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Citation: Huang, J.; Yin, J.; Xu, Z.; Li,
Y. Polarization Scattering Regions: A
Useful Tool for Polarization
Characteristic Description. Remote
Sens. 2025,17, 306. https://doi.org/
10.3390/rs17020306
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Article
Polarization Scattering Regions: A Useful Tool for Polarization
Characteristic Description
Jiankai Huang , Jiapeng Yin * , Zhiming Xu and Yongzhen Li
The State Key Laboratory of Complex Electromagnetic Environment Effects on Electronics and Information
System, College of Electronic Science and Technology, National University of Defense Technology,
Changsha 410073, China; huangjiankai18@nudt.edu.cn or huangjiankai08@163.com (J.H.);
zhimingxu@nudt.edu.cn (Z.X.); liyongzhen@nudt.edu.cn (Y.L.)
*Correspondence: yinjiapeng@nudt.edu.cn; Tel.: +86-13618498938
Abstract: Polarimetric radar systems play a crucial role in enhancing microwave remote
sensing and target identification by providing a refined understanding of electromagnetic
scattering mechanisms. This study introduces the concept of polarization scattering regions
as a novel tool for describing the polarization characteristics across three spectral regions:
the polarization Rayleigh region, the polarization resonance region, and the polarization op-
tical region. By using ellipsoidal models, we simulate and analyze scattering across varying
electrical sizes, demonstrating how these sizes influence polarization characteristics. The
research leverages Cameron decomposition to reveal the distinctive scattering behaviors
within each region, illustrating that at higher-frequency bands, scattering approximates
spherical symmetry, with minimal impact from the target shape. This classification pro-
vides a comprehensive view of polarization-based radar cross-section regions, expanding
upon traditional single-polarization radar cross-section regions. The results show that
polarization scattering regions are practical tools for interpreting polarimetric radar data
across diverse frequency bands. The applications of this research in radar target recog-
nition, weather radar calibration, and radar polarimetry are discussed, highlighting the
importance of frequency selection for accurately capturing polarization scattering fea-
tures. These findings have significant implications for advancing weather radar technology
and target recognition techniques, particularly as radar systems move towards higher
frequency bands.
Keywords: polarization scattering regions; radar cross-section (RCS); scattering mechanisms;
Cameron decomposition
1. Introduction
In radar systems, the detection, tracking, and identification of a target rely fundamen-
tally on the echo signal [
1
,
2
]. The ability to quantify or characterize echoes is essential in
radar operation. When an object is exposed to an electromagnetic wave, it disperses the
incoming energy in multiple directions. The object causing this scattering is often referred
to as a scatterer, and the process is known as scattering. To facilitate this, a parameter
known as the radar cross-section (RCS) is assigned to the target [
3
]. The RCS is used to
quantitatively describe the strength of echoes and is a widely recognized metric in radar
and electromagnetic research. The portion of energy that is scattered back toward the
source, known as backscattering, forms the radar echoes of the object [4].
An accurate understanding of the scattering mechanism is fundamental to polarimetric
radar detection and identification. For simple convex targets, single-polarization RCS
Remote Sens. 2025,17, 306 https://doi.org/10.3390/rs17020306
Remote Sens. 2025,17, 306 2 of 18
scattering regions can be classified into three spectral regions: the Rayleigh region, the
resonance region (or Mie region), and the optical region [
5
]. The monostatic backscatter
mechanism of spherical targets has been extensively investigated [
6
]. The resonance region
is typically defined by the electrical size
ka
, where
a
is the sphere radius, the wave number
k
is defined as
k=
2
π/λ
, and
λ
is the wavelength. The resonance region’s
ka
ranges from 1
to 10. The range
k×a=
1 to 10 implies that the scatterers are in a size range where they
are large enough to produce noticeable resonance effects but not so large that they cause
diffraction, which would occur at much higher values of
k×a
. Thus, the limits of
k×a
from 1 to 10 typically correspond to a region where the scatterer is large enough to exhibit
resonant scattering but small enough that the scattering is not dominated by geometric
diffraction [
3
]. The electrical size depends on the frequency of the radar system as well
as the size of the target. These theories are well established and form a cornerstone in
single-polarization radar science. However, few similar studies have been conducted in the
domain of polarization.
Radar polarimetry technology has significant applications in the fields of microwave
remote sensing, earth observation, meteorological measurement, anti-interference, and
target recognition [
7
–
9
]. In general, when a target has a larger projected area in one di-
rection, there is also a larger scattering polarization component in that direction. In other
words, when a target has a larger projected area in a particular polarization direction
(e.g., the horizontal polarization for a horizontally aligned dipole), the scattering polariza-
tion component in that direction is generally stronger, while the orthogonal polarization
component becomes negligible. In meteorology, the differential reflectivity
ZDR
, i.e., the
difference between the horizontal polarization echoes and the vertical polarization echoes.
ZDR
plays a significant role in identifying and estimating rainfall [
10
].
ZDR
is particu-
larly effective in distinguishing between rain and hail [
11
,
12
]. However, discrepancies
between
ZDR
values of the S band and X-band can arise during the data fusion process
when polarimetric weather radars observe cloud and rainfall areas [
13
]. The Colorado
State University–University of Chicago-Illinois State Water Survey (CSU-CHILL) radar
is designed to address the technical challenges posed by dual-frequency measurement.
