3D Bayesian Variational Full Waveform Inversion

2025-04-22 1 0 5.72MB 36 页 10玖币

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Xin Zhang1, Angus Lomas2, Muhong Zhou2,

York Zheng2and Andrew Curtis1

1School of GeoSciences, University of Edinburgh, UK

2BP p.l.c., London, UK

E-mail: x.zhang2@ed.ac.uk, andrew.curtis@ed.ac.uk

arXiv:2210.03613v2 [physics.geo-ph] 10 Oct 2022

SUMMARY

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 proba-

bilistic, Bayesian solutions efﬁciently using optimization. The method has been applied to 2D

FWI problems to produce full Bayesian posterior distributions. However, due to higher dimen-

sionality 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 high-

est 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 pro-

vides the most accurate results at intermediate computational cost. We thus conclude that 3D

variational full-waveform inversion is practically applicable, at least in small problems, and

can be used to image the Earth’s interior and to provide reasonable uncertainty estimates on

those images.

1 INTRODUCTION

A wide variety of academic studies and practical applications require that we interrogate the

Earth’s subsurface for answers to scientiﬁc questions. A common approach is to image subsurface

properties in three dimensions using data recorded on the Earth’s surface, and to interpret those

images to address questions of interest. In order to provide well justiﬁed and robust answers to

such interrogation problems, it is necessary to assess the uncertainty in property estimates (Arnold

& Curtis 2018).

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

et al. 2015), and global scale (French & Romanowicz 2014; Bozda˘

g 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 op-

timal 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; Bozda˘

g 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.

Bayesian inference provides a different way to solve inverse problems and quantify uncertain-

ties. The method uses Bayes’ theorem to update a prior probability density function (pdf) with

new information from the data to construct a so-called posterior probability density function. The

prior pdf describes information available about the parameters of interest prior to the inversion

(independently of the current data set), while the posterior pdf describes the resultant state of in-

formation after combining information in the prior pdf with information in the current data. In

principle, Bayesian inference thus provides accurate estimates of uncertainty.

Markov chain Monte Carlo (McMC) is one method to characterize the posterior pdf which

has been used widely in many ﬁelds. In McMC one constructs a set (chain) of successive sam-

ples generated from the posterior pdf by taking a structured random walk in parameter space

(e.g., Brooks et al. 2011); those samples can thereafter be used to infer the values of useful statis-

tics of that pdf (mean, standard deviation, etc.). The Metropolis-Hastings algorithm is one such

method (Metropolis & Ulam 1949; Hastings 1970) and has been applied to many applications in

geophysics, including gravity inversion (Mosegaard & Tarantola 1995; Bosch et al. 2006; Rossi

2017), vertical seismic proﬁle inversion (Malinverno et al. 2000), surface wave dispersion inver-

sion (Bodin et al. 2012; Shen et al. 2012; Young et al. 2013; Galetti et al. 2017; Zhang et al.

2018b), electrical resistivity inversion (Malinverno 2002; Galetti & Curtis 2018), electromagnetic

inversion (Minsley 2011; Ray et al. 2013; Blatter et al. 2019), travel time tomography (Bodin

& Sambridge 2009; Galetti et al. 2015, 2017) and more recently full-waveform inversion (Ray

et al. 2017; Sen & Biswas 2017; Guo et al. 2020). However, the basic Metropolis-Hastings al-

gorithm becomes computationally intractable in high dimensional space if the chain is attracted

to individual misﬁt minima rather than exploring all possible such minima. To reduce this issue,

more advanced McMC methods have been introduced to geophysics, such as Hamiltonian Monte

Carlo (Duane et al. 1987; Fichtner et al. 2018b; Gebraad et al. 2020; Kotsi et al. 2020), stochastic

Newton McMC (Martin et al. 2012; Zhao & Sen 2019), Langevin Monte Carlo (Roberts et al.

1996; Siahkoohi et al. 2020a) and parallel tempering (Hukushima & Nemoto 1996; Dosso et al.

2012; Sambridge 2013). However, the above studies mainly address 1D or 2D problems because

of the high computational expense of moving to 3D. Some studies have applied McMC meth-

ods to 3D inverse problems including body wave travel time tomography (Piana Agostinetti et al.

2015; Hawkins & Sambridge 2015; Burdick & Leki´

c 2017; Zhang et al. 2020b) and surface wave

dispersion inversion (Zhang et al. 2018b, 2020a; Ryberg et al. 2022), but they require enormous

computational cost even for small datasets. Thus, McMC methods are generally considered to be

intractable for large datasets and high dimensionality, such as occurs in 3D FWI problems.

Variational inference solves Bayesian inference problems in a different way: within a prede-

ﬁned family of (simpliﬁed) pdfs, the method seeks an optimal approximation to the posterior pdf

by minimizing the difference between the approximating pdf and the posterior pdf. A typical met-

ric used to measure this difference is the Kullback-Leibler (KL) divergence (Kullback & Leibler

1951). The method therefore solves an optimization problem rather than a stochastic sampling pro-

cess as in McMC methods. As a result, in some classes of problems variational inference may be

computationally more efﬁcient than McMC methods and provide better scaling to higher dimen-

sionality (Bishop 2006; Blei et al. 2017; Zhang et al. 2018a). The method can be applied to larger

datasets by dividing the dataset into small minibatches and using stochastic and distributed opti-

mization methods (Robbins & Monro 1951; Kubrusly & Gravier 1973). In addition, the method

can usually be parallelized at the individual sample level which makes the method even more ef-

ﬁcient in real time by taking advantage of modern high performance computational facilities. By

contrast, McMC methods cannot be parallelized at the sample level since each sample depends on

the previous sample, and cannot use minibatches as these break the detailed balance condition that

is required by common McMC methods (O’Hagan & Forster 2004).

In variational inference the choice of variational family is important as it determines the accu-

racy of the approximation and the complexity of the optimization problem. A good choice should

be rich enough to approximate complex distributions and simple enough such that the optimization

problem remains solvable. Difference choices of variational families lead to a variety of speciﬁc

methods. For example, a common choice is to use a mean-ﬁeld approximation in which the pa-

rameters are assumed to be mutually independent (Bishop 2006; Blei et al. 2017). In geophysics

the method has been applied to invert for geological facies distributions using seismic data (Nawaz

& Curtis 2018, 2019; Nawaz et al. 2020). While often leading to highly efﬁcient algorithms, this

method usually requires bespoke mathematical derivations which restricts its applicability to a lim-

ited range of problems. Based on a Gaussian variational family, Kucukelbir et al. (2017) proposed

a method called automatic differential variational inference (ADVI), which can be applied easily to

general problems. For example, the method has been used to solve seismic travel time tomography

(Zhang & Curtis 2020a) and earthquake slip inversion problems (Zhang & Chen 2022).

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摘要：

3DBayesianVariationalFullWaveformInversionXinZhang1,AngusLomas2,MuhongZhou2,YorkZheng2andAndrewCurtis11SchoolofGeoSciences,UniversityofEdinburgh,UK2BPp.l.c.,London,UKE-mail:x.zhang2@ed.ac.uk,andrew.curtis@ed.ac.uk1SUMMARYSeismicfull-waveforminversion(FWI)provideshighresolutionimagesofthesubsurfaceby...

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