  In time-distance helioseismology, information about the solar interior is
encoded in measurements of travel times between pairs of points on the solar
surface. Travel times are deduced from the cross-covariance of the random wave
field. Here we consider travel times and also products of travel times as
observables. They contain information about e.g. the statistical properties of
convection in the Sun. The basic assumption of the model is that noise is the
result of the stochastic excitation of solar waves, a random process which is
stationary and Gaussian. We generalize the existing noise model (Gizon and
Birch 2004) by dropping the assumption of horizontal spatial homogeneity. Using
a recurrence relation, we calculate the noise covariance matrices for the
moments of order 4, 6, and 8 of the observed wave field, for the moments of
order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and
4 of the travel times. All noise covariance matrices depend only on the
expectation value of the cross-covariance of the observed wave field. For
products of travel times, the noise covariance matrix consists of three terms
proportional to $1/T$, $1/T^2$, and $1/T^3$, where $T$ is the duration of the
observations. For typical observation times of a few hours, the term
proportional to $1/T^2$ dominates and $Cov[\tau_1 \tau_2, \tau_3 \tau_4]
\approx Cov[\tau_1, \tau_3] Cov[\tau_2, \tau_4] + Cov[\tau_1, \tau_4]
Cov[\tau_2, \tau_3]$, where the $\tau_i$ are arbitrary travel times. This
result is confirmed for $p_1$ travel times by Monte Carlo simulations and
comparisons with SDO/HMI observations. General and accurate formulae have been
derived to model the noise covariance matrix of helioseismic travel times and
products of travel times. These results could easily be generalized to other
methods of local helioseismology, such as helioseismic holography and ring
diagram analysis.
