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Green's functions for far-side seismic images: a polar expansion approach

We have computed seismic images of magnetic activity on the far surface of the Sun by using a seismic-holography technique. As in previous works, the method is based on the comparison of waves going in and out of a particular point in the Sun but we have computed here the Green's functions from a spherical polar expansion of the adiabatic wave equations in the Cowling approximation instead of using the ray-path approximation previously used in the far-side holography. A comparison between the results obtained using the ray theory and the spherical polar expansion is shown. We use the gravito-acoustic wave equation in the local plane-parallel limit in both cases and for the latter we take the asymptotic approximation for the radial dependencies of the Green's function. As a result, improved images of the far-side can be obtained from the polar-expansion approximation, especially when combining the Green's functions corresponding to two and three skips. We also show that the phase corrections in the Green's functions due to the incorrect modeling of the uppermost layers of the Sun can be estimated from the eigenfrequencies of the normal modes of oscillation.