Machine learning‐accelerated gradient‐based Markov chain Monte Carlo inversion applied to electrical resistivity tomography

Abstract

Expensive forward model evaluations and the curse of dimensionality usually hinder applications of Markov chain Monte Carlo algorithms to geophysical inverse problems. Another challenge of these methods is related to the definition of an appropriate proposal distribution that simultaneously should be inexpensive to manipulate and a good approximation of the posterior density. Here we present a gradient-based Markov chain Monte Carlo inversion algorithm that is applied to cast the electrical resistivity tomography into a probabilistic framework. The sampling is accelerated by exploiting the Hessian and gradient information of the negative log-posterior to define a proposal that is a local, Gaussian approximation of the target posterior probability. On the one hand, the computing time to run the many forward evaluations needed for both the data likelihood evaluation and the Hessian and gradient computation is decreased by training a residual neural network to predict the forward mapping between the resistivity model and the apparent resistivity value. On the other hand, the curse of dimensionality issue and the computational effort related to the Hessian and gradient manipulation are decreased by compressing data and model spaces through a discrete cosine transform. A non-parametric distribution is assumed as the prior probability density function. The method is first demonstrated on synthetic data and then applied to field measurements. The outcomes provided by the presented approach are also benchmarked against those obtained when a computationally expensive finiteelement code is employed for forward modelling , with the results of a gradient-free Markov chain Monte Carlo inversion, and also compared with the predictions of a deterministic inversion. The implemented approach not only guarantees uncertainty assessments and model predictions comparable with those achieved by more standard inversion strategies, but also drastically decreases the computational cost of the probabilistic inversion, making it similar to that of a deterministic inversion.

A computationally efficient Bayesian approach to full‐waveform inversion

“…In these contexts, the DCT can be replaced by non‐linear compression strategies (i.e. convolutional autoencoders) (Aleardi et al., 2022; Jiang & Jafarpour, 2021) that better preserve and recover the non‐linear features of the subsurface model. However, the downside of these strategies is related to the induced non‐linearities in the geometry of the posterior distribution (i.e.…”

“…Further details can be found in Zhao and Sen (2021) or Aleardi et al. (2022) who applied the method to probabilistically solve the electrical resistivity tomography. Gradient‐based deterministic inversions are designed for minimizing a particular misfit function, usually defined as a linear combination of data error and a model regularization term.…”

“…This means that the sampling of the model space is also guided by the model parameter correlation and not just by the data illumination. The inclusion of this information not only speeds up the probabilistic sampling, but also maximizes the independence of the samples maintaining, at the same time, high acceptance probabilities (Aleardi et al., 2022).…”

“…The model covariance matrix in the full model space C m can be computed as the double Kronecker product between the prior variance and the two spatial correlation functions $${\tau _{y\;}}$$ and $${\tau _x}$$: $$$$\begin{equation}{\mathrm{\;}}{{\bf{C}}_{\mathrm{m}}} = \left( {\left( {{\mathrm{var}}\left( {\bf{v}} \right) \otimes {\bf{T}}\left( {{\tau _x}} \right)} \right) \otimes {\bf{T}}\left( {{\tau _y}} \right)} \right),\end{equation}$$$$where var( v ) is the assumed prior variance of the velocity model (in our tests, we have considered the variance of the prior mean model m prior of Equation ), T indicates the Toeplitz matrix and ⊗ stands for the Kronecker product. Due to the large number of data points in the full data space, we have applied a numerical Monte Carlo simulation to project the data covariance matrix C d in the compressed DCT domain (Aleardi et al., 2022).…”