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

Abstract

S U M M A R YThe 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.

Gyres, jets and waves in the Earth’s core

“…The core-generated magnetic field and its time changes are today routinely monitored using measurements made by ESA's Swarm satellite mission 87,88 (Box 1), complemented by measurements at ground observatories [89][90][91][92] . On timescales of years and longer, for which the effects of electrical currents in the poorly conducting mantle are thought to be relatively small, the field and its time variations can be mapped down to the core surface 61,[93][94][95][96] (Fig. 1(a)).…”

Downward Continuation and Transformation of Total‐Field Magnetic Anomalies Into Magnetic Gradient Tensors Between Arbitrary Surfaces Using Multilayer Equivalent Sources

“…In past decades, airborne magnetic data sets have been built globally for various applications, such as geology studies, mineral exploration, and military detection (Kirkby & Duan, 2019; Krahenbuhl & Li, 2017; Liu et al, 2020; Wigh et al, 2020). Airborne surveys are flown with drape surfaces in most cases, and the acquired data sets are usually downward continued to other arbitrary surfaces close to the source (or ground) to increase the spatial resolution in the data interpretation for various applications (Fedi & Florio, 2002; Hagedoorn & Martinec, 2015; Plattner & Simons, 2017; Zhang et al, 2018). For example, downward continuation is utilized to generate high‐resolution magnetic data according to a large‐scale geophysical database in advanced GPS‐independent, all‐weather, passive navigation for air, ground, and underwater crafts (Zeng et al, 2013).…”