Acoustic VTI modeling using high-order finite differences

Stig Hestholm

doi:10.1190/1.3157242

Acoustic VTI modeling using high-order finite differences

Geophysics ◽

10.1190/1.3157242 ◽

2009 ◽

Vol 74 (5) ◽

pp. T67-T73 ◽

Cited By ~ 46

Author(s):

Stig Hestholm

Keyword(s):

Finite Differences ◽

Wave Equations ◽

Transversely Isotropic ◽

High Order ◽

Isotropic Layer ◽

Natural Form ◽

Near Surface ◽

First Order ◽

Seismic Reflectors ◽

Vti Media

Two second-order wave equations for acoustic vertical transversely isotropic (VTI) media are transformed to six first-order coupled partial differential equations for a more straighforward numerical implementation of the derivatives. The resulting first-order equations have a more natural form for discretization by any finite-difference, pseudospectral, or finite-element method. I discretized the new equations by high-order finite differences and used synthetic seismograms and snapshots for anisotropic and isotropic cases. The relative merits of placing the source deep and close to a free surface are assessed, illustrating advantages of exciting the source inside or outside of a near-surface, thin, isotropic layer. Results show that traveltimes from deep seismic reflectors can remain virtually unaffected when near-surface isotropic layers are included in acoustic VTI media.

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

Geophysics ◽

10.1190/geo2017-0472.1 ◽

2018 ◽

Vol 83 (3) ◽

pp. T139-T157 ◽

Cited By ~ 3

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.

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Staggered-Grid High-Order Differencemethod for the First-Order Elastic Wave Equations

Chinese Journal of Geophysics ◽

10.1002/cjg2.54 ◽

2000 ◽

Vol 43 (3) ◽

pp. 441-449 ◽

Cited By ~ 20

Author(s):

Liang-Guo DONG ◽

Zai-Tian MA ◽

Jing-Zhong CAO

Keyword(s):

Elastic Wave ◽

Wave Equations ◽

High Order ◽

Staggered Grid ◽

First Order ◽

Elastic Wave Equations

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

Geophysics ◽

10.1190/geo2016-0589.1 ◽

2017 ◽

Vol 82 (5) ◽

pp. T225-T235 ◽

Cited By ~ 6

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.

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Hybrid Fourier finite-difference 3D depth migration for anisotropic media

Geophysics ◽

10.1190/1.2828704 ◽

2008 ◽

Vol 73 (2) ◽

pp. S27-S34 ◽

Cited By ~ 25

Author(s):

Tong W. Fei ◽

Christopher L. Liner

Keyword(s):

Wave Equation ◽

Finite Difference ◽

Acoustic Wave ◽

Seismic Data ◽

Transversely Isotropic ◽

Acoustic Wave Equation ◽

Near Surface ◽

Depth Migration ◽

Surface Distortions ◽

Vti Media

When a subsurface is anisotropic, migration based on the assumption of isotropy will not produce accurate migration images. We develop a hybrid wave-equation migration algorithm for vertical transversely isotropic (VTI) media based on a one-way acoustic wave equation, using a combination of Fourier finite-difference (FFD) and finite-difference (FD) approaches. The hybrid method can suppress an additional solution that exists in the VTI acoustic wave equation, and it offers speed and other advantages over conventional FFD or FD methods alone. The algorithm is tested on a synthetic model involving log data from onshore eastern Saudi Arabia, including estimates of both intrinsic and layer-induced VTI parameters. Results indicate that VTI imaging in this region offers some improvement over isotropic imaging, primarily with respect to subtle structure and stratigraphy and to image continuity. These benefits probably will be overshadowed by perennial land seismic data issues such as near-surface distortions and multiples.

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First-order systems for elastic and acoustic variable-tilt TI media

Geophysics ◽

10.1190/geo2011-0249.1 ◽

2012 ◽

Vol 77 (5) ◽

pp. T157-T170 ◽

Cited By ~ 4

Author(s):

Kenneth P. Bube ◽

Tamas Nemeth ◽

Joseph P. Stefani ◽

Wei Liu ◽

Kurt T. Nihei ◽

...

Keyword(s):

Shear Wave ◽

Lower Order ◽

Elastic System ◽

Transversely Isotropic ◽

Order System ◽

Wave Speeds ◽

First Order ◽

Vti Media ◽

Ti Media ◽

Lower Order Terms

We derive and compare first-order wave propagation systems for variable-tilt elastic and acoustic tilted transversely isotropic (TTI) media. Acoustic TTI systems are commonly used in reverse-time migration. Starting initially with homogeneous vertical transversely isotropic (VTI) media, and then extending to heterogeneous variable-tilt TI media, we derive a pseudoacoustic [Formula: see text] first-order system of differential equations by setting the shear-wave speeds to zero and simplifying the full-elastic system accordingly. This [Formula: see text] system conserves a complete energy, but only when the anelliptic anisotropy parameter [Formula: see text]. For [Formula: see text] (including isotropic media), the system allows linearly time-growing and spatially nonpropagating nonphysical solutions frequently taken for numerical noise. We modified this [Formula: see text] acoustic first-order system by changing the stress variables to obtain a system that stays stable for [Formula: see text]. This system for homogeneous VTI media is generalized to heterogeneous variable-tilt TI media by rotating the stress and strain variables in the full elastic system before setting the shear-wave speeds to zero; the system obtained can be greatly simplified by combining the rotational terms, resulting in only one rotation and extra lower-order terms compared to the [Formula: see text] first-order acoustic system for VTI media. This new system can be simplified further by neglecting the lower-order terms. Both systems (with and without lower-order terms) conserve the same complete energy. Finally, the corresponding [Formula: see text] full elastic system for variable-tilt acoustic TI media can be used for the purposes of benchmarking.

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A one-way dual-domain propagator for scalar qP-waves in VTI media

Geophysics ◽

10.1190/1.1884826 ◽

2005 ◽

Vol 70 (2) ◽

pp. D9-D17 ◽

Cited By ~ 23

Author(s):

Qiyu Han ◽

Ru-Shan Wu

Keyword(s):

Dispersion Relation ◽

Phase Shift ◽

Wave Equations ◽

Large Angle ◽

Transversely Isotropic ◽

Transversely Isotropic Media ◽

Isotropic Media ◽

Elastic Wave Equations ◽

Dual Domain ◽

Vti Media

In this paper, we present an anisotropic one-way propagator for modeling and imaging quasi-P (qP) waves in transversely isotropic media with a vertically symmetric axis (VTI media). We derive the dispersion relation for a scalar qP-wave using elastic wave equations for anisotropic media. By applying a rational approximation to the dispersion relation, we obtain a one-way, dual-domain, scalar qP-wave propagator for heterogeneous VTI media. The propagator includes a phase-shift term and both phase-screen and large-angle correction terms. The phase-shift term is implemented in the wavenumber domain, while the other terms are implemented in the space domain. Fourier transformations are used to shuttle the wavefield between the two domains. This propagator can be used to propagate qP-wavefields within an isotropic or a VTI medium, with either medium containing lateral heterogeneities. Error analysis of the impulse response and dispersion relations demonstrates that the propagator is accurate and stable and has a wide-angle capability. The application of the propagator to the imaging of qP-wave data with VTI models which contain complex structures and large perturbations of velocity and anisotropy results in excellent image quality. This demonstrates the potential value of the propagator for use in modeling and imaging qP-wavefields within strongly heterogeneous VTI media.

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