Increasing stability and numerical methods for inverse acoustic and elastic medium problems

This paper investigates the inverse scattering problem of recovering the medium function in acoustic and elastic scattering equations from near-field or far-field data. We establish increasing stability estimates that comprise two components: a Hölder-type part and a logarithmic-type part, the latter vanishing as the frequency increases. The proof proceeds in two main steps. First, we derive an integral inequality with explicit frequency dependence. Next, by employing complex geometric optic solutions as test functions, we obtain the increasing stability estimates. Notably, the analysis requires measurement data at only a single frequency. Moreover, a numerical method is proposed to reconstruct a smooth compactly supported medium contrast from multi-static point-source measurements for both acoustic and elastic equation. Numerical examples demonstrate that the reconstruction error decreases as the wavenumber increases, confirming the improved resolving power of higher-frequency data.