Exact spherical-wave forward model for radio reflection from stratified media

Abstract: Radio detection of ultra-high energy particles ($\gtrsim10<sup>{18}$ eV) depends on how broadband radio pulses reflect from natural media boundaries. We extend the spherical-wave (Sommerfeld--Weyl) treatment of a single homogeneous interface to stratified media by replacing the Fresnel coefficient of each plane-wave component with the characteristic-matrix reflection coefficient of a layered medium, evaluated in the local tangent plane on the spherical surface. The calculation reduces to the single-boundary result at machine precision when the layer contrast is removed and agrees with the published spherical-surface calculation to better than $1.1\%$ at ten HiCal-2 elevation angles, with a mean deviation of $0.6\%$. We apply the formalism to shallow firn stacks proposed as explanations for anomalous-polarity ANITA events. For realistic firn contrasts, layering changes the reflected amplitude but does not reverse the pulse polarity over $150$--$850$ MHz for the elevations studied. In a reference $s$-polarized two-layer model, a coefficient sign change requires buried refractive index $n_2\simeq2.56$, $2.04$, and $1.79$ at local elevations of $8<sup>\circ$, $15<sup>\circ$, and $25<sup>\circ$, respectively. We also test the specular factorization used in fast propagation models, finding $0.2\%$ agreement with the full angular integral for high-altitude balloon geometries, while near-boundary sources require the full integral. The calculated reflected pulses reproduce the expected polarity inversion in $101$ of $106$ HiCal-1 direct/reflected pulse pairs. Because the medium enters through its complex refractive index, the framework applies to ice, lunar regolith, and conducting media.

• The paper develops an exact spherical-wave forward model that combines Sommerfeld–Weyl decomposition with characteristic-matrix reflection coefficients for stratified spherical media, including lossy and conducting layers.
• Validation reproduces HiCal-2 reflectivity within 1.1%, matches HiCal-1 reflected waveforms with a median correlation of 0.70, and finds expected polarity inversion in 95% of matched pairs.
• Realistic shallow-firn layering changes amplitude and phase but cannot reverse pulse polarity, while the model shows specular factorization is accurate to 0.21% at balloon altitude but fails for near-surface sources.

Motivation and scope

Radio detection of ultra-high energy (UHE) particles relies on broadband pulses that propagate to and reflect from natural media boundaries. The Antarctic Impulsive Transient Antenna (ANITA) and its calibration transmitters HiCal measure UHE cosmic-ray air showers via their geomagnetic emission reflected from the Antarctic ice surface, and surface reflectivity enters directly into reconstructed primary energy. A small number of ANITA events show reflected pulses with non-inverted polarity, contrary to the expectation for reflection off a higher-index medium, and shallow subsurface firn layering has been proposed as a possible explanation. The paper develops a forward model that extends the validated spherical-wave (Sommerfeld–Weyl) treatment of reflection from a single homogeneous spherical boundary (Prohira et al., 2018, Dasgupta et al., 2018) to stratified media, by replacing the Fresnel coefficient of each plane-wave component with the characteristic-matrix reflection coefficient of the layered stack evaluated in the local tangent plane.

Formalism

The source is a horizontal Hertz dipole whose spherical wave is decomposed exactly into plane waves via the Sommerfeld–Weyl identity; each component strikes the spherical surface of radius $R$ at a point $Q$ where the local incidence angle satisfies $\sin\alpha_0 = (R+h)\sin\alpha/R$. Curvature enters through $\alpha_0$, the reflected direction, and the path phase; the Fresnel coefficients retain their plane-wave form at each tangent plane. Surface roughness is included through the validated isotropic weight $$F_{\rm rough}=\exp[-2k^2\sigma_h^2(\rho_\perp)\cos^2\alpha_0^{\rm spec}]$$ with parameters fixed by HiCal-2 data ($L_0=150$ m, $\sigma_h=0.041$ m, $H=0.65$).

The stratified extension replaces $f_r^{s,p}$ with the characteristic-matrix coefficients $r_{\rm stack}^{s,p}$, which coherently sum the direct surface reflection and all internal multiple reflections. The matrix construction requires only Maxwell boundary conditions, admits complex refractive indices (so lossy and conducting media are handled identically), and reduces exactly to the single-boundary Fresnel coefficient when $Q$0. The only new geometric approximation is treating the concentric subsurface shells as parallel over the coherent footprint; the resulting phase error from shell sagitta is bounded by $Q$1 rad for the 37 km balloon geometry, independent of frequency. Birefringence contributes a differential delay of order 0.1 ns over the short in-medium path—too small to produce polarity reversal—and evanescent contour components are negligible in the far field.

Validation

The implementation is checked against three independent benchmarks. First, recomputing the band-averaged (200–650 MHz) reflected-to-direct power ratio $Q$2 reproduces the published HiCal-2 spherical-surface calculation to better than 1.1% at ten elevation angles, with mean deviation 0.6%, using no free normalization parameter. Second, removing the buried-layer contrast drives the layered calculation to the single-boundary result at machine precision (largest fractional deviation $Q$3). Third, applying the calculated transfer function to measured HiCal-1 direct waveforms reproduces the measured reflected pulses: the best pair has signed correlation 0.83, the population median is 0.70, and 101 of 106 matched pairs (95%) favor the expected polarity inversion, while the vertically polarized control channel gives 50/106, consistent with chance. Two anomalous pairs remain (1.9%), compared with the 9.4% anomalous fraction among ANITA cosmic-ray candidates; these two do not support the shallow-firn scenario, since reversal at those elevations requires indices well above realistic values.

