## Appendix B Derivation of the Inverse of the Jacobian in Equation (11)  
In this appendix, we derive the inverse of the Jacobian \(\Big{|}\frac{\partial(\mathbf{x}_{s},\mathbf{x}_{r},\omega)}{\partial( \mathbf{k},\theta)}\Big{|}\) following Appendix B from [9]. From the definition of the vector \(\mathbf{q}\) in Figure 11, we have \(\mathbf{q}=\mathbf{p}_{s}+\mathbf{p}_{r}\). The wavenumber vector \(\mathbf{k}\) is defined as \(\mathbf{k}=\omega\mathbf{q}=\omega(\mathbf{p}_{s}+\mathbf{p}_{r})\). In the isotropic case, we have \(|\mathbf{k}|=\frac{2|\omega|\cos\theta}{v}\), and \(|\mathbf{q}|=\frac{2\cos\theta}{v}\), where \(\theta\) is the reflection angle and \(v\) is the wave propagation velocity at the image point. By using the polar representation \((|\mathbf{k}|,\phi)\) for the vector \(\mathbf{k}\), we obtain:  
\[\begin{array}{c}\left|\frac{\partial(\mathbf{k},\theta)}{\partial(\mathbf{x }_{s},\mathbf{x}_{r},\omega)}\right|=\Big{|}\frac{\partial(\mathbf{k},\theta)} {\partial(\mathbf{k}|,\theta)}\Big{|}\Big{|}\frac{\partial(\mathbf{k},\theta) }{\partial(\mathbf{x}_{s},\mathbf{x}_{r},\omega)}\Big{|}\\ =|\omega||\mathbf{q}||\Big{|}\frac{\partial(\mathbf{k})}{\partial(\omega)} \Big{|}\Big{|}\frac{\partial(\mathbf{k},\theta)}{\partial(\mathbf{x}_{s}, \mathbf{x}_{r})}\Big{|}=|\omega||\mathbf{q}|^{2}\Big{|}\frac{\partial(\phi, \theta)}{\partial(\mathbf{x}_{s},\mathbf{x})}\Big{|} \tag{15}\]  
Using the relation:  
\[ \begin{cases}\phi=\frac{\alpha_{s}+\alpha_{s}}{2}\\ \theta=\frac{\alpha_{s}-2\alpha_{s}}{2}\end{cases}  \tag{16}\]  
We can have:  
\[\bigg{|}\frac{\partial(\phi,\theta)}{\partial(\mathbf{x}_{s},\mathbf{x}_{r})} \bigg{|}=\bigg{|}\frac{\partial(\phi,\theta)}{\partial(\alpha_{s},\alpha_{r}) }\bigg{|}\bigg{|}\frac{\partial(\alpha_{s},\alpha_{r})}{\partial(\mathbf{x}_{s}, \mathbf{x}_{r})}\bigg{|}=\bigg{|}\frac{\partial(\alpha_{s})}{\partial(\mathbf{x}_ {s})}\bigg{|}\bigg{|}\frac{\partial(\alpha_{r})}{\partial(\mathbf{x}_{r})}\bigg{|} \tag{17}\]  
Plugging Equation (17) into Equation (10), we can obtain:  
\[\bigg{|}\frac{\partial(\mathbf{k},\theta)}{\partial(\mathbf{x}_{s},\mathbf{x}_{ r},\omega)}\bigg{|}=|\omega||\mathbf{q}|^{2}\bigg{|}\frac{\partial(\alpha_{s})}{ \partial(\mathbf{x}_{s})}\bigg{|}\bigg{|}\frac{\partial(\alpha_{r})}{\partial( \mathbf{x}_{r})}\bigg{|} \tag{18}\]From Figure A1, we can see that the angle \(\phi\) is fixed for a given image point \(\mathbf{x}\). Thus, the angles \(\theta\) and \(\alpha_{s}\) differ only by a constant angle: \(\theta=\alpha_{s}-\phi\). So:  
\[\left|\frac{\partial\theta}{\partial\alpha_{s}}\right|=1\] (A12)  
\[\left|\frac{\partial\theta}{\partial\mathbf{x}_{s}}\right|=\left|\frac{ \partial\theta}{\partial\alpha_{s}}\right|\left|\frac{\partial\alpha_{s}}{ \partial\mathbf{x}_{s}}\right|=\left|\frac{\partial\alpha_{s}}{\partial \mathbf{x}_{s}}\right|\] (A13)  
The expression for the left-hand side of the above equation can be found in [9] as Equation (7):  
\[\left|\frac{\partial\theta}{\partial\mathbf{x}_{s}}\right|=8\pi A^{2}(\mathbf{ x}_{s},\mathbf{x})\frac{\cos\beta_{s}}{v(\mathbf{x}_{s})}\] (A14)  
Similarly, we can have:  
\[\left|\frac{\partial\theta}{\partial\mathbf{x}_{r}}\right|=8\pi A^{2}(\mathbf{ x}_{r},\mathbf{x})\frac{\cos\beta_{r}}{v(\mathbf{x}_{r})}\] (A15)  
From Equations (A11) and (A14)-(A16), we have:  
\[ \begin{array}{c}\left|\frac{\partial(\mathbf{k},\theta)}{\partial(\mathbf{x }_{s},\mathbf{x},\omega)}\right|=\left|\omega||\mathbf{q}^{2}\right|^{\frac{ \partial(\mathbf{q}_{s})}{\partial(\mathbf{x}_{s})}}\left|\right|\frac{ \partial(\alpha_{s})}{\partial(\mathbf{x}_{s})}\left|\right|\\ =\left|\omega||\mathbf{q}^{2}\right|^{2}\times 8\pi A^{2}(\mathbf{x}_{s}, \mathbf{x})\frac{\cos\beta_{s}}{v(\mathbf{x}_{s})}\times 8\pi A^{2}(\mathbf{x}_{r}, \mathbf{x})\frac{\cos\beta_{r}}{v(\mathbf{x}_{r})}\\ =64\pi^{2}|\omega||\mathbf{q}^{2}A^{2}(\mathbf{x}_{s},\mathbf{x})A^{2}( \mathbf{x}_{r},\mathbf{x})\frac{\cos\beta_{s}}{v(\mathbf{x}_{s})}\frac{\cos \beta_{r}}{v(\mathbf{x}_{r})}\end{array} \] (A16)  
Figure A1: Coordinates of the 2D ray approximation.