Resolution and Covariance in Generalized Least Squares Inversion

The Generalized Least Squares (GLS) method uses both data and prior information to solve for a best-fitting set of model parameters. We review the method and present simplified derivations of its essential formulas. Concepts of resolution and covariance – essential in all of inverse theory – are applicable to GLS, but their meaning, and especially that of resolution, must be carefully interpreted. We introduce derivations that show that the quantity being resolved is the deviation of the solution from the prior model, and that the covariance of the model depends on both the uncertainty in the data and the uncertainty in the prior information. On face value, the GLS formulas for resolution and covariance seem to require matrix inverses that may be difficult to calculate for the very large (but often sparse) linear systems encountered in practical inverse problems. We demonstrate how to organize the computations in an efficient manner and present MATLAB code that implements them. Finally, we formulate the well-understood problem of interpolating data with minimum curvature splines as an inverse problem, and use it to illustrate the GLS method.