Likelihood Function for a Simple Cardioid Sonobuoy

This paper considers the common sonobuoy configuration comprising two horizontal dipoles mounted orthogonally together with an omnidirectional sensor. Under the assumption that, within a narrow frequency band of interest, the acoustic power impinging on the sonobuoy consists of uncorrelated Gaussian noise processes, the paper examines how data collected by the sensor can be used to infer the directional distribution of the incident power, as characterised by its spherical harmonic coefficients. The results cover arbitrary directional distributions, and are applicable to small sample sizes as well as large. In particular, the paper examines the form of the likelihood function for the spherical harmonic coefficients, as required for Bayesian inference methods such as particle filtering. Specific attention is given to the case of a single point source superimposed on azimuthally isotropic noise. In this case it is shown that the maximum likelihood estimator for target bearing differs from, and can be considerably more efficient than, the bearing estimator traditionally used for this sensor configuration.