Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 58
Weather Radar: Operation and Phenomenology
where ζˆh and ζˆv are the linear estimated reflectivity values of the
horizontal and vertical channels, while Zˆ H and ZˆV are the estimated
reflectivity values reported in decibels relative to Z.
E. CO-POL CORRELATION COEFFICIENT
In the weather radar community, the co-pol correlation coefficient,
usually called simply the correlation coefficient, is the magnitude
of the statistical correlation between reflected horizontally and vertically polarized signals, i.e.,
ρ=
E S hh Svv∗
σ hhσ vv
,
(46)
2
2
hydrometeors, and σ hh = E S hh and σ vv = E Svv .
The co-pol correlation coefficient is considered a good indicator of the homogeneity or heterogeneity of scatterers in a resolution
volume. A value of ρ ≈ 1 indicates highly homogeneous scatterers,
whereas smaller values of ρ indicate more heterogeneous scatterers.
(51)
Thus, the co-pol correlation coefficient and differential reflectivity
provide an approximation of the first two moments of the DSD.
In general, differential reflectivity is considered a good indication
of the average oblateness of spheroidal scatterers. Still, differential
reflectivity does not provide information regarding the distribution
of scatterer shape and orientation. However, the correlation coefficient, when considered in conjunction with differential reflectivity,
is related to the variance of the ratios of the amplitude returns at
horizontal and vertical polarization [39].
Consider the amplitude ratios of the echoes from a single hydrometeor,
S hh
Svv
(47)
Svv
.
S hh
(48)
and
H
1
Np
ρˆ =
Np
i =1
( )
yih yiv
(49)
*
σˆ hhσˆ vv
,
(52)
v
h
where yi and yi are samples of the received signal in the horizontal
and vertical channels, respectively;
Np
1
Np
y
1
Np
y
h
i
i =1
2
;
(53)
.
(54)
and
σˆ vv =
Np
i =1
v
i
2
F. DIFFERENTIAL PHASE AND SPECIFIC DIFFERENTIAL
PHASE SHIFT
The differential phase product is the difference in phase between
the received signals in the horizontally and vertically polarized
channels [22]. It corresponds to the angle of the co-pol correlation
term at zero lag, i.e.,
E S hh Svv
.
φhv = arg
σ hhσ vv
*
Assuming a spheroidal scatterer, the variances of ϒ H and ϒV are a
function of the linear differential reflectivity ζdr and the correlation
coefficient ρ [39]:
σ ϒ2 = ζ dr (1 − ρ 2 )
The correlation coefficient is simply the magnitude of the zero-lag
correlation between the received signals in the horizontally and
vertically polarized channels. It is estimated as
σˆ hh =
1. Phenomenology
ϒV =
−1
σ D2 = 2.23ζ DR
(1 − ρ 2 ).
2. Calculation
where the superscript asterisk denotes the complex conjugate, Shh
and Svv are the zero-mean random-valued amplitude returns corresponding to the co-pol scattering coefficients of the ensemble of
ϒH =
Furthermore, in the case of equilibrium-shaped raindrops, i.e., theoretical raindrops whose shapes are no longer changing because
of a balance of forces, Jameson [39] has shown that the variance
of the volume-equivalent diameter of an ensemble of raindrops is
(55)
While the differential phase alone is not very informative, a derived product called the specific differential phase, which is defined as the range derivative of the differential phase, can be related to water content and the mass-weighted mean axis ratio of
drops in a resolution cell [40]:
and
σ ϒ2 = ζ dr−1 (1 − ρ 2 ) .
V
(50)
r=
∞
rD N ( D)dD .
D N ( D)dD
0
∞
3
3
(56)
0
58
IEEE A&E SYSTEMS MAGAZINE
JULY 2017, Part II of II
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