Tilted Transversely Isotropic Research Articles

Quasi-P-Wave Reverse Time Migration in TTI Media with a Generalized Fractional Convolution Stencil

In seismic modeling and reverse time migration (RTM), incorporating anisotropy is crucial for accurate wavefield modeling and high-quality images. Due to the trade-off between computational cost and simulation accuracy, the pure quasi-P-wave equation has good accuracy to describe wave propagation in tilted transverse isotropic (TTI) media. However, it involves a fractional pseudo-differential operator that depends on the anisotropy parameters, making it unsuitable for resolution using conventional solvers for fractional operators. To address this issue, we propose a novel pure quasi-P-wave equation with a generalized fractional convolution operator in TTI media. First, we decompose the conventional pure quasi-P-wave equation into an elliptical anisotropy equation and a fractional pseudo-differential correction term. Then, we use a generalized fractional convolution stencil to approximate the spatial-domain pseudo-differential term through the solution of an inverse problem. The proposed approximation method is accurate, and the wavefield modeling method based on it also accurately describes quasi-P-wave propagation in TTI media. Moreover, it only increases the computational cost for calculating mixed partial derivatives compared to those in vertical transverse isotropic (VTI) media. Finally, the proposed wavefield modeling method is utilized in RTM to correct the anisotropic effects in seismic imaging. Numerical RTM experiments demonstrate the flexibility and viability of the proposed method.

Efficient reverse time migration method in tilted transversely isotropic media based on a pure pseudoacoustic wave equation

In general, velocity anisotropy in shale media has been widely observed in laboratory and field work, which means that disregarding this characteristic can lead to inaccurate imaging locations when data are imaged with reverse time migration (RTM). Wavefields simulated with the conventional coupled pseudoacoustic wave equation may introduce S-wave noise and this equation is only valid in transversely isotropic media ([Formula: see text]). Certain decoupled qP-wave equations require the use of the pseudospectral method, which makes them computationally inefficient. To address these issues, we develop a new pure qP acoustic wave equation based on the acoustic assumption, which can be solved more efficiently using the finite-difference (FD) method. This equation can also be used in the forward-modeling process of RTM in tilted transversely isotropic (TTI) media. First, we perform a Taylor expansion of the root term in the pure qP-wave dispersion relation. This leads to an anisotropic dispersion relation that is decomposed into an elliptical anisotropic background factor and a circular correction factor. Second, we obtain the pure qP-wave equation in TTI media without a pseudodifferential operator. The new equation can be efficiently solved using FD methods and can be applied to RTM in TTI media with strong anisotropy. Our method indicates greater tolerance to numerical errors and is better suited for strong anisotropy, as compared with previously published methods. Numerical examples indicate the high kinematic and phase accuracy of our pure qP-wave equation along with its stability in TTI media characterized by ([Formula: see text]). By using a sag model and an overthrust TTI model, we determine the efficiency and accuracy of our TTI RTM.

Efficient least-squares reverse time migration in TTI media using a finite-difference solvable pure qP-wave equation

Abstract The anisotropic characteristics of underground media play a crucial role in seismic wave propagation and should be accounted for during seismic imaging. The least-squares reverse time migration (LSRTM) method achieves ideal imaging results by suppressing migration artifacts and balancing amplitude. Therefore, pure quasi-P (P-qP) wave equations in tilted transversely isotropic (TTI) media have been widely used to implement LSRTM, owing to their ability to produce stable and noise-free wavefields. However, solving the anisotropic P-qP-wave equations typically necessitates the use of spectral-based methods, making them computationally inefficient, especially in 3D applications. In this study, we first develop a P-qP-wave equation in TTI media that can be efficiently computed through the finite-difference (FD) approach. Numerical tests show that, in comparison to the previous TTI P-qP-wave equation, the newly derived FD solvable TTI P-qP-wave equation yields reasonably accurate and highly efficient wavefield simulations. Then, building on our newly derived wave equation, we derive its adjoint migration and demigration operator to implement TTI LSRTM. Two synthetic examples suggest that the newly presented P-qP-wave TTI LSRTM approach effectively correct for the anisotropy effects, providing high-quality imaging results. Additionally, our approach has superior computational efficiency over the conventional P-qP-wave TTI LSRTM technique based on a hybrid FD pseudo-spectral (HFDPS) solver.

