  Context: The cross-covariance of solar oscillations observed at pairs of
points on the solar surface is a fundamental ingredient in time-distance
helioseismology. Wave travel times are extracted from the cross-covariance
function and are used to infer the physical conditions in the solar interior.
Aims: Understanding the statistics of the two-point cross-covariance function
is a necessary step towards optimizing the measurement of travel times.
Methods: By modeling stochastic solar oscillations, we evaluate the variance of
the cross-covariance function as function of time-lag and distance between the
two points. Results: We show that the variance of the cross-covariance is
independent of both time-lag and distance in the far field, i.e., when they are
large compared to the coherence scales of the solar oscillations. Conclusions:
The constant noise level for the cross-covariance means that the
signal-to-noise ratio for the cross-covariance is proportional to the amplitude
of the expectation value of the cross-covariance. This observation is important
for planning data analysis efforts.

Solar and Stellar Astrophysics