## 3 Internal Multiple Elimination Using the Theory of Virtual Events
Substituting Formula (4) into Formula (3):  
\[P_{0}(x_{R},x_{S},\omega)=P(x_{R},x_{S},\omega)-P_{0}(x_{R},x_{S},\omega)\}_{ 0}AP_{0}(x_{S},x_{R},\omega) \tag{5}\]  
Figure 4: Construction of the virtual events and the internal multiples; (**a**) construction of the virtual events; (**b**) construction of the internal multiples. There pentagram represents convolution, the black line and black arrow represent the actual detectable seismic wave paths, red line and red arrow is used to assist in constructing virtual events and internal multiples. In terms of the Neumann series, Formula (5) can be written as follows:  
\[\{P_{0}(x_{R},x_{S},\omega)\}^{(n)}=P(x_{R},x_{S},\omega)-\{P_{0}(x_{R},x_{S}, \omega)\}_{0}^{(n-1)}A^{(n)}P(x_{R},x_{S},\omega), \tag{6}\]  
where \(n\) represents the number of iterations and the expression of surface factor \(A\) is as follows:  
\[ \begin{array}{l}A^{(n)}=(\{P_{0}(x_{R},x_{S},\omega)\}_{0}^{(n-1)})^{-1}(P(x_ {R},x_{S},\omega)\\ \phantom{P_{0}(x_{R},x_{S},\omega)\}^{n})(P(x_{R},x_{S},\omega))^{-1}\end{array}  \tag{7}\]  
In order to obtain more abundant internal multiple information, the multiple iteration theory can be used for multiple models. The multiples after iteration \(n+1\)th can be expressed as follows:  
\[\{M(x_{R},x_{S},\omega)\}^{(n+1)}=\{P_{0}(x_{R},x_{S},\omega)\}_{0}^{(n)}A^{(n )}P(x_{R},x_{S},\omega) \tag{8}\]  
The multiples after \(n\) iterations are expressed as follows:  
\[ \begin{array}{l}\{M(x_{R},x_{S},\omega)\}^{(n)}\ \ =\{P_{0}(x_{R},x_{S},\omega)\}_{0}^{(n-1)}[\{P_{0}(x_{R},x_{S}, \omega)\}_{0}^{(n-2)}]^{-1}\\ \times(P(x_{R},x_{S},\omega)-\{P_{0}(x_{R},x_{S},\omega)\}_{0}^{(n-1)})\end{array}  \tag{9}\]  
Bringing Formula (7) into Formula (9), the internal multiple model after \(n\) iterations is as follows:  
\[ \begin{array}{l}\{M(x_{R},x_{S},\omega)\}^{(n)}\ \ =\{P_{0}(x_{R},x_{S}, \omega)\}_{0}^{(n-1)}(\{P_{0}(x_{R},x_{S},\omega)\}_{0}^{(n-1)})^{T}\\ \times[\{P_{0}(x_{R},x_{S},\omega)\}_{0}^{(n-2)}(\{P_{0}(x_{R},x_{S},\omega) \}_{0}^{(n-2)})^{T}]^{-1}\\ \times(P(x_{R},x_{S},\omega)-\{P_{0}(x_{R},x_{S},\omega)\}_{0}^{(n-1)})\end{array}  \tag{10}\]  
where \(T\) represents matrix transpose. The primary focus of this article is on the suppression of internal multiples in post-stack profiles. In the post-stack profile, firstly, we can employ the tracing of horizons method to obtain the primary events. Secondly, by shifting the window up and down, we can acquire the primary data volume and other primary data volume below this primary used to construct the virtual events. Subsequently, we input the seismic data obtained, the iterative virtual event internal multiple suppression technique initially utilizes the virtual events internal multiple methods, such as Formulas (1) and (2), to construct the internal multiple models for the corresponding layer. Finally, through iterative updates, such as Formula (10), a more comprehensive multiple model is obtained, ultimately leading to the suppression of the multiple results. This approach not only enhances the computational efficiency, accuracy, and applicability but also reduces the requirements for the original input data, thereby improving imaging accuracy, which can make it widely applicable in the processing of field data.