Cepstral methods for the determination of reflection coefficients for dispersive systems

Cepstral methods have been used for the determination of reflection coefficients in situations where the propagating waves are nondispersive. However, if the propagating waves are dispersive then the reflection properties cannot be estimated by this technique and this paper describes how the cepstral method may be extended to accommodate this. This is achieved by incorporating the dispersion law into the procedure. Specifically, the ´usual´ operation of (inverse) Fourier transformation (with respect to frequency) of the log of the Fourier transform is changed to an (inverse) Fourier transform with respect to wave number. This results in a cepstrum in which the independent variable is space (and not time) such that, at a point determined by the distance between measurement point and reflector a function related to the reflection coefficient appears, which may then be recast as a usual reflection coefficient.