A Bayes factor discriminant is constructed for active sonar detection of a scattering body in an underwater refractive environment about which there is some depth uncertainty. The scenarios of interest here are associated with relatively high-frequency broadband waveforms and with reception along vertical arrays. The approach properly accounts for environmental information regarding the refractive media, as well as surface and volume reverberation models or in situ observations of the same. Uncertainty, both in the reverberation field and the scatterers' depth, is incorporated through proper marginalization rather than maximization as in more conventional generalized likelihood ratio tests. Bayes factor active sonar (BFAS) yields a set of time-varying quadratic forms in beam-delay space, optimally balancing uncertainty in the object of interest with reverberation and noise subspaces in the minimum average risk sense. By utilizing waveguide information, BFAS combines multi-path arrivals, optimally attenuating reverberation subspaces while preserving the target subspace, thereby effectively increasing signal-to-reverberation plus noise ratios despite uncertainty in target depth. Depth-invariant modes are leveraged to provide a valuable expansion of the discriminating information of the BFAS, thereby providing lower bounds on the performance of the BFAS. These bounds illustrate that even under depth uncertainty, the BFAS outperforms a single specular arrival detector with perfect knowledge of the scattering body's depth. Performance across various refractive and shallow-water environments is demonstrated, lending credence to the multi-path combining approach.
Gendron et al. (Sun,) studied this question.