  The flavor composition of ultra-high energy cosmic neutrinos (UHECN) carries
precious information about the physical properties of their sources, the nature
of neutrino oscillations and possible exotic physics involved during the
propagation. Since UHECN with different incoming directions would propagate
through different amounts of matter in Earth and since different flavors of
charged leptons produced in the neutrino-nucleon charged-current (CC)
interaction would have different energy-loss behaviors in the medium,
measurement of the angular distribution of incoming events by a neutrino
observatory can in principle be employed to help determine the UHECN flavor
ratio. In this paper we report on our investigation of the feasibility of such
an attempt. Simulations were performed, where the detector configuration was
based on the proposed Askaryan Radio Array (ARA) Observatory at the South Pole,
to investigate the expected event-direction distribution for each flavor.
Assuming $\nu_{\mu}$-$\nu_{\tau}$ symmetry and invoking the standard
oscillation and the neutrino decay scenarios, the probability distribution
functions (PDF) of the event directions are utilized to extract the flavor
ratio of cosmogenic neutrinos on Earth. The simulation results are summarized
in terms of the probability of flavor ratio extraction and resolution as
functions of the number of observed events and the angular resolution of
neutrino directions. We show that it is feasible to constrain the UHECN flavor
ratio using the proposed ARA Observatory.
