Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T02:26:01.141Z Has data issue: false hasContentIssue false

2 - Emerging Directions in Geophysical Inversion

from Part I - Introduction

Published online by Cambridge University Press:  20 June 2023

By
Andrew P. Valentine  and
Malcolm Sambridge
Edited by
Alik Ismail-Zadeh ,
Fabio Castelli ,
Dylan Jones  and
Sabrina Sanchez
Alik Ismail-Zadeh
Affiliation:
Karlsruhe Institute of Technology, Germany
Fabio Castelli
Affiliation:
Università degli Studi, Florence
Dylan Jones
Affiliation:
University of Toronto
Sabrina Sanchez
Affiliation:
Max Planck Institute for Solar System Research, Germany
Get access

Summary

Abstract: In this chapter, we survey some recent developments in the field of geophysical inversion. We aim to provide an accessible general introduction to the breadth of current research, rather than focusing in depth on particular topics. We hope to give the reader an appreciation for the similarities and connections between different approaches, and their relative strengths and weaknesses.

Keywords

Backus-Gilbert theoryBayes’ theoremGaussian processgenerative modelgrid-searchleast squaresMarkov chain Monte Carlo (MCMC)misfit functionsurrogate modellingvariational inference
Type
Chapter
Information
Applications of Data Assimilation and Inverse Problems in the Earth Sciences , pp. 9 - 26
Publisher: Cambridge University Press
Print publication year: 2023

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abadi, M., Barham, P., Chen, J. et al. 2016. Tensorflow: A system for large-scale machine learning. 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), pp. 265–83, USENIX Association, Savannah, GA.Google Scholar
Agarwal, S., Tosi, N., Breuer, D. et al. 2020. A machine-learning-based surrogate model of Mars’ thermal evolution. Geophysical Journal International, 222, 1656–70.Google Scholar
Aleardi, M., and Salusti, A. 2020. Hamiltonian Monte Carlo algorithms for target and interval-oriented amplitude versus angle inversions. Geophysics, 85, R177–R194.Google Scholar
Allmaras, M., Bangerth, W., Linhart, J. et al. 2013. Estimating parameters in physical models through Bayesian inversion: A complete example. SIAM Review, 55, 149–67.CrossRefGoogle Scholar
Ambrosio, L. 2003. Lecture notes on optimal transport problems. In Ambriosio, L., Deckelnick, K., Dziuk, G., Mimura, M., Solonnikov, V., and Soner, H, eds., Mathematical Aspects of Evolving Interfaces. Heidelberg: Springer, pp. 1–52.Google Scholar
Anderssen, R., Worthington, M., and Cleary, J. 1972. Density modelling by Monte Carlo inversion – I. Methodology. Geophysical Journal of the Royal Astronomical Society, 29, 433–44.Google Scholar
Aster, R., Borchers, B., and Thurber, C. 2013. Parameter estimation and inverse problems. Amsterdam: Academic Press.Google Scholar
Backus, G. 1970a. Inference from inadequate and inaccurate data, i. Proceedings of the National Academy of Sciences, 65, 1–7.CrossRefGoogle ScholarPubMed
Backus, G. 1970b. Inference from inadequate and inaccurate data, ii. Proceedings of the National Academy of Sciences, 65, 281–7.Google Scholar
Backus, G. 1970c. Inference from inadequate and inaccurate data, iii. Proceedings of the National Academy of Sciences, 67, 282–9.Google Scholar
Backus, G. 1988. Bayesian inference in geomagnetism. Geophysical Journal, 92, 125–42.Google Scholar
Backus, G., and Gilbert, F. 1968. The resolving power of gross Earth data. Geophysical Journal of the Royal Astronomical Society, 16, 169–205.Google Scholar
Bayes, T. 1763. An essay towards solving a problem in the doctrine of chances. Philosophical Transactions, 53, 370–418.Google Scholar
Bernal-Romero, M., and Iturrarán-Viveros, U. 2021. Accelerating full-waveform inversion through adaptive gradient optimization methods and dynamic simultaneous sources. Geophysical Journal International, 225, 97–126.Google Scholar
Betancourt, M. 2017. A conceptual introduction to Hamiltonian Monte Carlo. arXiv:1701.02434v1.CrossRefGoogle Scholar
