WMOD00471

• Title: Equivalent Radar Reflectivity Factors for Snow and Ice Particles
• Date: 8/8/1974
• Weather Modification - Doc Type: Report

1258

JOURNAL OF CLIMATE AND APPLIED METEOROLOGY

VOLUME 23

Equivalent Radar Reflectivity Factors for Snow and Ice Particles

PAUL L. SMITH

Institute of Atmospheric Sciences, South Dakota Scho%f Mines and Technology, Rapid City, SD 57701-3995
28 November 1983 and 2 May 1984

From time to time, questions arise concerning how
to compare radar observations of snow or ice particles
with corresponding particle size data or how to esti-
mate snowfall from radar observations. Weather radar
systems customarily measure the equivalent radar
reflectivity. factor Ze, so any calculations involving
particle sizes must ultimately be expressed in terms
of Ze. Persons skilled in radar meteorology , can (and
normally do) work out correct procedures, but stan-
dard works on radar meteorology (e.g., Battan, 1973)
do not discuss the matter explicitly. The purpose of
this note is to clarify the situation for the benefit of
those who may encounter difficulty with such prob-
lems.

1. Relationships between Z and Ze

Both physical factors and accepted conventions
complicate the subject, so it is best to work from the
basic definitions. The radar reflectivity factor Z for
rain can be expressed in terms of raindrop sizes as

Z = L D6/Vc = Ze. (1)

Here D represents .the drop diameter and the sum-
mation must be carried out over all the drops in the
radar contributing region of volume Vc. The last part
of ( 1) indicates that, for spherical drops with diameters
small compared to the radar wavelength, the equiv-
alent radar reflectivity factor Ze is equal to Z.
For snowflakes or other ice particles, the fact that
the particle shapes are generally far from spherical
complicates matters. That difficulty can be handled,
according to Marshall and Gunn (1952), by noting
that for particles small enough to fall in the Rayleigh .
scattering region (which requirement is reasonably
well fulfilled at the usual weather radar wavelengths
by snowflakes and small ice particles), the radar cross
section of an irregular particle composed of a weak
qielectric like ice is the same as that of a sphere of
the same mass. In other words, the exact shape of
the particle is immaterial. The general absence of
substantial depolarization (apart from that due to
propagation effects) in radar echoes from snow or ice
particles (e.g., Hendry and Antar, 1981) corroborates
this idea.

@ 1984 American Meteorological Society

One can therefore write the radar cross section of
such an ice particle as

(J = 1l"5IK\lD6/X4. (2)

(This and the other basic expressions used here can
be found in Chapter 4 of Battan, 1973). Here IKI2 is
the dielectric factor I(Er - l)/(Er + 2)12, Er being the
relative permittivity. (It is related to the index of
refraction n by Er = n2.) The subscript i indicates that
a value appropriate for ice should be used. The
diameter implied in (2) is that of a sphere having the
same mass as the particle in question. .
The radar reflectIvity '11 (or radar cross section per
unit volume) of the array of particles in the radar
contributing region of volume Vc is

(J 1l"5IKIT D6
'11 = L - = ~ L - . (3)
vcVc X vcVc

The equivalent radar reflectivity factor Ze is defined
as

X4'11
Ze = 1l"5IKI~'

where the subscript w indicates that the value appro-
priate for water (approximately 0.93 for the usual
meteorological radar wavelengths) is used by conven-
. tion. That convention is adopted because when radar
measurements are made, one is often not certain
whether the particles are water or ice. (Moreover,
particles of each type frequently occur in different
parts of the scanned volume.) Substituting the value
'of '11 from (3) into (4),

(4)

For ice particles:

. IKIT D6
Ze = IKI2 LV. (5)
w Vc c

When particle size data are analyzed to determine
radar variables, the quabtity usually calculated is the
radar reflectivity factor Zand not the equivalent
radar reflectivity factor Ze. The analysis yields values
of

D6 D6
L-=L-=Z,
VS Vs Vc Vc

(6)