## 2 Methodology  
### Theory of Shot-Domain Migration Aided by Attribute Migration to Output SOGs  
The equation using the Born equation to express the prestack shot-domain depth migration is:  
\[I(\mathbf{x},\,\mathbf{s})=\iint G(\mathbf{x},\,t|\mathbf{s},\,0)G(\mathbf{x},-t|\mathbf{r},\,0)* \ddot{d}(\mathbf{r},\,t|\mathbf{s},\,0)\mathrm{d}\mathbf{r}\mathrm{d}t \tag{1}\]  
where \(\mathbf{x}\) is the location of an imaging point; \(\mathbf{s}\) is the location of the shot; \(\mathbf{r}\) is the location of the receiver; \(I(\mathbf{x},\mathbf{s})\) is the imaging result of the shot \(\mathbf{s}\) at the point \(\mathbf{x}\); \(G(\mathbf{x},\,t|\mathbf{s},0)\) is [PERSON]'s function for forward propagation of shot-point data; \(G(\mathbf{x},-t|\mathbf{r},0)\) is [PERSON]'s function for backward propagation of receiver data; \(\ddot{d}(\mathbf{r},\,t\,|\,\mathbf{s},\,0)\) is the second derivative at the shot \(\mathbf{s}\) and the receiver \(\mathbf{r}\) versus time [24]. With a migration calculation performed twice, the quantitative parameters weighted on the data can be estimated, which is described as the following process:  
\[I_{\rm w}(\mathbf{x},\,\,\mathbf{s})=\iint G(\mathbf{x},\,\,t|\mathbf{s},\,\,0)G(\mathbf{x},-t|\bm {r},\,0)*(\ddot{d}(\mathbf{r},\,\,t|\mathbf{s},\,\,0)\cdot w(\mathbf{x}|\mathbf{s},\,\,\mathbf{r})) \mathrm{d}\mathbf{r}\mathrm{d}t, \tag{2}\]  
where \(w(\mathbf{x}\,|\,\mathbf{s},\,\mathbf{r})\) are the weighting parameters associated with the locations of the shot and the receiver at the imaging point, acting directly on the pre-migration data, which may be the angles of incidence and reflection and the coordinates of the shot and the receiver. The weighting parameter can be estimated with the ratio of the results from the two migrations using Equations (1) and (2). \[w(\mathbf{x}|\mathbf{s},\,\,\mathbf{r})\approx\frac{I_{\rm w}(\mathbf{x},\,\,\mathbf{s})}{I(\mathbf{x},\,\,\mathbf{s})}. \tag{3}\]  
If the weighting coefficient \(w(\mathbf{x}\,|\,\mathbf{s},\,\mathbf{r})\) is the offset attribute, the value of the offset at the location of the imaging point can be calculated, thus outputting the SOGs. Equations (1) to (3) are simplified to the cross-correlation imaging conditions of RTM and are specifically expressed as:  
\[I(\mathbf{x},\,\,\mathbf{s}) =\int u(\mathbf{x},\,\,\mathbf{s},\,\,t)\int v(\mathbf{x},\,\,\mathbf{s},\,\,\mathbf{ r},\,\,t)\mathrm{d}\mathbf{r}\mathrm{d}t, \tag{4}\]  
where \(u(\mathbf{x},\,\,\mathbf{s},\,\,t)\) is the shot wavefield at the time \(t\); \(v(\mathbf{x},\,\mathbf{s},\,\,\mathbf{r},\,\,t)\) is the receiver wavefield at \(\mathbf{s}\) at the time \(t\). Multiplying the shot-receiver distance \(\mathbf{h}=\mathbf{s}-\mathbf{r}\) and the shot-domain receiver data and then migrating the product results in:  
\[I_{\rm h}(\mathbf{x},\,\,\mathbf{s}) =\int u(\mathbf{x},\,\,\mathbf{s},\,\,t)\int\mathbf{h}v(\mathbf{x},\,\,\mathbf{s},\, \,\mathbf{r},\,\,t)\mathrm{d}\mathbf{r}\mathrm{d}t. \tag{5}\]  
Using the ratio between Equations (4) and (5), the shot-receiver distance obtained from a single shot at the imaging point \(\mathbf{x}\) is:  
\[h_{A}(\mathbf{x},\,\,\mathbf{s})=\frac{I_{\rm h}(\mathbf{x},\,\,\mathbf{s})}{I(\mathbf{x},\,\,\bm {s})}. \tag{6}\]  
According to the obtained values of the offset at various points in the imaging domain, imaging results are placed from every shot to the proper SOG. The final SOGs can be obtained by calculating the records at all shots successively and then iterating by  
\[I(\mathbf{x},\,\,\mathbf{h})=\int I(\mathbf{x},\,\,\mathbf{s})\delta(\mathbf{h}-h_{A}(\mathbf{x},\,\, \mathbf{s}))\mathrm{d}\mathbf{s}, \tag{7}\]  
where \(\delta\) is the pulse function. As for 3D data, Equations (4)-(6) can be used to calculate the horizontal and vertical offset, respectively, and then obtain the total value of the offset.