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.
Highlights
To assess the quality of the results we compare the outcomes provided by the EB-Electrical resistivity tomography (ERT) inversion with those achieved by a Differential Evolution Markov Chain (DEMC) algorithm sampling the same reduced space, and with a standard deterministic least-squares inversion running in the full data and model spaces
The discrete cosine transform (DCT) compression of the data and model space mitigated the ill-conditioning of the ERT problem and allowed running the inversion in lower-dimensional data and model spaces
This strategy reduced the computational cost of the ensemble-based inversion of ERT data (EB-ERT) inversion because reliable model estimations and uncertainty assessments can be obtained with smaller ensembles
Summary
Electrical resistivity tomography (ERT) is one of the most widely used geophysical methods that can successfully be employed for geotechnical characterization, monitoring earthen dams and embankments, landfill monitoring, groundwater exploration, and mapping of contaminant plumes (e.g., Aleardi et al, 2020; Arosio et al, 2017; Bievre et al, 2018; Chambers et al, 2006; Dahlin, 2020; Hermans & Paepen, 2020; Hojat et al, 2019a; Karimi-Nasab et al, 2011; Loke et al, 2020; Moradipour et al, 2016; Muller et al, 2010; Pollock & Cirpka, 2012; Supper et al, 2014; Tresoldi et al, 2019; Whiteley et al, 2017). For nonlinear problems and/or non-Gaussian assumptions, a complete characterization of the PPD is only possible through sampling, and in these contexts, Markov Chain Monte Carlo (MCMC; Sambridge & Mosegaard, 2002; Sen & Stoffa, 2013) algorithms can be used to numerically estimate the target posterior These methods provide accurate uncertainty assessments but require a considerable computational effort to converge toward stable PPD estimations, especially in large-dimensional model spaces and for expensive forward model evaluations (Aleardi & Salusti, 2020; Aleardi et al, 2017, 2020; Pradhan & Mukerji, 2020; Sajeva et al, 2014). To alleviate the computational workload for uncertainty assessment in nonlinear inverse problems, an approximate Bayesian method, the ensemblebased inversion can be applied This approach is a data assimilation algorithm in which the PPD is represented by an ensemble of model realizations obtained by assimilating the observed data. As far as the authors are aware, this is the first time that the EB and MCMC methods are compared in the context of ERT inversion
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