Seismic tomography with non-uniform ray coverage and autoregressive extrapolation

Seismic tomography as used in the laboratory and in the field is strongly affected by limited and non-uniform ray coverage, and this will generally result in ill-posed inverse problems. In this study several regularization methods to improve inversion for non-uniform ray coverage are described. Inversions based on variable damping are tested and compared with inversions using simple damping. A two-stage autoregressive extrapolation technique is then proposed that can be used to extend the observed data and provide better tomographic images. The algorithm is based on the principle that the extrapolated data adds minimal information to the existing data. The first stage of the extrapolation is to find the optimal prediction-error filter (PE filter). The second stage is to use the PE filter to find the values of the missing data. The missing data are estimated to have the same spectrum as the observed data and this is similar to maximizing an entropy criterion. In order to test the method, synthetic and laboratory tomography experiments for rock samples are used in which full ray coverage can be obtained. Autoregressive methods are then used to extrapolate for partial ray coverage and the tomographic results are compared with the full ray coverage case. These tests show that the autoregressive method can extrapolate known data to find missing data as well as provide improved tomographic images. The autoregressive extrapolation is also tolerant to noise. Autoregressive methods may have important applications to tomography experiments in the field where it is often impossible to set up tomography experiments with perfect ray coverage. A synthetic field example using a cross-borehole geometry is performed and shows that autoregressive extrapolation can be applied to a limited ray coverage field geometry for seismic tomography. We finally applied travel-time tomography techniques to observed datasets recorded in the laboratory for rock samples. The results for these experiments show that for laboratory tomographic experiments, simple as well as complicated internal structures can be reconstructed