VIP - Variational Inversion Package with example implementations of Bayesian tomographic imaging

Xin Zhang; Andrew Curtis

doi:10.26443/seismica.v3i1.1143

Authors

Xin Zhang https://orcid.org/0000-0003-0344-1453
Andrew Curtis University of Edinburgh https://orcid.org/0000-0003-1222-1583

DOI:

https://doi.org/10.26443/seismica.v3i1.1143

Abstract

Bayesian inference has become an important methodology to solve inverse problems and to quantify uncertainties in their solutions. Variational inference is a method that provides probabilistic, Bayesian solutions efficiently by using optimisation. In this study we present a Python Variational Inversion Package (VIP), to solve inverse problems using variational inference methods. The package includes automatic differential variational inference (ADVI), Stein variational gradient descent (SVGD) and stochastic SVGD (sSVGD), and provides implementations of 2D travel time tomography and 2D full waveform inversion including test examples and solutions. Users can solve their own problems by supplying an appropriate forward function and a gradient calculation code. In addition, the package provides a scalable implementation which can be deployed easily on a desktop machine or using modern high performance computational facilities. The examples demonstrate that VIP is an efficient, scalable, extensible and user-friendly package, and can be used to solve a wide range of low or high dimensional inverse problems in practice.

References

Afanasiev, M., Boehm, C., van Driel, M., Krischer, L., Rietmann, M., May, D. A., Knepley, M. G., & Fichtner, A. (2019). Modular and flexible spectral-element waveform modelling in two and three dimensions. Geophysical Journal International, 216(3), 1675–1692. https://doi.org/10.1093/gji/ggy469 DOI: https://doi.org/10.1093/gji/ggy469

Agata, R., Shiraishi, K., & Fujie, G. (2023). Bayesian seismic tomography based on velocity-space Stein variational gradient descent for physics-informed neural network. IEEE Transactions on Geoscience and Remote Sensing. https://doi.org/10.1109/TGRS.2023.3295414 DOI: https://doi.org/10.1109/TGRS.2023.3295414

Ahmed, Z., Yunyue, L., & Arthur, C. (n.d.). Regularized seismic amplitude inversion via variational inference. Geophysical Prospecting, n/a(n/a).

Aki, K., & Lee, W. (1976). Determination of three-dimensional velocity anomalies under a seismic array using first P arrival times from local earthquakes: 1. A homogeneous initial model. Journal of Geophysical Research, 81(23), 4381–4399. https://doi.org/10.1029/JB081i023p04381 DOI: https://doi.org/10.1029/JB081i023p04381

Andersen, K. E., Brooks, S. P., & Hansen, M. B. (2001). A Bayesian approach to crack detection in electrically conducting media. Inverse Problems, 17(1), 121. https://doi.org/10.1088/0266-5611/17/1/310 DOI: https://doi.org/10.1088/0266-5611/17/1/310

Arnold, R., & Curtis, A. (2018). Interrogation theory. Geophysical Journal International, 214(3), 1830–1846. https://doi.org/10.1093/gji/ggy248 DOI: https://doi.org/10.1093/gji/ggy248

Aster, R. C., Borchers, B., & Thurber, C. H. (2018). Parameter estimation and inverse problems. Elsevier. DOI: https://doi.org/10.1016/B978-0-12-804651-7.00015-8

Ba, J., Erdogdu, M. A., Ghassemi, M., Sun, S., Suzuki, T., Wu, D., & Zhang, T. (2022). Understanding the Variance Collapse of SVGD in High Dimensions. International Conference on Learning Representations. https://openreview.net/forum?id=Qycd9j5Qp9J

Behnel, S., Bradshaw, R., Citro, C., Dalcin, L., Seljebotn, D. S., & Smith, K. (2010). Cython: The best of both worlds. Computing in Science & Engineering, 13(2), 31–39. https://doi.org/10.1109/MCSE.2010.118 DOI: https://doi.org/10.1109/MCSE.2010.118

Bishop, C. M. (2006). Pattern recognition and machine learning. springer.

