# 3D In Situ Stress Estimation by Inverse Analysis of Tectonic Strains
## 2 Inverse Analysis for In Situ Stress Determination
### Tectonic Stress-Based Approach  
The in situ stress field has been considered as the superposition of gravity and tectonic stress field. Therefore, it can be expressed as Equation (1):  
\[\sigma\ =\ \sigma^{0}+\sigma^{t} \tag{1}\]  
where \(\sigma\) is the in situ stress at any point, \(\sigma^{0}\) is the gravity stress, which can be easily calculated by numerical analysis, and it can be recognized as a known quantity, and \(\sigma^{t}\) is the tectonic stress. Figure 1 demonstrates a simple calculating model, where the bottom is fixed and all lateral boundary conditions are normally restrained, and only the gravity load is applied. With this model, the gravity stress field can be calculated based on the theory of elasticity. Figure 2 shows a 2-D calculating model to calculate normal tectonic stress \(\sigma^{t}\), where in X-Z section, the normal tectonic stress on the boundary can be considered as a linear expression (Equation (2)), or quadratic expression (Equation (3)) of the coordinates on the boundaries. As the coordinate X is the constant, the coordinate Z changes with the location on the boundaries. Therefore, the normal tectonic stress can be expressed as the function of the coordinate Z (Equation (4)):  
\[\sigma_{n}\ =\ \mathbf{a}_{0}+\mathbf{a}_{1}\times\mathbf{z} \tag{2}\]  
\[\sigma_{n}\ =\ \mathbf{a}_{0}+\mathbf{a}_{1}\times\mathbf{z}+\mathbf{a}_{2}\times\mathbf{z}^{2} \tag{3}\]  
\[\sigma_{n}\ =\ \mathbf{f}(\mathbf{a}_{0},\mathbf{a}_{1},\mathbf{a}_{2}) \tag{4}\]  
where \(\mathbf{a}_{0},\mathbf{a}_{1},\mathbf{a}_{2}\) are just unknown coefficients which need to be solved by inverse analysis and have no physical meaning. Similarly, for the shear stress on boundaries (Figure 2), Equation (5) is the quadratic expression, in which the coordinate X is constant and the coordinate Z can be obtained on the basis of the location on the boundaries. Additionally, the shear stress should be zero at the intersection point between the ground surface and the boundary because of the equivalent theorem of shearing stress. For the direction of the shear stress on both sides in Figure 2, it needs to be clarified that it is just one possibility of tectonic stress in practical engineering, which is possible since they have the same direction on both sides of the model and the direction of the shear stress can be determined by optimizing the stress-based boundary conditions. During the optimization process, the direction of the shear stress is considered by the positive or negative value of the shear  
Figure 1: Sketch of 3-D calculating model. stress. Therefore, the shear stress can still be considered as the function of the coordinate \(Z\) (Equation (6)):  
\[\sigma_{\tau}\ =\ \mathbf{b_{1}}\times\mathbf{z}+\mathbf{b_{2}}\times\mathbf{z}^{2} \tag{5}\]  
\[\sigma_{\tau}\ =\ \mathbf{f}(\mathbf{b_{1}},\mathbf{b_{2}}) \tag{6}\]  
Here, just like in Equation (4), \(\mathbf{b_{1}}\) and \(\mathbf{b_{2}}\) are also just coefficients and need to be solved by inverse analysis. Therefore, combining the normal and shear stress boundary conditions, the boundary mode for this method can be summarized as Equation (7):  
\[\mathbf{f}\ =\ \mathbf{f}(\mathbf{a_{0}},\mathbf{a_{1}},\mathbf{a_{2}},\mathbf{b_{1}},\mathbf{b_{2}}) \tag{7}\]  
For the 2-D calculating model, it can be seen that there are three variables in linear boundary mode and five variables in quadratic boundary mode. However, there will be more variables in the 3-D calculating model. For the 3-D calculating model in Figure 1, on the boundary ABEF or OCDG, the normal stress can be linearly expressed as Equation (8). Nevertheless, the shear stress on boundary ABEF or OCDG, can be divided into two types, one direction is along axis Y, and the other is along axis Z.