1887
Volume 20, Issue 4
  • ISSN: 1569-4445
  • E-ISSN: 1873-0604
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Abstract

ABSTRACT

Expensive forward model evaluations and the curse of dimensionality usually hinder applications of Markov chain Monte Carlo algorithms to geophysical inverse problems. Another challenge of these methods is related to the definition of an appropriate proposal distribution that simultaneously should be inexpensive to manipulate and a good approximation of the posterior density. Here we present a gradient‐based Markov chain Monte Carlo inversion algorithm that is applied to cast the electrical resistivity tomography into a probabilistic framework. The sampling is accelerated by exploiting the Hessian and gradient information of the negative log‐posterior to define a proposal that is a local, Gaussian approximation of the target posterior probability. On the one hand, the computing time to run the many forward evaluations needed for both the data likelihood evaluation and the Hessian and gradient computation is decreased by training a residual neural network to predict the forward mapping between the resistivity model and the apparent resistivity value. On the other hand, the curse of dimensionality issue and the computational effort related to the Hessian and gradient manipulation are decreased by compressing data and model spaces through a discrete cosine transform. A non‐parametric distribution is assumed as the prior probability density function. The method is first demonstrated on synthetic data and then applied to field measurements. The outcomes provided by the presented approach are also benchmarked against those obtained when a computationally expensive finite‐element code is employed for forward modelling , with the results of a gradient‐free Markov chain Monte Carlo inversion, and also compared with the predictions of a deterministic inversion. The implemented approach not only guarantees uncertainty assessments and model predictions comparable with those achieved by more standard inversion strategies, but also drastically decreases the computational cost of the probabilistic inversion, making it similar to that of a deterministic inversion.

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2022-07-13
2022-08-18
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References

  1. Aguzzoli, A., Fumagalli, A., Scotti, A., Zanzi, L. and Arosio, D. (2021) Inversion of synthetic and measured 3D geoelectrical data to study the geomembrane below a landfill. In 4th Asia Pacific Meeting on Near Surface Geoscience & Engineering, 2021(1), 1–5.
    [Google Scholar]
  2. Aleardi, M. (2019) Using orthogonal Legendre polynomials to parameterize global geophysical optimizations: applications to seismic‐petrophysical inversion and 1D elastic full‐waveform inversion. Geophysical Prospecting, 67(2), 331–348.
    [Google Scholar]
  3. Aleardi, M. (2020) Combining discrete cosine transform and convolutional neural networks to speed up the Hamiltonian Monte Carlo inversion of pre‐stack seismic data. Geophysical Prospecting, 68(9), 2738–2761.
    [Google Scholar]
  4. Aleardi, M. and Salusti, A. (2020) Hamiltonian Monte Carlo algorithms for target‐and interval‐oriented amplitude versus angle inversions. Geophysics, 85(3), R177–R194.
    [Google Scholar]
  5. Aleardi, M., Vinciguerra, A., Hojat, A. and Stucchi, E. (2022) Probabilistic inversions of electrical resistivity tomography data with a machine learning‐based forward operator. Geophysical Prospecting. https://doi.org/10.1111/1365‐2478.13189.
    [Google Scholar]
  6. Aleardi, M., Vinciguerra, A. and Hojat, A. (2020) A geostatistical Markov chain Monte Carlo inversion algorithm for electrical resistivity tomography. Near Surface Geophysics, 19(1), 7–26.
    [Google Scholar]
  7. Aleardi, M., Vinciguerra, A. and Hojat, A. (2021) Ensemble‐based electrical resistivity tomography with data and model space compression. Pure and Applied Geophysics, 1–23.
    [Google Scholar]
  8. Aster, R.C., Borchers, B. and Thurber, C.H. (2018) Parameter Estimation and Inverse Problems. Cambridge: Elsevier.
    [Google Scholar]
  9. Bièvre, G., Oxarango, L., Günther, T., Goutaland, D. and Massardi, M. (2018) Improvement of 2D ERT measurements conducted along a small earth‐filled dyke using 3D topographic data and 3D computation of geometric factors. Journal of Applied Geophysics, 153, 100–112.
