Hu, 2019, Time-domain multiscale FWI based on a time-shift method and low-frequency data reconstruction: 89th Annual International Meeting, SEG, Expanded Abstracts, 1570–1574, doi: 10.1190/segam2019-3215657.1. GPYSA7 0016-8033 Abstract Web of Science Google Scholar SJOPE8 1095-7189 Crossref Web of Science Google Scholar Overton, 2005, A robust gradient sampling algorithm for nonsmooth, nonconvex optimization: SIAM Journal on Optimization, 15, 751–779, doi: 10.1137/030601296. MOREDQ 0364-765X Crossref Web of Science Google Scholar Overton, 2002, Approximating subdifferentials by random sampling of gradients: Mathematics of Operations Research, 27, 567–584, doi: 10.1287/moor.27.3.567.317. Simões, 2020, Gradient sampling methods for nonsmooth optimization, in Numerical nonsmooth optimization: State of the art algorithms: Springer, 201–225. Eckstein, 2011, Distributed optimization and statistical learning via the alternating direction method of multipliers: Foundations and Trends in Machine learning, 3, 1–122, doi: 10.1561/2200000016. Symes, 2004, Angle-domain common-image gathers for migration velocity analysis by wavefield-continuation imaging: Geophysics, 69, 1283–1298, doi: 10.1190/1.1801945. Almomin, 2014, Simultaneous inversion of full data bandwidth by tomographic full-waveform inversion: Geophysics, 79, no. 3, WA129–WA140, doi: 10.1190/geo2013-0340.1. Operto, 2019, Improving full-waveform inversion by wavefield reconstruction with the alternating direction method of multipliers: Geophysics, 84, no. 1, R139–R162, doi: 10.1190/geo2018-0093.1. Multiple numerical examples demonstrate that the proposed method alleviates the cycle-skipping problem of conventional FWI when starting from very crude initial velocity models without low-frequency data. This simplification provides an efficient realization in which the computational costs and memory requirements are the same as conventional FWI. For practical implementation, we only take one random space shift at each time step during the gradient calculation. The final descent search direction is obtained by summing all the shifted gradients. Theoretical derivation suggests that the two wavefields should be shifted in the same direction to obtain reasonable low-wavenumber updates. Based on the observation that a slight perturbation in the velocity model causes a small spatial shift of the wavefield, we have approximated the sampled gradients by crosscorrelating the space-shifted source- and receiver-side wavefields. The original implementation of GSA requires explicit calculation of the gradient at each sampled vector, which is prohibitively expensive. The search space is hugely expanded to have more freedom to accommodate large velocity errors in the starting model. To mitigate the dependence of FWI on the quality of starting model and on the low frequencies in the data, we apply the gradient sampling algorithm (GSA) introduced for nonsmooth, nonconvex optimization problems to FWI. Full-waveform inversion (FWI) is a highly nonlinear and nonconvex problem.
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