Space–time far-field representation of Green’s functions for cross-plane shear waves in general transversely isotropic media

1997 ◽  
Vol 102 (2) ◽  
pp. 733-740 ◽  
Author(s):  
Martin Spies
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Shear Waves ◽  
Transversely Isotropic ◽  
Space Time ◽  
Far Field ◽  
Plane Shear ◽  
Field Representation ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Three-dimensional analysis of cracks in layered transversely isotropic media

1989 ◽  
Vol 424 (1867) ◽  
pp. 307-322 ◽  
Keyword(s):  
Integral Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Crack Opening ◽  
Mathematical Formulation ◽  
Transversely Isotropic ◽  
Boundary Integral ◽  
Opening Displacement ◽  
Transversely Isotropic Media ◽  
Isotropic Media

A general mathematical formulation to analyse cracks in layered transversely isotropic media is developed in this paper. By constructing the Green’s functions, an integral equation is obtained to determine crack opening displacements when an applied crack face traction is specified. For the infinite body, the Green’s functions have solutions in a closed form. For layered media, a flexibility matrix in the integral transformed domain is formed that establishes the relation between the traction and the displacement for a single layer; the global matrix is formed by assembling all of the flexibility matrices constructed for each layer. The Green’s functions in the spatial domain are obtained by inversion of the Hankel transform. Finally, the crack opening displacement and the crack-tip opening displacement for a vertical planar crack in a layered transversely isotropic medium are obtained numerically by the boundary integral equation method.

A point dislocation in a layered, transversely isotropic and self-gravitating Earth — Part II: accurate Green's functions

2019 ◽  
Vol 219 (3) ◽  
pp. 1717-1728 ◽  
Author(s):  
J Zhou ◽  
E Pan ◽  
M Bevis
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Transversely Isotropic ◽  
Earth Model ◽  
Transversely Isotropic Media ◽  
Love Numbers ◽  
Isotropic Media ◽  
Point Dislocation ◽  
High Degree ◽  
Harmonic Degree

SUMMARY We present an accurate approach for calculating the point-dislocation Green's functions (GFs) for a layered, spherical, transversely-isotropic and self-gravitating Earth. The formalism is based on the approach recently used to find analytical solutions for the dislocation Love numbers (DLNs). However, in order to make use of the DLNs, we first analyse their asymptotic behaviour, and then the behaviour of the GFs computed from the DLNs. We note that the summations used for different GF components evolve at different rates towards asymptotic convergence, requiring us to use two new and different truncation values for the harmonic degree (i.e. the index of summation). We exploit this knowledge to design a Kummer transformation that allows us to reduce the computation required to evaluate the GFs at the desired level of accuracy. Numerical examples are presented to clarify these issues and demonstrate the advantages of our approach. Even with the Kummer transformation, DLNs of high degree are still needed when the earth model contains very fine layers, so computational efficiency is important. The effect of anisotropy is assessed by comparing GFs for isotropic and transversely isotropic media. It is shown that this effect, though normally modest, can be significant in certain contexts, even in the far field.

A Method to Evaluate Three-Dimensional Time-Harmonic Elastodynamic Green’s Functions in Transversely Isotropic Media

1992 ◽  
Vol 59 (2S) ◽  
pp. S96-S101 ◽  
Author(s):  
H. Zhu
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Isotropic Case ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Transversely Isotropic Media ◽  
Time Harmonic ◽  
Isotropic Media

The three-dimensional time-harmonic elastodynamic Green’s functions in infinite transversely isotropic media have been derived explicitly. The Green’s functions consist of the corresponding static Green’s functions and double integral representations over a finite domain with the integrands being continuous. The Green’s functions will reduce to those for the isotropic case when the isotropic elastic constants are substituted. The singular parts of the Green’s functions have been shown to be the same as those of the static ones. The far-field approximations have been obtained by using the stationary phase method. In addition, a simpler method to construct wave front curves has been presented.

Shear waves in acoustic transversely isotropic media

2004 ◽  
Author(s):  
Vladimir Grechka ◽  
Linbin Zhang ◽  
James W. Rector
Keyword(s):  
Shear Waves ◽  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Shear‐wave vibrator signals in transversely isotropic shale

Geophysics ◽  
1985 ◽  
Vol 50 (8) ◽  
pp. 1285-1293 ◽  
Author(s):  
Sheila Peacock ◽  
Stuart Crampin
Keyword(s):  
Shear Wave ◽  
Shear Waves ◽  
Transversely Isotropic ◽  
Synthetic Seismograms ◽  
Wave Polarization ◽  
Transversely Isotropic Media ◽  
Wave Splitting ◽  
Isotropic Media ◽  
Component Field ◽  
Theoretical Results

The experiments of Robertson and Corrigan (1983) on shale are among the first three‐component field observations of shear waves in transversely isotropic media to be published. Their data are reprocessed to highlight the effects of the shale’s anisotropy on shear waves. Two results emerge. First, shear‐wave splitting in a transversely isotropic substrate is most easily observed when the vibrator baseplate is oriented so that both SH‐ and SV‐waves reach the geophone. Second, the SV‐wave polarization deviates significantly from perpendicular to the raypath. Both results may significantly affect the interpretation. Both are found to agree with theoretical results and are modeled successfully by synthetic seismograms.

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
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