Linearized Seismic AVO Inversion: A Variational Bayesian Learning Method

Seismic reflectivity inversion problem can be formulated using a basis-pursuit method, aiming to generate a sparse reflectivity series of the subsurface media. In the basis-pursuit method, the reflectivity series is composed by large amounts of even and odd dipoles, thus the size of the seismic response matrix is huge and the matrix operations involved in seismic inversion are very time-consuming. In order to accelerate the matrix computation, a basis-pursuit method-based seismic inversion algorithm is implemented on Graphics Processing Unit (GPU). In the basis-persuit inversion algorithm, the problem is imposed with a L1-norm model constraint for sparsity, and this L1-norm basis-pursuit inversion problem is reformulated using a linear programming method. The core problems in the inversion are large-scale linear systems, which are resolved by a parallelised conjugate gradient method. The performance of this fully parallelised implementation is evaluated and compared to the conventional serial coding. Specifically, the investigation using several field seismic data sets with different sizes indicates that GPU-based parallelisation can significantly reduce the computational time with an overall factor up to 145. This efficiency improvement demonstrates a great potential of the basis-pursuit inversion method in practical application to large-scale seismic reflectivity inversion problems.

Highly non‐linear seismic trace inversion problems can be solved efficiently by an implementation of Tabu Search, a meta‐heuristic method related to artificial intelligence. The implementation under consideration is a deterministic, global search that combines the advantages ofa local search, giving a quick descent to local misfit minima, with an ability to cross misfit barriers in the model space. Once Tabu Search has found an area of low misfit, it performs an extensive exploration of its deepest points. This property makes it possible to use Tabu Search for a semiquantitative resolution and uncertainty analysis of the inverse problem.

In this paper, we consider continuous solution methods for migration deconvolution imaging in seismic inverse problems. Direct migration methods, using the adjoint operator L*, usually yield a lower resolution or blurred image. Linearized migration deconvolution requires solving a least‐squares migration (LSM) problem. However, we notice that the direct LSM method is unstable in computation which is a severe obstacle for visual explanation. We study regularized mathematical model. We fist formulate the problem by incorporating regularizing constraints, and then employ iterative gradient methods for migration deconvolution and inversion. A hybrid gradient technique for ill‐posed migration inverse problem is proposed. To show the potential for application of the proposed method, we perform synthetic one‐, two‐ and three‐dimensional seismogram for seismic migration inversion. Numerical performance indicates that the proposed method is very promising for practical seismic migration imaging.

Neural networks have been applied to seismic inversion problems since the 1990s. More recently, many publications have reported the use of Deep Learning (DL) neural networks capable of performing seismic inversion with promising results. However, when solving a seismic inversion problem with DL, each author uses, in addition to different DL models, different datasets and different metrics for performance evaluation, which makes it difficult to compare performances. Depending on the data used for training and the metrics used for evaluation, one model may be better or worse than another. Thus, it is quite challenging to choose the appropriate model to meet the requirements of a new problem. This work aims to review some of the proposed DL methodologies, propose appropriate performance evaluation metrics, compare the performances, and observe the advantages and disadvantages of each model implementation when applied to the chosen datasets. The publication of this benchmark environment will allow fair and uniform evaluations of newly proposed models and comparisons with currently available implementations.

SUMMARY The seismic waveform inversion problem is usually cast into the framework of Bayesian statistics in which prior information on the model parameters is combined with the data and physics of the forward problem to estimate the a posteriori probability density (PPD) in model space. The PPD is a function of an objective or fitness function computed from the observed and synthetic data. In general, the PPD or the fitness function is multimodal and its shape is unknown. Global optimization methods such as simulated annealing (SA) and genetic algorithms (GA) do not require that the shape of the fitness function be known. In this paper, we investigate GA to rapidly sample the most significant portion or portions of the PPD, when very little prior information is available. First, we use a simple three operator (selection, crossover and mutation) GA acting on a randomly chosen finite population of haploid binary coded models. We use plane wave transformed synthetic seismic data and a normalized cross-correlation function [E(m)] in the frequency domain as a fitness function. A moderate value of crossover probability, a low value of mutation probability, a high value of update probability and a proper population size are required to reach very close to the global maximum of the fitness function. Next, with an attempt to accelerate convergence we show that the concepts from simulated annealing can be used in stretching of the fitness function, i.e., we use exp [E(m)/T] rather than E(m) as the fitness function, where T is a control parameter analogous to temperature in simulated annealing. By a schemata analysis, we show that at low temperatures, schemata with above average fitness values are reproduced in large numbers causing a much more rapid convergence of the algorithm. A high value of temperature T assigns nearly equal selection probability to most of the schemata and thus retains diversity among the members of the population. Thus a GA with a step function type cooling schedule (very high temperature in the beginning followed by rapid cooling to a very low temperature) improves the performance dramatically: high values of the fitness function are obtained rapidly using only half as many models as would be required by a conventional GA. Similar performance could also be achieved by first using a high mutation probability and then decreasing the mutation probability to a very low value, while retaining the same low temperature throughout. We also address the problem of ‘genetic drift’ which causes the finite GAs to converge to one peak or the other when the algorithm is applied to a highly multimodal fitness function with several peaks of nearly the same height. A parallel genetic algorithm based on the concept of ‘punctuated equilibria’ is implemented to circumvent the problem. We run several GAs each with a finite subpopulation in parallel and collect many good models from each one of these runs. These are then used to grasp the most significant portion(s) of the PPD in model space. We then compute the weighted mean model and use the derived good models to estimate uncertainty in the derived model parameters.

