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

Saeed Vatankhah, Rosemary Renaut, Vahid E. Ardestani

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


The χ2 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 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 χ2 -distribution withm p n + . degrees of freedom for the model matrix G of sizem 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.

Original languageEnglish (US)
Article number085002
JournalInverse Problems
Issue number8
StatePublished - Aug 1 2014


  • gravity inversion
  • minimum support stabilizer
  • regularization parameter
  • unbiased predictive risk estimator

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Signal Processing
  • Mathematical Physics
  • Computer Science Applications
  • Applied Mathematics


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