Geoscience Reference
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characteristics, gain or half power beamwidth. In the simplest case of Rayleigh scattering ( D
< /4),  is expressed as:
5
(2)
K 2

Z
4
where K is the complex dielectric constant of the targets ( K =0.39 for ash (Adams et al.,
1996; Oguchi et al., 2009; Rogers et al., 2011) and Z the radar reflectivity factor. For spherical
targets, Z is given by:
(3)
Z
N
(
D
)
D
6
dD
0
where N(D)dD is the number of targets per unit volume whose diameters are between D
and D+dD . Z is generally expressed in dBZ units, defined by:
6-3
(4)
Z
(dBZ)
10
Log
Z
(mm .m
)
For Mie scattering (D  /4), one generally uses the so-called equivalent radar reflectivity
factor (Doviak and Zrnic', 1993), which is defined as the radar reflectivity factor of a small
particle population satisfying the Rayleigh approximation and that would return the same
received power. Interestingly, Z characterizes only the target and holds information on the
number and size of particles, and thus on the particle concentration. The reflectivity factor
obtained from radar measurements is usually calibrated using the dielectric factor of liquid
water (0.93) and must be corrected for volcanic ash as the dielectric factor of the latter is
lower (0.39):
2
K
(5)
Z
water
Z
2
.
38
Z
ash
water
water
2
K
ash
Finally, the radial velocity ( V r ) of the target is calculated from the frequency shift between
the transmitted and received signals. The velocity component along the antenna line of sight
(toward or away from the radar) causes the returned frequency f r to be different from the
transmitted frequency f t (Doppler effect), and is proportional to the Doppler shift f d :
2
V
2
V
r
r
f
  
f
f
f

(6)
d
r
t
t
c
Note that a negative Doppler shift ( f r < f t ) corresponds to a target with a radial component of
motion away from the antenna (positive radial velocity) and vice versa . Furthermore, if the
velocity vector is normal to the antenna direction, the Doppler shift is zero. In volcanic
soundings, the antenna beam can be set either to point upward, e.g. from the volcano slope
toward the summit, or downward, for instance aiming toward the eruptive vent from the
crater rim. In these cases, contributions of rising and falling particles to the spectra are
reversed (Fig. 6).
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