## 3 Numerical Examples
\[MSE(x,y)=\frac{1}{N}\sum_{i=1}^{N}\left(x_{i}-y_{i}\right)^{2} \tag{21}\]  
where \(x\) represents the disturbance model, \(y\) represents the imaging results of the three methods, and \(N\) represents the number of data matrix elements. Compared to the LSRTM method, LSRTM-MTV demonstrates a 30.4% enhancement in computation speed and an 21.1% reduction in MSE, as shown in Table 1. Additionally, the amplitude curve of track 85 was randomly selected and plotted in Figure 6. The amplitude curve in Figure 6 shows the disturbance model as a solid black line, while the LSRTM, LSRTM-TV, and LSRTM-MTV results are shown as solid green, dashed blue, and dashed red lines, respectively. Although the LSRTM results generally approximate the reflection interface positions in the disturbance model, they exhibit significant noise and artifacts in the uniform medium. After applying TV processing, the artifacts are somewhat reduced, and the oscillations are eliminated. After MTV processing, the red dashed line significantly reduces the noise and artifacts without sacrificing the interface information. Overall, the results are in good agreement with the disturbance model. \begin{table}
\begin{tabular}{c c c c} \hline  & **LSRTM** & **LSRTM-TV** & **LSRTM-MTV** \\ \hline MSE & 0.0649 & 0.0593 & 0.0512 \\ Iterations & 90 & 42 & 90 \\ Time (min) & 33.9255 & 11.9032 & 23.5991 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of calculation parameters between three methods. Figure 5: Convergence curve with LSRTM, LSRTM-TV, and LSRTM-MTV. Figure 6: Multiple curve comparison with disturbance model, LSRTM, LSRTM-TV, and LSRTM-MTV. To determine the effect of \(\lambda_{1}\) on the imaging results, we set \(\lambda_{1}\) as 0.0001, 0.001, and 0.01 to carry out comparative experiments, while \(\lambda_{1}\) values of 0.1, 10, and 100 resulted in unstable and inaccurate convergence. Figure 7 displays the imaging results for different \(\lambda_{1}\) values. It can be observed that the selection of \(\lambda_{1}\) significantly influences the imaging effect of LSRTM-MTV. Clutter is removed to the maximum extent and the interface information is retained when \(\lambda_{1}\) is set to 0.01, achieving the best effect. The outcomes presented in Table 2 further indicate that the lowest MSE is achieved when \(\lambda_{1}\) = 0.01. Subsequent experiments were developed based on this result, using \(\lambda_{1}\) = 0.01. It should be noted that \(\lambda_{2}\) in Equation (14) is determined by the data itself and is not the focus of our discussion. To evaluate the robustness of the proposed method, we added 10 dBW Gaussian white noise to the forward recording and tested its anti-noise performance. The forward recording after removing the direct wave is shown in Figure 8. As shown in Figure 9 and Table 3, the imaging result obtained using LSRTM-MTV is superior to that obtained using traditional TV, even after adding Gaussian white noise. This result further verifies the robustness and anti-noise performance of the proposed method. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\lambda_{1}\) = 0.0001 & \(\lambda_{1}\) = 0.001 & \(\lambda_{1}\) = 0.01 \\ \hline MSE & 0.0641 & 0.0563 & 0.0512 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of MSE between different \(\lambda_{1}\). Figure 7: Different \(\lambda\) imaging results with (**a**) \(\lambda\) = 0.0001, (**b**) \(\lambda\) = 0.001, (**c**) \(\lambda\) = 0.01.