Weighted least-squares criteria for seismic traveltime tomography

Methods are developed for design of linear tomographic reconstruction algorithms with specified properties. Assuming a starting model with constant slowness, an algorithm with the following properties is found: (1) the optimum constant for the starting model is determined automatically; (2) the weighted least-squares error between the predicted and measured traveltime data is as small as possible; (3) the variance of the reconstructed slowness from the starting model is minimized; (4) rays with the greatest length have the least influence on the reconstructed slowness, and (5) cells with most ray coverage tend to deviate least from the background value. The resulting algorithm maps the reconstruction problem into a vector space where the contribution to the inversion from the background slowness remains invariant, while the optimum contribution in orthogonal directions are found. For a starting model with constant slowness, the reconstruction algorithm has analogous properties.>