Measuring the rotation field before interpreting its cause

Purpose and status

This is a proposed follow-on to the current NASA airborne analysis, to begin after its numerical work is complete and reviewed. It asks what persistent rotation the original IMU measurements support, independently of choosing a physical explanation for it. An IMU (inertial measurement unit) measures turning and acceleration. GPS records position and motion over the ground. Survey-grade hardware makes slow rotation a credible measurement target; this study will test the vector pattern rather than assume its rate.

This page records the scope and sequence. It introduces no code, fits or new campaign allowance. The completed globe/flat comparisons and solver trial stay intact. The analysis roadmap adds the numerical, calibration-boundary, absolute-fit and recording-dependence checks this estimator will also need. Exact device characterization remains parallel work; it does not hold up development of the field estimator.

Observation, model and explanation

Gyroscopes measure the instrument's angular motion relative to an inertial reference. They do not directly observe the Earth's orbit, the motion of celestial matter or a mechanism that creates that reference. Aircraft motion, transport and instrument errors must be separated before estimating a persistent background field.

The current archive comparison uses the historical labels “rotating globe” and “still globe” for curved-surface models with a specified background angular rate and with zero background rate, respectively. A preference for the former supports that observable prediction over the latter. It does not compare all physical explanations of the field. In particular, a stationary-Earth hypothesis that proposes a nonzero inertial field is not the zero-field candidate. It needs explicit equations and testable predictions.

This project reports surface geometry and angular motion separately from causal interpretation. It does not claim that a gyro signal alone proves heliocentrism, or that an alternative mechanism has been established. Alternative explanations are tested where they predict distinguishable observations; agreement on one field does not show agreement with all other physical evidence.

What to measure

Represent the background field in local north, east and down directions. Begin with three freely fitted coefficients, expressed in degrees/hour for reporting:

where \(\phi\) is latitude. Do not impose \(A=-C\), set \(B=0\), or fix a rate near 15 degrees/hour. Those are relationships to test afterward. This particular latitude form is itself a model; compare it with alternatives instead of calling it assumption-free.

The main questions are whether a nonzero field is needed, its vector direction and magnitude, its variation with latitude and altitude, and its consistency across speeds, headings, dates, flights and instrument configurations. Test globe transport separately: turning caused by travel over a curved surface changes with trajectory, whereas a geographically fixed background field should not depend on the aircraft's velocity.

Separate measured rotation from aircraft motion

The gyro records rotation in its own axes. GPS ground track does not determine aircraft roll, pitch or body heading; wind and maneuvers matter. Reconstruct orientation and motion jointly from original gyro/acceleration increments and receiver observations, allowing for mounting, calibration and timing errors. Accelerometers support that reconstruction; the rotation field remains the measurement target. Applanix Group-1 fused attitude and angular rates must not supply the independent observation or prescribe its background field.

Declare the geometry used to calculate transport rotation. Test its influence explicitly where globe/flat remains unresolved. Keep frame-rotation and motion equations consistent; do not hide a fixed standard Earth rate in orientation initialization or motion correction. Conventional inertial navigation uses prior Earth-rate and gravity information, which is precisely why the existing fused answer cannot perform this test for us.

Fit the field and instrument errors together rather than subtracting an assumed background first. Use the same permitted errors in every field comparison. An arbitrary independent gyro offset for every short stretch can absorb a background signal; quantify that ambiguity before interpreting any fitted coefficients. Share calibration only within supported instrument/installation epochs. Matching filenames or system identifiers is insufficient.

Unknown onboard Earth-rate subtraction creates another ambiguity: the recorded field can be estimated, but a freely fitted original rate and subtraction amount may cancel each other exactly. Report the field retained in the logged measurements first. Infer a pre-correction field only where documentation or independent constraints separate those quantities. Routine noise filtering and removal of a persistent signal are different operations; neither the word "raw" nor a successful fit establishes the logging details.

