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.

电阻层析成像（ERT）数据的反演是一个不适定的问题，通常可以通过基于确定性梯度的方法来解决。这些方法保证了快速收敛，但阻碍了模型不确定性的准确评估。相反，可以使用马尔可夫链蒙特卡洛（MCMC）算法进行准确的不确定性评估，但是由于需要收敛许多前向模型评估，因此它们仍然是一项艰巨的计算任务。我们提出了ERT的替代方法，该方法不仅提供了最合适的电阻率模型，而且还提供了影响反解的不确定性的估计值。进一步来说，所实施的方法旨在通过基于预测的和测量的视在电阻率伪剖面之间的差异迭代更新模型的初始集合来提供地下电阻率值的多种实现。在对数-高斯分布电阻率值和高斯变异函数模型的假设下，使用地统计方法生成初始集合。有限元代码构成将电阻率值映射到关联的视电阻率伪截面上的正向算子。优化过程由具有多个数据同化的集成平滑器驱动，该集成平滑器是基于迭代的集成算法，在每次迭代时执行贝叶斯更新步骤。该方法的主要优点是可以应用于非线性逆问题，同时还提供了一组模型，可以从中推断出回收解决方案的不确定性。通过对数据空间和模型空间进行离散余弦变换重新参数化，可以减少反演过程的不良情况。首先对合成数据进行验证，然后将其应用于现场数据。我们还将拟议的方法与确定性最小二乘反演和MCMC算法进行比较。我们表明，基于集合的反演可估算电阻率模型和相关的不确定性，可与计算强度更高的MCMC采样所产生的不确定性相媲美。通过对数据空间和模型空间进行离散余弦变换重新参数化，可以减少反演过程的不良情况。首先对合成数据进行验证，然后将其应用于现场数据。我们还将拟议的方法与确定性最小二乘反演和MCMC算法进行比较。我们表明，基于集合的反演可估算电阻率模型和相关的不确定性，可与计算强度更高的MCMC采样所产生的不确定性相媲美。通过对数据空间和模型空间进行离散余弦变换重新参数化，可以减少反演过程的不良情况。首先对合成数据进行验证，然后将其应用于现场数据。我们还将拟议的方法与确定性最小二乘反演和MCMC算法进行比较。我们表明，基于集合的反演可估算电阻率模型和相关的不确定性，可与计算强度更高的MCMC采样所产生的不确定性相媲美。