An introduction to variational inference in geophysical inverse problems

ADVI automatic differential variational inference c a subset of variables (clique) C a set of cliques, i.e., c C det determinant dobs observed data vector ELBO evidence lower bound EM Expectation-Maximization F(q;Θ) evidence lower bound of probability distribution q defined as a function of parameters Θ Eq expectation with respect to probability distribution q Fθ a normalizing flow parameterized by θ FWI full-waveform inversion GM Gaussian mixture

[1]  Anandaroop Ray,et al.  Bayesian geophysical inversion with trans-dimensional Gaussian process machine learning , 2019, Geophysical Journal International.

[2]  H. Robbins.  A Stochastic Approximation Method , 1951 .

[3]  Thomas H. Jordan,et al.  Generalized seismological data functionals , 1992 .

[4]  D. M. Titterington,et al.  Variational approximations in Bayesian model selection for finite mixture distributions , 2007, Comput. Stat. Data Anal..

[5]  Dustin Tran,et al.  Variational Gaussian Process , 2015, ICLR.

[6]  R. Horgan,et al.  Statistical Field Theory , 2014 .

[7]  David J. C. MacKay,et al.  Information Theory, Inference, and Learning Algorithms , 2004, IEEE Transactions on Information Theory.

[8]  S. Walker.  Invited comment on the paper "Slice Sampling" by Radford Neal , 2003 .

[9]  Matthew D. Hoffman,et al.  A trust-region method for stochastic variational inference with applications to streaming data , 2015, ICML.

[10]  Judea Pearl,et al.  Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach , 1982, AAAI.

[11]  Thomas H. Jordan,et al.  Full 3D Tomography for the Crustal Structure of the Los Angeles Region , 2007 .

[12]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[13]  Felix J. Herrmann,et al.  Uncertainty quantification in imaging and automatic horizon tracking: a Bayesian deep-prior based approach , 2020, SEG Technical Program Expanded Abstracts 2020.

[14]  Romain Brossier,et al.  Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation , 2013 .

[15]  Malcolm Sambridge,et al.  Parallel tempering for strongly nonlinear geoacoustic inversion. , 2012, The Journal of the Acoustical Society of America.

[16]  David M. Blei,et al.  Variational Inference: A Review for Statisticians , 2016, ArXiv.

[17]  Geoffrey E. Hinton,et al.  A View of the Em Algorithm that Justifies Incremental, Sparse, and other Variants , 1998, Learning in Graphical Models.

[18]  Ernesto Della Rossa,et al.  Probabilistic petrophysical-properties estimation integrating statistical rock physics with seismic inversion , 2010 .

[19]  Hao Liu,et al.  Variational Inference with Tail-adaptive f-Divergence , 2018, NeurIPS.

[20]  Klaus Mosegaard,et al.  Informed proposal Monte Carlo , 2020, Geophysical Journal International.

[21]  Brian Baptie,et al.  Transdimensional Love-wave tomography of the British Isles and shear-velocity structure of the East Irish Sea Basin from ambient-noise interferometry , 2013 .

[22]  Brian Baptie,et al.  Seismic interferometry and ambient noise tomography in the British Isles , 2012 .

[23]  Eric Walter,et al.  Identification of Parametric Models: from Experimental Data , 1997 .

[24]  Dustin Tran,et al.  Automatic Differentiation Variational Inference , 2016, J. Mach. Learn. Res..

[25]  Xin Zhang,et al.  Seismic Tomography Using Variational Inference Methods , 2019, Journal of Geophysical Research: Solid Earth.

[26]  Shun-ichi Amari.  α-Divergence and α-Projection in Statistical Manifold , 1985 .

[27]  J. Tromp,et al.  Finite-Frequency Kernels Based on Adjoint Methods , 2006 .

[28]  M. Sambridge,et al.  Multiple reflection and transmission phases in complex layered media using a multistage fast marching method , 2004 .

[29]  Dustin Tran,et al.  Operator Variational Inference , 2016, NIPS.

[30]  Tom Minka,et al.  Expectation Propagation for approximate Bayesian inference , 2001, UAI.

[31]  P. Guo,et al.  Bayesian trans-dimensional full waveform inversion: synthetic and field data application , 2020 .

[32]  N. Yi,et al.  Bayesian mapping of quantitative trait loci for complex binary traits. , 2000, Genetics.

[33]  Huajian Yao,et al.  Analysis of ambient noise energy distribution and phase velocity bias in ambient noise tomography, with application to SE Tibet , 2009 .

[34]  James Martin,et al.  A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion , 2012, SIAM J. Sci. Comput..