Bianco, M., and Gerstoft, P. 2018. Travel time tomography with adaptive dictionaries. IEEE Transactions on Computational Imaging, 4, 499–511.Google Scholar
Bishop, C. 1995. Neural Networks for Pattern Recognition. Oxford: Oxford University Press.Google Scholar
Blei, D., Kucukelbir, A., and McAuliffe, J. 2017. Variational inference: A review for statisticians. Journal of the American Statistical Association, 112, 859–77.Google Scholar
Bodin, T., and Sambridge, M. 2009. Seismic tomography with the reversible jump algorithm. Geophysical Journal International, 178, 1411–36.Google Scholar
Bond-Taylor, S., Leach, A., Long, Y., and Willcocks, C. 2022. Deep generative modelling: A comparative review of VAEs, GANs, normalizing flows, energy-based and autoregressive models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(11), 7327–47. https://doi.org/10.1109/TPAMI.2021.3116668.Google Scholar
Box, G., and Muller, M. 1958. A note on the generation of random normal deviates. Annals of Mathematical Statistics, 29, 610–611.CrossRefGoogle Scholar
Bozdağ, E., Peter, D., Lefebvre, M. et al. 2016. Global adjoint tomography: First-generation model. Geophysical Journal International, 207, 1739–66.CrossRefGoogle Scholar
Brooks, S., Gelman, A., Jones, G., and Meng, X.-L. 2011. Handbook of Markov Chain Monte Carlo. Boca Raton, FL: CRC Press.Google Scholar
Burdick, S., and Lekić, V. 2017. Velocity variations and uncertainty from trans dimensional P-wave tomography of North America. Geophysical Journal International, 209, 1337–51.Google Scholar
Candès, E., and Wakin, B. 2008. An introduction to compressive sampling. IEEE Signal Processing Magazine, 25, 21–30.Google Scholar
Castruccio, S., McInerney, D., Stein, M. et al. 2014. Statistical emulation of climate model projections based on precomputed GCM runs. Journal of Climate, 27, 1829–44.CrossRefGoogle Scholar
Chandra, R., Azam, D., Kapoor, A., and Mu¨ller, R. 2020. Surrogate-assisted Bayesian inversion for landscape and basin evolution models. Geoscientific Model Development, 13, 2959–79.Google Scholar
Chou, C., and Booker, J. 1979. A Backus–Gilbert approach to inversion of travel-time data for three-dimensional velocity structure. Geophysical Journal of the Royal Astronomical Society, 59, 325–44.Google Scholar
Constable, S., Parker, R., and Constable, C. 1987. Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics, 52, 289–300.Google Scholar
Cook, A. 1990. Sir Harold Jeffreys. Biographical Memoirs of Fellows of the Royal Society, 36, 303–33.Google Scholar
Creswell, A., White, T., Dumoulin, V. et al. 2018. Generative adversarial networks: An overview. IEEE Signal Processing Magazine, 35, 53–65.Google Scholar
Curtis, A., and Lomax, A. 2001. Prior information, sampling distributions, and the curse of dimensionality. Geophysics, 66, 372–78.Google Scholar
Daley, T., Freifeld, B., Ajo-Franklin, J. et al. 2013. Field testing of fiberoptic distributed acoustic sensing (DAS) for subsurface seismic monitoring. The Leading Edge, 32, 699–706.Google Scholar
Das, S., Chen, X., Hobson, M. et al. 2018. Surrogate regression modelling for fast seismogram generation and detection of microseismic events in heterogeneous velocity models. Geophysical Journal International, 215, 1257–90.Google Scholar
de Wit, R., Käufl, P., Valentine, A., and Trampert, J. 2014. Bayesian inversion of free oscillations for Earth’s radial (an)elastic structure. Physics of the Earth and Planetary Interiors, 237, 1–17.Google Scholar
Dean, T., Tulett, J., and Barwell, R. 2018. Nodal land seismic acquisition: The next generation. First Break, 36, 47–52.Google Scholar
Dettmer, J., Benavente, R., Cummins, P., and Sambridge, M. 2014. Trans-dimensional finite-fault inversion. Geophysical Journal International, 199, 735–51.Google Scholar
Dinh, H., and Van der Baan, M. 2019. A grid-search approach for 4d pressure-saturation discrimination. Geophysics, 84, IM47–IM62.Google Scholar
Donoho, D. 2006. Compressed sensing. IEEE Transactions on Information Theory, 52, 1289–306.Google Scholar
