Analytical Models for Cross-Correlation Signal in Time-Distance Helioseismology

In time-distance helioseismology, the time signals (Doppler shifts) at two points on the solar surface separated by a fixed angular distance are cross-correlated, and this leads to a wave packet signal. Accurately measuring the travel times of these wave packets is crucial for inferring the subsurface properties in the Sun. The observed signal is quite noisy, and to improve the signal-to-noise ratio and make the cross-correlation more robust, the temporal oscillation signal is phase-speed filtered at the two points in order to select waves that travel a fixed horizontal distance. Hence a new formula to estimate the travel times is derived in the presence of a phase-speed filter, and it includes both the radial and horizontal component of the oscillation displacement signal. It generalizes the previously used Gabor wavelet that was derived without a phase-speed filter and included only the radial component of the displacement. This is important since it will be consistent with the observed cross-correlation that is computed using a phase-speed filter, and it also accounts for both the components of the displacement. The new formula depends on the location of the two points on the solar surface that are being cross-correlated and accounts for the travel time shifts at different locations on the solar surface.