Asymptotic inverse scattering

Several inverse techniques are developed for determining the shape of an unknown scattering surface by analyzing backscattered acoustic or electromagnetic waves. These techniques are based on asymptotic high frequency representations of the fields and may be divided into three categories. The first one is the geometrical imaging method where the surface is reconstructed by means of a travel-time analysis which is here specified to the far field by utilizing Minkowski's support function. Furthermore, a geometrical method is developed for localizing edges from mid field data measured along a curve. The second category is called quasigeometrical imaging and uses geometric optics or higher order amplitudes for the reconstruction. It is shown that cross-polarized electromagnetic far field amplitude measurements permit one to deduce the complete quadratic approximation of the surface at the specular points from which the surface can be reconstructed pointwise. The third category may be subsumed under ‘asymptotic inverse scattering identities’. Here, asymptotic relations between scattered fields and distributions associated with the geometry of the scatterer are established. It is shown that the physical optics far field inverse scattering identity is only a leading order asymptotic relation but as such is also valid for non-convex scatterers. Furthermore, asymptotic inverse scattering identities are deduced which relate the singular function of a closed surface to the backscattered field data measured on a sphere enclosing the scatterer. This generalizes far field results of Cohen and Bleistein (Wave Motion 1 (1979), p. 153) to the mid field.