Competing field descriptions

Specify a small comparison set before new observed-data fitting:

A standard rotation-axis pattern is the restriction \(A=-C\), \(B=0\). Test that restriction with magnitude fitted freely, rather than assuming it in the primary estimate. Compare models using uncertainty-aware profiles and predictive performance on genuinely held-out flights or installation groups. Extra field parameters must earn their complexity; a lower training residual alone does not establish a better description.

Work sequence

1. Audit available coverage without gyro-based selection. Reuse the saved eligible stretches and hashes. Tabulate latitude, longitude, altitude, speed, heading diversity, recorded IMU configuration and overlapping flight time. Establish which altitude and geographic comparisons the archive can actually support. Any broader selection needs a separately versioned motion/completeness policy, fixed before inspecting field outcomes.
2. Build and test the field estimator separately. Use the existing native-increment integration architecture in analysis/lll/ilvis0_shape.py, ilvis0_forward.py and ilvis0_tangent.cpp as references, preserving their frozen versions. Test known injected vectors, zero field, axis/sign transformations, transport-only motion, constant-offset ambiguity and unknown-subtraction ambiguity with cheap deterministic controls.
3. Check separation before an observed trial. Profile field coefficients against orientation, calibration, timing and transport freedoms. Report estimable combinations and unresolved directions. Near-polar latitude coverage can leave the north coefficient weakly constrained; high sample counts do not replace geographic diversity.
4. Freeze a bounded trial. Fix eligible identities, field models, error assumptions, validation split, stopping checks and a finite approved evaluation allowance. Reuse whole stretches; retain every failed or unfinished fit. Do not automatically expand the run.
5. Report measured patterns. Publish coefficient estimates, uncertainty, observable dependencies, cross-instrument agreement and held-out prediction errors. Avoid counting overlapping streams, repeated fits or short diagnostic sections as independent flights. Only then compare the measured pattern with proposed physical explanations.

The first implementation should establish the common vector and latitude pattern. Add altitude, speed or date dependence only where coverage separates those effects. The next precision stage will test a sidereal-compatible rate against a mean-solar-day rate, as specified below. These are separate milestones: finding a persistent field does not by itself resolve the much smaller difference between those rates.

Precision stage: sidereal versus solar rate

Fit the rate first, then compare it with both targets. A mean solar day is 24 hours; a sidereal day is about 23 hours, 56 minutes and 4 seconds in ordinary clock time. USNO explains the distinction. These use different astronomical references: solar time follows the Sun, while sidereal time uses the equinox, approximately the stellar sky. Their difference is established astronomical timing; the gyro question is which rate the freely estimated field matches. The present short-stretch fits have not measured that difference. A candidate containing a fixed sidereal-compatible rate cannot establish it merely by fitting better than zero. The comparison targets are approximately:

Freeze the exact constants and time conventions when implementing the trial. The IAU nominal mean angular velocity is 7.292115 × 10⁻⁵ radians/second. Sidereal-time conventions also involve precession, nutation and the distinction between Earth-rotation time and uniform time; use the USNO definitions rather than treating a rounded sidereal-day duration as an exact physical constant.

The four-minute minimum used for the existing shape study accumulates only about 9.9 arcseconds of separation between these targets. One continuous hour accumulates about 148 arcseconds. Raw increment resolution is only one part of the error budget; fine quantization does not establish this level of measurement accuracy.

Selection and estimator

1. Inventory longer continuous records using receiver coverage, timing, motion and configuration stability, before looking at fitted rotation. Save a new selection version; leave the existing 53-stretch selection and findings unchanged. Include heading and latitude diversity as observable design properties. Model maneuvers explicitly where the observation equations support them. Join files only after verifying sample, timing and installation continuity.
2. First test the freely fitted field's direction and latitude pattern. Where the geographic axis description is supported, estimate one common unknown magnitude Ω jointly with transport, orientation and instrument errors. Do not fix Ω to either target during estimation, or insert the standard rate into initialization or correction. Keep the unrestricted vector fit as an adequacy check on this reduced description.
3. Profile Ω with nuisance parameters reoptimized at each value. Compare the two fixed-rate targets under exactly the same error assumptions. Qualify this estimator with deterministic known-rate controls, zero field and calibration/clock degeneracies before observed fitting. Establish uncertainty coverage and a decision rule separately; a lower cost alone is a numerical preference.
4. Estimate supported installation-level calibration jointly across records, permitting justified changes between epochs. Hold out whole flights or installations for prediction and replication. Overlapping instrument streams, aliases and sections of one recording do not increase the count of independent flights.

