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研究生:韓樂懷
研究生(外文):LE HOAI HAN
論文名稱:馬爾可夫隨機場中使用的空間相關因子確定上的尺度效應
論文名稱(外文):Scale effect on the determination of spatial correlation factor used in Markov random field
指導教授:董家鈞董家鈞引用關係盧育辰
指導教授(外文):Jia-Jyun DongYu-Chen Lu
學位類別:碩士
校院名稱:國立中央大學
系所名稱:應用地質研究所
學門:自然科學學門
學類:地球科學學類
論文種類:學術論文
論文出版年:2024
畢業學年度:112
語文別:英文
論文頁數:148
中文關鍵詞:馬爾科夫隨機場地層模型不確定性尺度效應空間相關因子代表性 基本尺寸
外文關鍵詞:Markov random fieldStratigraphic model uncertaintyScale effectSpatial correlation factorRepresentative elementary size
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地質模型在地球科學、地質學和岩土工程中扮演了基礎性的角色。然而,並不總是能獲得100%正確的地質模型,而且地質模型的不確定性難以量化。最近,隨機馬爾可夫隨機場(MRF)被用於模擬一系列概率地質模型,這可以為評估地質不確定性提供信息。MRF 模擬中的主要參數之一是空間相關因子,記為a,它控制著地質模型的幾何形態並評估其不確定性。在實際操作中,為特定地點校準MRF模擬的a 是一個困難的過程,並且可能會因地質剖面的採樣尺度和鑽孔密度(定義為單位採樣剖面長度上的鑽孔數)的不同而有所不同。雖然一些研究已經探討了a 對MRF 模擬的影響並提出了校準方法,但很少有文獻討論改變採樣剖面大小和鑽孔密度對確定a 的影響。
本研究通過研究不同剖面大小和鑽孔密度對馬爾可夫隨機場(MRF)中空間相關因子(a)的尺度效應,呈現了其影響。在台北盆地,使用兩個地質剖面創建了合成種群剖面。一個是N-S 方向的地質剖面,另一個是E-W方向的地質剖面。首先,分別基於N-S 剖面和E-W剖面的校準鑽孔數據,使用MRF 生成了兩個合成地 質模型(SGM)。然後,通過從N-S 剖面和E-W剖面的SGM 中均勻切割相等大小的剖面,得到一系列採樣剖面。在採樣剖面中均勻分佈了一組條件和觀測鑽孔。條
件鑽孔用於MRF 模擬,觀測鑽孔用於通過比較在觀測鑽孔處不同a 值下MRF 預測的可能性來確定a。選擇具有MRF 預測最大可能性的a 作為該剖面的空間相關長度。對於每個a、鑽孔密度和採樣剖面,生成了一千個地質模型實現。然後,對每個鑽孔密度和採樣剖面計算了a 的平均值、變異係數(COV)和95%置信區間。
研究結果總結如下。首先,對於N-S 剖面和E-W剖面,不同採樣尺寸下a 的平均值變化不大。在兩個剖面中,這些值大致範圍在77 米到89 米之間。這表明在台北盆地,a 的分布可能呈現向異性。其次,對於兩個剖面,各種a 值的變異係數(COV)隨著採樣尺寸的增加而減小。可以進一步基於可接受的a 的COV 確定a 的代表性iv基本尺寸(RES)。第三,受鑽孔密度影響的a,隨著鑽孔密度減小而增加。根據結果,建議該地點的鑽孔密度為每公里7 到10 個鑽孔,相應於每個鑽孔之間的間隔為100 到140 米。
The geological model plays a fundamental role in earth science, geology, and geotechnical engineering. However, a 100% correct geological model is not always obtained, and geological model uncertainty is difficult to quantify. Recently, stochastic Markov random field (MRF) has been used to simulate a series of probabilistic geological models, which can provide information for evaluating geological uncertainty. One of the main parameters in MRF simulations is the spatial correlation factor, denoted as a, which
controls the geometry and evaluates the uncertainty of the geological model. In practice, calibrating a for MRF simulations for a given site is a difficult process, and it may be different depending on the sampling scales of the geological profiles and borehole densities (defined as the number of boreholes per unit length of a sampling profile). Although some studies have already investigated the influence of a on MRF simulations and proposed calibration methods for it, there is seldom literature discussing the effects of changing the sampling profile size and borehole density on determining a.