For the same meteorological frontal passage, S-band
ZDR
values are larger than those at
X-band [
14
,
15
]. However, certain special phenomena in radar polarimetry applications
require attention. In certain frequency ranges, for scatterers with the long axis in the vertical
direction, the
ZDR
can even exhibit reversals [
16
,
17
]. In 2015, the CSU-CHILL S-band radar
showed a negative
ZDR
column during a graupel-shower event [
18
]. Additionally, in the
field of SAR remote sensing, scatterers often do not conform strictly to spherical structures,
yet polarimetric decomposition yields a significant presence of spherical structures [
19
].
These occurrences indicate that the traditional assumptions may be ineffective in certain
scenarios. The complex interaction between polarization characteristics and target electrical
size requires further analysis.
Traditionally, we have attributed the polarization states of scattered electromagnetic
waves to factors such as the shape, material, and orientation of targets [
20
]. However, in
practice, beyond these factors, the electrical sizes of the targets also play a significant role
in shaping the target’s polarization scattering characteristics. The interplay between the
polarization properties and the electrical sizes needs to be subjected to a thorough analysis.
This paper establishes the concept of the polarization scattering regions. Emphatically, in
the polarization scattering optical region, the shapes of ellipsoids have minimal impact on
the polarization of scattering waves.
The remainder of the paper is structured as follows: Section 2offers a physical ex-
planation for the scattering regions. Section 3delves into the simulation mehtod and
mathematical modeling of polarized scattering from deformable ellipsoids. Section 4pro-
Remote Sens. 2025,17, 306 3 of 18
vides details of the polarization scattering regions. Section 5gives the discussion of the
results, and Section 6summarizes the key findings and their implications for future research
in radar technology.
2. Physical Mechanism
The single-polarization RCS represents the equivalent area of an idealized metal
sphere that would produce an identical echo signal if it were to replace the actual target.
This concept allows for a standardized way of expressing the target’s reflective properties,
allowing for more precise evaluation and comparison across different target types. An in-
depth study of the interaction between the electromagnetic field and a target is fundamental
to understanding the mechanisms of the RCS. In recent years, numerous researchers have
shown significant interest in exploring RCS mechanisms [
21
]. To facilitate RCS analysis,
the target scattering can be classified into three regions based on their electrical size. For
single-polarization RCSs, these regions include the Rayleigh region, resonance region,
and optical region. However, such classifications are rarely applied within the domain
of polarimetric radar. Unlike single-polarization RCSs, which only considers a single
scattering parameter, polarimetric radar requires a more sophisticated interpretation of
the scattering matrix, including co-polar and cross-polar components. This increases the
analytical challenge, making it less straightforward to directly extend traditional single-
polarization RCS classifications to the polarimetric domain. In the following, a simple
metal sphere is used as an example to introduce the classic single-polarization RCS regions
and explain the underlying mechanisms. Subsequently, the concept of single-polarization
RCS regions is extended to polarimetric RCS regions, conducting a detailed analysis of
their scattering characteristics.
2.1. Single-Polarization Scattering Region
The RCS of a target is derived using the incident electric field strength
Ei
of the wave
that strikes the target, and the electric-field strength of the scattered wave
Es
, measured at
the radar receiver. The RCS is formally defined as the norms of Eiand Es[1]:
σ=lim
R→∞4πR2|Es|2
|Ei|2, (1)
where
R
denotes the distance from the radar to the target, commonly known as the radar
range. Although transmission strength and distance are crucial for target detection, they
do not influence the calculation of the RCS, as the RCS is an inherent property of the
target’s reflectivity.
Figure 1shows the frequency-dependent RCS (in decibels per square meter, dBsm)
of an ideal metallic sphere with a radius of
a=
2.998 cm. The parameter
ka =
2
πa/λ
represents the circumference of the sphere in units of wavelength, where
a
is the radius
of the sphere and
λ
is the wavelength of the incident wave. The
x
-axis represents the
frequency
f
of the incident wave, spanning from 0.1 GHz to 100 GHz, where
f=c/λ
. The
speed of light is c, around 2.998 ×109m/s.
As shown in Figure 1, the RCS of the ideal metallic sphere increases sharply as the
frequency
f
increases from zero, reaching a peak near
ka =
1 (the green point “s”). After this
peak, the RCS exhibits a pattern of oscillations that gradually becomes flat (from the yellow
point “t”) in amplitude as the sphere becomes electrically larger. These oscillations are
caused by two distinct scattering mechanisms: (1) specular reflection from the sphere’s front
surface, and (2) a creeping wave that travels around the shadowed side of the sphere. The
interference between these contributions leads to alternating constructive and destructive
phases as the difference in their electrical path lengths increases continuously with f. The
Remote Sens. 2025,17, 306 4 of 18
amplitude of these oscillations diminishes at higher values of
f
because the creeping wave
loses energy over the extended path around the sphere’s shadowed side.
Figure 1. The RCS of an ideal metallic sphere with a radius of 2.98 cm, in decibel square meters, dBsm
(dB relative to 1 m2).
The transition from wavelength-dominated scattering behavior to size-dominated
behavior is reflected in three regions: Rayleigh, resonance, and optical. These regions
provide a comprehensive view of how the RCS varies with the sphere’s electrical size.
(1)
Rayleigh region (0
<ka <
1): In this low-frequency range, where the sphere is electri-
cally small, the normalized RCS scales with the fourth power of
ka
, i.e.,
RCS ∝(ka)4
.
This behavior is characteristic of small or thin structures, where the incident wave-
length is much larger than the object’s dimensions.
