Hamiltonian Monte Carlo algorithms for target- and interval-oriented amplitude versus angle inversions

Near Surface Geophysics

Vinciguerra

Hojat

2020

Self Cite

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.

Section: Discussionmentioning

confidence: 99%

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

Near Surface Geophysics

Vinciguerra

Hojat

2020

Self Cite

“…slower convergences towards the stationary regime and a stable posterior, and an increase of the correlation between successively sampled models). For this reason, the applicability of this strategy should be evaluated case by case (Aleardi and Salusti, 2020). On the other hand, the FD approach could be computationally prohibitive in the case of hundreds of parameters to be inferred from the data, even though a DCT compression of the elastic model space could be useful to partially reduce the overall computational effort (Aleardi, 2020).…”

Section: Methodsmentioning

confidence: 99%

“…In addition, the sampling ability of MCMC algorithms severely decreases in high‐dimensional model spaces due to the so‐called curse of dimensionality problem (Curtis and Lomax, 2001). Over the last decades, many MCMC approaches have been proposed to mitigate these issues, and in the very last years, the Hamiltonian Monte Carlo (HMC; Duane et al ., 1987) approach has also been used to solve geophysical inversions (Sen and Biswas, 2017; Fichtner and Simutè, 2018; Fichtner and Zunino, 2019; Fichtner et al ., 2019; Aleardi and Salusti, 2020; Gebraad et al ., 2020). Different from classical MCMC methods, this approach exploits the derivative information of the misfit function (the negative natural logarithm of the posterior) to explore the model space and to rapidly converge towards a stable posterior.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Geophysical Prospecting

2020

Self Cite

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.

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Geophysical Prospecting

2021

Self Cite

The main challenge of Markov chain Monte Carlo sampling is to define a proposal distribution that simultaneously is a good approximation of the posterior probability while being inexpensive to manipulate. We present a gradient‐based Markov chain Monte Carlo inversion of pre‐stack seismic data in which the posterior sampling is accelerated by defining a proposal that is a local, Gaussian approximation of the posterior model, while a non‐parametric prior is assumed for the distribution of the elastic properties. The proposal is constructed from the local Hessian and gradient information of the log posterior, whereas the non‐linear, exact Zoeppritz equations constitute the forward modelling engine for the inversion procedure. Hessian and gradient information is made computationally tractable by a reduction of data and model spaces through a discrete cosine transform reparameterization. This reparameterization acts as a regularization operator in the model space, while also preserving the spatial and temporal continuity of the elastic properties in the sampled models. We test the implemented algorithm on synthetic pre‐stack inversions under different signal‐to‐noise ratios in the observed data. We also compare the results provided by the presented method when a computationally expensive (but accurate) finite‐difference scheme is used for the Jacobian computation, with those obtained when the Jacobian is derived from the linearization of the exact Zoeppritz equations. The outcomes of the proposed approach are also compared against those yielded by a gradient‐free Monte Carlo sampling and by a deterministic least‐squares inversion. Our tests demonstrate that the gradient‐based sampling reaches accurate uncertainty estimations with a much lower computational effort than the gradient‐free approach.