1. Why Spatial Autocorrelation?

The magnetosphere-ionosphere (M-I) system is a driven dissipative continuous system. The solar wind drives energy into the magnetosphere; that energy dissipates through ionospheric currents, particle precipitation, and Joule heating. Between the driving and the dissipation, the system self-organizes into spatial patterns.

This framing comes directly from Busse & Kramer's "Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems" (NATO ASI Series B, Vol. 225, 1990). Their framework asks: given a spatially extended system with energy input and dissipation, what patterns emerge? How do those patterns depend on driving conditions? How do different observational windows reveal different aspects of the same underlying structure?

Spatial autocorrelation is the natural tool for this question. It measures whether nearby locations tend to have similar values (positive autocorrelation = patterns) or dissimilar values (negative = checkerboard), compared to what you'd expect from random placement. A geomagnetic storm that drives coherent large-scale currents should create positive spatial autocorrelation in ground magnetometer measurements. Quiet times should show noise — near-zero autocorrelation.

2. Moran's I: A Primer

Moran's I is the spatial analog of Pearson's correlation coefficient. Where Pearson measures association between two variables, Moran's I measures association between a variable and its spatially-lagged self — i.e., how similar is each location to its neighbors?

where N is the number of observations, wᵢⱼ are spatial weights (how "connected" locations i and j are), and S₀ = Σwᵢⱼ. The expected value under spatial randomness is E[I] = −1/(N−1), which approaches zero for large N.

Figure 1. Moran's I captures three regimes of spatial organization. In M-I physics: storms tend to create the left pattern (coherent perturbations), while quiet times produce the middle pattern (spatially uncorrelated noise).

Weight Matrices

The choice of weight matrix W determines what "neighbor" means. We use two types:

Inverse-distance weights (for irregular stations)

For SuperMAG and SWMF stations at irregular positions, we use wᵢⱼ = 1/dᵢⱼ (haversine distance), row-normalized so that each station contributes equally regardless of how many neighbors it has. This follows the PySAL convention and prevents dense clusters from dominating the statistic. Self-weights are zero (wᵢᵢ = 0).

Queen contiguity (for regular grids)

For TEC on the CODE GIM 2.5°×5° grid, we use 8-neighbor queen contiguity — each grid cell is connected to its 8 surrounding cells (N, S, E, W, and diagonals), with wrap-around at the dateline. Also row-normalized. This is the natural choice for gridded data where the grid spacing is the meaningful "neighbor" scale.

3. Three Windows into M-I Coupling

Figure 2. Three observational windows into the magnetosphere-ionosphere system. Each measures a different physical quantity at a different spatiotemporal resolution.

SuperMAG: Ground Magnetometers

SuperMAG is a worldwide collaboration of ground magnetometer networks providing standardized baseline-subtracted measurements of geomagnetic field variations. Our dataset includes 42+ stations across the Northern Hemisphere at 1-minute cadence, measuring north (N), east (E), and vertical (Z) components of δB in nanotesla.

Ground magnetometers measure the integrated effect of all current systems above them: magnetospheric currents (ring current, magnetopause, cross-tail), field-aligned currents connecting magnetosphere to ionosphere, and ionospheric Hall and Pedersen currents. The ground δB is the superposition of all these sources — which, as we'll see, creates a fundamental observability problem.

Figure 3. Ground magnetometer network showing 42 SuperMAG stations (blue) and 12 SWMF CCMC virtual stations (red stars). Green circles mark overlap stations. Northern high-latitude coverage spans North America, Scandinavia, and Siberia.

SWMF / BATS-R-US: Simulated Magnetometer Decomposition

The Space Weather Modeling Framework couples a global MHD magnetosphere model (BATS-R-US) with an ionospheric electrodynamics solver (Ridley). Using Biot-Savart integration, it computes the ground magnetic perturbation at virtual magnetometer locations and decomposes it into four current sources:

The total is: δB_total = δB_MHD + δB_FAC + δB_Hall + δB_Pedersen. Each column carries 15 values (N, E, D for each source). We use CCMC SWPCTEST runs with 12 virtual stations for 6 events spanning 2001–2011.

GPS TEC: Ionospheric Electron Density

CODE (Center for Orbit Determination in Europe) produces Global Ionosphere Maps (GIMs) from dual-frequency GPS observations, providing vertical total electron content (VTEC) on a 2.5° latitude × 5° longitude grid (71 × 73 cells) at 2-hour cadence in the IONEX format.

