# Why SWMF Can't See What SuperMAG Sees

A tutorial on decomposing spatial autocorrelation in MHD model output
Bruce Stephenson & Argus — Session 70, May 2026

## 1. The Question in One Sentence

When a geomagnetic storm hits, ground magnetometers across the auroral zone all swing in roughly the same direction at roughly the same time. Our Moran's I analysis quantifies this: I ≈ +0.35 to +0.40 during storms. The question is: can the leading global MHD model (SWMF) reproduce this spatial coherence?

The answer is no — and the decomposition tells us exactly where the model breaks.

## 2. Quick Refresher: What Moran's I Measures

Each dot is a magnetometer station. Arrows represent the magnetic perturbation vector. Moran's I measures whether nearby stations point the same way (high I) or not (low/negative I).

Moran's I is computed using inverse-distance weights: stations 500 km apart influence each other more than stations 5000 km apart. The formula is essentially "the correlation of each station's deviation from the mean with the distance-weighted average of its neighbors' deviations." Range: roughly −1 to +1.

## 3. The Four Current Systems Under Your Feet

When a storm hits, the ground magnetic field is perturbed by four distinct current systems, each at a different altitude. The SWMF model computes all four separately, which is what makes this decomposition possible.

The ground perturbation dBtotal = dBMHD + dBFAC + dBHall + dBPedersen. SWMF computes each term separately. A magnetometer on the ground sees only the sum.

What each current system is:

MHD currents flow in the magnetosphere itself — the ring current circling Earth at ~4 RE, the cross-tail current, magnetopause currents. These are far away and spatially smooth. They're why all stations see the field drop during a storm (negative Dst).

FAC (Birkeland currents) flow along magnetic field lines, connecting the magnetosphere to the ionosphere. Region 1 currents flow in on one side of the auroral oval and out the other; Region 2 close the circuit at lower latitude. These are large-scale sheets thousands of km across — they produce spatially coherent ground signatures.

Hall currents flow horizontally in the ionosphere (at ~110 km altitude), perpendicular to the electric field. They form the auroral electrojet. Critically, Hall currents depend on local ionospheric conductance — which varies sharply with particle precipitation, solar illumination, and composition. This makes them the most spatially variable current system.

Pedersen currents also flow horizontally in the ionosphere, but parallel to the electric field. They directly close the FAC circuit, so they inherit some of FAC's large-scale structure.

## 4. The Experiment

We did something nobody has done before: applied Moran's I spatial autocorrelation to the individual current-source components of SWMF output.

Data source: SWPCTEST validation repository (SWMFsoftware/SWPCTEST, Event 1: Halloween storm Oct 29–30, 2003). 12 real stations, 2863 timesteps, full 15-column decomposed output (dBN, dBE, dBD for each of 5 sources: total, MHD, FAC, Hall, Pedersen).

Comparison data: SuperMAG ground magnetometer observations for the same storm. 7 stations overlap between SWMF and SuperMAG (YKC, MEA, NEW, OTT, FRD, ABK, KIR). We also have a 42-station baseline from our earlier work.

Method: Same Moran's I computation on each: inverse-distance spatial weight matrix, computed at each timestep, producing an I(t) timeseries for each component.

## 5. The Result

The model produces zero spatial correlation where observations show strong positive correlation.

Source | I (North) | I (Horiz. mag.) | I (Vertical) | Stations
--- | --- | --- | --- | ---
42-sta baseline | — | +0.396 | — | 42
7-sta observed | +0.355 | +0.247 | +0.044 | 7
SWMF total | −0.029 | −0.036 | −0.022 | 12
SWMF FAC | +0.192 | +0.006 | +0.273 | 12
SWMF MHD | +0.096 | +0.092 | — | 12
SWMF Hall | +0.022 | +0.002 | — | 12
SWMF Pedersen | +0.114 | +0.021 | — | 12

The figure below shows the full timeseries comparison over 48 hours of storm:

Top: N-component observed (blue) vs SWMF total (red). Second: horizontal magnitude three-way (observed, baseline, SWMF). Third: SWMF decomposition by current source, with observed mean (dashed blue). Bottom: vertical component.

## 6. What's Happening — The Cancellation

The decomposition reveals a destructive interference between current systems in the model:

The model's structured FAC signal gets buried under noisy Hall currents. In reality, the total ground signal preserves spatial coherence.

The physical interpretation:

In the real ionosphere, Hall currents are organized by the large-scale convection pattern — the auroral electrojet flows in a coherent channel along the auroral oval. But in the SWMF's Ridley Ionosphere Model (RIM), the Hall current pattern appears to have too much small-scale spatial variability. This could be caused by:

- Conductance model: sharp boundaries between sunlit/dark hemispheres or precipitation zones create artificial conductance gradients, producing patchy Hall currents
- Grid resolution: the ionospheric grid (~1–2°) may alias smooth physical structures into jagged numerical patterns
- Coupling frequency: if the magnetosphere-ionosphere coupling updates too infrequently, temporal transients create spatial noise

## 7. Why This Matters

For model validation: Traditional validation compares observed vs modeled timeseries at individual stations (RMSE, correlation). That tests amplitude and timing at each point. Moran's I tests something these metrics cannot: does the model get the spatial relationships right? A model could score well on single-station metrics while completely failing the spatial coherence test. And that's exactly what we see.

For physics: The decomposition is a diagnostic tool. It tells modelers which subsystem to investigate. Not "your model is wrong" but "your ionospheric Hall current mapping introduces ~0.38 units of excess spatial incoherence." That's actionable.

For our research: The observed I ≈ +0.35 (from SuperMAG) is not an artifact of averaging or geometry — the model's FAC component independently confirms that large-scale current systems should produce spatial correlation. The question becomes: why does the real ionosphere preserve spatial coherence while the model destroys it? The answer likely involves how real ionospheric conductance is more spatially smooth than the model assumes.

The one-sentence version:

FAC builds spatial structure. Hall currents in the model destroy it. In reality, they don't. The model's ionospheric solver is the culprit.

## 8. The Vertical Component Tells Its Own Story

Weimer (2010) showed that the vertical (Z/D) ground perturbation is a good proxy for FAC intensity — it responds primarily to overhead field-aligned currents. Our decomposition confirms this:

SWMF FAC-only D component: I = +0.27 (strong spatial structure, as expected for large-scale Birkeland current sheets). But the total D is I = −0.02 — other current systems cancel even the vertical FAC signal. The observed Z: I = +0.04 (low, but at least not negative).

This suggests the cancellation problem affects all three components (N, E, D), not just the horizontal field. The Hall current noise is three-dimensional.

Data: SWMF SWPCTEST Event 1 (Halloween 2003), SuperMAG Oct 2003. Code: `scripts/swmf_morans_i.py`. Output: `results/ccmc_vmag/`. SCILOG: Entry 2026-05-08.

Open question: Would GAMERA/LFM with a higher-order ionospheric solver (e.g., REMIX, or a full electrodynamics solve) show better spatial correlation? Would increasing RIM grid resolution from 2° to 0.5° improve Moran's I? These are testable predictions.