Ray tracing in tilted transversely isotropic media: a group velocity approach

2004 ◽  
Author(s):  
Thomas A. Dickens
Keyword(s):  
Ray Tracing ◽  
Group Velocity ◽  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Isotropic Media

scholarly journals 3-D two‐point ray tracing for heterogeneous, weakly transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1883-1894 ◽  
Author(s):  
Vladimir Y. Grechka ◽  
George A. McMechan
Keyword(s):  
Ray Tracing ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Heterogeneous Media ◽  
Chebyshev Approximation ◽  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Derivatives Of

A two‐point ray‐tracing technique for 3-D smoothly heterogeneous, weakly transversely isotropic media is based on Fermat’s principle and takes advantage of global Chebyshev approximation of both the model and curved rays. This approximation gives explicit relations for derivatives of traveltime with respect to ray parameters and allows use of the rapidly converging conjugate gradient method to compute traveltimes. The method is fast because, for most smoothly heterogeneous media, approximation of rays by only a few polynomials and a few conjugate gradient iterations provide excellent precision in traveltime calculation.

Aberration‐free image for SH reflection in transversely isotropic media

Geophysics ◽  
1987 ◽  
Vol 52 (11) ◽  
pp. 1563-1565 ◽  
Author(s):  
J. M. Blair ◽  
J. Korringa
Keyword(s):  
Group Velocity ◽  
Isotropic Medium ◽  
Homogeneous Medium ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Elastic Symmetry ◽  
Source Point ◽  
Slowness Vector ◽  
Transversely Isotropic Media ◽  
Isotropic Media

This note is intended formulate and prove a theorem about shear (S) waves in a transversely isotropic medium for which we have found no reference in the literature. The theorem states the following: SH waves emanating from a point source in a homogeneous transversely isotropic medium are reflected from a planar interface between the transversely isotropic medium and another homogeneous medium in such a way that they define a reflective image that is free of aberrations, regardless of the relative orientation of the elastic symmetry axis and the interface. It is an image for rays in the direction of the group velocity vectors, not the slowness vectors. The image is located on a line through the source point in the direction of the group velocity of a wave for which the slowness vector is perpendicular to the interface. The distance, measured along this line, of the image behind the interface is equal to that of the source point in front. An analogous theorem for slowness vectors exists only for isotropic media, where it is trivial and coincides with the above.

scholarly journals Rational approximation of P-wave kinematics — Part 2: Orhorhombic media

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. C175-C186 ◽  
Author(s):  
Mohammad Mahdi Abedi
Keyword(s):  
Wave Propagation ◽  
Phase Velocity ◽  
Group Velocity ◽  
Transversely Isotropic ◽  
P Wave ◽  
Propagation Modeling ◽  
Wave Kinematics ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Wave Propagation Modeling

Orthorhombic anisotropy is a modern standard for 3D seismic studies in complex geologic settings. Several seismic data processing methods and wave propagation modeling algorithms in orthorhombic media rely on phase-velocity, group-velocity, and traveltime approximations. The algebraic simplicity of an approximate equation is an important factor in these media because the governing equations are more complicated than transversely isotropic media. To approximate the P-wave kinematics in acoustic orthorhombic media, we have developed a new 3D general functional equation that has a simple rational form. Using the general form, we adopt two versions of rational approximations for the phase velocity, group velocity, and traveltime. The first version uses a simpler functional form and parameter definition within the orthorhombic symmetry planes. The second version is more accurate, using one parameter that is defined out of the symmetry planes. For the phase velocity, we obtain another approximation that is no longer rational but is still algebraically simple, exact for 3D transversely isotropic media, and it is exact within the symmetry planes of orthorhombic media. We find superior accuracy in our approximations compared with previous ones, using numerical studies on multiple moderately anisotropic orthorhombic models. We investigate the effect of the negative anellipticity parameters on the accuracy and find that, in models in which the error of the existing most accurate approximations exceeds 2%, the error of the new approximations remains below 0.2%. The adopted approximations are algebraically simpler and stably more accurate than existing approximations; therefore, they may be considered as attractive alternatives for the existing approximations in many practical applications. We extend the applicability of our approximations by using them to obtain the equations of group direction as a function of phase direction and vice versa, which are useful in wave propagation modeling methods.

scholarly journals Rational approximation of P-wave kinematics — Part 1: Transversely isotropic media

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. C163-C173 ◽  
Author(s):  
Mohammad Mahdi Abedi
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Continued Fraction ◽  
Transversely Isotropic ◽  
P Wave ◽  
Algebraic Complexity ◽  
Wave Kinematics ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Phase Direction

In seismic data processing and several wave propagation modeling algorithms, the phase velocity, group velocity, and traveltime equations are essential. To have these equations in explicit form, or to reduce algebraic complexity, approximation methods are used. For the approximation of P-wave kinematics in acoustic transversely isotropic media, we have developed a new flexible 2D functional equation in a continued fraction form. Using different orders of the continued fraction, we obtain different approximations for (1) phase velocity as a function of phase direction, (2) group velocity as a function of group direction, and (3) traveltime as a function of offset. Then, we use them in the approximation of the group direction as a function of phase direction, and phase direction as a function of group direction. The proposed approximations have a rational form, which is considered algebraically simple and computationally efficient. The used continued fraction form rapidly converges to exact kinematics. By introducing the optimal ray into our approximations and using it for parameter definition, the convergence becomes faster, so the accuracy of the existing most accurate approximations is available by the third order, and new most accurate approximations are obtained by the fourth order of the proposed general form. The error of the most accurate version of the proposed approximations is below 0.001% for moderate anisotropic models with an anellipticity parameter up to 0.3. This high accuracy is considered to be attractive in practical implementations that use the kinematic equations and their derivatives.

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.

Variation of stacking velocity in transversely isotropic media

1995 ◽  
Vol 26 (2-3) ◽  
pp. 431-436 ◽  
Author(s):  
Patrick N.(Jr). Okoye ◽  
N. F. Uren ◽  
W. Waluyo
Keyword(s):  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Elastic Least-Squares Imaging in Tilted Transversely Isotropic Media for Multicomponent Land and Pressure Marine Data

2020 ◽  
Vol 41 (4) ◽  
pp. 805-833 ◽  
Author(s):  
Jidong Yang ◽  
Biaolong Hua ◽  
Paul Williamson ◽  
Hejun Zhu ◽  
George McMechan ◽  
...  
Keyword(s):  
Least Squares ◽  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Marine Data
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