臺灣博碩士論文加值系統

本研究將以圓錐貫入試驗 (cone penetration test, CPT) 的修正之錐尖阻抗 (qt) 資料做為討論對象，探討辨識趨勢函數的可行性與方法。現地調查所獲得的空間分布資料可以被分成兩項，趨勢函數以及沿著趨勢且平均值為零的變異性，而趨勢函數可以讓工程師更輕易的了解土壤性質隨空間的變化，空間變異性可以透過標準差(σ)及關聯性長度(δ)估計。除了垂直向的關聯性長度外，本研究探討三度空間問題，所以還需要估計水平方向的關聯性長度，相比於垂直向的關聯性長度，由於土層水平方向的變異性較小，且水平的資料數量遠少於垂直方向的資料，大大增加水平向參數估計的難度。
當進入三維分析時，所需要的計算量將大幅提升，導致運算時間太長甚至超過記憶體的負荷量，所以使用了Cholesky decomposition與克羅內克積 (Kronecker product) 等數學方法，大幅減少計算量。
本研究透過兩步驟的貝氏分析架構來辨識以及模擬空間中的趨勢函數，第一步是透過sparse Bayesian learning的架構來選擇真正需要的基函數 (basis function, BF)，不同種類的基函數形式在文中也會進行探討。第二步是透過漸進式馬可夫鏈蒙地卡羅法 (transitional Markov chain Monte Carlo, TMCMC; Ching and Chen, 2007) 作為估計隨機場參數的方法，透過上述兩個步驟就能模擬出代表現地趨勢函數，接下來則可以利用第二步所取得的趨勢與關聯性參數進一步進行隨機場的模擬。

This study investigated the modified cone tip resistance (qt) data from cone penetration tests (CPT). The feasibility and method of identifying the trend function were also discussed. The vertical spatial distribution is expressed as a depth-dependent trend function and a zero-mean spatial variation. Trend function can help us catch soil properties in space. Spatial variation can be estimated by standard deviation (σ) and scale of fluctuation (δ).
In addition to the vertical scale of fluctuation, in 3D case, horizontal scale of fluctuation is also important. However, the number of horizontal data is much less than that of the vertical data. Horizontal scale of fluctuation is hard to be estimated. The estimation of the horizontal parameter is difficult. Another problem is that when analyzing multiple data at a time, the matrix becomes very huge, increasing the computation and even exceeding the load of the memory. We use Cholesky decomposition and Kronecker product to simplify the matrix. In this way, we can greatly reduce the computation.
This study uses a two-step Bayesian analysis to identify trend functions. The first step is to select the basis functions we need by sparse Bayesian learning. In this study, we also consider the effects of different kinds of basis functions. The second step is to use transitional Markov chain Monte Carlo (TMCMC; Ching and Chen, 2007) as a method for estimating the parameters of the random field. Through the above two steps, we can fit the trend function and model the random field.

目錄
致謝 I
中文摘要 II
ABSTRACT III
目錄 IV
圖目錄 VII
表目錄 XV
第一章 緒論 1
1.1 研究動機與目的 2
1.2 研究方法 5
第二章 文獻回顧 8
2.1 隨機場 (RANDOM FIELD) 8
2.1.1 穩態隨機場 (STATIONARY RANDOM FIELD) 8
2.1.2 自關聯性函數 (AUTO-CORRELATION FUNCTION, ACF) 10
2.2 WHITTLE-MATÉRN模型 11
2.3 平滑性參數的影響 12
2.4 不同型態的基函數 15
2.5 一維及三維隨機場表示方式 19
2.6 SPARSE BAYESIAN LEARNING (SBL) 22
2.7 馬可夫鏈蒙地卡羅法 24
2.8 漸進式馬可夫鏈蒙地卡羅法 26
第三章 研究方法 29
3.1 資料模擬 30
3.1.1 隨機場參數對模擬資料的影響 30
3.1.2 三維空間分布資料之模擬 32
3.2 STEP1-選擇真正需要的基函數(BFS) 35
3.2.1最大化條件證據F(Y|S, Σ, Δ, Ν, M) 35
3.2.2 三維的計算問題與新的公式推導 39
3.2.3 模擬資料辨識結果 41
3.3 STEP2-貝氏分析 (BAYESIAN ANALYSIS) 42
3.3.1 (LNΣ, LNΔV, LNΔH, LNΝV, LNΝH) 的取樣 43
3.3.2 W’的取樣 44
3.4 不同型態的基函數擬合成果 50
3.5 STEP3-建立隨機場 55
第四章 現地案例討論 56
4.1 HOLLYWOOD, SOUTH CAROLINA 56
4.1.1 STEP1分析結果 58
4.1.2 STEP2分析結果 60
4.1.3 STEP3模擬結果 67
4.2 BAYTOWN, TEXAS 68
4.2.1 STEP1分析結果 72
4.2.2 STEP2分析結果 76
4.2.3 STEP3模擬結果 89
4.3 ADELAIDE, SOUTH AUSTRALIA 90
4.3.1 STEP1分析結果 93
4.3.2 STEP2分析結果 99
第五章 結論與建議 121
參考文獻 123
附錄A 128

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