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, and 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, and 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.