## 2 Forward Equations
### Causality Cones of Green's Functions and Focusing Functions
As illustrated in Figure 2b, the focusing functions and their time-reversed counterparts vanish outside \([-\tau_{d}^{S},\tau_{d}^{S}]\). Let \(\mathbf{F}_{Ud}^{+}= \begin{pmatrix}0&0\\ 0&F_{Ud}^{+SS}\end{pmatrix} \) and \(\mathbf{F}_{Ud}^{-}= \begin{pmatrix}0&0\\ 0&F_{Ud}^{-SS}\end{pmatrix} \) be the components of \(\mathbf{F}_{U}^{+}\) and \(\mathbf{F}_{L}^{-}\) that reside at the boundary of the interval \((-\tau_{d}^{S},\tau_{d}^{S})\) (corresponding to the direct non-converted \(S\)-wave transmissions). Now, we may partition the focusing function that we defined earlier in Equation (3) as  
\[\underbrace{ \begin{pmatrix}\mathbf{F}_{U}^{-}\\ \mathcal{Z}\mathbf{F}_{U}^{+}\\ \mathcal{F}_{L}^{+}\\ \mathcal{Z}\mathbf{F}_{L}^{-}\end{pmatrix} }_{\mathbf{F}_{L}^{-}}=\underbrace{ \begin{pmatrix}\mathbf{O}\\ \mathcal{Z}\mathbf{F}_{Ud}^{+}\\ \mathbf{O}\\ \mathcal{Z}\mathbf{F}_{L}^{-}\end{pmatrix} }_{\mathbf{F}_{d}}+\underbrace{ \begin{pmatrix}\mathbf{F}_{U}^{-}\\ \mathcal{Z}\mathbf{F}_{Ud}^{-}\\ \mathcal{Z}\mathbf{F}_{L}^{-}\end{pmatrix} }_{\mathbf{F}_{m}}, \tag{8}\]  
where \(\mathbf{F}_{Ud}^{+}=\mathbf{F}_{U}^{+}-\mathbf{F}_{Ud}^{+}\) and \(\mathbf{F}_{L}^{-}=\mathbf{F}_{L}^{-}-\mathbf{F}_{Ld}^{-}\) are referred to as the focusing function codas. We design a window matrix \(\mathbf{\Theta}^{(S)}\) that removes all data outside the interval \((-\tau_{d}^{S},\tau_{d}^{S})\). During the inversion that will be applied later in this paper, we wish to restrict \(\mathbf{F}_{m}\) to the interval \((-\tau_{d}^{S},\tau_{d}^{S})\). To enforce this in practice, we replace \(\mathbf{F}_{m}\) in Equation (8) with \(\mathbf{\Theta}^{(S)}\mathbf{F}_{m}\), leading to  
Figure 2: Symmetrized causality cone of (**a**) a Green’s function (**G** or \(\mathcal{Z}\mathbf{G}\)) and (**b**) a focusing function for the medium in Figure 1 at \(p=0.2\) ms\(\cdot\)m\({}^{-1}\), with the source/focal depth at \(z_{i}=0.5\) m (indicated by the magenta dashed line; the black dashed lines indicate layer boundaries). The blue lines denote the travel times \(\pm\tau_{\mu\mu}^{p}\) and \(\pm\tau_{\mu\mu}^{p}\) of the direct (non-converted) \(P\)-wave transmissions. The red lines denote the travel times \(\pm\tau_{\mu\mu}^{s}\) and \(\pm\tau_{\mu\mu}^{s}\) of the direct (non-converted) \(S\)-wave transmissions. All wavefields are stricly zero in the gray areas (whereas they may be non-zero in the yellow and orange areas). The areas where the Green’s functions and focusing functions can overlap are indicated in orange. \[\underbrace{ \begin{pmatrix}\mathbf{F}_{U}^{-}\\ \mathcal{Z}\mathbf{F}_{U}^{+}\\ \mathbf{F}_{L}^{+}\\ \mathcal{Z}\mathbf{F}_{\Gamma}\end{pmatrix} }_{\mathbf{F}}=\underbrace{ \begin{pmatrix}\mathbf{O}\\ \mathcal{Z}\mathbf{F}_{Ud}^{+}\\ \mathbf{O}\\ \mathcal{Z}\mathbf{F}_{\Gamma_{L}^{-}}^{-}\end{pmatrix} }_{\mathbf{F}_{d}}+ \underbrace{ \begin{pmatrix}\Theta_{U}^{(S)}&\mathbf{O}&\mathbf{O}&\mathbf{O} \\ \mathbf{O}&\Theta_{U}^{(S)}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\Theta_{L}^{(S)}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{O}&\Theta_{L}^{(S)}\end{pmatrix} }_{\mathbf{ \Theta}^{(S)}}\underbrace{ \begin{pmatrix}\mathbf{F}_{U}^{-}\\ \mathcal{Z}\mathbf{F}_{Um}^{+}\\ \mathcal{Z}\mathbf{F}_{Lm}^{-}\end{pmatrix} }_{\mathbf{F}_{m}}. \tag{9}\]  
In this formulation, \(\Theta_{U}^{(S)}\) and \(\Theta_{L}^{(S)}\) are operators that remove all data outside the intervals \((-\tau_{Ud}^{S},\tau_{Ud}^{S})\) and \((-\tau_{Ld}^{S},\tau_{Ud}^{S})\), respectively. Here, \(\tau_{Ud}^{S}\) and \(\tau_{Ld}^{S}\) are the direct \(S\)-wave travel times for propagation from \(z_{t}\) to \(z_{U}\) and \(z_{L}\), respectively.