Vector sensor cross-spectral density for surface noise in a stratified ocean-Formulas for arbitrary sensor geometries.

The cross-spectral density between pressure and particle velocity channels of two vector sensors with arbitrary orientation and separation is developed for a noise field consisting of directional sources distributed on a plane. Equations for arbitrary sensor orientations are necessary because cross-spectral density is an averaged product of two sensors' outputs that cannot be projected onto another direction after it is calculated. This prevents the use of existing methods, which calculate cross-spectral density for orthogonal particle velocity sensors, for sensors with different orientations. Analytic solutions to special cases for azimuthally symmetric environments and orthogonal velocity sensor orientations are developed and compared to previous results. The general equations are stated in a form suitable for use with any propagation model that produces Green's functions for pressure and orthogonal particle velocity components. Simulation examples demonstrate how sensor geometry and propagation conditions affect vector sensor array performance for wind-driven surface noise in deep and shallow water environments. Vector sensor beamforming is shown to provide additional gain in these directional noise fields as compared to isotropic noise.