Beamforming pointing error of a triaxial velocity sensor under gain uncertainties

A “triaxial velocity sensor” consists of three uniaxial velocity sensors, which are nominally identical, orthogonally oriented among themselves, and co-centered at one point in space. A triaxial velocity sensor measures the acoustic particle velocity vector, by its three Cartesian components, individually component-by-component, thereby offering azimuth-elevation two-dimensional spatial directivity, despite the physical compactness that comes with the collocation of its three components. This sensing system's azimuth-elevation beam-pattern has been much analyzed in the open literature, but only for an idealized case of the three uniaxial velocity sensors being exactly identical in gain. If this nominal identity is violated among the three uniaxial velocity sensors, as may occur in practical hardware, what would happen to the corresponding “spatial matched filter” beam-pattern's peak direction? How would this effective peak direction deviate from the nominal “look direction”? This paper, by modeling each uniaxial velocity sensor's gain as stochastic, derives this deviation's statistical mean and variance, analytically in closed mathematical forms. This analytical derivation is verified by Monte Carlo simulations.