## 3 Results and Discussion
### Case Study of Numerical Simulation
To illustrate the applicability of the proposed approach, we also examined its effectiveness in identifying microscopic seepage tunnels that contain water, as shown in Figure 4. The distribution of relative permittivity in this model is shown in Figure 4a; the water-containing seepage channels are represented by three anomalies in the lining, which have a relative permittivity of 81. The forward modeling results of this model are shown in Figure 4b. As compared to Figure 3b, the reflection responses of the anomalies are much easier to identify in this figure, as they are stronger and less susceptible to interference by residual direct waves and the host rock interfaces' reflections. However, the presence of multiple reflections below the hyperbolic reflections of the anomalies makes it difficult to determine the number of seepage channels in the lining, as well as their locations. Figure 4c shows the back-projection image of the model above. Even though the locations of the three seepage channels have been accurately inverted and are relatively clear, the multiple reflections created two false images below the real seepage channels, hindering the accurate identification of seepage channels. RPCA was used to extract the sparse matrix, as illustrated in Figure 4d. It is shown that the seepage channels were accurately extracted. The two false images have similar shapes with the real seepage channels, but their amplitudes are close to the diffracted waves on both sides, so they are weakened by RPCA. In a word, the ease and accuracy of GPR data interpretation were improved by the proposed RPCA algorithm to some extent. In order to illustrate the applicability of the proposed method to media with different reflectance, we designed a special model, as shown in Figure 5. The relative permittivity  
Figure 4: The water- bearing seepage channel model. (**a**) the distribution of relative permittivity; (**b**) 2-D GPR profile; (**c**) back projection imaging; (**d**) RPCA anomaly extraction. distribution in the model is shown in Figure 5a; There are three abnormal bodies with the same scale in the lining, and their relative dielectric constants are 1, 5 and 16, respectively. The forward simulation results of the model are shown in Figure 5b. The lowest anomaly with the highest reflectance has the strongest response, and the uppermost anomaly with the lowest reflectance has the weakest response. Figure 5c shows a back projection image of the above model. Similarly, the imaging of the lowest abnormal body is the strongest, and the imaging of the uppermost abnormal body is the weakest. RPCA is used to extract sparse matrix, as shown in Figure 5d. The results show that this method can accurately extract three abnormal bodies with different reflectance, but the imaging results of abnormal bodies with larger reflectance are clearer. In principle, RPCA can be used to extract the imaging results of any sparse anomaly. This makes RPCA especially well-suited for the identification of point-shaped anomalies such as seepage channels. In addition to seepage channels, the common tunnel lining defects are lining voids, lining-rock separation, fractures and concrete honeycomb [5]. As the features associated with lining voids, cracks and leakages in migration imaging also exhibit sparse characteristics, our approach should, in theory, be able to detect these defects. Of course, when the cracks and lining voids are very large, they could be harder to be detected with RPCA, as the imaging results of these defects are insufficiently sparse. However, this large lining defects are usually obvious in the original BP imaging result, so the lose effectiveness of RPCA has fewer negative effects.