## Appendix B Derivations
### Transmission-Based Representations  
We can derive a system of equivalent equations that are based on transmission data. First, we set \(m=U\) and \(n=I\). We choose the actual medium in state \(A\) and place a source just below \(z_{L}\), leading to \(\hat{\mathbf{P}}_{A}^{+}(p,z_{U},\omega)=\mathbf{O}\), \(\hat{\mathbf{P}}_{A}^{-}(p,z_{U},\omega)=-\hat{\mathbf{T}}_{UU}^{-}(p)\) (the transmission response from \(z_{l}\) to \(z_{U}\)), \(\hat{\mathbf{P}}_{A}^{+}(p,z_{U},\omega)=\hat{\mathbf{G}}_{U}^{+-}(p)\) and \(\hat{\mathbf{P}}_{A}^{-}(p,z_{U},\omega)=\hat{\mathbf{G}}_{U}^{--}(p)\). In state \(B\), we truncate the medium at \(z_{l}\) and choose a homogeneous halfspace below this level. For the wavefield in this state, we choose a focusing function that focuses at \(z_{l}\) and is purely downgoing below this level. This leads to \(\hat{\mathbf{P}}_{B}^{+}(p,z_{U},\omega)=\hat{\mathbf{F}}_{UU}^{-}(p)\), \(\hat{\mathbf{P}}_{B}^{-}(p,z_{U},\omega)=\hat{\mathbf{F}}_{UU}^{-}(p)\), \(\hat{\mathbf{P}}_{B}^{+}(p,z_{U},\omega)=\mathbf{I}\) and \(\hat{\mathbf{P}}_{B}^{-}(p,z_{l},\omega)=\mathbf{O}\). Substituting these quantities into Equation (A4) yields  
\[\hat{\mathbf{G}}_{U}^{++}(p)=\hat{\mathbf{T}}_{UU}^{+}(p)\hat{\mathbf{F}}_{UU }^{+}(p), \tag{11}\]  
where we have used \(\{\hat{\mathbf{T}}_{UU}^{-}(-p)\}^{\dagger}=\hat{\mathbf{T}}_{UU}^{+}(p)\) and \(\{\hat{\mathbf{G}}_{U}^{--}(-p)\}^{\dagger}=-\hat{\mathbf{G}}_{U}^{++}(p)\)[51]. We may substitute the same quantities into Equation (A5), leading to  
\[\{\hat{\mathbf{G}}_{U}^{+-}(-p)\}^{\star}=\{\hat{\mathbf{T}}_{UU}^{+}(-p)\}^{ \star}\hat{\mathbf{F}}_{UU}^{-}(p), \tag{12}\]  
where we have used \(\{\hat{\mathbf{T}}_{UU}^{-}(p)\}^{\dagger}=\{\hat{\mathbf{T}}_{UU}^{+}(-p)\}^ {\star}\) and \(\{\hat{\mathbf{G}}_{U}^{+-}(p)\}^{\dagger}=\{\hat{\mathbf{G}}_{U}^{+-}(-p)\}^ {\star}\)[51]. We can derive two more representations by setting \(m=I\) and \(n=L\), and placing a source just above \(z_{U}\). In state \(A\), we use the actual medium properties, leading to: \(\hat{\mathbf{P}}_{A}^{+}(p,z_{l},\omega)=\hat{\mathbf{G}}_{UU}^{++}(p)\), \(\hat{\mathbf{P}}_{A}^{-}(p,z_{l},\omega)=\hat{\mathbf{G}}_{UU}^{++}(p)\), \(\hat{\mathbf{P}}_{A}^{+}(p,z_{l},\omega)=\hat{\mathbf{T}}_{UU}^{+}(p)\) (the transmission response from \(z_{U}\) to \(z_{L}\)) and \(\hat{\mathbf{P}}_{A}^{-}(p,z_{L},\omega)=\mathbf{O}\). In state \(B\), we truncate the medium at \(z_{l}\) and choose a homogeneous halfspace above this level. For the wavefield in this state, we choose a focusing function that focuses at \(z_{l}\) and is purely upgoing above this level. This leads to \(\hat{\mathbf{P}}_{B}^{+}(p,z_{l},\omega)=\mathbf{O}\), \(\hat{\mathbf{P}}_{B}^{-}(p,z_{l},\omega)=\mathbf{I}\), \(\hat{\mathbf{P}}_{B}^{+}(p,z_{L},\omega)=\hat{\mathbf{F}}_{U}^{+}(p)\) and \(\hat{\mathbf{P}}_{B}^{-}(p,z_{L},\omega)=\hat{\mathbf{F}}_{U}^{-}(p)\). Substituting these quantities into Equation (A4) yields  
\[-\hat{\mathbf{G}}_{UU}^{--}(p)=\hat{\mathbf{T}}_{UU}^{-}(p)\hat{\mathbf{F}}_{U} ^{-}(p), \tag{13}\]  
where we have used \(\{\hat{\mathbf{T}}_{UU}^{+}(-p)\}^{\dagger}=\hat{\mathbf{T}}_{UU}^{-}(p)\) and \(\{\hat{\mathbf{G}}_{UU}^{++}(-p)\}^{\dagger}=-\hat{\mathbf{G}}_{UU}^{--}(p)\)[15]. Alternatively, the quantities can be substituted into Equation (A5), leading to  
\[-\{\hat{\mathbf{G}}_{UU}^{-+}(-p)\}^{\star}=\{\hat{\mathbf{T}}_{UU}^{-}(-p)\}^ {\star}\hat{\mathbf{F}}_{U}^{+}(p), \tag{14}\]  
where we have used \(\{\hat{\mathbf{T}}_{UU}^{+}(p)\}^{\dagger}=\{\hat{\mathbf{T}}_{UU}^{-}(-p)\}^ {\star}\) and \(\{\hat{\mathbf{G}}_{UU}^{-+}(p)\}^{\ddagger}=\{\hat{\mathbf{G}}_{UU}^{-+}(-p)\}^ {\star}\)[51]. The system of Equations (A11)-(A14) can be rewritten as  
\[ \begin{pmatrix}-\{\hat{\mathbf{G}}_{U}^{-+}(-p)\}^{\star}\\ -\hat{\mathbf{G}}_{U}^{--}(p)\\ \{\hat{\mathbf{G}}_{U}^{+-}(-p)\}^{\star}\\ \hat{\mathbf{G}}_{L}^{++}(p)\end{pmatrix} =