On the true velocity in full correlations analysis

Abstract

The usual assumption made of elliptical symmetry in the shape of the correlation functions for deriving algorithms to determine the "true" velocity of a time-changing random pattern is relaxed. We start, instead, by stating that in the frame of reference moving with the medium, the space-time autocorrelation function of any state parameter of the medium is invariant under an r = -r transformation, where r is the displacement vector. With no further assumptions, a simple formula for the velocity of the medium is found for statistically isotropic patterns and for anisotropic patterns when the sensors are aligned along the axis of symmetry. The simplicity of the derivation and the formula contrast with previous derivations under more restrictive conditions. For the case of unknown elongation direction, previously known algorithms for determining the velocity under the assumption that the locus of equal correlation in r space are elliptical are revised. They are derived as Taylor expausiou approximations of the functions specifying the shape of equal correlation shells. Furthermore, the assumption of fixed elliptical elongation parameters is relaxed. Three-dimensional cases are considered throughout, with one- and two-dimensional applications included as special cases.