Microwave scattering by rough polyhedral particles on a surface

The electromagnetic (EM) scattering by non-symmetric wavelength-scale particles on a planar surface has numerous applications in the remote sensing of planetary bodies, both in planetary and geo-sciences. We conduct numerical simulations of EM scattering by rough polyhedral particles (with 12 or 20 faces) using the discrete-dipole approximation and contrast the results to that of spheres. The particles have permittivities corresponding to common minerals in the microwave regime ($ε_r=4.7 + 0.016$i and $7.8 + 0.09$i), and a size-frequency distribution (SFD) consistent with the observed scattering properties (power-law distribution of size parameters between 0.5 and 8 with an index from $-2.5$ to $-3.5$). The assumed substrate permittivity $2.4 + 0.012$i corresponds to a powdered regolith. We present what roles the particle roundness, permittivity, and SFD for a realistic range of parameters play in the EM scattering properties as a function of incidence angle with a focus on backscattering in microwave-remote-sensing applications. The particle roundness and SFD have a clearly observable effect on the polarimetric properties, while the role of permittivity is relatively minor (in the studied range). Among various backscattering observables, the circular polarization ratio is the least sensitive to the decrease of the upper boundary (down to a size parameter of 3) and the index of the SFD.

💡 Research Summary

This paper presents a systematic numerical investigation of microwave (radar‑frequency) electromagnetic scattering from non‑symmetric, wavelength‑scale particles that rest on a planar substrate. Using the discrete‑dipole approximation (DDA) enhanced by the ADD‑A code, the authors are able to include the substrate analytically via a modified Green’s tensor, thereby overcoming the computational bottleneck that has limited previous studies of irregular particles on surfaces.

Two families of polyhedral particles are examined: a 12‑face (regular dodecahedron) and a 20‑face (regular icosahedron) shape, which serve as proxies for varying degrees of “roundness.” The particles are assigned complex permittivities typical of common minerals in the microwave band (εr = 4.7 + 0.016i and 7.8 + 0.09i). Their sizes are expressed by the dimensionless size parameter x = kr, ranging from 0.5 to 8, and the ensemble follows a power‑law size‑frequency distribution (SFD) N(x) ∝ x^p with exponent p between –2.5 and –3.5, reflecting realistic regolith particle populations. The substrate is modeled as a smooth powder‑regolith with ε = 2.4 + 0.012i.

The scattering problem is formulated in terms of the full 4 × 4 Mueller matrix. From the matrix elements the authors derive backscattering cross sections for circularly polarized incidence (same‑circular, SC, and opposite‑circular, OC), the circular polarization ratio (CPR = SC/OC), and linear‑polarization quantities (same‑linear, SL, and opposite‑linear, OL) together with the linear polarization ratio μL = σOL/σSL. The SFD weighting is performed using a non‑uniform Simpson’s rule to respect the variable step size in x (0.5‑step for x ≤ 3, 1‑step for larger x).

Key findings:

1. Roundness – The 12‑face polyhedra produce noticeably lower backscatter intensity and higher μL and CPR values than the 20‑face polyhedra, which behave more like spheres. As the number of faces increases, the scattering pattern approaches the Mie‑theory result for a sphere, and the polarimetric signatures diminish.

2. Size‑frequency distribution – Reducing the upper size bound from x = 8 to x = 3 has little effect on CPR, indicating that CPR is robust against truncation of the large‑particle tail. In contrast, μL is highly sensitive to both the exponent p and the upper bound; a steeper SFD (more negative p) or a lower upper bound reduces μL markedly. This demonstrates that linear polarization ratios are good diagnostics of the relative abundance of small particles.

3. Material permittivity – Changing the mineral permittivity between the two chosen values produces only modest variations in the backscatter and polarimetric ratios. The authors attribute this to the low loss (small imaginary part) of typical microwave‑band minerals, making geometric effects dominate over dielectric contrast.

4. Substrate influence – The presence of the substrate modifies both the incident field (direct plus reflected plane wave) and the scattered field (four possible paths: direct, single‑reflected, and double‑reflected). This breaks the symmetry of the Mueller matrix that holds in free space and enhances backscatter for large incidence angles. Nevertheless, for statistically isotropic ensembles the relation μC = 2μL/(1 – μL) derived for free‑space scattering remains approximately valid.

5. Observation geometry – The study examines both circular and linear incident polarizations, as used in planetary radar (e.g., Arecibo S‑band) and lidar. The results show that CPR is the least sensitive observable to variations in the SFD upper bound and exponent, making it a reliable indicator of surface roughness in radar remote sensing. Linear polarization ratios, however, provide finer discrimination of particle shape and size distribution.

The authors conclude that particle roundness and the power‑law size distribution are the dominant factors shaping microwave backscatter and polarimetric signatures, while mineral permittivity plays a secondary role within the examined range. The successful application of the DDA‑ADD‑A framework demonstrates that accurate, computationally efficient modeling of irregular particles on substrates is feasible, opening the door to more realistic simulations across the electromagnetic spectrum.

These insights have direct implications for interpreting radar observations of planetary bodies (Moon, asteroids, icy moons) and for designing Earth‑observation radar systems that must account for surface‑layer scattering. By linking measurable polarimetric ratios to physical parameters such as particle shape and size distribution, the work provides a valuable tool for inverse modeling of regolith properties from remote‑sensing data.