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

Abstract

Abstract. The χ 2 -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 2 -distribution with m + p − n degrees of freedom for the model matrix G of size m × n and regularizer L of size p × n.Here it is proved that the result holds for the underdetermined case, m < n provided that m + p ≥ n and that the null spaces of the operators do not intersect. 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, or singular value decomposition when L = I. 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 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 the Morozov Discrepancy principle. These experiments demonstrate the efficiency and robustness of the χ 2 -principle in this context, 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 2 -principle when implemented without the desirable knowledge of a mean value of the model.

Research Note: A unifying framework for the widely used stabilization of potential field inverse problems

“…These include generalized cross validation (Marquaridt ; Golub, Heath and Wahba ), the L ‐curve (Hansen ; Hansen ), unbiased predictive risk estimation (Vogel ) and the Morozov (Morozov ) and χ 2 ‐discrepancy principles (Mead and Renaut ; Vatankhah et al . ). The application of these parameter‐choice strategies in focusing inversion algorithms is discussed in Vatankhah et al .…”

“…The application of these parameter‐choice strategies in focusing inversion algorithms is discussed in Vatankhah et al . (2014a, , ).…”

“…Applying the L p ‐norm, for p = 0 or 1, to the gradient of the model parameters minimizes the number of discontinuous transitions of the reconstructed model. Alternatively, when applied directly to the model parameters, these L p ‐norms, with p = 0 or 1, provide sparsity in the solution, and are relevant when it can be assumed that the sources of interest are localized and compact (Last and Kubik ; Guillen and Menichetti ; Barbosa and Silva ; Portniaguine and Zhdanov ; Zhdanov and Tolstaya ; Ajo‐Franklin, Minsley and Daley ; Vatankhah, Ardestani and Renaut ; Vatankhah, Renaut and Ardestani ; Vatankhah, Ardestani and Renaut ; Vatankhah, Renaut and Ardestani ). Inversion algorithms that employ such sparsity constraints yield sparse and focused images of the subsurface structures.…”

Large-scale focusing joint inversion of gravity and magnetic data with Gramian constraint

“…The L 2 -norm leads to solutions which are smooth and have minimum structure (Constable et al 1987;Li & Oldenburg 1996;Pilkington 1997), and can be expected to only provide the important and large-scale features of subsurface target(s). To obtain models that are more consistent with true geologic structures, Farquharson (2008), it is necessary to apply stabilizers that can generate compact or blocky solutions, requiring the use of p = 0, or p = 1, for the model or its gradient, respectively, (Last & Kubik 1983;Barbosa & Silva 1994;Farquharson & Oldenburg 1998;Portniaguine & Zhdanov 1999;Boulanger & Chouteau 2001;Ajo-Franklin et al 2007;Farquharson 2008;Vatankhah et al 2014;Sun & Li 2014;Vatankhah et al 2017;Fournier & Oldenburg 2019). The generalized framework for the L p -norm stabilizer provided in Vatankhah et al (2020a) is easily extended for incorporation in a joint inversion, as was applied for the cross gradient inversion in Vatankhah et al (2022).…”

Regularized nonlinear inversion of magnetic anomalies of simple geometric models using Occam’s method: an application to the Morvarid iron-apatite deposit in Iran

“…Furthermore, the performance of the L-curve and weighted generalized cross validation (W-GCV) techniques are compared and contrasted. There have been a few successful applications of the L-curve (Haber 1997;Johnstone andGulrajani 2000, Farquharson andOldenburg 2004;Stefan 2008;Vatankhah et al 2014) and W-GCV (Chung et al 2008;Viloche Bazan and Borges 2010;Abedi et al 2013;Gholami and Sacchi 2012;Ghanati et al 2015) criteria to choose an optimum value of the regularization parameter in non-linear problems in geophysics. In this paper, due to the nonlinearity of inverse modeling of the magnetic simple-shaped structures, a nonlinear least squares constrained minimization problem based on the Occam's inversion is proposed.…”