Dziewonski, A., Chou, T.-A., and Woodhouse, J. 1981. Determination of earthquake source parameters from waveform data for studies of global and regional seismicity. Journal of Geophysical Research, 86, 2825–52.Google Scholar
Earp, S., Curtis, A., Zhang, X., and Hansteen, F. 2020. Probabilistic neural network tomography across Grane field (North Sea) from surface wave dispersion data. Geophysical Journal International, 223, 1741–57.Google Scholar
Enemark, T., Peeters, L., Mallants, D. et al. 2019. Hydrogeological Bayesian hypothesis testing through trans-dimensional sampling of a stochastic water balance model. Water, 11(7), 1463. https://doi.org/10.3390/w11071463.Google Scholar
Engquist, B., and Froese, B. 2014. Application of the Wasserstein metric to seismic signals. Communications in Mathematical Sciences, 12, 979–88.Google Scholar
Fernández-Martínez, J. L., and Fernández-Muñiz, Z. 2020. The curse of dimensionality in inverse problems. Journal of Computational and Applied Mathematics, 369, 112571. https://doi.org/10.1016/j.cam.2019.112571.Google Scholar
Fichtner, A., Zunino, A., and Gebraad, L. 2019. Hamiltonian Monte Carlo solution of tomographic inverse problems. Geophysical Journal International, 216, 1344–63.Google Scholar
Field, R., Constantine, P., and Boslough, M. 2011. Statistical surrogate models for prediction of high-consequence climate change, Tech. Rep. SAND2011-6496, Sandia National Laboratories.Google Scholar
Forrester, A., and Keane, A. 2009. Recent advances in surrogate-based optimization. Progress in Aerospace Sciences, 45, 50–79.Google Scholar
Galetti, E., Curtis, A., Baptie, B., Jenkins, D., and 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, 35–58.Google Scholar
Gallagher, K., Bodin, T., Sambridge, M. et al. 2011. Inference of abrupt changes in noisy geochemical records using transdimensional changepoint models. Earth and Planetary Science Letters, 311, 182–94.Google Scholar
Goodfellow, I., Pouget-Abadie, J., Mirza, M. et al. 2014. Generative adversarial nets. In Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N., and Weinberger, K. Q., eds., Advances in Neural Information Processing Systems, vol. 27. Red Hook, NY: Curran Associates. https://proceedings.neurips.cc/paper/2014/file/5ca3e9b122f61f8f06494c97b1afccf3-Paper.pdf.Google Scholar
Green, P. 1995. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711–32.Google Scholar
Green, W. 1975. Inversion of gravity profiles by a Backus–Gilbert approach. Geophysics, 40, 763–72.Google Scholar
Guo, P., Visser, G., and Saygin, E. 2020. Bayesian trans-dimensional full waveform inversion: Synthetic and field data application. Geophysical Journal International, 222, 610–27.CrossRefGoogle Scholar
Hassan, R., Hejrani, B., Medlin, A., Gorbatov, A., and Zhang, F. 2020. High-performance seismological tools (HiPerSeis). In Czarnota, K., Roach, I., Abbott, S., Haynes, M., Kositcin, N., Ray, A., and Slatter, E, eds., Exploring for the Future: Extended Abstracts. Canberra: Geoscience Australia, pp. 1–4.Google Scholar
Hawkins, R., and Sambridge, M. 2015. Geophysical imaging using trans-dimensional trees. Geophysical Journal International, 203, 972–1000.Google Scholar
He, Q., and Tartakovsky, A. 2021. Physics-informed neural network method for forward and backward advection-dispersion equations. Water Resources Research, 57, e2020WR029479.Google Scholar
He, W., Brossier, R., Métivier, L., and Plessix, R.-E. 2019. Land seismic multiparameter full waveform inversion in elastic VTI media by simultaneously interpreting body waves and surface waves with an optimal transport based objective function. Geophysical Journal International, 219, 1970–88.Google Scholar
Hedjazian, N., Bodin, T., and Métivier, L. 2019. An optimal transport approach to linearized inversion of receiver functions. Geophysical Journal International, 216, 130–47.Google Scholar
Hejrani, B., and Tkalčić, H. 2020. Resolvability of the centroid-moment-tensors for shallow seismic sources and improvements from modeling high-frequency waveforms. Journal of Geophysical Research, 125, e2020JB019643.Google Scholar
Herrmann, F., Erlangga, Y., and Lin, T. 2009. Compressive simultaneous full-waveform simulation. Geophysics, 74, A35–A40.Google Scholar
