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').

Also note, some sources do not have the minus sign between terms for $A_{2}$.