Cross correlations and cross spectra for spaced antenna wind profilers .1. Theoretical analysis

The presented theory ties the properties of a turbulently advected scattering medium to the cross correlation and cross spectrum of signals in a general configuration of receiving and transmitting antennas. The correlation length of Bragg scatterers and antenna diameter are the significant parameters determining the diffraction pattern's correlation length. We examine how vertical anisotropy of the scattering medium affects the diffraction pattern's correlation length. We demonstrate that the cross spectrum can be formulated in terms of a pair of spectral sampling functions (a one-dimensional Doppler and a three-dimensional wavenumber function), and closed form solutions are obtained. We give the conditions under which the scattering medium's statistical properties can be represented by a Gaussian correlation or spectral model, and the distance over which the diffraction pattern simply advects without significant change. We show that the diffraction pattern of a pair of scatterers can translate at the speed of the scatterers, not twice their speed as is commonly thought.