Geophysical Inverse Theory and Regularization Problems

In this paper we proposed an accurate and efficient numerical algorithm for solving the acoustic wave equation in three-dimensional heterogeneous media. Numerical solution of the wave equation has been used in various science and engineering applications, such as the seismic full waveform inversion (FWI) problem. FWI is a computationally intensive procedure, in which the acoustic wave equation is numerically solved (forward modelling) repeatedly during the iterative process. Therefore, efficiency and accuracy of the numerical method for solving the acoustic wave equation is critical in the success of seismic full waveform inversion. The new method is obtained by combining the Pade approximation and a novel algebraic manipulation with the Alternative Directional Implicit (ADI) method. Numerical experiments have shown that the new method is accurate, efficient and stable.

The Schur algorithm is an inverse scattering algorithm which computes the reflectivity function using the relation between incident and reflected waves on a medium. The use of the inverse scattering algorithm is fast and stable due to orthogonality. We show results of a number of experiments.

Seismic inversion, a nonlinear, non-unique-solution, and human-prone error problem, seeks the optimal solution in multivariate space, a task that AI algorithms perform better. So, inversion with AI becomes appropriate to extract information from post-stack data. A probabilistic neural network (PNN) and deep feedforward neural network (DFNN) inverted seismic data using multiattributes, and their results were compared with those provided by a model-based inversion (MBI). Applied to a post-stack volume of the Frontier Formation at the Teapot-Dome field, Wyoming-USA, the three methods provided porosity volumes (ϕ), gamma rays (GR), and Lamé impedances (µρ, λρ). The comparison between well-logs and the provided volumes revealed a reliable PNN prediction, while DFNN achieved the best geologically correlated parameter sections and a less reliable MBI prediction. The geomorphological time-slice analysis discriminated high ϕ-µρ areas and low GR–λρ areas, geologically interpreted as a deltaic environment, with distributary channels and beachfront lobes subject to the action of the tides, which agrees with the well-log interpretation and the basin geological evolution. The results enhanced the stacked section’s importance by the geological information contained, which would contribute to reducing the prospecting risk. Finally, PNN and DFNN encouraging results reaffirm their best searching-in-multivariate space ability.

The physics that describes the seismic response of an interval of saturated porous rocks with known petrophysical properties is relatively well understood and includes rock physics, petrophysics, and wave propagation models. The main goal of seismic reservoir characterization is to predict the rock and fluid properties given a set of seismic measurements by combining geophysical models and mathematical methods. This modeling challenge is generally formulated as an inverse problem. The most common geophysical inverse problem is the seismic (or elastic) inversion, i.e., the estimation of elastic properties, such as seismic velocities or impedances, from seismic amplitudes and traveltimes. The estimation of petrophysical properties, such as porosity, lithology, and fluid saturations, also can be formulated as an inverse problem and is generally referred to as rock-physics (or petrophysical) inversion. Several deterministic and probabilistic methods can be applied to solve seismic inversion problems. Deterministic algorithms predict a single solution, which is a “best” estimate or the most likely value of the model variables of interest. In probabilistic algorithms, on the other hand, the solution is the probability distribution of the model variables of interest, which can be expressed as a conditional probability density function or a set of model realizations conditioned on the data. The probabilistic approach provides a quantification of the uncertainty of the solution in addition to the most likely model. Our goal is to define the terminology, present an overview of probabilistic seismic and rock-physics inversion methods for the estimation of petrophysical properties, demonstrate the fundamental concepts with illustrative examples, and discuss the recent research developments.

Continuum model of fractured media in direct and inverse seismic problems

Geostatistical seismic inversion is an important method for establishing high‐resolution reservoir parameter models. There is no accurate representation method for reservoir structural features, and prior information about structural features cannot be incorporated into geostatistical inversion. Based on the assumption of the sparsity of stratigraphic sedimentary features, the same type of structural feature is used to represent the sedimentary pattern of reservoirs within the same facies. Different sparse representation patterns are used to represent the differences in sedimentary patterns between facies. Although changes in depositional environment might result in the multi‐scale characteristics of geological structures for varying sedimentary rhythms, this paper proposes a facies‐constrained geostatistical inversion method based on multi‐scale sparse representation to better accommodate such situation. Using the method of sparse representation combined with wavelet transform, the multi‐scale sedimentary structural features of reservoirs are learned from well‐logging data. Seismic facies and multi‐scale features are used as prior information for geostatistical inversion. Further, the likelihood function is constructed using seismic data to obtain the posterior probability distribution of reservoir parameters. Finally, the accurate inversion result is obtained by using multi‐scale sparse representation as a constraint in the posterior probability distribution of reservoir parameters. Compared with conventional geostatistical methods, this algorithm can better match the structural features of reservoir parameters with varying geological conditions. Field data tests have shown the effectiveness of this method in improving the accuracy and resolution of reservoir parameter structural features.

Summary A framework for probabilistic full waveform inversion is considered, to enable the assessment of uncertainty related to the estimated parameters. Inspired by the success of applying ensemble based methods in the area of reservoir characterisation, we consider the seismic inversion problem as a sequential filtering problem and uses the Ensemble Kalman Filter (EnKF) framework to sequentially condition on observations. The EnKF has beneficial properties for high-dimensional parameters spaces compared to other sampling based methods. It is however also known to suffer from problems of ensemble collapse and spurious updates, due high-dimensional data space - the full waveform data - and its linear update form. Here is presented some techniques to alleviate these issues namely dimension reduction and Kalman gain localisation. An example using a simple layered model is provided to illustrate the proposed method of sequentially processing of seismic data and inverting for unknown parameters.

Waveform-based microseismic location using stochastic optimization algorithms: A parameter tuning workflow

Bayesian uncertainty analysis of SA turbulence model for supersonic jet interaction simulations