Firm Conclusions from Electromagnetic Sounding

The problem of electromagnetic induction to recover electrical conductivity is a nonlinear inverse problem, ill-posed and based on inaccurate data. Because of the uncertainty, the information contained in the data can never give a precise description of the conductivity distribution, and so an essential part of an inversion must be an assessment of the variety of possible solutions. We present a method for obtaining bounds on the earth's conductivity in a way that permits the natural introduction of inequality constraints into the computational framework. The inequality constraints are then used to further narrow the bounds. Our method is based completely on optimization theory for an all-in-one approach in inverting frequency-domain electromagnetic data. The forward modeling equations are constraints in an optimization problem solving for the electric fields and the conductivity simultaneously. There is no regularization. We draw conclusions from the data. There are two complementary ways in which we can learn about the conductivity of the deep interior: the first is inversion of electromagnetic observations made at the surface, the second is by studying the properties of likely materials in the laboratory. There is broad agreement between the results of the two strategies, but whether that agreement is adequate or not must depend on the uncertainty ascribed to the conductivity models, something known only poorly if at all. We can provide a credible assessment of those uncertainties to find the reliable features of the electromagnetic response function.