Inversion formulas for time-distance helioseismology

Abstract

In this thesis we examine the inverse kinematic problem presented in time-distance helioseismology. Acoustic wave oscillations in the Sun travel along ray paths below the Sun's surface. These ray paths are de ned by Fermat's principle of least time. Di erences from the expected travel times of these oscillations to di erent points on the Sun's surface is indicative of inhomogeneities in the solar structure. Measuring these perturbations in the travel times allow the recovery of information about acoustic sources and ows within the Sun. Our goal is to accurately recover functions describing these perturbations, and hence information about solar features. An inverse problem is solved using the transport equation in conjunction with a rst order approximation of the ray path geometry. We obtain results that show this inversion process is unique and stable, as well as explicit formulas for the solutions to the scalar and vector tomography problems considered.