Combining discrete cosine transform and convolutional neural networks to speed up the Hamiltonian Monte Carlo inversion of pre‐stack seismic data

Abstract

Markov chain Monte Carlo algorithms are commonly employed for accurate uncertainty appraisals in non-linear inverse problems. The downside of these algorithms is the considerable number of samples needed to achieve reliable posterior estimations, especially in high-dimensional model spaces. To overcome this issue, the Hamiltonian Monte Carlo algorithm has recently been introduced to solve geophysical inversions. Different from classical Markov chain Monte Carlo algorithms, this approach exploits the derivative information of the target posterior probability density to guide the sampling of the model space. However, its main downside is the computational cost for the derivative computation (i.e. the computation of the Jacobian matrix around each sampled model). Possible strategies to mitigate this issue are the reduction of the dimensionality of the model space and/or the use of efficient methods to compute the gradient of the target density. Here we focus the attention to the estimation of elastic properties (P-, S-wave velocities and density) from pre-stack data through a non-linear amplitude versus angle inversion in which the Hamiltonian Monte Carlo algorithm is used to sample the posterior probability. To decrease the computational cost of the inversion procedure, we employ the discrete cosine transform to reparametrize the model space, and we train a convolutional neural network to predict the Jacobian matrix around each sampled model. The training data set for the network is also parametrized in the discrete cosine transform space, thus allowing for a reduction of the number of parameters to be optimized during the learning phase. Once trained the network can be used to compute the Jacobian matrix associated with each sampled model in real time. The outcomes of the proposed approach are compared and validated with the predictions of Hamiltonian Monte Carlo inversions in which a quite computationally expensive, but accurate finitedifference scheme is used to compute the Jacobian matrix and with those obtained by replacing the Jacobian with a matrix operator derived from a linear approximation of the Zoeppritz equations. Synthetic and field inversion experiments demonstrate that the proposed approach dramatically reduces the cost of the Hamiltonian Monte Carlo inversion while preserving an accurate and efficient sampling of the posterior probability.

A geostatistical Markov chain Monte Carlo inversion algorithm for electrical resistivity tomography

“…To this end, the adjoint‐state method can be employed. Alternatively, one can also exploit machine learning algorithms to approximate the Jacobian around each considered model (Aleardi, 2020).…”

A reduced-order variational Bayesian approach for efficient subsurface imaging

“…More specifically, we follow In the more general cases where the "inserted" a posteriori independence might not be relevant, the approximation accuracy is expected to degrade (Ait-El-Fquih et al 2019). One way to improve the approximation is to employ Stein Variational Gradient Descent using a kernelized Stein discrepancy as shown by Zhang & Curtis (2019, 2020, which is based on invertible transforms where a set of samples from the prior pdf are iteratively perturbed to represent samples of the posterior distribution through optimization. This method suffers however from the curse of dimensionality due to number of samples needed to describe the posterior .…”

“…For geophysical problems, the VB approach was recently demonstrated to provide comparable performances to MCMC in terms of estimation accuracy at significantly reduced computational cost (Nawaz & Curtis 2018;Ait-El-Fquih et al 2019;Zhang & Curtis 2019, 2020.…”

“…splitting the parameter vector to smaller partitions, while balancing between the estimation accuracy and the computational cost(Ait-El-Fquih & Hoteit 2015;Ait-El-Fquih et al 2019, 2020.…”

A gradient‐based Markov chain Monte Carlo algorithm for elastic pre‐stack inversion with data and model space reduction

“…As an alternative, gradient‐based MCMC (GB‐MCMC) (e.g. Hamiltonian Monte Carlo, Langevin Monte Carlo; Sen and Biswas; 2017; Fichtner and Simutè, 2018; Fichtner and Zunino, 2019; Fichtner et al ., 2019; Aleardi, 2020a; Aleardi and Salusti, 2020; Gebrad et al ., 2020) exploits the gradient information of the misfit function (the negative natural logarithm of the posterior) to efficiently explore the model space and to rapidly converge towards stable posterior uncertainties (MacKay, 2003; Neal, 2011). The main computational requirement of these methods is the need for computing derivatives, although this information is highly beneficial to speeding up the convergence of the sampling and to guarantee high independence of the samples while maintaining high acceptance rates.…”