Underground rock and iron structure warps Earth's magnetic field into a stable, mappable pattern of anomalies. A GPS-denied robot carrying a quantum (NV-diamond / SERF) magnetometer can localize itself by matching what it measures against a pre-surveyed magnetic anomaly map — no satellite signal required, and unlike inertial dead-reckoning the error does not grow without bound.
This is a particle filter (sequential Monte Carlo localization): a cloud of N pose hypotheses (particles) is propagated and re-weighted every step.
Predict: x_i ← x_i + Δu + N(0, σ_drift²) (odometry + slip noise)
Update: w_i ← exp( -(z - h(x_i))² / (2·σ_sensor²) ) (Bayes likelihood)
Estimate: x̂ = Σ w_i·x_i / Σ w_i (weighted mean)
Health: N_eff = 1 / Σ w_i² (effective sample size)
Resample: when N_eff falls below N/2, redraw particles proportional to w_i
- Predict — every particle is nudged by the same commanded motion as the real robot, plus independent random drift (wheel slip, no odometer is perfect). This alone makes the cloud spread out over time, exactly like pure dead-reckoning.
- Update — the real robot's noisy magnetometer reading z is compared to each particle's predicted field h(x_i) from the anomaly map. Particles sitting on a field value close to the measurement get a high weight; the rest are downweighted.
- Resample — periodically the filter discards low-weight particles and duplicates high-weight ones, concentrating computation where the true pose is likely to be.
Notice the cloud tightens fastest while crossing a steep magnetic gradient (a sharp anomaly edge) and drifts loosest over flat field regions — informativeness comes from the field's spatial variability, not its raw strength, exactly as it does for real magnetic-anomaly navigation systems used in submarines and GPS-denied aircraft.