Fusion Based Dynamic Platform Magnetic Compensation Beyond the Tolles Lawson Approach

Rong Yang; Yaakov Bar-Shalom

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Fusion Based Dynamic Platform Magnetic Compensation Beyond the Tolles Lawson Approach

Rong Yang, Yaakov Bar-Shalom

arXiv:2608.26525v1Aug 27, 2026eess.SY

TL;DR

Aeromagnetic compensation must estimate the dynamic external field despite platform interference, while existing approaches provide limited real-time field estimation and may require map or position aiding. The paper combines augmented linear calibration with map-less multi-sensor KFF, achieving a fused average RMSE of 12.3 nT and robustness to noisy channels.

Problem
Existing approaches focus primarily on calibration and sensor-specific compensation, with many online methods relying on map-aiding or platform position feedback.
Method
The framework uses augmented linear calibration followed by a map-less multi-sensor Kalman filter fusion scheme that jointly estimates the external field and compensates multiple sensors.
Results
12.3 nT fused average RMSE was achieved on the DAF-MIT MagNav dataset, despite relatively noisy sensors remaining in the fusion.
Takeaways & Limitations
The KFF estimator can prevent highly corrupted individual sensor channels from degrading global external-field tracking performance.

Abstract

from arXiv · show

Aeromagnetic compensation for airborne platforms is essential for real-time tracking of the dynamic external magnetic field free from platform interference, enabling robust magnetic navigation (MagNav), magnetic anomaly detection (MAD), and other geophysics applications. While the classical Tolles-Lawson (TL) framework focuses extensively on estimating compensation model parameters, it pays less attention to the real-time estimation of the dynamic external magnetic field. To bridge this gap, this paper proposes a two-stage calibration and compensation framework. First, an augmented linearized model is developed for sensor-based calibration parameter estimation, avoiding the information loss inherent in classical band-pass filtering (BPF). Second, utilizing these pre-estimated parameters, a map-less Kalman Filter Fusion (KFF) algorithm is developed to dynamically estimate the external field and compensate the platform interference directly from multi-sensor measurements corrupted by platform interference. The second step enables real-time joint dynamic estimation and multi-sensor compensation without requiring reference position data (which runs counter to the MagNav purpose) or prior anomaly maps. Validated on the public DAF-MIT MagNav dataset, the framework overcomes the non-causal limitations of existing approaches, reducing average compensation error from 359.2~nT to 12.3~nT while maintaining exceptional robustness against heavily corrupted sensor channels.

I. INTRODUCTION

Aeromagnetic compensation must remove platform interference while preserving the dynamic external field. The paper addresses gaps in classical TL compensation through augmented calibration and map-less multi-sensor KFF estimation.

Airborne magnetometer measurements are corrupted by interference from ferromagnetic materials, electronics, electrical currents, and aircraft maneuvers.
Classical Tolles–Lawson approach: The classical TL approach models permanent, induced, and eddy-current interference components and estimates calibration parameters efficiently using linear least squares.
Classical Tolles–Lawson approach: TL estimation is generally ill-conditioned, with practical calibration-matrix condition numbers reaching up to 10^7.
Research gap: Existing methods emphasize calibration and per-sensor interference removal while giving less attention to dynamic external-field estimation and explicit multi-sensor fusion.
Proposed framework: The proposed framework uses augmented linear calibration followed by map-less multi-sensor KFF for joint real-time external-field estimation and compensation.
Proposed framework: The framework avoids prior magnetic anomaly maps, IGRF projections, and external position aiding, while addressing information loss from conventional band-pass filtering.

III. LIMITATIONS OF THE TOLLES–LAWSON APPROACH

The TL framework has inherent limitations that become evident when its assumptions are evaluated against real aircraft magnetic-interference data.

The DAF-MIT MagNav dataset reveals a mismatch between theoretical TL assumptions and actual aircraft interference fields.

A. Lack of Real-Time Compensation Capability

Classical compensation is not rigorously suited to real-time operation and can structurally mismatch filtered measurements. Simulation shows that band-pass filtering suppresses meaningful low-frequency platform interference.

Lack of Real-Time Compensation Capability: Classical TL compensation can rely on a batch mean over an entire flight segment, creating non-causal dependence on future temporal data.
Structural Mismatch and Information Loss from Band-Pass Filtering: Band-pass filtering removes meaningful low-frequency platform-interference components, including permanent and induced fields.
Structural Mismatch and Information Loss from Band-Pass Filtering: Applying the unfiltered TL model directly to filtered data introduces a fundamental structural mismatch.
Structural Mismatch and Information Loss from Band-Pass Filtering: In the flight-1002.02 simulation, filtered total-field variation severely deviated from the true platform-interference target, confirming erroneous suppression of critical low-frequency interference.

C. Modeling Error from External Field Proxy Substitution

The classical TL formulation substitutes the corrupted total measured field for the unknown external field, introducing modeling error and biasing interference-parameter estimates. The proposed calibration instead models sensor-specific effects, including explicit biases and external-field projections, while linearization improves estimation accuracy.

