An analysis of the simultaneous joint inversion of compact operators

The first mention of joint inversion came in the 1975 paper “Joint Inversion of Geophysical Data,” by Vozoff and Jupp. There, the authors demonstrated through singular value decompositions that the decay rates of the singular values decreased when combining two different types of data and hence improved the inherent ill-conditioning of the discrete inverse problems they were investigating. Our work extends these ideas to compact linear operators defined by combining Green’s function solutions to multiple PDEs representing different types of data. Using a singular value expansion of the defined operators, we too measure the decay rate of the singular values to determine the effect of joint inversion on the ill-posedness of the continuous inverse problems. This method relies solely on the PDEs representing data and should therefore provide further mathematical justification for the effectiveness of existing joint inversion methods. As our work is refined, we seek to better inform the design of field experiments.