Iterative algorithm for the damped minimum norm, least-squares and constrained problem in seismic tomography

The ray tomography problem can be reduced to a minimum norm, least-squares constrained problem. As is well known, the generalized inverse matrix methods, such as singular value decomposition (SVD) and damped least squares residual inversion (DLSQR, WLSQR), which work well for dense matrix problems, become expensive for a large, sparse problem in ray tomography. Some iterative algorithms (ART, SIRT) are directly applied to solve the system without considering a priori information and the non-uniqueness of the problem. The conjugate gradient (CG) method and the Paige-Saunders′ algorithm are popular solutions only for the (undamped) LSQR problem. In this paper, the CG iterative algorithm is developed to solve the damped minimum norm, least squares constrained problem and the dual inversion problem in non-linear traveltime tomography. The suggested algorithms exhibit the following desirable features: (1) consideration is given to errors in both the primary estimation of the model and the traveltime data; (2) solutions are less sensitive to data error; (3) there is a reduction in the nonuniqueness of the solution using a priori information (‘hard’ and ‘soft’ bounds); and (4) the algorithms take advantage of the sparsitv of the matrix and consume less computer time and memory space for a large problem. The quantitative comparisons between most of the popular ray tomography methods and our algorithms were based on three measures: the misfit with the true model; solution stability; and image quality. Studies were made under different levels of noise. The results indicate that our algorithm is an accurate, flexible and efficient method for non-linear traveltime tomography and has greater tolerance to noise than any standard tomography algorithm.