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This work investigates the scattering coefficients for inverse medium scattering problems. It shows some fundamental properties of the coefficients such as symmetry and tensorial properties. The relationship between the scattering coefficients and the far-field pattern is also derived. Furthermore, the sensitivity of the scattering coefficients with respect to changes in the permittivity and permeability distributions is investigated. In the linearized case, explicit formulas for reconstructing permittivity and permeability distributions from the scattering coefficients is proposed. They relate the exponentially ill-posed character of the inverse medium scattering problem at a fixed frequency to the exponential decay of the scattering coefficients. Moreover, they show the stability of the reconstruction from multifrequency measurements. This provides a new direction for solving inverse medium scattering problems.

The Concept of Heterogeneous Scattering Coefficients and Its Application in Inverse Medium Scattering