Power-Law Frequency-Dependent Q Simulation and Reverse-Time Migration in Transversely Isotropic Acoustic Media

ABSTRACT Velocity and attenuation (Q) anisotropy are widely distributed in the Earth’s interior, significantly affecting the kinematic and dynamic characteristics of seismic-wave propagations. Previous studies developed to simulate these effects are mainly restricted to the constant-Q assumption. However, seismic attenuation in high-temperature and high-pressure regions is demonstrated to be frequency-dependent and usually follows a power-law formulation. To simulate this Q effect in transversely isotropic (TI) attenuating media, we derive a new pure-viscoacoustic wave equation with decoupled fractional Laplacians, which can simultaneously simulate amplitude dissipation and velocity dispersion effects. Based on the wavenumber relationship between the observation and physical coordinate systems, the tilted TI (TTI) wave equation is further derived. Compared with the pseudoviscoacoustic wave equation, the proposed pure-viscoacoustic equation can simulate stable P wavefields in complex geological structures without S-wave artifacts. To solve this new equation, two low-rank decompositions are introduced to approximate the real and imaginary parts and avoid the separation of wavenumbers and dip angles, making it much simpler in programming and implementation. We further use this equation to perform Q-compensated reverse-time migration to generate high-resolution migration images in anisotropic attenuating media. Numerical examples demonstrate the effectiveness of the proposed method for pure-viscoacoustic wavefield simulations and migrations in TTI attenuating media with power-law frequency-dependent Q effects.

Simplified TTI pure qP-wave equation implemented in the space domain and applied for reverse time migration in TTI media

The assumption that the earth’s subsurface is a homogeneous and isotropic medium enables the imaging of important geologic structures but results in information loss, especially in more complex geologic media. Therefore, considering anisotropy is crucial for seismic imaging, particularly the most common anisotropy in geophysics: the transversely isotropic medium. However, this also means a considerable increase in the computational cost of reverse time migration (RTM). Thus, a new pseudoacoustic wave equation for pure qP-wave in tilted transversely isotropic (TTI) media, which can also be efficiently implemented using the finite-difference (FD) method with the unit vector method (UVM), is proposed, aiming to reduce the computational cost of the RTM. The proposed equation solved with fast Fourier transforms is exact and faster for seismic migration than the base equations found in the literature; however, a greater efficiency is achievable by using FD to compute the second derivatives. In contrast, when solved with the UVM, it is kinematically accurate and significantly faster, although dynamically inaccurate as it is an acoustic approximation. The efficiency and efficacy of this new equation are demonstrated by modeling and migrating synthetic TTI data found in the literature.

Elastic full-waveform inversion in tilted transverse isotropy media using Gauss Newton optimization

In anisotropic full waveform inversion (FWI), the underground medium is commonly assumed to be vertical transverse isotropy (VTI). However, in reality, the presence of stress and tectonic movements often leads to a nonzero tilt angle θt of the strata's symmetry axis. This introduces variations in the direction of the fast axis during the propagation of seismic waves. Therefore, it is more practical and beneficial to consider tilted transverse isotropy (TTI) media when dealing with complex geological structures, as it enhances the accuracy of imaging and inversion. However, the consideration of TTI media also introduces a significant challenge of parameter coupling, commonly known as crosstalk or trade-off. This refers to the situation where the update of one parameter affects and projects onto other parameters. To address this issue and improve the inversion accuracy, we incorporate the Hessian within the inversion process. The off-diagonal elements of the Hessian matrix specifically provide important information about the correlations between different parameters, which aids in their decoupling and mitigates the effect of crosstalk. In numerical experiments, we applied the Gauss-Newton (GN) method to TTI elastic FWI and compared the results with those obtained using the preconditioned conjugate gradient (P-CG) method.The inversion results indicate that the GN method can better alleviate the parameter coupling, making simultaneous inversion of multiple parameters possible. θt