(2)
Resonance region: This intermediate region, where interference between specular and
creeping-wave components is prominent, extends approximately up to
ka =
10. In this
range, the RCS oscillates due to the phase interactions between the two contributions.
There is no sharply defined upper limit for this region, but
ka =
10 is generally
considered a practical boundary [3].
(3)
Optical region (
ka >
10): At high values of
ka
, the scattering is primarily governed by
the specular reflection from the sphere’s front surface. In this region, the geometric
optical approximation accurately predicts the RCS, as the sphere behaves as a larger
reflective surface with minimal influence from creeping waves.
2.2. Polarization Scattering Region
The polarization scattering regions are characterized in a similar way to the classical
single-polarization RCS regions, based on the size of the scattering object relative to the
wavelength of the incident electromagnetic wave. These polarization scattering regions can
be divided into the polarization Rayleigh region, the polarization resonance region, and the
polarization optical region. Understanding these regions can help analyze the scattering
behavior in different frequency bands and polarization states, which can improve detection,
classification, and understanding of the scattering objects [22].
Remote Sens. 2025,17, 306 5 of 18
The polarized electric field is often represented using the Jones vector, which consists
of a pair of orthogonal Jones vectors [
23
,
24
]. The Jones vectors for the incident and scattered
waves are given as
Ei="Ei
1
Ei
2#, (2)
and
Es="Es
1
Es
2#, (3)
respectively. The scattering relationship between these fields is described by a matrix
S
,
known as the scattering matrix. In this notation,
i
and
s
indicate incident and scattered
waves, while subscripts 1 and 2 represent any orthogonal polarization pair. The components
of
S
are generally complex. For horizontally (
H
) and vertically (
V
) polarized incident waves,
the scattered field is expressed as [25]
"Es
H
Es
V#=S"Ei
H
Ei
V#
="sHH sH V
sVH sVV #" Ei
H
Ei
V#.
(4)
In this scattering matrix, the diagonal elements (
sHH
and
sVV
) are referred to as co-polar
terms, as they describe interactions between identical polarization states. The off-diagonal
elements (
sHV
and
sVH
) are cross-polar terms, as they describe interactions between or-
thogonal states. Most targets satisfy the reciprocity condition, ensuring that the cross-polar
terms are symmetric [7].
The diagonal elements of the scattering matrix are termed co-polar terms because
they describe interactions involving the same polarization state for both the incident and
scattered fields. Conversely, the off-diagonal elements are referred to as cross-polar terms,
as they correspond to interactions between orthogonal polarization states. For most targets,
the scattering matrix
S
satisfies the reciprocity condition, implying that the cross-polar
term components are identical.
The polarization Rayleigh region corresponds to the scenario where the size of the
scattering object is much smaller than the wavelength of the incident wave. When an
electromagnetic wave encounters a small object, the object acts as a point scatterer. The
electric field of the incident wave induces dipole moments in the object, causing it to scatter
the wave isotopically. Polarization effects are significant because the induced dipoles align
with the electric field of the incident wave, affecting the scattered wave’s polarization state.
For instance, in the resonance region, which corresponds to a frequency of approximately
30 GHz, spherical drops with a radius of 1 cm are significantly affected. The polarization
resonance region corresponds to the scenario where the size of the scattering object is
comparable to the wavelength of the incident wave. Complex interactions lead to multiple
scattering peaks and nulls, influenced by the object’s geometry and polarization. Polar-
ization effects are also prominent as different parts of the object can scatter the incident
wave in different ways, depending on its polarization state. The polarization optical region
corresponds to the scenario where the size of the scattering object is much larger than
the wavelength of the incident wave. Scattering can be described using geometric optics
principles, such as reflection and refraction.
As mentioned before, when the target is located in the high-frequency optical region,
the result of polarimetric scattering decomposition is independent of the shape of the
scatterer. Why do targets typically demonstrate such scattering patterns in high-frequency
Remote Sens. 2025,17, 306 6 of 18
scenarios? This is because of the nature of electromagnetic wave backscattering. When
electromagnetic waves from a radar or other system hit the metal target, they cause electric
currents to be generated on the surface of the target. These currents then produce a counter-
acting scattering electromagnetic field. Both the electromagnetic radiation and scattering,
which re-radiates electromagnetic waves around the scatterer due to induced currents, are
considered secondary sources on the surface illuminated by the incident waves.
One of the boundary conditions for perfect conductors is
ˆ
n×E=
0, where vector
ˆ
n
is
a unit normal perpendicular to the boundary [
26
,
27
]. For large objects, surface currents can
be analyzed using the geometric optics method. The surface currents are represented by
the reflected and refracted fields according to boundary conditions, such as the continuity
of tangential electric and magnetic fields. This means that the parallel components of the
total electric field approach zero on the metal surface. In the polarization optical region,
the differential curvature on the target’s surface in various directions can be perceived
as uniform with respect to the wavelength, projecting nearly identically in a specific
polarimetric direction. This results in the components of the scattering electric field being
approximately equal in different directions. As a result, the target exhibits characteristics of
surface scattering or spherical scattering. This has been substantiated in physics and is used
in classical high-frequency approximation techniques in electromagnetic computations, as
validated in the physical optics method [28,29].
The following limitation of the polarization scattering region should be noted: it is only
applicable to convex bodies and does not account for multiple reflections between different
surface areas of the object. This limitation is the same as that of the single-polarization RCS
scattering regions, as mentioned in Section 1.
