Where Spatial CoherenceHow much nearby ground magnetometer stations agree with each other. High coherence (Moran's I near 1) means organized, large-scale structure. Near zero means random, uncorrelated perturbations. Lives: Altitude-Resolved Moran's I

Halloween 2003 — SuperMAGA worldwide collaboration of organizations providing ground-based magnetometer data. Over 300 stations across Earth's surface, measuring the magnetic field perturbations caused by ionospheric and magnetospheric currents. ground magnetometers + GPS TECTotal Electron Content — the total number of electrons along a path between a GPS satellite and ground receiver. Measured in TECU (10^16 electrons/m²). Reveals ionospheric density structure at different altitudes via tomographic inversion. tomographic decomposition
00:00 UT
CORRECTION — 28 JULY 2026

The altitude split on this page is not an independent altitude measurement. Both altitude channels, I(120 km) and I(300 km), are computed from the same ground TEC field through the two-layer Chapman model described below. The only altitude-dependent degree of freedom in that model is the scalar auroral fraction f, which is itself a deterministic function of the ground perturbation amplitude, applied through a fixed latitude mask. The decomposition is therefore a re-expression of the ground field, not a second, independent observation of it.

This matters for the independence check below. A low R² between the derived altitude indices and the ground index does not establish that the altitude channels carry independent information, because the mapping from ground field to altitude channels is deterministic. Two deterministic functions of one field can be weakly correlated with that field and still contain nothing it does not already contain.

The first write-up of this analysis overclaimed on exactly this point. Its conclusions should not be cited; the corrected version, v2.0.0, supersedes it. The dashboard below is retained because the decomposition itself is correctly implemented and its behaviour is worth being able to inspect — but it should be read as a model diagnostic, not as evidence about altitude structure.

What this page does still show. The Chapman decomposition is correctly implemented and internally consistent, and the dashboard is a faithful view of how that model behaves across a real storm. That is a statement about the model, and it is the only kind of statement this page supports. No critical exponent is claimed here, and none should be inferred from it.

Altitude Coherence Profile

Precipitation layer (120 km): auroral electron precipitationElectrons accelerated down magnetic field lines that slam into the upper atmosphere at 120 km altitude. They excite atmospheric atoms, producing the aurora. The energy deposition creates enhanced ionization at this specific altitude.
F-region (300 km): solar EUV ionizationExtreme ultraviolet radiation from the Sun ionizing atmospheric atoms at ~300 km altitude (the F-region). This is the 'quiet' source of ionospheric plasma — always present, varying with solar activity and local time.
Moran's IA spatial autocorrelation statistic ranging from -1 (anti-correlated) through 0 (random) to +1 (perfectly correlated neighbors). Values above ~0.3 indicate meaningful spatial organization. Named after Patrick Moran (1950). > 0 at an altitude = spatially organized at that height

Phase Space: I(120 km) vs I(300 km)

Above diagonal: precipitation dominant
Below diagonal: F-region dominant
The trajectory through this space tracks MI system state

Time Series: Ground I + Altitude Delta

Solid: ground I_N (integrated over all altitudes)
Dashed: Δ(120-300), positive=precipitation, negative=F-region
Background: regime classification

Storm Regime Timeline

The Two-Layer Chapman Model

GPS Total Electron Content (TEC) measures the column-integrated electron density — it cannot tell you where along the ray path those electrons sit. We decompose the vertical profile using a two-layer Chapman model: a solar EUV layer centered at 300 km (F-region, scale height H = 60 km) and an auroral precipitation layer at 120 km (E-region, H = 20 km).

ne(z) = nEUV · exp[½(1 − ζF − e−ζF)] + nprecip · exp[½(1 − ζE − e−ζE)]
where ζF = (z − 300)/60, ζE = (z − 120)/20
TEC-conserving formulation: The auroral fraction f is defined as the fraction of column TEC from precipitation, not the fraction of peak density. Because the E-region is 3× narrower (H=20 vs H=60), the precipitation peak density must be 3× higher to conserve the same column integral: nprecip = f · nmax · (Hsolar/Hprecip). This ensures that repartitioning TEC between layers does not create or destroy electrons.

The auroral fraction is storm-intensity-dependent: f = min(0.7, max(0.1, 0.1 + 0.6 × RMS/1000)), where RMS is the ground perturbation amplitude in nT. During quiet conditions (RMS ~ 20 nT), f ≈ 0.11; during intense storms (RMS ~ 700 nT), f ≈ 0.52. Only stations poleward of 60° magnetic latitude receive precipitation.

Chapman Profiles at Two Altitudes

Blue: Solar EUV (hmax=300 km, H=60 km). Red: Precipitation (hmax=120 km, H=20 km). Black dashed: sum. Moran's I is computed on ne at each altitude independently.

