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.

Tensorial elastodynamics for isotropic media

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. T359-T373
Author(s):  
Jeffrey Shragge ◽  
Tugrul Konuk
Keyword(s):  
Free Surface ◽  
Surface Topography ◽  
Numerical Solutions ◽  
Waveform Inversion ◽  
Real Life ◽  
Staggered Grid ◽  
Surface Boundary ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration

Numerical solutions of 3D isotropic elastodynamics form the key computational kernel for many isotropic elastic reverse time migration and full-waveform inversion applications. However, real-life scenarios often require computing solutions for computational domains characterized by non-Cartesian geometry (e.g., free-surface topography). One solution strategy is to compute the elastodynamic response on vertically deformed meshes designed to incorporate irregular topology. Using a tensorial formulation, we have developed and validated a novel system of semianalytic equations governing 3D elastodynamics in a stress-velocity formulation for a family of vertically deformed meshes defined by Bézier interpolation functions between two (or more) nonintersecting surfaces. The analytic coordinate definition also leads to a corresponding analytic free-surface boundary condition (FSBC) as well as expressions for wavefield injection and extraction. Theoretical examples illustrate the utility of the tensorial approach in generating analytic equations of 3D elastodynamics and the corresponding FSBCs for scenarios involving free-surface topography. Numerical examples developed using a fully staggered grid with a mimetic finite-difference formulation demonstrate the ability to model the expected full-wavefield behavior, including complex free-surface interactions.

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.

Efficient 3D elastic full-waveform inversion using wavefield reconstruction methods

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. R45-R55 ◽  
Author(s):  
Espen Birger Raknes ◽  
Wiktor Weibull
Keyword(s):  
Particle Velocity ◽  
Large Scale ◽  
Waveform Inversion ◽  
Transversely Isotropic ◽  
Reconstruction Method ◽  
Full Waveform Inversion ◽  
Reverse Time ◽  
Full Waveform ◽  
Time Migration ◽  
Reconstruction Methods

In reverse time migration (RTM) or full-waveform inversion (FWI), forward and reverse time propagating wavefields are crosscorrelated in time to form either the image condition in RTM or the misfit gradient in FWI. The crosscorrelation condition requires both fields to be available at the same time instants. For large-scale 3D problems, it is not possible, in practice, to store snapshots of the wavefields during forward modeling due to extreme storage requirements. We have developed an approximate wavefield reconstruction method that uses particle velocity field recordings on the boundaries to reconstruct the forward wavefields during the computation of the reverse time wavefields. The method is computationally effective and requires less storage than similar methods. We have compared the reconstruction method to a boundary reconstruction method that uses particle velocity and stress fields at the boundaries and to the optimal checkpointing method. We have tested the methods on a 2D vertical transversely isotropic model and a large-scale 3D elastic FWI problem. Our results revealed that there are small differences in the results for the three methods.

Reverse time migration in tilted transversely isotropic (TTI) media

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA179-WCA187 ◽  
Author(s):  
Robin P. Fletcher ◽  
Xiang Du ◽  
Paul J. Fowler
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
Transversely Isotropic ◽  
Shear Velocity ◽  
Vertical Axis ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Axis Of Symmetry ◽  
Tti Media

Reverse time migration (RTM) exhibits great advantages over other imaging methods because it is based on computing numerical solutions to a two-way wave equation. It does not suffer from dip limitation like one-way downward continuation techniques do, thus enabling overturned reflections to be imaged. As well as correctly handling multipathing, RTM has the potential to image internal multiples when the boundaries responsible for generating the multiples are present in the model. In isotropic media, one can use a scalar acoustic wave equation for RTM of pressure data. In anisotropic media, P- and SV-waves are coupled together so, formally, elastic wave equations must be used for RTM. A new wave equation for P-waves is proposed in tilted transversely isotropic (TTI) media that can be solved as part of an acoustic anisotropic RTM algorithm, using standard explicit finite differencing. If the shear velocity along the axis of symmetry is set to zero, stable numerical solutions can be computed for media with a vertical axis of symmetry and [Formula: see text] not less than [Formula: see text]. In TTI media with rapid variations in the direction of the axis of symmetry, setting the shear velocity along the axis of symmetry to zero can cause numerical solutions to become unstable. A solution to this problem is proposed that involves using a small amount of nonzero shear velocity. The amount of shear velocity added is chosen to remove triplications from the SV wavefront and to minimize the anisotropic term of the SV reflection coefficient. We show modeling and high-quality RTM results in complex TTI media using this equation.