Microseismic Data-Direct Velocity Modeling Method Based on a Modified Attention U-Net Architecture

In microseismic monitoring, the reconstruction of a reliable velocity model is essential for precise seismic source localization and subsurface imaging. However, traditional methods for microseismic velocity inversion face challenges in terms of precision and computational efficiency. In this paper, we use deep learning (DL) algorithms to achieve precise and efficient real-time microseismic velocity modeling, which holds significant importance for ensuring engineering safety and preventing geological disasters in microseismic monitoring. Given that this task was approached as a non-linear regression problem, we adopted and modified the Attention U-Net network for inversion. Depending on the degree of coupling among microseismic events, we trained the network using both single-event and multi-event simulation records as feature datasets. This approach can achieve velocity modeling when dealing with inseparable microseismic records. Numerical tests demonstrate that the Attention U-Net can automatically uncover latent features and patterns between microseismic records and velocity models. It performs effectively in real time and achieves high precision in velocity modeling for Tilted Transverse Isotropy (TTI) velocity structures such as anticlines, synclines, and anomalous velocity models. Furthermore, it can provide reliable initial models for traditional methods.

Seismic-electromagnetic projection attribute: Application in integrating seismic quantitative interpretation and 3D controlled-source electromagnetic-magnetotelluric broadband data inversion for robust ranking and sweet spotting of hydrocarbon prospects in offshore northwest Borneo

We introduce a new seismic-electromagnetic (EM) projection (SEMP) attribute to improve the characterization of pay sands by integrating seismic P impedance and S impedance with electrical resistivity measurements leveraging on the high sensitivity of resistivity to fluid saturation and the capability of seismic data to delineate thin beds. Porosity is predicted using an extended elastic impedance (EEI) approach to reduce the risk of encountering SEMP anomalies related to low porosity. First, we determine the seismic-EM integration concept using well logs from a well that encountered hydrocarbons in a deepwater fold belt in offshore Borneo. The SEMP logs supported by the EEI log successfully delineate the pay zones in the well. Second, we extend the approach in three dimensions and evaluate its effectiveness using large-size seismic controlled-source EM (CSEM) and magnetotelluric (MT) data from the same area. The seismic data are inverted to produce P- and S-impedance volumes. EEI volume calibrated to total porosity also is produced. We then perform 3D anisotropic cross-gradient joint CSEM and MT inversion of a large-size survey comprising 647 receivers with receiver spacing of 1.5–3 km at prospect to regional scale. The SEMP volumes computed using the seismic and CSEM-MT inversion results are then interpreted with the EEI volume to identify possible geobodies of pay sand. Resistive geobodies with low porosity are assigned higher risks because they are likely due to compaction rather than hydrocarbon presence, whereas geobodies with high porosity are assigned lower risks and identified as potential sweet spots. The geobodies need to be corroborated with independent geologic studies to mature them into exploration prospects and assess their associated risk and potential. Although a vertical transverse isotropy approximation is used in our CSEM-MT study, the SEMP concept also is applicable to cases considering tilted transverse isotropy or arbitrary anisotropy where available and operational for such large-size exploration data.

Numerical Fréchet derivatives of the displacement tensor for 2.5‐D frequency‐domain seismic full‐waveform inversion in viscoelastic TTI media

