Fig. 1: The two canonical setups considered for constructing a general solution to the scattering problem with a solid-fluid interface.

a The source S^e and receiver Rⁱ are located outside and inside the box, respectively. The superscripts g, e, and i on Ω represent the global, external, and internal domains. b Both the source S^e and the receiver R^e are located outside the box. The global domain Ω^g comprises a solid part \({{{\mathbf{\Omega }}}}_{{{{\boldsymbol{s}}}}^{{{\boldsymbol{g}}}}}\) and a fluid part \({{{\mathbf{\Omega }}}}_{{{{\boldsymbol{f}}}}^{{{\boldsymbol{g}}}}}\), separated by a solid-fluid coupling interface Γ_c. The local domain Ωⁱ, a confined box (yellow) within the global domain Ω^g, contains a local solid domain \({{{\mathbf{\Omega }}}}_{{{{\boldsymbol{s}}}}^{{{\boldsymbol{i}}}}}\) and a local fluid domain \({{{\mathbf{\Omega }}}}_{{{{\boldsymbol{f}}}}^{{{\boldsymbol{s}}}}}\), with a scattering object of interest (Δm). Leveraging the methodology derived in this paper, after initial global wavefield computations, synthetic seismograms can be computed from an external source to both external and internal receivers by modeling wave propagation only within a compact box that contains the scatterer(s).