## 2 Forward Equations
### Marchenko Equations  
In Appendix B.1, we derive the following system of [PERSON]'s function representations that are based on reflection data:  
\[\underbrace{ \begin{pmatrix}-\mathbf{G}_{U}^{-+}\\ -\mathbf{G}_{U}^{-}\\ \mathbf{G}_{L}^{+-}\end{pmatrix} }_{\mathbf{G}}= \begin{pmatrix}\mathbf{I}&- \mathcal{R}_{U}\mathcal{Z}&\mathbf{O}&\mathbf{O}\\ -\mathcal{R}_{U}\mathcal{Z}&\mathbf{I}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{I}&-\mathcal{R}_{L}\mathcal{Z}\\ \mathbf{O}&\mathbf{O}&-\mathcal{R}_{L}\mathcal{Z}&\mathbf{I}\end{pmatrix} }_{ \mathbf{F}}\underbrace{ \begin{pmatrix}\mathbf{F}_{U}^{-}\\ \mathcal{Z}\mathbf{F}_{L}^{+}\\ \mathcal{Z}\mathbf{F}_{L}^{-}\end{pmatrix} }_{\mathbf{F}}, \tag{10}\]  
where \(\mathbf{I}\) is a \(2\times 2\) identity matrix. When we apply the operator \(\mathbf{\Theta}^{[P]}\) to both sides of this equation, it follows, with the help of Equation (7), that  
\[\underbrace{ \begin{pmatrix}\mathbf{O}\\ -\mathbf{G}_{Ud}^{-+}\\ \mathbf{G}_{Ld}^{++}\end{pmatrix} }_{\mathbf{G}_{d}}=\mathbf{\Theta}^{[P]} \begin{pmatrix}\mathbf{I}&-\mathcal{R}_{U}\mathcal{Z}&\mathbf{O}&\mathbf{O} \\ -\mathcal{R}_{U}\mathcal{Z}&\mathbf{I}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{I}&-\mathcal{R}_{L}\mathcal{Z}\\ \mathbf{O}&\mathbf{O}&-\mathcal{R}_{L}\mathcal{Z}&\mathbf{I}\end{pmatrix} }_{ \mathbf{F}}\underbrace{ \begin{pmatrix}\mathbf{F}_{U}^{-}\\ \mathcal{Z}\mathbf{F}_{L}^{+}\\ \mathcal{Z}\mathbf{F}_{L}^{-}\end{pmatrix} }_{\mathbf{F}}. \tag{11}\]  
Next, we may substitute Equation (9) and rewrite the result as  
\[\underbrace{ \begin{pmatrix}\mathbf{G}_{U}^{[P]}\mathcal{R}_{U}\mathbf{F}_{Ud}^ {+}\\ -\mathbf{G}_{Ud}^{--}\\ \mathbf{G}_{L}^{+}\end{pmatrix} }_{\mathbf{B}_{Mar}}=\underbrace{\mathbf{ \Theta}^{[P]} \begin{pmatrix}\mathbf{I}&-\mathcal{R}_{U}\mathcal{Z}&\mathbf{O} &\mathbf{O}\\ -\mathcal{R}_{U}\mathcal{Z}&\mathbf{I}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{I}&-\mathcal{R}_{L}\mathcal{Z}\\ \mathbf{O}&\mathbf{O}&-\mathcal{R}_{L}\mathcal{Z}&\mathbf{I}\end{pmatrix} }_{ \mathbf{O}}\mathbf{\Theta}^{(S)}\underbrace{ \begin{pmatrix}\mathbf{F}_{U}^{-} \\ \mathcal{Z}\mathbf{F}_{Um}^{+}\\ \mathcal{Z}\mathbf{F}_{L}^{+}\\ \mathcal{Z}\mathbf{F}_{Lm}^{-}\end{pmatrix} }_{\mathbf{F}}. \tag{12}\]  
Here, we have used the fact that \(\tau_{d}^{S}\
otin[-\tau_{d}^{p},\tau_{d}^{p}]\), such that \(\Theta_{U}^{[P]}\mathcal{Z}\mathbf{F}_{Ud}^{+}=\mathbf{O}\) and \(\Theta_{U}^{[P]}\mathcal{Z}\mathbf{F}_{L}^{-}=\mathbf{O}\). We refer to Equation (12) as a system of reflection-based Marchenko equations, which could be inverted for the unknown components of the focusing function \(\mathbf{F}_{m}\). The block-diagonal structure of this system reveals that the Marchenko equations at \(z_{l}\) and \(z_{t}\) are decoupled. Operator \(\mathbf{\Theta}^{(5)}\) restricts the unknown focusing function \(\mathbf{F}_{m}\) to the interval \((-\tau_{d}^{S},\tau_{d}^{S})\). Matrix \(\mathbf{A}_{Mar}\) projects \(\mathbf{F}_{m}\) to the interval \([-\tau_{d}^{p},\tau_{d}^{p}]\). As \(\tau_{d}^{p}<\tau_{d}^{S}\), this leads to an underdetermined system of equations, which cannot be unconditionally inverted. We illustrate this by plotting the singular values of \(\mathbf{A}_{Mar}\) in Figure (a)a (red curve) for data from the model that we presented above in Figure 1. In this case, \(\mathbf{A}_{Mar}\) contains 2192 columns but only 928 independent rows. To increase the rank of this matrix, we propose to use auxiliary transmission data, for which we derive a similar system of equations in the following section. We refer to 15 as a system of auxiliary equations, which can be interpreted as a transmission-based inverse problem for \(\mathbf{F}_{m}\). In Figure 2(a) (green curve), we show that the governing matrix \(\mathbf{A}_{Aux}\) of this problem is rank-deficient (at least for the medium in Figure 1). Nevertheless, this matrix can provide complementary information to \(\mathbf{A}_{Mar}\), as illustrated in Figure 2(b) (orange curve).