Defending an inversion run: convergence diagnostics

Capstone of Part 6. The §6.2–§6.8 widgets all converged to truth on the toy 1D problem. In production FWI on real data, you do NOT have ground-truth velocity to compare against. You must judge whether the inversion succeeded from the available signals — and the data-fit residual reduction is necessary but not sufficient.

The four production-FWI diagnostics

• $J$ vs iteration. The misfit must DECREASE. In a well-behaved inversion, $J$ falls roughly exponentially in early iterations and plateaus near the optimum. A flat $J$ from iteration 1 means line search is failing (cycle-skipped or step too small). A growing $J$ means divergence (step too aggressive or sign error).
• $|c - c_{\mathrm{init}}|$ vs iteration. The model-update magnitude. Early iterations should show large updates; late iterations should taper. A model that doesn't move at all has a failed line search; a model that keeps moving never converges.
• $|\nabla J|$ vs iteration. The gradient norm. AT a local minimum $|\nabla J| = 0$. A non-zero plateau in $|\nabla J|$ means the inversion is at a saddle or step-size-limited (not at a minimum). A growing $|\nabla J|$ with shrinking $J$ usually means a parameter blow-up.
• Line-search step size vs iteration. The actual step taken per iteration. Healthy convergence: grows in early iterations, plateaus, may shrink near the optimum. Cycle-skipping: shrinks to zero immediately as line search fails. Bad step₀: oscillates between two scales.

Reading the four signals together

The diagnostic is not in any single trace; it is in their JOINT shape:

• Healthy convergence: J↓ exponentially, model-error↓, |∇J|↓, step plateaus or grows.
• Cycle-skipped: J flat, model-error flat, |∇J| flat (gradient sign wrong, magnitude unchanged), step → 0 (line search fails). The §6.4 / §6.5 cures apply.
• Stuck at saddle: J slightly down, model-error slightly down, |∇J| flat at non-zero, step shrinking. Need step-size-control adjustment or quasi-Newton (L-BFGS).
• Found wrong minimum: J↓, but model-error UP. Data fit improved by going AWAY from truth. The §6.6 cross-talk valley case. Multi-shot illumination needed.

Try it

Pick a starting c₂ and watch all four signals together:

• $c_2 = 1.8$ or 2.0: healthy. All four traces tell the same story.
• $c_2 = 0.7$ or 2.3: cycle-skipped. J flat or noisy, |∇J| flat, step → 0.
• $c_2 = 1.55$ (close to truth): healthy and fast. Three iterations, done.

The verdict text auto-classifies the inversion outcome from the joint signal shape. In production this kind of automated diagnostics IS the inversion-quality controller — automated codes flag suspect runs for human review.

The defence-of-inversion checklist

Before publishing or shipping an FWI result:

• $J$ reduced by ≥ 1 order of magnitude (robust); ideally ≥ 2 orders for clean acoustic data.
• Final |∇J| at least 1 order smaller than initial.
• Model-update trajectory smooth (no oscillations or jumps).
• Line-search step well-behaved (no shrink-to-zero cliffs).
• Final velocity model passes geological sanity check (no negative velocities, no high-frequency noise above seismic resolution, no impossibly fast layers).
• Synthetic shot records from the inverted model overlay the observed data well at all receivers.
• Holdout test: leave one shot out, invert on the rest, predict the holdout shot from the inverted model. Mismatch should be at the data-noise level.
• Resolution analysis: the model is only resolved at the spatial scales the data illuminates. State this.

Closing of Part 6

You now have the complete classical-FWI / PINN-FWI workflow:

• §6.1 The adjoint-state gradient. The atom of FWI.
• §6.2 The iteration loop. Plessix gradient + line search + Tikhonov, applied repeatedly.
• §6.3 Marmousi as the reference benchmark; smooth starting model construction.
• §6.4 Frequency continuation as the curriculum cure for cycle skipping.
• §6.5 Alternative misfits (envelope, Wasserstein, AWI) as the misfit-side cure.
• §6.6 Multi-parameter cross-talk and its remedies (multi-shot, joint inversion, impedance reparameterisation).
• §6.7 Source encoding: 100× compute saving via stochastic super-shots.
• §6.8 Loss-weight balance: the Tikhonov / NTK / SA-PINN auto-tuning trio.
• §6.9 Convergence diagnostics: how to know your inversion succeeded.

Part 7 (next) extends to TRAVEL-TIME and SURFACE-WAVE inversion: simpler PDE (eikonal), different physics, complementary FWI tooling. Part 8 covers OPERATOR LEARNING (DeepONet, FNO) — neural networks that learn the FORWARD wave-equation operator and provide GPU-friendly surrogates. Part 9 hybridises FWI + classical and adds Bayesian uncertainty quantification.

Expertise checkpoint — end of Part 6

You should now be able to:

• Derive the adjoint-state gradient $\partial J / \partial c(x) = -(2/c^3) \int u_{tt} \lambda dt$ from scratch.
• Implement a classical FWI iteration loop in 100 lines of code (you have, in §6.2).
• Read a Marmousi-class velocity model and recognise the structural ingredients that defeat naive FWI.
• Apply frequency continuation, envelope misfit, source encoding, and loss-weight tuning to specific failure modes you can diagnose from the convergence traces.
• Defend an FWI inversion to a sceptical audience using the four-diagnostic recipe.
• Read a PINN-FWI paper (Sun-Alkhalifah, Rasht-Behesht, Song-Alkhalifah, Yang et al.) and place its contribution in the §6.1–§6.9 taxonomy.

References

• Plessix, R.-E. (2006). A review of the adjoint-state method. GJI 167(2). Convergence indicators discussed throughout.
• Métivier, L., Brossier, R. (2016). The SEISCOPE optimization toolbox. Geophysics 81(2), F1–F15. Reference convergence-criterion implementations.
• Hanke, M. (1995). Conjugate Gradient Type Methods for Ill-Posed Problems. Pitman Research Notes. Discrepancy-principle theory.
• Brossier, R. (2009). Imagerie sismique à deux dimensions des milieux viscoélastiques par inversion des formes d'onde. PhD thesis. Production-FWI workflow with diagnostics.