A data-driven amplitude variation with offset inversion method via learned dictionaries and sparse representation

Solutions to the amplitude variation with offset (AVO) inverse problem are inherently nonunique. Thus, estimating the range of potential solutions is crucial. For this reason, uncertainty inversion is more appropriate than deterministic AVO inversion. However, most studies on deep learning (DL)-based inversion focus on deterministic prediction. In general, there are few studies on uncertainty inversion, mainly focused on poststack seismic data. In this study, we propose a DL-based AVO uncertainty inversion method based on an improved variational Bayesian neural network (VBNN) for predicting multiple elastic parameters. Moreover, to mitigate the issue of insufficient labeled data in inversion tasks, we combine the improved VBNN with a semi-supervised learning framework. Synthetic data experiments demonstrate that the proposed method exhibits higher accuracy and robustness to noisy seismic data than the well-known traditional Bayesian linearized inversion (BLI) method. The proposed method also has slightly higher inversion accuracy than the state-of-the-art improved-hybrid-seismic-prior-guided neural network (IHGNN). Moreover, the uncertainty estimation confirms that the proposed method can explore potential solutions to the AVO inverse problem more effectively and reasonably than the comparison methods. Furthermore, field data AVO inverse experiments verify that the proposed method can obtain reasonable predicted results and effectively reveal uncertainty in the inversion results.

It has been more than three years since The Leading Edge last published a special section on amplitude variation with offset (AVO) inversion, and interest in the subject remains strong. This past spring, SEG hosted a joint symposium in Houston, Texas, on the “Resurgence of seismic inversion,” and the body of talks and case studies demonstrated the method's continued relevance to making impactful drilling decisions. Despite this, and despite AVO inversion's position as a mature and well-established technique, there are an abundance of examples in which inaccurate AVO predictions have led to drastic failures at the drill bit. This highlights the challenges that still exist in the successful execution of such investigations and makes the subject occasionally controversial and certainly fraught with data-quality and best-practice considerations. In this vein, the special section presented here offers examples of the broad sweep of considerations and methods relevant to enabling successful AVO inversion and the interpretation of its products, as well as case studies that demonstrate how application of the technique can be impactful all the way through to appraisal and field development programs.

AVO inversion based on generalized extreme value distribution with adaptive parameter estimation

An improved artificial immune algorithm is proposed for geophysical P-wave amplitude variation with offset (AVO) inversion. In this paper, the algorithm is described and implemented. The orthogonal crossover is used to generate the initial population and the elitist-crossover is adopted to add the good patterns of the population. The hybrid mutation method is presented to increase the ability of local and global optimization. The improved immune algorithm is then applied to earth interface models of Mexican gulf for AVO inversion. The experimental results show that the improved algorithm is of high precision than the traditional immune algorithm.

Most existing amplitude variation with offset (AVO) inversion methods are based on the Zoeppritz’s equation or its approximations. These methods assume that the amplitude of seismic data depends only on the reflection coefficients, which means that the wave-propagation effects, such as geometric spreading, attenuation, transmission loss, and multiples, have been fully corrected or attenuated before inversion. However, these requirements are very strict and can hardly be satisfied. Under a 1D assumption, reflectivity-method-based inversions are able to handle transmission losses and internal multiples. Applications of these inversions, however, are still time-consuming and complex in computation of differential seismograms. We have evaluated an inversion methodology based on the vectorized reflectivity method, in which the differential seismograms can be calculated from analytical expressions. It is computationally efficient. A modification is implemented to transform the inversion from the intercept time and ray-parameter domain to the angle-gather domain. AVO inversion is always an ill-posed problem. Following a Bayesian approach, the inversion is stabilized by including the correlation of the P-wave velocity, S-wave velocity, and density. Comparing reflectivity-method-based inversion with Zoeppritz-based inversion on a synthetic data and a real data set, we have concluded that reflectivity-method-based inversion is more accurate when the propagation effects of transmission losses and internal multiples are not corrected. Model testing has revealed that the method is robust at high noise levels.

