Polarization Scattering Regions: A Useful Tool for Polarization Characteristic Description

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 optical 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 provides 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 recognition, weather radar calibration, and radar polarimetry are discussed, highlighting the importance of frequency selection for accurately capturing polarization scattering features. These findings have significant implications for advancing weather radar technology and target recognition techniques, particularly as radar systems move towards higher frequency bands.

1. Introduction

In radar systems, the detection, tracking, and identification of a target rely fundamentally 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 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 $k=2\pi /\lambda$, and $\lambda$ is the wavelength. The resonance region’s $ka$ ranges from 1 to 10. The range $k\times a=1$ 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 $k\times a$. Thus, the limits of $k\times a$ 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 [7,8,9]. In general, when a target has a larger projected area in one direction, 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 polarization 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 particularly 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 2 offers a physical explanation for the scattering regions. Section 3 delves into the simulation mehtod and mathematical modeling of polarized scattering from deformable ellipsoids. Section 4 provides details of the polarization scattering regions. Section 5 gives the discussion of the results, and Section 6 summarizes 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 $E_i$ of the wave that strikes the target, and the electric-field strength of the scattered wave $E_s$, measured at the radar receiver. The RCS is formally defined as the norms of $E_i$ and $E_s$ [1]:

$$\sigma =\underset{R\to \infty }{lim}4\pi R^2{\displaystyle \frac{|E_s{|}^2}{|E_i{|}^2}},$$

(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 1 shows 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 $ka=2\pi a/\lambda$ represents the circumference of the sphere in units of wavelength, where a is the radius of the sphere and $\lambda$ is the wavelength of the incident wave. The $x\text{-}$axis represents the frequency f of the incident wave, spanning from $0.1$ GHz to 100 GHz, where $f=c/\lambda$. The speed of light is c, around $2.998\times {10}^9$ m/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 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.

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 electrically small, the normalized RCS scales with the fourth power of $ka$, i.e., $\mathrm{RCS}\propto {\left(ka\right)}^4$. This behavior is characteristic of small or thin structures, where the incident wavelength 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].

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

$${\mathbf{E}}^{\mathbf{i}}=\left[\begin{array}{c}\hfill E_1^i\\ \hfill E_2^i\end{array}\right],$$

(2)

and

$${\mathbf{E}}^{\mathbf{s}}=\left[\begin{array}{c}\hfill E_1^s\\ \hfill E_2^s\end{array}\right],$$

(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]

$$\begin{array}{cc}\hfill \left[\begin{array}{c}\hfill E_H^s\\ \hfill E_V^s\end{array}\right]=& \mathbf{S}\left[\begin{array}{c}\hfill E_H^i\\ \hfill E_V^i\end{array}\right]\hfill \\ \hfill =& \left[\begin{array}{cc}{s}_{HH}\hfill & s_{HV}\hfill \\ s_{VH}\hfill & s_{VV}\hfill \end{array}\right]\left[\begin{array}{c}\hfill E_H^i\\ \hfill E_V^i\end{array}\right].\hfill \end{array}$$

(4)

In this scattering matrix, the diagonal elements ($s_{HH}$ and $s_{VV}$) are referred to as co-polar terms, as they describe interactions between identical polarization states. The off-diagonal elements ($s_{HV}$ and $s_{VH}$) are cross-polar terms, as they describe interactions between orthogonal 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 $\mathbf{S}$ 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. Polarization 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 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 counteracting 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 $\widehat{n}\times \mathbf{E}=0$, where vector $\widehat{n}$ 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 $x\text{-}$, $y\text{-}$, and $z\text{-}$axes by the equation

$${\displaystyle \frac{x^2}{R_x^2}}+{\displaystyle \frac{y^2}{R_y^2}}+{\displaystyle \frac{z^2}{R_z^2}}=1.$$

(5)

$R_x$, $R_y$, and $R_z$ represent the lengths of the $x\text{-}$, $y\text{-}$, and $z\text{-}$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 $L=R_x=R_z$ 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 $R_x=R_z<R_y$, it assumes the shape of a prolate spheroid, as depicted in Figure 2a. If all three axes are equal in length, with $R_x=R_z=R_y$, the ellipsoid transforms into a sphere, as shown in Figure 2b. Conversely, if the third axis is shorter, with $R_x=R_z>R_y$, 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 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.

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 $f_0=10$ GHz, and the corresponding wavelength is around ${\lambda }_0=c/f_0=2.998$ cm, where c represents the speed of light. The simulation frequency is defined as $F=b{f}_0$, where $b\in [0.1,2]$. The simulation frequencies range from the UHF band to the K-band. The radius of the ellipsoid’s $y\text{-}$axis (green coordinate axis) is defined as $R_y={\lambda }_0$, and those of the $x\text{-}$axis (blue coordinate axis) and the $z\text{-}$axis (red coordinate axis) are equivalently set to $L=R_x=R_z=l{\lambda }_0$, where the normalized axis ratio $l\in [0.1,2]$. 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 $L=0.1{\lambda }_0$, the simulation ellipsoid changes to a dipole-like shape, as shown in Figure 2a.

(2)

When $L=1{\lambda }_0$, the ellipsoid becomes a sphere, as depicted in Figure 2b.

(3)

When $L=2{\lambda }_0$, 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 $y\text{-}$axis length of the ellipsoid remains constant, the $z\text{-}$axis length changes as l varies.

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 $0.1\times 2.998\mathrm{cm}$ to $2\times 2.998\mathrm{cm}$, with a step size of $0.1\times 2.998\mathrm{cm}$. The frequency range was from $0.1\mathrm{GHz}$ to $20\mathrm{GHz}$, with a step size of $0.199\mathrm{GHz}$.

