Explicit expressions for the Fréchet derivatives in 3D anisotropic resistivity inversion

Geophysics ◽  
2009 ◽  
Vol 74 (3) ◽  
pp. F31-F43 ◽  
Author(s):  
S. A. Greenhalgh ◽  
B. Zhou ◽  
M. Greenhalgh ◽  
L. Marescot ◽  
T. Wiese
Keyword(s):  
Critical Factor ◽  
Transversely Isotropic ◽  
Special Cases ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Tti Media ◽  
Numerical Finite Element ◽  
Dip Angle ◽  
Axes Of Symmetry ◽  
Explicit Expressions

We have developed explicit expressions for the Fréchet derivatives or sensitivity functions in resistivity imaging of a heterogeneous and fully anisotropic earth. The formulation involves the Green’s functions and their gradients, and it is developed from a formal perturbation analysis and by means of a numerical (finite-element) method. A critical factor in the equations is the derivative of the electrical conductivity tensor with respect to the principal conductivity values and the angles defining the axes of symmetry. The Fréchet derivative expressions were derived for the 2.5D and 3D problems using constant-point and constant-block model parameterizations. Special cases such as an isotropic earth and tilted transversely isotropic (TTI) media emerge from the general solutions. Numerical examples were investigated for various sensitivities as functions of dip angle and strike of the plane of stratification in uniform TTI media.

Moveout approximations for P- and SV-waves in dip-constrained transversely isotropic media

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. C53-C59 ◽  
Author(s):  
Véronique Farra ◽  
Ivan Pšenčík
Keyword(s):  
Transversely Isotropic ◽  
Arbitrary Orientation ◽  
Transversely Isotropic Medium ◽  
Weak Anisotropy ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Dip Angle ◽  
P And Sv Waves ◽  
Axes Of Symmetry

We generalize the P- and SV-wave moveout formulas obtained for transversely isotropic media with vertical axes of symmetry (VTI), based on the weak-anisotropy approximation. We generalize them for 3D dip-constrained transversely isotropic (DTI) media. A DTI medium is a transversely isotropic medium whose axis of symmetry is perpendicular to a dipping reflector. The formulas are derived in the plane defined by the source-receiver line and the normal to the reflector. In this configuration, they can be easily obtained from the corresponding VTI formulas. It is only necessary to replace the expression for the normalized offset by the expression containing the apparent dip angle. The final results apply to general 3D situations, in which the plane reflector may have arbitrary orientation, and the source and the receiver may be situated arbitrarily in the DTI medium. The accuracy of the proposed formulas is tested on models with varying dip of the reflector, and for several orientations of the horizontal source-receiver line with respect to the dipping reflector.

Analysis of reflection traveltimes in 3-D transversely isotropic heterogeneous media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1884-1895 ◽  
Author(s):  
Vladimir Y. Grechka ◽  
George A. McMechan
Keyword(s):  
Ray Tracing ◽  
Heterogeneous Media ◽  
Chebyshev Approximation ◽  
Transversely Isotropic ◽  
Singular Value ◽  
Common Source ◽  
Model Parameters ◽  
Value Analysis ◽  
Frechet Derivatives ◽  
Fréchet Derivatives

A two‐point ray‐tracing technique for rays reflected from irregular, but smooth, interfaces in 3-D transversely isotropic heterogeneous media is developed. The method is based on Chebyshev parameterization of curved segments of the reflected rays, of the reflectors, and of the velocity and anisotropy distributions in the model. Chebyshev approximation also can describe the reflection traveltime surfaces to compress traveltime data by replacing them with coefficients of the corresponding Chebyshev series. The advantage of the proposed parameterization is that it gives traveltime as an explicit function of the model parameters. This explicitly provides the Frechét derivatives of the traveltime with respect to the model parameters. The Frechét derivatives are used in two ways. First, a two‐term Taylor series is constructed to relate variations in the model parameters to the corresponding perturbations in the traveltimes. This makes it possible, based on the results of a single ray tracing in a relatively simple model, to predict traveltimes for a range of more complicated models, without any additional ray tracing. Second, singular‐value decomposition of the Frechét matrix determines the influence of various model parameters on common‐source and common‐midpoint traveltimes. The singular‐value analysis shows that common‐source traveltimes depend mainly on the reflector position and shape. The common‐midpoint traveltimes also contain additional information about lateral velocity heterogeneity and anisotropy. However, both of these parameters affect the traveltimes in similar ways and so usually cannot be determined separately.

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.

Weak compactness of Fréchet-derivatives: application to composition operators

1980 ◽  
Vol 29 (4) ◽  
pp. 399-406
Author(s):  
Peter Dierolf ◽  
Jürgen Voigt
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Sobolev Spaces ◽  
Composition Operators ◽  
Weak Compactness ◽  
Nonlinear Integral Equations ◽  
Weakly Compact ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Compact Map

AbstractWe prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

scholarly journals Fréchet Derivatives For Curved Interfaces In the Ray Approximation

1989 ◽  
Vol 97 (3) ◽  
pp. 497-509 ◽  
Author(s):  
Robert L. Nowack ◽  
Jeff A. Lyslo
Keyword(s):  
Frechet Derivatives ◽  
Fréchet Derivatives

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.

Axisymmetric Boussinesq problem of a transversely isotropic half space with surface effects

2018 ◽  
Vol 24 (5) ◽  
pp. 1425-1437 ◽  
Author(s):  
Jing Jin Shen
Keyword(s):  
Half Space ◽  
Mixed Boundary ◽  
Contact Stiffness ◽  
Surface Elasticity ◽  
Transversely Isotropic ◽  
Surface Effects ◽  
Building Blocks ◽  
Material Anisotropy ◽  
Residual Surface Stress ◽  
Special Cases

A transversely isotropic half space with surface effects subjected to axisymmetric loadings is investigated in terms of the Lekhnitskii formulism. Surface effects including residual surface stress and surface elasticity are introduced by using the Gurtin–Murdoch continuum model. With the aid of the Hankel transforms, solutions corresponding to several different axisymmetic loadings are derived and used to determine the influence of surface effects on contact stiffness in nanoindentations. Numerical results are provided to show the influence of surface effects and material anisotropy on the material behaviours. Meanwhile, the obtained analytical Green’s functions for two special cases can be used as building blocks for further mixed boundary value problems.

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