Conclusions and Discussion

I have introduced total least-squares (TLS) optimization as a possible alternative to ``standard'' least-squares approaches. TLS approaches incorporate errors in both the data and in the modeling operator, to produce ``more accurate'' solutions. I put ``more accurate'' in quotes because in our real world, frequent appearences of nonempty nullspaces impose some subjectivity on any solution.

I implemented the conjugate gradient TLS solver (TLS-CG) published by Zhu et al. (1997), although in that paper, the authors omit a crucial model normalization step that leads to nonconvergence of the algorithm. I present a complete algorithm in Appendix A.

Tests on a synthetic 1-D deconvolution example seem to validate TLS as a tool. In those tests, when ideal noise was added to the filter and data, TLS resolved the true model better than normal least-squares or damped least-squares. Tests using the hyperbolic radon transform were inconclusive; no efforts were made to understand operator error in this case, and in summary, the TLS result looks somewhere in between LS and DLS.

Will TLS be a useful tool in geophysics? My suspicion is that TLS makes only a second order improvement in the quest to account for uncertainty in geophysical inverse problems. More interesting are efforts to perturb the nullspace of inverse problems to infer model statistics Chen and Clapp (2002); Clapp (2002).