We have developed a stochastic full-waveform inversion that uses genetic algorithms (GA FWI) to estimate acoustic macro models of the P-wave velocity field. Stochastic methods such as GA severely suffer the curse of dimensionality, meaning that they require unaffordable computer resources for inverse problems with many unknowns and expensive forward modeling. To mitigate this issue, we have proposed a two-grid technique with a coarse grid to represent the subsurface for the GA inversion and a finer grid for the forward modeling. We have applied this procedure to invert synthetic acoustic data of the Marmousi model, and we have developed three different tests. The first two tests use a velocity model derived from standard stacking velocity analysis as prior information and differ only for the parameterization of the coarse grid. Their comparison indicates that a smart parameterization of the coarse grid may significantly improve the final result. The third test uses a linearly increasing 1D velocity model as prior information, a layer-stripping procedure, and a large number of model evaluations. All three tests return velocity models that fairly reproduce the long-wavelength structures of the Marmousi. First-break cycle skipping related to the seismograms of the final GA-FWI models is significantly reduced compared with that computed on the models used as prior information. Descent-based FWIs starting from final GA-FWI models yield velocity models with low and comparable model misfits and with an improved reconstruction of the structural details. The quality of the models obtained by GA FWI plus descent-based FWI is benchmarked against the models obtained by descent-based FWI started from a smoothed version of the Marmousi and started directly from the prior information models. Our results are promising and demonstrate the ability of the two-grid GA FWI to yield velocity models suitable as input to descent-based FWI.

We have evaluated a two-step Bayesian algorithm for seismic-reservoir characterization, which, thanks to some simplifying assumptions, is computationally very efficient. The applicability and reliability of this method are assessed by comparison with a more sophisticated and computer-intensive Markov-chain Monte Carlo (MCMC) algorithm, which in a single loop directly estimates petrophysical properties and lithofluid facies from prestack data. The two-step method first combines a linear rock-physics model (RPM) with the analytical solution of a linearized amplitude versus angle (AVA) inversion, to directly estimate the petrophysical properties, and related uncertainties, from prestack data under the assumptions of a Gaussian prior model and weak elastic contrasts at the reflecting interface. In particular, we use an empirical, linear RPM, properly calibrated for the investigated area, to reparameterize the linear time-continuous P-wave reflectivity equation in terms of petrophysical contrasts instead of elastic constants. In the second step, a downward 1D Markov-chain prior model is used to infer the lithofluid classes from the outcomes of the first step. The single-loop (SL) MCMC algorithm uses a convolutional forward modeling based on the exact Zoeppritz equations, and it adopts a nonlinear RPM. Moreover, it assumes a more realistic Gaussian mixture distribution for the petrophysical properties. Both approaches are applied on an onshore 3D seismic data set for the characterization of a gas-bearing, clastic reservoir. Notwithstanding the differences in the forward-model parameterization, in the considered RPM, and in the assumed a priori probability density functions, the two methods yield maximum a posteriori solutions that are consistent with well-log data, although the Gaussian mixture assumption adopted by the SL method slightly improves the description of the multimodal behavior of the petrophysical parameters. However, in the considered reservoir, the main difference between the two approaches remains the very different computational times, the SL method being much more computationally intensive than the two-step approach.

We have developed a prestack 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 discrete cosine-transformed seismic data and the discrete cosine-transformed 2D 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 cosimulation algorithm with joint probability distribution is used to generate the training and validation data sets under the assumption of a stationary nonparametric prior and a Gaussian variogram model for the elastic properties. The DCT is an orthogonal transformation that we 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 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 evaluate 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 our 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.

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.

Conventional methods solve the full‐waveform inversion making use of gradient‐based algorithms to minimize an error function, which commonly measure the Euclidean distance between observed and predicted waveforms. This deterministic approach only provides a ‘best‐fitting’ model and cannot account for the uncertainties affecting the predicted solution. Local methods are also usually prone to get trapped into local minima of the error function. On the other hand, casting this inverse problem into a probabilistic framework has to deal with the formidable computational effort of the Bayesian approach when applied to non‐linear problems with expensive forward evaluations and large model spaces. We present a gradient‐based Markov Chain Monte Carlo full‐waveform inversion in which the posterior sampling is accelerated by compressing the data and model spaces through the discrete cosine transform, and by also defining a proposal that is a local, Gaussian approximation of the target posterior probability density. This proposal is constructed using the local Hessian and gradient informations of the log posterior, which are made computationally manageable thanks to the compression of the data and model spaces. We demonstrate the applicability of the approach by performing two synthetic inversion tests on portions of the Marmousi and BP acoustic model. In these examples, the forward modelling is performed using Devito, a finite difference domain‐specific language that solves the discretized wave equation on a Cartesian grid. For both examples, the results obtained by the implemented method are also validated against those obtained using a classic deterministic approach. Our tests illustrate the efficiency of the proposed probabilistic method, which seems quite robust against cycle‐skipping issues and also characterized by a computational cost comparable to that of the local inversion. The outcomes of the proposed probabilistic inversion can also play the role of starting models for a subsequent local inversion step aimed at improving the spatial resolution of the probabilistic result, which was limited by the model compression.