This study presents the scale effect of the spatial correlation factor (a) when using MRF by studying the impact of various profile sizes and borehole densities on its
determination. Two geological profiles are used to create the synthetic population profiles in the Taipei Basin. One is a geological profile in the N-S direction, and the other is a geological profile in the E-W direction. First, the two synthetic geological models (SGM) were generated using MRF, based on the calibrated borehole data for N-S profile and E-W profile, respectively. Then, a series of sampling profiles were obtained by cutting equalsized profiles uniformly from the SGMs of N-S profile and E-W profile, respectively. A set
of conditional and observational boreholes are uniformly dispersed in the sampling profiles. Conditional boreholes are used for MRF simulations, and observational boreholes are used for the determination of a by comparing the likelihoods of MRF predictions under various a values at the observational boreholes. The a with the maximum likelihood of MRF prediction was selected for the spatial correlation length for that profile. One-thousand geological model realizations were generated by MRF for each a, borehole density, and sampling profile. The mean, coefficient of variation (COV), and 95% confidence interval of a were then calculated for each borehole density and sampling profile.

The following findings are drawn. Firstly, for N-S profile and E-W profile, the means of a doesn't have significant changes for the different sampling sizes. In both profiles,
the values range approximately from 77m to 89m. This shows that the distribution of a may present isotropy in the Taipei Basin. Secondly, the coefficient of variation (COV) of various a values decreases with increasing sampling size for both profiles. The representative elementary sizes (RES) of a for both profiles could be further determined based on the acceptable COV of a. Thirdly, for a influenced by borehole density, a increasing with decreasing borehole density. According to the results, the recommended borehole density for this site is 7 to 10 boreholes per kilometer, corresponding to an interval of 100 to 140 meters between each borehole.
Abstract i
摘要 iii
Acknowledgements v
Table of Contents viii
List of Figures xi
List of Tables xviii
Explanation of Symbols xix
List of Abbreviations xxi
Chapter 1: Introduction 1
Chapter 2: Data Collection and Processing 8
2.1 Study site 8
2.2 Data processing 13
2.2.1 The Central Geological Survey (CGS) databank 13
2.2.2 Studied two profiles (NS and EW) and used boreholes 14
2.2.3 The input data for the Markov random field simulation 22
Chapter 3: Methodology 23
3.1 Simulating the Synthetic Geological Model (SGM) in N-S and E-W profiles via MRF by calibrating the spatial correlation factor 23
3.1.1 Setting the boreholes for MRF simulation in two profiles 23
3.1.2 Simulating potential geological models using MRF in the N-S and E-W profiles for a given spatial correlation factor 26
3.1.3 Calculating the accuracy of MRF simulation 27
3.1.4 Determining the most probable spatial correlation factor using the “maximum likelihood principle” 28
3.1.5 Establishing the SGMs based on the spatial correlation factor obtained 29
3.2 Simulating geological models of sampling window sizes from SGMs by using MRF via calibrating the spatial correlation factor 29
3.2.1 Setting the sampling window on SGMs 29
3.2.2 Drilling virtual boreholes in a sampling window 31
3.2.3 Simulating potential geological models using MRF for a sampling window for a given spatial correlation factor 31
3.2.4 Calculating the accuracy of MRF simulation 31
3.2.5 Determining the most probable spatial correlation factor for a sampling window using the “maximum likelihood principle” 31
3.3 Determining the Representative Elementary Size (RES) for various sampling window sizes in the calibration of the spatial correlation factor 31
3.4 Investigating the influence of borehole density on spatial correlation factors and recommending borehole densities for the E-W profile 32
Chapter 4: Research Results 35
4.1 The Synthetic Geological Model (SGM) in N-S and E-W profiles via MRF by calibrating the spatial correlation factor 35
4.1.1 The Synthetic Geological Model in N-S profile 35
4.1.2 The Synthetic Geological Model in E-W profile 40
4.2 The appropriate spatial correlation factor of SGMs with different sampling window sizes 43
4.2.1 The appropriate spatial correlation factor of SGM in the N-S profile 44
4.2.2 The appropriate spatial correlation factor of SGM in the E-W profile 55
4.3 The Representative Elementary Size (RES) for various sampling window sizes in the calibration of the spatial correlation factor 65
4.4 The influence of borehole densities and sampling window lengths to spatial correlation factor 67
Chapter 5: Discussions 70
Chapter 6: Conclusions 73
Reference 74
Appendix I: Markov random field 76
Appendix II: List of the “first type borehole” used for the analysis in two profiles 80
Appendix III: List of the “second type borehole” used to check the extension of soil boundaries in two profiles 84
Appendix IV: The realizations of MRF simulation and the log-likelihood of accuracy in the N-S profile 87
Appendix V: The realizations of MRF simulation and the log-likelihood of accuracy in the E-W profile 103
Appendix VIII: Conference achievements 119
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