FWI QC: synthetic-vs-recorded matching

How do you know your FWI model is right? You do not — no one ever does. What you have is a pile of metrics that, if all pass simultaneously across enough of the data, give you enough confidence to ship the model into production. This section catalogues the standard QC pack every FWI project uses and the failure modes each metric is designed to catch.

1. The central QC principle

FWI is an optimisation; it minimises a single scalar misfit. When the misfit reaches a plateau you have converged against the misfit you optimised, but not necessarily against the physics. QC checks the inversion from angles the optimiser did not directly see: different metrics, different data subsets, different frequency bands, different azimuths. Production acceptance requires passing every one.

2. The three time-domain metrics

The widget computes the three basic metrics on a single trace pair. Drag V_trial to sweep through "FWI output" states:

• Time-domain correlation (Pearson). $\text{corr}(d_{obs}, d_{syn}) = \langle d_{obs} \cdot d_{syn}\rangle / \sqrt{\langle d_{obs}^2\rangle \langle d_{syn}^2\rangle}$. Measures phase and wavelet-shape agreement but is insensitive to amplitude scaling. Threshold: > 0.9 for production acceptance.
• Normalised L2 residual. $|d_{obs} - d_{syn}|$2 / |d{obs}|_2. Measures both amplitude and phase agreement. Threshold: < 0.3 typically; < 0.1 for high-fidelity work.
• Spectral coherence. Overlap between the amplitude spectra of observed and synthetic. A correlation in the frequency domain that catches band-specific problems the time-domain correlation misses. Threshold: > 0.85 across the inversion band.

All three must pass. Each catches something the others miss:

• A synthetic shifted by exactly one wavelet period has a high amplitude-envelope correlation but a low L2 — L2 catches cycle skipping that correlation alone misses.
• A synthetic with correct phase but half the amplitude has a high correlation and a high L2 — only the normalised L2 residual catches the amplitude error.
• A synthetic missing a high-frequency band entirely (because the FWI was never run at that band) has a high time-domain correlation but a low spectral coherence — coherence catches bandwidth deficits that time-domain metrics average over.

3. Per-subset analysis

A single trace's QC is insufficient. Production QC slices the data along several axes and checks metrics on each subset:

• Per-offset. Near vs mid vs far. Far offsets probe deeper model structure; if they match worse than near offsets, the deep model is wrong.
• Per-azimuth. In wide-azimuth marine, matching one azimuth much better than another signals azimuthal anisotropy that was not modelled.
• Per-shot. Shots in one part of the survey matching worse than another signals a localised velocity error in that part of the model.
• Per-frequency band. If 3–5 Hz matches but 8–12 Hz does not, the inversion stopped short at the bottom of the spectrum and failed to refine at higher frequencies.
• Per-event. Match pre-critical reflections well but post-critical refractions poorly? The deep velocity structure has errors the reflection data did not expose.

4. Held-out data

The single most important safeguard against overfitting: never use all your data for inversion. Withhold 10–20 % of shots from the FWI loop and use them only for QC. If the withheld shots match the final model as well as the training shots did, the inversion has generalised. If the withheld shots match significantly worse, you have overfit the training set — common failure mode when regularisation is too weak or when the data has inconsistent noise.

Industrial best practice: reserve 3–5 representative shots per acquisition block, plus one or two well-tied traces where log data is available, as a permanent QC reference across all iterations.

5. Well-tie comparison

If the survey has wells, synthesise a trace at the well location from the FWI model (1D layered simulation from the model's velocity column) and compare to the well's sonic/density log or to the actual recorded trace at the well. The well tie is the gold standard of FWI QC — it compares the inverted model directly against an independent, high-resolution measurement of the earth. A good FWI should produce a synthetic that matches the well's sonic log within a few percent in interval velocity.

6. Common failure modes and their fingerprints

• Cycle skipping at low frequency. High amplitude spectrum coherence (looks right in frequency) but poor time-domain correlation (wrong cycle). Fix: start at lower frequency.
• Overfitting with bad regularisation. Training shots match well, held-out shots do not. Fix: increase model-space regularisation (smoothness, total-variation, well-tie constraint).
• Elastic mismatch with acoustic FWI. Near-offset match good, far-offset match fails. Residual has characteristic angle-dependent pattern. Fix: switch to elastic (§6.4).
• Source wavelet estimation error. Amplitude spectrum shape wrong, time-domain correlation unaffected. Fix: jointly invert for the source wavelet or use source-independent misfits.
• Missing near-surface or overburden complexity. Shot-to-shot misfit variance much higher than expected; per-shot time-shift is spatially systematic. Fix: improve the near-surface model (first-break tomography) before rerunning FWI.

7. Acceptance criteria for production

A reasonable sub-salt FWI project's acceptance criteria look like:

• Training-shot time-domain correlation > 0.92
• Training-shot normalised L2 residual < 0.20
• Training-shot spectral coherence over inversion band > 0.88
• Held-out shot time-domain correlation > 0.90 (within 2 % of training)
• Well-tie synthetic vs log peak correlation > 0.85
• Residual spatial distribution isotropic (no systematic azimuthal or spatial pattern)

Meeting all six is unusual — most projects release with one or two at threshold and the rest clearly above. Consistently failing one specific metric reliably points to the next piece of physics or pre-processing to add.

Part 6 closes here

You have the full FWI toolkit: the L2 objective function and adjoint-state gradient (§6.1), multi-scale frequency continuation (§6.2), encoded and computational strategies (§6.3), elastic and anisotropic physics (§6.4), and the QC pack that tells you when to ship (§6.5). Part 7 moves into quantitative interpretation: amplitude preservation, AVO, seismic inversion, and the workflows that turn a migrated image + velocity model into reservoir property estimates.

References

• Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics, 74, WCC1.
• Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, Part 1. Geophysics, 64, 888.
• Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49, 1259.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.