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.

中文翻译：

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用于电阻率层析成像的地统计马尔可夫链蒙特卡罗反演算法

电阻层析成像是一个不适定的非线性逆问题，通常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到最终解，但是逆算子的局部线性化阻碍了准确的不确定性评估。相反，数值马尔可夫链蒙特卡洛算法可进行准确的不确定性评估，但需要适当的马尔可夫链蒙特卡洛配方以减少计算量并使这些方法适合于现场数据。任何概率反演的一个关键方面是对模型参数的适当先验分布的定义，该模型参数还可以合并空间约束以减轻反问题的不适条件。通常，高斯先验过分地简化了模型参数的实际分布，这些模型参数由于存在多个岩石流体相而常常表现出多峰性。在这项工作中，我们开发了一种新颖的概率马尔可夫链蒙特卡罗方法，用于电阻率层析成像数据的反演。这种方法联合估计电阻率值，岩石流体相以及从测量的视电阻率伪剖面得到的相关不确定性。在我们的方法中，未知参数包括相模型以及连续电阻率值。在每个空间位置，电阻率值的分布被假定为多峰且非参数的，其模式与相数一样多。先进的马尔可夫链蒙特卡罗算法（差分进化马尔可夫链）用于高效地采样高维参数空间中的后验密度。高斯变异函数模型和一阶马尔可夫链分别说明了连续和离散模型属性的横向连续性，从而减少了解的零空间。直接顺序模拟地统计方法允许生成既符合假定的边际先验约束又具有空间约束的采样模型。在马尔可夫链蒙特卡罗走动过程中，我们通过从一种模式转换到另一种模式来迭代采样相，并通过在同一模式下进行采样来迭代电阻率值。首先通过对综合模型计算出的数据求逆来验证所提出的方法。然后，它适用于现场数据，并以标准的局部反演算法为基准。我们的实验证明，提出的马尔可夫链蒙特卡罗反演可以通过合理的计算工作来检索可靠的估计和准确的不确定性量化。