Parallax Seismology · Field Coherence Diagnostics

The field supports
multiple answers.
Measure it.

Ω is the mean pairwise disagreement among N physically independent structural reads of a seismic field. When the field collapses to one dominant organization, all reads agree. When they scatter, the field supports competing structures simultaneously — and a single-answer instrument may commit to the wrong one.

Coherence and correctness are separable variables. A field can be highly organized around the wrong answer. SNR does not see this. Ω does.

Primary result Pre-arrival Ω = 0.100 vs. P-window Ω = 0.075 · Cohen d = 1.78 · 4/5 events Independence gain 6.9× discriminating power from physical independence alone FK failures 15 cases, 4 arrays, 2 continents — 12 with P-wave present but not dominant Events 5 events · Mw 7.0–9.0 · USArray TA + IMS-class compact arrays
6.9×
Independence gain
Discriminating power increase when theories read genuinely different physical observables vs. different transforms of the same input.
1.78
Cohen's d
P-window vs. pre-arrival Ω separation under data-driven conditions. No source location or travel-time model required.
12
FK failures with P present
Cases where the direct P-wave was energetically real but did not win FK beamforming. 15 total cases, 4 arrays, 2 continents.
3
Ω decomposition layers
Quiet-day floor (~0.040) · ambient timing layer (+0.060) · geometric amplification layer (+0.069). Separable and reproducible.
01 — Two Papers, One Instrument

Physical independence
as a measurable quantity.

Structural Agreement Paper — Primary

What the signal is competing with

Ω is the mean pairwise disagreement among N physically independent structural reads of a seismic field. The paper demonstrates that the pre-arrival Ω signal decomposes into three separable layers: a quiet-day floor, an ambient timing layer recoverable without earthquake priors, and a geometric amplification layer added by event-geometry-informed picks. The geometric stagger does not create the pre-arrival elevation — it amplifies a genuine ambient signal approximately 2-fold. This decomposition is reproducible across five events spanning winter and summer, NW and SE source directions, and magnitudes Mw 7.0–9.0. The central finding: the discriminating power of Ω depends entirely on whether the N theories read genuinely different physical observables, not different mathematical transforms of the same input. That degree of independence is itself measurable.

FK Reliability Paper — Additional Context

When a single-answer instrument commits

FK beamforming failures on compact IMS-class arrays provide the operational consequence of the Ω framework: fifteen cases across four arrays on two continents where FK produced a confident, coherent, wrong answer. In twelve cases the direct P-wave was present and physically real yet did not win. SNR did not flag these failures. Station-removal robustness testing reveals three structurally distinct failure families — pre-existing ambient dominance, fragile knife-edge coherence, and a robust coherent competitor coupled to the P-wave onset — that produce the same operational symptom but require different remedies. The paper establishes that coherence is not evidence of correctness, and that a single-answer instrument encountering a multi-organization field will commit to one answer without reporting the competition.

Parallax Seismology — structural agreement field coherence diagnostic
Positions of five independent structural reads (theory centroids) in normalized field space, computed from USArray Transportable Array data. Pre-arrival: theories scatter. P-window: all five theories pull toward a common structural center. Coda: theories scatter again as scattered energy reorganizes the field. Ω is the mean pairwise distance among the colored points — the convergence during P, and the divergence before and after, is the measurement.
Five Physical Observables

The theories must read
different physics.

Observable Physical Property Role in Ω
Amplitude
Intensity distribution
Global intensity-weighted centroid of the wavefield amplitude surface. Reads where energy is concentrated — the dominant amplitude organization of the field at a given timestep.
STA/LTA
Onset timing
Short-term to long-term average ratio across stations. Reads where abrupt energy onsets occur — captures the arrival-time structure without reference to event geometry or travel-time models.
Frequency
Dominant spectral content
Dominant frequency centroid across the array. Different wave types (P, surface, coda) carry different frequency content — frequency reads a physically distinct property from amplitude or timing.
Similarity
Inter-station coherence
Pairwise waveform similarity across stations. Reads where the field is most coherent — not where energy is largest or where arrivals are earliest, but where the wavefield is most self-consistent across sensors.
Arrival-time
Geometric timing structure
Data-driven arrival-time picks projected into array geometry. Reads the spatial organization implied by relative arrival times — the direction from which the wavefront is arriving, without requiring source location or travel-time models.

Field State Visualization

Theory centroid positions across three field states — pre-arrival, P-window, coda
02 — Validation

Five events.
Same three-window structure.

USArray Transportable Array · Tōhoku Mw 9.0 + Japan 2021 Mw 7.1
Ω collapses during P-window and expands again — reproducibly across opposite source geometries.

