# Extended Analysis: Multi-Scale Spatial Coherence
in the Magnetosphere-Ionosphere System

Moran Correlograms, Component Decomposition, and Cross-Source Validation
Bruce Stephenson & Argus · May 2026

# 1. What's New: Six Extended Analyses

The baseline pipeline established four findings: storm-intensity dependence, FAC-Hall cancellation, station-count sensitivity, and grid vs station TEC behavior. This extended analysis goes deeper with six new techniques, each probing a different dimension of spatial coherence in the M-I system:

Analysis | Question | Key Finding
--- | --- | ---
Moran correlogram | At what spatial scale does coherence emerge? | Storms coherent out to 5000+ km
SWMF correlogram | Do FAC and Hall operate at different scales? | FAC: 5000 km; Hall: <2000 km
Component decomposition | Which B-field component carries most structure? | E-component > N-component
Latitude bands | Where in latitude is coherence concentrated? | Auroral zone (60–90°) only
Temporal dynamics | How does coherence evolve during a storm? | Oscillates on ~2–4 hour timescale
TEC day-night | Is TEC more coherent on the dayside? | Yes: ΔI ≈ 0.025 (dayside smoother)

# 2. Moran Correlograms: The Shape of Coherence

A Moran correlogram sweeps the distance cutoff in the inverse-distance weight matrix from 500 km to 10,000 km, computing Moran's I at each threshold. The result is the spatial analog of a temporal autocorrelation function — it reveals the characteristic scale of spatial organization.

In geostatistical terms, this is related to the variogram (which measures semivariance vs lag distance). Both probe the same question: at what separation distance do observations become independent? The Moran correlogram tells us this through the lens of spatial autocorrelation rather than variance.

Figure 1. Moran correlograms for all 24 events, colored by storm category. Each curve shows how spatial coherence varies with the distance scale being probed.

Finding 5: Storm coherence extends to ~5000 km. Major storms (red curves) show I > 0.3 out to distance cutoffs of 5000 km, with significant coherence persisting to 10,000 km. Quiet days (green) show I ≈ 0 or negative at all scales. The characteristic scale of storm coherence (~5000 km) matches the expected diameter of the Region 1/Region 2 FAC system projected onto the ground.

## Reading the Correlograms

Several features are physically informative:

- Short-range peak (500–1000 km): High I at short distances indicates that nearby stations see similar perturbations — the field is locally smooth. Major storms show I ≈ 0.7–1.0 at 500 km, meaning neighboring stations are highly correlated.
- Decay rate: How fast I drops with distance reveals the pattern scale. A slow decay (Halloween 2003, Nov 2004) means large-scale organization. A rapid decay (Apr 2010 with few stations) means the pattern is either small-scale or poorly sampled.
- Asymptotic value: What I approaches at large distances. For storms, this is typically +0.2 to +0.4, meaning that even stations 10,000 km apart share some common forcing. For quiet days, the asymptote is negative, indicating anti-correlation at large scales (stations on opposite sides of the globe see opposite perturbations from the weak tidal-scale forcing).

Connection to pattern formation theory: In Busse & Kramer's framework, the correlogram shape encodes the wavelength of the dominant spatial mode. A broad, slowly-decaying correlogram (storms) indicates a single large-scale mode dominates. A flat correlogram (quiet) indicates no dominant mode — the system is below the pattern-formation threshold.

# 3. SWMF Correlograms: FAC vs Hall at Every Scale

Figure 2. SWMF decomposition correlograms for all 6 CCMC events. Blue: FAC, Red: Hall, Black: Total, Green: Pedersen. Shaded bands show ±1σ temporal variability.

Finding 6: FAC and Hall operate at fundamentally different spatial scales. Across all 6 events, FAC (blue) maintains high I out to 5000+ km, while Hall (red) drops below FAC by ~1000–2000 km. This confirms that FAC carries the large-scale magnetospheric structure while Hall currents create structure at shorter scales — and the two partially cancel at the ground at ALL scales, not just on average.

## Scale-Dependent Cancellation

The most revealing feature of these correlograms is that the gap between FAC and Total increases with distance. At 1000 km cutoff:

Event | I_FAC(1km) | I_Total(1km) | Gap | I_FAC(5km) | I_Total(5km) | Gap
--- | --- | --- | --- | --- | --- | ---
Halloween 2003 | 0.80 | 0.20 | 0.60 | 0.40 | 0.01 | 0.39
Dec 2006 | 0.62 | 0.06 | 0.56 | 0.26 | −0.02 | 0.28
Aug 2011 | 0.67 | 0.19 | 0.48 | 0.31 | 0.01 | 0.30

(Distance cutoffs in ×1000 km)

The FAC component is substantially more coherent than the total at every scale. The cancellation by Hall is not a long-range phenomenon — it operates locally but has global consequences because it removes the large-scale structure that FAC creates.

