A priori estimates of attraction basins for velocity model reconstruction by time-harmonic full-waveform inversion and data-space reflectivity formulation

Abstract

The determination of background velocity by full-waveform inversion (FWI) is known to be hampered by the local minima of the data misfit caused by the phase shifts associated with background perturbations. Attraction basins for the underlying optimization problems can be computed around any nominal velocity model, and they guarantee that the misfit functional has only one (global) minimum. The attraction basins are further associated with tolerable error levels representing the maximal allowed distance between the (observed) data and the simulations (i.e., the acceptable noise level). The estimates are defined a priori, and they only require the computation of (possibly many) the first- and second-order directional derivatives of the (model to synthetic) forward map. The geometry of the search direction and the frequency influence the size of the attraction basins, and the complex frequency can be used to enlarge the basins. The size of the attraction basins for the perturbation of background velocities in classic FWI (global model parameterization) and the data-space reflectivity reformulation (migration-based traveltime [MBTT]) are compared: The MBTT reformulation substantially increases the size of the attraction basins (by a factor of 4–15). Practically, this reformulation compensates for the lack of low-frequency data. Our analysis provides guidelines for a successful implementation of the MBTT reformulation.

Diffraction Tomography, Fourier Reconstruction, and Full Waveform Inversion

“…In Algorithm 1, we further implement a progression in the frequency content, which is common to mitigate the ill-posedness of the nonlinear inverse problem, [5]. We further invert each frequency independently, from low to high, as advocated by [3,16]. For the implementation details using the HDG discretization, we refer to [18].…”

“…We could instead incorporate its specific representation as it is decomposed into a (known) smooth background and the perturbation (following relation k 2 " k 2 0 `f ). Then, FWI could certainly be improved if reformulated with such a model decomposition, which corresponds to the data-space reflectivity inversion of [7,16].…”

“…The difficulty of recovering a large inclusion with a strong contrast can be mitigated by the use of multifrequency data-sets, allowing a multi-scale reconstruction [5,16]. We now carry out the iterative reconstruction using increasing frequencies, starting with ν " 0.2 and up to ν " 1.…”

“…We now carry out the iterative reconstruction using increasing frequencies, starting with ν " 0.2 and up to ν " 1. Following [16], we use a sequential progression, that is, every frequency is inverted separately. The reconstructions for the inclusion of radius 4.5 and contrast f " ˘5 are pictured in Figure 8.…”

Adjoint-state method for Hybridizable Discontinuous Galerkin discretization, application to the inverse acoustic wave problem

“…Then, the scalar step is selected to obtain an appropriate amplitude of the updates, using linesearch algorithms, e.g., [59,17]. In our implementation, we also avoid the computation of the Hessian such that, at each iterations of the minimization, one must 1. solve the forward problem using the current model parameters, 2. compute the gradient of the misfit functional, Non-linear minimization suffers from local minima, which cannot be avoided with the deterministic approach, see, e.g., [15,69,8,33] in the context of seismic. We can mention the use of statistical-based methods but in the large-scale applications we have in mind, such approaches remain unusable at the moment.…”