Implications of Riemannian geometry in underwater acoustics

Abstract

Signal processing involving source localization and detection in underwater acoustics often depends on comparing covariance or cross-spectral density matrices (CSDMs) estimated for data, replica or noise source data from passively sensed acoustic fields incident on sensor arrays. Such comparisons often depend on a measure of similarity between matrix pairs and this typically involves a Euclidean metric. While Euclidean-based metrics are ubiquitous, convenient and useful, they are not fundamental to this task. By exploiting the facts that a CSDM is Hermitian and positive semi-definite, one can interpret such matrices as points constrained to a Riemannian, rather than a Euclidean manifold, with the implication that similarity between matrix pairs should be measured using a metric consistent with the manifold’s intrinsic structure. This geometric interpretation leads to alternative matched-field processors for source localization involving the Riemannian distance as a measure of similarity and can be used to solve this inverse problem. Some implications of this non-Euclidean approach in underwater acoustics, as well as a possible extension to source detection are discussed. [Work supported by the Office of Naval Research.]