Amplitude Variation With Offset Research Articles

Prediction of organic richness in shale gas reservoirs using a novel organic-inorganic decoupling seismic inversion method

Organic richness is associated with total hydrocarbon content and influences gas generation in shale reservoirs. The prediction of organic enrichment using seismic methods provides valuable information for identifying promising shale gas areas. Current seismic methods typically estimate organic richness using empirical relationships between organic matter (OM) content and elastic properties; these may be suitable for specific areas but cannot be generalized for universal applications. Therefore, developing a physics-based method for predicting OM content in shale gas reservoirs is critical. In this study, we proposed a novel organic-inorganic decoupling amplitude variation with offset (AVO) method by integrating the classical AVO equation with solid substitution theory. OM was considered solid pore-fillings saturated in the shale matrix. The proposed decoupling equation was extended to establish a frequency-dependent inversion framework for computing OM-related dispersion attributes, allowing for direct estimations of organic richness. Theoretical analyses suggested that the accuracy of the decoupling equation could be achieved via adjustable parameters determined using the rock physics method. Model tests demonstrated that the OM-related dispersion attributes were more sensitive to variations in OM content than the traditional P-wave velocity dispersion attribute and the commonly used elastic properties. The results of real data applications validated the superiority of our proposed OM-related dispersion attributes for reliable organic richness estimation, with the estimated results agreeing well with gas production in the study area. As a result, several promising shale gas zones were identified for further exploration. Our proposed method extends traditional fluid factors and can deal with the cases of both fluid and solid saturations in hydrocarbon resources, with widespread applications for hydrocarbon identification in various unconventional reservoirs.

AVO Inversion Based on Closed-Loop Multitask Conditional Wasserstein Generative Adversarial Network

Neural networks are commonly used for post-stack and pre-stack seismic inversion. With sufficient labelled data, the neural network-based seismic inversion results are more accurate than that use traditional seismic inversion methods. However, in the case of insufficient labeled data, the accuracy of neural networks-based seismic inversion results decreases and is even lower than those based on traditional inversion methods. In addition, the seismic inversion results based on neural networks generally suffer from lateral discontinuity. It further reduces the accuracy of the inversion results. To tackle these problems, we propose a pre-stack seismic amplitude variation with offset (AVO) inversion method based on Closed-Loop Multi-task conditional Wasserstein Generative Adversarial Network (CMcWGAN), which is a GAN-based AVO inversion method. CMcWGAN enables simultaneous and accurate inversion of P-wave velocity ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V_p</i> ), S-wave velocity ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V_s</i> ), and density ( ρ ). Moreover, it uses the low-frequency information of elastic parameters as a conditional input to alleviate the problem of lateral discontinuity in inversion results. Experimental results of simulated data show that the inversion results based on CMcWGAN have higher accuracy than those based on traditional AVO inversion methods. In addition, when the seismic angle gather is noisy, CMcWGAN has better robustness than the traditional methods. CMcWGAN can also obtain reasonable AVO inversion results in field seismic angle gather data.inversion results. Experimental results of simulated data show that the inversion results based on CMcWGAN have higher accuracy than those based on traditional AVO inversion method. In addition, when the seismic angle gather is noisy, CMcWGAN has better robustness than traditional method. CMcWGAN can also get reasonable AVO inversion results in field seismic angle gather data.

Direct Exact Nonlinear Broadband Seismic Amplitude Variations With Offset Inversion for Young’s Modulus

