Approximate containment relation between mode spaces in uncertain shallow water

Mode spaces are the subspaces that describe where the sound of interest impinging on the vertical line array (VLA) lives. It is spanned by the VLA sampled modes (modal depth functions), and is required in the implementation of some underwater source detection and localization algorithms. However, due to environmental uncertainty, especially in shallow water, the mode space is difficult to obtain. How the mode spaces derived from different realizations of uncertain shallow-water environments contain each other is investigated, in terms of how the mode space can contain the signal wavefronts lying in another, in an energy sense. The relative projection deviation (RPD) and the greatest relative signal-energy loss (GRSL) from one mode space to another are defined to quantify the containment degree. The simulation results in a benchmark uncertain shallow-water waveguide suggest that, due to good modal correlations, the mode space derived from one environmental realization is nearly a subset of those derived from others supporting the same or greater number of modes irrespective of the differences between the same-order sampled modes: Projecting the signal wavefronts in a certain mode space to another same or greater dimensional one will generally cause no more than 0.01 energy to be lost. This property holds increasingly well as the array aperture reduces, and is tolerant of environmental uncertainty degree. It is also demonstrated by SWellEx-96 experimental data. Several applications motivated from this property are also presented.