The Removal of Turbulent Broadening in Radar Doppler Spectra Using Linear Inversion with Double-Sided Constraints

Abstract

Remote sensing instruments have the ability to collect data over extensive temporal periods and spatial regions. A common thread between all these sensors is the need to relate the measured quantity to a meaningful observation of a system property. If the relationship between each measurement and the set of atmospheric quantities that influence that measurement is known, the problem can be reduced to a set of linear equations. Solving for the unknown atmospheric quantities then becomes a linear algebra problem, where the solution vector is equal to the inverse of the kernel matrix multiplied by the set of independent measurements. However, in most remote sensing applications, inversion of the kernel matrix is unstable, resulting in the amplification of measurement and computational uncertainties. Techniques to circumvent this error amplification have focused on methods of constraining the solution. In this paper, the authors adapt an existing technique to do such an inversion. Noise reduction is accomplished by the addition of double-sided inequality constraints for each unknown variable. The advantage of such a technique is the ability to individually adjust the solution space of each individual unknown, depending on a priori knowledge. The inversion algorithm is applied to the problem of retrieving radar Doppler spectra, which have been artificially broadened by turbulent air motions. First, to test the algorithm, radar Doppler spectra were simulated using known drop size and vertical air motion distributions. The simulated spectra were used as input to the retrieval algorithm, and the results were compared to the initial quiet-air spectrum. Results indicate that accurate retrievals can be performed despite the addition of moderate amounts of noise to the simulated spectra. Then, to demonstrate the practical retrieval of quiet-air Doppler spectra, the algorithm was used to process radar observations collected from continental stratocumulus. From these retrievals, a two-dimensional map of the large-scale vertical motions within the cloud was constructed as well as a profile of vertical velocity variance. In addition, a drop size distribution was also derived from an updraft region of the cloud.