Mapping 3-D mantle electrical conductivity from space: a new 3-D inversion scheme based on analysis of matrix Q-responses

We present a novel 3-D frequency-domain inversion scheme to recover 3-D mantle conductivity from satellite magnetic data, for example, provided by the Swarm mission. The scheme is based on the inversion of a new set of electromagnetic transfer functions, which form an array that we denote as matrix Q-response and which relate external (inducing) and internal (induced) coefficients of the spherical harmonic expansion of the time-varying magnetic field of magnetospheric origin. This concept overcomes the problems associated with source determination inherent to recent schemes based on direct inversion of internal coefficients. Matrix Q-responses are estimated from time-series of external and internal coefficients with a newly elaborated multivariate analysis scheme. An inversion algorithm that deals with matrix Q-responses has been developed. In order to make the inversion tractable, we elaborated an adjoint approach to compute the data misfit gradient and parallelized the numerical code with respect to frequencies and elementary sources, which describe the external part of the magnetic field of magnetospheric origin. Both parts of the scheme have been verified with realistic test data. Special attention is given to the issue of correlated noise due to undescribed sources.