# Reliability and Efficiency of Metamodel for Numerical Back Analysis of Tunnel Excavation
## 1 Introduction  
Tunnel engineers use numerical analysis methods, such as the finite element method, finite difference method, and distinct element method to predict the behaviours of underground and structures for support. However, it is difficult to obtain perfect information of ground due to its complexity and uncertainties. Additionally, it is impossible to conduct some experiments to obtain all of the ground properties due to costs and time. Back analysis has been employed to overcome the uncertain and limited information about the ground condition. Back analysis quantitatively assesses the ground properties via numerical analysis using measured displacements [1; 2] and stresses [3]. It comprises inverse and direct methods, and the direct method is generally used for the convenience of calculation. In the direct method, numerical analysis is performed, and the displacements or stresses of the analysis are compared with measured displacements or stresses. Here, errors between predicted and measured values are calculated. The numerical analysis is repetitively performed by tuning the target parameters until the mean of the errors is minimised or falls below the target of mean of errors. The object properties of a back analysis can be obtained from the properties that derive results satisfying the tolerance. In this paper, back analysis means the direct method of a back analysis. Back analysis has been widely used in geotechnical engineering. [PERSON] and [PERSON] [4] conducted a back analysis to assess the elastic modulus and confirmed that the design must consider a lower elastic modulus than the ground investigation results. [PERSON] et al. [5] evaluated the coefficient of lateral pressure and cohesion of the groundaround the tunnel excavation using measure displacements from two small gallery tunnels before the main tunnel construction. [PERSON] et al. [6] performed a three-dimensional back analysis to evaluate the elastic modulus and coefficient of lateral pressure; based on their research, they proposed a change in the length of the tunnel bench to ensure ground stability during construction. However, this method has a disadvantage in that it requires a lot of time because the numerical analyses must be performed repetitively until the allowable range of error is reached. Many researchers have been studying back analysis using artificial neural networks (ANNs) to overcome the above disadvantages. ANN models for back analysis are usually metamodels for numerical analysis. According to [PERSON] and [PERSON] [7], a metamodel is defined as \"a model to describe another model as an instance\". Once a metamodel is created, it has the advantage of reducing the overall back analysis time cause the repetitions of the numerical analyses are not required anymore. Using this method, [PERSON] and [PERSON] [8] assessed the elastic modulus and coefficient of lateral pressure of a tunnel excavation ground, based on 81 numerical analysis results. [PERSON] [9] evaluated the elastic modulus, cohesion, and coefficient of lateral pressure using 192 numerical analysis results. In addition, [PERSON] and [PERSON] [10] increased the number of data and created a metamodel with a total of 2187 numerical analysis results to estimate the ground properties more accurately than that reported by [PERSON] [9]. The common point in these previous studies is that the metamodels were generated using two or three values for each of the dependent variables without specific bases. Smaller numbers of values of each of the dependent variables incur less time for the generation of the initial database but cannot guarantee the reliability of a metamodel. On the other hand, the number of values of each dependent variable secures the reliability of a metamodel but additional time is required for the generation of the initial database. Therefore, a study on the number of values required for each dependent variable to guarantee the reliability and efficiency of a metamodel is necessary.