# Elastic Reverse Time Migration for Weakly Illuminated Structure
## 3 Numerical Examples  
In the first example, we employ a layered velocity model to probe the performance of the above equations. The P-wave migration velocity model is sketched in Figure 1a. The S-wave migration velocity is obtained using an empirical formula \(c_{S}=c_{P}/\sqrt{3}\). The density is set as 2000 kg/m\({}^{3}\). The model is dispersed by 650 \(\times\) 400 grid numbers, with a grid size of 10 \(\times\) 10 m\({}^{2}\). A Ricker wavelet with the dominant frequency of 25 Hz is set as the source signature at the location of (1 km, 0). The wavefields are simulated by the finite difference method with the second-order in time and tenth-order in space. In addition, we add a perfectly matched layer around the simulated area to absorb spurious reflections from the truncated boundary. Based on the VCWE, the \(x\)-component of the particle-velocity is shown in Figure 2a, in which the P- and S-waves are coupled together. When the coupled wavefield is extrapolated by solving Equation (4), it can be completely decoupled into pure P- and S-waves (Figure 2b,c). Notably, the seismic energy of the deep structure information is relatively weak in the decoupled wavefield. To reduce the shielding effect of the overburden rock on seismic waves, we simulate the wavefield based on SVDWE. Here, the second interface is set as the stained target. \(\widetilde{c_{p}}\) is obtained by \(\overline{\varphi}\) multiplied \(10^{-6}\) in the target area while is zero in other locations. The setting of \(\widetilde{c_{s}}\) is similar to that of \(\widetilde{c_{p}}\). Considering the similarity between \(\widetilde{c_{p}}\) and \(\widetilde{c_{s}}\) models, only one of them is shown in Figure 1b. By solving SVDWE, the stained wavefield \(\widetilde{\mathbf{V}}\) is generated after the \(\overline{\mathbf{V}}\) hits the stained target. The stained wavefield \(\widetilde{\mathbf{V}}\) only contains the reflection and transmission about the second interface as exhibited in Figure 3. Consequently, we infer that imaging with the stained wave can reduce interference from non-target structures. Furthermore, we utilize a contrastive approach to illustrate the adaptability of the ERTM for different equations. We evenly deploy 10 shots and 650 receivers on the surface of the layered velocity model. The total length of seismic records is 3 s, with a step time of 1 ms. Figure 4 shows the multi-shot stacked images based on VDWE. Moreover, we implement the ERTM using the SVDWE, the PP and PS images are sketched in Figure 5. The settings of the stained target are the same as those  
Figure 1: Migration model: (**a**) P-wave velocity; (**b**) Stained target. Figure 3: \(x\)-component of the stained wavefield: (**a**) Pure P-wave; (**b**) Pure S-wave. Figure 2: \(x\)-component of particle-velocity wavefield: (**a**) Coupled wave; (**b**) Pure P-wave; (**c**) Pure S-wave. of previous numerical experiments. Thanks to the stained wavefields, the primary information in the images is the second interface. However, some unwanted migration artifacts may interfere with subsequent interpretations. Next, we demonstrate that the proposed imaging condition plays an important role in eliminating migration artifacts. The filters in the imaging condition need to be prepared in advance, and their properties are determined by the direction vector and Poynting vector. For the snapshot shown in Figure 3a, the Poynting vector (Figure 6) can be obtained using Equation (15). Here, we exhibit four filtered wavefields (Figure 7) that, respectively, correspond to different direction vectors: \(\mathbf{n}_{So}^{p}=(1,\ 0)\), \(\mathbf{n}_{So}^{p}=(-1,\ 0)\), \(\mathbf{n}_{So}^{p}=(0,-1)\), and \(\mathbf{n}_{So}^{p}=(0,1)\). Figure 4: ERTM image obtained based on the VDWE: (**a**) PP imaging; (**b**) PS imaging. Figure 5: ERTM image based on the SVDWE: (**a**) PP imaging; (**b**) PS imaging. For flat strata, images with fewer artifacts can be obtained in the inner product of the upgoing stained source wavefield and the down-going stained receiver wavefield (Figure 8).