Accurate characterization of fracture attributes and in-situ stress fields is critical for optimizing horizontal well placement and hydraulic fracturing strategies in fractured shale gas reservoirs. This study introduces a novel model-constrained damped least-squares amplitude variation with angle of incidence and azimuth (AVAZ) inversion method to predict pore pressure and horizontal stresses. By integrating generalized Hooke’s law with Schoenberg’s linear-slip model, horizontal stress equations are derived for horizontally transversely isotropic (HTI) media. Pore pressure estimation is established through a vertical stress sensitivity parameter linked to porosity via a critical porosity model. A saturated stiffness matrix incorporating this parameter and fracture weakness parameters is developed. The P-wave reflection coefficient is formulated using a coupled scattering function and perturbed stiffness matrix, integrating five key parameters: fluid bulk modulus, porosity-dependent vertical stress sensitivity, rock P-wave modulus, fluid modulus-density product, and fracture density. A multi-parameter inversion algorithm is proposed to simultaneously estimate pore pressure, maximum, and minimum horizontal stresses. Validation with field data demonstrates that this method provides reliable estimates of pore pressure and horizontal stresses.

Deep learning techniques have seen widespread application in seismic inversion, yet they face significant challenges when applied to fracture property inversion. The limited availability of labeled data and the lack of robust geophysical constraints can severely impact the accuracy of predictions. To address these challenges, we propose a model-data dual-driven network system. To mitigate data scarcity, we introduce a semi-supervised learning framework enhanced by data augmentation techniques. This framework leverages unlabeled data to enhance the diversity of the training dataset, thereby improving the robustness of the learning process. Additionally, we embed initial model constraints into our inversion network, ensuring that the inversion process is guided by geophysically plausible starting points. Drawing on the principles of a horizontally transversely isotropic (HTI) medium, we develop a forward model that establishes a clear relationship between fracture properties and azimuth-dependent seismic data. This model is implemented through convolutional operations between the azimuthal PP-wave reflection coefficient equation and seismic wavelets, effectively linking fracture attributes to observable seismic responses. The integration of these components results in a model-data dual-driven network capable of performing fracture property inversion with high accuracy. To validate the effectiveness and superiority of our proposed method, we apply it to both synthetic overthrust model and real datasets, and the results demonstrate that the predicted fracture weaknesses values achieve high correlation coefficients of 0.97 and 0.84 with the ground truth, respectively. Furthermore, the training time is significantly reduced compared to conventional data-driven methods, effectively improving both inversion accuracy and computational efficiency.

Accurate source localization and source mechanism estimation in microseismic monitoring critically depend on precise subsurface velocity model reconstruction. Recent advances in data-driven approaches, such as neural network models, have demonstrated promising performance in seismic velocity inversion. However, purely data-driven methods are significantly constrained by their reliance on the completeness of the training dataset. Additionally, the performance of current neural network architectures remains suboptimal and requires further improvement. We propose a step-by-step training methodology that integrates the wave equation-based modeling with a modified U-shaped neural network (U-Net). This approach incorporates physical laws of wave propagation to constrain the updating process of the velocity model. The proposed method carries out the purely data-driven procedure first and then incorporates waveform modeling with finite-difference simulations. Besides, the new method defines dynamic weights within the loss function and introduces the Akaike information criterion (AIC) to detect and extract the P and S phases, which reduces the computational expenses as well as alleviates the artifacts in velocity models effectively. We conducted experiments with synthetic fault models and realistic layered models, and the results show that the wave equation-constrained method recovers the velocity model better than the purely data-driven algorithm. The approach leveraging physics-based and data-driven models can be easily adapted to other geophysical inversion tasks.

