On the unique reconstruction of induced spherical magnetizations

Abstract

Abstract. Recovering spherical magnetizations m from magnetic field data in the exterior is a highly non-unique problem. A spherical Hardy-Hodge decomposition supplies information on what contributions of the magnetization m are recoverable but it does not supply geophysically suitable constraints on m that would guarantee uniqueness for the entire magnetization. In this paper, we focus on the case of induced spherical magnetizations and show that uniqueness is guaranteed if one assumes that the magnetization is compactly supported on the sphere. The results are based on ideas presented in [4] for the planar setting.

On the reconstruction of inducing dipole directions and susceptibilities from knowledge of the magnetic field on a sphere

“…Theorem 2.2) of the magnetization, then onlym (2) is determined uniquely. If we know in advance that m is localized in a subregion ⊂ S r , thenm (1) andm (2) are determined uniquely. In other words, the question we are interested in can be reformulated as follows: Knowing only the uniquely determined components of m, what can be said about d and Q?…”

“…If we make the additional assumption that Q is locally supported in some subregion ⊂ S r , then the susceptibility is actually determined uniquely (cf. [2], based on results from [3,4] in a Euclidean set-up). Therefore, in the latter scenario, but under the condition that the dipole direction is not known, our goal is to find suitable candidates for the dipole direction d. If the magnetization m was known, then a standard procedure such as described in [5,Chapter 7] can be used to derive d from the direction of m or to see that m cannot be of the form (1.1).…”

“…However, just given the corresponding magnetic potential V [m] on the sphere S R , only certain components of m can be reconstructed uniquely (cf. [2,3,6]; a summary is provided in Section 2). Namely, if m =m (1) +m (2) +m (3) (1.3)…”

“…The latter classifies those components of the magnetization m (not necessarily of the form (1.1)) that are determined uniquely by knowledge of V [m] on S R . Namely, if m =m (1) +m (2) +m (3) is the Hardy-Hodge decomposition, then onlỹ m (2) is determined uniquely (e.g. [2,3,6]; we say that m andm (2) are 'equivalent from outside').…”

“…Namely, if m =m (1) +m (2) +m (3) is the Hardy-Hodge decomposition, then onlỹ m (2) is determined uniquely (e.g. [2,3,6]; we say that m andm (2) are 'equivalent from outside'). We also formulate the Helmholtz and Hardy-Hodge decompositions in terms of some well-known vector spherical harmonics, which will be of use for our considerations on band-limited magnetizations.…”