Hogg, D., and Foreman-Mackey, D. 2018. Data analysis recipes: Using Markov chain Monte Carlo. The Astrophysical Journal Supplement Series, 236:11 (18 pp.). https://doi.org/10.3847/1538-4365/aab76e.Google Scholar
Huang, G., Zhang, X., and Qian, J. 2019. Kantorovich–Rubinstein misfit for inverting gravity-gradient data by the level-set method. Geophysics, 84, 1–115.Google Scholar
Hussain, M., Javadi, A., Ahangar-Asr, A., and Farmani, R. 2015. A surrogate model for simulation-optimization of aquifer systems subjected to seawater intrusion. Journal of Hydrology, 523, 542–554.Google Scholar
Jeffreys, H. 1931. Scientific Inference. Cambridge: Cambridge University Press.Google Scholar
Jeffreys, H. 1939. The Theory of Probability. Oxford: Oxford University Press.Google Scholar
Jeffreys, H., and Bullen, K. 1940. Seismological Tables. London: British Association for the Advancement of Science.Google Scholar
Karniadakis, G., Kevrekidis, I., Lu, L. et al. 2021. Physics-informed machine learning. Nature Reviews Physics, 3, 422–40.Google Scholar
Kashinath, K., Mustafa, M., Albert, A. et al. 2021. Physics-informed machine learning: Case studies for weather and climate modelling. Philosophical Transactions, 379, 20200093.Google Scholar
Käufl, P. 2015. Rapid probabilistic source inversion using pattern recognition. Ph.D. thesis, University of Utrecht.Google Scholar
Käufl, P., Valentine, A., O’Toole, T., and Trampert, J. 2014. A framework for fast probabilistic centroid – moment-tensor determination – inversion of regional static displacement measurements. Geophysical Journal International, 196, 1676–93.Google Scholar
Käufl, P., Valentine, A., de Wit, R., and Trampert, J. 2016a. Solving probabilistic inverse problems rapidly with prior samples. Geophysical Journal International, 205, 1710–28.Google Scholar
Käufl, P., Valentine, A., and Trampert, J. 2016b. Probabilistic point source inversion of strong-motion data in 3-D media using pattern recognition: A case study for the 2008 Mw 5.4 Chino Hills earthquake. Geophysical Research Letters, 43, 8492–8.Google Scholar
Kennett, B., and Engdahl, E. 1991. Traveltimes for global earthquake location and phase identification. Geophysical Journal International, 105, 429–65.Google Scholar
Khan, A., Ceylan, S., van Driel, M. et al. 2021. Upper mantle structure of Mars from InSight seismic data. Science, 373, 434–8.Google Scholar
Kingma, D., and Welling, M. 2014. Auto-encoding variational Bayes, in 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, 14–16 April 2014, Conference Track Proceedings. https://arxiv.org/abs/1312.6114.Google Scholar
Kircher, A. 1665. Mundus subterraneus. Amsterdam: Joannem Janssonium & EliziumWegerstraten.Google Scholar
Knapmeyer-Endrun, B., Panning, M. P., Bissig, F. et al. 2021. Thickness and structure of the martian crust from In Sight seismic data. Science, 373, 438–43.Google Scholar
Kobyzev, I., Prince, S., and Brubaker, M. 2021. Normalizing flows: An introduction and review of current methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43, 3964–79.Google Scholar
Komatitsch, D., Ritsema, J., and Tromp, J. 2002. The spectral-element method, Beowulf computing, and global seismology. Science, 298, 1737–42.Google Scholar
Kornfeld, R., Arnold, B., Gross, M. et al. 2019. GRACE-FO: The gravity recovery and climate experiment follow-on mission. Journal of Spacecraft and Rockets, 56, 931–51.CrossRefGoogle Scholar
Koshkholgh, S., Zunino, A., and Mosegaard, K. 2021. Informed proposal Monte Carlo. Geophysical Journal International, 226, 1239–48.Google Scholar
Krischer, L., Smith, J., Lei, W. et al. 2016. An adaptable seismic data format. Geophysical Journal International, 207, 1003–11.Google Scholar
Kullback, S., and Leibler, R. 1951. On information and sufficiency. The Annals of Mathematical Statistics, 22, 79–86.Google Scholar
Lau, H., and Romanowicz, B. 2021. Constraining jumps in density and elastic properties at the 660 km discontinuity using normal mode data via the Backus-Gilbert method. Geophysical Research Letters, 48, e2020GL092217.Google Scholar
