  The ANITA balloon experiment was designed to detect radio signals initiated
by neutrinos and cosmic ray air showers. These signals are typically
discriminated by the polarization and phase inversions of the radio signal. The
reflected signal from cosmic rays suffer phase inversion compared to a direct
tau neutrino event. In this paper we study sub-surface reflection, which can
occur without phase inversion, in the context of the two anomalous up-going
events reported by ANITA. We find that subsurface layers and firn density
inversions may plausibly account for the events, while ice fabric layers and
wind ablation crusts could also play a role. This hypothesis can be tested with
radar surveying of the Antarctic region in the vicinity of the anomalous ANITA
events. Future experiments should not use phase inversion as a sole criterion
to discriminate between downgoing and upgoing events, unless the subsurface
reflection properties are well understood.
