Golub and Loan (1980) presented a numerically-stable TLS algorithm which utilizes the singular value decomposition (SVD). Subsequent refinements to the method predominantly use SVD, and much of the current literature emphasizes stabilization of the inverse and implicit model regularization by SVD truncation Fierro et al. (1997). Because it is numerically intensive, however, the SVD generally proves unrealistic for use in large-scale problems, which are the rule in exploration geophysics.
The TLS problem can be cast as an extremal eigenvalue/eigenvector estimation problem. Chen et al. (1986) present a conjugate gradient (CG) scheme to compute the minimum eigenvalue/eigenvector of a linear system. Zhu et al. (1997) extend Chen et al.'s algorithm to solve the TLS problem, in the context of optical tomography.
I begin with a short theoretical overview of the TLS problem. I implement the CG method described by Chen et al. (1986), adapted for the TLS problem in a similar fashion as the work of Zhu et al. (1997). I test the algorithm on two familiar geophysical problems: least-squares deconvolution of a 1-D signal, and velocity scan inversion with the hyperbolic Radon transform. Liu and Sacchi (2002) tested an SVD-based, regularized TLS approach on velocity scan inversion using the parabolic Radon transform.