  We evaluate the upward shower and muon event rates for two characteristic
four neutrino mixing models for extragalactic neutrinos, as well as for the
atmospheric neutrinos, with energy thresholds of 1 TeV, 10 TeV and 100 TeV. We
show that by comparing the shower to muon event rates, one can distinguish
between oscillation and no-oscillation models. By measuring shower and muon
event rates for energy thresholds of 10 TeV and 100 TeV, and by considering
their ratio, it is possible to use extragalactic neutrino sources to determine
the type of four-flavor mixing pattern. We find that one to ten years of data
taking with kilometer-size detector has a very good chance of providing
valuable information about the physics beyond the Standard Model.
