  We develop a general formalism to treat reflection of spherical
electromagnetic waves from a spherical surface. Our main objective is
interpretation of radio wave signals produced by cosmic ray interactions with
Earth's atmosphere which are observed by the Antarctica based ANITA detector
after reflection off the ice surface. 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. The
surface roughness effects are also included by using a simple model. 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 study
the properties of reflected radio pulses in order to study their phase
relationship with direct pulses. We find that for some roughness models the
pulse shape can be somewhat distorted and may be misidentified as a direct
pulse. However this is a rather small effect and is unable to provide an
explanation for the observed mystery events by ANITA.