Altitude Coherence Switch

Δ = I(120 km) − I(300 km)

Δ > 0: Precipitation-dominated spatial structure. Auroral electron precipitation creates E-region density enhancements at specific latitudes, producing organized spatial patterns at 120 km that differ from the smooth F-region.

Δ < 0: F-region dominant. During recovery or quiet conditions, the solar EUV gradient dominates the ionospheric density pattern, and the F-region at 300 km is more spatially organized than the E-region.

The switch oscillates by ±0.4 during storms and is consistent across all nine storms analysed (2000–2015). Nine events is a small sample; “consistent” is the strongest word the data supports.

Internal Consistency: 8 Checks

These are implementation and self-consistency checks on the decomposition: agreement with an independent library, analytical round-trips, and comparison against published empirical profiles. They establish that the model is coded correctly and conserves what it should. They do not establish that the altitude split carries independent altitude information — see the correction at the top of this page.

1. Moran's I vs PySAL PASS

Our implementation matches the PySAL reference (esda.Moran) to machine precision (max diff = 1.1×10−16) when both use haversine great-circle weights.

Systematic offset of up to 0.19 occurs with PySAL's default Euclidean weights — expected at 65° latitude where cos(λ)≈0.42 compresses east-west distances by 2.4×.

2. Chapman TEC Round-Trip PASS

Integrating the two-layer Chapman profile recovers input TEC. TEC-conserving formulation verified: redistributing TEC between layers (f = 0 to 0.7) preserves total column within machine precision. The non-conserving formulation loses up to 27% of TEC at f=0.4.

3. Permutation Calibration PASS

200 independent permutation tests (199 permutations each) at α=0.05. False positive rate = 4.5% (expected: 5.0%). KS test confirms uniform p-value distribution (D = 0.053). The permutation test is correctly calibrated.

4. Single-Layer Recovery PASS

Setting f=0 (no precipitation) produces identical Moran's I at all altitudes (spread = 2.2×10−16). With f=0.5, altitude spread increases to 0.307. The decomposition creates altitude variation only when precipitation is present.

5. Additive Consistency PASS

ne(solar) + ne(precip) = ne(total) at every altitude, every station. The Chapman profiles add linearly — no nonlinear coupling between layers.

6. Carpenter-Anderson PASS

Quiet-day V-shaped coherence profile minimum at L=6–7. Carpenter & Anderson (1992) plasmapause model: Lpp = 5.7 − 0.47·Kp predicts Lpp = 4.8–5.7 for typical quiet (Kp=0–2). The coherence minimum sits at or just outside the plasmapause — consistent.

7. Ground RMS Range PASS

Ground perturbation RMS ranges from 43 nT (quiet) to 341 nT (intense storm) across 9 events. This 8× dynamic range drives f from 0.11 to 0.52, exercising the full physical range.

8. Multi-Event Independence PASS

Across 9 events / 107 epochs: r(Ignd, I120) = +0.14, r(Ignd, I300) = −0.34, combined R² = 0.115. The derived indices are only weakly correlated with the ground index. Note the limit of this check: because the altitude channels are a deterministic function of the ground field, weak correlation does not imply independent information.

Sensitivity Analysis: Three Sweeps

The altitude coherence switch must be robust to three sources of uncertainty: (1) station count — does the signal survive with fewer magnetometers? (2) auroral fraction — how sensitive is Δ to the assumed f? (3) Chapman scale heights — what if the assumed layer thickness is wrong?

Bootstrap Station Subsampling

200 bootstrap resamples at each station count. Shaded bands show 5th–95th percentile. Convergence at N ≥ 20: std(Δ) drops below 0.10 and the sign of Δ stabilizes. The Halloween 2003 event has 41 stations (36 with TEC) — well above the threshold.

Auroral Fraction Sweep (f = 0.0 – 0.7)

Each curve shows Δ vs f at a different storm phase (UT). Key result: Δ sign is stable from f = 0.1 to 0.5 at every epoch except UT 06:00 (SSC — inherently ambiguous). The sign only flips at extreme f values outside the physically reasonable range.

Chapman Scale Height Sweep

Left: Hsolar swept 40–80 km (default 60). Right: Hprecip swept 10–30 km (default 20). Δ varies by < 0.01 across the full Hsolar range and by 0.03 across Hprecip. The altitude coherence switch is effectively independent of assumed scale heights.

Summary

The altitude coherence switch is robust.

✓ Converges with ≥ 20 stations (we have 30–41)
✓ fauroral stable over 0.1–0.5 (our range: 0.11–0.52)
✓ Hsolar sensitivity < 0.01 across 40–80 km
✓ Hprecip sensitivity < 0.03 across 10–30 km
✓ Sign of Δ preserved at all epochs except SSC onset
✓ TEC-conserving formulation vs original: max |ΔI| = 0.011

Sensitivity data from Halloween 2003 (29 Oct, 41 stations, 36 with TEC). Bootstrap uses 200 resamples per station count.