J = x H0 + Jr (1)

(Bossavit, 1998) where is magnetic susceptibility and H0 is the Earth’s magnetic field. The first term in (1), H0 , is the induced component of the magnetization and Jr is the remanent component. Remanent magnetization (or remanence) is a permanent magnetization that can be obtained by ferromagnetic material through several phenomena including thermo-, chemical and detrital remanence (Butler, 1992). Often, the remanence obtained in the past becomes oriented in a direction different from the Earth’s field today; this can occur through movement of the Earth’s magnetic poles or through tilting of the stratigraphic units containing the permanently magnetized material. Hence, the induced and remanent components can be oriented in different directions.

Typical magnetic inversion routines assume no remanent component exists and erroneous results can be obtained if this assumption is made incorrectly. To demonstrate this, consider the following synthetic. with an overlayed schematic showing the magnetization of the central body: the Earth’s field is oriented vertically downward and the remanent magnetization is oriented horizontally towards the east (right in this diagram) such that the total magnetization is oriented with a 45?dip. Other authors have approached the problem of remanence in magnetic inversions by assuming simple causative bodies with uniform magnetization directions (e.g. Choud-hury and Sarkar, 1990). Such an approach relies heavily on a priori information. Another option is to work with data that has minimal dependence on the magnetization direction: Shearer and Li (2004) invert total gradient data for the magnitude of the magnetization on a 3D mesh without knowing the direction of magnetization.

However, they still require a “nominal” magnetization direction within their forward modeling and this may introduce some error into their forward solution. In mineral exploration applications the remanence can be significant and the subsurface magnetization complicated; there may be different Earth regions containing quite different remanence. To approach such problems we consider the possibility of inverting magnetic data for the full, 3-component, magnetization vector. We investigate a simple, illustrative synthetic problem to improve our understanding of the inversion methods prior to applying them to real exploration problems.

Inversion for magnetization

To invert for magnetization we follow the methodology of Li and Oldenburg (1996, 2003) for inversion for susceptibility. The model region is split into an orthogonal 3D mesh of M rectangular prismatic cells, each with constant susceptibility.

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