以隱馬爾可夫模型進行土壤分層

Abstract

在工址調查的作業中，調查地下土層是相當重要的環節，因為了解土層分布 就能了解地層承載力的強弱，主導後續工程之設計。調查地下土層的主要內容為取得地下土層的厚度、層面之高程與土層之土壤種類，傳統做法是進行標準貫入試驗(standard penetration test, SPT)，取出土壤樣本來判別層面與土壤種類。而圓錐貫入試驗 (cone penetration test, CPT)雖然無法取樣，但是也可以進行土壤分層，而且其施作起來比SPT更簡單、方便。CPT資料經過計算後可以得到土壤行為指數(soil behavior type index, Ic)，它被證實能根據其數值大小而有效地區分土壤種類，因此許多學者致力發展以CPT資料為基礎的土壤分層方法。 本研究發展了以隱馬爾可夫模型(hidden Markov model, HMM)配合吉布斯抽樣法(Gibbs Sampling)應用於土壤分層的分析方法，稱為HMM土壤分層法。做法為將土壤種類視為隱馬爾可夫模型的隱含狀態，將Ic視為模型之輸出序列，並以Ic作為分析資料，用一維穩態高斯隨機場(stationary Gaussian random field)描述其空間變異性。接著以貝氏定理(Bayes' theorem)為基礎，使用吉布斯抽樣法估算Ic之平均值(μ)與標準差(σ)，再根據Ic與其平均值、標準差，使用前向—後向遞迴(forward-backward recursions, FB recursions)找出每一點最可能的土壤種類，將上述步驟進行疊代計算以得到收斂的結果，以此方法找出土壤的種類與層面。最後以概似遞迴 (likelihood recursions)計算每一種分群數的似然性(likelihood)，以此找出最佳分群數。 在案例分析方面，本研究使用南卡羅萊納州好萊塢(Hollywood, South Carolina)之現地CPT資料驗證HMM土壤分層法的結果，並與表現相當穩定的另一種一維土壤分層法——小波轉換調變極大值法(WTMM法，Ching et al., 2015)進行比較與討論。結論為HMM土壤分層法的優點為分群數量可做1~10群的變化，而且能分析Ic的變化，自動將類似的土層歸類為一層。然而也發現其具有孤兒層問題、分層分數問題等問題需要後續的研究來解決。 本研究的第二部分為嘗試將WTMM法與Park (2010)開發之廣義耦合馬爾可夫鏈(generalized coupled Markov chain, GCMC)模型結合，進行二維與三維土壤分層剖面預測之案例分析，藉此探索以CPT資料建立多維土壤分層模型之可行性，並分析了南卡羅萊納州好萊塢與南澳洲阿得雷德(Adelaide, South Australia)的South Parklands兩個案例，皆得到了合理的結果。

In the work of site investigation, investigating the underground soil layers is a quite important part. Once we know the distribution of the soil layers, we could understand the strength of stratum bearing capacity which leads the design of subsequent engineering project. The main contents of investigating underground soil layers are to obtain the thickness of the layers, the elevations of the interfaces and the types of the soils. The conventional way is to perform a standard penetration test (SPT) and take out the soil samples to identify the interfaces and the types of the soil layers. Although the cone penetration test (CPT) can not sample the soils, soil stratification still can be performed based on it. And CPT is more simple and convenient than SPT. Soil behavior type index (Ic), which has been proved to be able to distinguish soil types effectively according to its value, can be calculated from CPT data. Therefore, many scholars have devoted themselves to the development of soil stratification methods based in CPT data. This study developed a method of soil stratification using the hidden Markov model (HMM) and Gibbs sampling, which is called HMM soil stratification method. The approach is to regard the soil types as the hidden states of the hidden Markov model, and to regard Ic as the output sequence of the model. Using Ic as the analytical data, describe the spatial variability of Ic with one-dimensional stationary Gaussian random field. Then based on Bayes ' theorem, the mean (μ) and the standard deviation (σ) of Ic are estimated by Gibbs sampling. According to Ic and its mean and standard deviation, use forward-backward recursions (FB recursions) to find the most likely soil type at each point. The above steps are performed for iterative calculations to obtain convergent results, and the types and interfaces of the soil layers can be found by this method. Finally, the likelihood of each number of cluster is calculated by the likelihood recursions to find the optimal number of clusters. In terms of case studied, this study used the in-situ CPT data from Hollywood, South Carolina, to verify the results of HMM soil stratification method. And another stable 1D soil stratificaiton method—the wavelet transform modulus maxima method (WTMM method, Ching et al., 2015) was performed for comparison and discussion with HMM method. The conclusion is as following: the advantages of HMM soil stratification method is that the number of clusters can be changed from 1 to 10, and HMM can analyze the change of Ic and automatically classified similar soil layers into one layer. However, it was also found that the irrational thin layer problem and the cluster scores problem need to be addressed by subsequent studies. The second part of this study is trying to combine the WTMM method with the generalized coupled Markov chain (GCMC) model developed by Park (2010). We conducted the case studies of the predictions of 2D and 3D soil stratification profiles. And we explored the feasibility of using CPT data to build a multidimensional soil stratification model and analyzed two cases in Hollywood and South Parklands in Adelaide, South Australia, respectively. Both cases have received reasonable results.