[35]  R. Feynman.  Statistical Mechanics, A Set of Lectures , 1972 .

[36]  A. Curtis,et al.  Variational Bayesian inversion (VBI) of quasi-localized seismic attributes for the spatial distribution of geological facies , 2018 .

[37]  Yang Liu,et al.  Stein Variational Policy Gradient , 2017, UAI.

[38]  Andreas Fichtner,et al.  Hamiltonian Monte Carlo solution of tomographic inverse problems , 2019 .

[39]  Nicola De Cao,et al.  Block Neural Autoregressive Flow , 2019, UAI.

[40]  Carl Tape,et al.  Adjoint Tomography of the Southern California Crust , 2009, Science.

[41]  J. Virieux,et al.  Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion , 2016 .

[42]  Seokhoon Oh,et al.  Geostatistical approach to bayesian inversion of geophysical data: Markov chain Monte Carlo method , 2001 .

[43]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .

[44]  H. M. Iyer,et al.  Seismic tomography : theory and practice , 1993 .

[45]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[46]  Xin Zhang,et al.  Bayesian seismic tomography using normalizing flows , 2021, Geophysical Journal International.

[47]  Hongjian Fang,et al.  Wavelet-based double-difference seismic tomography with sparsity regularization , 2014 .

[48]  A. Tarantola,et al.  Generalized Nonlinear Inverse Problems Solved Using the Least Squares Criterion (Paper 1R1855) , 1982 .

[49]  Nicola Piana Agostinetti,et al.  Local three-dimensional earthquake tomography by trans-dimensional Monte Carlo sampling , 2015 .

[50]  David Duvenaud,et al.  Neural Ordinary Differential Equations , 2018, NeurIPS.

[51]  A. Tarantola.  Inversion of seismic reflection data in the acoustic approximation , 1984 .

[52]  Michael H. Ritzwoller,et al.  A 3-D shear velocity model of the crust and uppermost mantle beneath the United States from ambient seismic noise , 2009 .

[53]  Carlos S. Kubrusly,et al.  Stochastic approximation algorithms and applications , 1973, CDC 1973.

[54]  Radford M. Neal.  Pattern Recognition and Machine Learning , 2007, Technometrics.

[55]  S. M. Ali,et al.  A General Class of Coefficients of Divergence of One Distribution from Another , 1966 .

[56]  Iain Murray,et al.  Cubic-Spline Flows , 2019, ICML 2019.

[57]  Mrinal K. Sen,et al.  A gradient based MCMC method for FWI and uncertainty analysis , 2019, SEG Technical Program Expanded Abstracts 2019.

[58]  Alexandrine Gesret,et al.  New parameterizations for Bayesian seismic tomography , 2018 .

[59]  Mike Warner,et al.  Adaptive waveform inversion: Theory , 2014 .

[60]  M. Warner,et al.  Anisotropic 3D full-waveform inversion , 2013 .

[61]  M. Nawaz,et al.  Rapid Discriminative Variational Bayesian Inversion of Geophysical Data for the Spatial Distribution of Geological Properties , 2019, Journal of Geophysical Research: Solid Earth.

[62]  Richard J. Boys,et al.  Discussion to "Riemann manifold Langevin and Hamiltonian Monte Carlo methods" by Girolami and Calderhead , 2011 .

[63]  E. Galetti,et al.  Transdimensional Electrical Resistivity Tomography , 2018, Journal of Geophysical Research: Solid Earth.

[64]  Qiang Liu,et al.  Stein Variational Gradient Descent With Matrix-Valued Kernels , 2019, NeurIPS.

[65]  Hedvig Kjellström,et al.  Advances in Variational Inference , 2017, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[66]  Yi Li,et al.  A Robust-Equitable Measure for Feature Ranking and Selection , 2017, J. Mach. Learn. Res..

[67]  Stephen P. Brooks,et al.  A Bayesian approach to crack detection in electrically conducting media , 2001 .

[68]  Haijiang Zhang,et al.  Wavelet‐based time‐dependent travel time tomography method and its application in imaging the Etna volcano in Italy , 2015 .

[69]  Chong Wang,et al.  Asymptotically Exact, Embarrassingly Parallel MCMC , 2013, UAI.

[70]  Malcolm Sambridge,et al.  Transdimensional inversion of receiver functions and surface wave dispersion , 2012 .

[71]  Wim A. Mulder,et al.  A correlation-based misfit criterion for wave-equation traveltime tomography , 2010 .

[72]  Mingjun Zhong,et al.  Efficient Gradient-Free Variational Inference using Policy Search , 2018, ICML.

