Linear correlation constrained joint inversion using squared cosine similarity of regional residual model vectors

Abstract

Inversion of geophysical data is inherently non-unique, so joint inversion has been used to reduce the ambiguity of inversion. Among other things, the coupling between different physical parameters is crucial in geophysical joint inversion. In this paper, we improve the coupling constraint based on ‘Squared Cosine Similarity’ of regional residual model vectors for joint inversion. Compared with the widely used coupling constraints in joint inversion, such as the cross-gradients constraint and the gradients dot product constraint, whose coupling are constructed by the computing of model parameter's gradient using ‘local’ adjacent grids to calculate the derivatives in specified directions, the ‘region’ of the new proposed constraint is defined as a specified model space in which two vectors to compute the dot product of cosine similarity should satisfied the condition they are linear correlated, because the constraint we designed is to force the value of squared cosine similarity in this ‘region’ tends to 1. If the condition is satisfied, the ‘region’ may be any size in the model space, i.e. the whole model space or small as the ‘local’ grids. As for the ‘residual model vector’, it means the difference between the inversed model parameter with the reference background models, and usually the background is taken as the initial model of inversion in the defined region, that is the uniform half space. As for the ‘squared’, it means no matter the regional residual vector's varying direction is same or opposite, the value of squared cosine similarity always tends to 1. Then we compute the squared cosine similarity from regional residual model vectors and constrain all the region's summed value to be 1 to enforce the linear correlation of regional model variations during joint optimization. So the proposed constraint will result in enforcing the correlation between regional model parameters’ variations. We conduct model tests with an application of seismic traveltimes and gravity anomaly to verify the proposed constraint in joint inversions, and we can see the improvement of the joint inversion over the results of separated inversions and cross-gradients or gradient dot product constrained joint inversion. We calculate the values of cross-gradients, dot product and the rock's physical relationship to prove that joint inversion with this new constraint can improve the linear correlation between velocity and density variations, avoid computing the model gradients of the adjacent ‘local’ grids and reduce the local discontinuities. We also test the proposed constraint on field data with gravity and magnetic anomaly in Lower Yangtze region, China, and the result reveals the consistency of high density and high magnetic basement.