Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 72 / 1 / b / Solution

Let $\delta {\psi }_B={\psi }_1^{(B)}-{\psi }_i$ be the first-order scattered correction in the Born approximation for scalar wave scattering, using the paper's Green function convention. Comparing its outgoing integral with the preceding logarithmic correction givesTherefore the exact algebraic relation between the two first-order approximations isIf $V=O(\epsilon )$, then ${\chi }_1=O(\epsilon )$ on a controlled fixed region, andThus Born and Rytov agree to first order in the scattering potential, but differ as finite approximations. The Rytov approximation exponentiates the first logarithmic correction; the Born approximation adds the first field correction. The exponential's higher powers are not a calculation of all higher multiple-scattering terms in the Born series. This distinction explains why a smooth, appreciable wave phase accumulation can be represented more naturally by the Rytov approximation even when the corresponding linear field expansion is inaccurate.