Discrete cosine transform for parameter space reduction in Bayesian electrical resistivity tomography

Geophysical Prospecting

et al. 2022

Casting a geophysical inverse problem into a Bayesian setting is often discouraged by the computational workload needed to run many forward modelling evaluations. Here we present probabilistic inversions of electrical resistivity tomography data in which the forward operator is replaced by a trained residual neural network that learns the non‐linear mapping between the resistivity model and the apparent resistivity values. The use of this specific architecture can provide some advantages over standard convolutional networks as it mitigates the vanishing gradient problem that might affect deep networks. The modelling error introduced by the network approximation is properly taken into account and propagated onto the estimated model uncertainties. One crucial aspect of any machine learning application is the definition of an appropriate training set. We draw the models forming the training and validation sets from previously defined prior distributions, while a finite element code provides the associated datasets. We apply the approach to two probabilistic inversion frameworks: A Markov chain Monte Carlo algorithm is applied to synthetic data, while an ensemble‐based algorithm is employed for the field measurements. For both the synthetic and field tests, the outcomes of the proposed method are benchmarked against the predictions obtained when the finite element code constitutes the forward operator. Our experiments illustrate that the network can effectively approximate the forward mapping even when a relatively small training set is created. The proposed strategy provides a forward operator that is three orders of magnitude faster than the accurate but computationally expensive finite element code. Our approach also yields the most likely solutions and uncertainty quantifications comparable to those estimated when the finite element modelling is employed. The presented method allows solving the Bayesian electrical resistivity tomography with a reasonable computational cost and limited hardware resources.

Section: The Differential Evolution Markov Chainmentioning

confidence: 99%

Section: S Y N T H E T I C I N V E R S I O N Smentioning

confidence: 99%

Section: F I E L D Data a P P L I Cat I O Nmentioning

confidence: 99%

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confidence: 99%

See 2 more Smart Citations

Probabilistic inversions of electrical resistivity tomography data with a machine learning‐based forward operator

Geophysical Prospecting

et al. 2022

“…The employed MCMC recipe is described in Vinciguerra et al . (2021) with the only difference that the probabilistic sampling is here performed in DCVAE‐compressed data and model spaces. The MCMC method employed is the differential evolution Markov chain, a popular algorithm that employs interactive chains to improve the efficiency of probabilistic sampling (Vrugt, 2016).…”

Section: Introductionmentioning

confidence: 99%

Stochastic electrical resistivity tomography with ensemble smoother and deep convolutional autoencoders

Near Surface Geophysics

et al. 2022

To reduce both the computational cost of probabilistic inversions and the ill‐posedness of geophysical problems, model and data spaces can be reparameterized into low‐dimensional domains where the inverse solution can be computed more efficiently. Among the many compression methods, deep learning algorithms based on deep generative models provide an efficient approach for model and data space reduction. We present a probabilistic electrical resistivity tomography inversion in which the data and model spaces are compressed through deep convolutional variational autoencoders, while the optimization procedure is driven by the ensemble smoother with multiple data assimilation, an iterative ensemble‐based algorithm. This method iteratively updates an initial ensemble of models that are generated according to a previously defined prior model. The inversion outcome consists of the most likely solution and a set of realizations of the variables of interest from which the posterior uncertainties can be numerically evaluated. We test the method on synthetic data computed over a schematic subsurface model, and then we apply the inversion to field measurements. The model predictions and the uncertainty assessments provided by the presented approach are also compared with the results of a Markov Chain Monte Carlo sampling working in the compressed domains, a gradient‐based algorithm and with the outcomes of an ensemble‐based inversion running in the uncompressed spaces. A finite‐element code constitutes the forward operator. Our experiments show that the implemented inversion provides most likely solutions and uncertainty quantifications comparable to those yielded by the ensemble‐based inversion running in the full model and data spaces, and the Markov Chain Monte Carlo sampling, but with a significant reduction of the computational cost.

Machine learning‐accelerated gradient‐based Markov chain Monte Carlo inversion applied to electrical resistivity tomography

Near Surface Geophysics

et al. 2022

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 finiteelement 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.