# Depth-Extrapolation-Based True-Amplitude Full-Wave-Equation Migration from Topography
## 3 Examples
### Marmousi Model  
RTM is currently one of the best migration methods. It has excellent performance in complex structure imaging and amplitude preservation. In this section, we use the Marmousi model to compare our proposed method with the RTM method. In this experiment, we use a model with a flat surface because we want to eliminate the influence of terrain on the imaging effect and focus on the imaging differences between the two methods.  
The Marmousi model is shown in Figure 4. The model space is of \(1200\times 560\) in grid size (\(\mathrm{dx}=\mathrm{d}\mathrm{z}=6.25\) m). We place sources at shot locations \(\mathrm{x}_{1}^{\mathrm{s}}=\mathrm{i}*4\mathrm{dx},\ \mathrm{i}=0,\ 1,\dots,\ 239\) and \(\mathrm{z}=0\ \mathrm{m}\). The source wavelet is a Ricker of 30 Hz central frequency. For each shot, the receivers are put at \(\mathrm{x}_{\mathrm{j,i}}^{\mathrm{r}}=\mathrm{x}_{1}^{\mathrm{s}}+\mathrm{ jdx},\ \mathrm{j}=0,\ 1,\dots,\ 239\) and \(\mathrm{z}=0\ \mathrm{m}\). The imaging results of RTM and our proposed method are shown in Figure 5a,b, respectively. For a better comparison, we zoom in and display part of the imaging result in Figures 6 and 7. Figure 6 corresponds to the black rectangular part in Figure 4. Figure 7 corresponds to the white rectangular part. In order to compare the amplitude-preserving performance of the two methods, we extract the imaging amplitudes of the two logs (the location of log L1 and log L2 is shown in Figure 4) and compare them with the corresponding theoretical amplitudes. The theoretical amplitude here is obtained by convolution of reflection coefficient and wavelet. Figure 8 is the result of log L1. Figure 9 is the result of log L2.  
Figure 4: The velocity field of the Marmousi model. The red lines show the locations of log L1 and log L2.  
Figure 5: (**a**) The imaging result of reverse time migration (RTM); (**b**) The imaging result of our proposed method.  
Figure 6: (**a**) The theoretical reflection coefficient; (**b**) The imaging result of RTM; (**c**) The imaging result of our proposed method.  
Figure 8: **(a)** Comparison of the imaging amplitudes of RTM (blue line) and the theoretical amplitudes (red line) at log L1; **(b)** Comparison of the imaging amplitudes of our proposed method (blue line) and the theoretical amplitudes (red line) at log L1.  
Figure 7: **(a)** The theoretical reflection coefficient; **(b)** The imaging result of RTM; **(c)** The imaging result of our proposed method.  
Figure 9: **(a)** Comparison of the imaging amplitudes of RTM (blue line) and the theoretical amplitudes (red line) at log L2; **(b)** Comparison of the imaging amplitudes of our proposed method (blue line) and the theoretical amplitudes (red line) at log L2.  
From Figure 5, we can see that both methods can image the model well, but our proposed method is better than the RTM method in imaging of deep structures. Starting from a depth of about 3000 m, the imaging amplitude of the RTM method decreases significantly as the depth increases, and even some structures are missing (for example, the yellow rectangular area in Figure 5). For medium depth, we can see from Figures 6 and 7 that the imaging results of our method have better details than those of the RTM method. In Figures 8 and 9, we can find that the imaging amplitude curve of the RTM method agrees better with the theoretical curve than our method in the shallow part but at medium and deep depths, our method has obviously better performance than RTM. We can intuitively see that the imaging amplitude of the RTM will decrease as the depth increases, but our proposed method can maintain the amplitude very well. It proves that our method has better amplitude-preserving performance. This is why the imaging effect of our method is better than that of RTM in the deep part.