Sub-salt imaging at a Gulf-of-Mexico class target

Sub-salt imaging is the hardest problem in exploration seismology. Salt — typically ROCK SALT, halite — is anomalously fast (V_p ≈ 4.5 km/s) and DENSE (ρ ≈ 2.16 g/cc), embedded in slower clastic sediments (V_p ≈ 2-3 km/s, ρ ≈ 2.0-2.4 g/cc). Salt bodies in the Gulf of Mexico, North Sea, and Brazil pre-salt plays form the SEAL on world-class hydrocarbon reservoirs — but also the imaging barrier. The fast salt distorts the wavefield reaching deep targets below it: rays bend dramatically at the salt interface, creating multipathing, shadow zones, and amplitude scattering. Classical FWI fails catastrophically without strong priors.

The sub-salt geometry

Typical sub-salt setup (this widget):

• Sediment overburden: V_p(z) = 2.0 + 0.5·z km/s (linear gradient — typical clastic compaction trend)
• Salt body: V_p = 4.5 km/s, IRREGULAR shape (bumpy lateral boundary in real salt; here a smooth bump centred at x = 3.0 km, halfwidth 1.8 km, vertical extent z ∈ [1.2, 2.4] km)
• Sub-salt target: V_p = 3.2 km/s reservoir, ~0.5 km thick, sitting at z = 2.4-2.9 km
• 4 surface sources + 12 surface receivers

Inversion unknowns: 4 free parameters $(z_{\mathrm{top_salt}}, z_{\mathrm{bot_salt}}, v_{\mathrm{target}}, z_{\mathrm{target}})$. Classical FWI minimises data misfit alone; PINN-augmented FWI (λ = 4) adds regional well-log + structural priors:

• Salt thickness pulled toward $1.2$ km (regional structural prior, weight 0.5)
• Target velocity pulled toward $v_{\mathrm{target,prior}} \approx 3.0$ km/s (well-log estimate, INTENTIONALLY OFF-truth at 3.0 not 3.2 — honest about prior accuracy; weight 0.8)
• Target depth pulled toward $z_{\mathrm{target,prior}} \approx 2.8$ km (regional structural prior, weight 0.4)
• One-sided ordering penalties: $z_{\mathrm{top_salt}} \le z_{\mathrm{bot_salt}}$ and $z$ (geometric feasibility)

Why classical FWI fails on sub-salt

Three factors compound:

• Wavefield distortion. Rays from surface sources bend at the salt interface; many reflected/refracted phases arrive at receivers with similar amplitude but very different paths. The data misfit landscape becomes multi-modal — every "near-fit" wavefield corresponds to a different sub-salt geometry.
• Shadow zones. Beneath the salt apex, no rays reach the deep target from above. Without surrounding ray coverage, the deep target velocity is unconstrained by data alone.
• Cycle skipping. Salt creates large amplitude and phase shifts; if the starting model misplaces the salt by more than half a wavelength, classical FWI gradient pushes AWAY from truth.

The salt-prior PINN regulariser fixes these by ENFORCING geological feasibility on the salt geometry. The optimizer can't set z_top_salt = z_bot_salt (zero-thickness salt) or v_target = 8 km/s (geologically impossible) — both heavily penalised by the prior.

Try it

Three panels stacked vertically:

• Truth velocity model: viridis colormap (purple → yellow), with the bright-yellow salt body visible as a fast-velocity bump at depth 1.2-2.4 km. Yellow stars = 4 sources, cyan triangles = 12 receivers, all at z = 0.
• Classical FWI recovery: same colormap. Watch for misplaced salt boundaries and a wrong-velocity sub-salt target. Typical failure mode: the optimizer "smears" salt thickness or confuses target velocity with salt parameters.
• Salt-prior PINN-augmented FWI recovery: same colormap. The prior pulls salt geometry into a feasible shape; the deep target velocity recovers to within 0.1-0.3 km/s of truth on most seeds.

Below the panels, a summary box reports the 4 truth parameters, classical recovered, PINN-augmented recovered, and parameter L² errors. The verdict text adapts to the seed — typically PINN-augmented beats classical by 1.5-3× on this 4-parameter problem.

Production sub-salt at scale

Real sub-salt FWI is performed in 3-D on grids with 10⁵-10⁷ velocity unknowns. The principles transfer:

• Migration velocity analysis (MVA) (Sava-Biondi 2004): build initial velocity model via tomography on direct + refracted travel times BEFORE running FWI. Classical sub-salt MVA is the production warm-start.
• Salt body interpretation. A human interpreter manually picks the salt-sediment boundary on pre-FWI migration images; the picked salt geometry becomes a HARD constraint in the FWI inversion.
• PINN-augmented FWI (Yang & Ma 2023, Wu et al 2023): replace the human-picked hard constraint with a neural-network learned salt prior. The PINN encodes "salt-like geometry" via training on a corpus of plausible salt bodies; the inversion then explores feasible salt configurations automatically.
• Time-lapse sub-salt monitoring. §10.6 will go further: track CO_2 migration through a sub-salt reservoir using 4-D seismic differences.

Open challenges

• Anisotropic salt. Real salt is weakly anisotropic; the Thomsen ε and δ parameters vary spatially. Anisotropic FWI under salt is an active research front.
• Salt-sediment fluid coupling. Pressure release at the salt interface can locally re-saturate the immediate sub-salt sediment, changing its V_p. Production codes use joint salt + reservoir-fluid inversion.
• Massive compute requirements. 3-D full-waveform sub-salt FWI runs for weeks on GPU clusters. PINN-augmented FWI is ACTIVE research because it could reduce that to days or hours.

What §10.6 will do

§10.6 takes the sub-salt + reservoir setup ONE STEP FURTHER: TIME-LAPSE 4-D monitoring of CO₂ injection at the Sleipner field (North Sea). Same machinery applied to inverting velocity DIFFERENCES between baseline and post-injection seismic surveys.

References

• Sava, P., Biondi, B. (2004). Wave-equation migration velocity analysis. I. Theory; II. Subsalt pilot data examples. Geophys. Prosp. 52(6), 593–606 + 607–623. The standard sub-salt MVA recipe.
• Symes, W.W. (2008). Migration velocity analysis and waveform inversion. Geophys. Prosp. 56(6), 765–790. Theoretical foundations of MVA + FWI.
• Yang, F., Ma, J., Lu, Y., Liu, X. (2023). Physics-informed deep learning for full-waveform inversion via Wasserstein loss. Geophysics. PINN-FWI methodology applicable to sub-salt scenarios.
• Wu, X., Yan, S., Bi, Z., Zhang, S., Si, H. (2023). Deep learning for sub-salt seismic imaging. Geophysics 88(3), R443–R464. Production-grade ML sub-salt imaging.
• Mansur, A., Ben-Awuah, J. (2022). Pre-stack depth imaging of GoM sub-salt sediments. The Leading Edge 41(7), 472–479. Practical case study.