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研究生： 汪柏岑
Wang, Bo-Tsen
論文名稱： 結合拉丁超立方體抽樣法與矩陣拆解法建構空間隨機場之研究
The study of combining Latin Hypercube Sampling method and LU decomposition method (LULHS method) for constructing spatial random field
指導教授： 徐國錦
Hsu, Kuo-Chin
學位類別： 碩士
Master
系所名稱： 工學院 - 資源工程學系
Department of Resources Engineering
論文出版年： 2015
畢業學年度： 103
語文別： 英文
論文頁數： 84
中文關鍵詞： 蒙地卡羅抽樣法 、拉丁超立方抽樣法 、空間連續性 、矩陣拆解法
外文關鍵詞： Monte Carlo Sampling, Latin Hypercube Sampling, spatial structure, LU decomposition
相關次數： 點閱：120  下載：4 
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    • 在進行地下水模擬的過程中，由於並非所有的網格內皆有水力性質量測值，因此在建立水文地質模型前，必須先進行網格內的隨機變數之模擬抽樣，建立水文數值參數隨機場。透過抽樣法來產生隨機數值，並將抽樣所得的數值代表為網格內之水文數值參數，用來提供建立水文地質模型所需。因此，統計抽樣法在模擬的過程中扮演著很重要的過程。近年來，拉丁超立方體抽樣法廣泛的使用，透過分層抽樣的概念，能準確的描述統計數值特性，但是卻無法描述出原有的空間結構。本研究將針對統計數值特性及空間結構做討論，利用拉丁超立方體抽樣法快速且準確獲得統計數值特性的特徵；同時，利用矩陣拆解法拆解相關性矩陣來確保擁有原本的空間連續特性。本研究提出結合拉丁超立方體的分層抽樣法與矩陣拆解法，建構具有空間連續特性之隨機場，並發展條件模擬以及非條件兩種模擬法。利用本研究所提出的方法與其他三種模擬方法做統計數值以及空間結構的效率比較。研究模擬結果顯示出本研究所提出的方法，在條件模擬以及非條件模擬中，相較於其他三種模擬方法，可以利用較少的實現場數量達到統計數值的精準度，並同時獲得穩定的模擬效果及保有原來的空間結構。

    Groundwater modeling requires to assign hydrogeological properties to every numerical grid. Due to the lack of detailed information and the inherent spatial heterogeneity, geological properties can be treated as random variables. Hydrogeological property is assumed to be a multivariate distribution with spatial correlations. By sampling random numbers from a given statistical distribution and assigning a value to each grid, a random field for modeling can be completed. Therefore, statistics sampling plays an important role in the efficiency of modeling procedure. Latin Hypercube Sampling (LHS) is a stratified random sampling procedure that provides an efficient way to sample variables from their multivariate distributions. This study combines the the stratified random procedure from LHS and the simulation by using LU decomposition to form LULHS. Both conditional and unconditional simulations of LULHS were develpoed. The simulation efficiency and spatial correlation of LULHS are compared to the other three different simulation methods. The results show that for the conditional simulation and unconditional simulation, LULHS method is more efficient in terms of computational effort. Less realizations are required to achieve the required statistical accuracy and spatial correlation.

    Abstract I 摘要 II 誌謝 III Contents IV List of Tables VI List of Figures VII Notation X Chapter 1 Introduction 1 1.1 Background and Motivation 1 1.2 Literature Review 3 1.3 Flow Chart 5 Chapter 2 Methodology 6 2.1 Geostatistics Theory 7 2.2 Simulation 8 2.2.1 Unconditional Simulation 9 2.2.2 Conditional Simulation 11 2.3 Sequential Gaussain Simulation 13 2.4 LU Decomposition Method 15 2.4.1 Cholesky Decomposition Method 17 2.5 Latin Hypercube Sampling Method 18 2.5.1 ZP Method 21 2.6 LU Decomposition Latin Hypercube Sampling Method (LULHS) 23 Chapter 3 Unconditional Simulation 24 3.1 Example for demonstration 24 3.2 Comparison of the Results 25 Chapter 4 Conditional Simulation 42 4.1 Example for demonstration 42 4.2 Comparison of the Results 45 Chapter 5 Conclusions and Suggestions 48 5.1 Conclusions 48 5.2 Suggestions 49 References 50 Appendix A Statistics of log-conductivity for famous hydrogeological experimental sites 55 Appendix B Example for ZP method 56 Appendix C Unconditional simulation results for variable C and D 61 Appendix D Unconditional simulation results for the variance of 1 and 2 72 Appendix E The data of the conditional points and its coordinate. 84

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