Amplitude-variation-with-offset (AVO) inversion and neural networks (NN) are widely used to invert elastic parameters. With more constraints from well-log data, neural network-based inversion may estimate elastic parameters with greater precision and resolution than traditional AVO inversion; however, neural network approaches necessitate a massive number of reliable training samples. Furthermore, because the lack of low-frequency information in seismic gathers leads to multiple solutions of the inverse problem, both inversions rely heavily on proper low-frequency initial models. To mitigate the dependence of inversions on accurate training samples and initial models, we have adopted solving inverse problems with the recently developed invertible neural networks (INNs). Unlike conventional neural networks, which address the ambiguous inverse issues directly, INNs learn definite forward modeling and use additional latent variables to increase the uniqueness of solutions. Motivated by the newly developed neural networks, we adopt an INN-based AVO inversion method, which can reliably invert low- to medium-frequency velocities and densities with randomly generated easy-to-access data sets rather than trustworthy training samples or well-prepared initial models. Tests on synthetic and field data indicate that our method is feasible, antinoise capable, and practicable.

In 2008, WesternGeco acquired the 5000 sq/km Keystone multi-client survey located offshore between the giant Io/Jansz and Scarborough gas fields on Australia’s NW shelf, in order to image potential gas reservoir’s at multiple stratigraphic levels within complex geology. Searching for leads and prospects within such a large volume using seismic data alone is traditionally a time-consuming process. An efficient prospect screening tool has been produced in the form of a well log-calibrated gas sand probability cube derived from amplitude variation with offset (AVO) inversion outputs. Using advanced visualisation techniques the data can then be rapidly interrogated highlighting a number of direct hydrocarbon indicators (DHI's) and potential prospects. AVO and quantitative interpretation have played a key role in a number of discoveries in this area, particularly within the Triassic.

SUMMARY In this paper, we investigate the use of artificial neural networks (ANN) for amplitude variation with offset (AVO) analysis. Prospecting using AVO is dependent on the fact that the difference between near-offset and far-offset responses can be a hydrocarbon indicator. Neural networks are good at “learning” and “predicting” such relationships and can thus be trained to learn the relationship between, for example, a near-offset partially stacked trace and a far-offset partially stacked trace for background (non-hydrocarbon bearing rocks). Far traces can then be predicted by this learned relationship and the difference between observed and ANN predicted traces can potentially be used as a hydrocarbon indicator. An advantage of this method over conventional cross-plotting techniques is that it can be made insensitive to incorrect normal moveout corrections.

The estimation of petrophysical and fluid-filling properties of subsurface reservoirs from seismic data is a crucial component of reservoir characterization. Seismic amplitude-variation-with-offset (AVO) inversion driven by rock physics is an effective approach to characterize reservoir properties. In general, PP-wave reflection coefficients, elastic moduli, and petrophysical parameters are nonlinearly coupled, especially in multiple-type pore-space reservoirs, which makes seismic AVO petrophysics inversion ill-posed. We have developed a new approach that combines Biot-Gassmann’s poroelasticity theory with Russell’s linear AVO approximation, to estimate the reservoir properties including elastic moduli and petrophysical parameters based on multitrace probabilistic AVO inversion algorithm. We first derive a novel PP-wave reflection coefficient formulation in terms of porosity, stiff-pore volume fraction, rock-matrix shear modulus, and fluid bulk modulus to incorporate the effect of pore structures on elastic moduli by considering the soft and stiff pores with different aspect ratios in sandstone reservoirs. Through the analysis of the four types of PP-wave reflection coefficients, the approximation accuracy and inversion feasibility of the derived formulation are verified. Our stochastic inversion method aims to predict the posterior probability density function in a Bayesian setting according to a prior Laplace distribution with vertical correlation and prior Gaussian distribution with lateral correlation of model parameters. A Metropolis-Hastings stochastic sampling algorithm with multiple Markov chains is developed to simulate the posterior models of porosity, stiff-pore volume fraction, rock-matrix shear modulus, and fluid bulk modulus from seismic AVO gathers. The applicability and validity of our inversion method is illustrated with synthetic examples and a real data application.

