Null-space shuttles for targeted uncertainty analysis in full-waveform inversion

Full-waveform inversion (FWI) is an effective tool for recovering subsurface information, but many factors make this recovery subject to uncertainty. In particular, unwanted noise in measurements can bias results toward models that are not representative of the true subsurface and numerical optimization techniques used in the inversion only allow for approximate minimization of the objective function. Both factors contribute to the nonuniqueness of FWI solutions. Assessing the uncertainty that this nonuniqueness introduces can be difficult, due to the large dimensionality of the inversion problem. Fortunately, complete characterization of inversion uncertainty is seldom necessary for applications using an inversion result, meaning that the entire dimensionality of the problem may not be relevant for practical uncertainty quantification. Typically, it is only the uncertainty in a few specific aspects of the inversion that is important (for instance, confidence in a recovered anomaly). A targeted uncertainty quantification, characterizing only the confidence in a specific feature of the subsurface model, can greatly reduce the dimensionality of the uncertainty characterization problem, potentially making it tractable. We have adopted an approach for quantifying the confidence of the inversion in a chosen hypothesis about the recovered subsurface model. We tested each hypothesis through numerical optimization on the set of equal-objective model-space steps, called null-space shuttles. By approximating the null-space shuttle that maximally violates a given hypothesis about the inversion, this method establishes an effective approximation of the uncertainty in that hypothesis. We tested the use of this technique on several numerical examples for the case of viscoelastic inversion. These examples demonstrate that, at a reasonable computational cost, this method can generate estimates of the lower bound on the maximal uncertainty associated with incomplete numerical optimization. In the viscoelastic examples considered, the velocity variables are much better constrained than the Formula: see text and density variables according to this metric.