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.
Electrical resistivity tomography is a non‐linear and ill‐posed geophysical inverse problem that is usually solved through gradient‐descent methods. This strategy is computationally fast and easy to implement but impedes accurate uncertainty appraisals. We present a probabilistic approach to two‐dimensional electrical resistivity tomography in which a Markov chain Monte Carlo algorithm is used to numerically evaluate the posterior probability density function that fully quantifies the uncertainty affecting the recovered solution. The main drawback of Markov chain Monte Carlo approaches is related to the considerable number of sampled models needed to achieve accurate posterior assessments in high‐dimensional parameter spaces. Therefore, to reduce the computational burden of the inversion process, we employ the differential evolution Markov chain, a hybrid method between non‐linear optimization and Markov chain Monte Carlo sampling, which exploits multiple and interactive chains to speed up the probabilistic sampling. Moreover, the discrete cosine transform reparameterization is employed to reduce the dimensionality of the parameter space removing the high‐frequency components of the resistivity model which are not sensitive to data. In this framework, the unknown parameters become the series of coefficients associated with the retained discrete cosine transform basis functions. First, synthetic data inversions are used to validate the proposed method and to demonstrate the benefits provided by the discrete cosine transform compression. To this end, we compare the outcomes of the implemented approach with those provided by a differential evolution Markov chain algorithm running in the full, un‐reduced model space. Then, we apply the method to invert field data acquired along a river embankment. The results yielded by the implemented approach are also benchmarked against a standard local inversion algorithm. The proposed Bayesian inversion provides posterior mean models in agreement with the predictions achieved by the gradient‐based inversion, but it also provides model uncertainties, which can be used for penetration depth and resolution limit identification.
Electrical resistivity tomography is an ill‐posed and nonlinear inverse problem commonly solved through deterministic gradient‐based methods. These methods guarantee a fast convergence towards the final solution, but the local linearization of the inverse operator impedes accurate uncertainty assessments. On the contrary, numerical Markov chain Monte Carlo algorithms allow for accurate uncertainty appraisals, but appropriate Markov chain Monte Carlo recipes are needed to reduce the computational effort and make these approaches suitable to be applied to field data. A key aspect of any probabilistic inversion is the definition of an appropriate prior distribution of the model parameters that can also incorporate spatial constraints to mitigate the ill conditioning of the inverse problem. Usually, Gaussian priors oversimplify the actual distribution of the model parameters that often exhibit multimodality due to the presence of multiple litho‐fluid facies. In this work, we develop a novel probabilistic Markov chain Monte Carlo approach for inversion of electrical resistivity tomography data. This approach jointly estimates resistivity values, litho‐fluid facies, along with the associated uncertainties from the measured apparent resistivity pseudosection. In our approach, the unknown parameters include the facies model as well as the continuous resistivity values. At each spatial location, the distribution of the resistivity value is assumed to be multimodal and non‐parametric with as many modes as the number of facies. An advanced Markov chain Monte Carlo algorithm (the differential evolution Markov chain) is used to efficiently sample the posterior density in a high‐dimensional parameter space. A Gaussian variogram model and a first‐order Markov chain respectively account for the lateral continuity of the continuous and discrete model properties, thereby reducing the null‐space of solutions. The direct sequential simulation geostatistical method allows the generation of sampled models that honour both the assumed marginal prior and spatial constraints. During the Markov chain Monte Carlo walk, we iteratively sample the facies, by moving from one mode to another, and the resistivity values, by sampling within the same mode. The proposed method is first validated by inverting the data calculated from synthetic models. Then, it is applied to field data and benchmarked against a standard local inversion algorithm. Our experiments prove that the proposed Markov chain Monte Carlo inversion retrieves reliable estimations and accurate uncertainty quantifications with a reasonable computational effort.
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