## Appendix B Derivations
### Reflection-Based Representations  
First, we derive a convolution-based representation for reflection data at \(z_{ul}\). For this purpose, we set \(m=U\) and \(n=1\) in Equation (12). In state \(A\), we use the properties of the actual medium and we place a unit source just above \(z_{ul}\), such that \(\hat{\mathbf{P}}_{A}^{+}(p,z_{ul},\omega)=\mathbf{I}\) (a \(2\times 2\) identity matrix), \(\hat{\mathbf{P}}_{A}^{-}(p,z_{ul},\omega)=\hat{\mathbf{R}}_{U}^{\omega}(p)\) (the reflection response 'from above' at \(z_{ul}\)), \(\hat{\mathbf{P}}_{A}^{+}(p,z_{ul},\omega)=\hat{\mathbf{G}}_{ll}^{++}(p)\) (the downwgoing Green's function at \(z_{l}\) due to a downgoing source at \(z_{ul}\)) and \(\hat{\mathbf{P}}_{A}^{-}(p,z_{ul},\omega)=\hat{\mathbf{G}}_{ll}^{-+}(p)\) (the upgoing Green's function at \(z_{l}\) due to a downgoing source at \(z_{ul}\)). 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 with a focal point at \(z_{l}\), which is purely downgoing below this level [15]. This leads to \(\hat{\mathbf{P}}_{B}^{+}(p,z_{ul},\omega)=\hat{\mathbf{F}}_{ll}^{+}(p)\) (the downgoing focusing function at \(z_{ul}\)), \(\hat{\mathbf{P}}_{B}^{-}(p,z_{ul},\omega)=\mathbf{\hat{F}}_{ll}^{-}(p)\) (the upgoing focusing function at \(z_{ul}\)), \(\hat{\mathbf{P}}_{B}^{+}(p,z_{ul},\omega)=\mathbf{I}\) (the focused field at \(z_{l}\)) and \(\hat{\mathbf{P}}_{B}^{-}(p,z_{ul},\omega)=\mathbf{O}\). Substitution of these quantities into Equation (A4) yields  
\[-\hat{\mathbf{G}}_{ll}^{-+}(p)=\hat{\mathbf{F}}_{ll}^{-}(p)-\hat{\mathbf{R}}_{ ul}^{\omega}(p)\hat{\mathbf{F}}_{ll}^{+}(p),\] (A6)  
where we have used \(\{\hat{\mathbf{R}}_{ll}^{\omega}(-p)\}^{t}=\hat{\mathbf{R}}_{ll}^{\omega}(p)\) and \(\{\hat{\mathbf{G}}_{ll}^{-+}(-p)\}^{t}=\hat{\mathbf{G}}_{ll}^{-+}(p)\)[51]. We can derive an equivalent correlation-based representation by substituting the same quantities into Equation (A5), leading to  
\[-\{\hat{\mathbf{G}}_{ll}^{--}(-p)\}^{\star}=\hat{\mathbf{F}}_{ll}^{+}(p)-\{ \hat{\mathbf{R}}_{ll}^{\omega}(-p)\}^{\star}\hat{\mathbf{F}}_{ll}^{-}(p),\] (A7)  
where we have used \(\{\hat{\mathbf{R}}_{ll}^{\omega}(p)\}^{\star}=\{\hat{\mathbf{R}}_{ll}^{\omega }(-p)\}^{\star}\) and \(\{\hat{\mathbf{G}}_{ll}^{++}(p)\}^{\star}=-\{\hat{\mathbf{G}}_{ll}^{--}(-p)\}^ {\star}\)[51]. Two more representation can be derived for reflection data at \(z_{l}\) by choosing \(m=I\) and \(n=L\) in Equations (A4) and (A5) and placing a source just below \(z_{l}\). Once again, we choose the actual medium properties in state \(A\), such that \(\hat{\mathbf{P}}_{A}^{+}(p,z_{ul},\omega)=\hat{\mathbf{G}}_{ll}^{+-}(p)\) (the downgoing Green's function at \(z_{l}\) due to an upgoing source at \(z_{l}\)), \(\hat{\mathbf{P}}_{B}^{-}(p,z_{ul},\omega)=\hat{\mathbf{G}}_{ll}^{--}(p)\) (the upgoing Green's function at \(z_{l}\) due to an upgoing source at \(z_{l}\)), \(\hat{\mathbf{P}}_{A}^{+}(p,z_{ul},\omega)=-\hat{\mathbf{R}}_{U}^{-}(p)\) (the reflection response 'from below' at \(z_{l}\)) and \(\hat{\mathbf{P}}_{A}^{-}(p,z_{ul},\omega)=-\mathbf{I}\) (a \(2\times 2\) identity matrix). 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 with a focal point at \(z_{l}\), which is purely upgoing above this level. This leads to \(\hat{\mathbf{P}}_{B}^{+}(p,z_{l},\omega)=\mathbf{O}\), \(\hat{\mathbf{P}}_{B}^{-}(p,z_{l},\omega)=\mathbf{I}\) (the focused field at \(z_{l}\)), \(\hat{\mathbf{P}}_{B}^{+}(p,z_{l},\omega)=\hat{\mathbf{F}}_{ll}^{+}(p)\) (the downgoing focusing function at \(z_{l}\)) and \(\hat{\mathbf{P}}_{B}^{-}(p,z_{l},\omega)=\hat{\mathbf{F}}_{ll}^{-}(p)\) (the upgoing focusing function at \(z_{l}\)).