Young’s modulus and the Poisson ratio are critical for shale reservoir identification, and oil and natural gas detection since they are elastic parameters that can reflect the fracturing properties of subsurface rocks. The broadband inversion approach can make full use of the seismic ultralow-frequency information by combining it with the inversion results in the complex frequency domain. This plays an important role in the stability and reliability of the inversion, improving its resolution. Nevertheless, at present, the amplitude variation with offset (AVO) inversions of these parameters is mostly in the form of linear approximations, and there are few inversion methods of exact nonlinear equations or broadband. Considering that the nonlinear equation of reflection coefficient compared with the linear approximation has higher precision and fewer assumption conditions, and the low-frequency information of the complex frequency-domain inversion can improve the reliability of the inversion results, we derive an exact reflection coefficient equation for Young’s modulus, the Poisson ratio coefficient, and the density based on the exact Zoeppritz equation and apply it to the broadband complex domain nonlinear prestack inversion. Young’s modulus and the Poisson ratio coefficients estimated by the new inversion method provide a theoretical basis for evaluating the reservoir brittleness and compressibility, as well as a new approach for shale gas reservoir prediction and sweet spot identification. We tested the accuracy and rationality of this method with both synthetic and field data examples.

A Linearized Alternating Direction Method of Multipliers Algorithm for Prestack Seismic Inversion in VTI Media Using the Quadratic PP-Reflectivity Approximation

Periodic horizontal thin interlayers, or formations developing horizontally layered fractures in a homogeneous background, are commonly studied as a transverse isotropy with a vertical axis of symmetry (VTI) media. Anisotropy needs to be considered in predicting such special reservoirs, and can be inferred from the amplitude variation with offset (AVO) by the exact or linear reflectivity equations. However, the linear approximation is usually derived based on the assumptions of weak anisotropy and weak contrast, the accuracy suffers when the anisotropy or the contrast of two adjacent media increases. In addition, due to the high nonlinearity of the Graebner equation, the stability and efficiency of the exact inversion are limited. To address these issues, we propose a robust pre-stack seismic inversion for VTI media using quadratic PP-reflectivity approximation. Firstly, by deriving quadratic approximations for phase velocity and polarization, we analyze the advantage of the second-order terms at a higher-contrast interface. Then, based on perturbation theory, we derive a quadratic PP-reflectivity approximation from the exact equation. It consists of first- and second-order terms with isotropic and anisotropic components, and the accuracy is significantly improved even with enhanced anisotropy. Finally, we introduce the Hadamard product operator to linearly characterize the objective function, and develop an improved Alternating Direction Method of Multipliers (ADMM) algorithm to handle the nonlinearity of the quadratic equations. The feasibility and stability of the proposed method are tested on two datasets, that is, the logging curves used as numerical experiments and a field dataset from a shale reservoir.

Frequency-Dependent Nonlinear AVO Inversion for Q-Factors in Viscoelastic Media

Viscoelastic theory-based frequency-dependent amplitude variation with offset (AVO) inversions are a useful tool for identifying fluids based on their dispersion properties. When used to identify reservoir oil and gas qualities, the quality factors ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factors) derived from the inversion have produced good results. But currently, the majority of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factor inversions are linear inversions based on linear approximations, whereas nonlinear inversions based on exact equations with higher precision and fewer assumptions are only occasionally carried out. Meanwhile, in broadband seismic inversion, the low-frequency model estimated by complex frequency inversion can fully utilize seismic data’s low-frequency information and provide improved robustness and rationality for inversion. Therefore, we derive a frequency-dependent reflection coefficient equation that varies with angle by replacing the nearly constant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> model suggested by Aki and Richards into the exact Zoeppritz equation and simplifying. Additionally, using the Bayesian framework as a foundation, we created a novel two-stage broadband frequency-dependent nonlinear inversion approach for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factors that can estimate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factors, produce accurate fluid identification results, and serve as a solid foundation for oil and gas exploration and reservoir identification. Utilizing examples from both synthetic and real-world data, we verify the applicability of oil and gas indicators.

