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

Abstract. The χ-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 and, 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 χ-distribution with m + p − n degrees of freedom for the model matrix G of size m × n, 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, with m + p ≥ n. Numerical results first contrast and verify the Generalized Cross Validation, Unbiased Predictive Risk Estimation and χ algorithms when m < n, with regularizers L approximating zero to second order derivative approximations. The inversion of underdetermined 2D focusing gravity data produces models with non-smooth properties, for which typical solvers in the field use an iterative minimum support stabilizer, with both regularizer and regularizing parameter updated each iteration. The χ and unbiased predictive risk estimator of the regularization parameter are used for the first time in this context. For a simulated underdetermined data set with noise, these regularization parameter estimation methods, as well as the generalized cross validation method, are contrasted with the use of the L-curve and the Morozov Discrepancy principle. Experiments demonstrate the efficiency and robustness of the χ-principle and unbiased predictive risk estimator, moreover showing that the L-curve and Morozov Discrepancy Principle are outperformed in general by the three other techniques. Furthermore, the minimum support stabilizer is of general use for the χ-principle when implemented without the desirable knowledge of a mean value of the model.