Full-waveform inversion and uncertainty: a model is not the same as the truth

What FWI is solving

Full-waveform inversion estimates subsurface properties by minimizing disagreement between observed seismic data and synthetic waveforms generated from a numerical Earth model. Because it uses more of the wavefield than travel-time tomography, FWI can recover structure at finer scale. That power can also create false confidence: a detailed model is not automatically a well-constrained model.

The inverse problem is nonlinear. The starting model, source wavelet, frequency progression, regularization, and objective function shape the optimization path. Cycle skipping can align the wrong waveform cycles when prediction and observation differ too much. Low frequencies, long offsets, and a sensible initial model reduce this risk without guaranteeing uniqueness.

Resolution varies through the model

Resolution depends on bandwidth, acquisition geometry, noise, source characterization, and illumination. A shallow central region may be strongly constrained while deeper and peripheral regions inherit more from smoothing or the starting model.

Regularization and learned priors can create geologically plausible detail where the recorded data provide limited evidence. The relevant question is therefore which spatial wavelengths the data support at each location—not how sharp the final image appears.

Uncertainty is not one number

Data uncertainty includes noise, missing traces, geometry error, and acquisition limitation. Physics uncertainty includes an imperfect source wavelet and omitted attenuation, anisotropy, elasticity, or three-dimensional propagation. Parameterization uncertainty appears when velocity is allowed to change while density or anisotropy is fixed. Optimization uncertainty arises because different starting models and algorithms can reach different acceptable solutions.

These terms interact. A velocity update can compensate for a wavelet error, and structure can be introduced to explain physics absent from the simulator. A single posterior standard deviation does not automatically reveal which mechanism dominates.

Why calibration and physical audits matter

Ensembles, probabilistic networks, and conformal methods can estimate uncertainty, but their numerical values must be tested. Calibration asks whether nominal levels agree with empirical outcomes under a defined data-generating setting. That calibration does not automatically transfer to a new basin, geometry, or wavelet.

Held-out shots, perturbed source signatures, acquisition subsets, and posterior predictive simulation probe whether candidate models reproduce data excluded from inversion. Uncertainty should respond when evidence weakens. If interval width is unchanged after severe shot decimation or wavelet error, the estimate may reflect architecture more than physics.

A practical reporting standard

Synthetic tests provide known truth; field data provide real complexity. Responsible validation uses both, and reports more than the final data misfit. Two materially different models can fit the same observations.

• State which data components and frequencies were inverted.
• Explain how the starting model and physical assumptions were chosen.
• Report sensitivity to initialization, regularization, and acquisition coverage.
• Test calibration and prediction on information not used in optimization.