Modeling of pure qP- and qSV-waves in tilted transversely isotropic media with the optimal quadratic approximation

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. C71-C89 ◽  
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Peng Yong ◽  
Jinqiang Huang ◽  
Xu Guo ◽  
...  
Keyword(s):  
High Frequency ◽  
Wave Equations ◽  
Anisotropy Parameter ◽  
Pseudospectral Method ◽  
Transversely Isotropic ◽  
Quadratic Approximation ◽  
Isotropic Phase ◽  
Time Migration ◽  
Time Space ◽  
Tti Media

Seismic forward modeling in tilted transverse isotropic (TTI) media is crucial for the application of reverse time migration and full-waveform inversion. Modeling based on conventional coupled pseudoacoustic wave equations not only generates SV-wave artifacts, but it also suffers from instabilities in which the anisotropy parameter [Formula: see text]. To address these issues, we have started with the exact vertical transversely isotropic phase velocity formula and developed novel pure qP- and qSV-wave governing equations in TTI media by using the optimal quadratic approximation. For the convenience of using finite-difference (FD) method to solve the new pure qP- and qSV-wave wave equations, we decompose the equations into a combination of a time-space-domain wave equation that can be solved by the FD method and a Poisson equation that can be solved by the pseudospectral method. We find that the high-frequency errors caused by the pseudospectral method and the usual truncation errors in FD schemes may be responsible for the instability of the numerical simulations. To stabilize the computation, we design a 2D low-pass filtering operator to eliminate severe high-frequency numerical noise. Several numerical examples demonstrate that modeling using the new pure qP-wave equations does not have qSV-wave artifacts interference and is stable for [Formula: see text]. Our results indicate that our method can achieve highly accurate and stable modeling results even in extremely complex TTI media.

Reverse time migration in tilted transversely isotropic media

2020 ◽  
Vol 38 (2) ◽  
Author(s):  
Razec Cezar Sampaio Pinto da Silva Torres ◽  
Leandro Di Bartolo
Keyword(s):  
Seismic Anisotropy ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Computer Clusters ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Tti Media

ABSTRACT. Reverse time migration (RTM) is one of the most powerful methods used to generate images of the subsurface. The RTM was proposed in the early 1980s, but only recently it has been routinely used in exploratory projects involving complex geology – Brazilian pre-salt, for example. Because the method uses the two-way wave equation, RTM is able to correctly image any kind of geological environment (simple or complex), including those with anisotropy. On the other hand, RTM is computationally expensive and requires the use of computer clusters. This paper proposes to investigate the influence of anisotropy on seismic imaging through the application of RTM for tilted transversely isotropic (TTI) media in pre-stack synthetic data. This work presents in detail how to implement RTM for TTI media, addressing the main issues and specific details, e.g., the computational resources required. A couple of simple models results are presented, including the application to a BP TTI 2007 benchmark model.Keywords: finite differences, wave numerical modeling, seismic anisotropy. Migração reversa no tempo em meios transversalmente isotrópicos inclinadosRESUMO. A migração reversa no tempo (RTM) é um dos mais poderosos métodos utilizados para gerar imagens da subsuperfície. A RTM foi proposta no início da década de 80, mas apenas recentemente tem sido rotineiramente utilizada em projetos exploratórios envolvendo geologia complexa, em especial no pré-sal brasileiro. Por ser um método que utiliza a equação completa da onda, qualquer configuração do meio geológico pode ser corretamente tratada, em especial na presença de anisotropia. Por outro lado, a RTM é dispendiosa computacionalmente e requer o uso de clusters de computadores por parte da indústria. Este artigo apresenta em detalhes uma implementação da RTM para meios transversalmente isotrópicos inclinados (TTI), abordando as principais dificuldades na sua implementação, além dos recursos computacionais exigidos. O algoritmo desenvolvido é aplicado a casos simples e a um benchmark padrão, conhecido como BP TTI 2007.Palavras-chave: diferenças finitas, modelagem numérica de ondas, anisotropia sísmica.

