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

The $χ^2$-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 $χ^2$-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 $α$ which yields the optimal inverse covariance weighting in the case of a white noise assumption on the mapped model data. It is implemented for small-scale problems using the generalized singular value decomposition. Numerical results verify the algorithm for the case of regularizers approximating zero to second order derivative approximations, contrasted with the methods of generalized cross validation and unbiased predictive risk estimation. The inversion of underdetermined $2D$ focusing gravity data produces models with non-smooth properties, for which typical implementations in this field use the iterative minimum support (MS) stabilizer and both regularizer and regularizing parameter are updated each iteration. For a simulated data set with noise, the regularization parameter estimation methods for underdetermined data sets are used in this iterative framework, also contrasted with the L-curve and MDP. Experiments demonstrate efficiency and robustness of the $χ^2$-principle, moreover the L-curve and MDP are generally outperformed. Furthermore, the MS is of general use for the $χ^2$-principle when implemented without the knowledge of a mean value of the model.