Full-waveform inversion (FWI) is an algorithm (and a part of the measuring procedure in a wide sense) with the aim to find the governing law of a physical system using the partially measured physical fields with limited computational resources. A law is a forward theory equipped with the model parameters and the data parameters. The main characteristic of the law is the realizability assumption: the law explains all subsets of the measured data parameters and predicts all subsets of the unmeasured (in the given experiment) data parameters. To find the law, we have to guess a law (a forward theory and parametrization), measure some data parameters and check the realizability assumption.
To put it more precisely, I formulate a new probabilistic setting for inverse problems and full-waveform inversion. Instead of using the Bayes' theorem, the Tarantola-Valette conjunction or the principle of maximum entropy based on the prior information for the averaged quantities, I propose a principle of minimum relative information using the prior information for the non-averaged quantities. The Tarantola-Valette formula is obtained as a special ... mehrcase under the assumption that the theoretical and prior measures exist. Using the realizability assumption as a prior information, the principle of minimum relative information provides the parametric probabilistic solution with the arbitrary misfit functions. Maximization of the parametric probabilistic solution leads to a multiobjective minimization problem. All global Pareto optima are the sample points of the probabilistic solution with the highest values of the volumetric measure. Unfortunately, even a local multiobjective minimization problem is computationally intractable for FWI with many millions of model parameters.
To make it computationally attractive for large-scale FWI and to find at least a few local solutions of the multiobjective minimization problem, I implement the bilevel multiobjective waveform inversion (BMWI) using a single randomly chosen shot gather at each iteration. BMWI is a stochastic, nested algorithm with an adaptive parabolic line search and multiscale strategy. The computational cost per iteration is five forward modellings only. BMWI can worsen some of the single-shot misfit functions and the different random runs of BMWI converge to different points in the model manifold. I interpret these inverted models as the sample points of the probabilistic solution. I estimate the solution, uncertainty and sensitivity using the sample estimates of the mean, standard deviation and initial deviation of the sample points, respectively. Using the numerical examples with the Marmousi-2 model, I illustrate the potential of BMWI for automatic uncertainty and sensitivity analysis with just two-three sample points.
To test the idea with real-world data, I apply stochastic single-shot BMWI in a 2D acoustic finite-difference approximation to a 2D line of pressure data acquired in a shallow-water river delta with ocean bottom cables. I use minimal data preprocessing (only a new 3D-to-2D transform which is strictly valid in a linear-gradient medium), the linear gradient starting models and the diagonal preconditioners with a negligible regularization. I estimate the theoretical uncertainties due to the neglected 3D effects using the 3D-to-2D transforms. The uncertainties estimated by the random sequences of BMWI are higher than the uncertainties related to the 3D-to-2D transforms. I provide the estimates of the solution, uncertainty and sensitivity using up to fourteen sample points inverted with the different linear-gradient starting models, the differently 3D-to-2D-transformed real data sets and the different random sequences of descent directions. The uncertainty of sound velocities is the lowest in the central semicircle with the radius 3 km equal to half the length of the ocean bottom cable. The uncertainty of mass densities is the highest in the central semicircle. The sensitivity of the measuring procedure with respect to sound velocity and mass density is the highest in the central semicircle representing a footprint of the acquisition geometry. Outside the central semicircle the parameters are not falsifiable in the specified setting.
Full-waveform inversion is the quest for the unique governing law of the physical system under study. If the governing law is deterministic and the sample mean, standard deviation and initial deviation of the sample points represent the insufficient description of the solution, uncertainty and sensitivity, then the measuring procedure in a wide sense has to be improved.