地震，是目前最不可預測及有可能造成重大傷亡的天然災害，預防地震所造成的危害及了解地震是國人最重要的課題之一。地震又是如何發生的？當地底下應力累積超過岩石強度，造成岩石或岩層破裂進而產生錯動發生地震，然而實際地下應力累積方式，及岩石強度臨界值是否隨時間有所變化，何時能誘發地震皆無法確切得知，倘若能進一步了解地震，如獲得地震發生時的應力規模，可提供地下岩體材料係數額外之資訊，則可達到防災減災之目的。目前關於地震應力研究中，尚無有效地估算地下應力規模之方法，因此本研究之主要工作為發展新的應力規模計算方法，將不同深度的縮減應力張量(Reduced stress tensor)作為觀測資料，逆推三軸主應力規模與方向隨深度之分布，建立地下應力模型。
在本研究中，建立兩種應力模型：線性固定模型與線性旋轉模型，分別以地下不同深度之三軸主應力比值(Stress ratio, φ)與六個獨立方向的正應力比值(Normal stress ratio, RSi)作為觀測資料，使用網格搜尋(Grid search)方式進行逆推，計算三軸主應力規模與方向隨深度之變化關係。
以合成資料檢驗兩種模型在不同情境下之可行性，其結果顯示線性固定模型在不同情境之最小誤差解，因軸差應力(Differential stress)無法確定而出現多重解；線性旋轉模型在給定的情境，皆能獲得唯一的最小誤差解，且與觀測資料相符。
最後，以實際量測非彈性應變恢復法(Anelastic strain recovery, ASR)現地應力資料檢驗線性旋轉模型，在不同範數(norm)之誤差函數下進行測試。L1 norm 與L2 norm兩誤差函數之最小誤差結果，在各深度之三軸主應力方向並無明顯差異。模型使用L2 norm誤差函數進行計算，其模型預測最小誤差結果，在應力規模隨深度之分布與觀測資料不相符；而模型使用L1 norm誤差函數之測試中，模型預測最小誤差結果，在應力規模隨深度之分布在觀測資料95%信賴區間內。綜合上述分析，以線性旋轉模型，配合L1 norm誤差函數，使用不同深度之六方向正應力比值做觀測資料，可有效評估三軸主應力規模與方向隨深度之分布。

Earthquake, the most unpredictable natural disaster, may cause severe casualties and serious damage. Therefore, how to prevent the earthquake hazard becomes one of the most important issues around the world. Earthquake takes place theoretically when underground stress accumulated and exceed the rock strength, rock will rupture and slip. At the same time, rock failure will release the elastic strain energy to make earth shaking. However, we cannot predict when earthquake will happen since we still do not know how underground stress accumulate, and whether the rock strength would change with time. If we can further understand earthquake mechanism, such as obtaining the stress orientations, stress magnitudes, and gathering mechanical parameters when earthquake occurs, it will be possible to achieve the goal of preventing earthquake hazard or reducing earthquake damage. Nowadays, in term of stress research for earthquake study, we still don’t have a method to appropriately estimate the underground stress magnitude. The main task for this study is to develop a new method of determining underground stress magnitude with depth. Based on observation data of the reduced stress tensor with depth, this work develops an inversion method which is able to reconstruct 3D stress orientation and magnitude with the depth and establish the underground stress model.
In this study, we establish two stress models: Linear Constant Model and Linear Rotatable Model, respectively, which utilize three principal stress ratios (φ) and 6 normal stress ratios (RSi) in six independent directions as observational data, and conduct grid search to inverse the magnitude and orientation distribution of 3 principal stresses with depth.
We use synthetic data to test the feasibility of these two models. Test results show that Linear Constant Model has multiple solutions with same minimum error because the differential stress would not be constrained, but Linear Rotatable Model can retrieve an unique minimum error solution as same as synthetic data in all circumstances.
Finally, we use real in-situ ASR(Anelastic strain recovery) data to evaluate the Linear Rotatable Model, with error functions of different norm. The solution of 3 principal stress directions with minimum error for L1 norm and L2 norm error functions is pretty similar. In the solution with minimum error, the stress magnitude with depth predicted by the L2 norm error function disagrees with observation data. On the other hand, in the solution of minimum error, the values of stress magnitude with depth for L1 norm error function are fitted well within 95% confidence interval of observation data. In summary, based on observation data of 6 independent normal stress ratios with depth, Linear Rotatable Model with L1 norm error function is able to effectively reconstruct the orientation and magnitude distribution of 3 principal stresses with depth.

Anderson, E. M., 1951, The dynamics of faulting and dyke formation with applications to Britain, Hafner Pub. Co.
Angelier, J., 1979, Determination of the mean principal directions of stresses for a given fault population: Tectonophysics, v. 56, no. 3-4, p. T17-T26.
