Regularization parameter estimation for underdetermined problems by the χ ² principle with application to 2D focusing gravity inversion

The χ² principle generalizes the Morozov discrepancy principle to the augmented residual of the Tikhonov regularized least squares problem. For weighting of the data fidelity by a known Gaussian noise distribution on the measured data when the stabilizing or regularization term is considered to be weighted by unknown inverse covariance information on the model parameters the minimum of the Tikhonov functional becomes a random variable that follows a χ² -distribution withm p n + . degrees of freedom for the model matrix G of sizem m ×n, and regularizer L of size p×n. Then a Newton root-finding algorithm employing the generalized singular value decomposition or singular value decomposition when L = I can be used to find the regularization parameter α. Here the result and algorithm are extended to the underdetermined case m n.