## 2 Forward Equations
### Preliminaries  
Let \((x,y,z)\) be an Euclidean coordinate system with the \(z\)-axis pointing downwards, whereas \(t\) denotes time. We consider a layered lossless isotropic elastic medium, which is characterized by \(P\)-wave velocity \(c^{p}(z)\), \(S\)-wave velocity \(c^{s}(z)\) and mass density \(\rho(z)\). Let \(z_{U}\) and \(z_{L}\) be two depth levels, which are located above and below all heterogeneities in the medium, respectively (hence, constant medium properties are assumed above \(z_{U}\) and below \(z_{l}\)). Elastodynamic wave propagation is considered in the \((x,z)\)-plane, where wavefields are assumed to be constant in the \(y\)-direction. All wavefields are decomposed into flux-normalized up- and downgoing \(P\)-, \(Sv\)- and \(Sh\)-components in the \((p,\tau)\)-domain [35; 36; 37], where \(p\) is the rayparameter and \(\tau\) is the intercept time. The \(Sh\)-components are decoupled from the \(P\)- and \(Sv\)-components and will not be considered in this paper (for notational convenience, component \(Sv\) will be referred to as \(S\)). In Figure 1, we show a layered elastic medium, which will be used throughout our paper as a running example. For convenience, we have chosen the vertical dimension of the model to be \(1\) m. However, all quantities can be rescaled to fit a particular application in, e.g., ultrasound or seismology applications.More information on the design of the medium and the parameters that are used for modeling are provided in Appendix A. Our objective is to retrieve the Green's responses at \(z_{U}\) and \(z_{L}\) to a Green's source at a specified depth level \(z_{l}\) (see the dashed magenta line in Figure 1) from recorded (reflection and transmission) data. These Green's functions are represented by the following matrix:  
\[\mathbf{G}= \begin{pmatrix}-\mathbf{G}_{U}^{-+}\\ -\mathbf{G}_{U}^{-}\\ \mathbf{G}_{L}^{+-}\\ \mathbf{G}_{L}^{++}\end{pmatrix} . \tag{1}\]  
Here, \(\mathbf{G}_{U}^{-+}\) and \(\mathbf{G}_{U}^{--}\) are the upgoing (indicated by the first superscript \(-\)) Green's functions at \(z_{U}\) from a down- and upwards radiating (indicated by the second superscript \(+\) or \(-\)) virtual source at \(z_{l}\), respectively. These quantities contain distinguished \(PP\)-, \(PS\)-, \(SP\)- and \(SS\)-components and are organized as  
\[\mathbf{G}_{u}^{-\pm}= \begin{pmatrix}G_{u,PP}^{-\pm}(p,z,\tau)&G_{u,PS}^{-\pm}(p,z,\tau)\\ G_{u,PS}^{-\pm}(p,z,\tau)&G_{u,SS}^{-\pm}(p,z,\tau)\end{pmatrix} . \tag{2}\]  
Here, subscripts \(PP\) and \(SP\) indicate the \(P\)- and \(S\)-wave responses to a \(P\)-wave source, whereas subscripts \(PS\) and \(SS\) indicate the \(P\)- and \(S\)-wave responses to an \(S\)-wave source. Matrices \(\mathbf{G}_{l}^{+-}\) and \(\mathbf{G}_{l}^{++}\) in Equation (1) represent the downgoing [PERSON]'s functions at \(z_{l}\) from an up- and downwards-radiatingvirtual source at \(z_{l}\), and are organized akin to Equation (2). For the representations of [PERSON]'s functions, we make use of so-called focusing functions [13], which are represented by the matrix  
\[\mathbf{F}= \begin{pmatrix}\mathbf{F}_{u}^{-}\\ \mathcal{Z}\mathbf{F}_{u}^{+}\\ \mathcal{F}_{l}^{-}\\ \mathcal{Z}\mathbf{F}_{L}^{-}\end{pmatrix} . \tag{3}\]  
Here, \(\mathbf{F}_{u}^{-}\) and \(\mathbf{F}_{u}^{+}\) are the up- and downgoing focusing functions (organized akin to Equation (2) at \(z_{l}\). These functions are defined in a fictitious medium where the halfspace below \(z_{l}\) is homogeneous. They focus 'from above' at \(z_{l}\) and continue as a downgoing wavefield below this depth level (see [15] for details). Similarly, \(\mathbf{F}_{l}^{+}\) and \(\mathbf{F}_{L}^{-}\) are the down- and upgoing focusing functions at \(z_{L}\). These functions are defined in a medium where the halfspace above \(z_{l}\) is homogeneous. These functions focus 'from below' at \(z_{i}\) and continue as an upgoing wavefield above this depth level. Finally, \(\mathcal{Z}\) is an operator that reverses the signs of \(p\) and \(\tau\).