Uncertainty-aware velocity inversion

Part 9 capstone. The previous five sections gave us tools: (a) PINN as initialiser/regulariser/solver (§9.1); (b) λ-annealed PINN-augmented FWI (§9.2); (c) AE-based learned priors (§9.3); (d) generative priors for sampling (§9.4); (e) ensemble PINN UQ (§9.5). §9.6 puts them together to produce what production seismic-imaging teams actually need: a velocity model AND an uncertainty map.

Bootstrap-posterior FWI

The simplest production UQ recipe: bootstrap-resampling. For each of $N$ posterior samples:

• Re-sample noise on the data. $t_{\mathrm{obs}}^{(k)} = t_{\mathrm{obs}} + \varepsilon_k$ where $\varepsilon_k \sim \mathcal{N}(0, \sigma_{\mathrm{pick}}^2 \cdot I)$.
• Run a regularised inversion on the perturbed data: $p^{(k)} = \arg\min_p \bigl[ \mathcal{L}${\mathrm{data}}(p; t{\mathrm{obs}}^{(k)}) + \lambda , \mathcal{L}_{\mathrm{phys}}(p) \bigr].
• Record $p^{(k)}$ as one posterior sample.

The empirical distribution ${p^{(1)}, \ldots, p^{(N)}}$ is an approximation to the true posterior $p(p \mid t_{\mathrm{obs}})$. Mean is the central estimate; std gives uncertainty bars; 1.96·std is the 95% credible interval (under normality).

This is NOT a fully rigorous Bayesian posterior — it conditions on point estimates of the prior + regulariser + initial guess rather than marginalising over them. But it captures the dominant uncertainty source for clean-data FWI (data picking noise) and is the production simplest-thing-that-works.

Try it

The widget:

• Generates $t_{\mathrm{obs}}$ from the truth ($z_{\mathrm{int}} = 0.5$, $v_1 = 1.5$, $v_2 = 3.0$) at 8 surface receivers.
• For 8 posterior samples: re-noise $t_{\mathrm{obs}}$ with $\sigma_{\mathrm{pick}} = 0.02$ s AND jitter the smart initial guess $(0.45, 1.6, 2.8)$ by $\sigma_{\mathrm{init}} = (0.05, 0.15, 0.15)$, then run a 50-iteration mildly-regularised inversion (§9.2 hand-crafted prior, λ=0.3). The combined data-bootstrap + init-bootstrap is a simplified production-style "prior bootstrapping" that captures BOTH data uncertainty AND optimization-basin uncertainty (without it, every sample would cluster in the same partial-convergence well).
• Plot the posterior samples + mean ± 2σ band.

Two panels:

• Posterior c(z) profile: truth (yellow dashed), individual posterior samples (faint orange), posterior mean (orange bold), 2σ band (light orange shaded). Where the band is narrow, the inversion is confident; where it widens, the data is uninformative about that region.
• Joint (v₁, v₂) posterior cloud: 8 samples plotted as orange dots, truth as yellow star. The cloud's shape reveals correlations — for this 2-layer problem the cloud is elongated along the constraint $v_2 - v_1 \approx 1.5$ enforced by the §9.2 prior, with cross-cloud spread reflecting how much the data-misfit gradient + init jitter pushes individual samples off that ridge.

The summary box reports posterior mean ± 1σ for each parameter, the 95% credible interval, and a coverage check (does each CI include the truth?). For an honest posterior, all 3 of 3 CIs should cover truth.

What production codes do beyond this

The widget illustrates the simplest bootstrap-posterior recipe. Production seismic-imaging UQ pipelines layer on additional uncertainty sources:

• Prior bootstrapping. Run inversions from MANY different initial guesses to capture model-form uncertainty. Each initial guess samples a different basin of the loss landscape.
• Regulariser bootstrapping. Sample $\lambda$ from a hyperprior in each run — captures regularisation-strength uncertainty.
• Generative-prior bootstrapping. §9.4 generative-prior sampling: each posterior sample uses a different draw from the corpus prior as warm-start, ensuring the recovered model is on-manifold.
• Full Bayesian inference. HMC or SVGD over the FULL posterior $p(p \mid t_{\mathrm{obs}}) \propto p(t_{\mathrm{obs}} \mid p) p(p)$ using the generative prior $p(p)$. Most rigorous but costliest. Smith et al 2022 HypoSVI uses this approach for hypocentre location.

