joint inversion of gravity and magnetic data with petrophysical and structural coupling constraints using indicator functions

Joint inversion can utilize multiple geophysical datasets to supplement or enhance the information on subsurface structures, improve the resolution and certainty of recovered subsurface structures, and provide broad prospects for various geophysical application scenarios. Model coupling is crucial in joint inversion, and there are two main coupling methods based on structural similarity and petrophysical information. These two coupling methods have their own advantages and disadvantages. The structural similarity-based coupling method can obtain structurally similar models without prior information, but the assumption of structural similarity between models is not always valid. The petrophysics-based coupling method provides finer constraints on physical property values, and its difficulty lies in acquiring petrophysical information, which is usually imprecise and incomplete in the inversion region. Joint inversion using a single model coupling approach is insufficient to face complex joint inversion situations. Combining the two coupling methods can complement the structural similarity of the model in the inversion of incomplete petrophysical information.

We develop a novel joint inversion method based on the extended alternating direction method of multipliers (eADMM), which is compatible with multiple model coupling methods and reduces non-uniqueness and uncertainty more effectively. Multiple model coupling methods are contained in an indicator function, which requires the model to satisfy specific mathematical sets, allowing the various models to satisfy arbitrary relationships and ranges. The inequality constraints and linear and nonlinear relational equations extracted from the petrophysical information are expressed directly in mathematical sets, and the structural similarity coupling is implemented by a constraint set that requires a cross-gradient of zero between models. The solution of the indicator function in the eADMM framework is converted into a projection function, and we develop corresponding projection algorithms for multiple constraint sets of both model coupling strategies. The constraint sets are also spatially flexible. Regions with complete petrophysical information and regions requiring increased structural similarity can be constrained by the corresponding sets, respectively.

We apply the method to gravity and magnetic data to test its performance. We compare the performance of our method with that of the joint inversion using a single coupling method for incomplete petrophysical information, including petrophysical information for partial regions and partial geologic units. Synthetic examples show that regions and geologic units with known petrophysical information are recovered with accurate geometric boundaries and physical property values closer to the true values, and structural similarity coupling provides structural information for unknown regions or geologic units, recovers more accurate geometric structures and reduces model uncertainty. The new joint inversion method provides higher resolution models than the traditional joint inversion method, and the inversion results are closer to the true model.