[73]  Peter Gerstoft,et al.  Seismic interferometry-turning noise into signal , 2006 .

[74]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[75]  William T. Freeman,et al.  Understanding belief propagation and its generalizations , 2003 .

[76]  P. Deb.  Finite Mixture Models , 2008 .

[77]  Max Welling,et al.  Markov Chain Monte Carlo and Variational Inference: Bridging the Gap , 2014, ICML.

[78]  J. Virieux.  P-SV wave propagation in heterogeneous media: Velocity‐stress finite‐difference method , 1986 .

[79]  Mrinal K. Sen,et al.  Transdimensional seismic inversion using the reversible jump Hamiltonian Monte Carlo algorithm , 2017 .

[80]  Yoshua Bengio,et al.  NICE: Non-linear Independent Components Estimation , 2014, ICLR.

[81]  Malcolm Sambridge,et al.  A Parallel Tempering algorithm for probabilistic sampling and multimodal optimization , 2014 .

[82]  Lion Krischer,et al.  The Collaborative Seismic Earth Model: Generation 1 , 2018, Geophysical research letters.

[83]  M. Sambridge,et al.  Monte Carlo analysis of inverse problems , 2002 .

[84]  D. Grana.  Joint facies and reservoir properties inversion , 2018 .

[85]  Andrew Curtis,et al.  Bayesian full-waveform inversion with realistic priors , 2021, GEOPHYSICS.

[86]  R. Pratt.  Seismic waveform inversion in the frequency domain; Part 1, Theory and verification in a physical scale model , 1999 .

[87]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[88]  F. Feroz,et al.  Multimodal nested sampling: an efficient and robust alternative to Markov Chain Monte Carlo methods for astronomical data analyses , 2007, 0704.3704.

[89]  Variational full-waveform inversion , 2020 .

[90]  Radford M. Neal.  MCMC Using Hamiltonian Dynamics , 2011, 1206.1901.

[91]  Gary Martin,et al.  Marmousi2 An elastic upgrade for Marmousi , 2006 .

[92]  Andrew Curtis,et al.  3-D Monte Carlo surface wave tomography , 2018, Geophysical Journal International.

[93]  Eric Laloy,et al.  Inversion using a new low-dimensional representation of complex binary geological media based on a deep neural network , 2017, 1710.09196.

[94]  Yoram Singer,et al.  Adaptive Subgradient Methods for Online Learning and Stochastic Optimization , 2011, J. Mach. Learn. Res..

[95]  Mrinal K. Sen,et al.  2D Full-Waveform Inversion and Uncertainty Estimation using the Reversible Jump Hamiltonian Monte Carlo , 2017 .

[96]  B. Minsley.  A trans-dimensional Bayesian Markov chain Monte Carlo algorithm for model assessment using frequency-domain electromagnetic data , 2011 .

[97]  J. M. Hammersley,et al.  Markov fields on finite graphs and lattices , 1971 .

[98]  Sean Gerrish,et al.  Black Box Variational Inference , 2013, AISTATS.

[99]  S. Duane,et al.  Hybrid Monte Carlo , 1987 .

[100]  Jason Yosinski,et al.  Hamiltonian Neural Networks , 2019, NeurIPS.

[101]  E. Çinlar.  Probability and Stochastics , 2011 .

[102]  Keiiti Aki,et al.  Determination of three‐dimensional velocity anomalies under a seismic array using first P arrival times from local earthquakes: 1. A homogeneous initial model , 1976 .

[103]  Y. Marzouk,et al.  An introduction to sampling via measure transport , 2016, 1602.05023.

[104]  R. Plessix.  A review of the adjoint-state method for computing the gradient of a functional with geophysical applications , 2006 .

[105]  Albert Tarantola,et al.  Monte Carlo sampling of solutions to inverse problems , 1995 .

[106]  R. Arnold,et al.  Interrogation theory , 2018, Geophysical Journal International.

[107]  J. Tromp,et al.  Misfit functions for full waveform inversion based on instantaneous phase and envelope measurements , 2011 .

[108]  Nando de Freitas,et al.  Variational MCMC , 2001, UAI.

[109]  M. Sambridge,et al.  Markov chain Monte Carlo (MCMC) sampling methods to determine optimal models, model resolution and model choice for Earth Science problems , 2009 .

[110]  G. Mariéthoz,et al.  Multiple-point Geostatistics: Stochastic Modeling with Training Images , 2014 .

[111]  Andreas Fichtner,et al.  Bayesian Elastic Full‐Waveform Inversion Using Hamiltonian Monte Carlo , 2019, Journal of Geophysical Research: Solid Earth.

