以連續時間馬可夫鏈建立之序率輸砂模式

Abstract

為描述泥沙運動過程的間歇性特點，本研究應用序率方法，致力提出一周全的馬可夫鏈模式，以模擬水流中複雜的泥沙交換過程。我們著重於二部份，一為連續時間馬可夫鏈的應用，另一為三個狀態輸砂模式的建立。連續時間的馬可夫鏈能展現泥沙運動連續發生之行為過程，為確立連續時間馬可夫鏈應用於泥沙交換問題的數學處理方式，首先以一具兩個狀態的馬可夫鏈模擬泥沙起動過程，釐清連續時間馬可夫鏈之混淆定義；隨後，延續所確立之連續時間馬可夫鏈，提出三個狀態、連續時間之馬可夫輸砂模式，以描述泥沙運動中靜止、推移和懸移的三種狀態，由於一般推移質輸砂模式只考慮底床之泥沙交換，忽略懸移質對推移質輸砂量可能之影響，本研究增加懸移質之考慮，描述水流底部泥沙在三種不同狀態下交換之完整過程，進而發展一更準確之推移質輸砂模式。此連續時間、三個狀態之馬可夫鏈輸砂模式中，因其連續時間馬可夫鏈的應用，能夠求得狀態轉移機率對時間的函數之解析解，故不僅能推算達穩定平衡之推移質輸砂率，亦能求得達穩定狀態之前隨時間變換之輸砂率；另外，三個狀態之輸砂模式能夠表示一河段受推移質或懸移質的支配情形，以及三狀態泥沙的數量分佈。針對所提出的推移質輸砂模式所作分析驗證，其結果證明了此模式具有一定程度的有效性。

Based on the stochastic approach, a comprehensive Markov chain model for the sediment interchange process is performed in this study. The continuous-time Markov process is appropriate for describing the continuous behavior of particle movement under steady flow conditions. A certain mathematical formulation in employing the continuous-time Markov process is constructed by the presentation of a two-state Markov model for sediment entrainment. Then we accordingly propose a three-state continuous-time Markov model that completely simulates sediment exchange and thus can quantify the bedload transport rate more accurately. The three-state Markov model describes the particle movement across three motion states, i.e. bed material, bedload, and suspended load. We have added a third state to account for the influence of suspended load, which is different from general bedload transport studies. With the employment of the continuous-time Markov model, the bedload transport capacity in the long run and the varying transport rates with time before the steady state can be derived. On the other hand, the three-state model exhibits that the flow is subject to the bedload or suspended load and further the actual sediment distribution in three motion states. The proposed model is validated against the natural river data. The comparison has shown a reasonably good agreement and thus the validity is confirmed to some extent.