An Exact Zoeppritz Based Prestack Inversion Using Whale Optimization Particle Filter Algorithm Under Bayesian Framework

Conventional AVO (Amplitude Versus Offset) inversion methods are mainly based on various Zoeppritz approximations. The assumptions of small contrast and linear relationship lead the most inversion methods being difficult to have high inversion accuracy. In this paper, the exact Zoeppritz equation is used to establish the prestack inversion method under the Bayesian framework. It integrates multi-source information to generate posterior distributions of P-, S-wave velocity and density. In the Bayesian theory, the prior model works as the regularization term which has a strong effect on the inversion results. The strategy to obtain a relatively accurate prior model can improve the inversion accuracy. Therefore, an exact Zoeppritz equation based nonlinear AVO inversion algorithm combing whale optimization particle filtering (WOPF) is proposed. The WOPF method can generate a relatively stable and accurate initial model for the Bayesian prestack inversion. We validate the new method through two synthetic models and field data. Comparisons are made with the conventional linear and nonlinear AVO inversion methods. The results show that the proposed method can provide much more accurate inverted elastic parameters in different geological conditions.

Smart Seismic Modeling - Artificial Intelligence in the Petroleum Industry

Vertical seismic profile (VSP)&nbsp;data have many known advantages, if being compared to seismic data such as broader bandwidth, less attenuation, clearer under well true depth (TD)&nbsp;image, multiple noise differentiation, shear wave information, direct attenuation and anisotropy measurements, higher fidelity amplitude versus offset (AVO)&nbsp;effect, etc. In this paper, we&nbsp;share the bottom-up approach that aims to maximize the information extracting from borehole seismic, in conjunction with seismic and well log data, to generate a geological model by integrating full wave-field modeling (FWM) to deep learning techniques, called smart seismic modeling (SSM) project. The study starts using data at well locations, with&nbsp;vertical seismic profile&nbsp;(VSP)&nbsp;forward modeling process. The 2D rock physics model is thus built from the prior well logs and the major interpreted seismic horizons and faults. By altering rock-physics properties values and thickness of each layer, the model has been multiplied thousand times. The rock-physics models and their simulated seismic traces are fed, as labels and inputs,&nbsp;respectively, into a 2D deep learning network. The network then extracts the non-linear relationships between inputs and labels so that thickness and rock-physics properties can simultaneously be regressed with least squared error loss function. Optimal trained weight set (artificial intelligence&nbsp;engine) is finally used to predict wellbore’s surrounding geologic structures and rock properties for all the wells in the study area. To make sure machine learning can provide a good work, similar training and prediction process is carried out from surface seismic forward modeling. Not only is it used for rock property prediction at the area apart from wells, but also for cross checking the prediction at well location. All 2D predicted rock-property models are merged by multi-point statistics (MPS) method using hard data from wells’ prediction and trends from actual VSP processing,&nbsp;such as anisotropy, 2D Qp, Qs, for a complete 3D geologic model. By doing with two&nbsp;different modeling processes, i.e., VSP forward modeling and surface seismic forward to cross check the results, results can obtain from this study are:&nbsp;1)high quality static model for field development; 2)high quality dynamic model for field monitoring; 3)robust artificial intelligence&nbsp;engines for variety types of geology. In this study, we&nbsp;introduce the SMM approach and share promising successes in applying deep learning in rock property inversion process. The successful showcase&nbsp;has proved that deep learning technique can be used to extract any kind of rock properties from the big data of VSP, well log and of seismic.

Using 3-D seismic data to estimate stress based on the curvature attribute integrated mechanical earth model

SUMMARY The current state of the technology for unconventional gas and oil development integrates geology, geophysics and engineering into a comprehensive reservoir description. To construct a mechanical earth model (MEM) which takes the geological structure influence into account, we introduce the curvature attribute to express the horizontal strains and extend the classical uniform horizontal strains MEM. The horizontal strains are derived based on the theory of plates and depend on the curvature attributes which reflect the geological structure influence. To take seismic data into the unconventional reservoir characterization, we design a geophysical workflow to estimate the in-situ stress. Amplitude Versus Offset (AVO) inversion produces the elastic parameters and the finite-difference method calculates the curvature attributes. The Bowers’ method computes the pore pressure and the in-situ stresses computed from the well-log data are used for the calibration. Real data from the shale gas pilot development areas in the Zhaotong National Demonstration Zone are used to demonstrate the method and compare the new MEM results with the MEM modelling technique results. The curvature attribute integrated MEM shows its advantage.