Two teleseismic events recorded across the USArray TA provide the primary Ω demonstration. Tōhoku (Mw 9.0, NW source) and Japan 2021 (Mw 7.1, same general geometry): in both events, pre-arrival Ω is elevated relative to the P-window, and coda Ω expands again after the direct arrival. The three-window structure — diverge, converge, diverge — is reproducible. The convergence during P is not a threshold crossing. It is a continuous structural collapse visible across all five theory pairs simultaneously.

Pre-arrival Ω (data-driven) 0.100
P-window Ω 0.075
Cohen's d (4/5 events) 1.78
Quiet-day floor ~0.040
IMS-Class Compact Arrays · PDAR, TXAR, WRA, NVAR · 15 FK Failure Cases
Coherent, confident, wrong — twelve cases where P was present but did not win.

Fifteen FK beamforming failures across four IMS-class compact arrays on two continents. Twelve cases: the direct P-wave was present above noise, physically real, energetically detectable — and FK chose a different coherent answer. The Japan 2021 / PDAR case is the canonical example: P present at 71% of peak power; FK back-azimuth error 59.7°. SNR did not flag these cases. Station-removal robustness testing reveals three structurally distinct failure families, each with different diagnostic signatures and different required remedies.

Total failure cases 15
P present but lost 12
Canonical back-azimuth error 59.7° (Japan 2021 / PDAR)
SNR discrimination failed to separate families
Independence Experiment · Same Data, Different Theory Design
6.9× gain from physical independence — not from more theories, more data, or better math.

Five theories reading five different mathematical transforms of the same amplitude image: mean P/coda pairwise separation = 0.004. Five theories reading five genuinely different physical observables from the same wavefield: mean separation = 0.137. The gain is 6.9× under matched (data-driven) conditions. No new data. No parameter tuning. The entire gain came from replacing correlated mathematical transforms with physically independent observables. This is the central methodological finding: the degree of physical independence among theories is itself a measurable quantity, and it is the source of Ω's discriminating power.

Correlated transforms separation 0.004
Physical independence separation 0.137
Independence gain 6.9×
Data used identical in both conditions
Ω Decomposition · 5 Events Spanning Mw 7.0–9.0
Three separable layers — ambient floor, timing structure, geometric amplification — all reproducible.

The pre-arrival Ω elevation decomposes into: a quiet-day floor (~0.040) present on days with no earthquake; an ambient timing layer (+0.060) recovered by FK beamforming on pre-arrival data without earthquake priors; and a geometric amplification layer (+0.069) added by event-geometry-informed arrival-time picks. The geometric layer does not create the pre-arrival signal — it amplifies a genuine ambient coherent structure approximately 2-fold. This decomposition is reproducible across five events covering winter and summer, NW and SE source directions, and a 2-order-of-magnitude range in moment magnitude.

Quiet-day floor ~0.040
Ambient timing layer +0.060
Geometric amplification +0.069 (~2×)
Magnitude range Mw 7.0–9.0
The claim boundary, stated directly.

Ω is a diagnostic instrument, not a source-location algorithm. It does not replace FK beamforming, semblance, or array processing pipelines — it measures the structural conditions under which those pipelines are likely to produce unreliable answers. The FK failure cases are documented empirical examples, not a statistical sample from a known population; the three failure families are observational categories, not a complete taxonomy. The 6.9× independence gain is measured under specific data-driven conditions; the gain magnitude will vary with array geometry, event depth, and noise environment. The decomposition into three Ω layers is reproducible across five events at two arrays — broader replication across array types and tectonic environments remains the next required step. Ω outperforms mean pairwise cross-correlation (d = 0.17 → 1.78) and same-image Ω (d = 1.05 → 1.78) under matched conditions. The claim is structural field diagnostics and the coherence/correctness separation — not array processing replacement.

1.78 Cohen's d
vs. 0.17 for
cross-correlation
6.9× independence
gain from physical
observable design
not a
locator Ω measures field
conditions, not
source position
03 — Position

A different question
requires a different instrument.

FK beamforming answers
Where is the dominant coherent phase?

FK beamforming produces a single best-fit slowness and back-azimuth. It is the right answer to the question it asks. The problem is not that FK is wrong about what it measures — it is that the question assumes the field has a single coherent dominant structure. When the field supports multiple simultaneous coherent organizations, FK will commit to one and will not report the competition.

Ω answers
How many competing structural organizations does the field currently support?

Ω does not locate the source. It measures the structural state of the field — whether the field has collapsed to a single dominant organization or is currently supporting multiple competing ones. Low Ω means a single-answer instrument can be trusted. High Ω is a warning that coherence and correctness have separated, and that a committed answer should be held with more uncertainty.

The intended relationship is diagnostic and complementary. Ω is a pre-commitment quality metric for array processing pipelines. Run it before FK commits. Low Ω: proceed. High Ω: hold the answer with higher uncertainty, run station-removal robustness, or flag for review. The three FK failure families identified in this work are not edge cases — they are structurally distinct conditions that require structurally distinct responses.