Solution structure and stability for a two-parameter acoustic inverse scattering problem

This paper investigated the structure and stability of solutions for simultaneously recovering density and bulk modulus in acoustic inverse scattering. The nonlinear scattering model was linearized via the Born approximation, and the inverse problem was casted as a $2\times 2$ block operator equation. The off-diagonal blocks of this operator matrix captured the cross-coupling interactions between the two physical parameters. By employing the maximal Tseng generalized inverse, we developed an analytical formulation that remained valid even when diagonal blocks were non-invertible, overcoming limitations of traditional Schur complement methods. To address the inherent ill-posedness, we incorporated Tikhonov regularization into this linearized system, deriving explicit solution representations and establishing a rigorous stability estimate. Our analysis revealed that the strength of the off-diagonal interaction imposed a fundamental lower bound on the regularization parameter. As the parameter approached this bound, the reconstruction error exhibited singular behavior.