Raw helioseismic data are essentially time series of the motion or intensity fluctuations at different positions on the visible surface of the sun. These can be regarded as being the result of a superposition of normal modes of oscillation, or simply as the surface manifestation of a complicated wave field. The objective of helioseismology is to determine as much as possible about the internal structure and motion of the sun, with a view to answering pertinent questions in astrophysics.
This review will summarize the techniques that have been used in helioseismological investigations, in the hope that it will stimulate mathematicians to contemplate improving on what might be regarded as rather crude techniques. They fall into two classes: (i) by regarding the seismic disturbance as a superposition of normal modes of oscillation, whose frequencies can be represented as integrals over the volume of the star, find either averages of the structure over localized regions of the star or, perhaps, determine some representation of the structure as functions that satisfy some preascribed optimality condition (such as the
'smoothest') and which reproduce the frequencies within what are believed to be the observational errors, and (ii), treat the wave field asymptotically by ray theory, and represent the wave propagation time from one point on the observable surface of the sun to another by an appropriate integral along the ray path, attempt to determine that time observationally by cross-correlation with time lag signals from the two points in question, and then seek an underlying three-dimensional structure that is consistent with those ray-path integrals. There will be separate lectures on result from these methods in the minisymposium on - 'Applied Inverse Problems: Theoretical and Computational Aspects', to which this review serves as an introduction.
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