Joint inversion of gravity and vertical gradient data based on modified structural similarity index for the structural and petrophysical consistency constraint

Joint inversion is one of the most effective methods for reducing non-uniqueness for geophysical inversion. The current joint inversion methods can be divided into the structural consistency constraint and petrophysical consistency constraint methods, which are mutually independent. Currently, there is a need for joint inversion methods that can comprehensively consider the structural consistency constraints and petrophysical consistency constraints. This paper develops the structural similarity index (SSIM) as a new structural and petrophysical consistency constraint for the joint inversion of gravity and vertical gradient data. The SSIM constraint is in the form of a fraction, which may have analytical singularities. Therefore, converting the fractional form to the subtractive form can solve the problem of analytic singularity and finally form a modified structural consistency index of the joint inversion, which enhances the stability of the SSIM constraint applied to the joint inversion. Compared to the reconstructed results from the cross-gradient inversion, the proposed method presents good performance and stability. The SSIM algorithm is a new joint inversion method for petrophysical and structural constraints. It can promote the consistency of the recovered models from the distribution and the structure of the physical property values. Then, applications to synthetic data illustrate that the algorithm proposed in this paper can well process the synthetic data and acquire good reconstructed results.