  The temporal covariance between seismic waves measured at two locations on
the solar surface is the fundamental observable in time-distance
helioseismology. Above the acoustic cut-off frequency ($\sim$5.3~mHz), waves
are not trapped in the solar interior and the covariance function can be used
to probe the upper atmosphere. We wish to implement appropriate radiative
boundary conditions for computing the propagation of high-frequency waves in
the solar atmosphere. We consider the radiative boundary conditions recently
developed by Barucq et al. (2017) for atmospheres in which sound-speed is
constant and density decreases exponentially with radius. We compute the
cross-covariance function using a finite element method in spherical geometry
and in the frequency domain. The ratio between first- and second-skip
amplitudes in the time-distance diagram is used as a diagnostic to compare
boundary conditions and to compare with observations. We find that a boundary
condition applied 500 km above the photosphere and derived under the
approximation of small angles of incidence accurately reproduces the `infinite
atmosphere' solution for high-frequency waves. When the radiative boundary
condition is applied 2 Mm above the photosphere, we find that the choice of
atmospheric model affects the time-distance diagram. In particular, the
time-distance diagram exhibits double-ridge structure when using a VAL
atmospheric model.