Modeling error: The classical TL model uses total measured field as a proxy for the unknown external field, introducing non-negligible modeling error.Because the total field includes platform interference, the proxy does not equal the true driving field.
Modeling error: The proxy substitution biases the estimated aircraft interference parameters.
Sensor-specific calibration: Separate calibration parameter vectors are required because sensors at different physical locations experience distinct local magnetic environments.
Augmented model: An explicit bias term β is added because persistent magnetometer biases remain unmodeled by classical TL estimation.The augmented vector jointly represents this bias with interference parameters.
Estimation: The linearized scalar-intensity formulation yields significantly higher calibration-parameter estimation accuracy than the vector model.The vector approach is degraded by higher matrix conditioning and larger three-dimensional measurement errors.

V. EXTERNAL MAGNETIC FIELD DYNAMIC ESTIMATION WITH FUSION

The KFF method dynamically estimates the external magnetic-field magnitude by fusing multiple calibrated magnetometer measurements in a linear Kalman-filter framework. It uses at least one vector magnetometer and does not require IGRF or anomaly-map information.

Dynamic estimation: KFF fuses multi-sensor measurements to track the external magnetic-field magnitude in real time.Calibration parameters are estimated beforehand for each scalar or vector magnetometer.
Sensor fusion: The method requires at least one vector magnetometer to project scalar field states into sensor-frame vector components.
Measurement model: The fused measurement vector stacks measurements from all ns magnetometers, using vector-magnetometer norms when applicable.
Kalman filtering: A nearly constant-rate dynamic model and linear measurement model permit recursive estimation with a standard Kalman filter.The process and measurement noises are modeled as Gaussian with covariances Q and Ri.
Limitations: Errors in prior calibration parameters and directional-cosine measurements are not explicitly modeled in Q and R.The formulation assumes these unmodeled sources have a minor impact; addressing non-negligible effects is left for future study.

VI. EXPERIMENTAL RESULTS ON REAL DATA

The real-data evaluation uses the DAF-MIT MagNav dataset to assess both calibration-parameter estimation and dynamic external-field tracking. The validation proceeds in two stages using the proposed linearized calibration and dynamic state-estimation models.

Evaluation: The proposed algorithms are evaluated on real flight data from the DAF-MIT MagNav dataset.
Two-stage validation: Validation first estimates calibration parameters, then uses them to track the external magnetic field with the proposed dynamic state-estimation model.

A. Experimental Setup and Dataset Description

The study analyzes Flight 1002 from the public DAF-MIT MagNav dataset, using a dynamic calibration trajectory and multiple remaining trajectories for validation. The aircraft carried scalar and vector magnetometers, with specified data exclusions and a reference signal for evaluation.

Dataset: Flight 1002 was collected over Ottawa using a Cessna Grand Caravan and contains multiple trajectories with varying flight profiles.
Instrumentation: The instrumentation included five scalar magnetometers and four three-axis vector fluxgate magnetometers.
Calibration trajectory: Trajectory 1002.02 served as the calibration path, forming a closed box-like loop at nominal altitude 3000 m with dynamic pitch, roll, and yaw excitation.
Validation trajectories: Fourteen remaining trajectories were used to evaluate calibration and dynamic external-field-intensity estimation across varying flight dynamics and environmental conditions.

B. Calibration parameter Estimation Test

The augmented calibration framework estimates sensor-specific parameters using vector measurements and regularized least squares, then validates them through compensated-field RMSE and trajectory comparisons. Despite severe ill-conditioning and calibration-dependent parameter variation, forward compensation remains accurate, while cross-flight degradation motivates online adaptation.

Calibration setup: The augmented regressor requires sensor-frame orientation information and uses direct vector magnetometer mapping to avoid synthetic projection errors.The offline estimation also uses the reference external field magnitude from mag_1_c because no anomaly map was available for this dataset.
Calibration setup: The regularized LS formulation is used because the regressor condition number reaches approximately 10^7, causing numerical instability and overfitted standard-LS solutions.The parameter vector itself cannot be directly checked because its true physical values are unknown; validation therefore uses compensated-residual RMSE.
Calibration results: The compensated profiles for mag_3, mag_4, and mag_5 match the reference ground truth on calibration trajectory 1002.02 despite raw interference of several thousand nanoTeslas.The profiles compare raw measurements, the mag_1_c reference baseline, and augmented-model compensation during intense maneuvering.
Calibration results: Calibration on trajectory 1002.02 yields the smallest RMSE, whereas trajectory 1002.15 shows larger errors that mainly appear as small, bounded biases.The degradation reflects physical platform-environment changes and mathematical bias across flight paths.
Calibration limitations: Using trajectory 1002.20 for calibration produces similar average sensor RMSE but shifts errors across trajectories, demonstrating overfitting to the specific calibration dataset.Parameter estimates from trajectories 1002.02 and 1002.20 differ substantially, consistent with ill-conditioned estimation.
Calibration limitations: The framework preserves overall compensation accuracy under severe parameter-estimation instability, but real-time parameter adaptation remains necessary and is outside this paper’s scope.Offline calibration assumes a reliable anomaly map or equivalent reference and isolation from unmapped dynamic magnetic objects.

