Variable Density Linear Acoustic Inverse Problem

Abstract

The linear acoustic inverse problem is solved simultaneously for density (ρ) and compressibility (κ) using the basic ideas of diffraction tomography (DT). The key to solving this problem is to utilize frequency diversity to obtain the required independent measurements. The receivers are assumed to be in the far field of the object, and plane wave incidence is also assumed. The Born approximation is used to arrive at a relationship between the measured pressure field and two terms related to the spatial Fourier transform of the two unknowns, ρ and κ. The term involving compressibility corresponds to monopole scattering and that for density to dipole scattering. Measurements at several frequencies are used and a least squares problem is solved to reconstruct ρ and κ at the same time. It is observed that the low spatial frequencies in the spectra of ρ and κ produce inaccuracies in the results. Hence, a regularization method is devised to remove this problem. Several results are shown. Low contrast objects for which the above analysis holds are used to show that good reconstructions are obtained for both density and compressibility after regularization is applied.