FWI in practice: low-frequency strategies

§6.1 made the problem concrete: the L2 misfit $J(V)$ has a central basin around the true model whose width scales as $1/(2f)$. If your initial model is farther from truth than that half-period, gradient descent converges to a cycle-skipped solution. Every low-frequency strategy in production FWI is an engineering response to this constraint: either broaden the basin (by going lower in frequency, or by changing the misfit) or shrink the distance between initial model and truth (by starting from a better model).

1. Multi-scale continuation — basin chaining

The widget plots $J(V)$ at 3 Hz, 10 Hz, and 30 Hz for the same 1D toy. Pick $V_{guess} = 1500\ \text{m/s}$ (500 m/s off truth). At 3 Hz the basin runs [1714, 2400] m/s — 1500 is cycle-skipped at 3 Hz too, but barely; shift up to 1800 and it is inside the 3 Hz basin. At 10 Hz the basin has collapsed to [1905, 2105] — 1500 (and 1800) are deeply cycle-skipped. At 30 Hz the basin is only [1967, 2034] m/s wide. The implication is the entire multi-scale workflow:

• Filter the observed data to the lowest available frequency band.
• Invert at that low frequency until convergence — the initial model only needs to be within the wide 3 Hz basin.
• Use the converged low-f model as the starting model for the next frequency band. Because the low-f inversion moved the model closer to truth, you are now inside the narrower basin at the next frequency up.
• Repeat for 10 Hz, then 15 Hz, then 20 Hz, up to the maximum usable frequency of the data.

A production schedule typically uses 5–10 frequency bands, each inverted for 10–100 iterations before moving up. Computational cost grows linearly in the number of bands; the payoff is avoiding cycle skipping at every step.

2. Stretching the bottom of the bandwidth

The single most valuable piece of a marine survey for FWI is the few Hz at the lowest end of the spectrum. Every production technique below is about buying more of that:

• Long offsets. Near-offset traces record steep-angle reflections with narrow frequency content. Far-offset traces pick up refracted and diving waves that carry low-f energy deep into the earth. 8–12 km streamers (vs 4–6 km standard) improve FWI dramatically.
• Ocean-bottom nodes (OBN). Node receivers sit on the seafloor, avoiding the streamer notch from the ghost and picking up 1.5–3 Hz cleanly — a full octave below what streamers deliver. Many sub-salt FWI successes are from OBN data.
• Broadband marine sources. Low-frequency source arrays (tuned air guns, vibrators, marine vibroseis) extend the acquired spectrum down to ~2 Hz. Add nodes and you can FWI at 1.5 Hz.
• Accelerometer receivers. Conventional hydrophones measure pressure; accelerometers measure particle motion and have less low-f noise floor, extending the signal band downward by 0.5–1 Hz.
• Low-cut filter relaxation. Relaxing the low-cut of the processing flow (from, say, 5 Hz to 2.5 Hz) preserves the low-f information the later stages had been throwing away.

3. Misfits whose basin is wider than L2

A different line of attack: change the shape of $J(m)$ instead of changing the data. Two families are widely used:

Envelope FWI replaces the trace with its envelope (amplitude of its analytic signal) before computing the misfit. The envelope is a slowly-varying, non-oscillatory function — its misfit has a wide, near-convex basin even at high frequency. The cost: envelope is phase-blind, so an envelope-only inversion converges to a smooth, low-resolution velocity model. Use it as a preconditioner: run envelope FWI first to move the model close, then switch to L2 FWI to recover the high-resolution detail.

Adaptive Waveform Inversion (AWI) uses a Wiener-filter-based misfit: compute the filter that maps synthetic data to observed data and penalise its deviation from a delta function. This misfit is insensitive to half-wavelength misalignments that cycle-skip the L2 norm, yet recovers the same final model as L2 once close. Developed by Warner & Guasch (2016), it has become a common starting-stage misfit in deep-water sub-salt projects.

Other phase-coherent misfits include correlation-coefficient matching, deconvolution-based misfits, and optimal-transport (Wasserstein) distances — each with a different trade-off between basin width and final resolution.

4. Data preconditioning

• Time windowing. Early-arrival FWI uses only the first-break travel time, producing a robust low-resolution model. Reflection FWI uses the later reflected energy for high-resolution detail.
• Offset muting. Near-offset traces are dominated by reflections that contain less velocity information. Some production FWI starts with far-offset data only (the diving-wave regime) and adds near offsets later.
• Azimuth selection. In wide-azimuth marine, each azimuth samples different parts of the subsurface. Inverting one azimuth at a time can avoid some cycle-skipping pathologies.

5. Starting model quality

The other half of the cycle-skipping trade is starting closer. Investment in a better initial model directly reduces the need for aggressive low-f tricks:

• High-quality tomography (§5.9) as the starting point. If the tomography model is within a few percent of truth everywhere, even 10 Hz FWI has its guess in the basin.
• Well-log calibration. Tying the interval-velocity model to sonic logs at well locations constrains the model where logs exist; lateral extrapolation covers everywhere else.
• Prior FWI on nearby surveys. Regional velocity models from 4D or legacy work provide a rough starting point that a freshly acquired dataset refines.

6. A realistic production schedule

Putting it all together: a typical deep-water sub-salt FWI project might run:

• Tomography + well ties → initial model smooth to ~200 m scale.
• 1.5–2.5 Hz envelope FWI for 50 iterations → model closer to truth.
• 2.5–4 Hz L2 FWI on filtered OBN data, 100 iterations.
• 4–6 Hz L2 FWI, 100 iterations.
• 6–9 Hz L2 FWI, 100 iterations.
• 9–15 Hz L2 FWI, 50 iterations.
• 15–20 Hz L2 FWI if possible, 30 iterations.

Each stage's model feeds into the next. The total wall-clock time on a modern GPU cluster is several weeks per 3D survey.

Where this goes next

§6.3 turns to the other cost axis: computational. A naive FWI at 6 Hz on a 3D survey is thousands of GPU-hours per iteration. Source encoding, shot selection, stochastic FWI, and the encoded family of techniques collapse that cost by an order of magnitude.

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.