frequency analysis. In the first manuscript, the theoretical background for the Bayesian MCMC (Bayesian Markov Chain Monte Carlo) method and Metropolis-Hasting algorithm are studied. Two types of the prior distribution, a non-databased and a data-based prior distributions are developed and compared to perform the Bayesian MCMC method. It can be suggested that the results of a data-based prior distribution is more effective than those of a non-data-based prior distribution. The acceptance rate of the algorithm is computed to assess the effectiveness of the developed algorithm. In the second manuscript, the Bayesian MCMC method using a data-based prior distribution and MLE(Maximum Likelihood Estimation) using a quadratic approximation are performed for the at-site low flow frequency analysis. keywords : At-site low flow frequency analysis, Uncertainty, Bayesian MCMC, Prior distribution, Metropolis-Hastings algorithm, MLE, Quadratic approximation .............................................................................................................................................................................................. 요 지 저수분석(low flow analysis)은 수자원공학에서 중요한 분야 중 하나이며, 특히 저수량 빈도분석(low flow frequency analysis)의 결과는 저수(貯水)용량의 설계, 물 수급계획, 오염원의 배치 및 관개와 생태계의 보존을 위한 수량과 수질의 관리에 중요하게 사용된다. 그러므로 본 연구에서는 저수량 빈도분석을 위한 점 빈도분석을 수행하였으며, 특히 빈도분석에 있어서의 불확실 성을 탐색하기 위하여 Bayesian 방법을 적용하고 그 결과를 기존에 사용되던 불확실성 탐색방법과 비교하였다. 본 논문의I편에서는 Bayesian 방법 중 사전분포(prior distribution)와 우도함수(likelihood function)의 복잡성에 상관없 * 서울대학교 BK21 안전하고 지속가능한 사회기반건설 사업단 박사 후 연구원 Post-Doctor, Seoul National University BK21 SIR Group, Seoul National University, Seoul, 151-744, Korea (e-mail: plethor1@snu.ac.kr) ** 서울대학교 공과대학 건설.환경공학부 교수 Professor, Dept. of Civil and Environmental Engineering, Seoul National University, Seoul, 151-744, Korea (e-mail: kilselee@snu.ac.kr) DOI: 10.3741/JKWRA.2008.41.1.035

At-site Low Flow Frequency Analysis Using Bayesian MCMC: II. Application and Comparative Studies

The Bayesian MCMC(Bayesian Markov Chain Monte Carlo) and the MLE(Maximum Likelihood Estimation) methods using a quadratic approximation are applied to perform the at-site low flow frequency analysis at the 4 stage stations (Nakdong, Waegwan, Goryeonggyo, and Jindong). Using the results of two types of the estimation method, the frequency curves including uncertainty are plotted. Eight case studies using the synthetic flow data with a sample size of 100, generated from 2-parmeter Weibull distribution are performed to compare with the results of analysis using the MLE and the Bayesian MCMC. The Bayesian MCMC and the MLE are applied to 36 years of gauged data to validate the efficiency of the developed scheme. These examples illustrate the advantages of the Bayesian MCMC and the limitations of the MLE based on a quadratic approximation. From the point of view of uncertainty analysis, the Bayesian MCMC is more effective than the MLE using a quadratic approximation when the sample size is small. In particular, the Bayesian MCMC is a more attractive method than MLE based on a quadratic approximation because the sample size of low flow at the site of interest is mostly not enough to perform the low flow frequency analysis.

Improvement of Rating Curve Fitting Considering Variance Function with Pseudo-likelihood Estimation

This paper presents a technique for estimating discharge rating curve parameters. In typical practical applications, the original non-linear rating curve is transformed into a simple linear regression model by log-transforming the measurement without examining the effect of log transformation. The model of pseudo-likelihood estimation is developed in this study to deal with heteroscedasticity of residuals in the original non-linear model. The parameters of rating curves and variance functions of errors are simultaneously estimated by the pseudo-likelihood estimation(P-LE) method. Simulated annealing, a global optimization technique, is adapted to minimize the log likelihood of the weighted residuals. The P-LE model was then applied to a hypothetical site where stage-discharge data were generated by incorporating various errors. Results of the P-LE model show reduced error values and narrower confidence intervals than those of the common log-transform linear least squares(LT-LR) model. Also, the limit of water levels for segmentation of discharge rating curve is estimated in the process of P-LE using the Heaviside function. Finally, model performance of the conventional log-transformed linear regression and the developed model, P-LE are computed and compared. After statistical simulation, the developed method is then applied to the real data sets from 5 gauge stations in the Geum River basin. It can be suggested that this developed strategy is applied to real sites to successfully determine weights taking into account error distributions from the observed discharge data. keywords : Discharge rating curve, Heteroscedasticity, Pseudo-likelihood estimation, Non-linear regression, Simulated annealing, Segmentation of discharge rating curve * 한국수자원공사 조사관리처 차장 Manager, Investigation and Planning Department, Korea Water Resources Corporation, Daejeon, 306-711, Korea (e-mail: leews@kwater.or.kr) ** 서울대학교 BK21 안전하고 지속가능한 사회기반건설 사업단 박사 후 연구원(교신저자) Post-Doctor, SNU BK21 SIR Group, Seoul National University, Seoul, 151-744, Korea (e-mail: plethor1@snu.ac.kr) *** 서울대학교 공학연구소 선임연구원 Senior Researcher, Engineering Research Institute, Seoul National University, Seoul, 151-744, Korea (e-mail: cool77@snu.ac.kr) **** 서울대학교 공과대학 건설.환경공학부 교수 Professor, Dept. of Civil and Environmental Engineering, Seoul National University, Seoul, 151-744, Korea (e-mail: kilselee@snu.ac.kr) DOI: 10.3741/JKWRA.2008.41.8.807