Pseudoacoustic tilted transversely isotropic modeling with optimal k-space operator-based implicit finite-difference schemes

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. T139-T157 ◽  
Author(s):  
Shigang Xu ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Spatial Modeling ◽  
Wave Equations ◽  
Transversely Isotropic ◽  
Dispersion Relations ◽  
High Order ◽  
Space Operator ◽  
Modeling Accuracy ◽  
Tti Media ◽  
Temporal And Spatial

Current temporal high-order finite-difference (FD) stencils are mainly designed for isotropic wave equations, which cannot be directly extended to pseudoacoustic wave equations (PWEs) in tilted transversely isotropic (TTI) media. Moreover, it is difficult to obtain the time-space domain FD coefficients for anisotropic PWEs based on nonlinear dispersion relations in which anisotropy parameters are coupled with FD coefficients. Therefore, a second-order FD for temporal derivatives and a high-order FD for spatial derivatives are commonly used to discretize PWEs in TTI media. To improve the temporal and spatial modeling accuracy further, we have developed several effective FD schemes for modeling PWEs in TTI media. Through combining the [Formula: see text] (wavenumber)-space operators with the conventional implicit FD stencils (i.e., regular-grid [RG], staggered-grid [SG], and rotated SG [RSG]), we establish novel dispersion relations and determine FD coefficients using least-squares (LS). Based on [Formula: see text]-space operator compensation, we adopt the modified LS-based implicit RG-FD, implicit SG-FD, and implicit RSG-FD methods to respectively solve the second- and first-order PWEs in TTI media. Dispersion analyses indicate that the modified implicit FD schemes based on [Formula: see text]-space operator compensation can greatly increase the numerical accuracy at large wavenumbers. Modeling examples in TTI media demonstrate that the proposed FD schemes can adopt a short FD operator to simultaneously achieve high temporal and spatial modeling accuracy, thus significantly improve the computational efficiency compared with the conventional methods.

Acoustic wave propagation in tilted transversely isotropic media: Incorporating topography

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. C265-C278 ◽  
Author(s):  
Jeffrey Shragge
Keyword(s):  
Wave Propagation ◽  
Free Surface ◽  
Surface Topography ◽  
Domain Boundary ◽  
Waveform Inversion ◽  
Transversely Isotropic ◽  
Surface Boundary ◽  
Reverse Time ◽  
Time Migration ◽  
Tti Media

Simulating two-way acoustic wavefield propagation directly from a free-surface boundary in the presence of topography remains a computational challenge for applications of reverse time migration (RTM) or full-waveform inversion (FWI). For land-seismic settings involving heavily reworked geology (e.g., fold and thrust belts), two-way wavefield propagation operators should also handle commonly observed complex anisotropy including tilted transversely isotropic (TTI) media. To address these issues, I have extended a system of coupled partial differential equations used to model 3D acoustic TTI wave propagation in Cartesian coordinates to more generalized 3D geometries, including a deformed computational mesh with a domain boundary conformal to free-surface topography. A generalized curvilinear transformation is used to specify a system of equations governing 3D acoustic TTI wave propagation in the “topographic” coordinate system. The developed finite-difference time-domain numerical solution adapts existing Cartesian TTI operators to this more generalized geometry with little additional computational overhead. Numerical evaluations illustrate that 2D and 3D impulse responses are well-matched to those simulated on Cartesian meshes and analytic traveltimes for homogeneous elliptical TTI media. Accordingly, these generalized acoustic TTI propagators and their numerical adjoints are useful for undertaking most RTM or FWI applications using computational domains conforming to free-surface topography.