TEC measures the column-integrated electron density through the ionosphere — fundamentally a different observable than δB. While magnetometers see current structure, TEC sees density structure. The two are connected through ionospheric electrodynamics: FAC closure currents heat the ionosphere, driving density redistribution through enhanced recombination and plasma transport.

We have IONEX data for 1998–2006 and 2012–2016, giving us excellent overlap with both SuperMAG (continuous) and SWMF (6 events in 2001–2011).

4. The Comparable Buckets Problem

Comparing Moran's I across three different data sources requires aligning them in time, space, and observable. Each axis presents a distinct challenge:

Figure 4. Temporal alignment strategy. SuperMAG and SWMF operate at ~1-minute cadence; TEC at 2-hour. We aggregate SM and SWMF into 2-hour bins for three-way comparison, while retaining the native 1-minute cadence for the SM–SWMF pair.

Temporal alignment

SuperMAG: 1-minute. SWMF: ~1-minute (variable). TEC: 2-hour. For three-way comparison, we aggregate to the coarsest cadence (2 hours), using median for SuperMAG (robust to outliers) and mean for SWMF. For the SuperMAG–SWMF pair, we retain 1-minute resolution.

Spatial alignment

Two parallel comparison modes handle the spatial mismatch:

• Station-based: All sources evaluated at the same SuperMAG station positions. TEC values are bilinearly interpolated from the GIM grid to each station location. SWMF uses its 12 virtual stations (a subset of SuperMAG positions). All share the same inverse-distance weight matrix, making Moran's I values directly comparable.
• Grid-based: TEC analyzed at its native 2.5°×5° resolution using queen contiguity weights within the 50°–75°N auroral band (~627 cells). This captures spatial structure at the scale TEC actually resolves, without the sampling limitations of sparse station networks.

Observable alignment

δB (nT) and TEC (TECU) are fundamentally different quantities. Moran's I normalizes out amplitude (it's invariant to linear transformations), so the values are comparable as measures of spatial pattern regardless of the physical units. An I=0.3 in dB and I=0.3 in TEC both mean "moderate positive spatial autocorrelation" — nearby locations are more similar than chance would predict.

5. The Pipeline

Figure 5. Pipeline architecture. Four loader modules handle data heterogeneity; a shared spatial statistics module provides canonical Moran's I; the orchestrator runs per-event analysis and cross-event comparison.

Event Catalog

We analyze 24 event days across four geomagnetic activity categories:

Source availability varies by event: 4 events have all three sources (triple overlap: SWMF + SM + TEC), 2 have SWMF + SM, and the remainder have SM + TEC. The event catalog automatically verifies data availability across all sources before processing.

Per-Event Processing

For each event, the pipeline computes:

1. SuperMAG: 1-min and 2-hour Moran's I for N-component and horizontal magnitude (√(N²+E²))
2. TEC grid: Queen-contiguity Moran's I on 50°–75°N band at native 2-hour cadence
3. TEC at stations: Interpolated TEC at SuperMAG positions, same inverse-distance weights
4. SWMF: Moran's I for all 5 components (total, MHD, FAC, Hall, Pedersen) × 2 polarizations
5. Scaling analysis: Subsample SuperMAG stations (N = 6, 8, 10, 12, 15, 20, 25, 30, 40), 30 trials

All results are saved as compressed NumPy archives (.npz) with standardized keys for downstream analysis.

6. Results: Storm Intensity and Spatial Coherence

Figure 6. SuperMAG spatial coherence (2-hour Moran's I) by storm category. Major storms show clear positive autocorrelation; quiet days are near zero or negative. Individual events shown as jittered points.

This is the expected behavior for a driven dissipative system: weak driving produces localized, incoherent fluctuations; strong driving creates large-scale organized patterns. The transition occurs around Dst ≈ −100 nT, where Moran's I consistently crosses zero from negative to positive.

Figure 7. Four-panel overview. Top-left: SM coherence vs Dst, color-coded by category. Top-right: TEC grid I vs SM station I. Bottom-left: SWMF decomposition bars. Bottom-right: Station count sensitivity curves.

