19. The Forward and Adjoint Methods of Global Electromagnetic Induction forCHAMP Magnetic Data

Detailed mathematical derivations of the forward and adjoint sensitivity methods are presented for computing the electromagnetic induction response of a 2-D heterogeneous conducting sphere to a transient external electric current excitation. The forward method is appropriate for determining the induced spatiotemporal electromagnetic signature at satellite altitudes associated with the upper and mid-mantle conductivity heterogeneities, while the adjoint method provides an efficient tool for computing the sensitivity of satellite magnetic data to the conductivity structure of the Earth’s interior. The forward and adjoint initial boundary-value problems, both solved in the time domain, are identical, except for the specification of the prescribed boundary conditions. The respective boundary-value data at the satellite’s altitude are the X magnetic component measured by the CHAMP vector magnetometer along the satellite track for the forward method and the difference between the measured and predicted Z magnetic component for the adjoint method. Both methods are alternatively formulated for the case when the time-dependent, spherical harmonic Gauss coefficients of the magnetic field generated by external equatorial ring currents in the magnetosphere and the magnetic field generated by the induced eddy currents in the Earth, respectively, are specified. Before applying these methods, the CHAMP vector magnetic data are modeled by a two-step, track-by-track spherical harmonic analysis. As a result, the X and Z components of CHAMP magnetic data are represented in terms of series of Legendre polynomial derivatives. Four examples of the two-step analysis of the signals recorded by the CHAMP vector magnetometer are presented. The track-by-track analysis is applied to the CHAMP data recorded in the year 2001, yielding a 1-year time series of spherical harmonic coefficients. The output of the forward modeling of electromagnetic induction, that is, the predicted Z component at satellite altitude, can then be compared with the satellite observations. The squares of the differences between the measured and predicted Z component summed up over all CHAMP tracks determine the misfit. The sensitivity of the CHAMP data, that is, the partial derivatives of the misfit with respect to mantle conductivity parameters, are then obtained by the scalar product of the forward and adjoint solutions, multiplied by the gradient of the conductivity and integrated over all CHAMP tracks. Such exactly determined sensitivities are checked against the numerical differentiation of the misfit, and a good agreement is obtained. The attractiveness of the adjoint method lies in the fact that the adjoint sensitivities are calculated for the price of only an additional forward calculation, regardless of the number of conductivity parameters. However, since the adjoint solution proceeds backwards in time, the forward solution must be stored at each time step, leading to memory requirements that are linear with respect to the number of steps undertaken. Having determined the sensitivities, the conjugate gradient inversion is run to infer 1-D and 2-D conductivity structures of the Earth based on the CHAMP residual time series (after the subtraction of the static field and secular variations as described by the CHAOS model) for the year 2001. It is shown that this time series is capable of resolving both 1-D and 2-D structures in the upper mantle and the upper part of the lower mantle, while it is not sufficiently long to reliably resolve the conductivity structure in the lower part of the lower mantle.