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We have developed a novel probabilistic approach to the joint inversion of multi-modal geophysical data based on the Gramian constraint. The multi-modal geophysical survey is the most effective technique for geophysical exploration because different physical data reflect distinct physical properties of the different components of the geological system. The joint inversion of multi-modal data can produce enhanced subsurface images of the physical property distributions, which enhances our ability to explore natural resources. One effective method of joint inversion is based on the Gramian constraint. This technique enforces the relationships between different model parameters during the inversion process. We demonstrate that the Gramian can be interpreted as a determinant of the covariance matrix between different physical models representing subsurface geology in the framework of the probabilistic approach to inverse theory. This interpretation enables us to use all the power of the modern probability theory and statistics in developing new methods for the joint inversion of multi-modal geophysical data. We apply the developed joint inversion methodology to inversion of gravity gradiometry and magnetic data in the Nordkapp Basin, Barents Sea to image salt diapirs.