Estimators are developed for the mean and variance of the instantaneous velocity
of fish groups observed by continuous wide area Ocean-Acoustic Waveguide Remote
Sensing (OAWRS) [Makris et al., Science 311, 660 (2006)]. The estimators are
based on calculations of the Doppler shift and spread expected in long-range
scattering from fish groups on the continental shelf, which in turn are based
on the model for scattering from a moving target submerged in a stratified
ocean waveguide [Lai and Makris, J. Acoust. Soc. Am. 113, 223-244 (2003)]. We
show that the received field is a stationary random process in time due to
multiple scattering through the fish group. The moments of the spectral density
of mean field are shown to relate to the mean and variance of the instantaneous
fish velocity. Illustrative examples are presented on simulated data where the
spectral characteristics expected for randomly swarming and migrating fish
groups are used to successfully estimate the statistics of the instantaneous
velocity. The methods developed herein have important applications to
instantaneous estimation of velocity in long range imaging on the continental
shelf.