## 3 Inversion
### Initialization  
The direct arrival \(G_{Ud}^{--pp}\) that is required for the construction of \(\mathbf{G}_{Ud}^{--}\) can be expressed in terms of a delayed unit pulse \(\delta(\tau-\tau_{Ud}^{p})\) and an amplitude scaling factor, which we parameterize strategically as \(-\alpha^{\frac{1}{2}}\) for some (yet-unknown) \(\alpha\). This leads us to obtain  
\[\mathbf{G}_{Ud}^{--}=\begin{pmatrix}G_{Ud}^{--pp}&0\\ 0&0\end{pmatrix}=-\underbrace{\begin{pmatrix}\delta(\tau-\tau_{Ud}^{p})&0\\ 0&0\end{pmatrix}}_{\mathbf{E}_{Ud}^{p}}\alpha^{\frac{1}{2}}. \tag{20}\]  
The direct arrival, \(G_{Ud}^{++pp}\), that is required for the construction of \(\mathbf{G}_{Ud}^{++}\) can be related to \(G_{Ud}^{--pp}\) by the \(1D\) convolutional model \(T_{LUd}^{+PP}(\tau)=-\int_{-\infty}^{+\infty}G_{Ud}^{++PP}(\tau-\tau^{\prime})G _{Ud}^{--pp}(\tau^{\prime})\mathrm{d}\tau^{\prime}\). Here, \(T_{LUd}^{+pp}\) is the first event of the \(PP\)-component of the recorded transmission response \(\mathbf{T}_{LU}^{+}\) from \(z_{l}\) to \(z_{L}\). We assume that this event can be isolated from the transmission data by means of a time gate. Next, we define operator \(\mathcal{T}_{d}^{p}\) for the \(1D\) convolution of any signal with \(T_{LUd}^{+PP}\). The direct Green's function at the lower level \(G_{LU}^{++pp}\) may now be obtained by applying \(\mathcal{T}_{d}^{p}\) to the inverse of \(-G_{Ud}^{--pp}\), which can be expressed in our notation as \(\mathcal{Z}\mathbf{E}_{Ud}^{p}\alpha^{-\frac{1}{2}}\) (with \(\mathbf{E}_{Ud}^{p}\) as defined in Equation (20). This leads to  
\[\mathbf{G}_{LU}^{++}=\begin{pmatrix}G_{Ud}^{++pp}&0\\ 0&0\end{pmatrix}=\mathcal{T}_{d}^{p}\mathcal{Z}\mathbf{E}_{Ud}^{p}\alpha^{- \frac{1}{2}}. \tag{21}\]  
Similarly, the direct wave \(F_{Ud}^{+SS}\) that is required for the construction of \(\mathbf{F}_{Ud}^{+}\) can be expressed in terms of a time-advanced unit pulse \(\delta(\tau+\tau_{Ud}^{SS})\) and an amplitude scaling factor, which we parameterize strategically as \(\beta^{\frac{1}{2}}\) for some (yet-unknown) \(\beta\). This leads to  
\[\mathbf{F}_{Ud}^{+}=\begin{pmatrix}0&0\\ 0&F_{Ud}^{+SS}\end{pmatrix}=\underbrace{\begin{pmatrix}0&0\\ 0&\delta(\tau+\tau_{Ud}^{SS})\end{pmatrix}}_{Z\mathbf{F}_{Ud}^{SS}}\beta^{\frac{ 1}{2}}. \tag{22}\]  
The direct arrival \(F_{Ld}^{-SS}\) that is required for the construction of \(\mathbf{F}_{Ld}^{-}\) can be related to \(F_{Ud}^{+SS}\) by the \(1D\) convolutional model \(H_{LUd}^{+SS}(\tau)=\int_{-\infty}^{+\infty}F_{Ld}^{-SS}(\tau-\tau^{\prime})F_{ Ud}^{+SS}(\tau^{\prime})\mathrm{d}\tau^{\prime}\). Here, \(H_{LUd}^{+SS}\) isthe first event of the \(SS\)-component of the inverse transmission response \(\mathbf{H}^{+}_{LU}\) from \(z_{u}\) to \(z_{L}\), which can be obtained via the inversion of \(\mathbf{I}S(\tau)=\int_{\infty}^{+\infty}\mathbf{T}^{+}_{LU}(\tau-\tau^{\prime} )\mathbf{H}^{+}_{LU}(\tau^{\prime})\mathrm{d}\tau^{\prime}\), where \(S(\tau)=\int_{-\infty}^{\infty}s(\tau+\tau^{\prime})s(\tau^{\prime})d\tau^{\prime}\) is the autocorrelation of the source signal \(s(\tau)\), as defined in Appendix A. We assume that \(H^{+SS}_{LU}\) can be isolated from \(H^{+SS}_{LU}\) by means of a time gate. Next, we define an operator \(\mathcal{H}^{s}_{L}\) for \(1D\) convolution with \(H^{+SS}_{LU}\). The direct part of the focusing function at the lower level \(F^{-SS}_{LU}\) may now be obtained by applying this operator to the inverse of \(F^{+SS}_{LU}\), which can be expressed in our notation as \(\mathbf{E}^{\mathrm{s}}_{LU}\beta^{-\frac{1}{2}}\) (with \(\mathbf{E}^{\mathrm{s}}_{LU}=\mathcal{Z}\mathcal{Z}\mathbf{E}^{\mathrm{s}}_{ LU}\) as defined in Equation (22). This leads to  
\[\mathbf{F}^{-}_{LU}=\begin{pmatrix}0&0\\ 0&F^{-SS}_{LU}\end{pmatrix}=\mathcal{H}^{s}_{d}\mathbf{E}^{\mathrm{s}}_{LU} \beta^{-\frac{1}{2}}. \tag{23}\]  
We assume that the travel times \(\tau^{\prime}_{d}\) and \(\tau^{\mathrm{s}}_{d}\) are known. The amplitude scaling factors \(\alpha\) and \(\beta\) can be estimated from the data, as we discuss in the following section.