## 2 Method  
### Multiples and Artifacts Analysis  
The existence of multiples could cause artifacts in a LSRTM image. Referring to [PERSON] et al. [44], the discretized imaging conditions can be written as:  
\[\text{Image}(\mathbf{x})=\sum_{0}^{t\text{max}}u_{f}u_{b} \tag{1}\]  
The forward and backward-propagated wavefields can be written as:\[u_{f}=u_{f,0}+u_{f,1}+u_{f,2}+\ldots \tag{2}\]  
\[u_{b}=u_{b,0}+u_{b,1}+u_{b,2}+\ldots \tag{3}\]  
where, \(u_{f,0}\) is the primary of the forward-propagated wavefields, \(u_{f,1}\) is the first-order multiple of the forward-propagated wavefields, \(u_{f,2}\) is the second-order multiple of the forward-propagated wavefields; \(u_{b,0}\) is the backward-propagated wavefields of the primary, \(u_{b,1}\) is the backward-propagated wavefields of the first-order multiple, \(u_{b,2}\) is the backward-propagated wavefields of the second-order multiple.  
Substitute in Equation (1):  
\[\begin{array}{l}\text{Image}(\textbf{x})=\sum\limits_{0}^{t_{\text{max}}} \left\{u_{f,0}u_{b,0}+u_{f,1}u_{b,1}+u_{f,2}u_{b,2}+\ldots\right\}\\ \quad\quad\quad+\sum\limits_{0}^{t_{\text{max}}}\left\{\begin{array}{l}u_{f, 0}u_{b,1}+u_{f,0}u_{b,2}+u_{f,0}u_{b,3}+\ldots\\ +u_{f,1}u_{b,2}+u_{f,1}u_{b}^{3}+u_{f,1}u_{b,4}+\ldots\\ +u_{f,2}u_{b,3}+\ldots\end{array}\right\}\\ \quad\quad\quad+\sum\limits_{0}^{t_{\text{max}}}\left\{\begin{array}{l}u_{f, 1}u_{b,0}+\\ u_{f,2}u_{b,0}+u_{f,2}u_{b,1}+\\ u_{f,3}u_{b,0}+u_{f,3}u_{b,1}+u_{f,3}u_{b,2}+\ldots\end{array}\right\}\end{array} \tag{4}\]  
As Figure 1 shows, nth-order internal multiples propagate along the o-s direction and, after reflection, form (n + 1)th-order internal multiples along s-A-B-r1 and s-A-B', and after reflection, form (n + 2)th-order internal multiples along B-c-r2. The first term on the right side of Equation (4) is the cross-correlation between the forward-propagated wavefields and the backward-propagated wavefields of the primary and corresponding order internal multiples received by the receiver. As Figure 1 shows, it is the cross-correlation between the (n + 1)th-order internal multiple follows the path of s-A and the backward-propagated wavefields of the (n + 1)th-order follows the path of r1-B-A, and it is imaged on A, resulting in a correct image. The second term is the cross-correlation between the forward-propagated wavefields and the backward-propagated wavefields of the higher order internal multiple received by the receiver. As Figure 1 shows, it is the cross-correlation between the (n + 1)th-order internal multiple follows the path of s-A-B' and the backward-propagated wavefields of the (n + 2)th-order follows the path of r2-C-B'. This is imaged on B', making an artifact. The third term is the cross-correlation between the forward-propagated wavefields and the backward-propagated wavefields of the lower order internal multiple received by the receiver that was not imaged.  
Figure 1: The path of internal multiple and the generation of image artifacts. The nth-, (n + 1)th- and (n + 2)th-order internal multiples follow the path of o-s, s-A–B–r1, and B-C-r2. The nth-order internal  
multiple generates the (n + 1)th-order internal multiple, and the (n + 1)th-order internal multiple generates the (n + 2)th-order internal multiple. The dashed line r1-B-A and r2-C-B' represents the path of backward-propagated wavefield of the (n + 1)th-order internal multiple and (n + 2)th-order internal multiple, respectively.