Journal of Korea Water Resources Association (한국수자원학회논문집)

Korea Water Resources Association (한국수자원학회)
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At-site Low Flow Frequency Analysis Using Bayesian MCMC: I. Theoretical Background and Construction of Prior Distribution

Bayesian MCMC를 이용한 저수량 점 빈도분석: I. 이론적 배경과 사전분포의 구축

  • Kim, Sang-Ug (Seoul National University 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 : 2008.01.31
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Abstract

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-data- based 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.

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 불확실성을 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 상관없이 계산이 가능한 Bayesian MCMC(Bayesian Markov Chain Monte Carlo) 방법과 Metropolis-Hastings 알고리즘을 사용하기 위한 여러 과정의 이론적 배경과 Bayesian 방법에서 가장 중요한 요소인 사전분포를 구축하고 이를 비교 및 평가하였다. 고려된 사전분포는 자료에 기반하지 않은 사전분포와 자료에 기반한 사전분포로써 두 사전분포를 이용하여 Metropolis-Hastings 알고리즘을 수행하고 그 결과를 비교하여 저수량 빈도분석에 합리적인 사전분포를 선정하였다. 또한 알고리즘의 수행과정에서 필요한 제안분포(proposal distribution)를 적용하여 그에 따른 알고리즘의 효율성을 채택률(acceptance rate)을 산정하여 검증해 보았다. 사전분포의 분석 결과, 자료에 기반한 사전분포가 자료에 기반하지 않은 사전분포보다 정확성 및 불확실성의 표현에 있어서 우수한 결과를 제시하는 것을 확인할 수 있었고, 채택률을 이용한 알고리즘의 효용성 역시 기존 연구자들이 제시하였던 만족스러운 범위를 가지는 것을 알 수 있었다. 최종적으로 선정된 사전분포는 본 연구의 II편에서 Bayesian MCMC방법의 사전분포로 이용되었으며, 그 결과를 기존 불확실성의 추정방법의 하나인 2차 근사식을 이용한 최우추정(maximum likelihood estimation)방법의 결과와 비교하였다.

Keywords

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