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

$$ZDR=10\times {\log }_{10}{\parallel {\displaystyle \frac{HH}{VV}}\parallel }^2$$

(6)

in dB, and the differential phase

$${\varphi }_{DP}=\arg \left({\displaystyle \frac{HH}{VV}}\right)$$

(7)

in degrees.

The $ZDR$ and ${\varphi }_{DP}$ of the simulation results are depicted in Figure 4. A detailed analysis is outlined as follows:

(1)

When $L<0.2{\lambda }_0$, 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 $L<{\lambda }_0$, the phase difference ${\varphi }_{DP}$ remains less than $0^{\circ }$. 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 $L=1{\lambda }_0$, the ellipsoid becomes spherical, resulting in both the $ZDR$ and ${\varphi }_{DP}$ 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 $L>{\lambda }_0$, 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 $f<0.2f_0$, the $ZDR$ is sensitive to the ellipsoidal shape. When $L<1{\lambda }_0$, $ZDR<0$ dB; when $L>1{\lambda }_0$, $ZDR>0$ dB, exhibiting conventional polarization characteristics. Further exploration could involve establishing the relationship between L and $ZDR$. The phase difference ${\varphi }_{DP}$ 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 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 frequencies, the HH and VV echoes demonstrate a closer resemblance in both amplitude and phase.

4. Results

It can be seen from Figure 4 that there are differences in the $ZDR$ and ${\varphi }_{DP}$ 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 $\mathbf{S}$ is decomposed using Pauli matrices, enabling the identification of invariant target features [31]. This decomposition 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 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 $\mathbf{S}$:

(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. Scattering types of Cameron decomposition.

Number	Scatterer Type	Abbreviation	Scattering Matrix
1	Ball	B	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
2	Cylinder	C	$\left[\begin{array}{cc}1& 0\\ 0& {\displaystyle \frac{1}{2}}\end{array}\right]$
3	Dipole	DP	$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$
4	Quarter-Wave Reflector	QWR	$\left[\begin{array}{cc}1& j\\ j& 1\end{array}\right]$
5	Narrow Dihedral	NDP	$\left[\begin{array}{cc}1& 0\\ 0& -{\displaystyle \frac{1}{2}}\end{array}\right]$
6	Dihedral	DH	$\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$
7	Left Helix	LH	$\left[\begin{array}{cc}1& j\\ j& -1\end{array}\right]$
8	Right Helix	RH	$\left[\begin{array}{cc}1& -j\\ -j& -1\end{array}\right]$
9	Symmetric Scatterer	SS	–
10	Asymmetric Scatterer	AS	–

, the target can be divided into 10 scatterer types.

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.

5. Discussion

The boundaries (Boundary 1 and Boundary 2) of these regions can be determined based on Figure 4 and the Cameron decomposition results of Figure 6. The demarcation points are selected by considering the characteristics of the different $ZDR$ and ${\varphi }_{DP}$ 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

$$g_1:kL=1,$$

(8)

and

$$g_2:kL=10.$$

(9)

It can be observed that the boundaries defined by $g_1$ and $g_2$ align well with the results from Cameron decomposition. Boundary 1, represented by the red ∗, is in good agreement with $g_1$; and Boundary 2, represented by the blue ∆, aligns well with the $g_2$ 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 $L<l_0$, the long axis of the cylindrical-like body is along the $y\text{-}$axis) to a sphere (the two main axes are the same) and then to a cylindrical-like body (when $L>l_0$, the long axis of the cylindrical-like body is along the $z\text{-}$axis). Eventually, its projection becomes a dipole ($L=2l_0$). 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 $g_1$ and $g_2$ 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 $ka=10$. For a sphere with a radius of $\lambda =2.998$ 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 ($R^2$). These metrics are widely used to assess model accuracy and goodness of fit. Generally, SSE is defined as

$$\mathrm{SSE}=\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2,$$

(10)

where $y_i$ represents the observed values, and ${\widehat{y}}_i$ 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]

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2},$$

(11)

represents the standard deviation of residuals, where values below 0.05 are commonly considered indicative of high prediction accuracy. Finally, $R^2$, measuring the proportion of variance explained by the model, is defined as [37]

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}},$$

(12)

where $\overline{y}$ is the mean of the observed values. $R^2$ values closer to 1, typically exceeding 0.99, indicate an excellent fit [38].

The results of these evaluations are presented in

Table 2. Performance of the $g_1$ and $g_2$ fits.

Function	SSE	RMSE	${\mathit{R}}^2$
$g_1$	0.041228	0.047858	0.99277
$g_2$	0.0017329	0.012551	0.99879

. Both functions $g_1$ and $g_2$ achieved metrics within the boundaries, demonstrating a good model fit. Function $g_1$ had an SSE of 0.041228, an RMSE of 0.047858, and an $R^2$ value of 0.99277, suggesting a strong agreement between predictions and observations. Function $g_2$ performed slightly better, with an SSE of 0.0017329, an RMSE of 0.012551, and an $R^2$ of 0.99879. Based on these results, both $g_1$ and $g_2$ demonstrate excellent predictive performance, with $g_2$ slightly outperforming $g_1$. Therefore, both functions are suitable for modeling the relationship as a function of frequency.

In particular, when $L/{\lambda }_0=1$, the frequencies corresponding to the Cameron decomposition 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 classification 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 boundaries 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. Enhancing 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 characterization 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 properties, 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 effective frequency selection and target identification for polarimetric radar systems. Unlike traditional single-site RCS region divisions, which focus solely on single-polarization metrics, 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 electromagnetic 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 decomposition. 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 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) experience relatively lower attenuation in the atmosphere, making them suitable for long-range applications, such as weather observation or large-scale environmental monitoring. However, 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 effective 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 resonance 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.