Accuracy of Born and Ray Approximations in Time-Distance Helioseismology

Time-distance helioseismology attempts to infer localized departures from a nominal state of the solar interior from comparison of observed travel times of acoustic wave packets to those anticipated from the model. Until recently, such time-distance measurements have generally been modeled in the ray approximation, which neglects finite-wavelength effects. Concern that these effects can be important has prompted interest in the Born approximation, which is sensitive to them. In order to elucidate the ranges of validity of the ray and Born approximations and the nature of their limitations, we compare travel-time pertrubations calculated using these approximations to exact travel times for a simple problem, namely the propagation of adiabatic acoustic waves through the center of a spherically symmetric sound-speed perturbation to an otherwise uniform medium. We show that the Born and first-order ray approximations converge to the same result as the spatial scale of the medium perturbation becomes large compared to the first Fresnel zone, with a fractional error in the travel-time perturbation equal to the fractional strength of the perturbation, while a full ray calculation converges to the exact solution in this limit. For a uniform perturbation having a size the order of the first Fresnel zone, interference between direct and diffracted wave produces travel-time fluctuations that are entirely absent in the ray approximation; these fluctuations are only qualitatively replicated by the Born approximation for moderately weak (e.g., 5%) perturbation strengths. Such fluctuations are, however, largely suppressed for a smoothly-varying perturbation expected to be more representative of solar structures. The so-called banana-doughnut (here, ventilated cigar) form of the Born sensitivity kernels, i.e., a greatly reduced sensitivity of the travel-time perturbation to small-scale medium perturbations that fall near the unperturbed ray path that is absent in the ray approximation, is also shown to be consistent with the exact results.