Required precision checks

The existing ±1°/hour offset sensitivity case is about 24 times the target separation. Heading changes can help separate a sensor-fixed offset from a geographic vector, but they do not automatically identify gyro gain. An unconstrained common gain can trade directly against Ω. Integration-clock scale can create the same problem. Report joint profiles, estimable combinations and sensitivity to justified gain, offset, mounting and clock bounds; do not tighten bounds merely until the desired rate becomes distinguishable.

Unknown upstream subtraction creates a separate ambiguity between the original rate and what was removed. Estimate the retained field for those streams. An original-rate comparison requires independently constrained subtraction, or a clearly stated retained-rate assumption whose influence is reported. Exact hardware identification can remain parallel work; quantitative calibration and processing uncertainty still belong in this precision test.

Longer averaging reduces random noise, not shared gain errors or changing offsets. For illustration only, an angular random walk of 0.02°/√hour would give a one-hour angle error of 0.02°, versus a target separation of 0.0411°. This ideal white-noise calculation is not a measured raw-sensor uncertainty and excludes aircraft reconstruction errors, correlation and drift. Measure those effects from controls and held-out residuals rather than assigning the whole archive this illustrative specification.

Report the free Ω estimate and uncertainty, both target profiles, boundary behavior, absolute adequacy and held-out results. If both targets remain plausible, report solar versus sidereal unresolved. If neither fits adequately, report that discrepancy. An interval excluding one target is useful only after its uncertainty and systematic-error sensitivity are validated. Freeze a bounded trial and obtain its compute allowance before launch; this roadmap adds no new numerical starts.

Rate is different from astronomical phase

This stage asks whether the measured angular rate corresponds to a sidereal-compatible period. A constant geographic rotation vector does not reveal absolute sidereal phase or directly track stars. Testing astronomical phase would require an independently observed direction/time reference and a separate design. Also inspect residuals against local solar time and temperature where available, since daily instrument/environment cycles can imitate time dependence. A matching rate and vector pattern constrain explanations; they do not identify the field's ultimate cause by themselves.

Interpretation

A reproducible field near \(A=+15.04\), \(B=0\), \(C=-15.04\) degrees/hour would match the standard rotation-axis prediction in a specific, testable way. A different pattern would need explanation and measurement-model checks. Failure to separate a field from permitted instrument errors would be reported as a measurement limitation.

This experiment can distinguish explanations that predict different measurable fields. Two explanations predicting exactly the same gyro, acceleration and GPS observations cannot be distinguished by those observations alone. Measuring the field and identifying its ultimate cause are therefore separate tasks.

Report geometry, retained field and uncertainty, and interpretation limits as separate results. Do not turn “rotation measured” into “its physical source identified,” or “source not identified by this analysis” into “no evidence exists elsewhere.” Matching a sidereal-compatible rate would constrain explanations, not select a cosmology on its own. Any proposed celestial influence must supply quantitative predictions before the project can compare it; this analysis has not established such a mechanism.

Independent stationary companion

The crowdsourced ground experiment estimates signed horizontal reversal rates before checking latitude. It uses stationary consumer IMUs and a distinct protocol, not archival aircraft mounts or transport fitting. Its immediate target is geographic coherence of the horizontal field, with amplitude fitted freely. Consumer sensitivity and calibrated uncertainty remain to be demonstrated. Do not combine its estimates with airborne fits under a shared calibration policy or claim its horizontal component alone decides Earth's shape or the field's ultimate cause.