# Inversion Analysis Method for Tunnel and Underground Space Engineering: A Short Review
## 5 Inversion Analysis of Characteristic Parameter  
The types and quantities of characteristic parameters of the surrounding rock are closely related to the deformation behavior of the rock mass under external forces. When the rock mass is in an elastic stress state, Poisson's ratio and elastic modulus are the characteristic parameters of the surrounding rock. When in an elastic-plastic stress state, the internal friction angle and cohesion must be added to fully express the characteristics of the rock mass. Some soft rocks remain in a viscoelastic stress state due to their rheological properties, specifically creep and stress relaxation. Determining the characteristic parame \begin{table}
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**Inversion Analysis Method** & **Project Cases** & **Brief Description of Analysis Results** \\ \hline The particle swarm algorithm [59] & Subway tunnel construction by shield method & The calculation efficiency is high, but for complex rock mass structure, it is still difficult to determine the support load by this method \\ Inversion method based on probability density evolution theory [61] & Shanghai Metro Line M6 & Achieves the correct reflection of the objective system while greatly reducing the difficulty and workload of problem-solving \\ \hline \hline \end{tabular}
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Table 4: The statistical results of engineering cases for supporting structure load in intelligent inversion analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Inversion Analysis Method** & **Project Cases** & **Brief Description of Analysis Results** \\ \hline Least-squares method [65] & the Xu FuXiang Station to the Nan-jing Railway Station on subway line 1 in Nanjing City in eastern China & The external load of tunnel structure obtained by back analysis method is more accurate and convenient than that obtained by traditional direct monitoring or classical earth pressure calculation formula, and requires fewer monitoring sensors, but the effect is not good for deep-buried tunnels.  
\begin{tabular}{c c} Combining model test and numerical simulation [66] & Foshan Metro Line 2 (Green Island Lake-Liantang) & The consistency between the calculated values and the measured data is analyzed by using the comprehensive error function, and the uniform distribution of the vault load and the parabola distribution of the horizontal load are the most consistent with the test data. \\ Inversion analysis based on radial displacement and contact stress [67] & Lvjialiang Tunnel in Chongqing & The analysis is limited to the effect of symmetrical loads, and the internal forces of the main support for shallow-buried unsymmetrically-loaded tunnels need to be further explored. \\ Displacement back analysis [68] & Longsheng Tunnel in Guilin City & The regression analysis results are basically consistent with the measured results, and the relative error is controlled within 10\% \\ \hline Inversion analysis based on initial parameter method [56] & Multi-circular arch tunnel & Considering the time variation of tunnel construction process, and the analytical solution of main support internal force is feasible in tunnel construction \\ \hline \hline \end{tabular}
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Table 3: The statistical results of engineering cases for supporting structure load in traditional inversion analysis. ters is equivalent to identifying the constitutive model of the rock mass. Based on this, the problem of identifying various parameters is realized. Therefore, the inversion analysis of characteristic parameters can be considered an inverse parameter identification problem. For this type of problem, the suitability of the parameter solution must be solved first, including the results for uniqueness, stability, and identifiability. If the forward problem corresponding to the inverse problem has no unique solution, the inverse problem will have no unique solution.