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Square‐Root Variable Metric‐Based Nullspace Shuttle: A Characterization of the Nonuniqueness in Elastic Full‐Waveform Inversion
Author(s) -
Liu Qiancheng,
Peter Daniel
Publication year - 2020
Publication title -
journal of geophysical research: solid earth
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.983
H-Index - 232
eISSN - 2169-9356
pISSN - 2169-9313
DOI - 10.1029/2019jb018687
Subject(s) - hessian matrix , mathematics , eigenvalues and eigenvectors , inverse problem , inversion (geology) , linear subspace , inverse , square root , singular value decomposition , eigendecomposition of a matrix , metric (unit) , mathematical analysis , mathematical optimization , algorithm , geometry , physics , geology , paleontology , operations management , quantum mechanics , structural basin , economics
Full‐waveform inversion (FWI) is for most geophysical applications an ill‐posed inverse problem, with nonunique solutions. We examine its nonuniqueness by exploring the nullspace shuttle, which can efficiently generate an ensemble of data fitting solutions. We construct this shuttle based on a quasi‐Newton method, the square‐root variable‐metric (SRVM) method. The latter provides access to the inverse data‐misfit Hessian in FWI for large‐scale applications. Combining the SRVM method with a randomized singular value decomposition, we obtain the eigenvector subspaces of the inverse data‐misfit Hessian. Its primary eigenvalue and eigenvector are considered to determine the null space of inversion result. Using the SRVM‐based nullspace shuttle, we can modify the inverted result a posteriori in a highly efficient manner without corrupting the data misfit. Also, because the SRVM method is embedded through elastic FWI, our method can be extended to multiparameter problems. We confirm and highlight our approach with the elastic Marmousi example.