The spatial structure of subsurface media usually exhibits anisotropic characteristics due to geological factors such as depositional environment, tectonic movement and overlying pressure, and the spatial structure characteristics are different at different spatial points. The traditional Kriging interpolation method based on global spatial range is difficult to accurately describe the anisotropic spatial structure characteristics of each spatial point in the subsurface medium, and the obtained Kriging interpolation results have larger errors. In this paper, we estimate the statistical characteristic parameters of random media from seismic data, which correspond to the anisotropic spatial structure characteristics of each spatial point of the subsurface medium, so as to accurately describe the anisotropic spatial structure characteristics of each spatial point of the subsurface medium. Then the spatial correlation length of each spatial point of the underground medium in different directions is calculated from the statistical characteristic parameters of the random media, and then the anisotropic spatial covariance function is calculated, furthermore the anisotropic Kriging weight coefficients are obtained, and finally the final anisotropic spatial structure Kriging interpolation results are obtained. Compared with the traditional Kriging interpolation method, the Kriging interpolation results obtained by the method in this paper are more reliable, better match the actual anisotropic spatial structure characteristics of the subsurface medium, with smaller errors and less uncertainty. Model tests and field data application demonstrates the effect of this method and verify its feasibility and practicality.

In hydrocarbon exploration and development, fluid indicators that can directly identify reservoir hydrocarbons from seismic data are of great significance for seismic interpretation in the fracture-induced horizontal transversely isotropic (HTI) reservoirs. In this paper, based on the unified elastic wave theory of the medium, a new anisotropic fluid indicator is constructed incorporating squirt flow effect between the cracks of rocks. The novel anisotropic fluid indicator can better reflect the influence of pore fluid within the rock on wave propagation. Compared with conventional elastic parameters, the new established anisotropic fluid indicator is more sensitive to oil/gas. Subsequently, by combining the perturbation of the elastic stiffness component in fluid-saturated fractured porous media and the inverse scattering function, an anisotropic PP-wave reflection coefficient is derived in terms of an anisotropic fluid indicator incorporating squirt flow effect and fracture weaknesses. The comparison of Rüger’s equation and the new reflection coefficient equation confirms the validity of our equation for parameter estimation. Further our reflection coefficient equation is used to establish an anisotropic frequency-dependent azimuthal amplitude variation with offset (AVO) inversion method to estimate the anisotropic fluid indicator and fracture weaknesses. The feasibility of the inversion method is verified by the field data application in eastern China, which demonstrates that the anisotropic fluid indicator with the squirt flow effect is certainly sensitive to the gas-bearing fractured reservoirs, and can provide a more effective method for fluid identification in gas-fractured reservoirs.

Two sets of rotationally invariant, horizontal and vertical fractures permeated in a homogeneous isotropic background rock can yield a long-wavelength effective orthorhombic medium. Amplitude variation with angle and azimuth (AVAZ) inversion is an effective tool to invert azimuthal seismic data for orthorhombic medium parameters. Given the inherently band-limited nature of real seismic data, orthorhombic AVAZ inversion without reasonable model constraints is a challenging task to obtain stable and reliable estimated results. In this paper, we have developed a novel joint Laplace- and Fourier-domain seismic inversion method to estimate P- and S-wave velocities, density, and horizontal and vertical fracture densities. The orthorhombic AVAZ seismic forward modeling in the Laplace domain is established by introducing the Fourier and damping operators. We present a two-step strategy for joint Laplace- and Fourier-domain seismic inversion in a Bayesian inference framework. Firstly, we perform the Laplace-domain seismic inversion to obtain long-wavelength models of elastic parameters and fracture parameters by exploiting the low-frequency components of the damped azimuthal seismic data. Secondly, with the Laplace-domain inversion results as initial models, the Fourier-domain seismic inversion is implemented to obtain complete inversion results, where all frequency components of original azimuthal seismic data are employed. We demonstrate the feasibility of our method through a synthetic data example and an application of a field data set acquired over a fractured shale reservoir. Meanwhile, we compare our method with the Fourier-domain seismic inversion, results show that our method can yield more reasonable estimated results of orthorhombic medium parameters than the Fourier-domain seismic inversion.

