WMOD00471

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

AUGUST 1984

where Vs is some sampling volume, much smaller
than Vc. Comparing (5) and (6), one can see that

For ice particles:

IKIT
Ze = IKI~ z.

2. Values for the dielectric factor

The next point concerns the appropriate value for
IKIT in (7). To determine that requires recognition of
an artifice, first introduced by Marshall and Gunn
0952), which has become a generally used convention
in analyzing ice particle size data. They determined
the size of each snowflake by melting it and measuring
the diameter of the resulting water drop. This diameter
is smaller than that of the ice sphere of mass equivalent
to the original particle, by a factor 0.921/3 (0.92 being
the specific gravity of solid ice). If the melted diameter
were used in (2) with the actual dielectric factor for
ice, which has the value 0.176, the calculated radar
cross section of the particle would be too small by
the factor (0.92)2 = 0.846. The Marshall and Gunn
artifice consists of multiplying the true dielectric
factor by the quantity (YO.846) = U8, resulting in the
value 0.208.1 Then the melted drop diameter can be
used with that value for /KIT in (2) to obtain the
correct radar cross section for the particle. Of course,
the same result could be obtained by using 0.176 for
the dielectric factor and adjusting the melted drop
diameter by the factor 0.92-1/3 = 1.028.
Consequently, there are two possible "correct"
values of IKIT in the foregoing equations, depending
upon how the particle sizes are determined. If, in
calculating Z the particle sizes used are melted drop
diameters, as in the work of Gunn and Marshall
(1958), Sekhon and Srivastava (1970), and others,
the appropriate value for IKIT is 0.208 and

Ze = 0.224Z. (8)

In logarithmic form, this becomes

Ze (in dBz) = Z (in dBz) - 6.5 dB. (9)

If, on the other hand, the particle sizes are expressed
as equivalent ice sphere diameters, the appropriate
value for IKIT is 0.176 and

Ze = 0.189Z. (10)

In logarithmic form (10) becomes

Ze (in dBz) = Z (in dBz) - 7.2 dB. (11)

To determine the dielectric factor IKIT, Marshall

I The incorrect value 0.197 appeared in the original Gunn and
East (1954) paper. A corrigendum was pUblished in Vol. 81 of
Quart. J. Roy. Meteor. Soc. (p. 653), but the erroneous value has
been perpetuated by Battan (1973) and other authors.

NOTES

1259

(7)

and Gunn (1952) used a theory originated by Debye
to calculate the relative permittivity of the i~~-air
mixture making up a particle. The sJ-lbjectof the
dielectric properties of mixtures has recently received
considerable attention (cf. Evans, 1965; Bohren and
Battan, 1980, 1982) and questions have been raised
about the applicability of the Debye theory to snow.
In fact, it may not be possible to calculate exact
values for the permittivity of heterogeneous mixtures
(de Loor, 1983), although limiting boundaries can be
specified. Nevertheless, the expression given as Eq.
(10) in Bohren and Battan (1980) agrees better with
experimental data than the Debye function, and the
two can be compared to illustrate the magnitude of
possible differences.
The former expression gives values for the relative
permittivity of ice-air mixtures which are, at most,
about 5% higher than the values obtained from the
Debye function. The difference varies with the com-
position of the mixture, being zero when the "mix-
ture" is either 100% air or 100% ice and reaching the
maximum at an ice fraction of around 60%. The
corresponding differences in the dielectric factor IKIT
are larger, reaching a maximum of 18% at an ice
fraction of about 45%. This implies that the equivalent
radar reflectivity factors could be as much as 0.7 dB
higher than those indicated above. Because of uncer-
tainties in the theory and the fact that the difference
varies with the composition of the mixture (which is
seldom known for individual particles and probably
varies over the array of particles within the radar
contributing region), no more exact value can be
given.

3. Ze-R relationships for snow

As noted in the opening paragraph, weather radar
systems are customarily calibrated to measure the
"water equivalent" Ze defined by (4) with IKI~
= 0.93. The dielectric factor is incorporated into a
radar calibration constant, and that constant is 'not
altered when the precipitation form changes from
liquid to solid. This means that some care must be
used in employing published snow Z-R relationships
derived from particle size observations.
To illustrate this, consider the snow Z-R relation~
ship obtained by Sekhon and Srivastava (1970);. .

Z = 1780R2.21 (12)

with Z in mm6 m-3 and R in mm h-'. In logarithmic
form,
Z (in dBz) = 32.5 + 22.1 10gR. (13)

The snowflake size data used by Sekhon and Srivas-
tava were melted diameters, so (8) or (9) is the
appropriate relationship between Ze and Z. Thus, the
Sekhon and Srivastava result corresponds to

Ze (in dBz) = 26 + 22.1 10gR. (14)