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applsciletetrsa>vol-02>issue-01>3D Variational Bayesian Full Waveform Inversion
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Letter
3D Variational Bayesian Full Waveform Inversion
Xin Zhang1
Angus Lomas2
Applied Science Letters
2022 ° 10(10) ° 02-01
https://www.wikipt.org/applscilettersa
DOI: 10.1490/6980001.703applsci
Abstract
Seismic full-waveform inversion (FWI) provides high resolution images of the subsurface by exploiting information in the recorded seismic waveforms. This is achieved by solving a highly nonnlinear and nonunique inverse problem. Bayesian inference is therefore used to quantify uncertainties in the solution. Variational inference is a method that provides probabilistic, Bayesian solutions efficiently using optimization. The method has been applied to 2D FWI problems to produce full Bayesian posterior distributions. However, due to higher dimensionality and more expensive computational cost, the performance of the method in 3D FWI problems remains unknown. We apply three variational inference methods to 3D FWI and analyse their performance. Specifically we apply automatic differential variational inference (ADVI), Stein variational gradient descent (SVGD) and stochastic SVGD (sSVGD), to a 3D FWI problem, and compare their results and computational cost. The results show that ADVI is the most computationally efficient method but systematically underestimates the uncertainty. The method can therefore be used to provide relatively rapid but approximate insights into the subsurface together with a lower bound estimate of the uncertainty. SVGD demands the highest computational cost, and still produces biased results. In contrast, by including a randomized term in the SVGD dynamics, sSVGD becomes a Markov chain Monte Carlo method and provides the most accurate results at intermediate computational cost
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Introduction
Seismic full-waveform inversion (FWI) uses full seismic recordings to characterize properties of the Earth’s interior, and can provide high resolution images of the subsurface (Tarantola 1984; Gauthier et al. 1986; Tarantola 1988; Pratt 1999; Tromp et al. 2005; Fichtner et al. 2006; Plessix 2006). The method has been applied at industrial scale (Virieux & Operto 2009; Prieux et al. 2013; Warner et al. 2013), regional scale (Chen et al. 2007; Fichtner et al. 2009; Tape et al. 2009; Chen 2 et al. 2015), and global scale (French & Romanowicz 2014; Bozdag et al. 2016; Fichtner et al. ˘ 2018a; Lei et al. 2020). Due to the nonlinearity of relationships between model parameters and seismic waveforms, insufficient data coverage and noise in the data, FWI always has nonunique solutions and infinitely many sets of model parameters fit the data to within their uncertainty. It is therefore important to quantify uncertainties in the solution in order to better assess the reliability of inverted models (Tarantola 2005). FWI problems are traditionally solved using optimization methods in which one seeks an optimal set of parameter values by minimizing the difference or misfit between observed data and model-predicted data. The strong nonlinearity and nonuniqueness of the problem implies that a good starting model is required to avoid convergence to incorrect solutions (generally alternative modes or stationary points of the misfit function). Such models are not always available in practice. To alleviate this requirement a range of misfit functions that may reduce multimodality have been proposed (Luo & Schuster 1991; Gee & Jordan 1992; Fichtner et al. 2008; Brossier et al. 2010; Van Leeuwen & Mulder 2010; Bozdag et al. 2011; Métivier et al. 2016; Warner & Guasch 2016; ˘ Yuan et al. 2020; Sambridge et al. 2022). Nevertheless, none of the standard methods of solution using any of these misfit functions have been shown to allow accurate estimates of uncertainty to be made in realistic FWI problems.
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Conclusion
In this study we applied three different variational inference methods: automatic differential variational inference (ADVI), Stein variational gradient descent (SVGD) and stochastic SVGD (sSVGD) to 3D full-waveform inversion, and demonstrated feasibility of using these methods to solve largescale probabilistic inverse problems. The results show that ADVI with a mean-field approximation can provide rapid solutions but with systematically underestimated uncertainty. In practice, the method can therefore be used to provide a rapid initial estimate of the solution, or to provide a lower bound estimate of the uncertainty. SVGD appears to be the most expensive method, but still provides a biased solution because of the limited number of particles. In contrast, by adding a noise term in the dynamics of SVGD, sSVGD becomes a Markov chain Monte Carlo method 27 and provides the most accurate results. We thus conclude that variational inference methods can be used to solve real-world 3D full wave form inversion problems.
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