[112]  Barak A. Pearlmutter,et al.  Automatic differentiation in machine learning: a survey , 2015, J. Mach. Learn. Res..

[113]  A. Malinverno.  Parsimonious Bayesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem , 2002 .

[114]  Farhan Abrol,et al.  Variational Tempering , 2016, AISTATS.

[115]  A. Curtis,et al.  Variational Bayesian inversion of seismic attributes jointly for geologic facies and petrophysical rock properties , 2020, GEOPHYSICS.

[116]  Michel Campillo,et al.  High-Resolution Surface-Wave Tomography from Ambient Seismic Noise , 2005, Science.

[117]  David L. Alumbaugh,et al.  Robust and accelerated Bayesian inversion of marine controlled-source electromagnetic data using parallel tempering , 2013 .

[118]  Andrew Gelman,et al.  Handbook of Markov Chain Monte Carlo , 2011 .

[119]  Albert Tarantola,et al.  Theoretical background for the inversion of seismic waveforms including elasticity and attenuation , 1988 .

[120]  Richard E. Turner,et al.  Rényi Divergence Variational Inference , 2016, NIPS.

[121]  Shakir Mohamed,et al.  Variational Inference with Normalizing Flows , 2015, ICML.

[122]  P. Green,et al.  Reversible jump MCMC , 2009 .

[123]  Samy Bengio,et al.  Density estimation using Real NVP , 2016, ICLR.

[124]  O. Zobay.  Variational Bayesian inference with Gaussian-mixture approximations , 2014 .

[125]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[126]  Qinya Liu,et al.  Tomography, Adjoint Methods, Time-Reversal, and Banana-Doughnut Kernels , 2004 .

[127]  Iain Murray,et al.  Neural Spline Flows , 2019, NeurIPS.

[128]  Brian Baptie,et al.  SUPPLEMENTAL MATERIAL for Uncertainty Loops in Travel-Time Tomography from Nonlinear Wave Physics , 2015 .

[129]  Jun S. Liu,et al.  Sequential Monte Carlo methods for dynamic systems , 1997 .

[130]  A. L. Ramirez,et al.  Stochastic inversion of electrical resistivity changes using a Markov Chain Monte Carlo approach , 2005 .

[131]  Roel Snieder,et al.  Reconditioning inverse problems using the genetic algorithm and revised parameterization , 1997 .

[132]  A. Tarantola,et al.  Two‐dimensional nonlinear inversion of seismic waveforms: Numerical results , 1986 .

[133]  Anandaroop Ray,et al.  Frequency domain full waveform elastic inversion of marine seismic data from the Alba field using a Bayesian trans-dimensional algorithm , 2016 .

[134]  R. Weaver,et al.  On the precision of noise‐correlation interferometry. , 2009, 1103.1785.

[135]  Masa-aki Sato,et al.  Online Model Selection Based on the Variational Bayes , 2001, Neural Computation.

[136]  Dustin Tran,et al.  Hierarchical Variational Models , 2015, ICML.

[137]  Felix J. Herrmann,et al.  Faster Uncertainty Quantification for Inverse Problems with Conditional Normalizing Flows , 2020, ArXiv.

[138]  Michael H. Ritzwoller,et al.  Joint inversion of surface wave dispersion and receiver functions: a Bayesian Monte-Carlo approach , 2013 .

[139]  N. Metropolis,et al.  The Monte Carlo method. , 1949 .

[140]  David Duvenaud,et al.  FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models , 2018, ICLR.

[141]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[142]  Lester W. Mackey,et al.  Measuring Sample Quality with Stein's Method , 2015, NIPS.

[143]  Klaus Mosegaard,et al.  MONTE CARLO METHODS IN GEOPHYSICAL INVERSE PROBLEMS , 2002 .

[144]  R. Parker,et al.  Occam's inversion; a practical algorithm for generating smooth models from electromagnetic sounding data , 1987 .

[145]  Andreas Fichtner,et al.  Theoretical background for continental‐ and global‐scale full‐waveform inversion in the time–frequency domain , 2008 .

[146]  A. Dziewoński,et al.  Global Images of the Earth's Interior , 1987, Science.

[147]  M. Sambridge,et al.  Trans‐Dimensional Surface Reconstruction With Different Classes of Parameterization , 2019, Geochemistry, Geophysics, Geosystems.

[148]  Andreas Fichtner,et al.  Full seismic waveform tomography for upper-mantle structure in the Australasian region using adjoint methods , 2009 .