Blei, D. M., Kucukelbir, A., & McAuliffe, J. D. (2017). Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518), 859–877. https://doi.org/10.1080/01621459.2017.1285773 DOI: https://doi.org/10.1080/01621459.2017.1285773

Bodin, T., & Sambridge, M. (2009). Seismic tomography with the reversible jump algorithm. Geophysical Journal International, 178(3), 1411–1436. https://doi.org/10.1111/j.1365-246X.2009.04226.x DOI: https://doi.org/10.1111/j.1365-246X.2009.04226.x

Brooks, S., Gelman, A., Jones, G., & Meng, X.-L. (2011). Handbook of Markov chain Monte Carlo. CRC press. DOI: https://doi.org/10.1201/b10905

Chen, T., Fox, E., & Guestrin, C. (2014). Stochastic Gradient Hamiltonian Monte Carlo. In E. P. Xing & T. Jebara (Eds.), Proceedings of the 31st International Conference on Machine Learning (Vol. 32, Issue 2, pp. 1683–1691). PMLR. https://proceedings.mlr.press/v32/cheni14.html

Curtis, A., & Lomax, A. (2001). Prior information, sampling distributions, and the curse of dimensionality. Geophysics, 66(2), 372–378. https://doi.org/10.1190/1.1444928 DOI: https://doi.org/10.1190/1.1444928

Dosso, S. E., Holland, C. W., & Sambridge, M. (2012). Parallel tempering for strongly nonlinear geoacoustic inversion. The Journal of the Acoustical Society of America, 132(5), 3030–3040. https://doi.org/10.1121/1.4757639 DOI: https://doi.org/10.1121/1.4757639

Duane, S., Kennedy, A. D., Pendleton, B. J., & Roweth, D. (1987). Hybrid Monte Carlo. Physics Letters B, 195(2), 216–222. https://doi.org/10.1016/0370-2693(87)91197-X DOI: https://doi.org/10.1016/0370-2693(87)91197-X

Duchi, J., Hazan, E., & Singer, Y. (2011). Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research, 12(Jul), 2121–2159. http://jmlr.org/papers/v12/duchi11a.html

Fichtner, A, Bunge, H.-P., & Igel, H. (2006). The adjoint method in seismology: I. Theory. Physics of the Earth and Planetary Interiors, 157(1–2), 86–104. https://doi.org/10.1016/j.pepi.2006.03.016 DOI: https://doi.org/10.1016/j.pepi.2006.03.016

Fichtner, Andreas, Zunino, A., & Gebraad, L. (2018). Hamiltonian Monte Carlo solution of tomographic inverse problems. Geophysical Journal International, 216(2), 1344–1363. https://doi.org/10.1093/gji/ggy496 DOI: https://doi.org/10.1093/gji/ggy496

Galetti, E., Curtis, A., Baptie, B., Jenkins, D., & Nicolson, H. (2017). Transdimensional Love-wave tomography of the British Isles and shear-velocity structure of the East Irish Sea Basin from ambient-noise interferometry. Geophysical Journal International, 208(1), 36–58. https://doi.org/10.1093/gji/ggw286 DOI: https://doi.org/10.1093/gji/ggw286

Galetti, E., Curtis, A., Meles, G. A., & Baptie, B. (2015). Uncertainty loops in travel-time tomography from nonlinear wave physics. Physical Review Letters, 114(14), 148501. https://doi.org/10.1103/physrevlett.114.148501 DOI: https://doi.org/10.1103/PhysRevLett.114.148501

Gallagher, K., Charvin, K., Nielsen, S., Sambridge, M., & Stephenson, J. (2009). Markov chain Monte Carlo (MCMC) sampling methods to determine optimal models, model resolution and model choice for Earth Science problems. Marine and Petroleum Geology, 26(4), 525–535. https://doi.org/10.1016/j.marpetgeo.2009.01.003 DOI: https://doi.org/10.1016/j.marpetgeo.2009.01.003

Gallego, V., & Insua, D. R. (2018). Stochastic gradient MCMC with repulsive forces. ArXiv Preprint ArXiv:1812.00071.