3. Method
3.1. Simulation Method and Model
To quantitatively analyze the polarization scattering behavior from an ellipsoidal
model, as shown in Figure 2, a simulation experiment was conducted. The general ellipsoid,
a three-dimensional generalization of an ellipse, is defined in Cartesian coordinates along
the x-, y-, and z-axes by the equation
x2
R2
x
+y2
R2
y
+z2
R2
z
=1. (5)
Rx
,
Ry
, and
Rz
represent the lengths of the
x
-,
y
-, and
z
-semi-axes, respectively. An
ellipsoid’s shape varies depending on the relationship between these axes. When two of
these axes share the same length
L
, such as
L=Rx=Rz
as illustrated in Figure 2, the
ellipsoid takes the form of a spheroid. This is known as an ellipsoid of revolution. In this
scenario, the ellipsoid remains unchanged under rotation around the third axis. When the
third axis is the longest, with
Rx=Rz<Ry
, it assumes the shape of a prolate spheroid,
as depicted in Figure 2a. If all three axes are equal in length, with
Rx=Rz=Ry
, the
ellipsoid transforms into a sphere, as shown in Figure 2b. Conversely, if the third axis is
shorter, with
Rx=Rz>Ry
, the ellipsoid takes on the form of an oblate spheroid, as seen in
Figure 2c
. The front views of the simulated ellipsoid models are in the lower right corners
of Figure 2a–c.
This ellipsoid model is chosen for several reasons. Firstly, in the polarimetric radar
community, many real-world radar targets can be approximated by spherical or ellipsoidal
models when analyzing target characteristics. Secondly, by adjusting the axis ratio of
the ellipsoid, the model can be transformed from a dipole to a sphere and an ellipsoid,
encompassing a wide range of shapes. Thirdly, by changing the axial ratio, the main
axis of the target’s projection along the direction of the incident electromagnetic wave
Remote Sens. 2025,17, 306 7 of 18
can be altered from horizontal to vertical. This model can be used to analyze how the
target’s electrical size affects polarization scattering, providing valuable insights into the
polarization scattering mechanism.
(a) (b) (c)
Figure 2. The simulated ellipsoid model with the front view in the lower right corner. The direction
of incident wave propagation is indicated by the blue arrow along the
x
-axis to the origin of the
coordinate axis. (a) When
L=
0.1
λ0
, the model resembles a dipole; (b) at
L=
1
λ0
, the ellipsoid
becomes a sphere; and (c) at L=2λ0, the target takes on the shape of an oblate ellipsoid.
The simulation parameters are set as follows: the incident wave is fully polarized.
As shown in Figure 2, the direction of incident wave propagation is given by the blue
arrow symbol along the
x
-axis to the origin of the coordinate axis. The 10 GHz frequency
corresponds to the X-band. Without loss of generality, the center frequency of the simulation
is set as
f0=
10 GHz, and the corresponding wavelength is around
λ0=c/f0=2.998 cm
,
where
c
represents the speed of light. The simulation frequency is defined as
F=b f0
,
where
b∈[
0.1, 2
]
. The simulation frequencies range from the UHF band to the K-band. The
radius of the ellipsoid’s
y
-axis (green coordinate axis) is defined as
Ry=λ0
, and those of
the
x
-axis (blue coordinate axis) and the
z
-axis (red coordinate axis) are equivalently set
to
L=Rx=Rz=lλ0
, where the normalized axis ratio
l∈[
0.1, 2
]
. This simulation model
employs ellipsoidal shapes with adjustable aspect ratios and axis orientations to study the
polarization effects of scattering. By adjusting L, various shapes can be generated:
(1)
When
L=
0.1
λ0
, the simulation ellipsoid changes to a dipole-like shape, as shown in
Figure 2a.
(2)
When L=1λ0, the ellipsoid becomes a sphere, as depicted in Figure 2b.
(3)
When L=2λ0, it approaches a vertical flat body, as shown in Figure 2c.
The front views in the lower right corner of Figure 2a–c demonstrate that while the
y
-axis
length of the ellipsoid remains constant, the z-axis length changes as lvaries.
Electromagnetic computational simulations were performed using the software Altair
FEKO 2020 with the multilayer fast multipole algorithm (MLFMA) [
30
], a full-wave method.
In order to expedite the simulation process, a segmented simulation was utilized, with the
first frequency range being from 0.1 GHz to 10.5 GHz and the second range being from
10.5 GHz to 20 GHz. Specifically, the range of
L
in the simulation was from 0.1
×
2.998
cm
to 2
×
2.998
cm
, with a step size of 0.1
×
2.998
cm
. The frequency range was from 0.1
GHz
to 20 GHz, with a step size of 0.199 GHz.
Remote Sens. 2025,17, 306 8 of 18
3.2. Polarization Scattering of the Ellipsoid
The complex quad-polarization results concerning the frequency
f
and normalized
axis ratio
l
are illustrated in Figure 3. The simulated echoes of VH and HV are almost
identical, satisfying the reciprocity of the polarization scattering matrix. The amplitude
of the VH echoes and HV echoes are represented by Figure 3c,d, respectively. It should
be noted that the amplitude range in Figure 3c,d is from
−
180 dBsm to
−
100 dBsm,
which is much lower than those in Figure 3a,b. The amplitudes of the cross-polarization
components’ HV and VH echoes are relatively negligible, being around
−
100 dB lower
than the dominant-polarization components. The abrupt changes in both amplitudes and
phases at
10.5 GHz
are attributed to the use of segmented simulations, where relative errors
between computational simulations arise due to the diminutive values. The amplitudes of
the cross-polarization components approach zero in echoes, so there is no need for further
analysis. Subsequent detailed scrutiny is directed towards the dominant-polarization
components. It is evident that there are disparities in the amplitudes of the HH and VV
echoes at lower frequencies. These variations in amplitudes are linked to the target’s
dimensions across different polarization orientations, implying a potential relationship
between the discrepancies in polarized echoes and the targets’ electrical sizes.
