## 3 Numerical Tests
### Examples of Two-Dimensional Salt Hill Model  
The two-dimensional Salt Hill model was used in the numeric test in this paper, and the true velocity model and reflectivity model is shown in Figure 6. The model grid was 150 \(\times\) 644 and the mesh spacing was 12.5 m. A total of 70 shots and 645 receivers were evenly distributed on the surface. The shot interval was 112.5 m and the receiver interval was 12.5 m. The source wavelet was selected as the Ricker wavelet with a peak frequency of 15 Hz. Figure 6a is the velocity model and Figure 6b is the reflectivity model, which can be a comparison of the LSRTM image. Figure 7a shows the recorded data. Figure 7b shows the predicted internal multiples after five iterations. Figure 7c shows the primary of the data residual after using the weighting matrix. Figure 7d shows the internal multiples of data residual after using the weighting matrix. We can find the internal multiple in Figure 4a,b,d, that shows that it has been estimated, and is separated from the data residual. Figure 4e is the waveform comparison of the data residual and the data residual of the primary and internal multiple. Through comparison, we can find the multiple existing mainly in about 0.8 s, the energy of the multiple is low when the energy of the primary is strong, and the energy of the multiple is strong when the energy of the primary is weak. Therefore, we can say that the primary and multiple are separated in the data residual. Figure 5: (**a**) LSRTM image. (**b**) RWLSRTM image of the primary. (**c**) RWLSRTM image of the primary and internal multiples. (**d**) the comparison of 150 th trace (**e**) the comparison of 150 th trace deeper than 1.25 km. Figure 6: (**a**) True velocity model. (**b**) True reflectivity model. Figure 7: Cont. Figure 8a,c,e show the LSRTM image, the RWLSRTM image of the primary, and the RWLSRTM image of the primary and internal multiples. Figure 8b,d,f shows their close-up section. The red arrows of letter A and B indicate artifacts caused by the internal multiples in Figure 8a, and they are suppressed in Figure 8b,c. The artifacts caused by the internal multiples in the image by the primary were depressed. Without the interference of artifacts, the actual structure has better convergence and looks clearer, as the place that the letter C points out in Figure 8. However, the actual structure in the image by the primary was also depressed, as indicated below letter B. In the image by the primary and internal multiples, the actual structure was compensated. Because the artifacts were suppressed, the overlap between the actual structure and the artifact was also suppressed, which made the actual structure discontinuous. In addition, it did not introduce the artifacts in the image when the primary and the internal multiple were separated in the image, as indicated by the letter B.  
Figure 8: _Cont_. Figure 7: (**a**) Recorded data. (**b**) Predicted internal multiples after five iterations. (**c**) The primary of data residual after using the weighting matrix. (**d**) The internal multiple of data residual after using the weighting matrix. (**e**) Waveform comparison of data residual and data residual of primary and internal multiple. Figure 8: _Cont_. We chose the 290 th trace and the 320 th trace to compare their image, as Figure 9a,b show. The 290 th trace is indicated by the far left arrow in Figure 8f. In Figure 9a, the useful structure is at about 1.6 km. Only the L8 RTM without the internal multiple did not image here, and this shows that the internal multiples contain some information of useful structure and can be a good complement in image. The 320 th trace is about the second arrow in the left in Figure 8f. In Figure 9b, the artifact is about in 1.4 km. Only the L8 RTM contains the artifacts, and shows that it is effective in reducing the artifacts when separating the internal multiples. Figure 8: (**a**) L8 RTM image. (**b**) Close-up section of L8 RTM image. (**c**) RWLSRTM image of the primary. (**d**) Close-up section of RWLSRTM image of the primary.