Local and global statistics of clear‐air Doppler radar signals

A refined theoretical analysis of the clear‐air Doppler radar (CDR) measurement process is presented. The refined theory builds on the Fresnel‐approximated (as opposed to Fraunhofer‐approximated) radio wave propagation theory, and turbulence statistics like locally averaged velocities, local velocity variances, local dissipation rates, and local structure parameters are allowed to vary randomly within the radar's sampling volume and during the dwell time. A local version of the moments theorem and the random Taylor hypothesis are used to derive first‐principle formulations of all higher moments of the Doppler cross‐spectrum. The m th moment is written as a convolution integral of a spectral sampling function and a generalized, m th‐order refractive‐index spectrum or, alternatively, as a convolution integral of a lag‐space sampling function and a spatial cross‐covariance function of the local refractive‐index fluctuations and their local m th‐order time derivatives. Closed‐form expressions of the first three moments (i.e., m = 0, 1, 2) of the Doppler spectrum for the monostatic, single‐signal case are derived. This refined theory, or “local sampling theory,” enables one to correctly interpret CDR observations that are collected under conditions where the applicability of the traditional “global sampling theory” is questionable. The commonly used global sampling assumptions (Bragg‐isotropy, homogeneity, and stationarity of all turbulence statistics within the sampling volume and during the dwell time) may be invalid for small‐scale intermittency in the mixed layer, for refractive‐index sheets corrugated by gravity waves or instabilities, and for layered turbulence in the stably stratified atmosphere.