Decoupled equations for reverse time migration in tilted transversely isotropic media

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. T37-T45 ◽  
Author(s):  
Ge Zhan ◽  
Reynam C. Pestana ◽  
Paul L. Stoffa
Keyword(s):  
Broad Band ◽  
Transversely Isotropic ◽  
Rapid Expansion ◽  
P Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Coupled Equations ◽  
Time Migration ◽  
Tti Media ◽  
Wave Modes

Conventional modeling and migration for tilted transversely isotropic (TTI) media may suffer from numerical instabilities and shear wave artifacts due to the coupling of the P-wave and SV-wave modes in the TTI coupled equations. Starting with the separated P- and SV-phase velocity expressions for vertical transversely isotropic (VTI) media, we extend these decoupled equations for modeling and reverse time migration (RTM) in acoustic TTI media. Compared with the TTI coupled equations published in the geophysical literature, the new TTI decoupled equations provide a more stable solution due to the complete separation of the P-wave and SV-wave modes. The pseudospectral method is the most convenient method to implement these equations due to the form of wavenumber expressions and has the added benefit of being highly accurate and thus avoiding numerical dispersion. The rapid expansion method (REM) in time is employed to produce a broad band numerically stable time evolution of the wavefields. Synthetic results validate the proposed TTI decoupled equations and show that modeling and RTM in TTI media with the decoupled equations remain numerically stable even for models with strong anisotropy and sharp contrasts.

scholarly journals An explicit time evolution method for acoustic wave propagation

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. T117-T124 ◽  
Author(s):  
Huafeng Liu ◽  
Nanxun Dai ◽  
Fenglin Niu ◽  
Wei Wu
Keyword(s):  
Wave Propagation ◽  
Acoustic Wave ◽  
Time Evolution ◽  
Waveform Inversion ◽  
Cost Effective ◽  
Constant Density ◽  
Exact Nature ◽  
Reverse Time ◽  
Time Migration ◽  
Time Space

Cost-effective waveform modeling is the key to practical reverse time migration (RTM) and full-waveform inversion (FWI) implementations. We evaluated an explicit time evolution (ETE) method to efficiently simulate wave propagation in acoustic media with high temporal accuracy. We started from the constant-density acoustic wave equation and obtained an analytical time-marching scheme in the wavenumber domain. We then formulated an ETE scheme in the time-space domain by introducing a cosine function approximation. Although the ETE operator appears to be similar to the second-order temporal finite-difference (FD) operator, the exact nature of the ETE formula ensures high accuracy in time. We further introduced a set of optimum stencils and coefficients by minimizing evolution errors in a least-squares sense. Our numerical tests indicated that ETE can achieve similar waveform accuracy as FD with four times larger time steps. Meanwhile, the compact ETE operator keeps the computation efficient. The efficiency and capability to handle complex velocity field make ETE an attractive engine in acoustic RTM and FWI.

Tensorial Elastodynamics for Anisotropic Media

Geophysics ◽  
2021 ◽  
pp. 1-60
Author(s):  
Tugrul Konuk ◽  
Jeffrey Shragge
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Anisotropic Media ◽  
Essential Elements ◽  
Constitutive Relationship ◽  
Staggered Grid ◽  
General Strategy ◽  
Reverse Time ◽  
Time Migration

Elastic wavefield solutions computed by finite-difference (FD) methods in complex anisotropic media are essential elements of elastic reverse-time migration and full waveform inversion analyses. Cartesian formulations of such solution methods, though, face practical challenges when aiming to represent curved interfaces (including free-surface topography) with rectilinear elements. To forestall such issues, we propose a general strategy for generating solutions of tensorial elastodynamics for anisotropic media (i.e., tilted transversely isotropic (TTI) or even lower symmetry) in non-Cartesian computational domains. For the specific problem of handling free-surface topography, we define an unstretched coordinate mapping that transforms an irregular physical domain to a regular computational grid on which FD solutions of the modified equations of elastodynamics are straightforward to calculate. Our fully staggered grid with a mimetic finite-difference (FSG+MFD) approach solves the velocity-stress formulation of the tensorial elastic wave equation where we compute the stress-strain constitutive relationship in Cartesian coordinates and then transform the resulting stress tensor to generalized coordinates to solve the equations of motion. The resulting FSG+MFD numerical method has a computational complexity comparable to Cartesian scenarios using a similar FSG+MFD numerical approach. Numerical examples demonstrate that the proposed solution can simulate anisotropic elastodynamic field solutions on irregular geometries and is thus a reliable tool for anisotropic elastic modeling, imaging and inversion applications in generalized computational domains including handling free-surface topography.

Sign in / Sign up

Export Citation Format

Share Document