{"paper":{"title":"Regularization Parameter Estimation for Underdetermined problems by the $\\chi^2$ principle with application to $2D$ focusing gravity inversion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","stat.ME"],"primary_cat":"math.NA","authors_text":"Rosemary A Renaut, Saeed Vatankhah, Vahid E Ardestani","submitted_at":"2014-02-14T05:14:23Z","abstract_excerpt":"The $\\chi^2$-principle generalizes the Morozov discrepancy principle (MDP) to the augmented residual of the Tikhonov regularized least squares problem. Weighting of the data fidelity by a known Gaussian noise distribution on the measured data, when the regularization term is weighted by unknown inverse covariance information on the model parameters, the minimum of the Tikhonov functional is a random variable following a $\\chi^2$-distribution with $m+p-n$ degrees of freedom, model matrix $G:$ $m \\times n$ and regularizer $L:$ $p\\times n$. It is proved that the result holds also for $m