Doppler radar measurements of turbulence

A first-principle analysis of Doppler radar measurements of turbulence is presented. A set of limited, though practical, conditions are assumed to make the problem more tractable, with the primary condition being the discrete and finite temporal sampling of the radar signals. A theoretical derivation of the Doppler spectrum under these conditions is performed, and the distinction between what the radar actually measures and what results after a theoretical ensemble averaging is delineated. This is an important consideration, as all of the theoretical development in the literature is based upon ensemble averaging. It is shown that in the limit of an infinite number of samples, and after ensemble averaging, the Doppler spectrum can be represented by a sum of Dirac delta distributions and furthermore that the normalized spectrum will equal the probability distribution of the scatterer velocities. We show that the correlation structure of the velocity field manifests itself primarily in the square of the first moment. That is, a correlated field will have a certain degree of patchiness, which leads to variations in the first moments from realization to realization. These theoretical considerations are then studied via simulation. Parameters for a typical airborne X-band Doppler radar are used, and correlated von Kármán and uncorrelated random fields are employed. Energy dissipation rate estimates are calculated from the simulated Doppler spectra, and the performance based on moment averaging and spectral averaging is presented. Real-world application of the turbulence measurement methods is then shown with airborne X-band detection of convective turbulence.