Lei, W., Ruan, Y., Bozdağ, E. et al. 2020. Global adjoint tomography: Model GLAD-M25. Geophysical Journal International, 223, 1–21.CrossRefGoogle Scholar
Ley-Cooper, A., Brodie, R., and Richardson, M. 2020. AusAEM: Australia’s airborne electromagnetic continental-scale acquisition program. Exploration Geophysics, 51, 193–202.CrossRefGoogle Scholar
Liu, D., and Nocedal, J. 1989. On the limited memory BFGS method for large-scale optimization. Mathematical Programming, 45, 503–28.Google Scholar
Livermore, P., Fournier, A., Gallet, Y., and Bodin, T. 2018. Transdimensional inference of archeomagnetic intensity change. Geophysical Journal International, 215, 2008–34.Google Scholar
Logg, A., Mardal, K.-A., and Wells, G. N. 2012. Automated Solution of Differential Equations by the Finite Element Method. Heidelberg: Springer.Google Scholar
Lopez-Alvis, J., Laloy, E., Nguyen, F., and Hermans, T. 2021. Deep generative models in inversion: The impact of the generator’s nonlinearity and development of a new approach based on a variational autoencoder. Computers & Geosciences, 152, 104762.CrossRefGoogle Scholar
Luengo, D., Martino, L., Bugallo, M., Elvira, V., and Särkkä, S. 2020. A survey of Monte Carlo methods for parameter estimation. EURASIP Journal on Advances in Signal Processing, 2020, 25. https://doi.org/10.1186/s13634-020-00675-6.Google Scholar
Lythgoe, K., Loasby, A., Hidayat, D., and Wei, S. 2021. Seismic event detection in urban Singapore using a nodal array and frequency domain array detector: Earthquakes, blasts and thunderquakes. Geophysical Journal International, 226, 1542–57.Google Scholar
Meier, U., Curtis, A., and Trampert, J. 2007. Global crustal thickness from neural network inversion of surface wave data. Geophysical Journal International, 169, 706–722.Google Scholar
Meltzer, A., Rudnick, R., Zeitler, P. et al. 1999. USArray initiative. GSA Today, 11, 8–10.Google Scholar
Menke, W. 1989. Geophysical Data Analysis: Discrete Inverse Theory. New York: Academic Press.Google Scholar
Métivier, L., Brossier, R., Mérigot, Q., Oudet, E., and Virieux, J. 2016a. Increasing the robustness and applicability of full-waveform inversion: An optimal transport distance strategy. The Leading Edge, 35, 1060–7.Google Scholar
Métivier, L., Brossier, R., Mérigot, Q., Oudet, E., and Virieux, J. 2016b. Measuring the misfit between seismograms using an optimal transport distance: Application to full waveform inversion. Geophysical Journal International, 205, 345–77.Google Scholar
Métivier, L., Brossier, R., Mérigot, Q., Oudet, e., and Virieux, J. 2016c. An optimal transport approach for seismic tomography: Application to 3D full waveform inversion. Inverse Problems, 32, 115008.Google Scholar
Métivier, L., Brossier, R., Oudet, E., Mérigot, Q., and Virieux, J. 2016d. An optimal transport distance for full-waveform inversion: Application to the 2014 chevron benchmark data set. In Sicking, C. and Ferguson, J. SEG Technical Program Expanded Abstracts. Tulsa, OK: Society of Exploration Geophysicists, pp. 1278–83.Google Scholar
Montagner, J.-P., and Tanimoto, T. 1990. Global anisotropy in the upper mantle inferred from the regionalization of phase velocities. Journal of Geophysical Research, 95, 4797–819.Google Scholar
Montagner, J.-P., and Tanimoto, T. 1991. Global upper mantle tomography of seismic velocities and anisotropies. Journal of Geophysical Research, 96, 20337–51.Google Scholar
Moseley, B., Nissen-Meyer, T., and Markham, A. 2020. Deep learning for fast simulation of seismic waves in complex media. Solid Earth, 11, 1527–49.Google Scholar
Mosher, S., Eilon, Z., Janiszewski, H., and Audet, P. 2021. Probabilistic inversion of seafloor compliance for oceanic crustal shear velocity structure using mixture density networks. Geophysical Journal International, 227, 1879–92.Google Scholar
Mosser, L., Dubrule, O., and Blunt, M. 2020. Stochastic seismic waveform inversion using generative adversarial networks as a geological prior. Mathematical Geosciences, 52, 53–79.Google Scholar
