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The $\chi^2$-principle generalizes the Morozov discrepancy principle (MDP) tothe augmented residual of the Tikhonov regularized least squares problem.Weighting of the data fidelity by a known Gaussian noise distribution on themeasured data, when the regularization term is weighted by unknown inversecovariance information on the model parameters, the minimum of the Tikhonovfunctional 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 $m<n$ when $m+p\ge n$.A Newton root-finding algorithm is used to find the regularization parameter$\alpha$ which yields the optimal inverse covariance weighting in the case of awhite noise assumption on the mapped model data. It is implemented forsmall-scale problems using the generalized singular value decomposition.Numerical results verify the algorithm for the case of regularizersapproximating zero to second order derivative approximations, contrasted withthe methods of generalized cross validation and unbiased predictive riskestimation. The inversion of underdetermined $2D$ focusing gravity dataproduces models with non-smooth properties, for which typical implementationsin this field use the iterative minimum support (MS) stabilizer and bothregularizer and regularizing parameter are updated each iteration. For asimulated data set with noise, the regularization parameter estimation methodsfor underdetermined data sets are used in this iterative framework, alsocontrasted with the L-curve and MDP. Experiments demonstrate efficiency androbustness of the $\chi^2$-principle, moreover the L-curve and MDP aregenerally outperformed. Furthermore, the MS is of general use for the$\chi^2$-principle when implemented without the knowledge of a mean value ofthe model.

Regularization Parameter Estimation for Underdetermined problems by the $\chi^2$ principle with application to $2D$ focusing gravity inversion