For 2-D / 3-D production FWI with thousands of unknowns, computational tractability forces approximations. The cheapest is bootstrap-posterior (this section). Next cheapest is ensemble PINN over multiple inversions. Most expensive but most rigorous is full HMC over the generative-prior latent.

Calibration: are the uncertainty bars HONEST?

An uncertainty estimate is HONEST (or "calibrated") if its credible intervals contain the truth at the claimed rate. For 95% CIs, an honest method covers truth in 95% of independent runs. Calibration matters MORE than width — narrow intervals that miss the truth are worse than wide intervals that contain it.

Production calibration tests:

• Coverage probability test. Run $M$ inversions on synthetic data with KNOWN truth. Check what fraction of $M$ runs have CIs containing truth. Should match the claimed coverage (95% for 95% CIs).
• Reliability diagram. Plot empirical coverage vs claimed coverage across $\alpha \in [0, 1]$. Diagonal line = perfect calibration.
• Cross-validation. Hold out some receivers, predict their travel times via the posterior, check if held-out values fall in predicted intervals at the claimed rate.

Pure data-bootstrap (re-noise only, fixed init) typically UNDER-COVERS — credible intervals are narrower than the true posterior because all samples cluster in the same optimization basin. The widget here adds INIT-jitter on top of data-bootstrap, which inflates the cloud enough to give honest 95% coverage on this toy problem. Production codes layer in even more uncertainty sources (regulariser bootstrapping, prior sampling, cross-validation calibration) and inflate the bootstrap intervals by 1.5-3× when held-out data shows under-coverage.

Operational notes

• Number of samples. N=8 is the educational minimum. Production: N=100-1000 for tight CIs. Each sample is independent so embarrassingly parallel.
• Picking noise estimate matters. Set $\sigma_{\mathrm{pick}}$ from the actual pick variance. Too small → over-confident posterior; too large → useless wide intervals.
• Reporting standard. Always quote median + 5-95th percentile rather than just mean ± 1σ — captures asymmetry in the posterior. For seismic, the SEG-D (Society of Exploration Geophysicists Data) format includes uncertainty fields specifically.
• Decision-theoretic interpretation. A velocity model + uncertainty enables decisions about where to drill, where to acquire more data, and what oil-reserve confidence intervals to quote. Without uncertainty, FWI is just a point estimate that may or may not be reliable.

Part 9 wrap-up

Part 9 covered the production hybrid + UQ stack:

• §9.1 PINN's three roles in classical FWI: solver, initialiser, regulariser.
• §9.2 PINN-augmented FWI with the U-curve of regularisation strength + annealing.
• §9.3 Learned regulariser via autoencoder reconstruction loss.
• §9.4 Generative priors: AE-as-sampler, bridge to VAE / diffusion.
• §9.5 Ensemble PINNs for epistemic UQ (deep ensembles recipe).
• §9.6 Bootstrap-posterior FWI: velocity model + uncertainty map.

The reader who completes Part 9 has the toolkit to ship a production-grade UQ-aware seismic-inversion pipeline. Part 10 will walk eight real-field capstones — Marmousi, Sleipner, BP velocity benchmark, ANT (ambient-noise tomography), microseismic monitoring — applying the techniques from Parts 1-9 end-to-end on realistic data.

References

• Efron, B. (1979). Bootstrap methods: another look at the jackknife. Ann. Stat. 7(1), 1–26. The bootstrap paper that started the field.
• Sambridge, M., Mosegaard, K. (2002). Monte Carlo methods in geophysical inverse problems. Rev. Geophys. 40(3), 1009. Comprehensive review of MCMC for seismic inversion.
• Tarantola, A. (2005). Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM. The standard textbook on Bayesian inverse problems including FWI.
• Smith, J.D., Ross, Z.E., Azizzadenesheli, K., Muir, J.B. (2022). HypoSVI. GJI 228, 698–710. Production Bayesian PINN-FWI with SVGD.
• Zhang, X., Curtis, A. (2020). Variational full-waveform inversion. GJI 222(1), 406–411. Variational Bayesian FWI on real seismic data.
• Gebraad, L., Boehm, C., Fichtner, A. (2020). Bayesian elastic full-waveform inversion using Hamiltonian Monte Carlo. JGR 125(3), e2019JB018428. HMC over elastic-FWI posterior.