# Elastic Reverse Time Migration for Weakly Illuminated Structure
## 2 Methodology
### Stained VDWE  
To extract the wavefield associated with the weakly illuminated structure, we establish a governing equation. According to the principle of staining algorithm [9], the variables \(\mathbf{V}\), \(\mathbf{\sigma}\) and \(\mathbf{A}\) in VDWE are extended to complex domain, and possess the following forms:  
\[\mathbf{V}=\mathbf{\overline{V}}+i\mathbf{\widetilde{V}}=[\overline{v_{x}},\overline{v_{z }}]^{T}+i[\widetilde{v_{x}},\widetilde{v_{z}}]^{T}, \tag{5}\]  
\[\mathbf{\sigma}=\mathbf{\overline{\sigma}}+i\mathbf{\widetilde{\sigma}}=[\overline{\tau _{xx}},\overline{\tau_{zz}},\overline{\tau_{xz}}]^{T}+i[\widetilde{\tau_{xx}}, \widetilde{\tau_{zz}},\widetilde{\tau_{xz}}]^{T}, \tag{6}\]  
\[\mathbf{A}=\mathbf{\overline{A}}+i\mathbf{\widetilde{A}}=\left[ \begin{array}{cc}( \overline{c_{P}}+i\widetilde{c_{P}})^{2}&(\overline{c_{P}}+i\widetilde{c_{P} })^{2}-2(\overline{c_{S}}+i\widetilde{c_{S}})^{2}&0\\ (\overline{c_{P}}+i\widetilde{c_{P}})^{2}-2(\overline{c_{S}}+i\widetilde{c_{S} })^{2}&(\overline{c_{P}}+i\widetilde{c_{P}})^{2}&0\\ 0&0&(\overline{c_{S}}+i\widetilde{c_{S}})^{2}\end{array} \right] \tag{7}\]  
where \(i=\sqrt{-1}\), variables with \(-\) and \(\sim\) symbols, respectively, represent the real and imaginary parts. Note that setting \(c_{P}\) and \(c_{S}\) is the core step in extracting the wavefield. \(\overline{c_{P}}\) and \(\overline{c_{S}}\) are consistent with the P- and S-wave velocity. \(\widetilde{c_{P}}\) and \(\widetilde{c_{S}}\) are obtained by a dimensionless multiplied \(\overline{c_{P}}\) and \(\overline{c_{S}}\) near the exploration target, while \(\widetilde{c_{P}}\) and \(\widetilde{c_{S}}\) are equal to \(0\) in the unconcerned area. The location where \(\widetilde{c_{P}}\) and \(\widetilde{c_{S}}\) are non-zero is referred to as the stained target. By substituting Equations (5)-(7) into the VDWE, we obtain,  
\[\left\{ \begin{array}{c}\rho\frac{\partial(\overline{v}+i\widetilde{V})}{ \partial t}=\mathcal{L}(\mathbf{\overline{\sigma}}+i\mathbf{\widetilde{\sigma}})\\ \frac{\partial(\mathbf{\overline{\sigma}}+i\mathbf{\widetilde{\sigma}})}{\partial t}= \rho\Big{(}\mathbf{\overline{A}}+i\mathbf{\widetilde{A}}\Big{)}\mathcal{L}^{T}\Big{(} \mathbf{\overline{V}}+i\mathbf{\widetilde{V}}\Big{)}\end{array} \right. \tag{8}\]  
\[\left\{ \begin{array}{c}\frac{\partial\big{(}\overline{\tau}^{T}+i\widetilde {r}^{P}\big{)}}{\partial t}=\rho\Big{(}\overline{c_{P}}+i\widetilde{c_{P}} \Big{)}^{2}\Big{(}\frac{\partial\big{(}\overline{\tau}^{T}+i\widetilde{c_{P} }\big{)}}{\partial x}+\frac{\partial\big{(}\overline{\tau}^{T}+i\widetilde{c_{ P}}\big{)}}{\partial z}\Big{)}\\ \rho\frac{\partial\big{(}\overline{\tau}^{T}+i\widetilde{c_{P}}\big{)}}{ \partial t}=\frac{\partial\big{(}\overline{\tau}^{T}+i\widetilde{r}^{P}\big{)} }{\partial x}\\ \rho\frac{\partial\big{(}\overline{\tau}^{T}+i\widetilde{r}^{P}\big{)}}{ \partial t}=\frac{\partial\big{(}\overline{\tau}^{T}+i\widetilde{r}^{P}\big{)} }{\partial z}\end{array} \right. \tag{9}\]  
Taking the real parts of Equations (8) and (9), we acquire an equation expressed as,  
\[\left\{\begin{array}{c}\rho\frac{\partial\overline{V}}{\partial t}=\mathcal{ L}\mathbf{\overline{\sigma}}\\ \frac{\partial\overline{V}}{\partial t}=\rho\Big{(}\mathbf{\overline{A}}\mathbf{ \overline{C}}^{T}\mathbf{\overline{V}}-\mathbf{\widetilde{A}}\mathbf{\overline{C}}^{T} \mathbf{\overline{V}}\Big{)}\\ \rho\frac{\partial\overline{v}_{z}^{P}}{\partial t}=\rho\Big{(}\mathbf{\overline{A}} \mathbf{\overline{c_{S}}}+\frac{\partial\widetilde{v_{z}}}{\partial z}\Big{)}-2 \rho\Big{(}\frac{\partial\widetilde{c_{S}}}{\partial x}+\frac{\partial \widetilde{v_{z}}}{\partial z}\Big{)}\\ \rho\frac{\partial\overline{v}_{z}^{P}}{\partialThe imaginary parts of S-waves are defined as \(\widetilde{v_{x}^{S}}=\widetilde{v_{x}}-\widetilde{v_{x}^{P}}\) and \(\widetilde{v_{z}^{S}}=\widetilde{v_{z}}-\widetilde{v_{z}^{P}}\). Equations (10) and (11) jointly constitute the stained VDWE (SVDWE).