# Depth-Extrapolation-Based True-Amplitude Full-Wave-Equation Migration from Topography
## 3 Examples
### Horizontal Layer Model  
In order to achieve the full-wave-equation depth extrapolation for migration, we have three methods to provide the vertical partial derivative of the surface wavefield. The first is the traditional method of using the surface wavefield to calculate the vertical partial derivative under the premise that the surface velocity is uniform; the second is to use the double-layer wavefield recorded by the dual-sensor acquisition system to calculate the vertical partial derivative; the third is the method we proposed. In this experiment, three migration methods are used to compare the imaging results, and we call them Method 1, Method 2, and Method 3. The only difference between these three methods is that they use different ways to provide the vertical partial derivative of the surface wavefield. Method 1 is to use the traditional method to calculate the derivative, Method 2 is to use the dual-sensor acquisition system to provide the derivative, and Method 3 is our proposed depth-extrapolation-based full-wave-equation migration method which uses Equation (8) to calculate the derivative. The model has the dimension 2000 \(\times\) 1000 m, as presented in Figure 2a. The grid spacing is 5 m in both the vertical and horizontal directions. The lower layer of the model has velocity of 4000 m/s. The velocity of the upper layer is \(\mathrm{v}=\mathrm{x}*0.2+1500\) m/s. All sources are located on the surface (\(\mathrm{z}=0\;\mathrm{m}\)). We place sources at shot locations \(\mathrm{x}_{i}^{\mathrm{s}}=\mathrm{i}*4\mathrm{dx},\;\mathrm{i}=0,\;1,\dots,\; 69\). The source wavelet is a Ricker of 40 Hz central frequency. For each shot, the first receivers are put at \(\mathrm{x}_{\mathrm{j,i}}^{\mathrm{r}}=\mathrm{x}_{\mathrm{i}}^{\mathrm{s}}+ \mathrm{jdx},\;\mathrm{j}=0,\;1,\dots,\;119\) on the surface, the second receivers are placed 5 m directly below the first receivers (Method 2 uses a dual-sensor acquisition system, so we set up the second receivers). Figure 2b-d shows the imaging results obtained using three different methods. In order to study the amplitude-preserving performance, we extract the amplitude from the target reflector in the imaging result to estimate the reflection coefficient and compare the calculated reflection coefficient with its corresponding theoretical reflection coefficient. We plot the calculated reflection coefficient curves of the three imaging results and their corresponding theoretical coefficient curves in Figure 3a,c,e. Figure 3b,d,f show the relative error between the calculated reflection coefficient and the theoretical reflection coefficient. Figure 2: (**a**) The velocity model of the horizontal layer; (**b**) The imaging result of Method 1; (**c**) The imaging result of Method 2; (**d**) The imaging result of Method 3. From Figure 2, we can see that the imaging effects of the three methods are very close, and the position of the reflective layer is accurate. In Figure 3, we find that the calculated reflection coefficient curves of Method 2 and 3 are in good agreement with their corresponding theoretical curves, but the calculated reflection coefficient curve of Method 1 is inconsistent with the theoretical curve. Combining Figure 3 and Table 1, we can quantitatively analyze that the relative error of Method 1 is larger than that of the other two methods, and the performance of Method 2 and Method 3 are relatively close. These results indicate that Method 2 and 3 have better amplitude-preserving performance in this experiment. \begin{table}
\begin{tabular}{c c c c} \hline  & **Method 1** & **Method 2** & **Method 3** \\ \hline Mean-square error & 0.0384 & 0.0283 & 0.0274 \\ \end{tabular}
\end{table} 
Table 1: The mean-square error of the relative errors (400–1400 m).