Abstract Depth estimation from monocular panoramic image is a crucial step in 3D reconstruction, which is a close relationship with virtual reality and metaverse technologies. In recent years, some methods, such as HRDFuse, BiFuse++, and UniFuse, have employed a two‐branch neural network leveraging two common projections: equirectangular and cubemap projections (CMPs). The equirectangular projection (ERP) provides a complete field of view but introduces distortion, while the CMP avoids distortion but introduces discontinuity at the boundary of the cube. In order to address the issue of distortion and discontinuity, the authors propose an efficient depth estimation fusion module to balance the feature mapping of the two projections. Moreover, for the ERP, the authors propose a novel inflated network architecture to extend the receptive field and effectively harness visual information. Extensive experiments show that the authors’ method predicts more clear boundaries and accurate depth results while outperforming mainstream panoramic depth estimation algorithms.

Fracture weaknesses represent two critical elastic parameters utilized for characterizing fracture properties in naturally fractured reservoirs. Given the intricate seismic attributes associated with amplitude variation with angles of incidence and azimuth (AVAZ) in fractured reservoir, accurately delineating the mapping relationships between azimuthal seismic data and fracture weaknesses in analytic form poses a significant challenge. Leveraging neural networks offers a nonlinear mechanism to bridge this gap. Initially, we establish a forward model by employing convolution operations between the azimuthal PP-wave (incident and reflected P-wave) reflection coefficient equation in transversely isotropic (HTI) media with a horizontal axis of symmetry and seismic statistical wavelets. This foundation enables the synthesis of azimuth-dependent seismic data. Subsequently, a convolution neural network (CNN) is constructed to predict subsurface rock fracture properties from azimuthal pre-stack seismic data. To quantify the uncertainty associated with neural network estimation, we employ the approximate Bayesian computation (ABC) method to determine the posterior distribution of model parameters. Finally, we present the application of both synthetic and filed data. Our results indicate a correlation of 90% and 86.8% between the synthetic model and the blind well, respectively. Furthermore, the estimated posterior distribution serves to validate the constraint capability of the proposed method, thereby furnishing comprehensive evidence supporting the feasibility and robustness of our approach.

The inversion of fracture parameters, using the variation in prestack seismic amplitude with incident angle and azimuth (AVAZ), is a primary method for predicting subsurface fractures. However, the inversion for fracture parameter is inherently ill-posed, necessitating the integration of regularization constraints, such as the L1–2 norm, to enhance inversion accuracy. In this article, we hypothesize that the fractured reservoir exhibits horizontal transverse isotropy (HTI) and model it accordingly. Under this premise, we convolve the PP-wave reflection coefficient in HTI media, characterized by fracture parameters, with the seismic wavelet to establish a forward model. Subsequently, leveraging the Bayesian framework and assuming seismic noise follows a Gaussian distribution, and we introduce the L1–2 norm as a prior distribution model to constrain azimuthal prestack seismic inversion. To address the issue of insufficient lateral continuity in the inversion, we incorporate low-frequency information as a constraint in objective function formulation. Finally, we use the alternating direction method of multipliers (ADMM) to efficiently solve the objective function. The proposed method is validated through both synthetic and real datasets, demonstrating its efficacy in mitigating ill-posedness of fracture parameter inversion and enhancing inversion accuracy.

Extending the well-established theory of anisotropic poro-elasticity, we introduce a novel framework for low-frequency anisotropic poro-acoustoelastic behavior. It incorporates the paradigm of acousto-elasticity and the low-frequency anisotropic Gassmann equation to quantify how stress influences the elastic and fluid properties, along with seismic reflection coefficients, in fractured porous media under stress. We present a model-based Bayesian inversion methodology to extract key properties from wide-azimuth seismic data acquired in stressed, fluid-saturated fractured media with aligned fractures. These properties include fluid content, the influence of fractures (represented by normal and shear weaknesses), and stress (represented by two stress-dependent parameters). Our approach leverages the theory of anisotropic poro-acoustoelasticity and perturbation theory. We first derive a linearized approximation for the relationship between seismic amplitude versus offset and azimuth (AVOAz) using the stationary phase method. This approximation separates the influence of various parameters, allowing us to estimate the fluid/porosity term, fracture weaknesses, and stress-dependent parameters associated with third-order elastic constants (3oECs). Next, we integrate a convolution model with a Bayesian framework to propose a practical approach for model-regularized AVOAz inversion. This inversion employs an iteratively reweighted least-squares (IRLS) algorithm. Finally, the effectiveness of both the derived approximation and the proposed inversion method is validated using synthetic and real data from stressed, gas-saturated reservoirs with aligned fractures.

