This research is to develop further the theoretical basis for determining 3-D earth structure from long-period seismic data. The next major advance will be to invert a dataset that is an order of magnitude larger, of the order of 50,000 source-receiver pairs which will be developed elsewhere. Attention here will be focused on the accuracy and possible biases of the various approximations and techniques that will have to be employed in such a large-scale study. Specific topics to be addressed are the accurate computation of synthetic long-period seismograms, evaluation of approximate methods and inversion techniques, and theoretical studies of geometrical optics, anisotropy, and the Born approximation.