Muir, J., and Zhang, Z. 2021. Seismic wavefield reconstruction using a pre-conditioned wavelet-curvelet compressive sensing approach. Geophysical Journal International, 227, 303–15.Google Scholar
Neal, R. 2011. MCMC using Hamiltonian dynamics. In Brooks, S., Gelman, A., Jones, G., and Meng, X.-L, eds., Handbook of Markov Chain Monte Carlo. Boca Raton, FL: CRC Press.Google Scholar
Nicholson, T., Sambridge, M., and Gudmundsson, O. 2004. Three-dimensional empirical traveltimes: Construction and applications. Geophysical Journal International, 156, 307–28.Google Scholar
Nocedal, J., and Wright, S. 1999. Numerical Optimization. New York: Springer.Google Scholar
Nyquist, H. 1928. Certain topics in telegraph transmission theory. Transactions of the American Institute of Electrical Engineers, 47, 617–44.Google Scholar
Parker, R. 1994. Geophysical Inverse Theory. Princeton, NJ: Princeton University Press.Google Scholar
Parker, T., Shatalin, S., and Farhadiroushan, M. 2014. Distributed acoustic sensing – a new tool for seismic applications. First Break, 32, 61–69.Google Scholar
Paszke, A., Gross, S., Massa, F. et al. 2019. Pytorch: An imperative style, high-performance deep learning library. In Wallach, H. M., Larochelle, H., Beygelzimer, A., d’Alché-Buc, F., Fox, E. A., and Garnett, R., eds. Advances in Neural Information Processing Systems, vol. 32. Red Hook, NY: Curran Associates. https://proceedings.neurips.cc/paper/2019/file/bdbca288fee7f92f2bfa9f7012727740-Paper.pdf.Google Scholar
Petersen, K., and Pedersen, M. 2012. The matrix cookbook, Tech. rep. Technical University of Denmark, Kongens Lyngby. 72 p.Google Scholar
Pijpers, F., and Thompson, M. 1992. Faster formulations of the optimally localized averages method for helioseismic inversions. Astronomy and Astrophysics, 262, L33–L36.Google Scholar
Press, F. 1970. Earth models consistent with geophysical data. Physics of the Earth and Planetary Interiors, 3, 3–22.Google Scholar
Quiepo, N., Haftka, R., Shyy, W. et al. 2005. Surrogate-based analysis and optimization. Progress in Aerospace Sciences, 41, 1–28.Google Scholar
Raissi, M., Perdikaris, P., and Karniadakis, G. 2019. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707.Google Scholar
Ramgraber, M., Weatherl, R., Blumensaat, F., and Schirmer, M. 2021. Non-Gaussian parameter inference for hydrogeological models using Stein variational gradient descent. Water Resources Research, 57, e2020WR029339.Google Scholar
Rasmussen, C., and Williams, C. 2006. Gaussian Processes for Machine Learning. Cambridge, MA: MIT Press.Google Scholar
Ray, A. 2021. Bayesian inversion using nested trans-dimensional Gaussian processes. Geophysical Journal International, 226, 302–26.Google Scholar
Ray, A., and Myer, D. 2019. Bayesian geophysical inversion with trans-dimensional Gaussian process machine learning. Geophysical Journal International, 217, 1706–26.Google Scholar
Reuber, G., and Simons, F. 2020. Multi-physics adjoint modeling of Earth structure: combining gravimetric, seismic and geodynamic inversions. International Journal on Geomathematics, 11(30), 1–38.Google Scholar
Rezende, D., and Mohamed, S. 2015. Variational inference with normalizing flows. In Bach, F. and Blei, D., eds., Proceedings of the 32nd International Conference on Machine Learning, vol. 37; International Conference on Machine Learning, 7–9 July 2015, Lille, France, pp.1530–8. http://proceedings.mlr.press/v37/rezende15.pdf.Google Scholar
Rijal, A., Cobden, L., Trampert, J., Jackson, J., and Valentine, A. 2021. Inferring equations of state of the lower mantle minerals using mixture density networks. Physics of the Earth and Planetary Interiors, 319, 106784.Google Scholar
Ritsema, J., van Heijst, H., and Woodhouse, J. 1999. Complex shear wave velocity structure imaged beneath Africa and Iceland., Science, 286, 1925–31.Google Scholar
Ruthotto, L., and Haber, E. 2021. An introduction to deep generative modelling. GAMM-Mitteilungen, 44, e202100008.Google Scholar
Sambridge, M. 1998. Exploring multidimensional landscapes without a map. InverseProblems, 14, 427–40.Google Scholar
