Short note: Three dimensional deconvolution of helioseismic data

Solar deconvolution in 3D

Deconvolution in seismology typically seeks to remove the source signature from the recorded data. For 3D data this can be done in multiple ways: as a three way convolution integral in time, as a three way multiplication in the Fourier domain, or as 1D Fourier multiplication in helical coordinates. In this case the solar impulse response has been estimated using spectral factorisation, and by deconvolving the raw data with this response we can find information about the source - namely signature and location.

In frequency space the stochastic oscillation model can be described as the following multiplication

$$D(k_x,k_y, \omega) \ = S(k_x,k_y,\omega) G(k_x,k_y,\omega),$$ (1)

where $D$ is the raw data, $S$ the source function and $G$ the impulse response. The 3D deconvolution can then be applied as a division in frequency, and then transformed to time (Rickett, 2001).

This method suffers from the fact that by dividing the input data $D$ by $G$ , any small or zero values in $G$ will cause large perturbations in the solution for $S$ . This is a problem addressed many, many times in geophysics, and one solution to helping to constrain the estimation is to add a small amount of white noise (a constant in frequency space) to the denominator, ensuring a maximum possible value in the output (Claerbout, 2001).

$$S(k_x,k_y,\omega) \ = G(k_x,k_y,\omega) / ( D(k_x,k_y, \omega) + \epsilon ),$$ (2)

Part of the usual challenge of deconvolution is choosing an appropriate value for $\epsilon$ such that the final image has not been overly steered (Claerbout, 2001). To ensure the 3D Fourier deconvolution was working correctly a synthetic model was produced, convolved with the impulse response and transformed to the time domain. This was then deconvolved with the impulse response, and the initial model was recovered clearly, with the exception of some Gibbs' artifacts due to the domain transformations and truncation of the impulse response. Subsequently the deconvolution part of this process was applied to the raw solar data, and a 3D volume acquired.

Short note: Three dimensional deconvolution of helioseismic data