Space and Time Filtering of Remotely Sensed Velocity Turbulence

Abstract

It is well known that the width of a clear-air Doppler radar spectrum can be used to estimate the small-scale variability of the wind. To do this accurately requires that all contributions to the spectral width be accounted for. Recently, an approximate formula for correcting Doppler spectral widths for spatial and temporal filtering effects was proposed. The formula assumes independent additive contributions to the spectral width from the inherent volume averaging of the radar-sampling volume and the finite measurement interval. In the present paper the approximate formula is compared to an exact triple-integral formulation in which the spatial and temporal effects are shown to be coupled in a nonlinear fashion. The required integrations are evaluated numerically and are carried out over the full range of scales in wavenumber space, in contrast to earlier work, where truncated forms of isotropic, inertial-subrange spectral forms were used to obtain a simple, closed-form expression. Comparisons show that the approximate formula provides a good approximation to the integral solution over small to moderate length scales, but that it diverges from the integral solution at larger scales. Asymptotic limits to the exact integral formulation for large and small scales are presented. Finally, a double-integral solution that can be rapidly evaluated by any one of a number of commercial mathematics packages has been developed and shown to agree within 2% of the exact solution over all scales.