TLS Overview

Golub and Loan (1980) phrased the TLS problem as follows. Given a forward modeling operator L and measured data d, assume that both are contaminated with white noise of uniform variance; matrix N and vector n, respectively. Then the TLS solution is obtained by minimizing the Frobenius matrix norm of the augmented noise matrix:  

$$\mbox{min} \Vert [\bf N \;\; n ] \Vert _F,$$ (1)

subject to the constraint that the solution is in the nullspace of the combined augmented noise and input operators:  

$$\left([\bf L \;\; d ] + [\bf N \;\; n ]\right) \left[ \begin{array} {r} \bf m \\ -1 \end{array}\right] = \bold 0.$$ (2)

To solve the system of equations (1) and (2), Golub and Loan (1980) introduced a technique based on the Singular Value Decomposition (SVD). Although mathematically elegant, SVD-based approaches are generally unrealistic for the large-scale problems that are the norm in exploration geophysics.