以貝氏分析估計三度空間中的趨勢函數

Three-dimensional probabilistic site characterization by sparse Bayesian learning

黃玟翰(Wen-Han Huang)

指導教授 ： 卿建業

國立臺灣大學/工學院/土木工程學研究所/碩士(2019年)

https://doi.org/10.6342/NTU201902309

摘要

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。

關鍵字

大地工程 ； 圓錐灌入試驗 ； 趨勢函數 ； 空間變異性 ； 隨機場

並列摘要

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.

並列關鍵字

geotechnical engineering ； cone penetration test ； trend function ； site characterizarion ； spatial variability

參考文獻

王俊翔 (民105)。根據圓錐貫入試驗資料判識土壤層面與分析工址的機率特性 (碩士論文)。國立台灣大學，台北市。

吳采容 (民106)。以有限圓錐貫入試驗估計水平方向關聯性長度 (碩士論文)。國立台灣大學，台北市。

Abramowitz, M. and Stegun, I. (1970). Handbook of Mathematical Functions. Dover, New York.

Betz, W., Papaioannou, I., and Straub, D. (2016). Transitional Markov chain Monte Carlo: observations and improvements. J. Eng. Mech., 142(5), 04016016.

Bong, T. and Stuedlein, A.W. (2017). Spatial variability of CPT parameters and silty fines in liquefiable beach sands. ASCE Journal of Geotechnical and Geoenvironmental Engineering, 143(12), 04017093.