On the Recovery of Core and Crustal Components of Geomagnetic Potential Fields

In geomagnetism it is of interest to separate Earth’s core magnetic field from the crustal magnetic field. However, measurements by satellites can only sense the sum of the two contributions. In practice, the measured magnetic field is expanded in spherical harmonics, and separation into crust and core contributions is achieved empirically by a sharp cutoff in the spectral domain. In this paper, we derive a mathematical setup in which the two contributions are modeled by harmonic potentials Φ0 and Φ1 generated on two different spheres, SR0 (crust) and SR1 (core), with radii R1 < R0. Although it is not possible in general to recover Φ0 and Φ1 knowing their superposition Φ0 + Φ1 on a sphere SR2 with radius R2 > R0, we show that it becomes possible if the magnetization m generating Φ0 is localized in a strict subregion of SR0 . Beyond unique recoverability, we show in this case how to numerically reconstruct characteristic features of Φ0 (e.g., spherical harmonic Fourier coefficients). An alternative way of phrasing the results is that knowledge of m on a nonempty open subset of SR0 allows one to perform separation.