2D Marmousi-class inversion: setup

§6.2 closed the 1D loop on a clean toy problem. Real FWI is 2D or 3D, on velocity models with thousands of layers, and the data is a bank of multi-shot multi-receiver seismograms. The de-facto benchmark for testing FWI algorithms is Marmousi.

What is Marmousi?

The Marmousi model was constructed in 1988 by the Institut Français du Pétrole (IFP) for a 1990 EAGE-SEG benchmark workshop. The geological inspiration was the Cuanza Basin in Angola — a deltaic stratigraphy with strong velocity contrasts, faulted blocks, and a buried gas-charged sand. The original Marmousi was 9.2 km × 3 km with 240+ velocity layers and velocities from 1500 m/s near the seabed to 5500 m/s at depth (Versteeg 1994). Marmousi II (Martin, Wiley, Marfurt 2006) extended it to elastic and added P-wave velocity, S-wave velocity, density, plus a fluid model.

For thirty-five years, every published FWI algorithm has been benchmarked against Marmousi or one of its descendants (BP-2004, SEAM-Phase-I, Marmousi II, Marmousi III). It is to FWI what MNIST is to deep learning — solved enough that researchers complain, but unsolved enough that you cannot publish a serious FWI paper without it.

What makes it hard

Four structural ingredients combine to defeat naive FWI:

• Compaction trend. Velocity increases monotonically with depth (sediment compaction). The gradient is large — order 500 m/s per km — so seismic ray paths bend steeply. The forward problem couples velocities at every depth.
• Dipping layers with strong contrasts. Layer interfaces are not horizontal. Reflections come from off-vertical angles, and the relationship between recorded data and velocity is non-linear in a way that simple stratified-Earth models do not capture.
• Gas pocket / low-velocity anomaly. An embedded zone where $v$ drops 10–30%. Causes a "shadow zone" below the anomaly where rays do not penetrate, leading to nullspace-like ambiguity in FWI.
• Faulted blocks. Sharp lateral velocity jumps caused by tectonic offsets. Adjacent layers slip past each other, creating discontinuities the inversion must reconstruct from data that has been smoothed by wave propagation.

FWI without a good starting model fails on Marmousi every time. The data has so many high-frequency arrivals (reflections from every interface) that cycle skipping (§6.5) defeats the $L^2$ misfit at the shallow layers, propagating wrong updates downward.

The starting model

Production FWI workflows always begin with a SMOOTH starting model derived from another method:

• Refraction tomography: invert first-arrival travel times for a low-resolution velocity field. Resolves features at scales $\gtrsim$ a few wavelengths.
• Stacking velocity analysis: from CMP gathers, fit hyperbolic moveout to extract average velocities. Used since the 1960s; still standard.
• Check-shot and well-log calibration: drop a geophone in a well and time the source pulse to it. Gives a 1D velocity profile at well locations.
• Borehole sonic logs: high-resolution 1D velocity from a downhole sonic tool. Used to constrain the inversion at well locations.

The starting model is then HEAVILY SMOOTHED — typically with a Gaussian of sigma comparable to the dominant seismic wavelength (50–200 m for marine seismic). The job of FWI is to put back the high-spatial-frequency content using the seismic data. Smoothing is the standard FWI trick to avoid cycle skipping; §6.4 builds frequency continuation as the formal multi-scale extension.

Try it

The widget above builds a Marmousi-class proxy at 6 km × 3 km — the same structural ingredients (compaction, dipping layers, gas pocket, faulted block) at a coarse 121 × 61 grid that runs in the browser. Drag the smoothing-σ slider to see how progressively heavier smoothing of the truth produces increasingly low-resolution starting models. Pick a shot and click ▶ Forward-model: the FDTD solve runs in $\sim 0.5 \textrm{--} 2$ seconds and produces a 30-trace shot record (one trace per receiver, time on the y-axis). Notice the multiple wavefronts: direct wave first, then reflections from each interface, with travel-time and amplitude variations across the receiver line.

Compute scale: real Marmousi vs this widget

The widget is downsized by about $10^4 \times$ across every dimension. This is enough to convey the QUALITATIVE behaviour of FWI on Marmousi-class problems but does NOT quantitatively reproduce production FWI. Reference papers (Brossier 2009 PhD; Operto 2016; Borisov & Operto 2024) report production FWI runs at the full scale.

What §6.4 will do

§6.4 chains the §6.2 FWI iteration across MULTIPLE FREQUENCY BANDS via the Bunks-1995 frequency-continuation curriculum — the standard FWI remedy for the cycle-skipping problem. The widget races single-frequency FWI (§6.2 baseline) against a 3-stage curriculum on the §6.2 1D toy problem. The same recipe extends to Marmousi-class 2D production runs (Brossier 2009 PhD); the 1D toy makes the curriculum concept interactive in-browser without the production compute cost.

References

• Versteeg, R. (1994). The Marmousi experience: Velocity model determination on a synthetic complex data set. The Leading Edge 13(9), 927–936.
• Martin, G.S., Wiley, R., Marfurt, K.J. (2006). Marmousi2: An elastic upgrade for Marmousi. The Leading Edge 25(2), 156–166.
• Bunks, C., Saleck, F.M., Zaleski, S., Chavent, G. (1995). Multiscale seismic waveform inversion. Geophysics 60(5), 1457–1473. The frequency-continuation cure for cycle skipping.
• Brossier, R. (2009). Imagerie sismique à deux dimensions des milieux viscoélastiques par inversion des formes d'onde: Développements méthodologiques et applications. PhD thesis, Université Nice. Marmousi-class production FWI reference.
• Operto, S. (2016). Tutorial: 2-D and 3-D wave propagation modelling and inversion in time and frequency domains. SEG.