The Operator Factorization Method in Inverse Obstacle Scattering

Abstract

The standard factorization method from inverse scattering theory allows to reconstruct an obstacle pointwise from the normal far field operator F. The kernel of this method is the study of the first kind Fredholm integral equation (F * F)1/4 f = Φ z with the right-hand part $$\Phi _{z} {\left( \theta \right)} = \exp {\left( { - ikz \cdot \theta } \right)}.$$ In this paper we extend the factorization method to cover some kinds of boundary conditions which leads to non-normal far field operators. We visualize the scatterer explicitly in terms of the singular system of the selfadjoint positive operator F # = [(ReF)* (ReF)]1/2 + ImF. The following characterization criterium holds: a given point z is inside the obstacle if and only if the function Φ z belongs to the range of F # 1/2 . Our operator approach provides the tool for treatment of a wide class of inverse elliptic problems.