Magnetic Anomaly Maps Backup Material

IEEE/ION PLANS 2023, Monterey, CA

Aaron Nielsen

The views expressed in this article are those of the author and do not necessarily reflect the official policy or position of the United States Government, Department of Defense, United States Air Force or Air University.

Distribution A: Authorized for public release. Distribution is unlimited. Case No. 2023-0277.

Scalar magnetic potentials

Scalar Magnetic Potential \(\Phi_M\)

  • If \(\vec{J} = 0\), then \(\nabla \times \vec{H} = 0\)
  • \(\vec{H}(\vec{r}) = -\nabla \Phi_M(\vec{r})\)
  • and since \(\nabla \cdot \vec{B} = 0 \rightarrow \nabla^2 \Phi_M(\vec{r}) = 0\)
  • LaPlacian equation: \(\nabla^2 \Phi_M(\vec{r}) = 0\)
  • Solutions are often referred to as harmonic (sins and cosins)
  • This situation holds for many airborne problems

(Blakely 1995)

Scalar Potential Theory

A procedure to transform magnetic scalar potential \(\Phi_M\) in free space.

From Laplacian we can write (Green’s third identity) \[ \Phi_M(\vec{r}) = \frac{1}{4\pi} \int_S \left( \frac{1}{r} \frac{\partial \Phi_M}{\partial n} - \Phi_M \frac{\partial}{\partial n} \frac{1}{r} \right) dS \]

  • No sources in region \(R\)
  • Know \(\Phi_M\) on the surface \(S\) that bounds \(R\)
  • We can find \(\Phi_M\) anywhere in \(R\)
  • (Blakely 1995)

Hemispherical geometry - flat Earth model

  • Think of Earth’s surface as an infinite flat plane, all sources below
  • Measure the potential on a plane above the earth
  • Other side of hemisphere is \(\infty\) away and therefore zero potential
  • Potential anywhere inside can be computed

Fourier methods for potential theory

  • Goal: find the potential \(\Phi_M\) on a plane parallel to a known plane
  • Process: double integral over the surface of the known plane
  • Ends up looking like a 2-D Fourier transform over spatial coordinates
  • (Blakely 1995)

ION/IEEE PLANS MagNav Tutorial 2023

Blakely, Richard J. 1995. Potential Theory in Gravity and Magnetic Applications. Cambridge University Press.