The computation of the differential ratio between the horizontal and vertical reflectivity
is based on the Euclidean norm, given by
ZDR =10 ×log10∥H H
VV ∥2(6)
in dB, and the differential phase
ϕDP =argH H
VV (7)
in degrees.
The
ZDR
and
ϕDP
of the simulation results are depicted in Figure 4. A detailed
analysis is outlined as follows:
(1)
When
L<
0.2
λ0
, the
ZDR
assumes a positive value exceeding 0 decibels. With
the increase in frequency, this parameter gradually decreases from approximately
10 dB towards 0 dB, indicating a decrease in the difference between the reflection
characteristics of horizontally and vertically polarized waves.
(2) When
L<λ0
, the phase difference
ϕDP
remains less than
0◦
. This can be explained by
the fact that HH-polarization electromagnetic waves experience a longer diffraction
path compared to VV-polarization waves, resulting in a phase shift.
(3)
When
L=
1
λ0
, the ellipsoid becomes spherical, resulting in both the
ZDR
and
ϕDP
being 0 regardless of frequency. This indicates that the echo amplitudes and phases
in both channels are consistent with the physical reality. It aligns with the rotational
symmetry characteristic of an ideal spherical object with arbitrary radar-line-of-sight
rotation angles.
(4)
When
L>λ0
, the diffraction creeping paths of HH-polarization waves are shorter
than those of VV-polarization waves. However, as frequency increases, the diffraction
effect diminishes, leading to a decrease in the phase difference [7].
(5)
For the frequency
f<
0.2
f0
, the
ZDR
is sensitive to the ellipsoidal shape. When
L<
1
λ0
,
ZDR <
0 dB; when
L>
1
λ0
,
ZDR >
0 dB, exhibiting conventional polariza-
tion characteristics. Further exploration could involve establishing the relationship
between
L
and
ZDR
. The phase difference
ϕDP
approaches 0. The amplitude of the
scattering electromagnetic field echo is intrinsically linked to the extent of the object’s
projection along the given polarization orientation. Concurrently, the echo’s phase is
Remote Sens. 2025,17, 306 9 of 18
decisively influenced by the temporal lag resulting from its passage through a specific
trajectory. At extremely low frequencies, when the wavelength is much larger than
the target size, the phase difference produced at this time is negligible. At higher fre-
quencies, the HH and VV echoes demonstrate a closer resemblance in both amplitude
and phase.
(a)
(b)
(e)
(f)
(c)
(d)
(g)
(h)
Figure 3. The ellipsoid polarization scattering complex echoes at different frequencies and axial ratios.
(a) The amplitude of HH echoes; (b) the amplitude of VV echoes; (c) the amplitude of VH echoes;
(d) the amplitude of HV echoes; (e) the phase of HH echoes; (f) the phase of VV echoes; (g) the phase
of VH echoes; and (h) the phase of HV echoes. It should be noted that the amplitude range in (c,d) is
from −180 dBsm to −100 dBsm.
Remote Sens. 2025,17, 306 10 of 18
(a)
(b)
Figure 4. Differences in the two dominant-polarization echoes of ellipsoidal bodies concerning
frequencies and axial ratios. (a)ZDR; (b)ϕDP .
4. Results
It can be seen from Figure 4that there are differences in the
ZDR
and
ϕDP
across
various frequency and size ranges. The graph can be divided into distinct regions to more
accurately characterize the scattering mechanisms present in each region. In order to
achieve this, a classical Cameron decomposition was conducted on Figure 3.
Radar scatterers are characterized by two fundamental physical properties: reciprocity
and symmetry. A scatterer is considered reciprocal if it strictly adheres to the reciprocity
principle, which requires that its scattering matrix be symmetric. On the other hand,
a symmetric scatterer is defined as one possessing an axis of symmetry in the plane
orthogonal to the radar line of sight.
In the Cameron decomposition approach, the scattering matrix
S
is decomposed using
Pauli matrices, enabling the identification of invariant target features [
31
]. This decomposi-
tion provides a robust framework for analyzing the polarization characteristics of radar
targets [
32
]. A diagrammatic representation of this process is illustrated in Figure 5, which
highlights the physical and geometric interpretations of the decomposition [
33
]. Cameron
particularly emphasizes the importance of a specific category of targets termed symmetric
targets. These targets exhibit linear eigen-polarizations on the Poincaré sphere and have a
constrained parameterization of their target vectors, simplifying their characterization [
34
].
The Cameron decomposition [
31
,
32
] utilizes the target’s pointing angle invariance as
it rotates around the radar line of sight to extract the maximum symmetrical scattering
component. Through Cameron decomposition, the scattering mechanisms of the target
Remote Sens. 2025,17, 306 11 of 18
can be better understood and analyzed, leading to a more refined characterization of its
scattering characteristics. The Cameron decomposition provides a systematic approach
to understanding the scattering properties of radar targets by isolating the contributions
of reciprocity and symmetry. By categorizing scatterers into distinct types, this method
enables enhanced interpretation of radar measurements and facilitates the development of
robust target classification systems in applications such as remote sensing, surveillance,
and object recognition.
