Noise Statistics of a Higher Order Directional Sensor, Realized by Computing Finite Differences Spatially Across Multiple Isotropic Sensors

An acoustic "particle velocity sensor" (a.k.a. a geophone) exhibits a gain-response with a cosine-like directivity. The particle velocity sensor may be realized in hardware by two "pressure sensors" (of isotropic directivity) displaced in space, and by computing the spatial first-order finite difference between the data of the two isotropic component-sensors. As each component-sensor\´s data are degraded by additive noises (modeled here with much generality as stochastically distributed as "stable" (a.k.a. "alpha stable" or "$alpha$ stable"), and not restricted to being Gaussian), the particle velocity sensor as a whole would also experience noise, the statistics of which is analytically derived here. Furthermore, beyond this particle velocity sensor involving a first-order finite difference, the work presented here also derives the composite noise statistics of higher order difference realizations of sensors of higher order directivity in their gain responses.