On the convergence of the Rytov approximation for the reduced wave equation

The reduced wave equation Δu+k2n2(x)u=0 is treated where n(x) fluctuates about unity in a compact domain D, and is equal to unity in the region exterior to D. In the Rytov approximation the total field u(x) generated by an incident field ui(x) has the form u(x)∼ui(x)exp φ(x) where φ(x)ui(x)=(1/4π)∫D (eik‖x−y‖/‖x−y‖)k2(n2−1) ×ui(y)dτy. It is shown that under suitable conditions on n(x) and restrictions on ui(x), the approximation is a leading term of a convergent expansion holding for all x in D. This is in contrast to previous theory, which treated the Rytov approximation as an asymptotic expansion valid in the forward-scattered direction.