Analysis of lithospheric magnetization in vector spherical harmonics

Abstract

S U M M A R YThe lithospheric contribution to the geomagnetic field arises from magnetized rocks in a thin shell at the Earth's surface. The lithospheric field can be calculated as an integral of the distribution of magnetization using standard results from potential theory. Inversion of the magnetic field for the magnetization suffers from a fundamental non-uniqueness: many important distributions of magnetization yield no potential magnetic field outside the shell. We represent the vertically integrated magnetization (VIM) in terms of vector spherical harmonics that are new to geomagnetism. These vector functions are orthogonal and complete over the sphere: one subset (I) represents the part of the magnetization that produces a potential field outside the shell, the observed field; another subset (E) produces a potential field exclusively inside the shell; and a third, toroidal, subset (T ) produces no potential field at all. E and T together span the null space of the inverse problem for magnetization with perfect, complete data. We apply the theory to a recent global model of VIM, give an efficient algorithm for finding the lithospheric field, and show that our model of magnetization is dominated by E, the part producing a potential field inside the shell. This is largely because, to a first approximation, the model was formed by magnetizing a shell with a substantial uniform component by an potential field originating inside the shell. The null space for inversion of lithospheric magnetic anomaly data for VIM is therefore huge. It can be reduced if the magnetization is assumed to be induced by a known inducing field, but the null space for susceptibility is not so easily recovered.

A statistical spatial power spectrum of the Earth's lithospheric magnetic field

“…Gubbins et al 2011) due to source distributions that give no external field (the magnetic annihilators that were exemplified by Runcorn 1975). The deterministic studies explain many of the magnetic field structures observed at the satellite altitudes but they rarely account accurately for all the observed features in space.…”

On the unique reconstruction of induced spherical magnetizations

“…From Remark 2.2, it becomes clear that the spherical decomposition of Theorem 2.1 reveals the desired properties corresponding to the thin-plate case in Theorem 1.4(a),(b) (which has already been observed in [13]). …”

“…n,k (see, e.g., [1,13,18,19,22]). These vector spherical harmonics can be defined via a suitable connection to the inner harmonics H int n,k and the outer harmonics H ext n,k (i.e., the harmonic extensions of scalar orthonormalized spherical harmonics Y n,k into Ω int and Ω ext , respectively).…”

“…[4] for its Euclidean counterpart in R 2 ) and treat it in more detail later on. An illustration of this decomposition for recent magnetization models is supplied in [13]. Nonetheless, a characterization of m by (1.3) still states that the contributionsm (1) andm (3) cannot be reconstructed from knowledge of V in Ω ext R .…”

“…By dω we denote the surface element on the sphere Ω R . The non-uniqueness of recovering a vertically integrated magnetization m from the knowledge of V in Ω ext R can be characterized by a fairly well-known decomposition (see, e.g., [1,8,11,12,13,18,19,22]) m =m (1) +m (2) +m (3) , (1.3) which has the property that V ≡ 0 in Ω ext R if and only ifm (2) ≡ 0 (in other words, any magnetization of the form m =m (1) +m (3) produces no magnetic potential in the exterior Ω ext R ). We call such a decomposition a Hardy-Hodge decomposition (cf.…”