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

In Geomagnetism it is of interest to separate the 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 contribution 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 $\Phi_0$ and $\Phi_1$ generated on two different spheres $\mathbb{S}_{R_0}$ (crust) and $\mathbb{S}_{R_1}$ (core) with radii $R_1<R_0$. Although it is not possible in general to recover $\Phi_0$ and $\Phi_1$ knowing their superposition $\Phi_0+\Phi_1$ on a sphere $\mathbb{S}_{R_2}$ with radius $R_2>R_0$, we show that it becomes possible if the magnetization $\mathbf{m}$ generating $\Phi_0$ is localized in a strict subregion of $\mathbb{S}_{R_0}$. Beyond unique recoverability, we show in this case how to numerically reconstruct characteristic features of $\Phi_0$ (e.g., spherical harmonic Fourier coefficients). An alternative way of phrasing the results is that knowledge of $\mathbf{m}$ on a nonempty open subset of $\mathbb{S}_{R_0}$ allows one to perform separation.