Polarity thresholds and implications for ANITA

For the reference two-layer firn stack (2 m layers with $Q$4, $Q$5, substrate $Q$6), the layering ratio $Q$7 ranges from 0.14 to 1.68 in amplitude over 150–850 MHz with phase excursions up to 63.5°, but the pulse polarity is not reversed for realistic firn contrasts. A sign change of the $Q$8-polarized reflection coefficient first occurs at buried indices $Q$9, 2.04, and 1.79 at local elevations of 8°, 15°, and 25° respectively. The threshold depends mainly on the upper-layer index (varying $\sin\alpha_0 = (R+h)\sin\alpha/R$0 over 1.30–1.45 moves it from 1.87 to 2.41) and only weakly on thickness or frequency sampling; it is stable to about ±0.05 for fixed $\sin\alpha_0 = (R+h)\sin\alpha/R$1.

Applied to the six reported ANITA anomalous-polarity events, the calculation yields thresholds of $\sin\alpha_0 = (R+h)\sin\alpha/R$2 and 1.68 for the steep ANITA-I/III events, and $\sin\alpha_0 = (R+h)\sin\alpha/R$3–5.4 for the four near-horizon ANITA-IV events—all above realistic firn and dense-ice values, though near-horizon central values carry large angular-conversion uncertainty (e.g., roughly 4.2–9.0 for the shallowest event under ±1° elevation uncertainty). Feeding the reference-stack transfer function through all 106 HiCal-1 direct pulses at the six event angles produced zero non-inverted templates (0/636 tests). These results quantitatively constrain the shallow-layering mechanism proposed in Ref. (Shoemaker et al., 2019) and support its experimental disfavoring in Ref. (Smith et al., 2020), while leaving surface-topography explanations open.

Validity of specular factorization

Radar-sounding practice evaluates the boundary coefficient once at the specular direction. The paper derives the stationary-phase half-width $\sin\alpha_0 = (R+h)\sin\alpha/R$4 and tests the factorization directly: at balloon altitude it agrees with the full angular integral to 0.21% for the reference stack, so the fast approximation suffices for HiCal-like geometries. As the source approaches the surface, however, $\sin\alpha_0 = (R+h)\sin\alpha/R$5 grows rapidly—from 0.074° at $\sin\alpha_0 = (R+h)\sin\alpha/R$6 km to 6.26° at $\sin\alpha_0 = (R+h)\sin\alpha/R$7 m for the same geometry—so near-surface, low-altitude, and in-medium sources require the full angular integral. This matters because a specular model applied in that regime could misattribute geometric effects to subsurface structure.

Applications beyond Antarctic ice

Because the medium enters only through its complex refractive index, the same formalism describes lunar regolith ($\sin\alpha_0 = (R+h)\sin\alpha/R$8–3.0), conducting seawater ($\sin\alpha_0 = (R+h)\sin\alpha/R$9 S/m), and layered ice shells of ocean worlds without modification. The frequency-dependent layering ratio provides a discriminator between thin-layer interference and basal water interpretations of bright Martian radar reflectors, and a predictive alternative to treating lunar reflectivity as a free parameter in global 21-cm analyses. For orbital sounders (MARSIS, SHARAD, REASON, RIME) the tangent-plane sag remains below about 0.05 wavelengths, but lander-scale bistatic geometries have stationary-phase windows of tens of degrees, where the full calculation is essential.

Limitations and open questions

The paper states its limits explicitly. Transmission from in-medium sources, near-field source geometries, anisotropic ice, and lateral variation within the coherent footprint are outside the demonstrated scope, although the formalism accommodates laterally varying stacks $\alpha_0$0 provided structure varies slowly across the Fresnel radius (~178 m at 15° elevation). The polarity threshold is a deterministic output of an assumed stack, not a measurement; its dominant systematic is the assumed upper-layer index. Grazing-incidence and skin-depth behavior for conducting media is demonstrated but not separately validated. The result constrains one proposed mechanism—it does not explain any individual ANITA event—and the inverse problem of recovering layer parameters from measured waveforms remains degenerate over limited bandwidths, motivating efficient forward models such as this one as a prerequisite for quantitative inference.

Conclusion

The paper delivers a validated spherical-wave forward model for radio reflection from stratified spherical boundaries, combining the Sommerfeld–Weyl decomposition with characteristic-matrix layer response evaluated per plane-wave component. It reproduces published HiCal-2 reflectivity to within 1.1%, recovers the single-boundary limit at machine precision, matches measured HiCal-1 reflected pulses with expected polarity inversion in 95% of pairs, and shows that realistic shallow firn contrasts modulate amplitude and phase but cannot reverse pulse polarity—the required buried indices exceed firn and ice values at all tested elevations, most strongly near the horizon. It also quantifies when the standard specular factorization fails, showing it is adequate at 0.2% level for balloon altitudes but breaks down for near-surface sources. The framework applies generally to dielectric, lossy, and conducting stratified media and provides the forward basis for future sounding and inversion studies.