# 4. Component Decomposition: N vs E vs Z

Figure 3. Moran's I for each magnetic field component (N=north, E=east, Z=vertical, H=horizontal magnitude) across all 24 events.

Finding 7: The E-component is more spatially coherent than N. Across all events, mean I: E = 0.103, N = 0.069, H = 0.066, Z = −0.020. The east-west component carries more spatial structure than north-south.

## Why E > N?

This is a subtle but physically meaningful result. To understand it, consider what each component is sensitive to:

- N-component (δB_north): Dominated by east-west currents — primarily the auroral electrojets (Hall currents flowing in the ionospheric E-region). These create north-south perturbations via the right-hand rule (J × r). The electrojets are large-scale features but their ground signature includes substantial Hall current contribution.
- E-component (δB_east): Sensitive to north-south currents — primarily the ground projection of FAC and meridional ionospheric currents. FAC flow along field lines (roughly radial at high latitudes), creating east-west ground perturbations. This component sees more of the large-scale FAC pattern and less of the local Hall closure.

The E > N finding is consistent with the FAC-Hall cancellation story: the E-component preferentially samples the FAC system (which is spatially coherent), while the N-component preferentially samples the electrojet (which includes Hall cancellation). If Hall currents reduce spatial coherence in the total signal, the component most affected by Hall should show lower coherence — and that's exactly what we observe.

The Z-component (vertical) shows the least coherence (mean I = −0.020). This makes sense: vertical perturbations are dominated by local induction effects and ground conductivity variations, which are geologically determined rather than magnetospherically organized.

# 5. Latitude Band Stratification

Figure 4. Moran's I by latitude band. Auroral zone (60–90°) shows clear spatial coherence; subauroral and midlatitude bands show near-zero coherence.

Finding 8: Spatial coherence is concentrated in the auroral zone. Auroral stations (60–90°, median I ≈ 0.20) show dramatically higher coherence than subauroral (50–60°, median I ≈ −0.02) or midlatitude (<50°, median I ≈ 0.01).

This is expected: the R1/R2 FAC system closes in the auroral oval, typically at 65–75° magnetic latitude. Only stations within or near this oval see the organized FAC-driven perturbations. Subauroral stations see primarily the ring current (slow, azimuthally symmetric — not creating spatial pattern at the station-separation scale) and Sq current (smooth, not organized by storm driving).

The sharp drop from auroral to subauroral is itself informative — it means the pattern-forming region has well-defined spatial boundaries. During stronger storms, the auroral oval expands equatorward, which should shift the boundary. We don't have enough subauroral stations with high-N-station events to test this directly, but it's a prediction the pipeline could validate with more data.

# 6. Temporal Dynamics: Coherence Through a Storm

Figure 5. 1-hour sliding window Moran's I for the six events with ≥15 stations. Red: N-component, Blue: E-component, Green: horizontal magnitude.

Finding 9: Spatial coherence oscillates during storms on ~2–4 hour timescales. Rather than being constant during the storm main phase, Moran's I shows pronounced temporal modulation. Different components (N, E, H) often peak at different times.

## Storm Phases and Coherence

Looking at the Halloween 2003 panel (top): coherence rises during the storm main phase (hours 6–18), with peaks reaching I ≈ 0.5. But it doesn't maintain this level — it oscillates with ~2–4 hour period, suggesting substorm-cycle modulation.

The E-component (blue) and N-component (red) show partially correlated but distinct temporal patterns. During some intervals, E leads N; during others, they peak simultaneously. This is consistent with the FAC pattern establishing first (seen in E) and the electrojet response (seen in N) following with a short delay as ionospheric currents adjust.

Bastille Day 2000 shows particularly rich temporal structure, with three distinct coherence peaks at hours ~5, ~12, and ~18 — likely corresponding to three substorm injection events during the storm main phase.

# 7. TEC Day-Night Asymmetry

Figure 6. Left: TEC grid Moran's I for dayside vs nightside, per event. Right: Scatter showing most events fall below the diagonal (dayside more coherent).

Finding 10: Dayside TEC is consistently more spatially coherent than nightside. Mean I: day = 0.923, night = 0.898, difference = 0.025. Though both are high (the GIM is inherently smooth), the systematic day > night pattern is physically meaningful.

The dayside ionosphere is illuminated by solar EUV, creating a smooth, slowly-varying electron density gradient from the subsolar point outward. This produces high spatial autocorrelation — nearby grid cells have similar TEC because they're all controlled by the same smooth solar input.

