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On Generalized Ray Parameters For Vertically Inhomogeneous and Anisotropic Media

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... This property holds for both isotropic and anisotropic media regardless of the type of body wave generated at the boundary (longitudinal or transverse) and forms the basis for our strategy of calculating reflected and transmitted angles. The approach following Fermat's principle can be formulated through the calculus of variations (e.g., Slawinski and Webster, 1999). ...
... Third, Fermat's principle of stationary time must be satisfied (e.g., Helbig, 1994). This might not be obvious from a quick inspection, and it might require a reformulation in terms of the Euler-Lagrange equation (e.g., Slawinski and Webster, 1999). ...
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We have reformulated the law governing the refraction of rays at a planar interface separating two anisotropic media in terms of slowness surfaces. Equations connecting ray directions and phase-slowness angles are derived using geometrical properties of the gradient operator in slowness space. A numerical example shows that, even in weakly anisotropic media, the ray trajectory governed by the anisotropic Snell’s law is significantly different from that obtained using the isotropic form. This could have important implications for such considerations as imaging (e.g., migration) and lithology analysis (e.g., amplitude variation with offset). Expressions are shown specifically for compressional (qP) waves but they can easily be extended to SH waves by equating the anisotropic parameters (i.e., ε=δ⇒γ ) and to qSV and converted waves by similar means. The analytic expressions presented are more complicated than the standard form of Snell’s law. To facilitate practical application, we include our Mathematica code.
... This property holds for both isotropic and anisotropic media regardless of the type of body wave generated at the boundary (longitudinal or transverse) and forms the basis for our strategy of calculating reflected and transmitted angles. The approach following Fermat's principle can be formulated through the calculus of variations (e.g., Slawinski and Webster, 1999). Present exploration methods of data acquisition and processing offer the potential to investigate certain subtle characteristics of the subsurface. ...
... Third, Fermat's principle of stationary time must be satisfied (e.g., Helbig, 1994). This might not be obvious from a quick inspection, and it might require a reformulation in terms of the Euler-Lagrange equation (e.g., Slawinski and Webster, 1999). ...
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Snell’s law is a direct consequence of Fermat’s principle of stationary time. In horizontally layered media, it can be conveniently restated as a requirement for the horizontal component of the wave number, kx, to be continuous across the boundary. As a metter of fact, the horizontal component of the wave number remains constant for all layers, i.e., the ray parameter. This property must be preserved for both isotropic and anisotropic media regardless of the type of the wave generated at the boundary, e.g., longitudinal or transverse, and serves as a kernel for the strategy of calculating reflected and transmitted angles.
... Such an analysis is complicated even for relatively simple cases, such as horizontally layered media. Ray theory, which is invoked in this work, provides mathematical tools that simplify the analysis (e.g., Keller (1978);Červený (1985); Shearer and Chapman (1988); Slawinski and Webster (1999); Wang (2014); Slawinski et al. (2003Slawinski et al. ( , 2004). ...
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We present a strategy for selecting the values of elasticity parameters by comparing walk-away vertical seismic profiling data with a multilayered model in the context of Bayesian Information Criterion. We consider $P$-wave traveltimes and assume elliptical velocity dependence. The Bayesian Information Criterion approach requires two steps of optimization. In the first step, we find the signal trajectory and, in the second step, we find media parameters by minimizing the misfit between the model and data.
... The ray parameter is generally taken as horizontal phase slowness, p x ¼ sin ϕ∕vðz; ϕÞ, defined by the phase angle and the phase velocity. For the vertically variable 1D anisotropic velocity model, Vðx; z; θÞ ¼ Vðz; θÞ, the ray parameter can be expressed analytically in terms of the ray angle and the group velocity along a raypath (Slawinski and Webster, 1999), as follows ...
