**Appendix A Derivation of Equation (5)**

Equation (4) describes the relationship between the perturbation in the data and the perturbation in the model. Following [22], the inversion of the model perturbations from Equation (4) can be written as follows:

$$\frac{\delta\boldsymbol{\nu}}{\upsilon\_0} + \cos^2\theta \frac{\delta\rho}{\rho\_0} = \iiint \mathbf{B}(\mathbf{x}\_{\boldsymbol{\tau}}; \omega; \mathbf{x}\_{\boldsymbol{\sigma}}; \mathbf{x}; \theta) \delta d(\mathbf{x}\_{\boldsymbol{\tau}}; \mathbf{x}\_{\boldsymbol{\theta}}; \omega) e^{-i\omega \boldsymbol{\Gamma}(\mathbf{x}\_{\boldsymbol{\theta}}; \mathbf{x}; \mathbf{x})} d\mathbf{x}\_{\boldsymbol{\theta}} d\mathbf{x}\_{\boldsymbol{\theta}} d\omega \tag{A1}$$

where **B** is an inverse operator.

Substituting Equation (4) into Equation (A1), we have:

$$\begin{array}{lcl}\frac{\delta\nu}{v\_{0}} + \cos^{2}\theta\frac{\delta\rho}{\rho\_{0}} = -\iiint \mathbf{B}\left(\mathbf{x}\_{\mathbf{r}};\omega;\mathbf{x}\_{s};\mathbf{x};\theta\right) \times\\ \int \frac{2\omega\nu^{2}}{v(\mathbf{x}')^{2}} \left(\frac{\delta\nu(\mathbf{x}')}{v(\mathbf{x}')} + \cos^{2}\theta'\frac{\delta\rho(\mathbf{x}')}{\rho(\mathbf{x}')}\right)\\ \times A(\mathbf{x}\_{\mathbf{s}};\mathbf{x}\_{\mathbf{r}};\mathbf{x}')e^{i\omega\left(T(\mathbf{x}\_{\mathbf{s}};\mathbf{x}\_{\mathbf{r}},\mathbf{x}') - T(\mathbf{x}\_{\mathbf{s}};\mathbf{x},\mathbf{x})\right)} d\mathbf{x}' d\mathbf{x}\_{\mathbf{r}} d\mathbf{x}\_{\mathbf{s}} d\omega\end{array} \tag{A2}$$

In 2D, we change the variables from (**x***r*, **x***s*, *ω*) to (**k**, *θ*) and rewrite Equation (A2) as follows:

$$\begin{split} \frac{\delta\boldsymbol{v}}{\boldsymbol{v}\_{0}} + \cos^{2}\theta \frac{\delta\boldsymbol{\rho}}{\rho\_{0}} &= -\iint \mathbf{B}(\mathbf{x}\_{r};\boldsymbol{\omega};\mathbf{x}\_{s};\mathbf{x};\theta)\delta(\theta^{\prime}-\theta) \times \\ \int \frac{2\omega\mathbf{u}^{2}}{\boldsymbol{v}(\mathbf{x}^{\prime})^{2}} \left(\frac{\delta\boldsymbol{v}(\mathbf{x}^{\prime})}{\boldsymbol{v}(\mathbf{x}^{\prime})} + \cos^{2}\theta \frac{\delta\boldsymbol{\rho}(\mathbf{x}^{\prime})}{\boldsymbol{\rho}(\mathbf{x}^{\prime})}\right) \times A\left(\mathbf{x}\_{\sigma};\mathbf{x}\_{r};\mathbf{x}^{\prime}\right) \boldsymbol{e^{i}}\mathbf{k}(\mathbf{x}^{\prime}-\mathbf{x}) \left|\frac{\partial(\mathbf{x}\_{\nu},\mathbf{x}\_{r},\omega)}{\partial(\mathbf{k}\cdot\theta)}\right| d\mathbf{x}^{\prime}d\mathbf{k}d\theta^{\prime} \end{split} \tag{A3}$$

The inverse operator **B**(**x***r*; *ω*; **x***s*; **x**; *θ*) must satisfy the following equation according to Equation (A3):

$$\frac{2\omega^2}{v^2}A(\mathbf{x}\_{\mathbf{s}};\mathbf{x}\_{\mathbf{r}};\mathbf{x})\mathbf{B}(\mathbf{x}\_{\mathbf{r}};\omega;\mathbf{x}\_{\mathbf{s}};\mathbf{x};\theta)\left|\frac{\partial(\mathbf{x}\_{\mathbf{s}},\mathbf{x}\_{\mathbf{r}},\omega)}{\partial(\mathbf{k},\theta)}\right| = -\left(\frac{1}{2\pi}\right)^2\tag{A4}$$