    [Google Scholar]
  10. Brooks, S.P. and Gelman, A. (1998) General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7(4), 434–455.
    [Google Scholar]
  11. Curtis, A. and Lomax, A. (2001) Prior information, sampling distributions, and the curse of dimensionality. Geophysics, 66(2), 372–378.
    [Google Scholar]
  12. Dahlin, T. (2020) Geoelectrical monitoring of embankment dams for detection of anomalous seepage and internal erosion – experiences and work in progress in Sweden. Fifth International Conference on Engineering Geophysics (ICEG), Al Ain, UAE. https://doi.org/10.1190/iceg2019‐053.1.
  13. Dejtrakulwong, P., Mukerji, T. and Mavko, G. (2012) Using kernel principal component analysis to interpret seismic signatures of thin shaly‐sand reservoirs. In SEG Technical Program Expanded Abstracts 2012. Society of Exploration Geophysicists.
    [Google Scholar]
  14. Fichtner, A. and Simutė, S. (2018) Hamiltonian Monte Carlo inversion of seismic sources in complex media. Journal of Geophysical Research: Solid Earth, 123(4), 2984–2999.
    [Google Scholar]
  15. Fichtner, A. and Zunino, A. (2019) Hamiltonian nullspace shuttles. Geophysical Research Letters, 46(2), 644–651.
    [Google Scholar]
  16. Fichtner, A., Zunino, A. and Gebraad, L. (2019) Hamiltonian Monte Carlo solution of tomographic inverse problems. Geophysical Journal International, 216(2), 1344–1363.
    [Google Scholar]
  17. Gebraad, L., Boehm, C. and Fichtner, A. (2020) Bayesian elastic full‐waveform inversion using Hamiltonian Monte Carlo. Journal of Geophysical Research: Solid Earth, 125(3), e2019JB018428.
    [Google Scholar]
  18. Glorot, X. and Bengio, Y. (2010) Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, 249–256.
  19. Goodfellow, I., Bengio, Y. and Courville, A. (2016). Deep Learning. Cambridge: MIT Press.
    [Google Scholar]
  20. Haario, H., Saksman, E. and Tamminen, J. (2001) An adaptive Metropolis algorithm. Bernoulli, 7(2), 223–242.
    [Google Scholar]
  21. Haario, H., Laine, M., Mira, A. and Saksman, E. (2006) DRAM: efficient adaptive MCMC. Statistics and computing, 16(4), 339–354.
    [Google Scholar]
  22. Hansen, T.M. and Cordua, K.S. (2017) Efficient Monte Carlo sampling of inverse problems using a neural network‐based forward: applied to GPR crosshole traveltime inversion. Geophysical Journal International, 211(3), 1524–1533.
    [Google Scholar]
  23. Hermans, T. and Paepen, M. (2020) Combined inversion of land and marine electrical resistivity tomography for submarine groundwater discharge and saltwater intrusion characterization. Geophysical Research Letters, 47(3), e2019GL085877.
    [Google Scholar]
  24. Holmes, C., Krzysztof, L. and Pompe, E. (2017) Adaptive MCMC for multimodal distributions. Technical report. https://pdfs.semanticscholar.org/c75d/f035c23e3c0425409e70d457cd43b174076f.pdf.
  25. Hojat, A., Arosio, D., Di Luch, I., Ferrario, M., Ivanov, V.I., Longoni, L., et al. (2019a) Testing ERT and fiber optic techniques at the laboratory scale to monitor river levees. 25th European Meeting of Environmental and Engineering Geophysics, The Hague, Netherlands, https://doi.org/10.3997/2214‐4609.201902440.
  26. Hojat, A., Arosio, D., Longoni, L., Papini, M., Tresoldi, G. and Zanzi, L. (2019b) Installation and validation of a customized resistivity system for permanent monitoring of a river embankment. EAGE‐GSM 2nd Asia Pacific Meeting on Near Surface Geoscience and Engineering, Kuala Lumpur, Malaysia. https://doi.org/10.3997/2214‐4609.201900421.
