Full Waveform Inversion for Time-Distance Helioseismology
  Inferring interior properties of the Sun from photospheric measurements of the seismic wavefield constitutes the helioseismic inverse problem. Deviations in seismic measurements (such as wave travel times) from their fiducial values estimated for a given model of the solar interior imply that the model is inaccurate. Contemporary inversions in local helioseismology assume that properties of the solar interior are linearly related to measured travel-time deviations. It is widely known, however, that this assumption is invalid for sunspots and active regions, and likely for supergranular flows as well. Here, we introduce nonlinear optimization, executed iteratively, as a means of inverting for the sub-surface structure of large-amplitude perturbations. Defining the penalty functional as the $L_2$ norm of wave travel-time deviations, we compute the the total misfit gradient of this functional with respect to the relevant model parameters %(only sound speed in this case) at each iteration around the corresponding model. The model is successively improved using either steepest descent, conjugate gradient, or quasi-Newton limited-memory BFGS. Performing nonlinear iterations requires privileging pixels (such as those in the near-field of the scatterer), a practice not compliant with the standard assumption of translational invariance. Measurements for these inversions, although similar in principle to those used in time-distance helioseismology, require some retooling. For the sake of simplicity in illustrating the method, we consider a 2-D inverse problem with only a sound-speed perturbation. 
Shravan Hanasoge and Jeroen Tromp
astro-ph.SR astro-ph.EP