AbstractDerivatives of the displacement tensor with respect to the independent model parameters of the subsurface, also called Fréchet derivatives (or sensitivity kernels), are a key ingredient for seismic full‐waveform inversion (FWI) with a local‐search optimization algorithm. They provide a quantitative measure of the expected changes in the seismograms due to perturbations of the subsurface model parameters for a given survey geometry. Because 2.5‐D wavefield modelling involves a real point source in a 2‐D geological model with 3‐D (spherical) wave properties, it yields synthetic data much closer to the actual practical field data than the commonly used 2‐D wave simulation does, which uses an unrealistic line‐source in which the waves spread cylindrically. Based on our recently developed general 2.5‐D wavefield modelling scheme, we apply the perturbation method to obtain explicit analytic expressions for the derivatives of the displacement tensor for 2.5‐D/2‐D frequency‐domain seismic FWI in general viscoelastic anisotropic media. We then demonstrate the numerical calculations of all these derivatives in two common cases: (1) viscoelastic isotropic; and (2) viscoelastic tilted transversely isotropic (TTI) solids. Examples of the differing sensitivity patterns for the various derivatives are investigated and compared for four different homogeneous models involving 2‐D and 2.5‐D modelling. Moreover, the numerical results are verified against the analytic solutions for homogeneous models. We further validate the numerical derivatives in a 2‐D heterogeneous viscoelastic TTI case by conducting a synthetic data experiment of frequency‐domain FWI to individually recover the 12 independent model parameters (density, dip angle, 5 elastic moduli and 5 corresponding Q‐factors) in a simple model comprising an anomalous square box target embedded in a uniform background. Another 2.5‐D multi‐target model experiment presenting impacts from four common seismic surveying geometries validates the Fréchet derivatives again.

Sequential Bayesian seismic inversion for fracture parameters and fluid indicator in tilted transversely isotropic media

A rock permeated by tilted aligned fractures, which is common in the earth, can be considered as a tilted transversely isotropic (TTI) medium under the assumption of the long wavelength. We develop a feasible method to predict fracture parameters (namely weakness parameters and dip angle) and fluid type in TTI rock using azimuthal seismic data. Based on an approximate stiffness matrix, we first deduce a linear reflection coefficient of the gas-bearing TTI medium in terms of the new fluid indicator and fracture parameters. The reflection coefficient is then rewritten in the form of the Fourier series to decouple the fracture and skeleton-fluid information. Next, the sequential Bayesian inversion method is proposed, which consists of three steps. The first two steps conquer inversion instability owing to coupling of the fracture parameters by constructing a linear relationship between the second- and fourth-order Fourier coefficients. The last step aims at estimating the skeleton-fluid parameters. The sequential Bayesian inversion method alleviates the inversion ill-posedness that is caused by large differences in contributions of the fracture and skeleton-fluid parameters to the reflection coefficient. Synthetic and field cases prove our method is stable and rational in fracture and fluid detections. Finally, we draw the following conclusions from numerical experiments. The approximate stiffness coefficients and derived reflection coefficient are of satisfactory accuracy for the gas-bearing reservoir with low fracture density. The new fluid indicator is sensitive to fluid type but very weakly dependent on mineral composition and porosity. Deconvolution processing can improve the accuracies of different seismic components calculated using the discrete Fourier transformation.

A fast anisotropic decoupled operator of elastic wave propagation in the space domain for P-/S-wave-mode decomposition and angle calculation, with its application to elastic reverse time migration in a TI medium

The conventional anisotropic P-/S-wave-mode decomposition and angle calculation usually are performed in the wavenumber domain and require multiple Fourier transforms and rely on strong model assumptions. Therefore, it is difficult to use in production due to the prohibitively high computational cost. To tackle this problem, we develop a fast anisotropic P-/S-wave-mode decomposition approach in the space domain and further apply this decomposition to the vertical transversely isotropic (VTI)/tilted transversely isotropic (TTI) elastic reverse time migration (ERTM) and the associated angle-domain common-image gathers (ADCIGs). To start with, we solve the Christoffel equation of the VTI media and obtain an anisotropic-Helmholtz (AH) operator that represents the P- and S-wave polarizations in the wavenumber domain. The decoupled operators for the P-/S-wave decomposition are then derived in terms of the AH operator. To improve the efficiency and model adaptability, we convert the decoupled operators from the wavenumber domain to the space domain as a function of the model parameters and phase angle. By further eliminating the phase-angle term, such space-domain decoupled operators lead to the approximate P- and S-wave components which are proven to achieve the same effect as the accurate P and S wavefields applied in the angle calculation and ERTM images. A fast space-domain AH decomposition approach is thus obtained and can be used for P-/S-wave-mode decomposition, ADCIGs, and ERTM. In addition, we extend our approach to the TTI media by using the coordinate transform. Three examples are used to demonstrate the effectiveness and feasibility of our method.

