# Inversion and Analysis of the Initial Ground Stress Field of the Deep-Buried Tunnel Area
## 2 Multiple Linear Regression Inversion Principle of Initial Ground Stress Field
### Multiple Linear Regression of Ground Stress Field  
#### 2.2.1 Coordinate Transformation of Ground Stress Measured Results  
The geological survey results of the initial ground stress are expressed in the direction of East, West, South, and North and the corresponding dip angle, which do not correspond to the coordinate system of the three-dimensional geological model and cannot be directly substituted into the multiple linear regression model for calculation. Therefore, while using the measured ground stress to carry out multiple linear regression for the equivalent ground stress field of the tunnel, it is necessary to conduct coordinate transformation [39] on the measured ground stress of the on-site measuring points. The coordinates of the corresponding points in the coordinate system of the three-dimensional geological model can be obtained. Then, decomposing the measured ground stress, nine stress components in the three-dimensional spatial coordinate system are transformed and obtained. While the ground stress measurement is conducted with the hydraulic fracturing method, it is generally considered that the measured vertical stress is the vertical principal stress [40], that is, \(\tau_{xz}=\tau_{zx}=0\) and \(\tau_{yz}=\tau_{zy}=0\). Therefore, the principal stress in the XOY coordinate plane is only needed to decompose in the plane coordinate system. \(\sigma_{x}\), \(\sigma_{y}\), \(\tau_{xy}\) and \(\tau_{yx}\) are accordingly obtained. Taking the due north direction as the polar coordinate X-axis direction, the transformation formula of stress component from polar coordinate to Cartesian coordinate is as follows.  
\[\begin{cases}\sigma_{x}=\frac{\sigma_{x}+\sigma_{\theta}}{2}+\frac{\sigma_{x} -\sigma_{\theta}}{2}\cos 2\theta-\tau_{\theta}\sin 2\theta\\ \sigma_{y}=\frac{\sigma_{x}+\sigma_{\theta}}{2}-\frac{\sigma_{x}-\sigma_{ \theta}}{2}\cos 2\theta+\tau_{\theta}\sin 2\theta\\ \tau_{xy}=\tau_{yx}=\frac{\sigma_{x}-\tau_{\theta}}{2}\sin 2\theta+\tau_{ \theta}\cos 2\theta\end{cases} \tag{1}\]  
In the formula:  
\(\sigma_{r}\)**--**Radial stress value of initial ground stress at measuring point in polar coordinate system;  
\(\sigma_{\theta}\)**--**Circumferential stress value of the initial ground stress at the measuring point in the polar coordinate system;  
\(\tau_{r\theta}\)**--**Shear stress value of initial ground stress at measuring point in polar coordinate system;  
\(\theta\)**--**The angle between the initial ground stress direction of the measuring point and the x-axis in the polar coordinate system;  
\(\sigma_{x}\)**--**x direction stress component of initial ground stress at measuring point in Cartesian coordinate system;  
\(\sigma_{y}\)**--**y direction stress component of initial ground stress at measuring point in Cartesian coordinate system;  
Figure 2: Different stress states. (**a**) Applying gravity stress; (**b**) Applying horizontal squeezing stress in the X direction; (**c**) Applying horizontal squeezing stress in the Y direction; (**d**) Applying shear stress in the XOY coordinate plane; (**e**) Applying shear stress in XOZ coordinate plane.  
\(\tau_{xy}\), \(\tau_{yx}\)--Shear stress component of initial ground stress at measuring point in Cartesian coordinate system.