The spectrum of weather echoes

When the term “spectrum” arises, radar meteorologists usually think of the Doppler spectrum of the weather echo from a given point in the atmosphere. However, the spectrum of the echo from a given transmitted pulse, as received in range-time, is the one of concern in designing receiver filters to enhance discrimination of signal from noise, or in determining the power loss in such filters. The echo from any one particle in the atmosphere is a replica of the transmitted pulse, ideally with the sinx/x spectrum of a rectangular pulse. Superposition arguments suggest that the total echo from all the particles contributing to the echo at each instant should have the same spectrum (under ideal conditions of reflectivity uniform with range, and range-corrected echo intensity). Section 5.5.1 of Bringi and Chandrasekar (2001) shows that the autocorrelation function of the echo under such idealized conditions is triangular, and a triangular ACF corresponds to a sinx/x spectrum.

But the video envelope of weather echoes bears little resemblance to the transmitted pulse, and it is far from obvious that the spectra should be identical. We set out to demonstrate this equivalence, first by conducting computer simulations of an idealized weather echo (allowing us to control the properties of the simulated signal) and computing its spectrum, and then by analyzing some data collected with the CSU-CHILL radar (now available in the form of digitized IF with sub-pulse-duration sampling rates). The simulations demonstrate that analysis of a very long length of record (hundreds of times the pulse duration) is needed to begin to bring out the sinx/x character of the spectrum. Averaging of multiple spectra, which can be based on shorter records, turns out to be a more effective way to bring out the nature of this spectrum.

Processing the digitized IF data collected with CHILL during stratiform precipitation substantiated the basic conclusions of the simulations. However, reflectivity variations along the beam, coupled with the limited lengths of record available and the limited size of the data set acquired, prevented a clear demonstration of the underlying sinx/x nature of the echo spectrum.