First-order approximate analytical expressions of oblique incident elastic wave at an interface of porous media saturated with a non-viscous fluid

The analysis technology of Amplitude Variation with Offset (AVO) is one of the important methods for oil and gas reservoir prediction. Zoeppritz equation and its approximations are the theoretical basis of AVO analysis, which assumes that the upper and lower media of a horizontal interface are single-phase media. Limited by this assumption, AVO analysis has limited prediction and identification accuracy for complex porous reservoirs. In view of this, the first-order approximate analytical expressions of oblique elastic wave at an interface of porous media are derived. Firstly, the incident and scattering characteristics of various waves at the interface of porous media are analyzed, and the displacement vectors generated by these elastic waves are described by exponential function. Secondly, the kinematic and dynamic boundary conditions at the interface of porous media are discussed. Thirdly, by substituting the displacement vectors of incident and scattered waves into boundary conditions, the exact analytical equation is derived. Then, considering the symmetry of scattering matrix in the equation, the exact analytical expressions of each scattered wave are obtained. Furthermore, under the assumptions of small incident angle, weak elasticity at an interface of porous media, and ignoring the second- and higher-order terms, the first-order approximate analytical expressions are derived. Establishing a model of sandstone porous media with different porosity in upper and lower media, the correctness of the approximate analytical expressions is verified, and the elastic wave response characteristics of lithology and pore fluids are analyzed.

Seismic inversion and fracture prediction in tilted transversely isotropic media

Abstract The conventional Amplitude variation with offset (AVO) inversion has been mainly developed for the isotropic media and therefore it is generally inapplicable to the anisotropic fractured formations. A set of tilted fractures in isotropic medium can be regarded as a transversely isotropic (TTI) medium. The reflection coefficient equation in TTI media contains many parameters such as anisotropic parameters, velocities, azimuth and dip angles. The selection of objective functions can significantly affect its performance in searching for the optimal solution. The seismic inversion of the TTI medium remains challenging because of its many parameters, and is very complicated after conversion into solving function problems. The genetic algorithm (GA) provides a universal framework to solve the optimization problem in complex systems, and it is independent of the types and disciplines of problems. In this study, the GA-based seismic prestack inversion of the TTI medium is implemented by constructing the objective functions of anisotropic parameters based on the TTI medium reflection coefficient approximation. The inversion performance is shown to be satisfactory on the one-dimensional well logging model, modified Hess model and the field datum. Furthermore, fracture filling materials are well predicted according to the theories of the equivalent medium for fractured materials and fracture weakness. The inversion results demonstrate the proposed method to be feasible and robust. The findings of this research are expected to provide theoretical guidance for the inversion of elastic parameters of anisotropic media and fracture type identification in hydrocarbon reservoirs.

Frequency-dependent AVO inversion applied to physically based models for seismic attenuation

SUMMARYSeismic inversion of amplitude versus offset (AVO) data in viscoelastic media can potentially provide high-resolution subsurface models of seismic velocities and attenuation from offset/angle seismic gathers. P- and S-wave quality factors (Q), whose inverse represent a measure of attenuation, depend on reservoir rock and pore fluid properties, in particular, saturation, permeability, porosity, fluid viscosity and lithology; however, these quality factors are rarely taken into account in seismic AVO inversion. For this reason, in this work, we aim to integrate quality factors derived from physically based models in AVO inversion by proposing a gradient descent optimization-based inversion technique to predict the unknown model properties (P- and S-wave velocities, the related quality factors and density). The proposed inversion minimizes the non-linear least-squares misfit with the observed data. The optimal solution is iteratively obtained by optimizing the data misfit using a second-order limited-memory quasi-Newton technique. The forward model is performed in the frequency–frequency-angle domain based on a convolution of broad-band signals and a linearized viscoelastic frequency-dependent AVO (FAVO) equation. The optimization includes the adjoint-state-based gradients with the Lagrangian formulation to improve the efficiency of the non-linear seismic FAVO inversion process. The inversion is tested on synthetic seismic data, in 1-D and 2-D, with and without noise. The sensitivity for seismic quality factors is evaluated using various rock physics models for seismic attenuation and quality factors. The results demonstrate that the proposed inversion method reliably retrieves the unknown elastic and an-elastic properties with good convergence and accuracy. The stability of the inverse solution especially seismic quality factors estimation relies on the noise level of the seismic data. We further investigate the uncertainty of the solution as a function of the variability of the initial models.

