Geometric and Level Set Tomography using Ensemble Kalman Inversion
Tomography is one of the cornerstones of geophysics, enabling detailed spatial descriptions of otherwise invisible processes. However, due to the fundamental ill-posedness of tomography problems, the choice of parametrizations and regularizations for inversion significantly affect the result. Parametrizations for geophysical tomography typically reflect the mathematical structure of the inverse problem. We propose, instead, to parametrize the tomographic inverse problem using a geologically motivated approach. We build a model from explicit geological units that reflect the a priori knowledge of the problem. To solve the resulting large-scale nonlinear inverse problem, we employ the efficient Ensemble Kalman Inversion scheme, a highly parallelizable, iteratively regularizing optimizer that uses the ensemble Kalman filter to perform a derivative-free approximation of the general iteratively regularized Levenberg–Marquardt method. The combination of a model specification framework that explicitly encodes geological structure and a robust, derivative-free optimizer enables the solution of complex inverse problems involving non-differentiable forward solvers and significant a priori knowledge. We illustrate the model specification framework using synthetic and real data examples of near-surface seismic tomography using the factored eikonal fast marching method as a forward solver for first arrival traveltimes. The geometrical and level set framework allows us to describe geophysical hypotheses in concrete terms, and then optimize and test these hypotheses, helping us to answer targeted geophysical questions.
Additional Information© 2019 The Author(s). Published by Oxford University Press on behalf of The Royal Astronomical Society. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model). Accepted 2019 October 17. Received 2019 October 13; in original form 2019 June 11. Published: 21 October 2019. The authors would like to thank Nicholas Rawlinson and an anonymous reviewer for providing useful commentary that has significantly improved the quality of the manuscript. We would also like to thank Editor Michael Ritzwoller and the anonymous assistant editor for managing the review process. JBM would like to thank Andrew Stuart (Caltech Computational and Mathematical Sciences) and the 2018 Gene Golub SIAM summer school for useful discussions regarding this study. Data from Carrizo Plains was collected during the 2017 Caltech Applied Geophysics Field Course, for which JBM was a Teaching Assistant. JBM would like to thank the instructors Rob Clayton and Mark Simons, and co-TA Voon Hui Lai, as well as the students, for the course. JBM would also like to thank the General Sir John Monash Foundation and the Origin Energy Foundation for financial support. This study was supported by NSF grant EAR-1453263. All calculations were computed using the Julia language (Bezanson et al.2017). Code for our model specification language can be found at https://github.com/jbmuir/EarthModels.jl. Code for the EKI optimizer can be found at https://github.com/jbmuir/EnsembleKalmanInversion.jl. Code for a Julia 1.0+ compliant factored Eikonal fast marching method forward solver can be found at https://github.com/jbmuir/FEFMM.jl.
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