The adjoint-state method for the downward continuation of the geomagnetic field

hagedoorn, j. m. ; martinec, z.;

The downward continuation of the observed geomagnetic field from the Earth’s surface to the core–mantle boundary (CMB) is complicated due to induction and diffusion processes in the electrically conducting Earth mantle, which modify the amplitudes and morphology of the
geomagnetic field. Various methods have been developed to solve this problem, for example, the perturbation approach by Benton & Whaler, or the non-harmonic downward continuation by Ballani et al.
In this paper, we present a new approach for determining the geomagnetic field at the CMB by reformulating the ill-posed, one-sided boundary-value problem with time-variable boundary-value function on the Earth’s surface into an optimization problem for the boundary
condition at the CMB. The reformulated well-posed problem is solved by a conjugate gradient technique using the adjoint gradient of a misfit. For this purpose,we formulate the geomagnetic adjoint-state equations for efficient computations of the misfit gradient. Beside the theoretical description of the new adjoint-state method (ASM), the first applications to a global geomagnetic field model are presented. The comparison with other methods demonstrates the capability of the new method to determine the geomagnetic field at the CMB and allows us to investigate the variability of the determined field with respect to the applied methods. This shows that it is necessary to apply the ASM when investigating the effect of the Earth’s mantle conductivity because the difference between the results of approximate methods (harmonic downward continuation, perturbation approach) and the rigorous ASM are of the same order as the difference between the results of the ASM applied for different mantle conductivities.