  A particle cascade (shower) in a dielectric, for example as initiated by an
ultra-high energy cosmic ray, will have an excess of electrons which will emit
coherent \v{C}erenkov radiation, known as the Askaryan effect. In this work we
study the case in which such a particle shower occurs in a medium just below
its surface. We show, for the first time, that the radiation transmitted
through the surface is independent of the depth of the shower below the surface
when observed from far away, apart from trivial absorption effects. As a direct
application we use the recent results of the NuMoon project, where a limit on
the neutrino flux for energies above $10^{22}$\,eV was set using the Westerbork
Synthesis Radio Telescope by measuring pulsed radio emission from the Moon, to
set a limit on the flux of ultra-high-energy cosmic rays.
