Anderson function

Overview

Anderson Functions describe the projection of a magnetic field in a given direction at points along a line passing a single magnetic dipole. They are useful in the study of magnetic anomaly detection, with historical applications in submarine hunting and underwater mine detection. They approximately describe the signal detected by a total field sensor as the sensor passes by a target (assuming the targets signature is small compared to the Earth's magnetic field).

Definition

The magnetic field from a magnetic dipole along a given line, and in any given direction can be described by the following basis functions:

${\frac {\theta ^{~i-1}}{(\theta ^{2}+1)^{\frac {5}{2}}}},{\text{ for }}i=1,2,3$

which are known as Anderson functions.

Definitions:

• ${\vec {m}}$ is the dipole's strength and direction
• ${\vec {B}}_{E}$ is the projected direction (often the earth's magnetic field in a region)
• $x$ is the position along the line
• ${\hat {v}}$ points in the direction of the line
• ${\vec {r}}$ is a vector from the dipole to the point of closest approach (CPA) of the line
• $\theta =x/r$, a dimensionless quantity for simplification

The total magnetic field along the line is given by

$B(\theta )={\frac {\mu _{0}}{4\pi }}{\frac {|m|}{r^{3}}}\left({\frac {A_{1}}{(\theta ^{2}+1)^{\frac {5}{2}}}}+{\frac {A_{2}\theta ^{1}}{(\theta ^{2}+1)^{\frac {5}{2}}}}+{\frac {A_{3}\theta ^{2}}{(\theta ^{2}+1)^{\frac {5}{2}}}}\right)$

where $\mu _{0}$ is the magnetic constant, and $A_{1,2,3}$ are the Anderson coefficients, which depend on the geometry of the system. These are

${\begin{aligned}A_{1}&=3({\hat {m}}\cdot {\hat {r}})({\hat {r}}\cdot {\hat {B}}_{E})-~~({\hat {m}}\cdot {\hat {B}}_{E})\\A_{2}&=3({\hat {m}}\cdot {\hat {r}})({\hat {v}}\cdot {\hat {B}}_{E})+3({\hat {m}}\cdot {\hat {v}})({\hat {r}}\cdot {\hat {B}}_{E})\\A_{3}&=3({\hat {m}}\cdot {\hat {v}})({\hat {v}}\cdot {\hat {B}}_{E})-~~({\hat {m}}\cdot {\hat {B}}_{E})\end{aligned}}$

where ${\hat {m}},{\hat {r}},$ and ${\hat {B}}_{E}$ are unit vectors (given by ${\frac {\vec {m}}{|{\vec {m}}|}},{\frac {\vec {r}}{|{\vec {r}}|}},$ and ${\frac {{\vec {B}}_{E}}{|{\vec {B}}_{E}|}}$, respectively).

Note, the antisymmetric portion of the function is represented by the second function. Correspondingly, the sign of $A_{2}$ depends on how ${\vec {v}}$ is defined (e.g. direction is 'forward').

Total Field Measurements

The total field measurement resulting from a dipole field ${\vec {B}}_{D}$ in the presence of a background field ${\vec {B}}_{E}$ (such as earth magnetic field) is

${\begin{aligned}|B|&={\sqrt {{\vec {B}}_{D}+{\vec {B}}_{E})\cdot ({\vec {B}}_{D}+{\vec {B}}_{E})}}\\&=|B_{E}|{\sqrt {1+{\frac {2{\vec {B}}_{D}\cdot {\vec {B}}_{E}}{|B_{E}|^{2}}}+{\frac {{\vec {B}}_{E}^{2}}{|B_{E}|^{2}}}}}\\&\approx |B_{E}|+{\frac {{\vec {B}}_{D}\cdot {\vec {B}}_{E}}{|B_{E}|}},&|B_{E}|>>|B_{D}|\end{aligned}}$

which shows that the measured total is approximately just the sum of the magnitudes of the background field and the projection of the dipole field onto the background field (which can be described by the equations above). This is true so long as the dipole's contribution to the magnetic field is small in comparison to the background field in the region of interest.