Short note: Three dimensional deconvolution of helioseismic data

Introduction

Deconvolution is a technique that the majority of geophysicists are very familiar with, since the presence of a source function inherent within recorded seismic data is an issue that permeates all areas of seismology. Deconvolution simply seeks to remove this source function from the recorded data, leaving the response of the Earth (and any other instrument responses from the recording system.) The main problem within deconvolution is the estimation of the source function, since often these are non-stationary and non-minimum phase, and as such typical 'batch processing' methods such as spiking and predictive deconvolution will make these assumptions, along with several others.

Helioseismology attempts to determine certain aspects of the solar interior by studying acoustic oscillations and disturbances on the surface of the sun. The source function for these oscillations is attributed to turbulence within the convective envelope of the outer sun, and as such is stochastic. Generally helioseismologists work with these source functions by cross-correlating passively observed seismograms and calculating time-distance curves by picking lags.

Rickett and Claerbout (2001) showed that is possible to estimate the solar impulse response by using spectral factorisation techniques, notably by applying one-dimensional Kolmogorov spectral factorisation theory to the 3D oberserved data by using helical boundary conditions (Claerbout, 1998). This short note is an extension of Rickett's work, whereby the postulation is that if we deconvolve the raw data with the three dimensional solar impulse response information about the location and separation of these solar source regions may be revealed.

The helioseismic data used was the SOHO/MDI dataset. The data was transformed to Cartesian coordinates by projecting the high resolution data from an approximate 18 degree square onto a tangent plane. Time sampling for this survey is 60 seconds, and after pre-processing the sampling in space over the solar surface is largely regular and set at 824,800 m.

Short note: Three dimensional deconvolution of helioseismic data