# Inversion Analysis Method for Tunnel and Underground Space Engineering: A Short Review
## 4 Inversion Analysis of Supporting Load
### Traditional Inversion Analysis Method  
The concept of load on supporting structures has undergone three stages, as shown in Figure 8. From the late 19 th century until the first half of the 20 th century, the support structure was considered to be a structure that only bore loads, without taking into account the magnitude of the action load distributed on it by the surrounding rock resistance and the bearing capacity of the surrounding rock itself. In the 1930s, the load on the support was divided into active load (active surrounding rock pressure) and passive load (the result of surrounding rock pressure constraint on the support). After the 1960s, with the introduction of the New Austrian Tunnelling Method (NATM), the surrounding rock became an integral part of the support system. Based on further understanding of the structural nature of the surrounding rock and its loading effects, many scholars have interpreted the nature of the support structure in detail and proposed the tunnel surrounding rock-support synergistic action system. Figure 8: The development process of the load concept on supporting structure [46; 47; 48]. In 1977, Professor [PERSON] [49] from the Department of Rock and Tunnel Engineering at the Zurich University of Applied Sciences conducted the first study on structural load inversion analysis. The main idea was to divide the entire support structure into several parts, establish the relationship between the measured curvature and strain of the support and between the axial force and bending moment of the support, and calculate the distribution of the tangential and normal stress values of the support load. In 1981, the Italian scholar [PERSON] [50] published an inverse calculation method based on the finite element method, and discussed the inversion analysis of rock pressure considering the effect on retaining structure and the characterization of in situ rock mass material parameters in 1987. The inversion analysis problem is still non-linear, regardless of the linear representation of load and material characteristics, and must be solved by appropriate optimization methods. These articles laid the theoretical foundation for early structural load inversion analysis. In the 1990s, theoretical research on the inversion of structural support loads began in China, starting with the load inversion of roadway supports and conducting a lot of basic work, followed by in-depth research on the inversion analysis method of tunnel lining load. [PERSON] [51] conducted the inversion analysis of the surrounding rock action pressure on the support lining and concluded that the site measuring point arrangement and boundary conditions directly affect the results of the inversion analysis. [PERSON] [52] proposed a general numerical analysis method for identifying tunnel lining load. This method uses a group of in situ displacement measurements of the sprayed concrete layer for back analysis and calculation, which avoids the complex characteristics of rock mass, and the load value after inversion calculation can effectively reflect the combined effect of tunnel-surrounding rock. To solve the problem of instability and non-uniqueness of the inverse solution of underground structures, [PERSON] et al. [53] proposed three kinds of back analysis methods: generalized least square solution estimation, generalized minimum norm solution estimation, and generalized minimum variance solution estimation. These methods greatly improved the accuracy of load inversion of supporting structures. In terms of stability analysis of supporting structures, [PERSON] et al. [54] proposed a tunnel structure stability analysis method based on ready-made displacement monitoring data, considering the influence of tunnel buried depth and cross-section. The loads of supporting structures in horseshoe tunnels [55], multi-center circular arch tunnels [56], and shallow eccentric tunnels [57; 58] were also studied.