The SM–Dst Relationship

Panel (a) of Figure 7 shows a clear trend: the most intense storms (Dst < −300 nT) produce the highest spatial coherence (I = 0.2–0.4). But the relationship isn't linear — some moderate storms achieve high coherence while the most extreme (Halloween 2003, Dst = −383) shows I ≈ 0.30, not the maximum. This suggests that storm geometry (e.g., the spatial extent and orientation of the electrojets) matters as much as raw intensity.

Detailed Timeseries: Halloween 2003

Figure 8. Halloween 2003 Moran's I timeseries across all three sources. (a) SuperMAG 1-min with 30-min running mean, (b) SWMF decomposition showing FAC dominance, (c) TEC grid vs TEC-at-stations comparison.

The Halloween 2003 event illustrates several key features:

• SuperMAG (panel a): Moran's I oscillates between 0 and 0.5, with a ~30-minute modulation visible in the running mean. The 1-minute raw values are noisy but the structure is persistent — spatial coherence is maintained throughout the storm's main phase.
• SWMF (panel b): The FAC component (blue) consistently shows higher I than the total (black) or Hall (red). This is the FAC-Hall cancellation in action — see Section 7.
• TEC (panel c): Grid-based I (purple, ~0.94) is always high because the TEC field is inherently smooth on the GIM grid. Station-based I (green, ~0.30) is much lower and more variable — comparable to SuperMAG, as expected when using the same spatial sampling.

7. The SWMF Decomposition: FAC-Hall Cancellation

Figure 9. The FAC-Hall cancellation mechanism. Field-aligned currents create large-scale spatial structure; Hall currents locally cancel it. Ground magnetometers see only the residual.

Physical Interpretation

Field-aligned currents form the Region 1 / Region 2 system — a large-scale pattern driven by magnetospheric convection. This pattern spans thousands of kilometers and connects dawn-dusk asymmetry in the magnetosphere to the high-latitude ionosphere. It is inherently spatially coherent.

Hall currents, by contrast, flow locally in the ionospheric E-region (~110 km altitude) as horizontal closure currents. They respond to the local electric field and conductance, creating structure on smaller scales that partially cancels the large-scale FAC signal when projected to the ground.

This has profound implications for ground-based studies: the spatial coherence visible at the ground is significantly less than the actual magnetospheric driving structure. Ground magnetometers are not wrong — they accurately measure the total δB — but the total carries less spatial information than its largest constituent.

8. Station Count Sensitivity

Figure 10. Moran's I as a function of the number of SuperMAG stations used. Each curve represents one event; error bars show trial-to-trial variability from random subsampling. The N=25 threshold marks where coherence estimates stabilize.

The scaling curves show two regimes:

• N < 15: Rapid increase and high variance. The sparse network misses large-scale structure and amplifies noise. Moran's I is unreliable in this regime.
• N > 25: Curves plateau and error bars shrink. The network is dense enough to resolve the dominant spatial patterns.

This explains why SWMF's 12 virtual stations show I ≈ 0 for the total component even during intense storms: it's a combination of FAC-Hall cancellation and inadequate spatial sampling. The scaling analysis lets us disentangle the two effects — at N = 12, even pure FAC (no cancellation) would show I ≈ 0.1 rather than the I ≈ 0.2 we'd measure with 40 stations.

Correction Factors

The scaling curves provide empirical correction factors: if you observe I = 0.18 at N = 12 stations during Halloween 2003, the full-network value (N = 40) is I = 0.31 — about 1.7× higher. These correction factors are event-dependent (stronger storms have steeper scaling curves because there's more large-scale structure to resolve), but the general rule is:

This is an empirical approximation; the actual correction should use the full I(N) curve for each event.

9. TEC: The View from Above

Grid-Based vs Station-Based

TEC Moran's I reveals a striking contrast depending on the spatial sampling strategy:

Interestingly, quiet days show slightly higher TEC grid I (0.97) than storm days (0.94). This makes physical sense: during quiet times, the TEC field is dominated by smooth solar-controlled ionization with a simple day-night gradient. Storms disrupt this smoothness by driving irregular density enhancements (storm-enhanced density, or SED) and depletions (negative storm effects), creating local structure that slightly reduces the overall spatial autocorrelation.

TEC–SuperMAG Correlation

The Spearman correlation between TEC-at-stations Moran's I and SuperMAG Moran's I at 2-hour cadence is weak and variable across events (r = −0.66 to +0.60, median ≈ 0). This suggests that while both respond to storm driving, they track different aspects of the M-I system:

• SuperMAG δB reflects current structure — dominated by electrojet position and intensity
• TEC reflects density structure — driven by precipitation, convection, and neutral winds

The two can decouple because TEC responds to processes (e.g., neutral composition changes, diffusion) that operate on longer timescales than the current systems driving δB.