Automatic Calibration of Rainfall-runoff Model Using Multi-objective Function

A rainfall-runoff model should be calibrated so that the model simulates the hydrological behavior of the basin as accurately as possible. In this study, to calibrate the five parameters of the SSARR model, a multi-objective function and the genetic algorithm were used. The solution of the multi-objective function will not, in general, be a single unique set of parameters but will consist of the so-called Pareto solution according to various trade-offs between the different objectives. The calibration strategy using multi-objective function could decrease calibrating time and effort. From the Pareto solution, a single solution could be selected to simulate a specific flow condition.

Analysis of uncertainty of rainfall frequency analysis including extreme rainfall events.

극치사상을 예측하기 위한 기존의 빈도분석 결과의 이용에 대한 많은 문제점들이 부각되고 있다. 특히, 통계적 모형을 이용하기 위해서 흔히 사용되는 점근적 모형 (asymptotic model)의 합리적인 검토 없는 외삽 (extrapolation)은 산정된 확률 값을 과대 또는 과소평가하는 문제를 일으켜, 예측결과에 대한 불확실성을 과다하게 산정함으로써 불확실성에 대한 신뢰도를 감소시키는 문제가 있다. 그러므로 본 연구에서는 국내에서 극치강우사상을 포함한 강우자료의 빈도분석에 대한 연구사례를 제공하고 점근적 모형을 사용하는 경우 발생되는 불확실성을 감소시키기 위한 방법론을 제시하였다. 이를 위하여 본 연구에서는 극치강우사상의 빈도분석을 수행하는 데 있어서 최근 들어 여러 분야에서 다양하게 적용되고 있는 Bayesian MCMC (Markov Chain Monte Carlo) 방법을 사용하였으며, 그 결과를 최우추정방법 (Maximum likelihood estimation method)과 비교하였다. 특히 강우사상의 점 빈도분석에 흔히 이용되는 확률밀도함수로 GEV (Generalized Extreme Value) 분포와 Gumbel 분포를 모두 고려하여 두 분포의 결과를 비교하였으며, 이 과정에서 각각의 산정결과 및 불확실성은 근사식을 이용한 최우추정방법과 Bayesian 방법을 이용하여 각각 비교 및 분석되었다. 【There is a growing dissatisfaction with use of conventional statistical methods for the prediction of extreme events. Conventional methodology for modeling extreme event consists of adopting an asymptotic model to describe stochastic variation. However asymptotically motivated models remain the centerpiece of our modeling strategy, since without such an asymptotic basis, models have no rational for extrapolation beyond the level of observed data. Also, this asymptotic models ignored or overestimate the uncertainty and finally decrease the reliability of uncertainty. Therefore this article provide the research example of the extreme rainfall event and the methodology to reduce the uncertainty. In this study, the Bayesian MCMC (Bayesian Markov Chain Monte Carlo) and the MLE (Maximum Likelihood Estimation) methods using a quadratic approximation are applied to perform the at-site rainfall frequency analysis. Especially, the GEV distribution and Gumbel distribution which frequently used distribution in the fields of rainfall frequency distribution are used and compared. Also, the results of two distribution are analyzed and compared in the aspect of uncertainty.】

Effect of Climate Change and Urbanization on Flow and BOD Concentration Duration Curves

This study developed an integrated approach to climate change and urbanization impact assessment by linking models of SDSM (statistical downscaling model), HSPF (hydrological simulation program?Fortran) and ICM (impervious cover model). A case study of the Anyangcheon watershed illustrated how the proposed framework can be used to analyze the impacts of climate change and urbanization in terms of flood control, water security and water quality. The evaluation criteria were the variations of flow and pollutant concentration duration curves. In this study, nine scenarios including three climate (present condition, A1B and A2) and three urbanization scenarios were analyzed using HSPF model. As a result, climate change is a large influence on the flowrate and the urbanization affects the pollutant concentration. Therefore, the impacts of both climate change and urbanization must be included into the watershed management and water resources planning for sustainable development.

Effectiveness Analysis of Alternatives for Water Resources Management Considering Climate Change and Urbanization

This study derived the analysis results of alternatives for integrated watershed management under urbanization and climate change scenarios. Climate change and urbanization scenarios were obtained by using SDSM (Statistical Downscaling Method) model and ICM (Impervious Cover Model), respectively. Alternatives for the Anyangcheon watershed are reuse of wastewater treatment plant effluent, and redevelopment of existing reservoir. Flow and BOD concentration duration curves were derived by using HSPF (Hydrological Simulation Program Fortran) model. As a result, low flow (Q99, Q95, Q90) and BOD concentration (C30, C10, C1) were very sensitive to the alternatives comparing to high flow(Q10, Q5, Q1). Although urbanization makes the hydrological cycle distorted, effective alternatives can reduce its damage. The numbers of days to satisfy the instreamflow requirements and target water quality were also sensitive to urbanization. This result showed that the climate change and urbanization should be considered in the water resources/watershed and environmental planning. keywords : Alternative, Anyangcheon, Climate change, HSPF, Urbanization ..............................................................................................................................................................................................