Ensemble-Based Electrical Resistivity Tomography with Data and Model Space Compression

Abstract

Inversion of electrical resistivity tomography (ERT) data is an ill-posed problem that is usually solved through deterministic gradient-based methods. These methods guarantee a fast convergence but hinder accurate assessments of model uncertainties. On the contrary, Markov Chain Monte Carlo (MCMC) algorithms can be employed for accurate uncertainty appraisals, but they remain a formidable computational task due to the many forward model evaluations needed to converge. We present an alternative approach to ERT that not only provides a best-fitting resistivity model but also gives an estimate of the uncertainties affecting the inverse solution. More specifically, the implemented method aims to provide multiple realizations of the resistivity values in the subsurface by iteratively updating an initial ensemble of models based on the difference between the predicted and measured apparent resistivity pseudosections. The initial ensemble is generated using a geostatistical method under the assumption of log-Gaussian distributed resistivity values and a Gaussian variogram model. A finite-element code constitutes the forward operator that maps the resistivity values onto the associated apparent resistivity pseudosection. The optimization procedure is driven by the ensemble smoother with multiple data assimilation, an iterative ensemble-based algorithm that performs a Bayesian updating step at each iteration. The main advantages of the proposed approach are that it can be applied to nonlinear inverse problems, while also providing an ensemble of models from which the uncertainty on the recovered solution can be inferred. The ill-conditioning of the inversion procedure is decreased through a discrete cosine transform reparameterization of both data and model spaces. The implemented method is first validated on synthetic data and then applied to field data. We also compare the proposed method with a deterministic least-square inversion, and with an MCMC algorithm. We show that the ensemble-based inversion estimates resistivity models and associated uncertainties comparable to those yielded by a much more computationally intensive MCMC sampling.

Stochastic electrical resistivity tomography with ensemble smoother and deep convolutional autoencoders

“…See Aleardi et al . (2021b) for a more detailed discussion on how the ensemble size affects the uncertainty estimation in electrical resistivity tomography inversion solved via ensemble‐based algorithms. For comparison, the Markov chain Monte Carlo (MCMC) employs 30 chains and runs in the compressed model space for 3000 iterations, with a burn‐in period of 500.…”

“…For this reason, the reduction of the parameter space should be a compromise between the expected model resolutions, and the accuracy of the uncertainty assessments. Also, note that the posterior uncertainty is underestimated in the ES‐MDA if the number of ensemble members is not sufficient to statistically represent the model space (Aleardi et al ., 2021b). Reducing the data space partially mitigates the underestimation because it makes the problem more underdetermined, thus increasing its condition number and consequently the posterior uncertainties (Grana et al ., 2019).…”

“…As demonstrated in Aleardi et al . (2021b), linear compression methods are very effective to reduce model and data spaces in 2D ERT inversion. However, the popularity that machine learning compression methods have recently gained in the geophysical community, motivated us to contemporarily assess the applicability of DCVAEs to solve the same problem.…”

“…Indeed, smaller ensembles resulted in underestimated posterior uncertainties, while larger ensembles (e.g., 500 and 1000 models) provided uncertainty similar to the one obtained with 250 models. See Aleardi et al (2021b) for a more detailed discussion on how the ensemble size affects the uncertainty estimation in electrical resistivity tomography inversion solved via ensemble-based algorithms. For comparison, the Markov chain Monte Carlo (MCMC) employs 30 chains and runs in the compressed model space for 3000 iterations, with a burnin period of 500.…”

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

“…In particular, the number of ensemble members should be large enough to get an accurate estimate of the $${\stackrel{\mathbf{\sim}}{\mathbf{C}}}_{\mathbf{dd}}^{p}$$ and $${\stackrel{\mathbf{\sim}}{\mathbf{C}}}_{\mathbf{md}}^{p}$$ matrices but small enough not to make the forward evaluations computationally impractical. Usually, the number of ensemble members needed to get accurate uncertainty assessments increases with the dimension of the model space (Aleardi et al ., 2021b).…”