## 2 Forward Equations
### Causality Cones of Green's Functions and Focusing Functions  
In moderately inhomogeneous acoustic media, the Green's function and the focusing function are separated in time, except for a single overlapping event, which is commonly referred to as the _direct wave_. In the derivation of the acoustic Marchenko equation, this observation is exploited by truncating wavefields either before [2,11] or after [38,39] the direct wave. In elastic media, there can be a multitude of overlapping events, making the situation significantly more cumbersome [16]. To illustrate this problem, we show the (symmetrized) causality cones of multi-component Green's functions and focusing functions in Figure 2. In particular, we refer the reader to the orange areas, where the Green's functions and focusing functions may overlap. Because of this potential overlap, we have designed two distinct time window operators: one for [PERSON]'s functions, which is based on the (non-converted) direct \(P\)-wave travel time \(\tau_{d}^{p}\) and one for focusing functions, which is based on the (non-converted) direct \(S\)-wave travel time \(\tau_{d}^{S}\). First, we discuss the window operator for Green's functions, which is based on the travel time \(\tau_{d}^{p}\) of the (non-converted) \(P\)-wave, propagating from \(z_{l}\) outwards. In Figure 2a, we can see that the Green's function and its time-reversed counterpart vanish in the interval \((-\tau_{d}^{p},\tau_{d}^{p})\). Let \(\mathbf{G}_{Ud}^{--}= \begin{pmatrix}G_{Ud,PP}^{--}&0\\ 0&0\end{pmatrix} \) and \(\mathbf{G}_{Ld}^{++}= \begin{pmatrix}G_{Ud,PP}^{++}&0\\ 0&0\end{pmatrix} \) be the components of \(\mathbf{G}_{U}^{--}\) and \(\mathbf{G}_{L}^{++}\) that reside at the boundary of the interval \([-\tau_{d}^{p},\tau_{d}^{p}]\) (corresponding to the direct non-converted \(P\)-wave transmissions). Now, we can partition the [PERSON]'s function that we defined earlier in Equation (1) as  
\[\underbrace{ \begin{pmatrix}-\mathbf{G}_{U}^{-+}\\ -\mathbf{G}_{U}^{--}\\ \mathbf{G}_{L}^{--}\\ \mathbf{G}_{L}^{++}\end{pmatrix} }_{\mathbf{G}}=\underbrace{ \begin{pmatrix} \mathbf{O}\\ -\mathbf{G}_{Ud}^{--}\\ \mathbf{O}\\ \mathbf{G}_{d}^{++}\end{pmatrix} }_{\mathbf{G}}+\underbrace{ \begin{pmatrix}- \mathbf{G}_{U}^{-+}\\ -\mathbf{G}_{Um}^{--}\\ \mathbf{G}_{L}^{++}\end{pmatrix} }_{\mathbf{G}_{L}^{+-}}, \tag{6}\]  
where \(\mathbf{G}_{Um}^{--}=\mathbf{G}_{U}^{--}-\mathbf{G}_{Ud}^{--}\) and \(\mathbf{G}_{Lm}^{++}=\mathbf{G}_{L}^{++}-\mathbf{G}_{Ld}^{++}\) are referred to as the [PERSON]'s function codasand \(\mathbf{O}\) is a zero matrix. We design a window matrix \(\mathbf{\Theta}^{[P]}\) that removes all data outside the interval \([-\tau_{d}^{p},\tau_{d}^{p}]\). When we apply this matrix to the [PERSON]'s function in Equation (6), it follows that  
\[\underbrace{ \begin{pmatrix}\mathbf{\Theta}_{U}^{[P]}&\mathbf{O}&\mathbf{O}& \mathbf{O}\\ \mathbf{O}&\mathbf{\Theta}_{U}^{[P]}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{\Theta}_{L}^{[P]}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{\Theta}_{L}^{[P]}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{\Theta}_{L}^{[P]}\end{pmatrix} }_{\mathbf{G}} \underbrace{ \begin{pmatrix}-\mathbf{G}_{U}^{-+}\\ -\mathbf{G}_{U}^{--}\\ \mathbf{G}_{L}^{+-}\end{pmatrix} }_{\mathbf{G}}=\underbrace{ \begin{pmatrix} \mathbf{O}\\ -\mathbf{G}_{Ud}^{--}\\ \mathbf{O}\\ \mathbf{G}_{d}^{++}\end{pmatrix} }_{\mathbf{G}_{d}}. \tag{7}\]In this formulation, \(\Theta_{U}^{[p]}\) and \(\Theta_{L}^{[p]}\) are operators that remove all data outside the intervals \([-\tau_{\mu\mu}^{p},\tau_{\mu\mu}^{p}]\) and \([-\tau_{\mu\mu}^{p},\tau_{\mu\mu}^{p}]\), respectively. Here, \(\tau_{\mu\mu}^{p}\) and \(\tau_{\mu\mu}^{p}\) are the direct \(P\)-wave travel times for propagation from \(z_{\mu}\) to \(z_{\mu}\) and \(z_{L}\), respectively. We proceed with the window operator for focusing functions, which is based on the travel time \(\tau_{d}^{S}\) of the (non-converted) direct \(S\)-wave, propagating from \(z_{l}\) outwards.