Three-term AVO inversion using group total variation regularization

The seismic amplitude-versus-offset (AVO) inversion based on the total variation (TV) regularization is used to estimate blocky solutions for elastic parameters. It is achieved by minimizing the l1-norm of the first-order difference for model parameters. As the parameters in the subsurface are highly correlated with each other, a group optimization strategy was applied to the TV regularized AVO inversion. A group TV-norm that is a composition of the TV-norm and the l2-norm was designed. In this way, the TV regularization is performed among groups to promote sparsity of the gradients for the model parameters. By contrast, l2 regularization is performed within each group to constrain uniformity. The new three-term AVO inversion algorithm based on group TV regularization helps estimate the elastic parameters with desired blocky structure. Also, this algorithm ensures correlated resulting solutions even when no covariance matrix is used. It has proved to obtain good results in numerical and field examples.

High-Resolution Seismic Characterization of Gas Hydrate Reservoir Using Wave-Equation-Based Inversion

The high-resolution seismic characterization of gas hydrate reservoirs plays an important role in the detection and exploration of gas hydrate. The conventional AVO (amplitude variation with offset) method is based on a linearized Zoeppritz equation and utilizes only the reflected wave for inversion. This reduces the accuracy and resolution of the inversion properties and results in incorrect reservoir interpretation. We have studied a high-resolution wave-equation-based inversion method for gas hydrate reservoirs. The inversion depends on the scattering integral wave equation that describes a nonlinear relationship between the seismic wavefield and the elastic properties of the subsurface medium. In addition to the reflected wave, it considers more wavefields including the multiple scattering and transmission during inversion to improve the subsurface illumination, so as to enhance the accuracy and resolution of the inversion properties. The results of synthetic data from Pearl River Mouth Basin, South China Sea, demonstrate the validity and advantages of the wave-equation-based inversion method. It can effectively improve the resolution of inversion results compared to the conventional AVO method. In addition, it has good performance in the presence of noise, which makes it a promising method for field data.

Characterizing Gas Hydrate-Bearing Marine Sediments Using Elastic Properties—Part 2: Seismic Inversion Based on a Pore-Filling–Solid Matrix Decoupling Scheme

Characterizing gas hydrate-bearing marine sediments using seismic methods is essential for locating potential hydrate resources. However, most existing pre-stack seismic inversion methods estimate the properties of sediments containing gas hydrates without considering specific characteristics associated with gas hydrate occurrences. In the present study, a pore-filling–solid matrix decoupling amplitude variation with offset (AVO) formula is proposed to represent seismic reflectivity in terms of properties associated with gas hydrates. Based on the rock physics relationships of solid substitution, the parameters introduced into the decoupling AVO equation estimate the concentration of gas hydrates with different occurrences, including pore fillings mixed with water and solid components forming part of the dry sediment frame. A theoretical model test indicates that seismic attributes obtained with the decoupling AVO inversion are superior to the conventional wave velocities-related properties in predicting gas hydrate saturations. A realistic model test further validates the applicability of the proposed method in characterizing a gas hydrate system with varying concentrations and layer thickness. By adjusting the tuning parameters, the configurations and concentrations of the gas hydrate system can be identified using the obtained attributes. Therefore, the presented method provides a useful tool for the characterization of gas hydrate-bearing sediments.

