Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies

Abstract

We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are given by the far field patterns u∞(xˆ,k) for all observation directions xˆ in some neighborhood of a fixed direction xˆ0 and for all wave numbers k belonging to a finite interval (0,K). In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval K>1. In three dimensions we establish stability estimates of the L2-norm and H−1-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of Hölder type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.