Bayesian inverse scattering theory for multiparameter full-waveform inversion in transversely isotropic media

Full-waveform inversion (FWI) is an effective method that uses kinematic and dynamic information to provide an accurate image of the subsurface target. By applying the Bayesian approaches to FWI workflow, the traditional deterministic solution is replaced with a maximum a posteriori solution. This provides not only a reliable subsurface model for the nonlinear inverse scattering problem but also an uncertainty quantification. The high computational cost of the Bayesian FWI and the complexity of a large-scale anisotropy inversion, however, restrict its application in industrial settings. We have developed an FWI approach for tilted transversely isotropic (TTI) media. The strain field and the particle displacement of the anisotropic media are expressed by an integral method based on the scattering theory and an explicit multiscattering modified Green’s function. To reduce the computational cost, the wavefield is calculated by using the preconditioned generalized minimal residual iterative solver, which approximates the solution in the Krylov subspace. The Fréchet derivatives are expressed by using the distorted Born iterative method. An iterated extended Kalman filter is applied to solve the inverse problem and the posterior covariance matrix frequency-by-frequency. Numerical experiments indicate that the presented methodology can reasonably reconstruct the elastic moduli of the TTI medium. Furthermore, the numerical experiments demonstrate the accuracy and feasibility of our methodology.

Explicit finite-difference modeling for the acoustic scalar wave equation in tilted transverse isotropic media with optimal operators

We have developed an explicit finite-difference modeling scheme for the scalar acoustic wave equation in tilted transverse isotropic (TTI) media, which is computationally efficient in terms of speed and memory and can be used for reverse time migration. Our method uses local spatial operators optimized in the least-squares sense by matching the discrete operator in the wavenumber domain. The calculated operator is applied locally to the wavefield in the space-time domain when the medium is TTI and turns into a conventional finite-difference operator when the medium is isotropic. Our wavefield modeling is stable with no limits such as [Formula: see text] on anisotropic parameters and does not generate S-wave artifacts. The accuracy and the efficiency of our modeling scheme are demonstrated for homogeneous, two half-space TTI, and the BP 2007 TTI models.

Fracture detection, fluid identification and brittleness evaluation in the gas-bearing reservoir with tilted transverse isotropy via azimuthal Fourier coefficients

Seismic characterization of the sweet spot plays a crucial role in the exploration and exploitation of oil and gas reservoirs. Therefore, we focused on fluid discrimination, natural fracture detection and brittleness evaluation in the reservoir permeated by an inclined set of parallel fractures using pre-stacked seismic inversion method. To this end, we proved by numerical experiments that the suggested fluid indicator is available for identifying the gas-bearing reservoir due to its high sensitivity to the gaseous hydrocarbon and extremely weak dependences on the porosity and the fracture density of the gas-bearing formation. To reduce the cumulative errors induced by indirect calculations, we derived the reflection coefficient of the TTI (tilted transverse isotropy) rock in terms of Young's modulus, Poisson's ratio, the fluid indicator and the fracture parameters and then rewrote it in the form of the Fourier series. By comparing with the published results, the derived reflectivity equation was proven rational and of satisfactory accuracy. Furthermore, we proposed a hierarchical inversion method incorporating the discrete Fourier transformation (DFT) and the Bayesian inversion strategy, which consists of three steps. Firstly, the DFT method is utilized to decouple anisotropic seismic components from the isotropic seismic component. In the last two steps, the Bayesian inversion approach is taken to acquire the stable and reliable estimations of the fluid and brittleness indicators and the fracture parameters from the second-order and zeroth-order seismic components, respectively. The synthetic example proved that we can realize rational and reliable predictions of the unknown parameters using the hierarchical method even for the case with moderate noise. The field case demonstrated that the proposed method is feasible and stable, and the suggested sweet spot indicators worked well for gaseous hydrocarbon discrimination, natural fracture detection and brittleness evaluation.

