Journal of Korea Water Resources Association (한국수자원학회논문집)

Korea Water Resources Association (한국수자원학회)
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Comparison Study on the Various Forms of Scale Parameter for the Nonstationary Gumbel Model

다양한 규모매개변수를 이용한 비정상성 Gumbel 모형의 비교 연구

  • Jang, Hanjin (School of Civil and Environmental Engineering, Yonsei Univ.) ;
  • Kim, Sooyoung (School of Civil and Environmental Engineering, Yonsei Univ.) ;
  • Heo, Jun-Haeng (School of Civil and Environmental Engineering, Yonsei Univ.)
  • 장한진 (연세대학교 대학원 토목환경공학과) ;
  • 김수영 (연세대학교 대학원 토목환경공학과) ;
  • 허준행 (연세대학교 사회환경공학부 토목환경공학과)
  • Received : 2015.02.06
  • Accepted : 2015.03.24
  • Published : 2015.05.31
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Abstract

Most nonstationary frequency models are defined as the probability models containing the time-dependent parameters. For frequency analysis of annual maximum rainfall data, the Gumbel distribution is generally recommended in Korea. For the nonstationary Gumbel models, the time-dependent location and scale parameters are defined as linear and exponential relationship, respectively. The exponentially time-varying scale parameter of nonstationary Gumbel model is generally used because the scale parameter should be positive. However, the exponential form of scale parameter occasionally provides overestimated quantiles. In this study, various forms of time-varying scale parameters such as exponential, linear, and logarithmic forms were proposed and compared. The parameters were estimated based on the method of maximum likelihood. To compare the accuracy of each scale parameter, Monte Carlo simulation was performed for various conditions. Additionally, nonstationary frequency analysis was conducted for the sites which have more than 30 years data with a trend in rainfall data. As a result, nonstationary Gumbel model with exponentially time-varying scale parameter generally has the smallest root mean square error comparing with another forms.

비정상성 빈도해석을 위해 개발된 비정상성 확률분포 모형들은 대부분 매개변수에 시간항을 포함하는 형태로 정의된다. 이 중에서도 우리나라에 널리 사용되고 있는 Gumbel 모형에 대해 살펴보면, 비정상성 Gumbel 모형의 위치 및 규모매개변수는 시간에 대해 선형(linear) 및 지수(exponential) 함수의 관계를 보이는 형태로 가정한다. 규모매개변수의 지수함수의 형태는 음(-)의 값이 추정되는 것을 방지하기 위해 제안되어 널리 사용되고 있으나 이로 인해 확률수문량이 과다산정되는 문제가 발생하기도 한다. 본 연구에서는 이러한 문제를 해결하기 위해 비정상성 Gumbel 모형의 규모매개변수의 다양한 형태를 비교하고자 한다. 이를 위해 비정상성 Gumbel 모형의 규모매개변수를 지수함수, 선형, 로그 형태로 가정하여 비교하였다. 각 모형의 매개변수의 추정은 최우도법을 적용하였고 규모매개변수의 형태별 정확도 비교를 위해 모의실험을 수행하였으며, 실제 자료에 대한 적용으로 자료기간 30년 이상을 보유하면서 경향성을 가지는 강우량 자료들을 대상으로 비정상성 빈도해석을 수행하였다. 그 결과, 지수함수 형태를 가정한 규모매개변수를 가지는 비정상성 Gumbel 모형이 가장 작은 오차를 가지는 것으로 분석되었다.

Keywords

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