Using data collected by the Askaryan Radio Array (ARA) experiment at the South Pole, we have used long-baseline propagation of radio-frequency signals to extract information on the radio-frequency index-of-refraction in South Polar ice. Owing to the increasing ice density over the upper 150--200 meters, rays are observed along two, nearly parallel paths, one of which is direct and a second which refracts through an inflection point, with differences in both arrival time and arrival angle that can be used to constrain the neutrino properties. We also observe indications, for the first time, of radio-frequency ice birefringence for signals propagating along predominantly horizontal trajectories, corresponding to an asymmetry of order 0.1% between the ordinary and extra-ordinary paths, numerically compatible with previous measurements of birefringent asymmetries for vertically-propagating radio-frequency signals at South Pole. Taken together, these effects offer the possibility of redundantly measuring the range from receiver to a neutrino interaction in Antarctic ice, if receiver antennas are deployed at shallow (25 m<z<100 m) depths. Such range information is essential in determining both the neutrino energy, as well as the incident neutrino direction.
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