C. External Field Be Dynamic Estimation Test

The dynamic-estimation test evaluates map-less KFF using pre-estimated parameters and multi-sensor measurements across 15 flight trajectories. KFF provides fused, real-time external-field estimates, outperforming classical TL and single-sensor AUG while remaining robust to corrupted channels.

Evaluation setup: The KFF evaluation fuses mag_3, mag_4, and mag_5 using calibration parameters estimated from trajectory 1002.02 and compares estimates against mag_1_c ground truth.Testing uses a 0.1 s interval with specified process and sensor measurement-noise covariances.
Benchmark comparison: Unlike TL and AUG, KFF generates a fused estimate from all three sensors, whereas the benchmarks report sensor-by-sensor averages.The comparison evaluates bias error, bias-removed standard deviation, and total RMSE.
Benchmark comparison: 359.2 nT total RMSE is reported for classical TL, including a 356.9 nT bias error, while the three-sensor KFF fused error stabilizes at 12.3 nT.KFF combines mag_5 at 11.6 nT with heavily disturbed mag_3 and mag_4 at 90.6 nT and 80.9 nT, respectively.
Sensor configurations: Single-sensor KFF tracking remains close to idealized AUG accuracy without using AUG’s true external-field baseline from mag_1_c.The trajectory plots also show KFF tracking the ground truth almost perfectly across calibration and worst-case trajectories.
Sensor configurations: 60.5 nT combined RMSE is obtained for dual-sensor mag_3, mag_4 fusion, demonstrating accuracy improvement over both individual sensors.The reported findings characterize KFF as robust multi-sensor state tracking and fusion.

VII. OTHER POSSIBLE APPROACHES

The section evaluates nonlinear and alternative calibration approaches against the proposed linearized approach. Severe ill-conditioning makes iterated least squares less reliable despite reducing measurement residuals.

The alternative methods were theoretically attractive, but experiments found none outperformed the proposed linearized approach.
Lower measurement residual RMS from iterated least squares does not translate into better compensation accuracy than linear least squares.The nonlinear iterations increasingly fit the estimation dataset rather than improving parameter accuracy.
The ill-conditioning is driven mainly by the flight trajectory's maneuvering profile, although iterated least squares is more sensitive because of repeated inversions.The evaluated trajectory still produced poor observability despite following conventional calibration guidelines.

B. Vector Model for Calibration Parameter Estimation

The vector calibration model uses all three vector-magnetometer components without scalar linearization, but it performs worse than the proposed scalar model on trajectory 1002.02. The broader framework combines linearized scalar calibration with robust multi-sensor KFF compensation.

Vector Model for Calibration Parameter Estimation: The vector model incorporates all three magnetic-field components simultaneously, increasing observations and avoiding scalar-model linearization.Its parameters are estimated using regularized least squares.
Vector Model for Calibration Parameter Estimation: The vector-model and scalar-model estimators were compared on flux_b, flux_c, and flux_d using flight trajectory 1002.02.The corresponding results are presented in Figures 9 and 10 and summarized in Table VIII.
Vector Model for Calibration Parameter Estimation: The scalar model consistently outperforms the vector model, reducing RMSE by more than 86% for flux_c and flux_d.
Vector Model for Calibration Parameter Estimation: The vector model has poorer numerical stability and noisier individual channels, with a condition number of 10^8 versus 10^7 for the scalar model.
Framework Implications: The framework's KFF stage fuses multiple sensors to jointly estimate the external field and compensate platform interference in real time.The conclusions report robust compensation despite ill-conditioned calibration and corrupted channels.
Framework Implications: Future work targets real-time online calibration to address parameter overfitting to calibration data and parameter drift over long flights.Optimized flight paths are also planned to improve parameter excitation and observability.

APPENDIX A DETAILED RESULTS OF THE KFF

The appendix reports detailed KFF evaluation metrics across fifteen DAF-MIT MagNav flight trajectories. The proposed stages substantially reduce average compensation RMSE and remain robust to noisy sensors.

The appendix reports bias, bias-removed error standard deviation, and RMSE for classical TL, augmented calibration, and KFF across fifteen trajectories.
356.9 nT average RMSE occurs for classical TL on individual sensors, particularly mag_3 and mag_4.Large DC biases omitted by band-pass filtering drive this degradation and cannot be recovered during real-time compensation.
359.2 nT to 58.0 nT is the reduction in total average RMSE after Stage 1 augmented calibration across all sensors.The augmented model removes most of the previously unmodeled sensor biases.
12.3 nT is the final fused average RMSE from KFF, using clean mag_5 characteristics despite relatively noisy mag_3 and mag_4.The result demonstrates robustness against highly corrupted individual sensor channels.