Gebraad, L., Boehm, C., & Fichtner, A. (2020). Bayesian Elastic Full-Waveform Inversion Using Hamiltonian Monte Carlo. Journal of Geophysical Research: Solid Earth, 125(3), e2019JB018428. https://doi.org/10.1029/2019JB018428 DOI: https://doi.org/10.1029/2019JB018428

Gong, C., Peng, J., & Liu, Q. (2019). Quantile Stein Variational Gradient Descent for Batch Bayesian Optimization. In K. Chaudhuri & R. Salakhutdinov (Eds.), Proceedings of the 36th International Conference on Machine Learning (Vol. 97, pp. 2347–2356). PMLR. https://proceedings.mlr.press/v97/gong19b.html

Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Byesian model determination. Biometrika, 711–732. https://doi.org/10.1093/biomet/82.4.711 DOI: https://doi.org/10.1093/biomet/82.4.711

Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1), 97–109. https://doi.org/10.1093/biomet/57.1.97 DOI: https://doi.org/10.1093/biomet/57.1.97

Hawkins, R., & Sambridge, M. (2015). Geophysical imaging using trans-dimensional trees. Geophysical Journal International, 203(2), 972–1000. https://doi.org/10.1093/gji/ggv326 DOI: https://doi.org/10.1093/gji/ggv326

Hukushima, K., & Nemoto, K. (1996). Exchange Monte Carlo method and application to spin glass simulations. Journal of the Physical Society of Japan, 65(6), 1604–1608. https://doi.org/10.1143/JPSJ.65.1604 DOI: https://doi.org/10.1143/JPSJ.65.1604

Izzatullah, M., Alkhalifah, T., Romero, J., Corrales, M., Luiken, N., & Ravasi, M. (2023). Plug-and-Play Stein variational gradient descent for Bayesian post-stack seismic inversion. 84th EAGE Annual Conference & Exhibition, 2023(1), 1–5. https://doi.org/10.3997/2214-4609.202310177 DOI: https://doi.org/10.3997/2214-4609.202310177

Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. ArXiv Preprint ArXiv:1412.6980.

Komatitsch, D., Tromp, J., Garg, R., Gharti, H. N., Nagaso, M., Oral, E., & et al. (2023). SPECFEM/specfem3d: SPECFEM3D v4.1.0 (v4.1.0) [Computer software]. Zenodo. https://doi.org/10.5281/zenodo.10413988

Kubrusly, C., & Gravier, J. (1973). Stochastic approximation algorithms and applications. 1973 IEEE Conference on Decision and Control Including the 12th Symposium on Adaptive Processes, 763–766. https://doi.org/10.1109/CDC.1973.269114 DOI: https://doi.org/10.1109/CDC.1973.269114

Kucukelbir, A., Tran, D., Ranganath, R., Gelman, A., & Blei, D. M. (2017). Automatic differentiation variational inference. The Journal of Machine Learning Research, 18(14), 1–45. http://jmlr.org/papers/v18/16-107.html

Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 79–86. http://www.jstor.org/stable/2236703 DOI: https://doi.org/10.1214/aoms/1177729694

Kumar, R., Kotsi, M., Siahkoohi, A., & Malcolm, A. (2021). Enabling uncertainty quantification for seismic data preprocessing using normalizing flows (NF)—An interpolation example. First International Meeting for Applied Geoscience & Energy, 1515–1519. https://doi.org/10.1190/segam2021-3583705.1 DOI: https://doi.org/10.1190/segam2021-3583705.1

Li, X., Bürgi, P. M., Ma, W., Noh, H. Y., Wald, D. J., & Xu, S. (2023). DisasterNet: Causal Bayesian Networks with Normalizing Flows for Cascading Hazards Estimation from Satellite Imagery. Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 4391–4403. https://doi.org/10.1145/3580305.3599807 DOI: https://doi.org/10.1145/3580305.3599807

Lions, J. L. (1971). Optimal control of systems governed by partial differential equations (Vol. 170). Springer. DOI: https://doi.org/10.1007/978-3-642-65024-6

Liu, Q., & Wang, D. (n.d.). In D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, & R. Garnett (Eds.), Advances in Neural Information Processing Systems.

Lomas, A., Luo, S., Irakarama, M., Johnston, R., Vyas, M., & Shen, X. (2023). 3D Probabilistic Full Waveform Inversion: Application to Gulf of Mexico Field Data. 84th EAGE Annual Conference & Exhibition, 2023(1), 1–5. https://doi.org/10.3997/2214-4609.202310720 DOI: https://doi.org/10.3997/2214-4609.202310720

Ma, Y.-A., Chen, T., & Fox, E. (n.d.). Advances in Neural Information Processing Systems.

