The Stokes relations linking time reversal and the inverse filter

Abstract

Our aim in this paper is to show how two adaptive focusing techniques, Time Reversal (TR) and the Spatio Temporal Inverse Filter (STIF), are related by the Stokes equations linking waves transmitted and reflected through a medium. For that purpose a model experiment has been investigated: a solid plate located between two arrays of transducers. When sending a wave from an array to the other through the plate, multiple transmitted waves are induced. TR and STIF are used to cancel these echoes. The echoes can be suppressed by TR, using the two arrays cavity surrounding the plate. They can also be cancelled by STIF, inverting the transmission operator between the arrays. The STIF achieves echoes cancellation by using only the transmitted fields through the plate, whereas TR also requires the reflected fields. The STIF's strategy is analyzed in light of the Stokes relations: thanks to the reflections in the medium, it is able to simulate a TR cavity with only one array. A mathematical analysis of the matrix expression of the Stokes relations then leads to two iterative ways to invert the transmission operator. Finally, this general technique is applied to a more complex medium: a human skull bone..