3D time-domain induced polarization modelling considering anisotropy and topography

Induced polarization (IP) method is one of the most widely used techniques in mineral exploration, hydrogeology, and environmental monitoring, becoming indispensable especially for metallic ore exploration. Topography of the surface and anisotropy of the subsurface plays an important role in the IP data interpretation and modelling leading to wrong resistivity and chargeability models if they are not considered. We developed a 3D IP forward modelling considering arbitrary anisotropy and topography using the finite element method (FEM). The newly developed algorithm is compared with the numerical solutions and analytical results for classical two-layer anisotropic models successfully. By using unstructured mesh to consider complex topography, we implement 3D IP forward modelling for arbitrary anisotropic medium under the condition of uneven terrain and investigate the effect of anisotropy and topography on IP data interpretation. Topography has a decreased or reinforcing effect on time-domain induced polarization (TDIP) response. Vertical transverse isotropic (VTI) medium has a tiny effect and can accurately locate the anomaly in the subsurface while tilted transverse isotropic (TTI) medium has a very large effect and cannot be explored. Under the comprehensive influence of complex topography and anisotropy with an angle larger than 30 degrees, it is very difficult to interpret TDIP results. The 3D inversion of the synthetic data derived by the 3D anisotropic chargeability model indicates that topography weakens the TDIP response. However, VTI anisotropic anomaly can be detected accurately in the horizontal direction, the TTI anisotropic anomaly is difficult to be explored.

Elastic anisotropic finite-difference full-wave modeling and imaging of 2D tilted transversely isotropic (TTI) media

Imaging below the tilted transversely isotropic (TTI) media pose serious problems and distortions of the subsurface geological targets of interest for hydrocarbon exploration, which is very common in fold-thrust belts or active tectonic areas like the Rocky Mountain foothills of the Canada or foothills of the Himalaya. To model and image below the TTI media having symmetry axis orthogonal to dipping anisotropic layers, a 2D TTI thrust sheet is considered for elastic anisotropic finite-difference full-wave modeling and generation of synthetic seismic data using the robust staggered-grid scheme of wave propagation. Free-surface boundary conditions at the top and absorbing boundary conditions for the other three sides of the model have been imposed to reduce undesirable edge effects and suppress dispersions. The synthetic seismic data generated for the 2D TTI thrust model are migrated using the anisotropic Kirchhoff pre-stack time (PSTM) and depth migration (PSDM) technique followed by the migration velocity analysis (MVA) algorithm. The traveltime computations are made using robust eikonal equations with fourth-order elastic anisotropic finite-difference solutions to handle large tilts (ν) along with anisotropic ray-tracing to image below the TTI thrust sheet. The MVA algorithm adopted also estimates the anisotropic parameters ε and δ accurately within the acceptable error limit through successive iterations of inversion and parameter update, which results in sufficient flattening of the reflection events in the common image gathers (CIGs). This is a powerful and stable methodology to handle large tilt (ν = 60°) of the steeply dipping reflectors in complex geological structure like the TTI thrust model without much distortion of CIGs for which the symmetry axis of each TTI block is set orthogonal to the bottom of the reflectors having different tilts.

Q-compensated least-squares reverse time migration in TTI media using the visco-acoustic TTI wave equation based on the SLS model

The anelasticity and anisotropy widely exist in real subsurface media. Strong anelasticity will lead to phase dispersion and amplitude attenuation during seismic wave propagation. Anisotropy cause seismic waves to have obviously different kinematic characteristics from that of isotropy. For seismic migration, ignoring the anelasticity and anisotropy of subsurface media will significantly reduce the quality of migration images, even cause imaging failure. We propose a Q-compensated least-squares reverse time migration (Q-LSRTM) in tilted transversely isotropic (TTI) media to correct these effects. According to the Born approximation of seismic wave equation, a linearized visco-acoustic TTI pure qP-wave modeling operator is derived using a new visco-acoustic TTI wave equation for one standard linear solid (SLS) model, which can deal with the anelasticity and can simulate pure qP-wave steadily in attenuating anisotropic media without qSV-wave artifacts. Then, the corresponding adjoint equation is formulated using the adjoint-state method to calculate the gradient sensitivity kernel for the visco-acoustic TTI media. Because of the least-squares inversion, the Q compensation can be achieved during the iterations, so that the over-amplification of noises can be avoided naturally. In addition, compared with conventional LSRTM, the proposed method compensates for the anelasticity and corrects the anisotropy, so as to produce images with better spatial resolution and amplitude fidelity. Numerical examples demonstrate the feasibility and advantages of the proposed method for the data including strong attenuation effects over conventional LSRTM.