Malinverno, A. (2002). Parsimonious Byesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem. Geophysical Journal International, 151(3), 675–688. https://doi.org/10.1046/j.1365-246X.2002.01847.x DOI: https://doi.org/10.1046/j.1365-246X.2002.01847.x

Malinverno, A., Leaney, S., & others. (2000). A Monte Carlo method to quantify uncertainty in the inversion of zero-offset VSP data. 2000 SEG Annual Meeting. https://doi.org/10.1190/1.1815943 DOI: https://doi.org/10.1190/1.1815943

Martin, J., Wilcox, L. C., Burstedde, C., & Ghattas, O. (2012). A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion. SIAM Journal on Scientific Computing, 34(3), A1460–A1487. https://doi.org/10.1137/110845598 DOI: https://doi.org/10.1137/110845598

McKean, S., Priest, J., Dettmer, J., Fradelizio, G., & Eaton, D. (2023). Separating Hydraulic Fracturing Microseismicity From Induced Seismicity by Bayesian Inference of Non-Linear Pressure Diffusivity. Geophysical Research Letters, 50(14), e2022GL102131. https://doi.org/10.1029/2022GL102131 DOI: https://doi.org/10.1029/2022GL102131

Metropolis, N., & Ulam, S. (1949). The Monte Carlo method. Journal of the American Statistical Association, 44(247), 335–341. https://doi.org/10.1080/01621459.1949.10483310 DOI: https://doi.org/10.1080/01621459.1949.10483310

Mosegaard, K., & Sambridge, M. (2002). Monte Carlo analysis of inverse problems. Inverse Problems, 18(3), R29. https://doi.org/10.1088/0266-5611/18/3/201 DOI: https://doi.org/10.1088/0266-5611/18/3/201

Mosegaard, K., & Tarantola, A. (1995). Monte Carlo sampling of solutions to inverse problems. Journal of Geophysical Research: Solid Earth, 100(B7), 12431–12447. https://doi.org/10.1029/94JB03097 DOI: https://doi.org/10.1029/94JB03097

Nicolson, H., Curtis, A., & Baptie, B. (2014). Rayleigh wave tomography of the British Isles from ambient seismic noise. Geophysical Journal International, 198(2), 637–655. https://doi.org/10.1093/gji/ggu071 DOI: https://doi.org/10.1093/gji/ggu071

Nicolson, H., Curtis, A., Baptie, B., & Galetti, E. (2012). Seismic interferometry and ambient noise tomography in the British Isles. Proceedings of the Geologists’ Association, 123(1), 74–86. https://doi.org/10.1016/j.pgeola.2011.04.002 DOI: https://doi.org/10.1016/j.pgeola.2011.04.002

O’Hagan, A., & Forster, J. J. (2004). Kendall’s advanced theory of statistics, volume 2B: Bayesian inference (Vol. 2). Arnold.

Oksendal, B. (2013). Stochastic differential equations: an introduction with applications. Springer Science & Business Media.

Orozco, R., Louboutin, M., Siahkoohi, A., Rizzuti, G., van Leeuwen, T., & Herrmann, F. (2023). Amortized Normalizing Flows for Transcranial Ultrasound with Uncertainty Quantification. ArXiv Preprint ArXiv:2303.03478. https://doi.org/10.48550/arXiv.2303.03478

Parisi, G. (1988). Statistical field theory. Addison-Wesley. https://doi.org/10.1063/1.2811677 DOI: https://doi.org/10.1063/1.2811677

Pinder, T., Nemeth, C., & Leslie, D. (2020). Stein variational Gaussian processes. ArXiv Preprint ArXiv:2009.12141. https://doi.org/10.48550/arXiv.2009.12141

Plessix, R.-E. (2006). A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophysical Journal International, 167(2), 495–503. https://doi.org/10.1111/j.1365-246X.2006.02978.x DOI: https://doi.org/10.1111/j.1365-246X.2006.02978.x