As shown in Figure 5, the Cameron decomposition method involves multiple stages
of analysis based on the input polarization scattering matrix S:
(1)
Verification of reciprocity: The first step is to verify the reciprocity of the input target
scattering matrix. For most practical radar targets, reciprocity holds true, which
validates the correctness of the polarization scattering matrix measurements.
(2)
Symmetry assessment: In the second step, the symmetry of the target is analyzed.
If the target is asymmetric, it can be further categorized as exhibiting either left-
handed helicity or right-handed helicity, depending on its polarization behavior.
Symmetric targets, in contrast, exhibit invariance under specific transformations,
providing unique structural insights.
(3)
Calculation of symmetric components: The third stage focuses on decomposing
the symmetric components of the scattering matrix. This includes determining the
symmetry properties, identifying the rotation angle, and classifying the target based
on its scattering characteristics.
(4)
Classification of scattering structures: Finally, the target is classified into one of ten
predefined scattering structure types based on its physical geometric and microwave
polarization features. As shown in Table 1, the target can be divided into 10 scat-
terer types.
Table 1. Scattering types of Cameron decomposition.
Number Scatterer Type Abbreviation Scattering Matrix
1 Ball B 1 0
0 1
2 Cylinder C 1 0
01
2
3 Dipole DP 1 0
0 0
4 Quarter-Wave Reflector QWR 1j
j1
5 Narrow Dihedral NDP 1 0
0−1
2
6 Dihedral DH 1 0
0−1
7 Left Helix LH 1j
j−1
8 Right Helix RH 1−j
−j−1
9 Symmetric Scatterer SS –
10 Asymmetric Scatterer AS –
Remote Sens. 2025,17, 306 12 of 18
Reciprocity
test
Symmetry
test
Helix
test
Calculate
1. Symmetric component
2. Target rotation angle
3. Scatterer type
Scatterer
type
test
Nonreciprocal
scatterer
Right helix
(RH)
Ball
(B)
Symmetric
Scatterer
(SS)
Dipole
(DP)
Dihedral
(DH)
Cylinder
(C)
Quarter-Wave
Reflector
(QWR)
Narrow
Dihedra
(NDP)
No
Yes
No
Yes
No
Asymmetric
Scatterer
(AS)
No
Left helix
(LH)
Figure 5. Cameron decomposition and classification scheme. Different colors represent different
decomposition results. The colors are the same as the target-type colors in Figure 6.
The results of the Cameron decomposition are shown in Figure 6, which can be divided
into three regions based on the distribution characteristics of the scattering characteristics.
In single-polarization scattering region classification, regions are distinguished based on
the size of
ka
. As described in Section 2.1, the Rayleigh region is defined by
ka <
1, while
the optical region is defined by
ka >
10; the region between them is the resonance region. In
the simulation for ellipsoidal targets, we extended this concept by using
kL
as a criterion to
differentiate the polarization Rayleigh, resonance, and optical regions. Specifically,
kL <
1
indicates the polarized Rayleigh region,
kL >
10 denotes the polarization optical region,
and intermediate values correspond to the polarization resonance region. As shown in
Figure 6, region A represents the polarization Rayleigh region; region B is the polarization
resonance region; and region C is the polarization optical region.
Remote Sens. 2025,17, 306 13 of 18
Figure 6. The result of the Cameron decomposition reveals the polarization scattering regions, with
A representing the polarization Rayleigh region, B representing the polarization resonance region,
and C representing the polarization optical region. These regions can be distinguished by Boundary 1
and Boundary 2, with
g1
and
g2
. The meanings of the abbreviations and colors in the color bar can be
found in Figure 5.
5. Discussion
The boundaries (Boundary 1 and Boundary 2) of these regions can be determined
based on Figure 4and the Cameron decomposition results of Figure 6. The demarcation
points are selected by considering the characteristics of the different
ZDR
and
ϕDP
values
and the results of the Cameron decomposition. To better characterize these regions and their
polarization scattering mechanisms, the functions of the two boundary lines are given by
g1:kL =1, (8)
and
g2:kL =10. (9)
It can be observed that the boundaries defined by
g1
and
g2
align well with the results
from Cameron decomposition. Boundary 1, represented by the red
∗
, is in good agreement
with
g1
; and Boundary 2, represented by the blue
△
, aligns well with the
g2
curve. In the
polarization Rayleigh region, the polarization characteristics of the target highly correlate
with its shape. As the frequency decreases, the echo scattering characteristics change from
a dipole to a cylindrical-like body (when
L<l0
, the long axis of the cylindrical-like body
is along the
y
-axis) to a sphere (the two main axes are the same) and then to a cylindrical-
like body (when
L>l0
, the long axis of the cylindrical-like body is along the
z
-axis).
Eventually, its projection becomes a dipole (
L=
2
l0
). In the polarization resonant region,
the polarization characteristics of the target change abruptly with the change in scale and
frequency, which is similar to the jumps in RCS values with frequency in the resonant region
of radar. As the target enters the polarization optical region, the two primary polarization
channels gradually become less discrepant. The polarimetric scattering characteristics of
the target become less dependent on frequencies and shapes, tending towards a simplistic
spherical target. This is consistent with the partitioning of radar echo scattering. At this
point, high-frequency polarized electromagnetic waves are unable to accurately determine
the shape of electrically large target.