Inversion of azimuthally dependent NMO velocity in transversely isotropic media with a tilted axis of symmetry

Geophysics ◽  
2000 ◽  
Vol 65 (1) ◽  
pp. 232-246 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Wave Velocity ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Model Parameters ◽  
Symmetry Direction ◽  
Time Migration ◽  
Inversion Procedure ◽  
Tti Media

Just as the transversely isotropic model with a vertical symmetry axis (VTI media) is typical for describing horizontally layered sediments, transverse isotropy with a tilted symmetry axis (TTI) describes dipping TI layers (such as tilted shale beds near salt domes) or crack systems. P-wave kinematic signatures in TTI media are controlled by the velocity [Formula: see text] in the symmetry direction, Thomsen’s anisotropic coefficients ε and δ, and the orientation (tilt ν and azimuth β) of the symmetry axis. Here, we show that all five parameters can be obtained from azimuthally varying P-wave NMO velocities measured for two reflectors with different dips and/or azimuths (one of the reflectors can be horizontal). The shear‐wave velocity [Formula: see text] in the symmetry direction, which has negligible influence on P-wave kinematic signatures, can be found only from the moveout of shear waves. Using the exact NMO equation, we examine the propagation of errors in observed moveout velocities into estimated values of the anisotropic parameters and establish the necessary conditions for a stable inversion procedure. Since the azimuthal variation of the NMO velocity is elliptical, each reflection event provides us with up to three constraints on the model parameters. Generally, the five parameters responsible for P-wave velocity can be obtained from two P-wave NMO ellipses, but the feasibility of the moveout inversion strongly depends on the tilt ν. If the symmetry axis is close to vertical (small ν), the P-wave NMO ellipse is largely governed by the NMO velocity from a horizontal reflector Vnmo(0) and the anellipticity coefficient η. Although for mild tilts the medium parameters cannot be determined separately, the NMO-velocity inversion provides enough information for building TTI models suitable for time processing (NMO, DMO, time migration). If the tilt of the symmetry axis exceeds 30°–40° (e.g., the symmetry axis can be horizontal), it is possible to find all P-wave kinematic parameters and construct the anisotropic model in depth. Another condition required for a stable parameter estimate is that the medium be sufficiently different from elliptical (i.e., ε cannot be close to δ). This limitation, however, can be overcome by including the SV-wave NMO ellipse from a horizontal reflector in the inversion procedure. While most of the analysis is carried out for a single layer, we also extend the inversion algorithm to vertically heterogeneous TTI media above a dipping reflector using the generalized Dix equation. A synthetic example for a strongly anisotropic, stratified TTI medium demonstrates a high accuracy of the inversion (subject to the above limitations).

Approximation of pure acoustic seismic wave propagation in TTI media

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB97-WB107 ◽  
Author(s):  
Chunlei Chu ◽  
Brian K. Macy ◽  
Phil D. Anno
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Computational Cost ◽  
P Wave ◽  
Anisotropy Axis ◽  
S Wave ◽  
Time Migration ◽  
Tti Media

Pseudoacoustic anisotropic wave equations are simplified elastic wave equations obtained by setting the S-wave velocity to zero along the anisotropy axis of symmetry. These pseudoacoustic wave equations greatly reduce the computational cost of modeling and imaging compared to the full elastic wave equation while preserving P-wave kinematics very well. For this reason, they are widely used in reverse time migration (RTM) to account for anisotropic effects. One fundamental shortcoming of this pseudoacoustic approximation is that it only prevents S-wave propagation along the symmetry axis and not in other directions. This problem leads to the presence of unwanted S-waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudoacoustic wave equations become unstable for anisotropy parameters [Formula: see text] and for heterogeneous models with highly varying dip and azimuth angles in tilted transversely isotropic (TTI) media. Pure acoustic anisotropic wave equations completely decouple the P-wave response from the elastic wavefield and naturally solve all the above-mentioned problems of the pseudoacoustic wave equations without significantly increasing the computational cost. In this work, we propose new pure acoustic TTI wave equations and compare them with the conventional coupled pseudoacoustic wave equations. Our equations can be directly solved using either the finite-difference method or the pseudospectral method. We give two approaches to derive these equations. One employs Taylor series expansion to approximate the pseudodifferential operator in the decoupled P-wave equation, and the other uses isotropic and elliptically anisotropic dispersion relations to reduce the temporal frequency order of the P-SV dispersion equation. We use several numerical examples to demonstrate that the newly derived pure acoustic wave equations produce highly accurate P-wave results, very close to results produced by coupled pseudoacoustic wave equations, but completely free from S-wave artifacts and instabilities.