Ramgraber, M., Weatherl, R., Blumensaat, F., & Schirmer, M. (2021). Non-Gaussian Parameter Inference for Hydrogeological Models Using Stein Variational Gradient Descent. Water Resources Research, 57(4), e2020WR029339. https://doi.org/10.1029/2020WR029339 DOI: https://doi.org/10.1029/2020WR029339

Ramirez, A. L., Nitao, J. J., Hanley, W. G., Aines, R., Glaser, R. E., Sengupta, S. K., Dyer, K. M., Hickling, T. L., & Daily, W. D. (2005). Stochastic inversion of electrical resistivity changes using a Markov Chain Monte Carlo approach. Journal of Geophysical Research: Solid Earth, 110(B2). https://doi.org/10.1029/2004JB003449 DOI: https://doi.org/10.1029/2004JB003449

Rawlinson, N. (2005). FMST: fast marching surface tomography package–Instructions. Research School of Earth Sciences, Australian National University, Canberra, 29, 47.

Rawlinson, N., & Sambridge, M. (2004). Multiple reflection and transmission phases in complex layered media using a multistage fast marching method. Geophysics, 69(5), 1338–1350. https://doi.org/10.1190/1.1801950 DOI: https://doi.org/10.1190/1.1801950

Robbins, H., & Monro, S. (1951). A stochastic approximation method. The Annals of Mathematical Statistics, 400–407. https://doi.org/10.1214/aoms/1177729586 DOI: https://doi.org/10.1214/aoms/1177729586

Roberts, G. O., Tweedie, R. L., & others. (1996). Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli, 2(4), 341–363. https://doi.org/10.2307/3318418 DOI: https://doi.org/10.2307/3318418

Rocklin, M., & others. (2015). Dask: Parallel computation with blocked algorithms and task scheduling. Proceedings of the 14th Python in Science Conference, 130, 136. https://doi.org/10.25080/majora-7b98e3ed-013 DOI: https://doi.org/10.25080/Majora-7b98e3ed-013

Rücker, C., Günther, T., & Wagner, F. M. (2017). pyGIMLi: An open-source library for modelling and inversion in geophysics. Computers & Geosciences, 109, 106–123. https://doi.org/10.1016/j.cageo.2017.07.011 DOI: https://doi.org/10.1016/j.cageo.2017.07.011

Sambridge, M. (2013). A parallel tempering algorithm for probabilistic sampling and multimodal optimization. Geophysical Journal International, ggt342. https://doi.org/10.1093/gji/ggt342 DOI: https://doi.org/10.1093/gji/ggt342

Sambridge, M., & Mosegaard, K. (2002). Monte Carlo methods in geophysical inverse problems. Reviews of Geophysics, 40(3), 3–1. https://doi.org/10.1029/2000RG000089 DOI: https://doi.org/10.1029/2000RG000089

Sen, M. K., & Biswas, R. (2017). Transdimensional seismic inversion using the reversible jump Hamiltonian Monte Carlo algorithm. Geophysics, 82(3), R119–R134. https://doi.org/10.1190/geo2016-0010.1 DOI: https://doi.org/10.1190/geo2016-0010.1

Shen, W., Ritzwoller, M. H., Schulte-Pelkum, V., & Lin, F.-C. (2012). Joint inversion of surface wave dispersion and receiver functions: a Byesian Monte-Carlo approach. Geophysical Journal International, 192(2), 807–836. https://doi.org/10.1093/gji/ggs050 DOI: https://doi.org/10.1093/gji/ggs050

Siahkoohi, A., Orozco, R., Rizzuti, G., & Herrmann, F. J. (2022). Wave-equation-based inversion with amortized variational Bayesian inference. ArXiv Preprint ArXiv:2203.15881. https://doi.org/10.48550/arXiv.2203.15881

Siahkoohi, A., Rizzuti, G., & Herrmann, F. J. (2022). Deep Bayesian inference for seismic imaging with tasks. Geophysics, 87(5), S281–S302. https://doi.org/10.1190/geo2021-0666.1 DOI: https://doi.org/10.1190/geo2021-0666.1

