## 3 Numerical Examples  
A model is established to verify the applicability of the LSRTM-MTV method to complex models [36], the size of which is 4.0 m \(\times\) 8.0 m, as shown in Figure 2a. Two-layer undulating interfaces are designed in the model, with two lithologic contrast units on the right and two irregular cavities in the lower layer. The simulation is performed under the common-offset mode of the monostatic system, and the transmitting and receiving antennas are placed in the air layer. The smoothed true model is used as the initial model for migration. To obtain the initial model, a Gaussian filter with a template size of 50 \(\times\) 50 and a standard deviation of 3 was used for smoothing; note that these two parameters determine the degree of smoothness compared to the true model. The disturbance model, calculated by the true model and the smoothed model, is shown in Figure 2b. Figure 1: Flowchart of the proposed LSRTM-MTV algorithm. The simulation was carried out using the finite difference time domain (FDTD) method [2; 37] with a Riker wavelet source at the central frequency of 200 MHz and a time window of 108 ns. The time and space intervals were set at 0.09 ns and 0.04 m, respectively. The simulation results of the true model forward record and the initial model forward record are shown in Figure 3a,b, respectively. The residual record of the true model and initial model are shown in Figure 3c, confirming the feasibility and appropriateness of the smoothed model as an initial model. Figure 3: True model forward record (**a**), initial model forward record (**b**), and residual record (**c**). Figure 2: True model (**a**) and the corresponding disturbance model (**b**). The imaging results are shown in Figure 4. LSRTM imaging results with 20, 40 and finial iterations are shown in Figure 4a-c, respectively. It can be seen that imaging results after the 20 th iteration are relatively vague, as the interface and abnormal position cannot be precisely localized, while with increasing iterations the interface and abnormal position can be pre-determined after the 40 th iteration. As shown in the finial iteration, except for significant noise and artifacts, the image result clearly shows the fluctuating interface, the interface of lithological units, and the location of two irregular anomalies. To this end, MTV regularization is introduced as an optimization scheme, and the conventional TV is also presented to compare performance after processing the iterative results of Figure 4a-c. Imaging results of LSRTM-TV with 20, 40 and finial iterations are shown in Figure 4d-f and LSRTM-MTV with 20, 40 and finial iterations are shown in Figure 4g-i, respectively. Compared to LSRTM results, the imaging results of traditional TV show some noise suppression capabilities while artifacts remain, blurring the interfaces. In contrast, MTV imaging results demonstrate enhanced noise suppression performance. For example, see the comparative imaging results of the 20 th iteration shown in Figure 4f;i the imaging resolution is significantly improved as the reflection interface is clearer, and the abnormal position is localized precisely, allowing verification of the effectiveness of the LSRTM-MTV scheme. The convergence curve using different methods is shown in Figure 5. As a result of the highly unstable convergence encountered in the LSRTM-TV problem, the convergence of the calculation process is not assured, leading to termination after a mere 42 iterations. In contrast, the MTV method proposed in this paper addresses the limitations of the TV method. To assess the quality of the imaging results, the mean square error (MSE) was calculated between each of the three results and the disturbance model. MSE is the most common estimator of image quality measurement metrics. It is a full reference metric, and values closer to zero are better [38]. MSE between two images is defined as:  
Figure 4: LSRTM imaging results (**a–c**), LSRTM-TV imaging results (**d–f**) and LSRTM-MTV imaging results (**g–i**) with different iterations.