SVD preprocessing of helioseismic data for solar structure inversion

Helioseismic inversion to determine solar structure is based on the analysis of very substantial numbers of modes and hence may involve considerable computational expense. This is particularly true for inversions using methods of optimally localised averages, which require inversion of matrices whose order is the number ofmodes in the set; yet suchmethods are desirable to make the full use of the data. On the other hand, there is considerable redundancy in the data, different modes carrying almost the same information about solar structure. Thus, in the unavoidable presence of data errors, the number of independent pieces of information is generally much less than the number ofmodes. This suggests that themodeset can be reduced by forming suitable combinations of the data before the more computationally intensive inversion is performed.We show that such combinations may be based on singular value decomposition (SVD) of the problem, leading to a drastically reduced inverse problem whose solution is essentially indistinguishable from the solution of the original problem. In addition, the results of the SVD provide insight into the information content of the helioseismic data.