On different possibilities offered by the Born approximation in inverse scattering problems

One of the most productive approximations used to obtain analytic results in inverse scattering problems is the Born approximation. When the exciting wave is a monochromatic plane wave, in an n-dimensional problem, this approximation allows the author to obtain the Fourier transform of the unknown function on a certain (2n-1)-dimensional complex manifold M. The inverse transform performed on each submanifold of M, which can be bijectively mapped to Rⁿ, results in a different method capable of determining the unknown function. In this paper, the author first establishes a unified theory which permits one to correctly grasp the differences and relationships between various methods and then, by introducing the concepts of accessible and inaccessible parts of the submanifolds, he suggests a basis for comparing the effectiveness of possible methods in the case of n=2. This comparison reveals that a rather simple method which has not attracted sufficient attention until now is one of the best methods.