On multipolar magnetic anomaly detection: multipolar signal subspaces, an analytical orthonormal basis, multipolar truncature and detection performance

In this paper, we consider the magnetic anomaly detection problem which aims to find hidden ferromagnetic masses by estimating the weak perturbation they induce on local Earth's magnetic field. We consider classical detection schemes that rely on signals recorded on a moving sensor, and modeling of the source as a function of unknown parameters. As the usual spherical harmonic decomposition of the anomaly has to be truncated in practice, we study the signal vector subspaces induced by each multipole of the decomposition, proving they are not in direct sum, and discussing the impact it has on the choice of the truncation order. Further, to ease the detection strategy based on generalized likelihood ratio test, we rely on orthogonal polynomials theory to derive an analytical set of orthonormal functions (multipolar orthonormal basis functions) that spans the space of the noise-free measured signal. Finally, based on the subspace structure of the multipole vector spaces, we study the impact of the truncation order on the detection performance, beyond the issue of potential surparametrization, and the behaviour of the information criteria used to choose this order.