Seismic deconvolution is a typical ill-posed inverse problem. The regularization technique in terms of different prior information is used for a unique and stable solution. Due to the difference between prior information and the actual subsurface situation, it is hard to obtain a solution with satisfactory accuracy and resolution. This letter presents the dictionary learning as an efficient adaptive deconvolution method for the reflectivity reconstruction problem. Considering the curse of dimensionality of conventional dictionary learning and the suboptimal solution of the patch-based dictionary learning, we take the convolutional sparse coding (CSC) model as the dictionary learning method. In this method, the prior information can be obtained from the well-log data in the form of sparse CSC dictionary of reflectivity. On the assumption that the deposition of the subsurface layers is stable, the CSC dictionary extracted from the well-log data can also be applied in the whole work area. The CSC-based deconvolution can be seen as the adaptive deconvolution due to the independence of the assumption made about the reflectivity and seismic data. The process of the adaptive CSC-based deconvolution is divided into three parts. First, the CSC dictionary is learned from the well-log data. Then, the objective function is formulated by combining the CSC dictionary and the single-channel seismic record misfit term for the reconstruction of reflectivity. Finally, the objective function is efficiently solved with the coordinate descent approach. We illustrate the performance of our adaptive deconvolution with synthetic and field seismic data.

Field records for small source-receiver offsets often contain intensive converted PS-waves that may be caused by the influence of anisotropy on either side of the reflector. Here, we study the small-angle reflection coefficients of the split converted [Formula: see text]- and [Formula: see text]-waves ([Formula: see text] and [Formula: see text]) for a horizontal interface separating two transversely isotropic (TI) media with arbitrary orientations of the symmetry axis. The normal-incidence reflection coefficients [Formula: see text] and [Formula: see text] vanish when both half-spaces have a horizontal symmetry plane, which happens if the symmetry axis is vertical or horizontal (i.e., if the medium is VTI or HTI). For a tilted symmetry axis in either medium, however, the magnitude of the reflection coefficients can reach substantial values that exceed 0.1, even if the anisotropy strength is moderate. To study the influence exerted by the orientation of the symmetry axis and the anisotropy parameters, we develop concise weak-contrast, weak-anisotropyapproximations for the PS-wave reflection coefficients and com-pare them with exact numerical results. In particular, the analytic solutions show that the contributions made by the Thomsen parameters [Formula: see text] and [Formula: see text] and the symmetry-axis tilt [Formula: see text] to the coefficients [Formula: see text] and [Formula: see text] can be expressed through the first derivative of the P-wave phase velocity at normal incidence. If the symmetry-axis orientation and anisotropy parameters do not change across the interface, the normal-incidence reflection coefficients are insignificant, regardless of the strength of the velocity and density contrast. The AVO (amplitude variation with offset) gradients of the PS-waves are influenced primarily by the anisotropy of the incidence medium that causes shear-wave splitting and determines the partitioning of energy between the [Formula: see text] and [Formula: see text] modes. Because of their substantial amplitude, small-angle PS reflections in TI media contain valuable information for anisotropic AVO inversion of multicomponent data. Our analytic solutions provide a foundation for linear AVO-inversion algorithms and can be used to guide nonlinear inversion that is based on the exact reflection coefficients.

This paper uses a Bayesian approach for inverting seismic amplitude versus offset (AVO) data to provide estimates and uncertainties of the viscoelastic physical parameters at an interface. The inversion is based on Gibbs' sampling approach to determine properties of the posterior probability distribution (PPD), such as the posterior mean, maximum a posteriori (MAP) estimate, marginal probability distributions, and covariances. The Bayesian formulation represents a fully nonlinear inversion; the results are compared to those of standard linearized inversion. The nonlinear and linearized approaches are applied to synthetic test cases which consider AVO inversion for shallow marine environments with both unconsolidated and consolidated seabeds. The result of neglecting attenuation in the seabed is investigated, and the effects of data factors such as independent and systematic errors and the range of incident angles are considered. The Bayesian approach is also applied to estimate the physical parameters and uncertainties from AVO data collected at two sites along a seismic line in the Baltic Sea with differing sediment types; it clearly identifies the distinct seabed compositions. Data uncertainties (independent and systematic) required for this analysis are estimated using a maximum‐likelihood approach.

