Exact analytic reconstruction in multidimensional inverse problems

Abstract

This presentation will review some recent mathematical breakthroughs in inverse scattering theory. Consider the pressure wave p(r,s,ω) generated by a point source s oscillating harmonically with frequency ω ∇r⋅{[1/ρ(r)]∇rp(r,s,ω)} + ω2K(r)p(r,s,ω) = −δ(r−s). The problem of determining the density ρ(r) and the compressibility K(r) from knowledge of p(r,s,ω) for sources s and receivers r located on a given surface S has been considered notoriously difficult. Traditional approaches rely on Born or Rytov approximations. A new constructive method for recovering ρ and K from measurements on a surface S with arbitrary geometry, at two frequencies ω1, ω2, will be presented. The procedure involves the solution of certain integral equations on S. No simplifying approximations are made, and the reconstruction is, in principle, exact.