事件最大值數列與混合分布在降雨頻率分析之應用

Abstract

在水利工程與風險評估中降雨頻率分析是不可或缺的一步，其中年最大值數列法因為其簡單快速的步驟而最廣為使用。年最大值數列法以極端值理論 (extremal types theorem) 為基礎，使用廣義極端值分布 (Generalized Extreme Value Distribution, GEV) 配適年最大值數列並進行估計。然而年最大值數列法每年僅選取一筆資料的作法可能讓樣本數過少，不但增加估計的不確定性也使其容易受到極端值影響。除此之外，年最大值數列法在抽樣過程中沒有考慮降雨事件與降雨類型，因此實際上並不符合極端值理論的假設，不應以廣義極端值分布進行配適。考量以上因素，本研究提出基於降雨事件來進行估計的事件最大值數列法作為年最大值數列法的替代。事件最大值數列法將臺灣的降雨事件分為颱風、梅雨、對流雨和鋒面雨，各類型降雨的年最大值分布可以由事件數分布與事件降雨量分布混合而成，而考慮所有事件下的年最大值降雨分布則為各類型降雨年最大值分布的混合分布 (mixture distribution)。本研究首先以蒙地卡羅模擬比較年最大值數列法和事件最大值數列法所估計的設計降雨，結果顯示事件最大值數列法在偏誤 (bias) 和均方根誤差 (Root Mean Squared Error, RMSE) 的表現上均較佳。除此之外在五堵、頭汴坑與嘉義雨量站的頻率分析中，事件最大值數列法因為使用所有降雨事件進行估計而受極端值的影響較低，也不會選取跨事件樣本而高估設計降雨。尖峰流量分析結果顯示事件最大值數列法的年最大值降雨樣本之尖峰流量將大於等於年最大值數列法的樣本之尖峰流量，因此事件最大值數列法所估計的年最大值降雨更符合設計降雨的需求。兩種方法以拔靴法 (bootstrap) 所建立的信賴區間在最後進行模擬與實際資料比較，模擬結果顯示雖然均無法達到理論覆蓋率，事件最大值數列法的信賴區間覆蓋率仍然比年最大值數列法更接近理論值，並且隨著重現期 (return period) 而變化的程度較低。

The Annual Maximum Series (AMS) method is a conventional way of conducting rainfall frequency analysis, which plays a crucial role in hydrology engineering in terms of hydrological risk assessment. Given any design duration, the method retrieves only the maximum rainfall within a year and approximate the Annual Maximum Rainfall (AMR) distribution by the Generalized Extreme Value (GEV) distribution according to the Extremal Types Theorem. However, the GEV approximation is inappropriate since AMS is prone to have insufficient sample size and does not take storm events and storm types into account. To overcome the above problems, the Event Maximum Series (EMS) method is proposed. The EMS method classifies storm events in Taiwan into Typhoon, Meiyu, frontal rain and convective storm. The AMR distribution of a given storm type can be derived from the corresponded event occurrence distribution and event rainfall distribution, and the AMR distribution of all events is a mixture distribution of different types of AMR distribution. As a result, the EMS method provides a more suitable and effective design-rainfall than the traditional approach. The EMS method outperforms the AMS approach in many ways. In Monte Carlo simulation, the EMS method is superior to AMS method in terms of the bias and Root Mean Squared Error (RMSE). Three stations in Taiwan are selected for frequency analysis and peak flow analysis, the results show that EMS method can avoid overestimation, capture larger peak flow events and is less affected by outliers. Finally, simulation and real data analyses of confidence interval (CI) through bootstrap method are performed. Although CI of both method does not achieve the theoretical coverage rate, the coverage rate of EMS method is more stable in different return period.