Sambridge, M. 1999a. Geophysical inversion with a neighbourhood algorithm – I. Searching a parameter space. Geophysical Journal International, 138, 479–94.Google Scholar
Sambridge, M. 1999b. Geophysical inversion with a neighbourhood algorithm – II. Appraising the ensemble Geophysical Journal International, 138, 727–46.Google Scholar
Sambridge, M., and Kennett, B. 1986. A novel method of hypocentre location. Geophysical Journal International, 87, 679–97.Google Scholar
Sambridge, M., and Mosegaard, K. 2002. Monte Carlo methods in geophysical inverse problems. Reviews of Geophysics, 40, 3-1–3-29. https://doi.org/10.1029/2000RG000089.Google Scholar
Sambridge, M., Gallagher, K., Jackson, A., and Rickwood, P. 2006. Trans-dimensional inverse problems, model comparison and the evidence. Geophysical Journal International, 167, 528–42.Google Scholar
Sambridge, M., Rickwood, P., Rawlinson, N., and Sommacal, S. 2007. Automatic differentiation in geophysical inverse problems. Geophysical Journal International, 170, 1–8.Google Scholar
Sambridge, M., Bodin, T., Gallagher, K., and Tkalcic, H. 2012. Transdimensional inference in the geosciences. Philosophical Transactions of the Royal Society, 371.Google Scholar
Santambrogio, F. 2015. Optimal Transport for Applied Mathematicians. Basel: Birkhäuser.Google Scholar
Scales, J., and Snieder, R. 1997. To Bayes or not to Bayes. Geophysics, 62, 1045–46.Google Scholar
Scheiter, M., Valentine, A., and Sambridge, M. 2022. Upscaling and downscaling Monte Carlo ensembles with generative models. Geophysical Journal International, 230(2), 916–31. https://doi.org/10.1093/gji/ggac100.Google Scholar
Sen, M., and Biswas, R. 2017. Transdimensional seismic inversion using the reversible jump Hamiltonian Monte Carlo algorithm. Geophysics, 82, R119–R134.Google Scholar
Shahriari, B., Swersky, K., Wang, Z., Adams, R., and de Freitas, N. 2016. Taking the human out of the loop: A review of Bayesian optimization. Proceedings of the IEEE, 104, 148–75.Google Scholar
Siahkoohi, A., and Herrman, F. 2021. Learning by example: Fast reliability-aware seismic imaging with normalizing flows. arXiv:2104.06255v1. https://arxiv.org/abs/2104.06255Google Scholar
Simons, F., Loris, I., Nolet, G. et al. 2011. Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity. Geophysical Journal International, 187, 969–88.Google Scholar
Smith, J., Azizzadenesheli, K., and Ross, Z. 2020. EikoNet: Solving the eikonal equation with deep neural networks. IEEE Transactions on Geoscience and Remote Sensing, 59(12), 10685–96. https://doi.org/10.1109/TGRS.2020.3039165.Google Scholar
Smith, J., Ross, Z., Azizzadenesheli, K., and Muir, J. 2022. HypoSVI: Hypocenter inversion with Stein variational inference and physics informed neural networks. Geophysical Journal International, 228, 698–710.Google Scholar
Snieder, R. 1991. An extension of Backus–Gilbert theory to nonlinear inverse problems. Inverse Problems, 7, 409–433.Google Scholar
Song, C., Alkhalifah, T., and Bin Waheed, U. 2021. Solving the frequency-domain acoustic VTI wave equation using physics-informed neural networks. Geophysical Journal International, 225, 846–59.Google Scholar
Spurio Mancini, A., Piras, D., Ferreira, A., Hobson, M., and Joachimi, B. 2021. Accelerating Bayesian microseismic event location with deep learning. Solid Earth, 12, 1683–705.Google Scholar
Stähler, S., Khan, A., Banerdt, W. et al. 2021. Seismic detection of the Martian core. Science, 373, 443–48.Google Scholar
Steinmetz, T., Raape, U., Teßmann, S. et al. 2010. Tsunami early warning and decision support. Natural Hazards and Earth System Sciences, 10, 1839–50.Google Scholar
Tanimoto, T. 1985. The Backus–Gilbert approach to the three-dimensional structure in the upper mantle – I. Lateral variation of surface wave phase velocity with its error and resolution. Geophysical Journal International, 82, 105–23.Google Scholar
Tanimoto, T. 1986. The Backus–Gilbert approach to the three-dimensional structure in the upper mantle – II. SH and SV velocity. Geophysical Journal International, 84, 49–69.Google Scholar
