Optimal modal beamforming of bandpass signals using an undersized sparse vertical hydrophone array: theory and a shallow-water experiment

Conventional methods for modal beamforming of underwater acoustic signals using a vertical-line hydrophone array (VLA) can suffer significant degradation in resolution when the array is geometrically deficient, i.e., consists of sparsely spaced elements and spans the water column partially or is poorly navigated. Designed for estimating the coefficients of the normal modes, these conventional methods include the direct projection (DP) of the data on the calculated mode shapes and least-squares (LS) fitting of the mode sum to the data. The degradation, in the form of modal cross talk or sidelobes, is a result of an undersampling in depth. This cross talk may be mitigated with the application of proper space-time filter constraints in the case of a pulse transmission. In this paper, a generalized least-squares (GLS) mode beamformer, capable of incorporating physical space-time constraints on the propagation of sound, is presented. The formulation is based on the well-known theorem of Gauss and Markov. Initialized by a model prediction of the basic arrival structure of the normal modes and incorporating, iteratively, refined estimates of the statistics of the modal fluctuations, this GLS technique strives to boost the resolution of a geometrically deficient VLA. The improvement is demonstrated using the VLA data collected during a shallow-water tomography experiment in the Barents Sea. The superiority of the GLS method over the conventional DP and LS methods is evident, providing a high-quality time series of modal arrivals as a function of geophysical time, which, in turn, reveals the dominant time scales of the oceanic processes associated with the Barents Sea Polar Front.