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Total-variation regularization strategies in full-waveform inversion
Ernie Esser, Lluis Guasch, Tristan van Leeuwen, Aleksandr Y. Aravkin, Felix J. Herrmann
TL;DR
Full-waveform inversion suffers from parasitic local minima and missing spatial frequencies, especially when reconstructing high-contrast salt structures from poor starting models. The paper introduces a constrained inversion framework using relaxed, warm-started convex subproblems with box, TV, and asymmetric TV constraints. Experiments show improved artifact removal and salt delineation, with asymmetric TV addressing the bottom of salt bodies for poor initializations.
Problem
Full-waveform inversion must recover large- and small-scale velocity structure despite local minima, bandwidth limitations, finite aperture, and coupled wave-equation scales.
Method
The framework solves relaxed and warm-started constrained subproblems using box, total-variation, and one-sided asymmetric total-variation constraints.
Results
The scheme removes inversion artifacts and improves salt delineation for accurate starting models, while asymmetric TV improves salt recovery from poor starting models.
Takeaways & Limitations
Successively relaxed asymmetric TV discourages early velocity lows and helps prevent the inversion from becoming trapped when initialization is poor.
Abstract
from arXiv · showhide
We propose an extended full-waveform inversion formulation that includes general convex constraints on the model. Though the full problem is highly nonconvex, the overarching optimization scheme arrives at geologically plausible results by solving a sequence of relaxed and warm-started constrained convex subproblems. The combination of box, total-variation, and successively relaxed asymmetric total-variation constraints allows us to steer free from parasitic local minima while keeping the estimated physical parameters laterally continuous and in a physically realistic range. For accurate starting models, numerical experiments carried out on the challenging 2004 BP velocity benchmark demonstrate that bound and total-variation constraints improve the inversion result significantly by removing inversion artifacts, related to source encoding, and by clearly improved delineation of top, bottom, and flanks of a high-velocity high-contrast salt inclusion. The experiments also show that for poor starting models these two constraints by themselves are insufficient to detect the bottom of high-velocity inclusions such as salt. Inclusion of the one-sided asymmetric total-variation constraint overcomes this issue by discouraging velocity lows to buildup during the early stages of the inversion. To the author's knowledge the presented algorithm is the first to successfully remove the imprint of local minima caused by poor starting models and band-width limited finite aperture data.
I. Introduction
The paper frames full-waveform inversion as a large, nonconvex and ill-conditioned problem whose local minima and missing spatial frequencies complicate recovery of realistic subsurface structure. It proposes convex model constraints and extended formulations to control the feasible space and improve recovery, especially for high-contrast salt bodies.
- Problem setting: Local minima and missing spatial frequencies make full-waveform inversion sensitive to initialization and hinder recovery of both large- and small-scale velocity variations.Surface acquisition, bandwidth limits, and finite aperture remove spatial-frequency information.
- Reported outcome: For accurate starting models, the final scheme removes inversion artifacts and improves delineation of high-velocity, high-contrast salt inclusions.Poor starting models require the asymmetric constraint to prevent detrimental artifacts from building up.
- Illustrative examples: The extended formulation enlarges the search space and mitigates local minima relative to the conventional objective in the one-dimensional example.Figure 1c shows local minima in the conventional objective, while the extended objective reduces them.
- Illustrative examples: In the monotonic-profile example, unconstrained inversion fits the observations but misses the global velocity trend, whereas constrained inversion recovers the ground-truth model exactly.The constraints impose a lower velocity bound and monotonic increase with depth.
- Motivation and contribution: The proposed framework combines bounds, total variation, and one-sided total-variation constraints to control the model space in PDE-constrained inversion.The overarching problem remains nonconvex, but the framework applies multiple convex constraints to the model.
II. Scaled Gradient Projection for Large-Scale Programs
This section develops scaled gradient projection for optimizing nonconvex PDE-constrained objectives under convex model constraints. The method keeps iterates feasible, supports intersections of simple constraint sets, and uses Hessian approximations and adaptive damping to obtain computationally tractable descent steps.
- Large-scale optimization: The framework separates simple convex constraints from expensive PDE constraints, making constrained optimization suitable for large-scale problems.Gradient and Hessian approximation costs remain fixed within the constrained subproblem solve.
