On the ill-posedness of the 2.5-dimensional linearized density inversion model

This paper considers a linearized inverse problem of recovering the medium density rho (x, y) in an acoustic space Omega = {(x, y, z) : z > 0}, under the assumptions that the wave speed is constant and the acoustic medium is weakly inhomogeneous. We take wave fields measured on a line as the inversion input. By making full use of the property of the 2.5D acoustic medium, we gel an integral equation for the 2D density perturbation. Applying the Fourier transform technique, we prove that the integral equation has a unique solution, but that the solution is unstable in essence. Thus the ill-posedness of the linearized inverse problem is clarified.