# Algorithm for Extraction of Reflection Waves in Single-Well Imaging Based on MC-ConvTasNet
## Appendix A Expressions for Direct and Reflected Waves  
Excited by a dipole source along the x-axis, the displacement expression in the fluid is [26]  
\[u_{1}=\
abla(\phi_{d}+\phi_{r}), \tag{1}\]  
where \(\phi_{d}\) and \(\phi_{r}\) are the displacement potentials associated with the source and reflected from the borehole wall, respectively; and \(\
abla\) is the Laplacian operator in the cylindrical coordinate system. The displacement potential function in the borehole can be expressed as  
\[\phi_{d}(r,\varphi,z,\omega)=\frac{-V_{0}T(\omega)}{4\pi^{2}}\int \limits_{-\infty}^{+\infty}r_{0}fK_{1}(fr)\mathrm{e}^{\mathrm{i}kz}\cos\varphi \mathrm{d}k, \tag{2}\] \[\phi_{r}(r,\varphi,z,\omega)=\frac{-V_{0}T(\omega)}{4\pi^{2}} \int\limits_{-\infty}^{+\infty}\left(\frac{r_{0}f}{2}\right)A(k,\omega)I_{1}( fr)\mathrm{e}^{\mathrm{i}kz}\cos\varphi\mathrm{d}k,\]where \(f^{2}=k^{2}-\frac{\omega^{2}}{a_{f}^{2}};f\) and \(k\) are the radial wavenumber and axial wavenumber of fluid; \(w\) is the angular frequency; \(\alpha_{f}\) is the fluid velocity in the borehole; \(I_{1}\) and \(K_{1}\) are the first-order modified Bessel function of the first and second kinds, respectively; \(r_{0}\) is the radius of the dipole source; \(T(\omega)\) represents the spectrum of the time function of the acoustic source; \(V_{0}\) represents the peak amplitude of the source volume change; \(\varphi\) indicates the azimuth angle; and the coefficient \(A\) can be obtained from borehole boundary conditions. The direct wave excited by a dipole source is a flexural wave whose waveform can be calculated by the real axis integration method [27] according to Formula (A1). Excited by a dipole source along the x-axis, the reflected displacement along the x-axis received in the borehole is [28]  
\[u_{2}=u_{\text{SH}}+u_{\text{SV}}+u_{\text{P}}+u_{\text{SP}}+u_{\text{PS}},\] (A3)  
where  
\[ \begin{array}{l}u_{SH}=T(\omega)R_{SH}\text{sin}^{2}\varphi,\\ u_{SV}=T(\omega)R_{SV}\text{cos}^{2}\varphi,\\ u_{P}=T(\omega)R_{P}\text{cos}^{2}\varphi,\\ u_{PS}=T(\omega)R_{PS}\text{cos}^{2}\varphi,\\ u_{SP}=T(\omega)R_{SP}\text{cos}^{2}\varphi,\end{array} \] (A4)  
\[ \begin{array}{l}R_{SH}=\frac{P_{0}R_{0}}{L\theta^{2}}\text{e}^{i\omega L/ \beta}G_{SH}F_{SH}[k_{S}(\theta_{1}),\omega]F_{SH}[k_{S}(\theta_{2}),\omega],\\ R_{SV}=\frac{P_{0}R_{0}}{L\theta^{2}}\text{e}^{i\omega L/\beta}G_{SV}F_{SV}[k_ {S}(\theta_{1}),\omega]F_{SV}[k_{S}(\theta_{2}),\omega],\\ R_{P}=\frac{P_{0}R_{0}}{L\theta^{2}}\text{e}^{i\omega L/\alpha}G_{P}F_{P}[k_ {P}(\theta_{1}),\omega]F_{P}[k_{P}(\theta_{2}),\omega],\\ R_{PS}=\frac{P_{0}R_{0}}{L\theta^{2}}\text{e}^{i\omega L/\beta}G_{PS}F_{P}[k_ {P}(\theta_{1}^{\text{P}}),\omega]F_{SV}\left[k_{S}(\theta_{2}^{\text{S}}), \omega\right],\\ R_{SP}=\frac{P_{0}R_{0}}{L\theta^{2}}\text{e}^{i\omega L_{P}/\alpha}G_{SP}F_{ SV}\left[k_{S}(\theta_{1}^{\text{P}}),\omega\right]F_{P}\left[k_{P}( \theta_{2}^{\text{P}}),\omega\right],\end{array} \] (A5)  
\[ \begin{array}{l}F_{\text{P}}(k_{P},\omega)=\frac{fC(k_{P},\omega)}{2\rho/ \omega}j\rho\alpha,\\ F_{\text{SH}}(k_{S},\omega)=-\frac{fD(k_{S},\omega)}{2\rho_{f}}j\rho\sin\theta,\\ F_{\text{SV}}(k_{S},\omega)=-\frac{fE(k_{S},\omega)}{2\rho_{f}}\rho\sin\theta.\end{array}