Natural fractures and rock brittleness play an important role in the fracturing and development of shale gas reservoirs. Amplitude variation with angle and azimuth (AVAZ) inversion provides an effective tool for estimating brittleness and fracture properties. However, AVAZ inversion generally employs a linearized reflection coefficient approximation, commonly not applicable for cases with strong contrasts in rock properties and long-offset data. Meanwhile, the large number of model parameters poses great challenges for multiparameter simultaneous inversion. To overcome these limitations, we propose a novel optimal AVAZ inversion approach to estimate the brittleness indicator and fracture density in horizontal transversely isotropic (HTI) media. We first utilize the anisotropic Zoeppritz equation and the convolution model to generate reference AVAZ gathers from well-log data. The singular value decomposition (SVD) method is then employed to obtain optimal basis functions and optimal coefficients that directly link band-limited elastic and fracture reflectivities to observed AVAZ data. This enables the direct estimation of band-limited elastic and fracture reflectivities from observed AVAZ data. Finally, Bayesian poststack seismic inversion, constrained by modified Cauchy prior and low-frequency models, is utilized to invert these band-limited reflectivities for elastic and fracture parameters individually. The feasibility of our method is demonstrated on a synthetic example and a field dataset from the southern Sichuan Basin. Results reveal the ability of our method to produce more satisfactory estimated results of the brittleness indicator and fracture density than the AVAZ simultaneous inversion method, which could aid in the seismic characterization of rock brittleness and natural fractures.

Pre-stack seismic inversion is an effective means of using seismic data to achieve prediction of physical parameters of underground media. However, pure PP wave inversion suffers from high inversion multiplicity solutions and limited prediction accuracy. Therefore, we propose a multi-wave stochastic inversion physical parameters prediction method driven by linearized rock physics. Firstly, this method establishes a robust relationship between elastic and physical parameters based on a statistical rock physics model. Secondly, the linearized AVO approximations for PP wave and SH-SH wave expressed by porosity, clay volume and water saturation are derived. Numerical simulations of the reflection characteristics of the two model interfaces show that both new formulations have high accuracy. On this basis, we construct a joint inversion equation of PP wave and SH-SH wave for porosity, clay volume and water saturation. A stochastic inversion method based on the ensemble smoother with multiple data assimilation (ES-MDA) for the multi-wave joint physical parameters is proposed. Stanford VI-E model tests indicate that the physical parameters obtained from the joint PP wave and SH-SH wave inversion have higher identification accuracy and smaller relative errors compared to the pure PP wave inversion. Moreover, field data tests demonstrate the good practicality of the method for seismic prediction of underground media physical parameters.

Deep learning is a data-driven technique that demands network models trained using big datasets. In seismic structure interpretation, it is very difficult, time-consuming, and relatively economic costs to prepare training datasets by directly annotating real seismic data. Seismic convolution method is an efficient way to synthesize seismic data, which can easily and quickly generate large amount of training datasets. However, there are large differences in feature space between synthetic seismic data and real seismic data. That results in the poor performance of network models trained using synthetic training datasets on real seismic data. In this paper, we propose to use Fourier domain adaptation (FDA) to achieve domain transfer. First, amplitude spectrum of synthetic seismic data is replaced with those of real seismic data to make feature space mapping. Then synthetic seismic data is used for transfer training of network models to improve its performance on real seismic data. The experimental results demonstrate that the FDA performs the domain transfer of synthetic seismic data to real seismic data, which improves the generalization of network models trained based on synthetic seismic data. Meanwhile, the FDA is a promising method for deep learning model transfer training in seismic structure interpretation.

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.

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).

Seismic data interpolation is an effective way to reconstruct missing seismic traces and to improve the quality of the seismic data set. In the field of deep learning, generative adversarial networks are capable of data generation and interpolation and have been widely used for high-quality image generations and image interpolations. In this letter, we propose a dual-domain conditional generative adversarial network (DD-CGAN) for seismic data interpolation. The DD-CGAN consists of a generator network and a discriminator network and uses the seismic data set and discrete Fourier transformed data set in the frequency domain as input vectors. The loss function of the DD-CGAN is defined by the generative-adversarial loss, the data loss function, and the total variation loss. Thus, the DD-CGAN can be trained more accurately. The discriminator is used to calculate the feature differences between the interpolated seismic data set and the complete seismic data set to drive the generator network for learning optimal parameters. The numerical results on the test data set and field seismic data set demonstrate the effectiveness of the proposed DD-CGAN.

To better solve the passenger assignment problem, which is a subproblem of the transit network optimization problem, we build an artificial urban transit system (AUTS) and adopt a day-to-day learning mechanism to describe passengers' route and departure-time-choice behaviors. With the support of AUTS to handle the lower level assignment problem, we are able to solve the upper level transit network design problem. Compared with other bilevel models, our approach better accommodates passengers' dynamic learning behavior and their heterogeneity. Based on AUTS, we solve the frequency optimization problem and compare the results with an analytical method. We also perform some numerical experiments on AUTS and discover some interesting issues on the capacity of public transportation system and passengers' heterogeneity.