Elastic moduli parameterization and inversion considering nanopores effects in shale

Nanopores are widely developed in organic kerogen in shale. Due to the large surface-bulk ratio, nanopores cause significant surface effects that affect the overall elastic properties of shale. It is essential to consider the effects of nanopores when applying the prestack seismic inversion technique for shale reservoir prediction. Based on classical elastic theory, the technique of amplitude variation with offset (AVO) can usually extract velocity of the longitudinal wave and shear wave, elastic modulus, and density in shale reservoir from prestack seismic amplitudes but often ignores the surface effects of nanopores. For this reason, we have derived a new AVO parameterization combining the nanoporoelasticity theory and AVO technique. The newly derived parameterization contains four parameters (shear modulus of matrix, nanopores-related parameter, shear modulus of saturated rock, and density) which can be used to estimate the nanopore properties of shale. The model test and real seismic data examples verify the feasibility and suitability of the proposed method by applying the Bayesian inversion technique with smooth background constraint.

Model-data-driven seismic inversion method based on small sample data

As sandstone layers in thin interbedded section are difficult to identify, conventional model-driven seismic inversion and data-driven seismic prediction methods have low precision in predicting them. To solve this problem, a model-data-driven seismic AVO (amplitude variation with offset) inversion method based on a space-variant objective function has been worked out. In this method, zero delay cross-correlation function and F norm are used to establish objective function. Based on inverse distance weighting theory, change of the objective function is controlled according to the location of the target CDP (common depth point), to change the constraint weights of training samples, initial low-frequency models, and seismic data on the inversion. Hence, the proposed method can get high resolution and high-accuracy velocity and density from inversion of small sample data, and is suitable for identifying thin interbedded sand bodies. Tests with thin interbedded geological models show that the proposed method has high inversion accuracy and resolution for small sample data, and can identify sandstone and mudstone layers of about one-30th of the dominant wavelength thick. Tests on the field data of Lishui sag show that the inversion results of the proposed method have small relative error with well-log data, and can identify thin interbedded sandstone layers of about one-15th of the dominant wavelength thick with small sample data.

Integrated machine learning and physics-based workflows for rapid qualitative and quantitative insights on monitoring carbon capture and sequestration

With the onset of extensive development and focus on clean and sustainable sources of energy, our subsurface geoscience tools and technologies can be effectively repurposed and evolved to support the energy transition. These sustainable energy sources are easily replenishable and low carbon. Some of the emerging sectors are geothermal energy, carbon capture and sequestration (CCS), energy storage, offshore wind energy, sustainable battery-grade lithium extraction, and hydrogen production. These sectors can benefit immensely from economically viable geophysical tools that are available at our disposal to deliver better and faster business outcomes. In this paper we will discuss the integrated machine learning (ML) and physics-based workflows that can be used for enabling effective and rapid energy transition solutions. In our study we will focus on CCS subsurface monitoring workflows, but the same methods have been applied to many other energy transition areas (for example, offshore wind and geothermal energy). The integrated workflows mentioned above were applied on the Sleipner gas field in the North Sea, where CCS operations have been ongoing since 1996. Rapid qualitative workflows like amplitude analysis and seismic blending were used to build an initial understanding of the carbon dioxide (CO2) plume and its spread. Geoscientist-driven ML-assisted horizon mapping facilitated rapid interpretation of structure across different time vintages. Then, advanced techniques like amplitude vs. offset (AVO) analysis and time-lapse seismic inversion were used for quantitative analysis. These analyses gave comprehensive information on CO2 spread in each subformation, as well as an estimate of the changing CO2 volume through time. These workflows can be applied in CCS operations to successfully monitor CO2 in the subsurface and quickly detect and assess the risk of leaks.

