FWI in practice: low-frequency strategies

Section 6.1 made the problem concrete: the L2 misfit $J(V)$ has a central basin around the true model. Gradient descent reaches it only while the traveltime error stays under about half a period, $|\\Delta t| < T/2 = 1/(2f)$, which in velocity is $|\\Delta V/V| \\lesssim 1/(2 f t\_0)$. If your initial model is farther from truth than that, 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 figure runs the 1D toy of Section 6.1 in three frequency bands at once: one reflector, one unknown velocity, true value 2000 m/s. Choose a starting velocity and a frequency schedule, run the inversion, and see which bands can reach the true velocity from that start.

Start from $V\_{\\text{start}} = 1800\\ \\text{m/s}$, 200 m/s slow, so the modelled reflection arrives 0.111 s late. At 3 Hz that is 0.33 of a period, inside the 3 Hz basin, which runs from 1748 to 2336 m/s in (b). At 10 Hz the basin has narrowed to 1917-2090 m/s and at 30 Hz to 1972-2029 m/s, so 1800 m/s is cycle-skipped in both, and 30 Hz alone stalls at 1798 m/s. Inverted in order, the 3 Hz stage gets close to 2000 m/s, the 10 Hz and 30 Hz stages start inside their basins, and the schedule ends at 2001 m/s after 10 iterations. The basins end where the traveltime error reaches 0.43 of a period, a property of the Ricker wavelet; half a period is the rule of thumb. Now try 1600 m/s: its error is 0.75 of a 3 Hz period, and even the lowest band fails until the lowest usable frequency drops to 1.5 Hz. The implication is the entire multi-scale workflow:

1. Filter the observed data to the lowest available frequency band.
2. Invert at that low frequency until convergence. The initial model only needs to be within the wide low-frequency basin.
3. 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.
4. 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. The low bands are robust but imprecise: in the figure the 3 Hz answer varies by 4.1 m/s (one standard deviation) over eight noise draws, and the 30 Hz stage tightens it to 0.4 m/s. They are also cheap, because they run on coarse grids; in 3D the per-iteration cost grows roughly as $f^4$, so the highest band dominates the bill. 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 are dominated by reflections, which constrain impedance contrasts more than velocity. Far-offset traces record diving and refracted waves, transmitted arrivals whose traveltimes pin down the long-wavelength velocity that FWI must get right first. 8-12 km streamers (vs 4-6 km standard) improve FWI dramatically.
• Ocean-bottom nodes (OBN). Node receivers sit on the quiet seafloor, away from swell noise and the streamer receiver ghost that suppresses the lowest frequencies, and record 1.5-3 Hz cleanly, about an octave below conventional streamers. 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. On land, MEMS accelerometers keep a flat response below the roughly 10 Hz natural frequency of conventional geophones, extending the usable band downward.
• 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, so its misfit has a wide, near-convex basin even at high frequency. In the figure, with 30 Hz only and the envelope first, a start 1.25 periods out (1920 m/s) reaches 2001 m/s, where L2 alone stops in a side minimum at 1942 m/s; at 4 % noise the envelope's reach ends near two periods, where the envelope misfit sinks into the noise floor; without noise it reaches much farther. 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 (Section 5.9) as the starting point. If the tomography model's traveltimes are within half a period of truth, $|\\Delta V/V| < 1/(2 f t\_0)$ (about 2.5% for 10 Hz energy that has travelled 2 s), even 10 Hz FWI starts inside 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:

1. Tomography + well ties → initial model smooth to ~200 m scale.
2. 1.5-2.5 Hz envelope FWI for 50 iterations → model closer to truth.
3. 2.5-4 Hz L2 FWI on filtered OBN data, 100 iterations.
4. 4-6 Hz L2 FWI, 100 iterations.
5. 6-9 Hz L2 FWI, 100 iterations.
6. 9-15 Hz L2 FWI, 50 iterations.
7. 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

Section 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.
• Bunks, C., Saleck, F. M., Zaleski, S., Chavent, G. (1995). Multiscale seismic waveform inversion. Geophysics, 60, 1457.
• Warner, M., Guasch, L. (2016). Adaptive waveform inversion: Theory. Geophysics, 81, R429.