馬可夫鏈蒙地卡羅法應用於河川砂礫啟動模式參數不確定性之降低

Abstract

本研究探討馬可夫鏈蒙地卡羅法應用於河川砂礫啟動模式參數不確定性降低之效果。馬可夫鏈蒙地卡羅法是一種具高計算效率且適用在高維度問題的數值貝氏推論，本研究建立模式參數與輸出結果之貝氏架構，並利用實測資料及馬可夫鏈蒙地卡羅數值模擬進行參數更新。研究結果顯示參數事後分布的範圍較事前分布為小，表示不確定性降低。本研究探討不同鏈數及起始值對收斂性的影響，結果顯示採用雙鏈與三鏈的事後分布差異不大；本研究亦探討不同數量資料進行更新之效應，結果顯示當資料愈多時，不確定性降低的效果愈顯著；而分別更新單一參數與多個參數時，發現後者之事後分布較不會受到模式誤差的影響，且其結果與其它經驗公式所計算的值相近。為提升河川砂礫啟動模式的計算速度，本研究採用蒙地卡羅積分取代傳統數值積分，結果顯示蒙地卡羅積分能有效縮減模式計算時間，當樣本數愈多時，誤差愈小，但計算時間較長，本研究利用妥協規劃優選最佳樣本數。

This study investigates the application of Markov Chain Monte Carlo (MCMC) methodology to reduction of parameter uncertainty in a sediment entrainment model. Markov Chain Monte Carlo methodology is efficient for numerical Bayesian inference, particularly in high-dimension problems. In this work, a Bayesian framework of model parameters and outputs has been developed for updating model parameters by Markov Chain Monte Carlo methodology using the measured data. The results show that the parameter posterior distribution has a narrower range than the prior distribution, which means the parameter uncertainty is reduced. This study investigates the effect of different chain numbers and starting values on the convergence of MCMC. Little difference was observed between the results of two chains and three chains. This study also investigates the effect of different amount of data. Results show that the more data available, the more effective by the uncertainty is reduced. It is found that the posterior obtained by multiple-parameter updating is similar to those calculated by empirical formula and less prone to model inaccuracy than the posterior by single-parameter updating. To accelerate the computation speed, this study applies Monte Carlo integration to replace the traditional numerical adaptive quadrature. Results show that Monte Carlo integration effectively reduces computation time within the tolerance. As the sample number increases, the error decreases but the run time increases. This study uses the compromise programming to optimize the sample number.