The improvement of resolution and efficiency of seismic inversion is one of the key problems in seismic reservoir characterization. We first express the PP-wave (incident P wave, reflected or scattered P wave) reflection coefficient of transversely isotropic media with a horizontal symmetry axis (HTI) in the form of Fourier coefficients. Then, we analyze the sensitivity of Fourier coefficients to elastic parameters and fracture weaknesses. The results demonstrate that the zeroth-order Fourier coefficient is sensitive to elastic parameters but insensitive to normal and shear fracture weakness, and the second-order Fourier coefficient is more sensitive to fracture weaknesses. To reliably estimate isotropic and fracture parameters, we next construct the seismic forward solver characterized by elastic parameters and fracture weaknesses by Fourier coefficient decomposition, and the objective function is constructed via Bayesian framework. Based on the automatic decoupling of multi-frequency components for seismic signals in frequency domain, the optimal solution of inverse problem is searched by the iterative solution method using seismic multi-frequency components. Finally, we use the synthetic data and real data to verify the effectiveness and stability of the proposed method. The experimental examples show that the proposed method can realize the inversion of elastic parameters and fracture parameters simultaneously by using the fractional expansion of Fourier coefficients, and improve the accuracy of seismic inversion by using the multi-frequency component iteration method in the frequency domain.

Fracture weaknesses are two important parameters to describe the characteristics of reservoir fracture-induced anisotropy. Accurate estimation for fracture weaknesses is of great significance for seismic exploration and characterization of naturally fractured reservoir, especially for gas-bearing reservoir. Linear seismic inversion based on Bayesian theory can fuse prior information. And the inverse problem is solved in the form of the posterior probability density function (PDF) to obtain the maximum posterior solution. Based on the PP-wave reflection coefficient in a transversely isotropic medium with a horizontal symmetry axis (HTI), we first construct forward models in different domains by using the convolution model. Then, we select a relatively good initial model to realize the seismic inversion in time domain, frequency domain, and joint time-frequency domain, respectively. Next, a relatively poor initial model is selected to estimate the low-frequency components in Laplace-Fourier domain, and the final estimation is compared with the traditional approach. Finally, both synthetic and real data are used to verify the feasibility of inversion approaches in different domains. The anti-noise ability of estimated fracture weaknesses in time domain is stronger than those of frequency-domain inversion. The joint time-frequency inversion inherits the advantages of strong anti-noise ability of time-domain inversion and retains the excellent characteristics of high resolution of frequency-domain inversion, it balances between improving the resolution of seismic inversion and suppressing random noise. The estimation of low-frequency components for fracture weaknesses in Laplace-Fourier domain can reduce the dependence of seismic inversion on the initial model and improve the estimated accuracy.

Fracture detection and characterization remains challenging for azimuthal seismic inversion for effective elastic anisotropic media, especially for the more complex orthorhombic (ORT) anisotropic media formed by two orthogonal sets of vertical, aligned fractures in a homogenously isotropic background rock. We first review multiple modes of rock-physics characterization for orthorhombic anisotropy. We also review the inversion methods or strategies in seismic characterization for the complicated naturally fractured reservoirs. Next, we derive the effective elastic stiffness matrix in such a weak ORT anisotropic medium, and also obtain the linearized PP-wave reflection coefficient based on scattering theory and perturbated stiffness tensor. Then, we propose a method of Bayesian azimuthal seismic inversion for the effective elastic ORT anisotropic media based on the anisotropic perturbations in PP-wave reflection coefficient and the iteratively reweighted least-squares algorithm. Finally, the applications of synthetic and real data sets acquired from a gas-bearing fractured reservoir with orthorhombic symmetry demonstrate the effectiveness of the proposed inversion approach, and our method may provide an alternative way to describe the effective elastic orthorhombic anisotropic media formed by two orthogonal sets of vertical fractures.