  27. Hojat, A., Tresoldi, G. & Zanzi, L. (2021) Correcting the effect of the soil covering buried electrodes for permanent electrical resistivity tomography monitoring systems. 4th Asia Pacific Meeting on Near Surface Geoscience & Engineering, Ho Chi Minh, Vietnam, Online, DOI: 10.3997/2214‐4609.202177070.
  28. Karaoulis, M., Revil, A., Tsourlos, P., Werkema, D.D. and Minsley, B.J. (2013) IP4DI: a software for time‐lapse 2D/3D DC‐resistivity and induced polarization tomography. Computers & Geosciences, 54, 164–170.
    [Google Scholar]
  29. Jiang, A. and Jafarpour, B. (2021) Deep convolutional autoencoders for robust flow model calibration under uncertainty in geologic continuity. Water Resources Research, 57(11), e2021WR029754.
    [Google Scholar]
  30. Laloy, E., Hérault, R., Lee, J., Jacques, D. and Linde, N. (2017) Inversion using a new low‐dimensional representation of complex binary geological media based on a deep neural network. Advances in Water Resources, 110, 387–405.
    [Google Scholar]
  31. Laloy, E., Linde, N., Ruffino, C., Hérault, R., Gasso, G. and Jacques, D. (2019) Gradient‐based deterministic inversion of geophysical data with generative adversarial networks: is it feasible?. Computers & Geosciences, 133, 104333.
    [Google Scholar]
  32. Lieberman, C., Willcox, K. and Ghattas, O. (2010) Parameter and state model reduction for large‐scale statistical inverse problems. SIAM Journal on Scientific Computing, 32(5), 2523–2542.
    [Google Scholar]
  33. Liu, M. and Grana, D. (2020) Time‐lapse seismic history matching with an iterative ensemble smoother and deep convolutional autoencoder. Geophysics, 85(1), M15–M31.
    [Google Scholar]
  34. Lochbühler, T., Breen, S.J., Detwiler, R.L., Vrugt, J.A. and Linde, N. (2014) Probabilistic electrical resistivity tomography of a CO2 sequestration analog. Journal of Applied Geophysics, 107, 80–92.
    [Google Scholar]
  35. Loke, M.H., Papadopoulos, N., Wilkinson, P.B., Oikonomou, D., Simyrdanis, K. and Rucker, D.F. (2020) The inversion of data from very large three‐dimensional electrical resistivity tomography mobile surveys. Geophysical Prospecting, 68(8), 2579–2597.
    [Google Scholar]
  36. 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.
    [Google Scholar]
  37. MacKay, D.J. (2003) Information Theory, Inference and Learning Algorithms. Cambridge: Cambridge University Press.
    [Google Scholar]
  38. Martin, J., Wilcox, L.C., Burstedde, C. and 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.
    [Google Scholar]
  39. Menke, W. (2018) Geophysical Data Analysis: Discrete Inverse Theory. Orlando, FL: Academic Press.
    [Google Scholar]
  40. Mo, S., Zabaras, N., Shi, X. and Wu, J. (2020) Integration of adversarial autoencoders with residual dense convolutional networks for estimation of non‐Gaussian hydraulic conductivities. Water Resources Research, 56(2), e2019WR026082.
    [Google Scholar]
  41. Moghadas, D. and Vrugt, J.A. (2019) The influence of geostatistical prior modeling on the solution of DCT‐based Bayesian inversion: a case study from chicken creek catchment. Remote Sensing, 11(13), 1549.
    [Google Scholar]
  42. Monajemi, H., Donoho, D.L. and Stodden, V. (2016) Making massive computational experiments painless. In 2016 IEEE International Conference on Big Data, 2368–2373.
  43. Moradipour, M., Ranjbar, H., Hojat, A., Karimi‐Nasab, S. and Daneshpajouh, S. (2016) Laboratory and field measurements of electrical resistivity to study heap leaching pad no. 3 at Sarcheshmeh copper mine. 22nd European Meeting of Environmental and Engineering Geophysics, https://doi.org/10.3997/2214‐4609.201602140.
