On the Iterative Solution of Ill-Conditioned Normal Equations by the Use of Lanczos Methods

The determination of the earth’s exterior gravity field from noisy satellite observations lacks stability, due to the downward continuation process and due to data gaps. This property translates into an ill-conditioned least-squares problem. It is routinely solved after forming the normal equations and adding a constraint or covariance matrix which reflects the prior knowledge about the gravity field and accounts for regularization. In this contribution we investigate a modification: the use of Lanczos methods for the iterative solution of the problem, and, by controlled stopping and in combination with Tikhonov regularization, for suppressing the amplification of data noise. There are essentially two regularization parameters to be chosen properly: the Tikhonov parameter, and the number of iterations. Our simulations show that this can be done effiiciently by applying a cross-validation procedure on the satellite gravity data. However, further investigations are needed.