Nullspace shuttles

In seismic tomography the problem is generally underdetermined. The solution to the tomographic problem depends on the specific optimization condition used and is inherently distorted due to noise in the data and approximations in the theory. Smoothing is often applied to reduce inversion artefacts with short correlation lengths. However, a posteriori smoothing generally affects the data fit. For more sophisticated, non-linear filters this effect can be severe. We present a technique to conserve data fit for filters of arbitrary complexity. The difference between the ‘optimal’ solution and a filtered version is projected onto the nullspace of the model space in order to preserve the data fit. Thus, we only allow changes to the image that do not conflict with the data. We demonstrate the benefits of such conservative filters using several different non-linear filters to reduce noise, smooth the image, and highlight edges. The method is exact in small-scale experiments, where we can use the method of singular value decomposition: eigenvectors with large eigenvalues are used to project the difference between the original model and the filtered version onto the nullspace. With large-scale tomographic problems, calculation of all of the large eigenvectors is unrealistic. We show how to use the iterative method of conjugate gradients to apply conservative filters to large-scale tomographic problems with minimum computational effort.