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Seismic ray tracing with a path bending method leads to a nonlinear system that has much higher nonlinearity in anisotropic media than the counterpart in isotropic media. Any path perturbation causes changes in directional velocity which depends on not only the spatial position but also the local velocity direction in anisotropic media. To combat the high nonlinearity of the problem, Newton-type iterative algorithm is modified by enforcing Fermat’s minimum-time principle as a constraint for the solution update, instead of conventional error minimization in the nonlinear system. As the algebraic problem is integrated with the physical principle, the solution is robust for such a high-nonlinear problem as ray tracing in realistically complicated anisotropic media. This modified algorithm is applied to two ray-tracing schemes. The first scheme is newly derived raypath equations. The latter are approximate for anisotropic media, but the minimum-time constraint will ensure the solution steadily converges to the true solution. The second scheme based on the minimal variation principle is more efficient, as it solves a tridiagonal system in each iteration and does not need to compute the Jacobian and its inverse. Even in this second scheme Fermat’s minimum-time constraint is still employed for the solution update, as same as the other, to guarantee a robust convergence of the iterative solution in anisotropic media.
... as shown by Slawinski and Webster [12]. For isotropic, vertically nonuniform media, the second term in expression (11) vanishes and the raypath parameter is immediately reduced to the standard form of Snell's law. ...
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This study is based on Fermat’s principle of stationary traveltime in the context of perfect elasticity wherein the velocity of a signal is solely a function of its direction and position. A general formulation of raypaths as parametric curves in horizontally uniform and arbitrarily anisotropic media is used to calculate raypaths and traveltimes. (In this paper, uniformity is equivalent to homogeneity. The former term is used in order to distinguish between the physical property of a medium and the mathematical property of a function.) The nonuniformity of the media exhibits horizontal symmetry, which is associated with the mathematically convenient and physically insightful concept of conserved quantities. The anisotropy of the media is described by elementary wavefronts whose size, orientation and shape can change from point to point along the vertical axis. The general formulation is exemplified by a horizontally uniform, elliptically anisotropic medium.
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We discuss, improve, and apply the slowness-polarization method for estimating local anisotropy from VSP data. Although the idea of fitting a given anisotropic model to the apparent slownesses measured along a well and polarization vectors recorded by three-component downhole geophones is hardly new, we extend the area of applicability of the technique and make the anisotropic inversion more robust by eliminating the most operationally difficult and noisy portion of the data, the shear waves. We show that the shear-wave velocity is actually unnecessary for fitting the slowness-of-polarization dependence of P-wave VSP data. For the most common geometry of a vertical borehole in a vertically transversely isotropic subsurface, such data are governed by the P-wave vertical velocity V(P0) and two quantities, delta(VSP) and eta(VSP), that describe the influence of anisotropy. These quantities depend on conventional anisotropic coefficients delta and eta and absorb the S-wave velocity. We apply the developed theory to a 2D walkaway VSP acquired over a subsalt prospect in the Gulf of Mexico. Our data set contains geophones placed both inside the salt and beneath it, allowing us to estimate the anisotropy of different rock formations. We find the: salt to be nearly isotropic in the examined 1200 ft (360 m) depth interval. In contrast, the sediments below the salt exhibit substantial anisotropy. While the physical origins of subsalt anisotropy are still to be fully understood, we observe a clear correlation between lithology and the values of delta(VSP) and delta: both anisotropic coefficients are greater in shales and smaller in the sandier portion of the well.
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The recent advances in determination of fracture strike and crack density from P-wave seismic data was presented. It was shown that the amplitude variation with angle and azimuth (AVAZ) analysis technique correctly identifies the orientation and relative intensity of open, fluid filled fractures at depth in many cases. The technique uses prestacked data for analyzing horizontal transverse isotropic (HTI) media for seismic anisotropy. The information gathered using the technique is used to determine reservoir parameters like fracture permeability and for increasing production efficiency of naturally fractured reservoirs.
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Contenido: Primera variación; Segunda variación; Generalización de los resultados de los capítulos anteriores; Máxima y mínima relativas y problemas isoperimetricales; Principio de Hamilton y el principio de la última acción; Principio de Hamilton en la teoría especial de la relatividad; Métodos aproximativos con aplicaciones a problemas de elasticidad; Integrales con puntos finales vairables. Inegral de Hilbert; Variables fuertes y la función E weierstrassiana.
Lessons in seismic computing: published by SEG
  • M M Slotnick
Ray Parameters and the modeling of complex features: CSEG National Convention; Abstracts
  • M Epstein
  • M A Slawinski