Making sound inferences from geomagnetic sounding

We examine the nonlinear inverse problem of electromagnetic induction to recover electrical conductivity. As this is an ill-posed problem based on inaccurate data, there is a critical need to find the reliable features of the models of electrical conductivity. We present a method for obtaining bounds on Earth’s average conductivity that all conductivity profiles must obey. Our method is based completely on optimization theory for an all-at-once approach to 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 required to solve the problem. The computational framework easily allows additional inequality constraints to be imposed, allowing us to 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. If the upper mantle is assumed to be volatile free we find it has an average temperature of 1409–1539  ° C. For the top 1000 km of the lower mantle, we find an average temperature of 1849–2008  ° C. These are in agreement with generally accepted mantle temperatures. Our conclusions about water content of the transition zone disagree with previous research. With our bounds on conductivity, we calculate a transition zone consisting entirely of Wadsleyite has < 0.27  wt.% water and as we add in a fraction of Ringwoodite, the upper bound on water content decreases proportionally. This water content is less than the 0.4 wt.% water required for melt or pooling at the 410 km seismic discontinuity.