10. Physical Interpretation

The M-I System as Pattern Formation

Returning to the Busse & Kramer framework: the magnetosphere-ionosphere system exhibits classic behavior of a driven dissipative continuous system:

1. Below threshold (quiet: Dst > −30 nT): No large-scale organization. Moran's I ≈ 0 or negative. The system is subcritical — local fluctuations don't couple into coherent patterns.
2. Above threshold (storms: Dst < −100 nT): Emergence of large-scale patterns. Moran's I ≈ 0.2–0.4. The driving (solar wind Bz < 0) exceeds the critical threshold for coherent convection patterns, and the R1/R2 FAC system organizes the high-latitude ionosphere.
3. Strong driving (extreme: Dst < −300 nT): Patterns are intense but not necessarily more coherent than moderate storms. I plateaus around 0.3–0.4 even as amplitude increases, suggesting a saturation in the spatial organization even as the amplitude continues to grow.

What Creates Structure?

The SWMF decomposition provides the answer hierarchy:

1. FAC (Region 1/2 system) — primary source of large-scale spatial structure (I ≈ 0.10–0.19). Driven directly by magnetospheric convection, this is the pattern-forming mechanism.
2. MHD (ring current, etc.) — contributes smooth large-scale background but varies slowly, contributing little to spatial pattern on station-separation scales.
3. Hall — locally generated, partially cancels FAC structure at the ground. This is the key destroyer of observable coherence.
4. Pedersen — weak, little spatial pattern.

The Observability Gap

This decomposition reveals a fundamental observability gap: the most physically meaningful spatial structure (FAC, driven by magnetospheric dynamics) is partially hidden from ground magnetometers by ionospheric closure currents. Ground-based Moran's I of ~0.3 during major storms represents the residual coherence after cancellation, not the full structure. The FAC alone shows I ≈ 0.19 at N = 12 stations — which, corrected for station-count effects, would be I ≈ 0.30 at N = 40. The true magnetospheric structure is more organized than the ground can reveal.

11. Future Directions

Higher-resolution TEC

The 2-hour/2.5°×5° CODE GIMs are the most accessible global TEC product, but higher-resolution alternatives exist: MADRIGAL provides 1°×1° maps at 5-minute cadence using GPS phase observations. These would better resolve storm-time TEC structure and enable more meaningful comparison with the 1-minute SuperMAG cadence.

AMPERE: A True Fourth Spatial Leg

The Active Magnetosphere and Planetary Electrodynamics Response Experiment (AMPERE) derives field-aligned current maps from the Iridium constellation's engineering magnetometers. This provides a direct observation of FAC spatial structure — the very quantity we infer from SWMF decomposition. AMPERE data from 2010–present would overlap with our 2012–2016 SuperMAG+TEC events and provide the missing observational constraint on FAC coherence.

GAMERA: Next-Generation Simulation

GAMERA (Grid Agnostic MHD for Extended Research Applications) replaces BATS-R-US with a sixth-order numerical scheme that dramatically reduces numerical diffusion. If GAMERA preserves more small-scale structure in FAC, we'd expect higher Moran's I in the FAC component and possibly different cancellation behavior. This is a direct test of whether the FAC-Hall cancellation we observe is physical or partially numerical.

Multi-Scale Analysis

Moran's I with a distance cutoff can probe different spatial scales. By sweeping the cutoff from 500 km to 5000 km, we can build a Moran correlogram — the analog of a variogram in geostatistics — revealing the characteristic scale of spatial organization in each source. This connects directly to the correlation length in Busse & Kramer's pattern formation theory.

Temporal Dynamics

How does spatial coherence evolve through a storm? Our 1-minute SuperMAG data could reveal whether coherence builds gradually, appears suddenly at onset, or oscillates with substorm cycles. The ~30-minute modulation visible in the Halloween 2003 timeseries (Figure 8a) hints at substorm-scale organization.

Summary of Key Findings

Generated by the unified Moran's I pipeline · 24 events · 3 data sources · May 2026
Pipeline: scripts/unified_morans_pipeline.py · Results: results/unified_morans/