Exact spherical-wave forward model for radio reflection from stratified media and implications for the anomalous-polarity events observed by ANITA

Radio detection of ultra-high energy particles relies on the propagation and reflection of broadband radio pulses at boundaries between natural media. In the Sommerfeld--Weyl treatment, a spherical wave is decomposed into plane-wave components and their reflection from a single homogeneous interface is calculated exactly. We extend that treatment to an arbitrary number of laterally uniform spherical layers. Both the spherical-wave decomposition of the source and the spherical geometry of the boundary are retained, while the reflection and transmission coefficients of the plane-wave components are replaced by the exact characteristic-matrix coefficients of the layered medium, evaluated at the local incidence angle on the spherical boundary. When the layer contrast is removed, the formalism recovers the single-boundary result to machine precision, and it reproduces the published spherical-surface reflectivity calculation to better than 1.1% at ten HiCal-2 elevation angles, with a mean deviation of 0.6%. We reproduce the measured HiCal-1 reflected pulses from their measured direct partners, with a best signed correlation of 0.83 and a median of 0.70 across 106 pairs, of which 101 show the expected polarity inversion. Applied to the six reported ANITA anomalous-polarity event geometries, the buried-layer refractive index required for a sign change of the reflection coefficient ranges from 1.68 at the steepest event to 3.8 -- 5.4 at the four near-horizon events. The full waveform calculation gives no non-inverted reflected pulse at any of these angles, showing that shallow, laterally uniform firn layering does not account for the polarity of the anomalous ANITA events. Because the formalism depends only on the complex refractive index of the medium, it applies more generally to isotropic, nonmagnetic stratified media, including ice, lunar regolith, and conducting layers.