In the absence of fracture, the strata with horizontal interbedding structure can be approximately equivalent to transversely isotropic media with vertical axis of symmetry (VTI) in the sedimentary basin. Accurate estimation of Young’s modulus, Poisson’s ratio, and weak anisotropy (WA) parameters can provide basic information for further prediction of shale reservoir rock brittleness andin situstress. Based on the scattering theory and Born approximation, we derive the P-wave reflection coefficients involving Young’s modulus and Poisson’s ratio and WA parameters for an interface separating two elastic VTI media. Assuming in the case of shale, we modify this reflection coefficient to involve only three model parameters through a series of integration and simplification for stabler inversion. A Bayesian amplitude versus offset (AVO) inversion method is implemented to estimate the three attributes, which are then converted to calculate the brittleness-related and WA parameters. The numerical simulation results show that for AVO of III in the case of gas-bearing shale, the derived approximation has more accuracy than the Young’s modulus, Poisson’s ratio and density (YPD) equation, especially at large angle. Tests on synthetic and real seismic data verify that the established inversion strategy for VTI shale reservoirs is stable and accurate in the estimation of brittleness-related and WA parameters.

Fracture weakness is one of the most important anisotropic parameters used to characterize the fractures and identify the fluids. The model of horizontal transversely isotropic (HTI) medium is usually utilized in seismic azimuthal inversion for the fracture weaknesses. Low-frequency component of fracture weaknesses plays a significant role in seismic fracture characterization and fluid identification due to the deficiency of low-frequency component in acquired seismic azimuthal data. The commonly used approaches to estimate low-frequency component include the smoothing model constraints and the damped wave field in complex-frequency domain. Following the Bayesian framework, we propose a novel approach used for compensating the low-frequency component of fracture weaknesses in Laplace-Fourier domain. Firstly, we reconstruct the seismic forward solver in Laplace-Fourier domain to obtain the low-frequency component of fracture weaknesses. Then, we propose a method of seismic azimuth-difference inversion for fracture weaknesses in Laplace-Fourier domain in a Bayesian framework. Finally, both synthetic and field data examples are used to demonstrate the superiority and stability of the proposed inversion approach. Compared with the conventional inversion approach, the proposed approach can reduce the dependence on the initial model of model parameters and weaken the effect of missing low-frequency components of fracture weaknesses in azimuthal seismic data, and it may help to improve the inversion accuracy of fracture weaknesses and reduce the uncertainty of inversion results.

Knowledge ofin situstress is of great significance for hydraulic fracture stimulating of unconventional reservoirs. Quantitative estimation ofin situstress, especially in the far field, is a major challenge. This research mainly focuses on developing a novel Bayesian AVAZ inversion approach to estimatein situstress of fractured shale-gas reservoirs with horizontal transversely isotropic (HTI) symmetry. Using the generalized Hooke’s law and Schoenberg’s linear slip model, we first deduce the anisotropic horizontal stress equation in an HTI medium formed by a single set of vertically aligned fractures embedded in an isotropic background rock. Based on the critical porosity model, we then obtain the saturated stiffness matrix with vertical effective stress-sensitive parameters and dry fracture weaknesses using the relationship between vertical effective stress and porosity. Combining the scattering function and the perturbed stiffness matrix, we deduce a linearized PP-wave reflection coefficient as a function of fluid bulk modulus, vertical effective stress-sensitive parameter, dry-rock P- and S-wave moduli, density, and fracture density. Finally, we propose a novel Bayesian amplitude variation with azimuth (AVAZ) inversion constrained by the Cauchy regularization and low-frequency regularization to estimate these model parameters, which are further used to calculatein situstress. The analysis of synthetic data demonstrates thatin situstress can be reasonably estimated even with moderate noise. A field data set test reveals that the inversion results agree well with the well log interpretation, and the proposed approach can generate meaningful results that are useful for seismic identification of potential fracturing barriers during fracturing stimulation of shale-gas reservoirs.

