Joint Inversion of Compact Operators

Joint inversion of multiple data types was studied as early as 1975 in [K. Vozoff and D. L. Jupp, Joint inversion of geophysical data, Geophys. J. Internat. 42 1975, 3, 977–991], where the authors used the singular value decomposition to determine the degree of ill-conditioning of joint inverse problems. The authors demonstrated in several examples that combining two physical models in a joint inversion, and by effectively stacking discrete linear models, improved the conditioning as compared to 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 determine the well-posedness 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. The expression also quantifies the benefits of including repeated measurements in an inversion. We give an example of joint inversion with two moderately ill-posed Green’s function solutions, and quantify the improvement over individual inversions. This work provides a framework in which to identify data types that are advantageous to combine in a joint inversion.