Three-dimensional inversion of sound speed below a sunspot in the born approximation

We revise the inversion of acoustic travel times for the three-dimensional sound-speed structure below the solar NOAA Active Region 8243 of 1998 June. We benefit from recent progress in time-distance helioseismology that provides us with more reliable tools to infer subsurface solar properties. Among the improvements we implement here are the use of Born approximation-based travel-time sensitivity kernels that take into account finite-wavelength effects and thus are more accurate than the previously employed ray-path kernels, the inclusion of solar noise statistical properties in the inversion procedure through the noise covariance matrix, and the use of the actual variance of the noise in the temporal cross-covariances in the travel-time fitting procedure. Of these three improvements, the most significant is the application of the Born approximation to time-distance helioseismology. This puts the results of this discipline at the same level of confidence as those of global helioseismology based on inversion of normal-mode frequencies. Also, we compare inversion results based on ray-path and Born approximation kernels. We show that both approximations return a similar two-region structure for sunspots. However, the depth of inverted structures may be offset by 1 or 2 Mm, and the spatial resolution of the results is more accurately estimated with the more realistic Born sensitivity kernels. Finally, using artificial realizations of Doppler velocities of the quiet Sun, we are now able to estimate the statistical uncertainties of these inversion results.