Direct and inverse scattering in the time domain via invariant imbedding equations

This paper examines the implementation of a new approach to direct and inverse scattering problems in the time domain, which is applied here to the case of a one-dimensional lossless medium. The method is based on an integro-differential equation which is satisfied by the reflection kernel and has been derived elsewhere using invariant imbedding techniques. An example of an exact solution to this equation is given. Numerical schemes for solving the direct and inverse problems are derived by discretizing this equation. The behavior of the resulting algorithms is then tested on several examples. The results suggest that the inversion method is stable in the presence of additive noise.