Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 49

Reed, Lanterman, and Trostel
tive transmitted pulses in a CPI is called the pulse repetition interval (PRI), and its reciprocal is the pulse repetition frequency
(PRF):
PRF =

1
.
PRI

(14)

The range of Doppler frequencies that may be measured without
ambiguity is determined by the Nyquist criterion:
−

PRF
PRF
≤ fd ≤
.
2
2

(15)

The magnitude of the discrete Fourier transform of the slow-time
sequence provides an estimate of the Doppler frequencies of targets in that particular resolution volume. In the case of a single
scatterer that has constant RCS and constant radial velocity over
the duration of the CPI, the samples of the slow-time sequence are
perfectly correlated and the magnitude of the Doppler spectrum
has a narrow peak at the Doppler frequency of the target. The case
in which the target echoes have an exponential power distribution
and are perfectly correlated from pulse to pulse is called a Swerling I target [13]. A Swerling II target model refers to the case of
exponentially distributed target echoes that are decorrelated from
pulse to pulse, resulting in a white Doppler spectrum. At the PRIs
used in most weather radars, raindrop scatter only partially decorrelates from pulse to pulse and cannot be modeled as a Swerling I
(completely correlated pulse to pulse) or Swerling II (completely
decorrelated pulse to pulse) target. The Doppler spectrum of rain
scatter is usually modeled as Gaussian, with the spectral width related to the rate of decorrelation.

G. POLARIZATION
EM waves, traveling along a path, are associated with a vector
quantity E, which lies in a plane orthogonal to the direction of
propagation. Assuming the EM waves travel along the z-axis of a
Cartesian coordinate system, the electric field vector, varying sinusoidally in time and space, has the form
ˆ x cos ( 2π f t t − k w z ) +
E = xE

ˆ y cos ( 2π f t t − kw z + Ψ t ) ,
yE

(16)

where xˆ and yˆ are unit vectors in the horizontal (i.e., parallel to
Earth's surface and orthogonal to the direction of propagation) and
vertical (i.e., orthogonal to both the Earth's surface and the direction
of propagation) directions, Ex and Ey are the amplitudes of sinusoidal variation in the x and y directions, kw is the wavenumber (spatial
frequency in radians per meter), ft is the transmitted frequency, and
Ψt is the relative phase of the x and y components upon transmission. The orientation of the electric field vector defines the polarization of the wave. If the wave only has a horizontal (x) component,
it is horizontally linearly polarized. Similarly, if only a vertical (y)
component is present, the wave is vertically linearly polarized. The
wave may also be linearly polarized at an angle other than horizontal or vertical. Other forms of polarization include circular, elliptical, or unpolarized. A dual-polarization (dual-pol) radar, such as the
JULY 2017, Part II of II

WSR-88D, simultaneously transmits and receives horizontally and
vertically polarized waves [10]. For a more in-depth discussion on
polarization and its applications in radar, see [14].
The backscatter RCS of a target depends on the polarization
of the incident EM wave. The RCS of spheres and spheroids are
of particular interest because these are often used as shape models
for reflecting hydrometeors (e.g., small raindrops and some hail).
In the case of a wave incident on a perfect sphere, the RCS is independent of polarization and can be related to the sphere diameter.
However, given an oblate or prolate spheroid, the RCS is a function
of polarization. Denoting the transmit polarization as t type and the
received polarization as r type, the polarization-dependent scattering coefficient Srt is related to the RCS σrt through the equation2
2

S rt =

σ rt
.
4π

(17)

This polarization dependence can be expressed in a scatter matrix:
 Esh   S hh
 v = 
 Es   Svh

Shv   Eih 
  ,
Svv   Eiv 

(18)

where Ei· is an incident electric field component, Es· is a scattered
electric field component, ·h indicates horizontal polarization, and ·v
indicates vertical polarization.3 The scattering matrix reveals how
the reflecting object affects the incident wave in terms of polarization, amplitude, and phase shifts.
When a sphere's radius is significantly smaller than the wavelength of the incident EM field, it lies in the Rayleigh scattering
regime [15] and the RCS is directly related to the sphere's radius.
Similarly, Rayleigh theory describes the RCS of an oblate spheroidal4 raindrop, in which case the effective RCS is largest if the plane
of polarization is the same as the plane of the semimajor axes. This
phenomenon is used in meteorological applications to infer information regarding the size, shape, composition, and orientation of
detected hydrometeors, as discussed in more detail in Section V.

H. ATTENUATION
When EM waves are transmitted through the atmosphere, the
waves are attenuated by gases, water vapor, and hydrometeors because of absorption and scattering. In contrast to traditional radar
applications, the target of weather radar is also the source of attenuation. At the frequency band of most weather radars (S band),
2

In general, the scattering coefficient is a function of the direction of scatter. For the purposes of this article, unless otherwise
stated, the scattering coefficients refer to backscatter in the direction of the radar.
For simplicity, as is common notation, a range-dependent spherical wave factor term is suppressed in (18).
In the weather radar literature, the Rayleigh scattering of spheroids is termed Rayleigh-Gans scattering after Gans who derived
the equations for Rayleigh scattering of spheroids [16]. This unfortunate terminology is not used in this article to avoid confusion with actual Rayleigh-Gans scattering, an approximation for
the scattering of small particles with an index of refraction near
unity [15].

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