Modeling of pure visco-qP-wave propagation in attenuating tilted transversely isotropic media based on decoupled fractional Laplacians

The pseudoviscoacoustic anisotropic wave equation is widely used in the oil and gas industry for modeling wavefields in attenuating anisotropic media. Compared to the full viscoelastic anisotropic wave equation, it can greatly reduce the computational cost of wavefield modeling while maintaining the visco-qP-wave kinematics very well. However, even if we place the source in a thin isotropic layer, there will be some unwanted S-wave artifacts in the qP wavefield simulated by the pseudoviscoacoustic anisotropic wave equation due to the stepped approximation of inclined layer interfaces. Furthermore, the wavefield simulated by the pseudoviscoacoustic anisotropic wave equation may suffer from numerical instabilities when the anisotropy parameter epsilon is less than delta. To overcome these problems, we derive a pure-viscoacoustic tilted transversely isotropic (TTI) wave equation in media with anisotropy in velocity and attenuation based on the exact complex-valued phase velocity formula. The pure-viscoacoustic TTI wave equation has decoupled amplitude dissipation and phase dispersion terms, which is suitable for further reverse time migration with Q compensation. For numerical simulations, we adopt the second-order Taylor series expansion to replace the mixed-domain spatially variable fractional Laplacian operator, which guarantees the decoupling of the wavenumber from the space-related fractional order. Then, we use an efficient and stable hybrid finite-difference and pseudospectral method (HFDPSM) to solve the pure-viscoacoustic TTI wave equation. Numerical tests indicate that the simulation results of the newly derived pure-viscoacoustic TTI wave equation are stable, free from S-wave artifacts, and accurate. We further demonstrate that HFDPSM outperforms the pseudospectral method in terms of numerical simulation stability and computing efficiency.

Detection of Natural Tilted Fractures From Azimuthal Seismic Amplitude Data Based on Linear-Slip Theory

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).

Fracture Detection With Azimuthal Seismic Amplitude Difference Inversion in Weakly Monoclinic Medium

The rock with a vertical group of aligned fractures in the tilted transverse isotropy (TTI) background medium is regarded as an effective monoclinic medium. Thomsen parameters and fracture weaknesses can reflect the development degree of TTI background anisotropy and vertical fractures, respectively. Thus, this paper first establishes the approximate stiffness matrix of the monocline medium through the Bond transformation from Thomsen weak anisotropy theory and effective medium theory. Moreover, considering the Born scattering theory, a linearized approximate formula of PP wave reflection coefficient is further deduced. This approximate formula contains the parameters of incident angle, azimuth, tilted angle, compressional and shear moduli, density, fracture weaknesses, and Thomsen parameters. Then, the contribution of each parameter on the reflection coefficient by numerical simulation is analyzed respectively. Finally, from the perspective of Bayesian inference, an azimuthal seismic amplitude difference inversion approach incorporating Cauchy prior information and low-frequency regularization of model parameters is proposed to realize the simultaneous estimation of fracture weaknesses and Thomsen parameters. The synthetic test demonstrates that the inversion approach proposed in this article can obtain information on vertical fractures and TTI background medium and can be applied to azimuthal seismic data with moderate noise well. Furthermore, the inversion results of the synthetic test and field data illustrate the stability and feasibility of the inversion approach and also confirm the fracture weaknesses and Thomsen parameters of the monoclinic medium can be reasonably estimated by the inversion equation derived in this paper.