Stability for inverse potential scattering with attenuation

Abstract

This paper is concerned with inverse potential scattering problem for Helmholtz equation with constant attenuation. We first derive a logarithmic stability estimate for determining the potential at a single wavenumber by point-source boundary measurements. The proof utilizes the construction of complex geometric optics (CGO) solutions. Further, given the multi-wavenumber data, we derive a stability estimate which consists of two parts: one part is a Lipschitz data discrepancy and the other part is a logarithmic stability. The latter decreases as the wavenumber increases, which exhibits the phenomenon of increasing stability. The proof employs the physical asymptotic behavior of the radiated field and the properties of the Radon transform. Moreover, as multi-wavenumber data is available, the proof does not resort to the commonly used unphysical CGO solutions. We trace the dependence of the upper bound of the stability estimate on the constant attenuation through an analysis of the resolvent estimates. Both of the stability estimates show exponential dependence on the attenuation coefficient, which illustrates the poor resolution of the inverse scattering with attenuation.