Lift and Relax for PDE-Constrained Inverse Problems in Seismic Imaging
Name
2006.16492.pdf
Description
Accepted version
Size
3.35 MB
Format
Adobe PDF
Checksum (MD5)
3d006fbe2e5355a95104d03d82b6e40a
Author(s)
Fang, Zhilong
Demanet, Laurent
Date Issued
2020
Journal
IEEE Transactions on Geoscience and Remote Sensing
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Version
Original manuscript
Abstract
IEEE We present lift and relax for waveform inversion (LRWI), an approach that mitigates the local minima issue in seismic full waveform inversion (FWI) via a combination of two convexification techniques. The first technique (Lift) extends the set of unknown variables to their products, arranged as a moment matrix. This algebraic idea is a celebrated way to replace a hard polynomial optimization problem by a semidefinite programming approximation. Concretely, both the model and the wavefield are lifted from vectors to rank-2 matrices. The second technique (Relax) invites to relax the strict wave-equation constraint--a technique known as wavefield reconstruction inversion (WRI), which introduces wave-equation misfits as a weighted penalty term in the objective function. The relaxed penalty formulation enables balancing the data and wave-equation misfits by tuning a penalty parameter. Together, ``Lift'' and ``Relax'' help reformulate the inverse problem as a set of constraints on a rank-2 moment matrix. Such a lifting strategy permits good data and wave equation fits throughout the inversion process while leaving the numerical rank of the rank-2 moment matrix to be minimized down to one. Moreover, LRWI does not require adjoint wavefield to compute the gradient, which mitigates computational burdens. Numerical examples indicate that starting with a poor initial model, LRWI can conduct successful inversions with a starting frequency that is higher than that required by FWI and WRI.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Earth Resources Laboratory
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1109/TGRS.2020.3037856