Siahkoohi, A., Rizzuti, G., Witte, P. A., & Herrmann, F. J. (2020). Faster Uncertainty Quantification for Inverse Problems with Conditional Normalizing Flows. ArXiv Preprint ArXiv:2007.07985. https://doi.org/10.48550/arXiv.2007.07985

Smith, J. D., Ross, Z. E., Azizzadenesheli, K., & Muir, J. B. (2022). HypoSVI: Hypocentre inversion with Stein variational inference and physics informed neural networks. Geophysical Journal International, 228(1), 698–710. https://doi.org/10.1093/gji/ggab309 DOI: https://doi.org/10.1093/gji/ggab309

Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49(8), 1259–1266. https://doi.org/10.1190/1.1441754 DOI: https://doi.org/10.1190/1.1441754

Tarantola, A. (1988). Theoretical background for the inversion of seismic waveforms, including elasticity and attenuation. In Scattering and Attenuations of Seismic Waves, Part I (pp. 365–399). Springer. https://doi.org/10.1007/bf01772605 DOI: https://doi.org/10.1007/978-3-0348-7722-0_19

Tarantola, A. (2005). Inverse problem theory and methods for model parameter estimation (Vol. 89). SIAM. https://doi.org/10.1137/1.9780898717921 DOI: https://doi.org/10.1137/1.9780898717921

Team, S. D., & others. (2016). Stan modeling language users guide and reference manual. Technical Report.

Tromp, J., Tape, C., & Liu, Q. (2005). Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophysical Journal International, 160(1), 195–216. https://doi.org/10.1111/j.1365-246X.2004.02453.x DOI: https://doi.org/10.1111/j.1365-246X.2004.02453.x

Valentine, A. P., & Sambridge, M. (2020). Gaussian process models—I. A framework for probabilistic continuous inverse theory. Geophysical Journal International, 220(3), 1632–1647. https://doi.org/10.1093/gji/ggz520 DOI: https://doi.org/10.1093/gji/ggz520

Wang, D., Tang, Z., Bajaj, C., & Liu, Q. (2019). Stein variational gradient descent with matrix-valued kernels. Advances in Neural Information Processing Systems, 7836–7846. https://doi.org/10.48550/arXiv.1910.12794

Wang, W., McMechan, G. A., & Ma, J. (2023). Re-weighted variational full waveform inversions. Geophysics, 88(4), 1–61. https://doi.org/10.1190/geo2021-0766.1 DOI: https://doi.org/10.1190/geo2021-0766.1

Wathelet, M., Chatelain, J.-L., Cornou, C., Giulio, G. D., Guillier, B., Ohrnberger, M., & Savvaidis, A. (2020). Geopsy: A user-friendly open-source tool set for ambient vibration processing. Seismological Research Letters, 91(3), 1878–1889. https://doi.org/10.1785/0220190360 DOI: https://doi.org/10.1785/0220190360

Welling, M., & Teh, Y. W. (2011). Bayesian learning via stochastic gradient Langevin dynamics. Proceedings of the 28th International Conference on Machine Learning (ICML-11), 681–688.

Zaroli, C. (2016). Global seismic tomography using Backus–Gilbert inversion. Geophysical Supplements to the Monthly Notices of the Royal Astronomical Society, 207(2), 876–888. https://doi.org/10.1093/gji/ggw315 DOI: https://doi.org/10.1093/gji/ggw315

Zaroli, C. (2019). Seismic tomography using parameter-free Backus–Gilbert inversion. Geophysical Journal International, 218(1), 619–630. https://doi.org/10.1093/gji/ggz175 DOI: https://doi.org/10.1093/gji/ggz175

Zaroli, C., Koelemeijer, P., & Lambotte, S. (2017). Toward seeing the Earth’s interior through unbiased tomographic lenses. Geophysical Research Letters, 44(22), 11–399. https://doi.org/10.1002/2017GL074996 DOI: https://doi.org/10.1002/2017GL074996

Zeiler, M. D. (2012). Adadelta: an adaptive learning rate method. ArXiv Preprint ArXiv:1212.5701. https://doi.org/10.48550/arXiv.1212.5701

Zhang, Chendong, & Chen, T. (2022). Bayesian slip inversion with automatic differentiation variational inference. Geophysical Journal International, 229(1), 546–565. https://doi.org/10.1093/gji/ggab438 DOI: https://doi.org/10.1093/gji/ggab438

