Geophysical Imaging of Subsurface Structures with Least Squares Estimates

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Electrical resistivity tomography (ERT) is a useful tool for subsurface imaging. However, the resulting nonlinear inverse problem is severely under-determined. Smoothness constraints are commonly implemented with least squares to make the problem solvable, but the constraints limit the ability to produce discontinuities in the model parameters. In practice, sharp delineations have been recovered with these constraints by applying appropriate weights or covariance matrices which relax the constraint at different regions. For example in medical imaging, anatomical information has been included in electrical impedance tomography through matrix valued fields. More closely related to this work, prior structural information was added to ERT inversion through matrix valued fields. In this work, we analyze the binary matrix that relaxes the smoothness constraint in locations where there is a known boundary. Analysis of the recovered parameters using this matrix gives insight as to the type of heterogeneities that can be recovered. We conclude that it is more effective to include structural information with 1st and 2nd order derivative constraints than with initial parameter estimates. In addition, we show that the 1st derivative constraint produces models with piecewise constant variability, while a 2nd derivative will yield linear variability. These conclusions are verified on synthetic ERT inversions in seven different subsurface structures.