scholarly journals The offset‐midpoint traveltime pyramid in transversely isotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1316-1325 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Fourth Power ◽  
Weak Anisotropy ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Equivalent Equation ◽  
Two Parameters ◽  
Vti Media

Prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an offset‐midpoint traveltime equation, Cheop’s pyramid equivalent equation for VTI media. The derivation of such an equation for VTI media requires approximations that pertain to high frequency and weak anisotropy. Yet the resultant offset‐midpoint traveltime equation for VTI media is highly accurate for even strong anisotropy. It is also strictly dependent on two parameters: NMO velocity and the anisotropy parameter, η. It reduces to the exact offset‐midpoint traveltime equation for isotropic media when η = 0. In vertically inhomogeneous media, the NMO velocity and η parameters in the offset‐midpoint traveltime equation are replaced by their effective values: the velocity is replaced by the rms velocity and η is given by a more complicated equation that includes summation of the fourth power of velocity.

High-order finite-difference approximations to solve pseudoacoustic equations in 3D VTI media

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T225-T235 ◽  
Author(s):  
Leandro Di Bartolo ◽  
Leandro Lopes ◽  
Luis Juracy Rangel Lemos
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Computational Cost ◽  
Transversely Isotropic ◽  
High Order ◽  
Staggered Grid ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Grid Scheme

Pseudoacoustic algorithms are very fast in comparison with full elastic ones for vertical transversely isotropic (VTI) modeling, so they are suitable for many applications, especially reverse time migration. Finite differences using simple grids are commonly used to solve pseudoacoustic equations. We have developed and implemented general high-order 3D pseudoacoustic transversely isotropic formulations. The focus is the development of staggered-grid finite-difference algorithms, known for their superior numerical properties. The staggered-grid schemes based on first-order velocity-stress wave equations are developed in detail as well as schemes based on direct application to second-order stress equations. This last case uses the recently presented equivalent staggered-grid theory, resulting in a staggered-grid scheme that overcomes the problem of large memory requirement. Two examples are presented: a 3D simulation and a prestack reverse time migration application, and we perform a numerical analysis regarding computational cost and precision. The errors of the new schemes are smaller than the existing nonstaggered-grid schemes. In comparison with existing staggered-grid schemes, they require 25% less memory and only have slightly greater computational cost.

Acoustic VTI Modeling Using an Optimal Time-Space Domain Finite-Difference Scheme

2016 ◽  
Vol 24 (04) ◽  
pp. 1650016 ◽  
Author(s):  
Hongyong Yan ◽  
Lei Yang ◽  
Xiang-Yang Li ◽  
Hong Liu
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Taylor Series Expansion ◽  
Transversely Isotropic ◽  
High Accuracy ◽  
Seismic Exploration ◽  
Numerical Dispersion ◽  
Optimal Time ◽  
Space Domain ◽  
Time Space

Finite-difference (FD) schemes have been used widely for solving wave equations in seismic exploration. However, the conventional FD schemes hardly guarantee high accuracy at both small and large wavenumbers. In this paper, we propose an optimal time-space domain FD scheme for acoustic vertical transversely isotropic (VTI) wave modeling. The optimal FD coefficients for the second-order spatial derivatives are derived by approaching the time-space domain dispersion relation of acoustic VTI wave equations with the combination of the Taylor-series expansion and the sampling interpolation. We perform numerical dispersion analyses and acoustic VTI modeling using the optimal time-space domain FD scheme. The numerical dispersion analyses show that the optimal FD scheme has smaller dispersion than the conventional FD scheme at large wavenumbers, and also preserves high accuracy at small wavenumbers. The acoustic VTI modeling examples also demonstrate that the optimal time-space domain FD scheme has greater accuracy compared with the conventional time-space domain FD scheme for the same modeling parameters.

Sign in / Sign up

Export Citation Format

Share Document