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.

中文翻译：

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用于贝叶斯电阻率层析成像中参数空间缩减的离散余弦变换

电阻率层析成像是一种非线性和不适定的地球物理反问题，通常通过梯度下降方法解决。这种策略计算速度快且易于实施，但会妨碍准确的不确定性评估。我们提出了二维电阻率断层扫描的概率方法，其中使用马尔可夫链蒙特卡罗算法对后验概率密度函数进行数值评估，该函数完全量化了影响恢复解的不确定性。马尔可夫链蒙特卡罗方法的主要缺点与在高维参数空间中实现准确后验评估所需的大量采样模型有关。因此，为了减少反演过程的计算负担，我们采用差分进化马尔可夫链，非线性优化和马尔可夫链蒙特卡罗采样之间的混合方法，它利用多个交互链来加速概率采样。此外，采用离散余弦变换重新参数化来降低参数空间的维数，去除电阻率模型中对数据不敏感的高频分量。在这个框架中，未知参数成为与保留的离散余弦变换基函数相关联的一系列系数。首先，合成数据反演用于验证所提出的方法并证明离散余弦变换压缩提供的好处。为此，我们将实施方法的结果与完整运行的差分进化马尔可夫链算法提供的结果进行比较，未减少的模型空间。然后，我们应用该方法对沿河堤采集的现场数据进行反演。所实现的方法产生的结果也与标准的局部反演算法进行了基准测试。所提出的贝叶斯反演提供了与基于梯度反演实现的预测一致的后验均值模型，但它也提供了模型不确定性，可用于穿透深度和分辨率极限识别。