Layer Stripping

We describe a rigorous layer stripping approach to inverse scattering for the Helmholtz equation in one dimension. In section 3, we show how the Ricatti ordinary differential equation, which comes from the invariant embedding approach to forward scattering, becomes an inverse scattering algorithm when combined with the principle of causality. In section 4 we discuss a method of stacking and splitting layers. We first discuss a formula for combining the reflection coefficients of two layers to produce the reflection coefficient for the thicker layer built by stacking the first layer upon the second. We then describe an algorithm for inverting this procedure; that is, for splitting a reflection coefficient into two thinner reflection coefficients. We produce a strictly convex variational problem whose solution accomplishes this splitting. Once we can split an arbitrary layer into two thinner layers, we proceed recursively until each reflection coefficients in the stack is so thin that the Born approximation holds (i.e. the reflection coefficient is approximately the Fourier transform of the derivative of the logarithm of the wave speed). We then invert the Born approximation in each thin layer.