Commonly-used seismic fracture characterization method assumes one single fracture set, and this article aims to relax the limitation and demonstrate the feasibility of seismic characterization for two sets of orthorhombic fractures. Elastic properties of orthorhombic fractures can be studied using a model of horizontal and vertical fractures as linear-slip interfaces embedded in an isotropic background. High fracture density in hydrocarbon reservoirs is the “sweet spot” of effective permeability for fluid flow, and seismic characterization of orthorhombic fractures using the variations in reflection amplitude versus offset and azimuth (AVOAz) helps to optimize the production of naturally fractured reservoirs. Based on an effective orthorhombic linear-slip model, we first construct the effective elastic stiffness matrix in terms of elastic moduli and decoupled horizontal and vertical fracture densities under the assumption of weak anisotropy (WA) or small fracture densities. Using the scattering function and first-order perturbation in effective elastic stiffness components, we derive a WA and linearized PP-wave reflection coefficient containing decoupled orthorhombic fracture densities to describe the characteristics of horizontal and vertical fracture sets. We then propose an efficient AVOAz inversion method to perform the seismic characterization of decoupled orthorhombic fracture densities based on observed surface azimuthal amplitude data in a Bayesian framework. We finally apply the inversion approach to synthetic and field datasets to demonstrate its feasibility and reasonability in orthorhombic fracture characterization.

Tilted transverse isotropy (TTI) is common for naturally fractured reservoir when the fractures exist a tilted axis of symmetry. Seismic characterization of natural tilted fractures from observable azimuthal reflected amplitude data, however, is quite difficult. We focus on the detection of natural tilted fractures from azimuthal seismic amplitude data, especially for low- and high-angle tilted fractures. Using the linear-slip theory, we first express the effective elastic stiffness matrix of a TTI medium as a function of background elastic moduli, fracture density, and three angles, including the incident angle, the azimuth angle, and the tilt angle. Based on the scattering function and the first-order perturbations in stiffness components across a weak-contrast reflection interface separating two weak-anisotropy TTI media, we then derive a weak-anisotropy and linearized PP-wave reflection coefficient containing the tilt angle and fracture density. For low- and high-angle aligned fracture sets, we formulate the low- and high-angle approximate PP-wave reflection coefficient equations, respectively, and propose a Bayesian seismic inversion approach to estimate the tilt angle and the corresponding fracture densities using the azimuthal differences in seismic reflected amplitude data. Synthetic and real datasets are used to demonstrate the feasibility and reliability of the proposed inversion approach. Test results make us believe that our proposed inversion approach allow us to obtain the tilted fracture properties of hydrocarbon reservoirs in a more accurate manner than previous cases of vertically transverse isotropy (VTI) or horizontally transverse isotropy (HTI).

The probabilistic prediction approach can be used not only for obtaining the maximum posterior probability solution but also for uncertainty evaluation. Its prior distribution has a significant impact on the prediction result. An improper prior assumption may lead to prediction deviation. To improve the prediction accuracy of elastic parameters, a Laplace prior with total variation (TV) constraint is introduced in the probabilistic prediction. First, the effect of TV constraint on the probability distribution of elastic parameters is analyzed in detail. Then, two approaches are proposed to handle the cases where the elastic parameters have blocky boundaries and no blocky boundaries: probabilistic prediction scheme for elastic parameters with blocky boundaries and probabilistic prediction scheme with blocky lithology prior constraint. The former imposes a sparse constraint on the elastic parameters, while the latter imposes a sparse constraint on the TV processing lithology. Their posterior probabilities are re-derived. Considering that the discrete lithology is more likely to be blocky compared with the continuous elastic parameters, the sparse lithology constraint can handle more general cases. In addition, this approach allows for lithology prediction. The applications of numerical examples and field seismic data verify the feasibility of the proposed approaches.