Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 57

Reed, Lanterman, and Trostel

2. Calculation
The average radial velocity of the power spectrum for a given resolution volume is estimated using pulse-pair processing [33], which
assumes a Gaussian-shaped power spectrum. The mean velocity is
estimated from the phase angle of the first lag of the autocorrelation of the slow-time signal:
Vˆr = −

λ
4π PRI

arg Rˆ y 1,

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where Rˆ y is the estimated correlation function of the measured
slow-time sequence and arg denotes the angle of a complex number.

C. SPECTRAL WIDTH
Spectral width is a measure of dispersion of velocities within the
radar sample volume and is defined as the standard deviation11 of
the velocity power spectrum [22]. It is also a measure of the reliability of the estimated radial velocity. Before the advent of dualpol, this was a useful parameter to determine areas of intense activity related to debris involved in tornadoes.

1. Phenomenology

σ V2 = σ s2 + σ α2 + σ 2f + σ o2 + σ t2 ,

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where σ s2 is due to radial shear, σ α2 is due to antenna motion, σ f
is due to variation in hydrometeor fall speed, σ o2 is due to oscillation and orientation changes of hydrometeors, and σ t2 is due to
turbulence.
2

2. Calculation
Similar to the computation for average radial velocity, the estimate
of the spectral width of the slow-time power spectrum is computed
using pulse-pair processing [33]:
 ˆ
Ry 0 − σˆ n2
ln   
σˆVr =
2π PRI 2  Rˆ y 1


1. Phenomenology
While small raindrops are approximately spherical, aerodynamic
and viscous forces cause larger raindrops to take on an oblate (i.e.,
oriented with the major axis in the horizontal dimension) spheroidal shape. Several numerical models have been suggested for
relating raindrop size to the axis ratio,
r=

b
,
a

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where b is the semiminor vertical axis and a is the semimajor horizontal axis. Among the most well known are the linear fit model of
Pruppacher and Beard [35],
r = 1.03 − 0.0062 D,

0 ≤ D ≤ 9 mm ,

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and the polynomial fit model of Beard and Chuang [36],
r = 1.0048 + 5.7 × 10−4 D − 2.628 × 10−2 D 2

The spectral width is modeled using a sum of independent broadening mechanisms [11], [34]:

cell [22]. It is considered a good indicator of raindrop shape, which
is highly correlated with drop size. Differential reflectivity is helpful in estimating rainfall rate, because reflectivity does not have a
one-to-one relation with rain rate.


.



+ 3.682 × 103 D 3 − 1.677 × 10−4 D 4 ,

∞

 rS
r =
 S
0

2
hh

∞

pr ( r )dr

pr (r )dr

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and the differential reflectivity, ζdr, of an ensemble of oblate spheroidal raindrops is [38]
7

If the term inside the logarithm is negative, then the estimate is
not valid.

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Another model, proposed by Andsager et al. [37], combines the
Beard and Chuang equilibrium model with experimental observations to account for an apparent upward shift in the mean axis ratio
of raindrops relative to their theoretical equilibrium axis ratios.
As discussed in Section II.G, when reflecting spheroids are in
the Rayleigh scattering region, the increased horizontal radius of
the oblate raindrop results in an increased reflectivity in the horizontally polarized channel relative to the vertically polarized channel. A good but biased approximation relating the power-weighted
average axis ratio,

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0 ≤ D ≤ 7 mm.

ζ dr = r 3 .

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2. Calculation
D. DIFFERENTIAL REFLECTIVITY
Differential reflectivity is the ratio of the reflected horizontally and
vertically polarized power returns from a given volume resolution
11

Here, we use the term "standard deviation" as a mathematical description of the spread of the power spectrum; the power
spectrum itself is not a probability density, although the power
spectrum is expected to be related to the probability density of
the underlying hydrometeor velocities.

JULY 2017, Part II of II

Differential reflectivity is reported in decibels relative to Z and is
estimated as the ratio of the horizontal and vertical linear reflectivity estimates, i.e.,
Zˆ DR = 10log10 ζˆdr
ζˆ
= 10log10 h
ζˆ
= Zˆ H − ZˆV ,

IEEE A&E SYSTEMS MAGAZINE

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Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine July 2017 Tutorial XI