Calculation of the sun's acoustic impulse response by multi-dimensional spectral factorization

Abstract

Calculation of time-distance curves in helioseismology can be formulated as a blind-deconvolution (or system identification) problem. A classical solution in one-dimensional space is Kolmogorov's Fourier domain spectral-factorization method. The helical coordinate system maps two-dimensions to one. Likewise a three-dimensional volume is representable as a concatenation of many one-dimensional signals. Thus concatenating a cube of helioseismic data into a very long 1-D signal and applying Kolmogorov's factorization, we find we can construct the three-dimensional causal impulse response of the Sun by deconcatenating the Kolmogorov result. Time-distance curves calculated in this way have the same spatial and temporal bandwidth as the original data, rather than the decreased bandwidth obtained obtained by cross-correlating traces. Additionally, the spectral factorization impulse response is minimum phase, as opposed to the zero phase time-distance curves produced by cross-correlation.