  44. Moseley, B., Nissen‐Meyer, T. and Markham, A. (2020) Deep learning for fast simulation of seismic waves in complex media. Solid Earth, 11(4), 1527–1549.
    [Google Scholar]
  45. Neal, R.M. (2011) MCMC Using Hamiltonian Dynamics. In: Brooks, S., Gelman, A., Jones, G. and Meng, X. (Eds) Handbook of Markov Chain Monte Carlo.. New York: Chapman and Hall, pp. 113–162.
    [Google Scholar]
  46. Norooz, R., Olsson, P.I., Dahlin, T., Günther, T. and Bernston, C. (2021) A geoelectrical pre‐study of Älvkarleby test embankment dam: 3D forward modelling and effects of structural constraints on the 3D inversion model of zoned embankment dams. Journal of Applied Geophysics, 191, 104355.
    [Google Scholar]
  47. 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.
    [Google Scholar]
  48. Rucker, D.F., Fink, J.B. and Loke, M.H. (2011) Environmental monitoring of leaks using time‐lapsed long electrode electrical resistivity. Journal of Applied Geophysics, 74(4), 242–254.
    [Google Scholar]
  49. Sambridge, M. (2014) A parallel tempering algorithm for probabilistic sampling and multimodal optimization. Geophysical Journal International, 196(1), 357–374.
    [Google Scholar]
  50. Sambridge, M. and Mosegaard, K. (2002) Monte Carlo methods in geophysical inverse problems. Reviews of Geophysics, 40(3), 3–1.
    [Google Scholar]
  51. Sen, M.K. and Biswas, R. (2017) Transdimensional seismic inversion using the reversible jump Hamiltonian Monte Carlo algorithm. Geophysics, 82(3), R119–R134.
    [Google Scholar]
  52. Sen, M.K. and Stoffa, P.L. (2013) Global Optimization Methods in Geophysical Inversion. Cambridge: Cambridge University Press.
    [Google Scholar]
  53. Soares, A. (2001) Direct sequential simulation and cosimulation. Mathematical Geology, 33(8), 911–926.
    [Google Scholar]
  54. Song, C., Alkhalifah, T. and Waheed, U.B. (2021) Solving the frequency‐domain acoustic VTI wave equation using physics‐informed neural networks. Geophysical Journal International, 225(2), 846–859.
    [Google Scholar]
  55. Tarantola, A. (2005) Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics.
    [Google Scholar]
  56. Turner, B.M. and Sederberg, P.B. (2012) Approximate Bayesian computation with differential evolution. Journal of Mathematical Psychology, 56(5), 375–385.
    [Google Scholar]
  57. Uhlemann, S., Wilkinson, P.B., Chambers, J.E., Maurer, H., Merritt, A.J., Gunn, D.A. and Meldrum, P.I. (2015) Interpolation of landslide movements to improve the accuracy of 4D geoelectrical monitoring. Journal of Applied Geophysics, 121, 93–105.
    [Google Scholar]
  58. Vinciguerra, A., Aleardi, M., Hojat, A. and Stucchi, E. (2021) Discrete cosine transform for parameter space reduction in bayesian electrical resistivity tomography. Geophysical Prospecting. 70(1), 193–209. https://doi.org/10.1111/1365‐2478.13148.
    [Google Scholar]
  59. Vrugt, J.A. (2016) Markov chain Monte Carlo simulation using the DREAM software package: theory, concepts, and MATLAB implementation. Environmental Modelling & Software, 75, 273–316.
    [Google Scholar]
  60. Whiteley, J., Chambers, J.E. and Uhlemann, S., (2017) Integrated monitoring of an active landslide in lias group mudrocks, north yorkshire, UK. In: Hoyer, S. (Ed.) GELMON 2017: Fourth International Workshop on GeoElectrical Monitoring, 27.
    [Google Scholar]
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  • Article Type: Research Article
Keyword(s): Electrical resistivity; Inversion; Tomography
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