Obtaining vector magnetic field maps from single-component measurements of geological samples

Maxwell's equations can be used to demonstrate that the components of a static magnetic field in a region of space devoid of sources are not independent. This means that magnetometers that measure a single component of the magnetic field can potentially obtain all of three components of the field external to a source. Here we present an improved technique in the Fourier domain which can obtain the complete vector field planar map from just the planar map of one component. This technique is fast, robust, does not rely on any specific source type or configuration, and does not require the formulation of an inverse problem. An in-depth analysis of the advantages and shortcomings of the technique is presented, demonstrating that high-quality three-component field maps with virtually no information loss can be obtained when proper sensor and mapping configurations are used. Several results derived from both synthetic and experimental data are presented. In particular, practical cases are shown where vector maps can assist the analysis of magnetic properties of geological samples. MATLAB (R) routines implementing the basic vector map calculation algorithm are available as auxiliary materials and can be readily adapted for processing magnetic data obtained from a variety of magnetic sensors.