The division of the polarization scattering regions into
g1
and
g2
is consistent with
the demarcation points of the single-polarization RCS scattering regions. As mentioned in
Section 1
, the boundary between the Rayleigh region and the resonance region is
ka =
1,
and the boundary between the resonance region and the optical region is approximately
Remote Sens. 2025,17, 306 14 of 18
ka =
10. For a sphere with a radius of
λ=
2.998 cm (shown in Figure 2b), the frequencies
at the boundaries between the Rayleigh region, the resonance region, and the optical region
are 1.69 GHz and 16.9 GHz, respectively.
The performance of the models was evaluated using three key metrics: the sum of
squared errors (SSE), the root mean squared error (RMSE), and the R-squared (
R2
). These
metrics are widely used to assess model accuracy and goodness of fit. Generally, SSE is
defined as
SSE =
n
∑
i=1
(yi−ˆ
yi)2, (10)
where
yi
represents the observed values, and
ˆ
yi
the predicted values, and
n
the number
of observations. Lower SSE values indicate a better fit, with values below 0.05 generally
suggesting good alignment between predictions and observations [
35
]. RMSE, calculated
as [36]
RMSE =s1
n
n
∑
i=1
(yi−ˆ
yi)2, (11)
represents the standard deviation of residuals, where values below 0.05 are commonly
considered indicative of high prediction accuracy. Finally,
R2
, measuring the proportion of
variance explained by the model, is defined as [37]
R2=1−∑n
i=1(yi−ˆ
yi)2
∑n
i=1(yi−¯
y)2, (12)
where
¯
y
is the mean of the observed values.
R2
values closer to 1, typically exceeding 0.99,
indicate an excellent fit [38].
The results of these evaluations are presented in Table 2. Both functions
g1
and
g2
achieved metrics within the boundaries, demonstrating a good model fit. Function
g1
had an SSE of 0.041228, an RMSE of 0.047858, and an
R2
value of 0.99277, suggesting a
strong agreement between predictions and observations. Function
g2
performed slightly
better, with an SSE of 0.0017329, an RMSE of 0.012551, and an
R2
of 0.99879. Based on
these results, both
g1
and
g2
demonstrate excellent predictive performance, with
g2
slightly
outperforming
g1
. Therefore, both functions are suitable for modeling the relationship as a
function of frequency.
Table 2. Performance of the g1and g2fits.
Function SSE RMSE R2
g10.041228 0.047858 0.99277
g20.0017329 0.012551 0.99879
In particular, when
L/λ0=
1, the frequencies corresponding to the Cameron decom-
position results are around 1.72 GHz (the green point “p” in Figure 6, corresponding to the
green point “s” in Figure 1) and 16.45 GHz (the yellow point “q” in Figure 6, corresponding
to the green point “t” in Figure 1), respectively, with errors of 1.7% and 2.7%. These errors
may have two sources: firstly, the classification of each region is gradual and there is no
absolute standard point; secondly, the simulation sampling points are not dense enough.
However, these errors are acceptable in practical engineering applications. Firstly, the clas-
sification of each region is inherently gradual, without sharply defined boundaries, as the
transitions between the Rayleigh, resonance, and optical regions occur continuously. This
lack of absolute standard points can introduce variability when aligning theoretical bound-
Remote Sens. 2025,17, 306 15 of 18
aries with the observed scattering characteristics, particularly in regions where changes in
scattering behavior are subtle. Secondly, the density of simulation sampling points may
not be sufficient to capture these fine transitions accurately. Coarser frequency steps or
axis ratio intervals can lead to small but impactful discrepancies near the boundaries. En-
hancing the sampling resolution or refining the classification criteria would help minimize
these errors and yield more precise boundary determinations. Therefore, the proposed
polarization scattering regions is an extension of the concept of single-polarization RCS
scattering regions.
The polarization scattering region concept extends the traditional single-site RCS
region division by introducing polarization-specific analysis, allowing for a detailed char-
acterization of scattering behaviors. In different polarization scattering region zones, the
scattering properties vary, while within the same zone, the properties exhibit similarities.
This enables a comprehensive analysis of polarization scattering characteristics, addressing
the limitations of the traditional single-site RCS framework, which only describes single-
polarization metrics and cannot account for polarization-specific features. The polarization
scattering region enhances the interpretation of target shapes and electromagnetic proper-
ties, particularly in applications such as weather radar calibration, polarimetric SAR image
analysis, and advanced target recognition. A major limitation of the polarization scattering
region concept is its dependency on polarimetric radar systems, which involve increased
complexity and cost compared to single-polarization systems. The need for additional
transmission and reception channels poses challenges for widespread adoption, especially
in cost-sensitive applications. What is more, the proposed polarization scattering region
framework assumes convex targets, meaning it does not account for multiple reflections or
interactions between non-convex surfaces.
The polarization scattering region framework provides a structured methodology
for characterizing polarization scattering across different frequency bands, enabling effec-
tive frequency selection and target identification for polarimetric radar systems. Unlike
traditional single-site RCS region divisions, which focus solely on single-polarization met-
rics, the polarization scattering region concept addresses a broader range of polarization
properties, offering a more holistic understanding of target behavior in diverse remote
sensing applications.
