A reliable assessment of the posterior uncertainties is a crucial aspect of any amplitude versus angle (AVA) inversion due to the severe ill-conditioning of this inverse problem. To accomplish this task, numerical Markov chain Monte Carlo algorithms are usually used when the forward operator is nonlinear. The downside of these algorithms is the considerable number of samples needed to attain stable posterior estimations especially in high-dimensional spaces. To overcome this issue, we assessed the suitability of Hamiltonian Monte Carlo (HMC) algorithm for nonlinear target- and interval-oriented AVA inversions for the estimation of elastic properties and associated uncertainties from prestack seismic data. The target-oriented approach inverts the AVA responses of the target reflection by adopting the nonlinear Zoeppritz equations, whereas the interval-oriented method inverts the seismic amplitudes along a time interval using a 1D convolutional forward model still based on the Zoeppritz equations. HMC uses an artificial Hamiltonian system in which a model is viewed as a particle moving along a trajectory in an extended space. In this context, the inclusion of the derivative information of the misfit function makes possible long-distance moves with a high probability of acceptance from the current position toward a new independent model. In our application, we adopt a simple Gaussian a priori distribution that allows for an analytical inclusion of geostatistical constraints into the inversion framework, and we also develop a strategy that replaces the numerical computation of the Jacobian with a matrix operator analytically derived from a linearization of the Zoeppritz equations. Synthetic and field data inversions demonstrate that the HMC is a very promising approach for Bayesian AVA inversion that guarantees an efficient sampling of the model space and retrieves reliable estimations and accurate uncertainty quantifications with an affordable computational cost.

In geophysical inverse problems, the posterior model can be analytically assessed only in case of linear forward operators, Gaussian, Gaussian mixture, or generalized Gaussian prior models, continuous model properties, and Gaussian‐distributed noise contaminating the observed data. For this reason, one of the major challenges of seismic inversion is to derive reliable uncertainty appraisals in cases of complex prior models, non‐linear forward operators and mixed discrete‐continuous model parameters. We present two amplitude versus angle inversion strategies for the joint estimation of elastic properties and litho‐fluid facies from pre‐stack seismic data in case of non‐parametric mixture prior distributions and non‐linear forward modellings. The first strategy is a two‐dimensional target‐oriented inversion that inverts the amplitude versus angle responses of the target reflections by adopting the single‐interface full Zoeppritz equations. The second is an interval‐oriented approach that inverts the pre‐stack seismic responses along a given time interval using a one‐dimensional convolutional forward modelling still based on the Zoeppritz equations. In both approaches, the model vector includes the facies sequence and the elastic properties of P‐wave velocity, S‐wave velocity and density. The distribution of the elastic properties at each common‐mid‐point location (for the target‐oriented approach) or at each time‐sample position (for the time‐interval approach) is assumed to be multimodal with as many modes as the number of litho‐fluid facies considered. In this context, an analytical expression of the posterior model is no more available. For this reason, we adopt a Markov chain Monte Carlo algorithm to numerically evaluate the posterior uncertainties. With the aim of speeding up the convergence of the probabilistic sampling, we adopt a specific recipe that includes multiple chains, a parallel tempering strategy, a delayed rejection updating scheme and hybridizes the standard Metropolis–Hasting algorithm with the more advanced differential evolution Markov chain method. For the lack of available field seismic data, we validate the two implemented algorithms by inverting synthetic seismic data derived on the basis of realistic subsurface models and actual well log data. The two approaches are also benchmarked against two analytical inversion approaches that assume Gaussian‐mixture‐distributed elastic parameters. The final predictions and the convergence analysis of the two implemented methods proved that our approaches retrieve reliable estimations and accurate uncertainties quantifications with a reasonable computational effort.

We develop a pre-stack inversion algorithm that combines a Discrete Cosine Transform (DCT) reparameterization of data and model spaces with a Convolutional Neural Network (CNN). The CNN is trained to predict the mapping between the DCT-transformed seismic data and the DCT-transformed 2-D elastic model. A convolutional forward modeling based on the full Zoeppritz equations constitutes the link between the elastic properties and the seismic data. The direct sequential co-simulation algorithm with joint probability distribution is used to generate the training and validation datasets under the assumption of a stationary non-parametric prior and a Gaussian variogram model for the elastic properties. The DCT is an orthogonal transformation that is here used as an additional feature extraction technique that reduces the number of unknown parameters in the inversion and the dimensionality of the input and output of the network. The DCT reparameterization also acts as a regularization operator in the model space and allows for the preservation of the lateral and vertical continuity of the elastic properties in the recovered solution. We also implement a Monte Carlo simulation strategy that propagates onto the estimated elastic model the uncertainties related to both noise contamination and network approximation. We focus on synthetic inversions on a realistic subsurface model that mimics a real gas-saturated reservoir hosted in a turbiditic sequence. We compare the outcomes of the implemented algorithm with those provided by a popular linear inversion approach and we also assess the robustness of the CNN inversion to errors in the estimated source wavelet and to erroneous assumptions about the noise statistic. Our tests confirm the applicability of the proposed approach, opening the possibility of estimating the subsurface elastic parameters and the associated uncertainties in near real-time while satisfactorily preserving the assumed spatial variability and the statistical properties of the elastic parameters.