Shared measurement errors through motion correction

A GPS error can affect several parts of this experiment at once: estimated heading, acceleration correction and the wind constraint. Those effects must travel through the calculation together. Counting them as independent errors can give the wrong uncertainty.

This research equation evaluates aircraft motion, the three Earth-model terms and fitted IMU drift at every GPS timestamp. It replaces interpolation of minute-level predictions. GPS and accelerometer inputs receive the existing 30-second filter; the complete prediction and observed gyro receive the same additional 30-second filter before minute averaging. Gaps and IMU turns sever support. All six checks retain the same 79 minutes and hold the saved fitted parameters fixed.

These are existing simulated IMU recordings following an observed airline route, not measurements from a real IMU. No new flight, fit, bootstrap, calibration or scientific decision was run.

The equation gives almost the same residuals as the earlier filtering study. The important change is uncertainty propagation: GPS contributes about 90% of the calculated variance under the present assumptions. Gyro errors account for almost all the rest. The accelerometer contribution and its cross covariance with the gyro are small in these particular recordings.

The predicted variation exceeds the observed residual variation. It is therefore not an established measurement of physical uncertainty. GPS accuracy fields are provisional error scales; seconds and some GPS channels are assumed independent. Real GPS errors can persist over time. Recovered mounting and forward directions are held fixed; calibration uncertainty, instrument systematics and model-dependent acceleration corrections remain open. Evaluating filtered nonlinear attitude also does not guarantee an exact physical observation equation.

IMU covariance preserves accelerometer/gyro correlations. Samples are matched one-to-one within one tenth of a sample period, within the same second and mounting interval. Unmatched samples remain represented in the mean-error scaling. Empirical covariance includes aircraft motion, so it is not device white-noise characterization. GPS velocity covariance preserves the north/east correlation arising from speed and course errors. Position marginals are provisional: treating horizontal accuracy as equal component scales is an explicit assumption.

The local fit-response calculation also preserves shared GPS errors in the physical wind/airspeed constraint. It describes how small input errors would move parameters in a linearized fit with the original weights and fixed penalties. No new optimum was fitted. This is sampling-error propagation, not a posterior interval, calibrated test, or proof of model identifiability.

Independent checks compare complete-equation perturbations with the chained input derivative, verify covariance components and positive semidefiniteness, and reproduce the parameter response by a separate least-squares solve. Relative discrepancies are at most 1.53×10⁻⁹ and 1.69×10⁻¹⁴, respectively. All 29 focused tests pass. These checks verify arithmetic, not the physical uncertainty assumptions.

Next test sensitivity to time-correlated GPS and IMU errors before choosing weights or defining further matched refits. Real-device characterization is still needed. Production analysis and decisions remain unchanged.

The documented corpus contains plan.json, six task-named case reports and NPZ archives, review.json, verification.json, status.json and run.lock. The initial strict equal-timestamp preflight failed before any case evaluation; its historical plan/status are preserved separately. The current plan freezes bounded time matching, source/input/checkpoint hashes and the numerical environment. Completed results are immutable; a resume check skipped all six cases. No worker remains active.

To resume an interrupted evaluation under unchanged source/input/environment:

env OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 MKL_NUM_THREADS=1 /home/sweisman/venv/bin/python analysis/tests/review_continuous_measurement_motion.py docs/observed-pair-replay-20261007/run docs/joint-acceleration-motion-20261007/refits --output docs/continuous-measurement-motion-20261007