Using the waveguide invariant to analyze Lofargrams

Brekhovskikhi and Lysanov introduced the concept of a "waveguide invariant" to a broader audience. Assuming a horizontal array and incoherent processing, they showed how the measured acoustic intensity produced by a distant source would exhibit striations when plotted versus range and frequency. The striations are a consequence of interference between the propagating acoustic modes. A similar interference effect can also be observed when the measured data are processed coherently as with a beamformer. In practice, a passive sonar system often displays its beamformed output as a spectrum evolving in time. These plots, called LOFARgrams, are typically generated in multiple look-directions of the beamformer. In the present work, the striations observed in LOFARgrams are related to those observed with incoherent processing. Equations are derived for the trajectories of the striations. Numerical examples are presented. Sources located both above and below a thermocline are considered.