Zhang, Cheng, Bütepage, J., Kjellström, H., & Mandt, S. (2018). Advances in variational inference. IEEE Transactions on Pattern Analysis and Machine Intelligence, 41(8), 2008–2026. https://doi.org/10.1109/TPAMI.2018.2889774 DOI: https://doi.org/10.1109/TPAMI.2018.2889774

Zhang, X., & Curtis, A. (2020a). Seismic tomography using variational inference methods. Journal of Geophysical Research: Solid Earth, 125(4), e2019JB018589. DOI: https://doi.org/10.1029/2019JB018589

Zhang, X., & Curtis, A. (2020b). Variational full-waveform inversion. Geophysical Journal International, 222(1), 406–411. https://doi.org/10.1093/gji/ggaa170 DOI: https://doi.org/10.1093/gji/ggaa170

Zhang, X., & Curtis, A. (2021). Bayesian Full-waveform Inversion with Realistic Priors. Geophysics, 86(5), 1–20. https://doi.org/10.1190/geo2021-0118.1 DOI: https://doi.org/10.1190/geo2021-0118.1

Zhang, X., & Curtis, A. (2022). Interrogating probabilistic inversion results for subsurface structural information. Geophysical Journal International, 229(2), 750–757. https://doi.org/10.1093/gji/ggab496 DOI: https://doi.org/10.1093/gji/ggab496

Zhang, X., Curtis, A., Galetti, E., & de Ridder, S. (2018). 3-D Monte Carlo surface wave tomography. Geophysical Journal International, 215(3), 1644–1658. https://doi.org/10.1093/gji/ggy362 DOI: https://doi.org/10.1093/gji/ggy362

Zhang, X., Lomas, A., Zhou, M., Zheng, Y., & Curtis, A. (2023). 3-D Bayesian variational full waveform inversion. Geophysical Journal International, 234(1), 546–561. https://doi.org/10.1093/gji/ggad057 DOI: https://doi.org/10.1093/gji/ggad057

Zhang, Z., & Curtis, A. (2023). Variational Inversion Package. https://doi.org/10.5281/zenodo.10036815

Zhao, X., & Curtis, A. (2024). Bayesian inversion, uncertainty analysis and interrogation using boosting variational inference. Journal of Geophysical Research: Solid Earth, 129(1), e2023JB027789. https://doi.org/10.1029/2023JB027789 DOI: https://doi.org/10.1029/2023JB027789

Zhao, X., Curtis, A., & Zhang, X. (2022a). Bayesian seismic tomography using normalizing flows. Geophysical Journal International, 228(1), 213–239. DOI: https://doi.org/10.1093/gji/ggab298

Zhao, X., Curtis, A., & Zhang, X. (2022b). Interrogating Subsurface Structures using Probabilistic Tomography: an example assessing the volume of Irish Sea basins. Journal of Geophysical Research: Solid Earth, 127(4), e2022JB024098. https://doi.org/10.1029/2022JB024098 DOI: https://doi.org/10.1029/2022JB024098

Zhao, Z., & Sen, M. K. (2019). A gradient based MCMC method for FWI and uncertainty analysis. In SEG Technical Program Expanded Abstracts 2019 (pp. 1465–1469). Society of Exploration Geophysicists. https://doi.org/10.1190/segam2019-3216560.1 DOI: https://doi.org/10.1190/segam2019-3216560.1

Zhdanov, M. S. (2002). Geophysical inverse theory and regularization problems (Vol. 36). Elsevier. https://doi.org/10.1016/s0076-6895(02)x8037-8 DOI: https://doi.org/10.1016/S0076-6895(02)X8037-8

Zunino, A., Gebraad, L., Ghirotto, A., & Fichtner, A. (2023). HMCLab: a framework for solving diverse geophysical inverse problems using the Hamiltonian Monte Carlo method. ArXiv Preprint ArXiv:2303.10047. https://doi.org/10.1093/gji/ggad403 DOI: https://doi.org/10.1093/gji/ggad403

VIP - Variational Inversion Package with example implementations of Bayesian tomographic imaging

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