Examining an Alternative Normalization of Drop Size Distributions and Its Utility for Spaceborne Radar Retrievals of Rain

The normalized distribution advanced by Testud et al (2001) has enjoyed much popularity since its introduction. This normalization is achieved through the definition and use of volume-weighted mean diameter, or mean volume diameterin short, which is the ratio of the fourth to the third moments of the drop size distribution (DSD). However, Kuo et al (2004) reports that the equivalent reflectivity and specific attenuation at Ku and Ka bands can be characterized, with decreasing importance, by liquid water content, effective radius, and effective variance of the DSD. Effective radius is defined as the ratio of the third to second moments of the DSD in radius and is half of effective diameter, whereas effective variance is defined as the ratio of the product of fourth and second moments to the square of the third moment of DSD minus one. This inspires us to define an alternative DSD normalization through effective diameter (or mean cross-section diameter). The effective variance thus simplifies to the ratio of mean volume diameter to effective diameter minus one.

(For derivation please see the accompanying Image.)

It is thus logical to examine the effectiveness of using both normalizations for the characterization of equivalent reflectivity at frequencies used for spaceborne radars, i.e. Ku, Ka, and W bands, and in turn corresponding retrievals of rain, because the characterizing parameters may now be more concisely expressed. DSDs derived from 2D video disdrometer (2DVD) observations in various field campaigns provide the DSD population for our investigation. The DSD moments and corresponding equivalent reflectivity and specific attenuation at frequencies are first calculated and tabulated. We first analyze the information content and remove redundancy. We then attempt to relate equivalent reflectivity and specific attenuation to liquid water content, effective diameter, and mean volume diameter in a functional form. We also derive functional form for the inverse, i.e. relating equivalent reflectivity and specific attenuation at multiple frequencies to liquid water content, effective diameter, and mean volume diameter, essentially a retrieval scheme.

Kuo, K. S., E. A. Smith, Z. Haddad, E. Im, T. Iguchi, and A. Mugnai, 2004: Mathematical–physical framework for retrieval of rain DSD properties from dual-frequency Ku–Ka-band satellite radar. J Atmos Sci, doi:10.1175/1520-0469(2004)061<2349:MFFROR>2.0.CO;2.

Testud, J., S. Oury, R. A. Black, P. Amayenc, and X. Dou, 2001: The concept of normalized distribution to describe raindrop spectra: A tool for cloud physics and cloud remote sensing. J. Appl. Meteorol, 40, 1118–1140.