Radar Targets and Pauli Matrices: Why Is the Canonical 45° Tilted Dihedral So Rare?

And Why HV Rarely Comes Alone

Introduction to radar polarimetry

Radar polarimetry is an advanced field in remote sensing that leverages the polarization of electromagnetic waves to extract detailed information about the physical and geometric properties of targets. Unlike traditional radar systems, which measure only a single polarization component, polarimetric radars analyze the interaction of waves with a target across multiple polarization orientations. This analysis is commonly represented through mathematical matrices, (Sinclair Matrices) among which the Pauli matrices play a fundamental role in describing scattering mechanisms.

In radar polarimetry, the Pauli matrices enable the modeling of target responses as combinations of idealized scattering behaviors, often referred to as “canonical targets.” There are four Pauli matrices, which provide a framework for decomposing and interpreting radar responses based on specific types of scattering. Each Pauli matrix is generally associated with a simple canonical target in radar backscattering, serving as valuable reference points for analysis.

However, this association is not as straightforward as people often think, and this is where the community sometimes misses the mark. While two of the four Pauli matrices have clear canonical targets in backscattering, a third matrix is often grouped with them — this one has zero copolarization components. Its associated target, however, represents an exceptionally rare and specific geometric configuration, seldom encountered in practice.

That’s exactly the perception I’m here to clarify.

Radar and optics convention (or BSA and FSA conventions)

Before delving into the Pauli matrices and their interpretations in radar polarimetry, it’s essential to understand the distinction between radar and optical conventions. In radar polarimetry, the convention chosen uses backscattering as the reference configuration. This means that the radar is configured to receive the returning wave from the same direction as the emitted wave, which is known as a monostatic configuration.

In optics, by contrast, measurements can be made in various configurations. However, the standard setup is forward scattering, where the wave is observed after passing through the target in a direction opposite to that of incidence.

This shift in orientation between the two conventions causes an inversion of certain polarimetric parameters in the scattering matrix calculations, particularly when transitioning from the Jones matrix (used in optics) to the Sinclair matrix (used in radar). This distinction must therefore be clarified before analyzing the meaning of the Pauli matrices in radar backscattering, as it directly influences their interpretation.

In other words, what we call the Jones matrix in optics (defined in a FSA convention) and the Sinclair matrix in radar (defined in a so-called BSA convention) are related by:

Definition of Pauli Matrices

The four Pauli matrices, denoted I, σx, σy, and σz, allow us to decompose the polarimetric response into a series of components associated with idealized scattering behaviors, or “canonical targets.”

Reciprocity and Symmetry in the Sinclair Matrix

Another key concept in radar polarimetry is reciprocity, which places specific constraints on the observed scattering matrix. In radar backscattering, the reciprocity assumption requires that the cross-polarization coefficients be equal, meaning HV=VH. This implies that, for passive targets (i.e., those without active components that alter polarization, such as Faraday rotation induced by the ionosphere), the Sinclair matrix must be symmetric.

As a result, the σy ​ matrix is out of the picture!

Generally, we build the polarimetric vector by projecting the Sinclair matrix onto the three remaining matrices: I, σx ​, and σz ​.

Demystifying the Purely Oriented Element and the “Disoriented Double-Bounce”

In the context of radar backscattering polarimetry, these matrices are typically interpreted as follows:

• Matrix I: The identity matrix, I, is associated with a mirror, a sphere, or a trihedral corner reflector. These targets produce a symmetric scattering response that does not depend on the polarization of the incident wave.

• Matrix σz ​: The σz ​ matrix represents a perfect dihedral, a target that produces a phase reversal in a specific direction. A dihedral is a common reflector in radar, such as a building corner or an “L”-shaped metal structure. This matrix thus represents a typical canonical backscattering behavior, which underscores its significance in radar polarimetry.

So, what does the σx ​ matrix correspond to?

First hypothesis: The perfectly oriented dipole.

The σx matrix is sometimes interpreted as corresponding to a target oriented at 45° relative to the incident polarization. However, this is not actually the case.

Let’s consider a horizontally oriented dipole that reflects part of the incident wave in a defined polarization, with maximum intensity in the direction parallel to its orientation (i.e., horizontal, noted as HH) and very weak response in the perpendicular direction (i.e., vertical, noted as VV). In this initial configuration, the Sinclair matrix for a purely horizontal dipole can be represented as:

To calculate the effect of this 45° rotation on the Sinclair matrix, we use a rotation matrix, denoted R(-45∘), to transform the target’s scattering matrix. Sinclair’s new Srotated matrix, after rotating the target by 45°, is calculated by :

A little matrix calculation provides

The diagonal elements don’t cancel each other out! Some of the H polarization is converted to V and vice versa, but not all of it.

Second hypothesis: the random volume

You’ll sometimes hear, “The matrix with maximum HV occurs in a volume of random scatterers” — that is, an environment where scatterer orientations are completely disordered, with no preferred direction. This model is especially relevant for heterogeneous natural targets, like forests or clouds, where the orientation of individual scatterers (branches, leaves, particles) is assumed to be statistically random. In this case, the resulting coherence matrix shows isotropic behavior, meaning it has no preferred polarization direction.

But here’s the thing: “no preferred orientation” implies we’ll keep observing copolar components (HH or VV). Getting a zero average for the first Pauli component, HH+ VV, is impossible since it will always reflect a non-zero sum of copolar energies.

Third hypothesis: the inclined dihedral

By applying the same rotation formulas as before to the matrix of the perfect dihedron, we obtain:

Yes, almost, but… keep in mind that achieving this dihedral scattering pattern requires a complex orientation. This orientation simultaneously involves both a rotation around the vertical axis and a tilt relative to the horizontal plane, a configuration that is indeed rare. Consequently, the corresponding scattering behavior is also very uncommon!

It’s important to note that this matrix cannot be obtained with just a simple tilt or an azimuthal misalignment. (look also this post)

Summary

In conclusion, using the Pauli matrices, radar polarimetry provides a powerful framework for analyzing the interactions of electromagnetic waves with various targets. However, to interpret these matrices correctly in radar backscattering, it’s essential to understand the limitations imposed by:

• reciprocity and the symmetry of the Sinclair matrix,
• and the fact that we’re interpreting 3D targets through the lens of a 2D projection in polarimetry!

The matrices I, σz, and σx correspond to canonical targets (mirror, dihedral and 45° tilted dihedral), and σy has no canonical counterpart in radar backscattering. However, the 45° tilted dihedral represents a very rare occurrence.

Additionally, canonical target analysis often assumes ideal, perfectly conducting metallic targets. In reality, materials are often dielectric, which disrupts polarimetric coefficients. For example, a dielectric “mirror” surface does not have a Sinclair matrix equal to the identity.

Beware of the myths: the “disoriented double-bounce” and perfectly conducting metallic targets.

Historically, radar polarimetry was developed using measurements of deterministic military targets (planes, tanks). Applying these interpretations to natural environments inevitably leads to misinterpretations. The Pauli basis is a tremendously powerful tool from a mathematical standpoint, but its physical interpretation remains complex and non-trivial.

Of course, this is my personal view, and I’m more than open to discussion!