# Inversion Analysis Method for Tunnel and Underground Space Engineering: A Short Review
## 3 Inversion Analysis of Initial Ground Stress
### Traditional Inversion Analysis Method  
The 20-year period spanning from 1970 to 1990 marked the first stage of inversion research in China on the initial geo-stress of rock mass. During this stage, the Institute of Geology of the Chinese Academy of Sciences conducted the first research on initial stress inversion analysis. They proposed a finite element atlas method [23] for inversion analysis of initial stress based on mapping. The atlas method approximates the displacement and stress of surrounding rock in practical engineering based on finite element calculation results. By preparing a series of atlases according to commonly used section forms in engineering, the stress and displacement in actual engineering surrounding rock can be projected. The atlas model includes vertical load \(q\), horizontal ground stress \(P\), horizontal lateral pressure \(\xi\), and weight of surrounding rock. The model structure is shown in Figure 4. While the mapping method is still a good method for conducting rapid inference of elastic stresses and displacements in early studies, it is not suitable for tunnel and underground space engineering with special section shapes and complex geological environments. In such cases, the finite element method or other methods are used to obtain more reasonable and accurate results. In 1983, Professor [PERSON] of Tianjin University proposed the method of using regression analysis to inverse the initial stress field [24]. The method first establishes a three-dimensional calculation domain coordinate system, as shown in Figure 5, and establishes a finite element calculation model based on the determined geological and topographic survey data and using the factors that may form the initial ground stress field of the rock mass, such as the self-weight of the rock mass, geological tectonic movement, and temperature as the undetermined analysis value, from which the multiple regression relationship between the undetermined analysis value and the measured data is established, as shown in Equation (1):  
\[\sigma=f(x,y,z,E,\mu,\gamma,\Delta,U,V,W,T) \tag{1}\]  
where \(\sigma\) is the initial stress value, two-dimensional problems represent three stress components, and three-dimensional problems represent six stress components, \(x,y,z\) are the coordinate system of the spatial location of topography and geological bodies can be obtained from survey data, \(E\), \(\mu\), \(\gamma\) are elastic modulus, Poisson's ratio, and bulk density of rock mass, respectively, which can be obtained by physical testing, \(\Delta\) is the self-weight factor, \(U,V\), and \(W\) are factors of geological structure, and \(T\) is the temperature factor. Figure 4: Finite element atlas model [23]. The multiple regression equation of Equation (1) is linear due to the complex composition of the initial stress, which is only a gradual approximation method. If the initial stress can be estimated according to the nonlinear regression equation, it can better meet the needs of practical engineering. The general nonlinear regression model is expressed as Equation (2) [25]:  
\[y_{k}=f\big{(}x_{k},\beta_{p}\big{)}+\varepsilon_{k} \tag{2}\]  
where \(y_{k}\) is the observed value; \(x_{k}\) is a non-random regression coefficient, \(\beta_{p}\) is a unknown parameter vector and \(\varepsilon_{k}\) is a random vector. In Equation (2), the value of \(k\) does not correspond to the value of p.  
[PERSON] [26] adopts the concept of \"releasing load\" to analyze the stress present in the current underground cavern excavation. It is considered that the disturbed stress field and displacement field are caused by the unloading effect resulting from the release of the initial ground stress around the tunnel, and the load released is completely determined by the initial ground stress. In addition, in 1983, [PERSON] et al. [27] carried out back analysis of the in situ stress of the dam powerhouse through the disc fracture phenomenon and rockburst phenomenon of the core. In 1989, [PERSON] et al. [28] proposed the geostress function analysis method.