where and are two measured fields. By using normalized values of each data field, the cross-gradient function vanishes where the two fields are similar in structure or either one of them is very smooth. () and Maysami and Clapp (2009) note that cross-gradient functions can be used as a measure of similarity. The cross-gradient function can be a suitable choice for the regularization (model-shaping) term of an optimization problem that imposes the structure of the auxiliary field on the estimated model. The differentiating nature of these functions, however, raises a sensitivity issue where there is a large gap between the frequency bands of different types of data. It is also expected that cross-gradient function is not a very effective choice for low-frequency data, since the smooth behavior causes the gradient to vanish.
Geophysical data integration and its application to seismic tomography