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The Accuracy of the Born and Ray Approximations in Time-Distance Helioseismology

Time-distance helioseismology measures the time for acoustic wave packets to travel, through the solar interior, from one location on the solar surface to another. Interpretation of travel times requires an understanding of their dependence on subsurface inhomogeneities. Traditionally, time-distance measurements have been modeled in the ray approximation. Recent efforts have focused on the Born approximation, which includes finite-wavelength effects. In order to understand the limitations and ranges of validity of the ray and Born approximations, we apply them to a simple problem—adiabatic acoustic waves in a uniform medium with a spherical inclusion—for which a numerical solution to the wave equation is computationally feasible. We show that, for perturbations with length scales large compared to the size of the first Fresnel zone, both the Born and first-order ray approximations yield the same result and that the fractional error in the travel time shift, computed by either approximation, is proportional to the fractional strength of the sound speed perturbation. Furthermore, we demonstrate that for perturbations with length scales smaller than the first Fresnel zone the ray approximation can substantially overestimate travel time perturbations while the Born approximation gives the correct order of magnitude. The main cause of the inaccuracy of the Born approximation travel times is the appearance of a diffracted wave. This wave, however, has not yet been observed in the solar data.