We consider seismic amplitude variation with offset (AVO) inversion for prediction of the reservoir properties porosity and water saturation. An oil reservoir at the initial state is studied; hence, gravitational effects dominate and keep hydrocarbons from mixing with water. Histograms of observations of water saturation along wells are consequently clearly bimodal, which is challenging to model. The seismic AVO inversion is cast into a Bayesian framework. The prior spatial model for porosity and water saturation is specified to be a selection Gaussian random field (S-GRF), which is capable of representing spatial variables with multimodal histograms. By using linear models for the seismic and rock-physics likelihoods, the posterior model is also an S-GRF. Hence, the Bayesian seismic inversion can be solved analytically, and the bimodal characteristics of water saturation can be reproduced. The methodology is defined and demonstrated on two synthetic cases inspired by real data from an oil reservoir, and thereafter applied to the real case. The well observations are fairly accurately reproduced and the inversion results are considered to be substantial improvements compared to standard spatial Gaussian models.

The conventional amplitude-variation-with-offset (AVO) inversion method based on the amplitude attribute of prestack gathers calculates the elastic parameters of underground media through the variation of amplitude with offset. However, when the underground medium is a thin interlayer, the tuning effect will occur, which is the aliasing phenomenon of amplitude at the reflection interface. The tuning effect makes the conventional AVO inversion method based on amplitude attributes difficult to solve the problem of thin interlayer recognition. In addition, the same reflection interface will have different AVO characteristics at different frequencies, whereas the frequency factor is not included in conventional AVO inversion methods. Two-stage neural network approaches based on deep learning are combined to improve the resolution of thin interlayers and to accurately invert the elastic parameters. For the first-stage neural network, a fully connected network is used to solve the inversion spectral decomposition problem. It can eliminate the thin interlayer tuning effect, effectively improve the resolution, and obtain reflection coefficients at different frequencies. For the second-stage neural network, a multichannel convolutional neural network is used to establish the mapping relationship between multifrequency reflection coefficients and elastic parameters, so that the multifrequency joint inversion of the elastic parameters could be realized. This procedure is applied to synthetic data (with and without noise) to indicate the resistance to noise interference of the two-stage deep-learning method. Compared with the method of directly predicting elastic parameters using seismic data and the conventional AVO inversion method, the two-stage deep-learning method can describe the elastic parameters of thin interbeds more accurately. The same procedure is applied to the field data, and the inversion results indicate that they can well match with the well-logging data. Hence, it is promising for practical applications.

Deep‐learning has achieved good performance and demonstrated great potential for solving forward and inverse problems. In this work, two categories of innovative deep‐learning‐based inverse modeling methods are proposed and compared. The first category is deep‐learning surrogate‐based inversion methods, in which the Theory‐guided Neural Network (TgNN) is constructed as a deep‐learning surrogate for problems with uncertain model parameters. By incorporating physical laws and other constraints, the TgNN surrogate can be constructed with limited simulation runs and accelerate the inversion process significantly. Three TgNN surrogate‐based inversion methods are proposed, including the gradient method, the Iterative Ensemble Smoother method, and the training method. The second category is direct‐deep‐learning‐inversion methods, in which TgNN constrained with geostatistical information, named TgNN‐geo, is proposed as the deep‐learning framework for direct inverse modeling. In TgNN‐geo, two neural networks are introduced to approximate the random model parameters and the solution, respectively. In order to honor prior geostatistical information of the random model parameters, the neural network for approximating the random model parameters is first trained by using observed or generated realizations. Then, by minimizing the loss function of TgNN‐geo, the estimation of model parameters and the approximation of the model solution can be simultaneously obtained. Since the prior geostatistical information can be incorporated, the direct‐inversion method based on TgNN‐geo works well, even in cases with sparse spatial measurements or imprecise prior statistics. Although the proposed deep‐learning‐based inverse modeling methods are general in nature, and thus applicable to a wide variety of problems, they are tested with several subsurface flow problems. It is found that satisfactory results are obtained with high efficiency. Moreover, both the advantages and disadvantages are further analyzed for the proposed two categories of deep‐learning‐based inversion methods.