Angelier, J., 1984, Tectonic analysis of fault slip data sets: Journal of Geophysical Research: Solid Earth, v. 89, no. B7, p. 5835-5848.
Angelier, J., 1989, From orientation to magnitudes in paleostress determinations using fault slip data: Journal of Structural Geology, v. 11, no. 1, p. 37-50.
Baumann, H., 1993, In situ stress in the German KTB pilot hole deduced from differential strain analysis: Scientific Drilling, v. 3, p. 263-270.
Bell, J., and Gough, D., 1979, Northeast-southwest compressive stress in Alberta evidence from oil wells: Earth and planetary science letters, v. 45, no. 2, p. 475-482.
Brudy, M., and Zoback, M. D., 1999, Drilling-induced tensile wall-fractures: implications for determination of in-situ stress orientation and magnitude: International Journal of Rock Mechanics and Mining Sciences, v. 36, no. 2, p. 191-215.
Gephart, J. W., 1990, Stress and the direction of slip on fault planes: Tectonics, v. 9, no. 4, p. 845-858.
Gephart, J. W., and Forsyth, D. W., 1984, An improved method for determining the regional stress tensor using earthquake focal mechanism data: Application to the San Fernando Earthquake Sequence: Journal of Geophysical Research, v. 89, no. B11.
Grotzinger, J. P., and Jordan, T. H., 2014, Understanding Earth, New York, NY : W. H. Freeman and Company.
Hubbert, M. K., and Willis, D. G., 1972, Mechanics of hydraulic fracturing.
Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P., 1983, Optimization by simulated annealing: science, v. 220, no. 4598, p. 671-680.
Lawson, A. C., and Reid, H. F., 1908, The California Earthquake of April 18, 1906: Report of the State Earthquake Investigation Commission, Carnegie institution of Washington, v. 87.
Matsuki, K., 1991, Three-dimensional In-situ Stress Measurement With Anelastic Strain Recovery of a Rock Core, 7th ISRM Congress: Aachen, Germany, International Society for Rock Mechanics and Rock Engineering, p. 4.
Menke, W., 2012, Geophysical Data Analysis: Discrete Inverse Theory, Academic Press.
Michael, A. J., 1984, Determination of stress from slip data: faults and folds: Journal of Geophysical Research: Solid Earth, v. 89, no. B13, p. 11517-11526.
Michael, A. J., 1987, Use of focal mechanisms to determine stress: A control study: Journal of Geophysical Research, v. 92, no. B1.
Ren, N., and Roegiers, J.-C., Differential Strain Curve Analysis-A New Method for Determining the Pre-Existing In-Situ Stress State from Rock Core Measurements, in Proceedings 5th ISRM Congress1983, International Society for Rock Mechanics and Rock Engineering.
Simmons, G., Siegfried, R. W., and Feves, M., 1974, Differential strain analysis: a new method for examining cracks in rocks: Journal of geophysical Research, v. 79, no. 29, p. 4383-4385.
Suppe, J., 2014, Fluid overpressures and strength of the sedimentary upper crust: Journal of Structural Geology, v. 69, p. 481-492.
Teufel, L., Determination of in-situ stress from anelastic strain recovery measurements of oriented core, in Proceedings SPE/DOE Low Permeability Gas Reservoirs Symposium1983, Society of Petroleum Engineers.
Wu, Y.-M., Zhao, L., Chang, C.-H., and Hsu, Y.-J., 2008, Focal-mechanism determination in Taiwan by genetic algorithm: Bulletin of the Seismological Society of America, v. 98, no. 2, p. 651-661.
Zoback, M., Barton, C., Brudy, M., Castillo, D., Finkbeiner, T., Grollimund, B., Moos, D., Peska, P., Ward, C., Wiprut, D. J. I. J. o. R. M., and Sciences, M., 2003a, Determination of stress orientation and magnitude in deep wells, v. 40, no. 7-8, p. 1049-1076.
Zoback, M. D., Barton, C. A., Brudy, M., Castillo, D. A., Finkbeiner, T., Grollimund, B. R., Moos, D. B., Peska, P., Ward, C. D., and Wiprut, D. J., 2003b, Determination of stress orientation and magnitude in deep wells: International Journal of Rock Mechanics and Mining Sciences, v. 40, no. 7, p. 1049-1076.
李偉誠, 2015, 非彈性應變回復法評估花蓮和平地區及彰濱工業區之現地應力場 [碩士: 國立臺灣師範大學, 228 p.
劉建麟, 2014, 利用電測資料推估台灣彰濱地區鑽井場址的地下應力場 [碩士: 國立中央大學, 105 p.
嚴珮綺, 2012, 利用鑽井資料推估台灣新竹至台中地區的地下現地應力狀態 [碩士: 國立中央大學, 125 p.