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: v1 · 2026-08-30 · History
Domain-specific #: 1284
Origin domain: physics
Subdomain: magnetometry
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.^[1]

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. This entry is mathematical and descriptive, not an operational target-detection procedure.

Structural Signature¶

The point-dipole source model. A magnetic moment generates the ideal dipolar field.
The straight sampling line. Sensor position varies along one declared path direction.
The closest-approach geometry. A perpendicular offset supplies the distance scale.
The normalized coordinate θ. Along-path displacement is divided by closest-approach distance.
The projection direction. A scalar sensor observes one component of the vector field.
The three rational basis shapes. Even and odd functions span the ideal line-sampled projection.
The geometry coefficients. Dot products encode orientations without changing basis shape.
The weak-anomaly linearization. Total-field magnitude reduces approximately to background plus projected anomaly.
The residual test. Departure from the span diagnoses model or background mismatch.

What It Is Not¶

Not the Anderson localization model. The shared name is unrelated.
Not a general magnetic-field solution. The basis assumes a dipole and line sampling geometry.
Not exact total-field linearization for a large anomaly. Small signal relative to background is required.
Not a unique source inversion by itself. Coefficient and geometry ambiguities can remain.
Not insensitive to path direction. Reversing the coordinate changes the odd component's sign.
Not an operational detection workflow. It is a signal representation requiring separate sensing, noise, and inference design.

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. Normalize the along-track coordinate and identify the three basis functions before estimating coefficients. The coefficients encode components of one magnetic dipole under the assumed geometry; they are not arbitrary curve-fitting weights with independent physical meanings. A change of origin, sign convention, path direction, or sensor axis changes the coefficient interpretation even when the sampled curve looks similar. State whether the background field has been removed and whether the far-field dipole approximation is adequate. If multiple sources, extended magnetization, curved motion, or attitude variation are material, the three-function form may fit locally while losing its claimed identity.^[1]

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.

The function reduces a vector field sampled along a path to a small linear basis. Geometry supplies nonlinear-looking rational dependence on position, while the unknown dipole contribution enters through three scalar coefficients. This separation makes least-squares estimation and comparison tractable and reveals which parts of the signal are identifiable from the chosen path. It also exposes degeneracy: closely correlated basis columns, a short traversal interval, uncertain closest approach, or a strong slowly varying background can make coefficients unstable. Residual structure should be inspected by position, and conditioning should accompany the fitted values. A low residual is insufficient if different geometric assumptions produce radically different dipole estimates. The abstraction manages the forward model; it does not guarantee a unique or physically valid inverse solution.

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. Generic polynomial regression or Fourier decomposition can approximate the same trace but does not preserve that mechanism. Likewise, the name should not be conflated with Anderson functions in unrelated mathematics or physics. Transfer to another sensor is literal only when its measurement is the same projection under compatible geometry; using three convenient shapes without the dipole derivation creates a new empirical model.

Examples¶

Canonical¶

A scalar sensor traverses a straight line past an ideal magnetic dipole. Writing along-track displacement as θ times closest-approach distance factors the trace into three shared rational shapes. Geometry-specific coefficients combine those shapes into symmetric and antisymmetric waveform components; reversing the path reverses the odd component's sign.^[1]

Mapped back: vector dipole field → straight-path restriction → directional projection → normalized three-function basis → scalar trace.

Applied / In Practice¶

In a controlled laboratory scan, a known magnetic dipole is measured along parallel tracks. Analysts fit the three-function span and compare residuals across offsets. Systematic residual lobes suggest that the source is not point-like or the platform path is curved, so the basis becomes a diagnostic model rather than an unquestioned source estimate.

A sensor traverses a nearly straight line past a compact magnetic source. After estimating closest approach and subtracting a background trend, the analyst constructs the three Anderson basis columns over normalized along-track position and fits their coefficients. The reconstructed trace is compared with held-out portions of the pass, while coefficient sensitivity is tested against plausible changes in path height and orientation. A systematic two-lobed residual suggests an extended or multiple source rather than permission to reinterpret one coefficient. A second pass on a different line can break degeneracy and test whether both traces correspond to one dipole. This workflow maps each fitted term back to geometry and projection, preserving the function's physical identity instead of treating it as a named generic regression curve.

Mapped back: controlled source/path → basis fit → coefficient comparison → structured residual → model refinement.

Structural Tensions¶

Compact basis vs. source realism. Three functions are efficient because the source is idealized. Diagnostic: Are residuals consistent with a point dipole?
Scalar convenience vs. vector information loss. One projection is easy to sense but discards components. Diagnostic: Are source parameters identifiable from the available path?
Universal shapes vs. geometric coefficients. Basis curves are fixed while orientations control weights. Diagnostic: Are coordinate and sign conventions explicit?
Weak-anomaly linearization vs. strong signals. First-order projection simplifies total field. Diagnostic: Is anomaly magnitude small relative to background?
Autonomous function family vs. generic projection. Projection travels; dipole line geometry defines Anderson functions. Diagnostic: Does the signal retain the three rational dipole basis shapes?

Structural–Framed Character¶

Anderson functions are structural-leaning. Maxwell dipole geometry and scalar projection are physical; coordinate direction, path parameterization, and basis normalization are conventions. It is evaluatively neutral. The construct remains domain-specific because it requires a magnetic dipole, straight-line sampling, and the specific rational basis.

The model's autonomy lies in the constrained basis, not in the surname or a single plotted curve. A diagnostic fit should compare the Anderson basis with an unconstrained smooth alternative and inspect whether the physically structured coefficients remain stable across the traversal window. If the unconstrained model improves fit only by absorbing background drift, preprocessing may be at fault; if it reveals repeatable structure orthogonal to all three basis functions, the dipole/path assumptions may be wrong. This comparison keeps model criticism separate from coefficient estimation and prevents a convenient three-term regression from masquerading as the derived magnetic function.

Structural Core vs. Domain Accent¶

The skeleton is rich field → path restriction + directional projection → low-dimensional basis expansion. The accent is dipole decay, closest-approach scaling, magnetic background, and three Anderson shapes. Removing them yields generic projection or basis fitting.

Projection is the strict parent because the observed trace is the dipole vector field projected onto a chosen direction and restricted to a line. Signal Extraction is related as a possible downstream use.

The prospective workspace queue contains one strict upward edge to prime:projection. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Anderson Function Domain-specific

Parents (1) — more general patterns this builds on

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 → Basis → Set and Membership

Neighborhood in Abstraction Space¶

Anderson Function sits in a sparse region of the domain-specific corpus (92^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Anderson localization. Wave localization in disordered media.
Magnetic dipole field. The full vector field before path restriction and projection.
Matched filter. A detection operator that may use a template but is not the basis family itself.
Multipole expansion. A broader source-field representation beyond a single dipole.
X-ray transform. Line integrals of a field rather than pointwise values sampled along a line.

References¶

[1] Edward P. Loane, Speed and Depth Effects in Magnetic Anomaly Detection, EPL Analysis technical report, 1976, DTIC accession ADA081329. registry ↩a ↩b ↩c