Small amounts of water can significantly affect the physical properties of mantle materials, including lowering of the solidus, and reducing effective viscosity and seismic velocity. The amount and distribution of water within the mantle thus has profound implications for the dynamics and geochemical evolution of the Earth. Electrical conductivity is also highly sensitive to the presence of hydrogen in mantle minerals. The mantle transition zone minerals wadsleyite and ringwoodite in particular have high water solubility, and recent high pressure experiments show that the electrical conductivity of these minerals is very sensitive to water content. Thus estimates of the electrical conductivity of the mantle transition zone derived from electromagnetic induction studies have the potential to constrain the water content of this region. Here we invert long period geomagnetic response functions to derive a global-scale three-dimensional model of electrical conductivity variations in the Earth's mantle, revealing variations in the electrical conductivity of the transition zone of approximately one order of magnitude. Conductivities are high in cold, seismically fast, areas where slabs have subducted into or through the transition zone. Significant variations in water content throughout the transition zone provide a plausible explanation for the observed patterns. Our results support the view that at least some of the water in the transition zone has been carried into that region by cold subducting slabs.

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SUMMARY The Jacobian of the non‐linear mapping from model parameters to observations is a key component in all gradient‐based inversion methods, including variants on Gauss–Newton and non‐linear conjugate gradients. Here, we develop a general mathematical framework for Jacobian computations arising in electromagnetic (EM) geophysical inverse problems. Our analysis, which is based on the discrete formulation of the forward problem, divides computations into components (data functionals, forward and adjoint solvers, model parameter mappings), and clarifies dependencies among these elements within realistic numerical inversion codes. To be concrete, we focus much of the specific discussion on 2‐D and 3‐D magnetotelluric (MT) inverse problems, but our analysis is applicable to a wide range of active and passive source EM methods. The general theory developed here provides the basis for development of a modular system of computer codes for inversion of EM geophysical data, which we summarize at the end of the paper.

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