The Jacobian is derived in Appendix B. Substituting the Jacobian in Equation (A16) into Equation (A4), we can obtain the following:

$$\begin{split} \mathbf{B}(\mathbf{x}\_{r};\omega;\mathbf{x}\_{s};\mathbf{x};\theta) &= -\left(\frac{1}{2\pi}\right)^{2} \frac{\upsilon^{2}}{2\omega^{2}} \times 64\pi^{2} |\omega| \left|\mathbf{q}\right|^{2} A(\mathbf{x}\_{s};\mathbf{x}\_{r};\mathbf{x}) \frac{\cos\beta\_{s}}{v(\mathbf{x}\_{s})} \frac{\cos\beta\_{r}}{v(\mathbf{x}\_{r})} \\ &= -\frac{32\cos^{2}\theta}{|\omega|} A(\mathbf{x};\mathbf{x}\_{s}) A(\mathbf{x}\_{r};\mathbf{x}) \frac{\cos\beta\_{s}}{v(\mathbf{x}\_{r})} \frac{\cos\beta\_{r}}{v(\mathbf{x}\_{r})} \end{split} \tag{A5}$$

By substituting Equation (A5) into Equation (A1), and by using the following relation:

$$\frac{\delta(\rho v)}{\rho\_0 v\_0} = \frac{\delta v}{v\_0} + \frac{\delta \rho}{\rho\_0} \tag{A6}$$

We can obtain the desired relationship in Equation (5):

$$\sin^2\theta \frac{\delta v}{v\_0} + \cos^2\theta \frac{\delta(\rho v)}{\rho\_0 v\_0} = -\iint\iint \frac{32\cos^2\theta'}{|\omega|} \frac{\cos\beta\_r}{v(\mathbf{x}\_\mathbf{r})} \frac{\cos\beta\_\mathbf{s}}{v(\mathbf{x}\_\mathbf{s})} A(\mathbf{x};\mathbf{x}\_\mathbf{s}) A(\mathbf{x}\_\mathbf{r};\mathbf{x}) \tag{A7}$$

### **Appendix B Derivation of the Inverse of the Jacobian in Equation (A4)**

In this appendix, we derive the inverse of the Jacobian *∂*(**x***s*,**x***r*,*ω*) *∂*(**k**,*θ*) following Appendix B from [9].

From the definition of the vector **q** in Figure A1, we have **q** = **p***<sup>s</sup>* + **p***r*. The wavenumber vector **k** is defined as **k** = *ω***q** = ω(**p***<sup>s</sup>* + **p***r*). In the isotropic case, we have |**k**| = <sup>2</sup>|*ω*|cos *<sup>θ</sup> <sup>v</sup>* , and <sup>|</sup>**q**<sup>|</sup> <sup>=</sup> 2cos*<sup>θ</sup> <sup>v</sup>* , where *θ* is the reflection angle and *v* is the wave propagation velocity at the image point. By using the polar representation (|**k**|, *φ*) for the vector **k**, we obtain:

$$\begin{array}{c|c|c|c} & \left| \frac{\partial(\mathbf{k},\theta)}{\partial(\mathbf{x}\_{\ast},\mathbf{x}\_{\ast},\omega)} \right| = & \left| \frac{\partial(\mathbf{k},\theta)}{\partial(\mathbf{k},\phi,\theta)} \right| \left| \frac{\partial(|\mathbf{k}|,\phi,\theta)}{\partial(\mathbf{x}\_{\ast},\mathbf{x}\_{\ast},\omega)} \right| \\ = & |\omega||\mathbf{q}| \left| \frac{\partial(\mathbf{k})}{\partial(\mathbf{x}\_{\ast})} \right| \left| \frac{\partial(\mathbf{k})}{\partial(\mathbf{x}\_{\ast})} \right| \left| \frac{\partial(\boldsymbol{\phi},\theta)}{\partial(\mathbf{x}\_{\ast},\mathbf{x}\_{\ast})} \right| = & |\omega||\mathbf{q}|^{2} \left| \frac{\partial(\boldsymbol{\phi},\theta)}{\partial(\mathbf{x}\_{\ast},\mathbf{x}\_{\ast})} \right| \end{array} \tag{A8}$$

Using the relation:

$$\begin{array}{l} \phi = \frac{\alpha\_s + \alpha\_r}{2} \\ \theta = \frac{\alpha\_s - \alpha\_r}{2} \end{array} \tag{A9}$$

We can have:

$$\left|\frac{\partial(\boldsymbol{\phi},\boldsymbol{\theta})}{\partial(\mathbf{x}\_{\boldsymbol{s}},\mathbf{x}\_{r})}\right| = \left|\frac{\partial(\boldsymbol{\phi},\boldsymbol{\theta})}{\partial(\boldsymbol{\alpha}\_{\boldsymbol{s}},\boldsymbol{\alpha}\_{r})}\right| \left|\frac{\partial(\boldsymbol{\alpha}\_{\boldsymbol{s}},\boldsymbol{\alpha}\_{r})}{\partial(\mathbf{x}\_{\boldsymbol{s}},\mathbf{x}\_{r})}\right| = \left|\frac{\partial(\boldsymbol{\alpha}\_{\boldsymbol{s}})}{\partial(\mathbf{x}\_{\boldsymbol{s}})}\right| \left|\frac{\partial(\boldsymbol{\alpha}\_{r})}{\partial(\mathbf{x}\_{r})}\right|\tag{A10}$$