Petrophysical inversion based on f-s-r amplitude-variation-with-offset linearization and canonical correlation analysis

The prediction of petrophysical properties, such as porosity and rock-fluid volumes, from partially stacked seismic data typically requires a rock-physics model that often is lithology dependent and difficult to calibrate. We adopt canonical correlation analysis (CCA) to infer the underlying relation between petrophysical properties and elastic attributes estimated from seismic data. We develop a two-step inversion approach: first, we predict elastic properties from partially stacked seismic data using a Bayesian linear inverse method based on an amplitude-variation-with-offset (AVO) linearization in terms of fluid, rigidity, and density factors, and then we predict petrophysical properties from the estimated AVO attributes using CCA. The novelty of our approach is the application of CCA to the fluid and rigidity factors, which avoids the calibration of an explicit rock-physics model by automatically deriving a linear relation in the lower dimensional space of the canonical variables. The parameterization of the linearization in terms of fluid, rigidity, and density factors maximizes the correlation with respect to the petrophysical properties of interest. Furthermore, the probabilistic approach is extended to the petrophysical inversion using Bayesian linear theory and the posterior distribution of petrophysical properties conditioned by seismic data is computed by combining the probability distributions obtained from seismic and petrophysical inversion to propagate the uncertainty from the seismic to the petrophysical domain. The inversion is validated on a synthetic case that finds high accuracy of our formulation. A case study with synthetic and real partially stacked seismic data also is presented and compared to a traditional inversion with an explicit rock-physics model.

Amplitude variation with incidence angle inversion of fluid parameters in porous media using an accurate estimation of the Jacobian matrix

Amplitude variation with incidence angle (AVA) or amplitude variation with offset (AVO) inversion provides remarkable information in discriminating lithology in hydrocarbon reservoirs. Fluid parameters (e.g., porosity, water, and oil saturations) are normally obtained through two steps and difficult to obtain from seismic data directly and accurately. Theoretically, the exact Zoeppritz equations and Biot-Gassmann equations can be combined together to invert fluid parameters using seismic data directly. We carry out AVA inversion to invert porosity, water, and oil saturations from seismic data using the exact Zoeppritz equations and accurate estimation of the Jacobian matrix. This matrix is composed of the partial derivatives of P- and S-wave reflection coefficients with respect to fluid parameters. Characteristics of these partial derivatives are analyzed. In addition, the iterative equations for AVA inversion are derived based on the Taylor series expansion. The numerical tests indicate that this algorithm can invert fluid parameters directly and accurately using seismic gathers. Unlike the conventional AVA using the approximations, the inversion using the exact Zoeppritz equations and accurate estimation of the Jacobian matrix is not limited by the selection of incidence angles and reflection interfaces. This method also is applicable to seismic data collected from formations with strong elastic contrast interfaces or large incidence angles.

Sparse Radon transform in the mixed frequency-time domain with ℓ1-2 minimization

Due to the finite acquisition aperture and sampling of seismic data, the Radon transform (RT) suffers from a smearing problem which reduces the resolution of the estimated model. In addition, inverting the RT is typically an ill-posed problem. To address these challenges, a sparse RT mixing the [Formula: see text] and [Formula: see text] norms of the RT coefficients in the mixed frequency-time domain is developed, and it is denoted as SRTL1-2. In most conventional sparse RTs, the sparse constraint term often is the [Formula: see text] norm of the Radon model. We prove that the sparsity effect of the [Formula: see text] minimization is better than that of the [Formula: see text] norm alone by comparing and analyzing their 2D distribution patterns and threshold functions. The difference of the convex functions algorithm and the alternating direction method of multipliers algorithm are modified by combining the forward and inverse Fourier transforms to solve the corresponding sparse inverse problem in the mixed frequency-time domain. Our method is compared with three RT methods, including a least-squares RT (LSRT), a frequency-domain sparse RT (FSRT), and a time-invariant RT in the mixed frequency-time domain based on an iterative 2D model shrinkage method (SRTIS). Furthermore, we modify the basis function in SRTL1-2 by including an orthogonal polynomial transform to fit the amplitude-variation-with-offset (AVO) signatures found in seismic data, and we denote this as high-order SRTL1-2. Compared to the SRTL1-2, the high-order SRTL1-2 performs better when processing seismic data with AVO signatures. Synthetic and real data examples indicate that our method has better performance than the LSRT, FSRT, and SRTIS in terms of attenuation of multiples, noise mitigation, and computational efficiency.