Calibration of the pass-through magnetometer-I. Theory

Abstract

By studying a simple model of a pass-through magnetometer we show that there are circumstances in which misleading results might arise if the spatial sensitivity of the instrument is not properly corrected. For example, if the core sample is not correctly centred, or the magnetometer itself is misaligned, serious distortion can appear in the inferred inclination distribution. The possibility of such errors warrants a thorough study of laboratory instruments and, as a first step, we require a spatial calibration, that is, an estimate of the sensitivity of the various coils to samples placed anywhere in the sensing region. Only when this information is available for laboratory magnetometers will it be possible to calculate suitable corrections. The fact that laboratory magnetometers employ superconducting material makes inferring the response from the geometry of the coils impractical because the field from a specimen is modified inside the instrument by image currents flowing in the superconducting elements. To overcome this obstacle we treat a very general calibration problem. We show that the sensitivity of a particular coil as a function of position obeys Laplace's equation, and therefore the description in space of the sensitivity is mathematically exactly the same as modelling the geomagnetic field. A calibration experiment consists of several hundred measurements performed on a tiny dipole sample, systematically positioned throughout the sensing volume of the instrument. From such observations we aim to construct a harmonic interpolating function that represents the response in the measurement region. The natural geometry for the problem is that of a cylinder, so we work from the cylindrical harmonic expansion of an equivalent magnetic field. Cylindrical harmonic expansions take the form of an infinite set of unknown functions, not just a collection of coefficients as with spherical harmonics. To build a suitable interpolating function from them we appeal to the principles of spline interpolation by constructing a model that minimizes some measure of response complexity. We examine in detail two such measures. The first corresponds to magnetic field energy; the second is a more abstract norm that smoothes more heavily than the energy norm, and whose Gram matrix elements can be found without recourse to lengthy numerical procedures. The second norm promises to form the basis of a practical programme of calibration.