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

Abstract

Time-distance helioseismology measures the time for acoustic wavepackets to travel, through the solar interior, from one locationon the solar surface to another. Interpretation of travel times requiresan understanding of their dependence on subsurface inhomogeneities.Traditionally, time-distance measurements have been modeled in theray approximation. Recent efforts have focused on the Born approximation,which includes finite-wavelength effects. In order to understandthe limitations and ranges of validity of the ray and Born approximations,we apply them to a simple problem-adiabatic acoustic waves in a uniformmedium with a spherical inclusion-for which a numerical solutionto the wave equation is computationally feasible. We show that, forperturbations with length scales large compared to the size of thefirst Fresnel zone, both the Born and first-order ray approximationsyield the same result and that the fractional error in the traveltime shift, computed by either approximation, is proportional tothe fractional strength of the sound speed perturbation. Furthermore,we demonstrate that for perturbations with length scales smallerthan the first Fresnel zone the ray approximation can substantiallyoverestimate travel time perturbations while the Born approximationgives the correct order of magnitude. The main cause of the inaccuracyof the Born approximation travel times is the appearance of a diffractedwave. This wave, however, has not yet been observed in the solardata.