- Scaled gradient projection: Scaled gradient projection updates preserve feasibility while solving constrained model-update subproblems for nonconvex objectives.The constraint set can represent prior information such as box constraints and intersections with TV constraints.
- Hessian approximations: Hessian choices range from scaled gradient descent to Newton-like methods, trading approximation quality against computational efficiency.The conventional formulation has a dense Hessian requiring PDE solves, whereas simpler approximations reduce cost.
- Implicit trust-region approach: An implicit trust-region strategy adaptively damps the Hessian approximation and rejects steps that fail to sufficiently decrease the objective.This avoids expensive line searches in PDE-constrained optimization.
- Convergence: Under Lipschitz continuity of the gradient and bounded feasible level sets, every limit point of the algorithm is a stationary point of the constrained problem.The stationarity condition is expressed over all feasible model vectors.
III. Total-Variation constraints
Total-variation constraints restrict model complexity while preserving discontinuities, helping reduce artifacts and shape feasible models for full-waveform inversion. The section motivates these constraints, defines their discrete form, and describes projection-based solution methods for the resulting convex subproblems.
- The TV norm is defined as the sum of the ℓ2 norms of discrete gradient vectors across model locations, using a finite-difference operator with Neumann boundary conditions.
- Example: Projecting the Marmousi model on the intersection of box and TV-norm constraints: Combining TV-norm and box constraints forms an intersection set onto which the Marmousi model is orthogonally projected under velocity bounds and decreasing TV radii.The example uses velocity bounds corresponding to 1500 ms^-1 and 5500 ms^-1, with τ values of .6τ0 and .3τ0.
- Example: Projecting the Marmousi model on the intersection of box and TV-norm constraints: TV-norm constraints reduce model complexity while still allowing discontinuous unconformities, with smaller constraint radii producing more cartoon-like feasible models.The parameter τ controls the complexity of models close to the original model in the 2-norm.
- Solving the convex subproblems: The convex constrained subproblems are solved using primal-dual hybrid gradient after forming a quadratic approximation with a gradient and Hessian approximation.The method uses a dualized formulation and modified PDHG iterations for the TV-constrained subproblems.
- Solving the convex subproblems: Projection onto the TV constraint requires projecting spatial gradient norms onto a simplex, for which an O(n) implementation is available.
IV. Numerical Experiments
The experiments evaluate constrained waveform inversion on BP 2004 models with accurate and poor starting models. Box and TV constraints improve reconstructions from accurate starts, while poor starts require additional asymmetric constraints to avoid persistent local-minimum artifacts.
- Accurate starting model: Without TV constraints, repeated frequency passes reduce some oscillations but leave noisy artifacts, poor illumination effects, and unclear salt boundaries.The second pass removes much of the oscillation below the left top salt, but the result remains noisy near model boundaries.
- Accurate starting model: Box and TV constraints significantly improve BP 2004 reconstructions from an accurate smooth start by removing artifacts and sharpening salt boundaries.The TV-constrained result better resolves the top salt and, after a second pass, improves the bottom while preserving low-velocity inclusions.
- Poor starting model: For poor starting models, bound constraints alone produce noisy results that do not improve substantially across frequency passes because early velocity decreases persist inside salt.This behavior is associated with data missing low frequencies and parasitic stationary points.
- Poor starting model: TV constraints recover the top salt from a poor start but remain trapped by the velocity low, failing to improve after three passes.Relaxing the TV bound slightly between passes does not remove the dominant low-velocity imprint.
V. Asymmetric TV-norm Constraints
The asymmetric TV constraint penalizes downward depthwise velocity jumps and is introduced through continuation from a strong restriction to a relaxed one. This addition recovers major salt features from poor starts and prevents constrained methods from stagnating at poor solutions.
- Constraint design: The one-sided TV constraint penalizes positive depthwise differences in slowness squared, targeting velocity decreases with depth in sedimentary-basin models.It is combined with box and ordinary TV constraints because depth-only penalization can otherwise create vertical artifacts.
- Continuation strategy: Continuation starts with a small asymmetric-TV level and progressively relaxes it across frequency passes, discouraging early downward velocity jumps before allowing better data fits.The constraint is motivated by velocities that generally increase with depth while remaining relatively constant in salt.
