The Stokes relations linking time reversal and the inverse filter

When focusing ultrasonic waves through plates or tubes, the presence of multiple interfaces induces reflected wavefronts that follow the main wavefront. Adaptive focusing techniques can be used to cancel the echoes. To illustrate this problem and suggest a solution, an experiment was performed using two linear arrays of transducers placed on each side of a titanium plate. Three propagation operators are acquired - a transmission one from one array to the other, and two reflection operators associated to each array. Two adaptive focusing methods are used to cancel the echoes. The first is time reversal, using the two arrays cavity surrounding the plate. The second is the spatio temporal inverse filter (STIF), inverting the transmission operator. The inverse filter achieves echo cancellation using only the transmitted fields, whereas time reversal also requires the reflected fields. In fact, transmission and reflection operators are linked by the Stokes relations in a matrix formalism. These relations help understanding the inverse filter strategy, that deduces the wave front to be emitted from one array to imitate the time reversal cavity using both arrays. The matrix formulation of the Stokes relations also inspires a way to invert the transmission operator by an iterative method, showing a physical link between iterative time reversal and the inverse filter.