應用雙變數截斷迦瑪分布於序率暴雨模擬

Abstract

雙變數截斷迦瑪分布(Bivariate truncated gamma distribution,簡稱BTG)為一考慮兩隨機變數之相關性和截斷值之機率分布。本研究將颱風降雨事件隨時間之變化視為一非平穩性迦瑪馬可夫歷程，並提出雙變數截斷迦瑪分布之模擬方法，應用於序率模擬繁衍颱風事件之時雨量。 首先，本研究整理常見之單變數截斷分布型式(截斷常態分布、截斷迦瑪分布)及其參數推估法。其二，提出雙變數截斷迦瑪分布之模擬流程，其是根據雙變數截斷迦瑪分布與雙變數截斷標準常態分布的一對應轉換關係，透過模擬雙變數截斷標準常態分布不同之截斷值，建立雙變數截斷標準常態分布之相關係數和雙變數標準常態分布之相關係數，其存在的一關係式(其可表示為截斷值之函數)。並結合Cheng 等人 (2001) 提出之雙變數迦瑪分布與雙變數常態分布之相關係數轉換關係，考慮在給定雙變數迦瑪分布之相關係數下，尋找一對應之雙變數截斷迦瑪分布，並可推估其參數和進行序率模擬。 其三，將上述模擬流程應用到北台灣之宜蘭雨量站，進行該站之颱風事件時雨量模擬。模擬結果顯示本研究所提出之方法可繁衍颱風事件之時雨量，其保留了歷史雨量資料之統計特性(平均值、標準差、偏度、一階自相關係數)。其方法也可作為在極端降雨下對氣候變遷之衝擊評估。

Bivariate truncated gamma (BTG) distribution is a probability distribution considering the correlation between two gamma random variables with truncation points. In this study, the temporal variation of typhoon rainfalls is modeled as a non-stationary gamma process. A bivariate truncated gamma simulation approach was proposed and, under the Markovian assumption, used for stochastic simulation of hourly rainfalls of individual typhoon events. A summarized description of parameter estimation and stochastic simulation of univariate truncated normal and gamma distributions was given. The proposed BTG simulation approach is based on a transformation between the BTG and a corresponding bivariate truncated standard normal distribution. Through stochastic simulation of bivariate truncated standard normal distribution with various truncation points, an empirical relationship, as a function of truncation points, between correlation coefficient of the bivariate standard normal distribution and correlation coefficient of bivariate truncated standard normal distribution were established. By coupling this empirical relationship and a correlation coefficients conversion between bivariate gamma distribution and bivariate standard normal distribution derived by Cheng et al. (2001), correlation coefficient of the bivariate gamma distribution, which corresponds to the bivariate truncated gamma distribution under investigation, can be estimated and used for stochastic simulation of the bivariate truncated gamma distribution. The proposed BTG simulation approach was applied to simulation of hourly rainfalls of individual typhoon events at Yilan rainfall station in northern Taiwan. The simulation results demonstrate that the proposed approach is capable of generating hourly rainfall realizations of typhoons which preserve statistical properties (mean, standard deviation, skewness and lag-1 autocorrelation coefficient) of historical rainfall data. The proposed approach can also be used for assessing the impact of climate change on rainfall extremes.