- Results: With a poor initial model, asymmetric-TV continuation recovers the main ground-truth features, although boundary recovery remains poor because of limited illumination.The authors attribute left- and right-boundary errors to sources beginning about 1000 m away from the boundaries.
- Results: The constrained adjoint-state method produces results nearly as good as WRI, while WRI achieves significantly lower model error relative to ground truth.Adjoint-state results are visually slightly worse near the top and sides, but both methods continue improving across passes.
- Conclusion: Continuation in the asymmetric-TV parameter appears promising for preventing constrained WRI and adjoint-state FWI from stagnating at poor solutions.Model error and data misfit decrease during the passes as relaxation permits additional structure and matched events.
VI. Discussion
The discussion presents an automatic continuation strategy for difficult salt inversion and identifies methodological limitations involving validation realism, parameter selection, and convex-subproblem optimization.
- Numerical behavior: WRI reduced model misfit below 30%, whereas FWI stalled below 50% in the reported 72-iteration experiments.The WRI model error initially increased under strict constraints before descending after relaxation; FWI showed less reduction.
- Validation and implementation caveats: The study identifies inversion crime, boundary conditions, continuation strategy, and convex-subproblem algorithms as remaining issues.Experiments with more sophisticated boundary conditions preserved the reported improvements, while continuation choices and convex-subproblem solvers remain areas for improvement.
- Validation and implementation caveats: The experiments showed improved results with all sources, although source encoding reduces wave-equation solves while introducing noisy crosstalk.
- Automatic continuation strategy: Successive asymmetric-TV relaxations with warm-started frequency-batch passes replace labor-intensive manual salt flooding and help steer away from parasitic stationary points.The approach was presented as an automatic heuristic for challenging salt-inversion settings.
- Practical considerations: The lowest frequency must be sufficiently low for progress during each frequency pass.
- Practical considerations: Constraint sets must balance freedom for model updates against sufficient restriction to avoid parasitic stationary points.
VII. Conclusions and Future Work
The paper develops a constrained optimization framework for WRI and FWI and reports that asymmetric TV constraints improve salt recovery from poor initial models.
- Conclusions: The framework minimizes WRI and FWI formulations with bound, TV, and asymmetric-TV constraints using scaled gradient projection.The convex constraints may be replaced or augmented provided their feasibility regions remain nonempty.
- Conclusions: Asymmetric TV constraints produced major improvements from poor starting models by preventing velocity lows from forming after the inversion entered salt.A continuation strategy gradually weakened the asymmetric constraint to avoid stalling at the constrained solution.
IX. In memoriam: John “Ernie” Esser† (May 19, 1980 – March 8, 2015)
The in memoriam section commemorates Ernie Esser and depicts his personal connection to total-variation minimization.
- In memoriam: Ernie Esser is remembered as a promising, positive, generous, and talented scientist who died while the manuscript was being prepared.
- In memoriam: A figure memorializes Ernie with a photograph and a total-variation-constrained version of the image.
X. Appendix: PDE-constrained optimization
The appendix contrasts reduced adjoint-state FWI with an extended formulation that retains wavefields as optimization variables and uses PDE solves for derivatives and approximations.
- PDE-constrained optimization: The forward operator applies a measurement operator to wavefields obtained by solving the discretized PDE, which dominates objective-evaluation cost.
- Reduced adjoint-state method: The reduced adjoint-state method computes forward and adjoint wavefields through PDE solves to obtain model derivatives.
- Gradient and Hessian computations: The full Hessian is impractical to store or evaluate at scale, so the method uses approximations whose action or entries require fewer additional PDE solves.The pseudo-Hessian approximation is diagonal in the stated discretization, while the extended formulation's approximation is typically sparse.
- Extended formulation: Extended formulations retain model and state variables jointly, giving them more flexibility to fit data and making them less prone to local minima than eliminated PDE-constrained FWI.The extended objective can be evaluated through a closed-form quadratic optimization over the wavefields.
- Numerical discretization: In the numerical discretization, one system solve separates across frequencies, with block-diagonal wavefield, source, and Jacobian structures.