# 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 the knowledge of practical engineering, a new approach is proposed in this paper for the inverse analysis of the in situ stress field, and only the tectonic strain on boundaries can be assumed to be uniform. Then, the tectonic displacement on the boundary can be expressed in the form of engineering-scale tectonic strain as in Equation (20):  
\[\left( \begin{array}{c}u^{t}\\ v^{t}\\ w^{t}\end{array} \right)\ =\ \left[ \begin{array}{ccc}\varepsilon_{x}^{t}&\varepsilon_{xy}^{t}& \varepsilon_{xz}^{t}\\ \varepsilon_{yx}^{t}&\varepsilon_{y}^{t}&\varepsilon_{yz}^{t}\\ \varepsilon_{zx}^{t}&\varepsilon_{zy}&\varepsilon_{z}^{t}\end{array} \right] \left( \begin{array}{c}x\\ y\\ z\end{array} \right) \tag{20}\]  
where \(\left[ \begin{array}{ccc}\varepsilon_{x}^{t}&\varepsilon_{xy}^{t}&\varepsilon _{xz}^{t}\\ \varepsilon_{yx}^{t}&\varepsilon_{y}^{t}&\varepsilon_{yz}^{t}\\ \varepsilon_{zx}^{t}&\varepsilon_{zy}&\varepsilon_{z}^{t}\end{array} \right]\) is the regional tectonic strain tensor. As it cannot be directly applied on boundaries for strain condition, it has to be transformed to other conditions, such as stress boundary condition and displacement boundary condition. Due to the easiness of applying displacement boundary, here the engineering-scale tectonic strain can be transformed into displacement condition. Then, if the engineering-scale strain on the boundary is known, the total displacement on boundaries can be determined in Equation (21). Therefore, the calculating process of this approach can be demonstrated as in Figure 4:  
Figure 3: Sketch of displacements on boundaries ABEF and BCDE. \[\left( \begin{array}{c}\mathbf{u}\\ \mathbf{v}\\ \mathbf{w}\end{array} \right)\ =\ \left( \begin{array}{c}\mathbf{u}^{0}\\ \mathbf{v}^{0}\\ \mathbf{w}^{0}\end{array} \right)+\left( \begin{array}{c}\mathbf{u}^{t}\\ \mathbf{v}^{t}\\ \mathbf{w}^{t}\end{array} \right)\ =\ \left( \begin{array}{c}\mathbf{u}^{0}\\ \mathbf{v}^{0}\\ \mathbf{w}^{0}\end{array} \right)+\left( \begin{array}{c}\mathbf{\varepsilon}_{x}^{t }&\mathbf{\varepsilon}_{xy}^{t}&\mathbf{\varepsilon}_{xz}^{t}\\ \mathbf{\varepsilon}_{yx}^{t}&\mathbf{\varepsilon}_{y}^{t}&\mathbf{\varepsilon}_{yz}^{t}\\ \mathbf{\varepsilon}_{xx}^{t}&\mathbf{\varepsilon}_{zy}^{t}&\mathbf{\varepsilon}_{z}^{t} \end{array} \right)\left( \begin{array}{c}\mathbf{x}\\ \mathbf{y}\\ \mathbf{z}\end{array} \right) \tag{21}\]  
where \(\left(u^{0},v^{0},w^{0}\right)\) and \(\left(u^{t},v^{t},w^{t}\right)\) are defined in Equation (20), \(\left(u^{0},v^{0},w^{0}\right)\) is the displacement at any point on the lateral boundary induced by gravity stress field in the 3-D calculating model, and \(\left(u^{t},v^{t},w^{t}\right)\) is the displacement induced by the tectonic stress field. Taking the 3-D model in Figure 3 as an example, on the lateral boundaries, the tectonic displacements along the axis X, Y and Z, were, respectively, applied as the Equations (22)-(24) based on the proposed approach:  
\[\mathbf{u}^{t} = \mathbf{\varepsilon}_{x}^{t}\mathbf{x}+\mathbf{\varepsilon}_{xy}^{t}\mathbf{y}+ \mathbf{\varepsilon}_{xz}^{t}\mathbf{z} \tag{22}\] \[\mathbf{v}^{t} = \mathbf{\varepsilon}_{yx}^{t}\mathbf{x}+\mathbf{\varepsilon}_{y}^{t}\mathbf{y}+ \mathbf{\varepsilon}_{yz}^{t}\mathbf{z}\] (23) \[\mathbf{w}^{t} = \mathbf{\varepsilon}_{xx}^{t}\mathbf{x}+\mathbf{\varepsilon}_{zy}^{t}\mathbf{y}+ \mathbf{\varepsilon}_{z}^{t}\mathbf{z} \tag{24}\]  
From Equations (21)-(24), it was demonstrated that the coordinates X, Y, and Z are known based on the location of the boundaries and only the tectonic strain tensor was unknown in Equations (21)-(24), and based on the symmetry characteristic of the tectonic strain tensor, the strain \(\mathbf{\varepsilon}_{xx}^{t}=\mathbf{\varepsilon}_{xz}^{t}\), \(\mathbf{\varepsilon}_{xy}^{t}=\mathbf{\varepsilon}_{yz}^{t}\) and \(\mathbf{\varepsilon}_{yx}^{t}=\mathbf{\varepsilon}_{xy}^{t}\), therefore, there are only six unknown variables in Equations (21)-(24), and they need to be solved by inverse analysis.