The nightside ionosphere lacks this organizing illumination. Instead, electron density is maintained by precipitation (patchy, organized by magnetospheric structure), diffusion (smooth but weak), and neutral winds (variable). The result is a less smooth TEC field — still highly autocorrelated (this is a 2.5°×5° grid, which smooths out most small-scale structure), but measurably less so than the dayside.

The scatter plot (right panel) confirms the pattern: nearly all events fall below the 1:1 line, meaning nightside I < dayside I regardless of storm intensity.

# 8. Cross-Source Validation: Do SM and TEC Agree?

Figure 7. Left: Epoch-by-epoch SM vs TEC Moran's I (each dot is one 2-hour window). Right: Per-event Spearman correlation between SM and TEC coherence timeseries.

Finding 11: SM and TEC spatial coherence are largely independent. Mean Spearman r = 0.04 ± 0.31, range [−0.66, +0.60]. The two sources do not track each other epoch-by-epoch. This is not a pipeline failure — it reflects genuine physical independence of current structure and density structure.

## Why Don't They Correlate?

This seems counterintuitive — shouldn't storm driving create coherent patterns in both δB and TEC? The answer is yes, but on different timescales and through different mechanisms:

- δB responds immediately to current changes. When the R1/R2 FAC system strengthens, ground δB increases within minutes. The spatial pattern of δB tracks the instantaneous current distribution.
- TEC responds with delay and inertia. When FAC heating increases, TEC changes through recombination (τ ≈ minutes in E-region, hours in F-region), composition changes (τ ≈ hours), and plasma transport (τ ≈ hours). The TEC spatial pattern at time t reflects the integrated history of drivers over the preceding hours, not the instantaneous state.

The one significantly positive correlation (Feb 2014, r = 0.60, p = 0.039) is a moderate storm (Dst = −116) — possibly a case where the storm was steady enough that instantaneous and time-integrated drivers aligned. The strong anti-correlations (Aug 2005, r = −0.66; Aug 2001, r = −0.52) may reflect cases where storm intensification created coherent δB while simultaneously disrupting the smooth quiet-time TEC field.

# 9. Synthesis: What the Data Tells Us

## The Emerging Picture

Combining the baseline and extended analyses, a coherent physical picture emerges:

1. The M-I system has a pattern-formation threshold around Dst ≈ −50 to −100 nT. Below this, ground magnetometer observations are spatially incoherent (I ≈ 0). Above it, coherent large-scale patterns emerge (I ≈ 0.2–0.4).
2. The dominant pattern is the R1/R2 FAC system, with a characteristic scale of ~5000 km. Moran correlograms show this directly: storm coherence extends to 5000+ km, and SWMF FAC correlograms confirm the FAC component carries structure at this scale.
3. Hall currents act as a spatial low-pass filter in reverse — they preferentially destroy the large-scale structure created by FAC. The ground sees the residual: much less coherent than the driving FAC pattern. The E-component of δB retains more FAC information than the N-component, providing a partial workaround.
4. Coherence is concentrated in the auroral zone (60–90°), where FAC close through the ionosphere. Subauroral and midlatitude stations contribute noise, not signal, to the spatial coherence measurement.
5. TEC and δB track independent aspects of the M-I system. Both respond to storm driving, but through different mechanisms and on different timescales. Cross-source correlation is weak epoch-by-epoch, confirming that current structure ≠ density structure.
6. Spatial coherence oscillates during storms on ~2–4 hour timescales, likely modulated by substorm cycles. The coherence is not a static property of the storm but a dynamic quantity that tracks the evolving current system.

## Summary Table: All 11 Findings

# | Finding | Evidence | Physical Interpretation
--- | --- | --- | ---
1 | SM coherence tracks Dst | 24 events, I vs Dst | Driven dissipative pattern formation
2 | FAC creates structure, Hall destroys it | 6 SWMF events | Ionospheric closure cancels ground signal
3 | N ≥ 25 stations needed | Scaling analysis | Sparse networks miss large-scale structure
4 | TEC grid I always ~0.95 | 22 events with TEC | GIM is smooth by construction
5 | Storm coherence extends to 5000+ km | Correlograms | Scale of R1/R2 FAC system
6 | FAC scale > Hall scale | SWMF correlograms | Magnetospheric vs ionospheric scales
7 | E-component > N-component | Component analysis | E sees FAC, N sees Hall-contaminated
8 | Coherence in auroral zone only | Latitude bands | FAC close in auroral oval
9 | Coherence oscillates (~2–4 hr) | Temporal dynamics | Substorm-cycle modulation
10 | TEC dayside > nightside | Day-night split | Solar illumination smooths dayside
11 | SM and TEC independent | Cross-source r ≈ 0 | Current ≠ density structure

Extended analysis pipeline: `scripts/extended_analysis.py` · 24 events · 27.8 seconds · May 2026