crossgradient

Crossgradient is a regularization constraint used in the joint inversion of multiple geophysical or geological datasets to promote coherent structural features between different model parameters. The underlying idea is that geological boundaries often produce correlated changes in different physical properties, so enforcing alignment of their gradients can yield more consistent and sharper interfaces across datasets.

Mathematically, for two model fields m1(x) and m2(x) defined on a common domain, with gradients ∇m1 and ∇m2, the cross-gradient term is typically written as the magnitude of their cross product, C = ∫Ω ||∇m1(x) × ∇m2(x)|| dx or, in a squared form, C = ∫Ω ||∇m1(x) × ∇m2(x)||^2 dx. Minimizing this term encourages the gradients to be parallel almost everywhere, so edges in one model tend to coincide with edges in the other. In practice, the cross-gradient term is added to a conventional objective function in joint inversion: J = ||d − F(m1, m2)||^2 + λ C, where λ controls the weight of the cross-gradient constraint.

Applications include joint inversion of gravity and magnetic data, electrical resistivity tomography with other modalities, and other combinations where shared structural information is expected. The approach can be extended to more than two datasets by including pairwise cross-gradient terms.

Limitations include the risk of forcing incorrect feature alignment if the datasets respond to different geological controls, sensitivity to noise and grid alignment, and the need for careful tuning of regularization weight. Crossgradient remains a common tool for enhancing structural consistency in multi-physics inversions.