Joint inversion of compact operators

Published in Journal of Inverse and Ill-posed Problems, 2020

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Abstract: Joint inversion of multiple data types was studied as early as 1975, where authors used the singular value decomposition to determine the degree of illconditioning of joint inverse problems. The authors demonstrated in several examples that combining two physical models in a joint inversion, and effectively stacking discrete linear models, improved the conditioning of individual inversions. This work extends the notion of using the singular value decomposition to determine the conditioning of discrete joint inversion to using the singular value expansion to determining the wellposedness of joint operators. We provide a convergent technique for approximating the singular values of continuous joint operators. In the case of self-adjoint operators, we give an algebraic expression for the joint singular values in terms of the singular values of the individual operators. This expression allows us to show that while rare, there are situations where ill-posedness may be not improved through joint inversion and in fact can degrade the conditioning of an individual inversion. We give an example of improving inversion with two moderately ill-posed Green’s function solutions, and quantify the improvement over individual inversions. Results from this work show that analysis of singular values of compact operators describing different data types before an inversion helps identify which types of data are advantageous to combine in a joint inversion.

  year = 2020,
  month = {feb},
  publisher = {Walter de Gruyter {GmbH}},
  volume = {28},
  number = {1},
  pages = {105–118},
  author = {Jodi L. Mead and James F. Ford},
  title = {Joint inversion of compact operators},
  journal = {Journal of Inverse and Ill-posed Problems}