[149]  Prafulla Dhariwal,et al.  Glow: Generative Flow with Invertible 1x1 Convolutions , 2018, NeurIPS.

[150]  A. Curtis,et al.  Bayesian inversion of seismic attributes for geological facies using a Hidden Markov Model , 2017 .

[151]  S. Operto,et al.  Which data residual norm for robust elastic frequency-domain full waveform inversion? , 2010 .

[152]  Martin J. Blunt,et al.  Stochastic Seismic Waveform Inversion Using Generative Adversarial Networks as a Geological Prior , 2019, Mathematical Geosciences.

[153]  R. Tweedie,et al.  Exponential convergence of Langevin distributions and their discrete approximations , 1996 .

[154]  A. Curtis,et al.  Imaging the subsurface using induced seismicity and ambient noise: 3-D tomographic Monte Carlo joint inversion of earthquake body wave traveltimes and surface wave dispersion , 2020, Geophysical Journal International.

[155]  Daniel Hernández-Lobato,et al.  Black-Box Alpha Divergence Minimization , 2015, ICML.

[156]  K. Hukushima,et al.  Exchange Monte Carlo Method and Application to Spin Glass Simulations , 1995, cond-mat/9512035.

[157]  A. Curtis,et al.  Rayleigh wave tomography of the British Isles from ambient seismic noise , 2014 .

[158]  Matthew J. Beal.  Variational algorithms for approximate Bayesian inference , 2003 .

[159]  Dilin Wang,et al.  Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm , 2016, NIPS.

[160]  David M. Blei,et al.  Population Empirical Bayes , 2014, UAI.

[161]  R. E. Wengert,et al.  A simple automatic derivative evaluation program , 1964, Commun. ACM.

[162]  Shurtleff.  Introduction to sampling , 2009 .

[163]  Sam Kaplan,et al.  Low frequency full waveform seismic inversion within a tree based Bayesian framework , 2018 .

[164]  Gerard T. Schuster,et al.  Wave-equation traveltime inversion , 1991 .

[165]  N. Rawlinson,et al.  Transdimensional inversion of ambient seismic noise for 3D shear velocity structure of the Tasmanian crust , 2013 .

[166]  Clifford H. Thurber,et al.  Parameter estimation and inverse problems , 2005 .

[167]  David Duvenaud,et al.  Invertible Residual Networks , 2018, ICML.

[168]  Qiang Liu,et al.  A Kernelized Stein Discrepancy for Goodness-of-fit Tests , 2016, ICML.

[169]  David Pugmire,et al.  Global adjoint tomography: first-generation model , 2016 .

[170]  Alberto Malinverno,et al.  A Monte Carlo Method to Quantify Uncertainty in the Inversion of Zero-Offset VSP Data , 2000 .

[171]  M. Sambridge,et al.  Seismic tomography with the reversible jump algorithm , 2009 .

[172]  Qiang Liu,et al.  Stein Variational Gradient Descent as Gradient Flow , 2017, NIPS.

[173]  Gerhard Lakemeyer,et al.  Exploring artificial intelligence in the new millennium , 2003 .

[174]  A. Malcolm,et al.  Time-lapse full-waveform inversion using Hamiltonian Monte Carlo: A proof of concept , 2020 .

[175]  Andreas Fichtner,et al.  The adjoint method in seismology – I. Theory , 2006 .

[176]  Rhys Hawkins,et al.  Geophysical imaging using trans-dimensional trees. , 2015 .

[177]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[178]  Matthew J. Beal,et al.  The variational Bayesian EM algorithm for incomplete data: with application to scoring graphical model structures , 2003 .

[179]  G. A. Young,et al.  High‐dimensional Statistics: A Non‐asymptotic Viewpoint, Martin J.Wainwright, Cambridge University Press, 2019, xvii 552 pages, £57.99, hardback ISBN: 978‐1‐1084‐9802‐9 , 2020, International Statistical Review.

[180]  Manfred Opper,et al.  Perturbative Black Box Variational Inference , 2017, NIPS.

[181]  Yongxin Pan,et al.  Physics of the Earth and Planetary Interiors , 2015 .

[182]  C. Stein.  A bound for the error in the normal approximation to the distribution of a sum of dependent random variables , 1972 .

[183]  Barbara Romanowicz,et al.  Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography , 2014 .

[184]  Joost van der Neut,et al.  Marchenko redatuming, imaging and multiple elimination, and their mutual relations , 2021, GEOPHYSICS.

[185]  Michael W Deem,et al.  Parallel tempering: theory, applications, and new perspectives. , 2005, Physical chemistry chemical physics : PCCP.