Anderson Function¶

Express the scalar projection of a magnetic dipole field sampled along a straight path as a linear combination of three dimensionless rational basis functions.

Version: v2 · 2026-09-19 · History
Domain-specific #: 1284
Domain group: Natural Sciences
Origin domain: Physics
Subdomain: Magnetometry → Physics
Aliases: Anderson basis functions, Magnetic-dipole Anderson functions

Core Idea¶

Anderson functions are three basis functions proportional to 1/(1+θ²)^(5/2), θ/(1+θ²)^(5/2), and θ²/(1+θ²)^(5/2). A scalar projection of a point magnetic-dipole field measured along a straight line can be written as their linear combination after distance along the path is normalized by closest-approach distance. The coefficients encode dipole orientation, path direction, closest-approach geometry, and projection direction.

For a total-field magnetometer in a much stronger background field, the small dipole anomaly is approximately the projection of the dipole field onto the background direction, plus an offset. The functions provide a compact signal-shape representation, not a universal detector: point-dipole validity, straight constant-geometry passage, weak-anomaly linearization, background stability, and sensor bandwidth delimit use.

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Three Magnet Bump Shapes

A magnet-sensor that moves past a small hidden magnet in a straight line draws a wiggly bump as it goes by. Anderson Functions are three standard bump shapes. Mix the right amounts of the three and you can match the wiggle, and the amounts tell you how the magnet is turned and how the sensor went past it.

Three Curves for a Magnet Pass

A small magnet makes an invisible magnetic field around it. If a sensor travels past it in a straight line, the reading goes up and down in a pattern. Anderson Functions are three fixed curve shapes, and that pattern can be written as a mix of the three. How much of each curve you need depends on which way the magnet points, which way the sensor travels, how close it gets, and which direction the sensor measures. They describe the shape of a signal; they are not by themselves a machine for finding hidden objects.

Dipole Pass Basis Functions

Anderson functions are three basis functions of a normalized distance θ: 1/(1+θ²)^(5/2), θ/(1+θ²)^(5/2), and θ²/(1+θ²)^(5/2). Here θ is your position along a straight path divided by the closest-approach distance to a point magnetic dipole. Any single-direction component (scalar projection) of the dipole's field measured along that path can be written as a weighted sum of these three functions. The weights encode the dipole's orientation, the direction of travel, the closest-approach geometry, and the projection direction. For a total-field magnetometer in Earth's much stronger field, the small anomaly is approximately the projection onto the background direction plus an offset, so it fits this form. The representation only holds if the source acts like a point dipole, the path is straight at constant geometry, and the anomaly is weak.

Anderson Functions are the three basis functions proportional to 1/(1+t^2)^(5/2), t/(1+t^2)^(5/2), and t^2/(1+t^2)^(5/2), with t the along-track distance normalized by the closest-approach distance. The scalar projection of a point magnetic-dipole field sampled along a straight line is exactly a linear combination of these three shapes; the coefficients encode dipole moment orientation, path direction, closest-approach geometry, and the projection direction. For a total-field magnetometer in a background field much stronger than the anomaly, the measured change in total field is, to first order, the projection of the dipole field onto the background field direction, plus an offset, which makes the decomposition applicable to such sensors. The functions therefore give a compact description of expected signal shape. Their validity is bounded by the point-dipole approximation, a straight passage at constant geometry, the weak-anomaly linearization, a stable background, and adequate sensor bandwidth. They are a mathematical, descriptive representation, not a complete target-detection procedure.

Scope of Application¶

Anderson functions are literal in magnetic-dipole line-scan modeling and benign magnetic-anomaly signal analysis under the stated geometric approximation.

Magnetometer calibration. Checking line-scan response to a known dipole-like source.
Geophysical reconnaissance. Fitting compact dipolar anomalies along a traverse.
Laboratory field mapping. Representing projected fields along controlled straight paths.
Sensor simulation. Generating idealized basis responses for algorithm validation.
Residual diagnosis. Detecting multipole, trajectory, background, or bandwidth departures.
Signal-feature study. Separating symmetric and antisymmetric components of a pass-by waveform.

Clarity¶

State the dipole approximation, path line and direction, closest-approach vector and distance, projection direction, normalized coordinate, coefficient convention, background-field magnitude, and whether the total-field linearization is used. Report residuals and do not infer unique source properties unless identifiability has been established separately.

Declare the dipole coordinate frame, the straight traversal line, the closest-approach distance, the direction of motion, and the measured projection axis.

Manages Complexity¶

Three fixed shapes compress a continuous pass-by waveform into a small coefficient vector and separate universal path geometry from orientation-specific weights. This supports fitting and model checking. The compression hides extended sources, curved trajectories, uncertain speed, colored background, and sensor response; a good fit is evidence for adequacy within the model, not proof of a unique source.

Abstract Reasoning¶

Fix a dipole source and scalar projection model.
Parameterize the straight path about closest approach.
Normalize displacement by closest-approach distance.
Evaluate the three basis functions over θ.
Map orientation geometry into coefficients.
Fit coefficients and offset to the observed scalar trace.
Check weak-anomaly and sensor-response assumptions.
Interpret residuals and identifiability before assigning source parameters.

Knowledge Transfer¶

The strict parent is Projection: a vector dipole field is mapped onto one observed direction along a lower-dimensional path, discarding unobserved components. Basis representation is related, but the projection operation generates the scalar signal that the basis spans.

Projection is the strict parent because the measured scalar is obtained by resolving a vector magnetic field onto a declared observation direction. The transferable pattern is vector field + observation path + projection axis -> scalar basis expansion. The Anderson residual fixes a dipole field, straight-line geometry, nondimensional coordinate, and three rational basis functions.

Relationships to Other Abstractions¶

Current abstraction Anderson Function Domain-specific

Parents (1) — outgoing typed relations

Anderson Function is a kind of Basis Prime

The accepted reference-grade review places Anderson Function under Basis because the child instantiates or depends on the parent's broader structure while retaining its own constitutive identity.

Hierarchy path (1) — routes to 1 parentless root

Anderson Function → is a kind of Basis → is a kind of Set and Membership

Neighborhood in Abstraction Space¶

Anderson Function sits in a sparse region of the domain-specific corpus (93^rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

Ziggurat Algorithm — 0.79
Curvelet Transform — 0.79
Blind deconvolution — 0.78
Seismic wide-angle reflection and refraction — 0.77
Variogram — 0.77

Computed from structural-signature embeddings · 2026-10-25