The Factorization Method In Inverse Scattering Problems

In this paper, we consider the inverse scattering problems for plane wave incidence uⁱ=e^ikx·d in a homogeneous medium to determine the shape of the scattering obstacle D from far field measurement u^∞— of the scattered fields u^s.u = uⁱ + u^s satisfiesλ is the parameters, complex-valued numbers or real numbers. In order to reconstruct the shape of scatterer D, we establish the relation between the scatterer and the far field pattern. The relation can be fixed by so called far field operator F. Therefore it's very important to analyze some properties of F, such as the factorization of the far field operator F in the form F = -4Ï€GS^*(N^*, A^*)G^* and the conditions S(N, A) satisfies. With the operator F, we derive that z belongs to D if and as if (Fψ, ψ) must satisfy some conditions. Thus the shape of the scatterer D can be defined accordingly. Actually, the results can be said to be a further development of A.Kirsch's sampling methods in a sense.