3D-electromagnetic imaging using adjoint fields

We present a new approach to the inverse problem of electromagnetic induction tomography. The main application we have in mind is the monitoring of conducting contaminant plumes out of surface and borehole electromagnetic imaging data. The essential feature of the method developed here is the use of adjoint fields for the reconstruction task, combined with a splitting of the data into smaller groups which define subproblems of the inversion problem. This method is based on the so-called "adjoint technique," which (for the present application) has the very useful property that the inverse problem can be solved approximately by making two uses of the same forward modeling code. Using a somewhat oversimplified description of our technique, the updates to the electrical conductivity are obtained by first making one pass through the code using the latest best guess of the nature of the conducting medium, and then another pass with the adjoint operator applied to the differences in computed and measured data. Then the results of these two calculations are combined to determine updates to the original conductivity model. The resulting procedure is iterative and can be applied successively to parts of the data, e.g., data associated with one transmitter location can be used to update the model before other transmitter locations are considered. Numerical experiments are performed for a layered background medium in which one or two localized (3D) inclusions are immersed. These have to be monitored out of surface to borehole and cross-borehole EM data. We show that the algorithm is able to recover a single inclusion in the Earth which has high contrast to the background, and to distinguish between two separated inclusions in the Earth given certain borehole geometries.