Doppler effects for randomly moving scatterers and platforms

Abstract

Previous studies of scattering from randomly moving media, interfaces (and platforms) generally do not provide an adequate or correct treatment of the ensuing Doppler effects. This has occured chiefly because of the failure to apply a suitably explicit Doppler analysis, which recognizes that the scattering elements act like randomly moving sources of (re-)radiation, with path delays which are critical functions of the dynamic geometry between the moving source (T), scattering element (S), and receiver (R), and which accordingly require appropriately modified Langevin equations (ensembles of propagation equations). The principal random Doppler (i.e., motional) effects are O (c−1) and appear as (space–time) random displacements about the equilibrium positions of the scatterers, thus producing phase modulations (or equivalent frequency modulations, proportional to velocity), with, however, the proper time delays of T,S,vis-à-visR. Not only have incorrect path delays been employed in most previous studies, but also an explicit Doppler analysis has been lacking, which further makes it impossible to include the effects of platform motion adequately. The present analysis remedies these defects and, in addition, provides many new results of interest for problems in underwater acoustics specifically here (under conditions of zero velocity gradients, ∇c=0). These include platform and scatterer Doppler effects, for the practical cases of small Dopplers and farfield operation; iteration procedures for these relations and high-order corrections, and conditions for their validity; finite aperture solutions; mono- and bistatic operation; deterministic velocity components, which generate scanning effects, attributable to this type of platform motion. Results for the approximate first- and second-order statistics of these Doppler phenomena needed in scattering analysis are also presented, for both cases of broadband- and narrow-band-injected signals. Particular attention is given to Doppler phenomena from moving wave surfaces. The treatment is generally canonical, however, in that all modes (surface, volume, and bottom) are included in the formal analysis, both from the viewpoint of a fixed reference system (in the motionless medium) and from the viewpoint of the observer on a moving platform.