# 3D In Situ Stress Estimation by Inverse Analysis of Tectonic Strains
## 2 Inverse Analysis for In Situ Stress Determination
### Proposed Tectonic-Strain-Based Approach
Based on aforementioned analysis, the tectonic strain tensor containing six unknown variables can be expressed as Equation (25). Therefore, for the proposed approach, the displacement boundary conditions can be applied on each lateral boundary as Equations (26)-(28), Equation (26) is for the displacement along the axis X, Equation (27) is along the axis Y, and the Equation (28) is along the axis Z on each lateral boundary ABEF, BCDE, CODG and  
Figure 4: Calculating process of the proposed method. AOGF in Figure 3. Therefore, in this approach the displacement boundary mode for the 3-D calculating model can be summarized as Equation (29):  
\[\left[ \begin{array}{ccc}\epsilon_{\mathbf{x}}^{\mathbf{t}}&\epsilon_{\mathbf{y}}^{\mathbf{t}}& \epsilon_{\mathbf{z}x}^{\mathbf{t}}\\ \epsilon_{\mathbf{yx}}^{\mathbf{t}}&\epsilon_{\mathbf{y}}^{\mathbf{t}}&\epsilon_{\mathbf{yz}}^{\mathbf{ t}}\\ \epsilon_{\mathbf{z}x}^{\mathbf{t}}&\epsilon_{\mathbf{zy}}&\epsilon_{z}^{\mathbf{t}}\\ \end{array} \right]=\left[ \begin{array}{ccc}a_{0}&\mathbf{a}_{1}&\mathbf{a}_{2}\\ a_{1}&\mathbf{b}_{0}&\mathbf{b}_{1}\\ a_{2}&\mathbf{b}_{1}&\mathbf{c}_{0}\\ \end{array} \right] \tag{25}\]  
\[\mathbf{u} = u_{0}+\mathbf{a}_{0}x+\mathbf{a}_{1}\mathbf{y}+\mathbf{a}_{2}\mathbf{z} \tag{26}\] \[\mathbf{v} = v_{0}+\mathbf{a}_{1}x+\mathbf{b}_{0}\mathbf{y}+\mathbf{b}_{1}\mathbf{z}\] (27) \[\mathbf{w} = w_{0}+\mathbf{a}_{2}x+\mathbf{b}_{1}\mathbf{y}+\mathbf{c}_{0}\mathbf{z}\] (28) \[\mathbf{f} = f(\mathbf{a}_{0},\mathbf{a}_{1},\mathbf{a}_{2},\mathbf{b}_{0},\mathbf{b}_{1},\mathbf{c}_ {0}) \tag{29}\]  
where \(\mathbf{a}_{0}\), \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\), \(\mathbf{b}_{0}\), \(\mathbf{b}_{1}\), \(\mathbf{c}_{0}\) are just unknown coefficients, i.e., tectonic strains, and need to be solved by inverse analysis using the optimization algorithm. In this approach, the engineering-scale strain on lateral boundaries is assumed as the variables, and Equations (26)-(28) are considered as the displacement boundary conditions applied in the 3-D simulating model. In order to find the reasonable tectonic strain on boundaries, the objective function needs to be established. In practical engineering, the objective function is generally based on the measured values at limited test points and calculated values at the corresponding points, and the minimum of the 2-norm of a stress tensor composed of the difference values between the measured values and calculated values can be made the objective function. Additionally, using the optimization algorithm, the tectonic strain would be automatically optimized until the objective function is converged as the best-fit function. For the optimization algorithm, the mathematic software MATLAB has a global optimization toolbox including a simplex algorithm, global search algorithm, pattern search method, genetic algorithm and simulated annealing method. Due to the detailed explanation in the global optimization toolbox of the MATLAB software, here, these optimization algorithms would not be introduced. Additionally, it is convenient to use these optimization algorithms to find the best-fit tectonic strain. Compared with traditional approaches in simulating a in situ stress field, there are many advantages for this approach: (1) In this approach, it does not need to select a boundary mode, which can overcome the key drawback of the traditional approach. Additionally, the proposed approach directly connects the normal and shear boundary conditions, and demonstrates the internal connection between both boundary conditions, which make sense in mechanics. (2) There are only six variables that need to be optimized, which overcome the drawbacks of time-consuming and global convergence using the optimization algorithm. Whatever the optimization algorithm is, it is very easy to quickly obtain the in situ stress field through optimizing six variables; (3) In traditional inverse analysis, the variables are only coefficients (Equations (14)-(19)), which does not make sense in mechanics.