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The importance of ray parameters as first integrals of the ray equation does not appear to have been sufficiently recognized in Geophysical theory or applications. In fact, the existence of ray parameters in fairly general propagation situations can be exploited to obtain explicit exact solutions for the rays by integration of first- (rather than second-) order differential equations. Making use of Noether’s theorem, which associates conserved quantities with symmetries with the Lagrangian afforded by Fermat’s variational principle, we derive ray parameters for a class of inhomogeneous media, denoted as pseudo-linear, whose characteristics are adjustable to represent a wide variety of structural features in a realistic and computationally convenient way. Exact solutions for the rays are obtained in a surprisingly wide variety of speed profiles, this permitting a qualitative and quantitative investigation of certain phenomena, such as the existence of a definite zone of accessibility from a given source, the exact location of the reflexion points in certain cases, and other general features. In particular, a sub-class of pseudo-linear media provide exact ray solution for complex geological profiles that can be interpreted as dipping layers, synclines or anticlines, according to the numerical value of controllable parameters in the mathematical formulas.
... In geometrical optics, the conserved quantity (9) is a generalized raypath parameter (e.g. [3]) while in classical mechanics, this conserved quantity is the generalized momentum (e.g. [9]). ...
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.
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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