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.
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.
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) — more general patterns this builds on
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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 (92nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Ziggurat Algorithm — 0.79
- Curvelet Transform — 0.79
- Variogram — 0.77
- Dim Spot — 0.77
- Finite Difference Coefficient — 0.77
Computed from structural-signature embeddings · 2026-09-08