Plugging Equation (A10) into Equation (A11), we can obtain:

$$\left|\frac{\partial(\mathbf{k},\theta)}{\partial(\mathbf{x}\_{s},\mathbf{x}\_{r},\omega)}\right| = |\omega||\mathbf{q}|^{2} \left|\frac{\partial(\mathbf{a}\_{s})}{\partial(\mathbf{x}\_{s})}\right| \left|\frac{\partial(\mathbf{a}\_{r})}{\partial(\mathbf{x}\_{r})}\right|\tag{A11}$$

From Figure A1, we can see that the angle *φ* is fixed for a given image point **x**. Thus, the angles *θ* and *α<sup>s</sup>* differ only by a constant angle: *θ* = *α<sup>s</sup>* − *φ*. So:

$$\left|\frac{\partial\theta}{\partial a\_s}\right| = 1\tag{A12}$$

$$
\left|\frac{\partial\theta}{\partial\mathbf{x}\_{\rm s}}\right| = \left|\frac{\partial\theta}{\partial\alpha\_{\rm s}}\right| \left|\frac{\partial\alpha\_{\rm s}}{\partial\mathbf{x}\_{\rm s}}\right| = \left|\frac{\partial\alpha\_{\rm s}}{\partial\mathbf{x}\_{\rm s}}\right|\tag{A13}
$$

The expression for the left-hand side of the above equation can be found in [9] as Equation (7): 

$$\left|\frac{\partial\theta}{\partial\mathbf{x}\_{\mathsf{s}}}\right| = 8\pi A^2(\mathbf{x}\_{\mathsf{s}}, \mathbf{x}) \frac{\cos\beta\_{\mathsf{s}}}{\upsilon(\mathbf{x}\_{\mathsf{s}})} \tag{A14}$$

Similarly, we can have:

$$\left|\frac{\partial\theta}{\partial\mathbf{x}\_{r}}\right| = 8\pi A^{2}(\mathbf{x}\_{r},\mathbf{x})\frac{\cos\beta\_{r}}{v(\mathbf{x}\_{r})}\tag{A15}$$

From Equations (A11) and (A14)–(A16), we have:

 

$$\begin{split} \left| \frac{\partial(\mathbf{k}, \boldsymbol{\theta})}{\partial(\mathbf{x}\_{\boldsymbol{\theta}}, \mathbf{x}\_{\boldsymbol{\theta}}, \omega)} \right| &= |\omega| |\mathbf{q}|^{2} \left| \frac{\partial(\boldsymbol{u}\_{\boldsymbol{\theta}})}{\partial(\mathbf{x}\_{\boldsymbol{\theta}})} \right| \left| \frac{\partial(\boldsymbol{u}\_{r})}{\partial(\mathbf{x}\_{r})} \right| \\ = |\omega| |\mathbf{q}|^{2} \times 8\pi A^{2}(\mathbf{x}\_{\boldsymbol{\theta}}, \mathbf{x}) \frac{\cos\beta\_{\boldsymbol{\theta}}}{\overline{v(\mathbf{x}\_{\boldsymbol{\theta}})}} \times 8\pi A^{2}(\mathbf{x}\_{r}, \mathbf{x}) \frac{\cos\beta\_{r}}{\overline{v(\mathbf{x}\_{\boldsymbol{\theta}})}} \\ = 64\pi^{2} |\omega| |\mathbf{q}|^{2} A^{2}(\mathbf{x}\_{\boldsymbol{\theta}}, \mathbf{x}) A^{2}(\mathbf{x}\_{r}, \mathbf{x}) \frac{\cos\beta\_{\boldsymbol{\theta}}}{\overline{v(\mathbf{x}\_{\boldsymbol{\theta}})}} \frac{\cos\beta\_{r}}{\overline{v(\mathbf{x}\_{r})}} \end{split} \tag{A16}$$

**Figure A1.** Coordinates of the 2D ray approximation. **x** is the image point, and **x**s and **x**r are the source and receiver position, respectively; vectors **p**<sup>s</sup> and **p**<sup>r</sup> define the specular ray parameters at the image point from the source and receiver, respectively; vector **q** is defined as the sum of the source ray parameter **p**<sup>s</sup> and receiver ray parameter **p**r; in the specular cases, **q** coincides with the reflector normal vector;αs, αr, and ∅ are the angles with respect to the vertical of the vectors **p**s, **p**<sup>r</sup> and **q**, respectively; θ is the reflection angle with respect to the normal, and 2θ is the angle between **p**<sup>s</sup> and **p**r; β<sup>s</sup> and β<sup>r</sup> are the takeoff angles at the source and receivers, respectively (figure adapted from [9]).