Tarantola, A. 2005. Inverse Problem Theory and Methods for Model Parameter Estimation. Philadelphia, PA: Society of Industrial and Applied Mathematics.Google Scholar
Tarantola, A., and Nercessian, A. 1984. Three-dimensional inversion without blocks. Geophysical Journal of the Royal Astronomical Society, 76, 299–306.Google Scholar
Tarantola, A., and Valette, B. 1982. Generalized nonlinear inverse problems solved using the least squares criterion. Reviews of Geophysics and Space Physics, 20, 219–32.Google Scholar
Tibshirani, R. 1996. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society B, 58, 267–88.Google Scholar
Trampert, J., and Snieder, R. 1996. Model estimations biased by truncated expansions: Possible artifacts in seismic tomography. Science, 271, 1257–60.Google Scholar
Tran, D., Kucukelbir, A., Dieng, A. B. et al. 2016. Edward: A library for probabilistic modeling, inference, and criticism. arXiv:1610.09787.Google Scholar
Valentine, A., and Davies, D. 2020. Global models from sparse data: A robust estimate of earth’s residual topography spectrum. Geochemistry, Geophysics, Geosystems, e2020GC009240.Google Scholar
Valentine, A., and Sambridge, M. 2020a. Gaussian process models – I. A framework for probabilistic continuous inverse theory. Geophysical Journal International, 220, 1632–47.Google Scholar
Valentine, A., and Sambridge, M. 2020b. Gaussian process models – II. Lessons for discrete inversion. Geophysical Journal International, 220, 1648–56.Google Scholar
Valentine, A., and Trampert, J. 2016. The impact of approximations and arbitrary choices on geophysical images. Geophysical Journal International, 204, 59–73.Google Scholar
van Herwaarden, D. P., Boehm, C., Afansiev, M. et al. 2020. Accelerated full-waveform inversion using dynamic mini-batches. Geophysical Journal International, 221, 1427–38.Google Scholar
Waddell, M. 2006. The world, as it might be: Iconography and probabalism in the Mundus subterraneus of Athanasius Kircher. Centaurius, 48, 3–22. https://doi.org/10.1111/j.1600-0498.2006.00038.x.Google Scholar
Wang, Y., Cao, J., and Yang, C. 2011. Recovery of seismic wavefields based on compressive sensing by an -norm constrained trust region method and the piecewise random subsampling. Geophysical Journal International, 187, 199–213.Google Scholar
Wang, Z., Hutter, F., Zoghi, M., Matheson, D., and de Freitas, N. 2016. Bayesian optimization in a billion dimensions via random embeddings. Journal of Artificial Intelligence Research, 55, 361–87.Google Scholar
Wiggins, R. 1972. The general linear inverse problem: Implication of surface waves and free oscillations for Earth structure. Reviews of Geophysics and Space Physics, 10, 251–85.Google Scholar
Woodhouse, J. 1980. The coupling and attenuation of nearly resonant multiplets in the Earth’s free oscillation spectrum. Geophysical Journal of the Royal Astronomical Society, 61, 261–83.Google Scholar
Woodhouse, J., and Dziewonski, A. 1984. Mapping the upper mantle: Three-dimensional modelling of Earth structure by inversion of seismic waveforms. Journal of Geophysical Research, 89, 5953–86.Google Scholar
Worthington, M., Cleary, J., and Anderssen, R. 1972. Density modelling by Monte Carlo inversion – II. Comparison of recent Earth models. Geophysical Journal of the Royal Astronomical Society, 29, 445–57.Google Scholar
Zaroli, C. 2016. Global seismic tomography using Backus–Gilbert inversion. Geophysical Journal International. 207, 876–88.Google Scholar
Zhang, C., Bütepage, J., Kjellström, H., and Mandt, S. 2019. Advances in variational inference. IEEE Transactions on Pattern Analysis and Machine Intelligence, 41, 2008–26.Google Scholar
Zhang, X., and Curtis, A. 2020. Seismic tomography using variational inference methods. Journal of Geophysical Research, 125, e2019JB018589.Google Scholar
Zhao, X., Curtis, A., and Zhang, X. 2022. Bayesian seismic tomography using normalizing flows. Geophysical Journal International, 228, 213–39.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×