Discrete cosine transform for parameter space reduction in linear and non-linear AVA inversions

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.

“…We refer the reader to Grana et al . (2019) and Aleardi (2020) for a more in‐depth discussion of this topic in the context of post‐ and pre‐stack seismic inversion.…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

“…In this case, the full state space is projected onto a limited number of basis functions and the algorithm generates samples in this reduced domain. This technique must be applied taking in mind that part of the information in the original (unreduced) parameter space could be lost in the reduced space, and, for this reason, the model parameterization must always constitute a compromise between model resolution and model uncertainty (Dejtrakulwong et al ., 2012; Lochbühler et al ., 2014; Aleardi, 2019; Grana et al ., 2019; Aleardi, 2020b).…”

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

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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.

Stochastic electrical resistivity tomography with ensemble smoother and deep convolutional autoencoders

Near Surface Geophysics

Vinciguerra

Stucchi

et al. 2022

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To reduce both the computational cost of probabilistic inversions and the ill‐posedness of geophysical problems, model and data spaces can be reparameterized into low‐dimensional domains where the inverse solution can be computed more efficiently. Among the many compression methods, deep learning algorithms based on deep generative models provide an efficient approach for model and data space reduction. We present a probabilistic electrical resistivity tomography inversion in which the data and model spaces are compressed through deep convolutional variational autoencoders, while the optimization procedure is driven by the ensemble smoother with multiple data assimilation, an iterative ensemble‐based algorithm. This method iteratively updates an initial ensemble of models that are generated according to a previously defined prior model. The inversion outcome consists of the most likely solution and a set of realizations of the variables of interest from which the posterior uncertainties can be numerically evaluated. We test the method on synthetic data computed over a schematic subsurface model, and then we apply the inversion to field measurements. The model predictions and the uncertainty assessments provided by the presented approach are also compared with the results of a Markov Chain Monte Carlo sampling working in the compressed domains, a gradient‐based algorithm and with the outcomes of an ensemble‐based inversion running in the uncompressed spaces. A finite‐element code constitutes the forward operator. Our experiments show that the implemented inversion provides most likely solutions and uncertainty quantifications comparable to those yielded by the ensemble‐based inversion running in the full model and data spaces, and the Markov Chain Monte Carlo sampling, but with a significant reduction of the computational cost.