6. Conclusions
Our study aims to reveal the phenomena and polarization scattering characteristics
for practical polarimetric radar applications. Specifically, when the wavelength of electro-
magnetic waves is small enough, the ellipsoids exhibit spherical scattering behavior. This
means that the polarized scattering echoes are minimally influenced by the macroscopic
shapes of the objects. This idea is supported by the validation using Cameron decom-
position. This study presents the concept of the polarization scattering region through
simulation experiments. As an extension of the single-polarization RCS scattering regions,
the proposed concept is well compatible with previous work.
The conclusions drawn from these foundational studies on polarization scattering
mechanisms can support our future applications. For instance, metal spheres are usually
used to externally calibrate the
ZDR
of meteorological radars. In practical scenarios, there
are manufacturing errors and deformations in the spheres. Employing larger spheres can
help mitigate the impacts of varying curvatures on the sphere surface. Polarimetric weather
radars often operate in the Rayleigh region when observing raindrops, snowflakes, or small
ice crystals. The size of raindrops is much smaller than the radar wavelength (typically
in the centimeter range). By analyzing the differential reflectivity and differential phase,
meteorologists can differentiate between rain, snow, and mixed precipitation. To determine
Remote Sens. 2025,17, 306 16 of 18
the shape of a target using polarimetric radars, lower-frequency bands are recommended
for determining the shape of a target using polarimetric radars, as polarization scattering
in this region provides valuable information on particle size and shape. The frequency
bands of weather radars are increasing, from the early S-band and C-band radars to the
current X-band and a few Ku-band ones. There may be higher-frequency radar bands,
possibly even terahertz meteorological radar in the future. However, based on our findings
of this study, higher-frequency polarimetric radars may be less capable of measuring the
ZDR
. For example, hail identification methods depend on the
ZDR
; therefore, efforts to
estimate the size and shape distributions work best when Rayleigh scattering is present
at lower frequencies (e.g., at an S-band). While polarimetric synthetic-aperture radar is
used for high-resolution mapping of urban environments. Buildings, bridges, and other
large structures have dimensions much larger than the radar wavelength, placing them in
the optical region. By using polarimetric decomposition techniques, different scattering
mechanisms can be identified. What is more, polarization-specific scattering helps to
classify different types of urban structures and monitor their condition over time.
The impact of attenuation across different radar bands for the proposed polarization
scattering region concept is different. Lower-frequency radar bands (e.g., S-band) experi-
ence relatively lower attenuation in the atmosphere, making them suitable for long-range
applications, such as weather observation or large-scale environmental monitoring. How-
ever, these bands may not capture fine-scale features due to their longer wavelengths.
Higher-frequency radar bands (e.g., X-band and Ku-band) are more prone to attenuation
caused by atmospheric absorption and scattering by hydrometeors. This limits their effec-
tive range but allows for high-resolution imaging, making them valuable for applications
such as urban mapping and small-target identification. The attenuation of different radar
bands influences the effective size and utility of the proposed polarization scattering region
framework. For example, at higher frequencies where attenuation is more significant, the
boundaries of the polarization optical region (where scattering becomes less sensitive to
target shape) might be affected by reduced signal strength. Conversely, in lower-frequency
bands, with minimal attenuation, the boundaries of the polarization Rayleigh and reso-
nance regions remain stable over longer ranges, enhancing their reliability for applications
like weather radar calibration.
The proposed polarization scattering regions serve as an extension of traditional
single-polarization radar RCS regions. They provide a framework for understanding how
target size, shape, and electromagnetic wave frequency interact to influence scattering
behavior in different polarimetric radar systems. This framework has various applications,
including optimized radar calibration, frequency selection, meteorological analysis, and
urban mapping through advanced remote sensing techniques. Future research will validate
and refine the polarization scattering regions framework through microwave darkroom
measurement, providing empirical support to complement the simulation-based findings.
Additionally, we aim to expand its applicability to non-convex targets and investigate the
effects of environmental factors on polarization scattering. These efforts will ensure the
practical implementation of polarization scattering regions in advanced radar technologies,
such as ultra-wideband, terahertz, and space-based radar systems, addressing challenges
in modern polarimetric radar applications.
Author Contributions: Conceptualization, J.H. and Y.L.; methodology, J.H. and J.Y.; software, J. H.
and Z.X.; validation, J.H. and Z.X.; formal analysis, J.H., J.Y. and Z.X.; investigation, J.H. and J.Y.;
resources, Y.L.; data curation, J.H. and Z.X.; writing—original draft preparation, J.H.; writing—review
and editing, J.H. and J.Y.; visualization, J.H.; supervision, Y.L.; project administration, J.Y. and Y.L.;
funding acquisition, J.Y. and Y.L. All authors have read and agreed to the published version of
the manuscript.
Remote Sens. 2025,17, 306 17 of 18
Funding: This research was supported in part by the National Natural Science Foundation of China
under Grants 62231026, 61971429, and 62171447; and in part by the Hunan Province Graduate
Research Innovation Project under Grant QL20220012.
Data Availability Statement: The raw data supporting the conclusions of this article are made
available by the authors on request.
Acknowledgments: The authors extend their gratitude to Yuqing Zheng for her valuable insights and
technical expertise, as well as for her contributions to the successful execution of the experimental
setup. In addition, the authors wish to acknowledge the professional editors and reviewers for their
astute constructive comments and suggestions, which significantly enriched the quality and clarity of
this manuscript.
Conflicts of Interest: The authors declare no conflicts of interest.
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