254 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 48, NO. 2, FEBRUARY 2000 Radar Reflection from Clouds: Gigahertz Backscatter Cross Sections and Doppler Spectra David A. de Wolf, Herman W. J. Russchenberg, and Leo P. Ligthart Abstract—This work deals with reflections of gigahertz-fre- quency radar signals from typical clouds over The Netherlands. Four principal mechanisms of reflection are identified. While the backscatter cross sections for these are mostly well known, there is a need to identify which, if any, are dominant in each frequency range. Numerical studies of superpositions of the main backscatter mechanisms are presented for a range of parameter values thought to occur commonly. These studies confirm previous results, but are generalized to incorporate gamma-function particle drop-size distributions. The results are relatively insensitive to the power of the diameter in the distribution function. The Doppler spectra of the reflected signals sometimes exhibit a bimodal form. One possible mechanism investigated here is the observation of reflec- tions that occur simultaneously from turbulently moving globules of particles and from incoherent reflections from particles with diameter-dependent spreads in velocities. Index Terms—Clouds, doppler radar, radar cross section. I. INTRODUCTION T HE purpose of this study is to quantify and compare the di- verse contributions to the radar backscatter cross section from clouds. Cloud altitudes and locations are such that contri- butions from ice particles are not expected to play a significant role. The major contributions come from distributions of liquid particles and possibly also from clear-air turbulence. As there is a large and well-documented body of literature on the interaction of electromagnetic waves with various media, we give only a few key references to texts [1]–[3] for interac- tions with distributions of discrete particles [4], [5], with con- tinuously varying random refractive-index media and [6], [7] specifically on applications to rain and clouds.The equations to be discussed and their backgrounds are to be found in the above references. Fig. 1 depicts a typical radar backscatter experiment: a cloud particle at is at distance from the radar. The effective halfwidth of the radar beam is defined by angle and we assume quasi-monochromatic pulses centered around wavenumber . The radar pulse defines a range cell of volume around . Here, is the effective length of a range cell defined, Manuscript received December 24, 1997; revised November 3, 1999. D. A. de Wolf is with the Center for Stochastic Processes in Science and Engi- neering, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061-0111 USA. H. W. J. Russchenberg and L. P. Ligthart are with the International Research Centre for Telecommunications-Transmission and Radar Faculty of Information Technology and Systems, Delft Universityof Technology, 2628 CD Delft, The Netherlands. Publisher Item Identifier S 0018-926X(00)01653-7. Fig. 1. Sketch of radar-cloud geometry. for example, by the 10-dB power level of an individual pulse. Typical values for the 3.3-GHz DARR radar [8] are m and . If absorption of electromagnetic energy to and from a range cell is ignored, then the instantaneous radar backscatter cross section is defined as with (1) if is the backscatter amplitude of a single particle at with effective diameter . It is reasonable to assume the Rayleigh backscatter cross section (which requires diameter wavelength ) for cloud particles at 1–10 GHz, i.e., (2) Here, is the relative dielectric permittivity of the particle. In a continuum description, and (1) is replaced by a spatially averaged quantity (3) where represents the average number of particles per unit volume with diameters between and . Expe- rience with convection fogs [9] has shown that modified gamma functions are useful in modeling (4) where is a gamma function, is the usual particle density in m and where and also can be functions of 0018–926X/00$10.00 © 2000 IEEE