## 3 Results and Discussion  
### Case Study of Numerical Simulation  
Four kinds of common lining defects are simulated here to show the effects of the combined method, which are microscopic seepage channels, lining voids and lining-rock separation, lining fractures and concrete honeycomb [10]. Here the microscopic seepage channels refer to the leakage channels with smaller scale (centimeter scale) and perpendicular to the GPR detection line in tunnel lining. The GPR detection of this microscopic seepage channel perpendicular to the detection line is approximate hyperbola, and its BP imaging result is approximate a point, which has strong sparsity. Therefore, it could be extracted by RPCA with best effects. We will discuss the numerical simulation result of the microscopic seepage channels most detailly. We first illustrate the effect of the processing method on the GPR detection data proposed in this study through a numerical simulation case. The synthetic data was simulated by an FDTD (Finite-Difference Time-Domain, the most used GPR detection simulation method) program with a grid size of 0.01 m, an antenna spacing of 0.1 m and a time window size of 21 ns. A Ricker wavelet with a center frequency of 400 MHz was used as the source wavelet. The simulation of one trace of data is carried out every two grids, and totally 74 traces of data are obtained from each model. Figure 3 illustrates the first numerical experiment of this work, where microscopic seepage channels were simulated in the lining of a tunnel. The size of this model was 1.7 m \(\times\) 0.9 m. Figure 3a shows the distribution of relative permittivity in the model. The upper and lower parts of the model are the lining and host rock, which have average relative permittivity of 7 and 9, respectively. There are four microscopic anomalies with a relative permittivity of 1 in the middle of the lining; these are meant to simulate water-less microscopic seepage channels. To ensure that the model is realistic, an elliptical autocorrelation function [24] was used to construct a random two-dimensional medium which simulates the non-uniformity of the latter. Figure 3b shows the forward-modeling results of the abovementioned numerical experiment. Direct waves were removed by cepstral mean subtraction. Here, it is shown that the responses of the four seepage channels are hyperbolic events. The direct waves were not fully removed by cepstral mean subtraction, as horizontal perturbations remain in the results. Background noise is also high in these results, and the unevenness of the host rock interface led to undulating reflection events. Due to the combined effects of these issues, it is challenging to identify the responses of the seepage channels. Figure 3: The water-free seepage channel model. **(a)** the distribution of relative permittivity; **(b)** 2-D GPR profile; **(c)** back projection imaging; **(d)** RPCA anomaly extraction. Figure 3c shows the smoothed back-projection image, where the hyperbolic responses of the seepage channels have been inverted into points. Even though the seepage channels were correctly projected in Figure 3c, it is still difficult to visually identify the locations of the seepage channels, due to the interference caused by residual direct waves and host rock interfaces. The imaging results of the residual direct waves and host rock interfaces are relatively regular and exhibit low-rank features, whereas the imaging results of the seepage channels are clustered into points and are sparse. Therefore, the RPCA algorithm proposed in this paper may be used to extract the elements of the image that correspond to seepage channels, as shown in Figure 3d. In the sparse matrix produced by the RPCA algorithm, the pixel values of the direct waves and host rock interfaces were eliminated, and the ghost images of the microscopic seepage channels were also suppressed. Clear images of the (sparse) microscopic seepage channels were thus obtained. Therefore, the RPCA algorithm could improve the ease and accuracy of GPR data interpretation to some extent.