The magnetotelluric inverse problem

The nonlinear inverse problem of electromagnetic induction to recover electrical conductivity is examined. As this is an ill-posed problem based on inaccurate data, there is a strong need to find the reliable features of the models of electrical conductivity. By using optimization theory for an all-at-once approach to inverting frequency-domain electromagnetic data, we attempt to make conclusions about Earth structure under assumptions of one-dimensional and two-dimensional structure. The forward modeling equations are constraints in an optimization problem solving for the fields and the conductivity simultaneously. The computational framework easily allows additional inequality constraints to be imposed.;Under the one-dimensional assumption, we develop the optimization approach for use on the magnetotelluric inverse problem. After verifying its accuracy, we use our method to obtain bounds on Earth's average conductivity that all conductivity profiles must obey. There is no regularization required to solve the problem. With the emplacement of additional inequality constraints, we further narrow the bounds. We draw conclusions from a global geomagnetic depth sounding data set and compare with laboratory results, inferring temperature and water content through published Boltzmann-Arrhenius conductivity models.;We take the lessons from the 1-D inverse problem and apply them to the 2-D inverse problem. The difficulty of the 2-D inverse problem requires that we first examine our ability to solve the forward problem, where the conductivity structure is known and the fields are unknown. Our forward problem is designed such that we are able to directly transfer it into the optimization approach used for the inversion. With the successful 2-D forward problem as the constraints, a one-dimensional 2-D inverse problem is stepped into a fully 2-D inverse problem for testing purposes. The computational machinery is incrementally modified to meet the challenge of the realistic two-dimensional magnetotelluric inverse problem. We then use two shallow-Earth data sets from different conductivity regimes and invert them for bounded and regularized structure.