A rigorous formalism to unfold the mystery of unusual events detected by NASA sponsored balloon borne ANITA experiment.

We develop a general formalism to treat reflection of spherical electromagnetic waves from a spherical surface applicable to ANITA experiment. The incident wave is decomposed into plane waves and each plane wave is reflected off the surface using the standard Fresnel formalism. For each plane wave the reflected wave is assumed to be locally a plane wave. This is a very reasonable assumption and there are no uncontrolled approximations in our treatment of the reflection phenomenon. We apply our formalism to the radiation produced by the balloon-borne HiCal radio-frequency (RF) transmitter. Our final results for the reflected power are found to be in good agreement with data for all elevation angles. We also applied our procedure to investigate whether the mystery events observed by ANITA can be explained by conventional physics. Due to reflection of radio pulse at the air-ice interface, the reflected pulse is expected to be 180⁰ out of phase with the incident pulse. However, observations show that in some cases the observed pulse does not show this phase difference. Using our formalism, we point out that local surface deformity can give rise to unusual phase shift in the reflected pulses.