Analysis of the time reversal operator for a scatterer undergoing small displacements

The method of the time reversal operator decomposition is usually employed to detect and characterize static targets using the invariants of the time reversal operator. This paper presents a theoretical and experimental investigation into the impact of small displacements of the target on these invariants. To find these invariants, the time reversal operator is built from the multistatic response matrix and then diagonalized. Two methods of recording the multistatic response matrix while the target is moving are studied: Acquisition either element by element or column by column. It is demonstrated that the target displacement generates new significant eigenvalues. Using a perturbation theory, the analytical expressions of the eigenvalues of the time-reversal operator for both acquisition methods are derived. We show that the distribution of the new eigenvalues strongly depends on these two methods. It is also found that for the column by column acquisition, the second eigenvector is simply linked to the scatterer displacements. At last, the implications on the Maximum Likelihood and Multiple Signal Classification detection are also discussed. The theoretical results are in good agreement with numerical and 3.4 MHz ultrasonic experiments.