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

TL;DR

This work addresses magnetic anomaly detection (MAD) under multipolar sources by introducing a direct multipolar signal-subspace framework and an analytical Multipolar Orthonormal Basis (MOBF) built from orthogonal polynomials. It reveals that finite truncations yield overlapping subspaces, motivating careful truncation order selection and the design of GLRT detectors that project observations onto well-conditioned bases. The paper derives the MOBF analytically, demonstrates its numerical robustness under sampling, and analyzes how the detector order $M$ relative to the source order $N$ affects detection performance via SNR and ROC analyses. It also proposes information-criterion-based methods (AIC/BIC) to select $M$ and discusses their trade-offs, supported by simulations of quadrupolar sources. Overall, the results offer physically grounded guidance for choosing truncation and receiver orders in multipolar MAD, with practical implications for robust, real-time detection systems.$

Abstract

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.

Figures (14)

• Figure 1: Geometry of the problem. Center of the source is located at $O$ and sensor's position $P$ moves along the dashed line. $D$ is the distance between the CPA and the source location, and is reached at time $t_0$ by the sensor.
• Figure 2: The set of basis functions $\mathcal{F}_2 = \{f_{2,n}\}_{n=0}^4$ (left) and set $\mathcal{G}_2 = \{g_{2,n}\}_{n=0}^4$ of MOBF (right).
• Figure 3: The set of basis functions $\mathcal{F}_3 = \{f_{3,n}\}_{n=0}^6$ (left) and set $\mathcal{G}_3 = \{g_{3,n}\}_{n=0}^6$ of MOBF (right).
• Figure 4: Relative error in terms of Frobenius distance between Gram matrix of $\boldsymbol{G}_N$ and identity under different multipolar order $N$. Left: w.r.t. $K$ where $u$ is taken in $\left[ -10 \, , \, 10 \right]$ ($R = 20$); Right: w.r.t. $R$ with constant sampling step $\dfrac{R}{K-1} = 2/100$.
• Figure 5: Frobenius distance $\varepsilon(\boldsymbol{E}_N)$ between Gram matrix of $\boldsymbol{E}_N$ and identity w.r.t. $N$, with respectively $\boldsymbol{E}_N = \boldsymbol{G}_N$, $\boldsymbol{F}_N^{\mathrm{gs}}$, $\boldsymbol{G}_N^{\mathrm{gs}}$ for parameters given in Table \ref{['Tab:params']}.