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 Volume 41, Issue 4, 1993
Geophysical Prospecting  Volume 41, Issue 4, 1993
Volume 41, Issue 4, 1993


ANISOTROPIC TRAVELTIME TOMOGRAPHY
Authors R. J. MICHELENA, F. MUIR and J. M. HARRISAbstractVelocity estimation technique using seismic data is often based on time/distance equations which are independent of direction, and even though we now know that many rocks are quite anisotropic, useful results have been obtained over the years from these isotropic estimates. Nevertheless, if velocities are significantly direction‐dependent, then the isotropic assumption may lead to serious structural interpretation errors. Additionally, information on angle‐dependence may lead to a better understanding of the lithology of the rocks under measurement. VSP and cross‐well data may each lack the necessary aperture to estimate more than two velocity parameters for each wave type, and if the data straddle a symmetry axis, then these may be usefully chosen to be the direct velocities (from time‐and‐distance measurements along the axis) and NMO velocities (from differential time‐offset measurements). These sets of two parameters define ellipses, and provide intermediate models for the variation of velocity with angle which can later be assembled and translated into estimates of the elastic moduli of the rocks under scrutiny.
If the aperture of the measurements is large enough e.g. we have access to both VSP and cross‐well data, we divide the procedure into two independent steps, first fitting best ellipses around one symmetry axis and then fitting another set around the orthogonal axis. These sets of four elliptical parameters are then combined into a new, double elliptical approximation. This approximation keeps the useful properties of elliptical anisotropy, in particular the simple relation between group and phase velocities which simplifies the route from the traveltimes measurements to the elastic constants of the medium.
The inversion proposed in this paper is a simple extension of well‐known isotropic schemes and it is conceptually identical for all wave types. Examples are shown to illustrate the application of the technique to cross‐well synthetic and field P‐wave data. The examples demonstrate three important points: (a) When velocity anisotropy is estimated by iterative techniques such as conjugate gradients, early termination of the iterations may produce artificial anisotropy. (b) Different components of the velocity are subject to different type of artifacts because of differences in ray coverage, (c) Even though most rocks do not have elliptical dispersion relations, our elliptical schemes represent a useful intermediate step in the full characterization of the elastic properties.



THE MAGNETIC ANOMALY OF 3D SOURCES HAVING ARBITRARY GEOMETRY AND MAGNETIZATION
More LessAbstractMagnetic anomalies of complicated 3D sources can be calculated by using a combination of analytical and numerical integration. Two surfaces and the magnetization parameters (the amplitudes of the induced and remanent components and the direction cosines) of the source can be defined by arbitrary functions or by discrete data points in a plane. When combined with a polynomial magnetization function in the direction of the third axis, 3D magnetization distribution can also be modelled.
The method gives very general equations for anomaly calculation. It can be used for direct modelling of sources interpreted by seismic or other methods and also for interactive interpretation with fast computers. It is possible to calculate anomalies of, for example, intrusives or folded sedimentary beds whose surfaces are functions of horizontal coordinates and which have polynomial magnetization variations in the vertical direction due to gravitational differentiation and arbitrarily varying magnetization in the horizontal direction due to regional metamorphosis.
If the distribution of magnetization parameters in the vertical direction cannot be described satisfactorily by polynomials, models can be used whose surfaces are functions of the vertical coordinate and which can then have any arbitrary magnetization distribution in the vertical direction.



POROSITY AND PORE STRUCTURE FROM ACOUSTIC WELL LOGGING DATA^{1}
More LessAbstractWyllie's time‐average equation and subsequent refinements have been used for over 20 years to estimate the porosity of reservoir rocks from compressional (P)‐wave velocity (or its reciprocal, transit time) recorded on a sonic log. This model, while simple, needs to be more convincingly explained in theory and improved in practice, particularly by making use of shear (S)‐wave velocity. One of the most important, although often ignored, factors affecting elastic velocities in a rock is pore structure, which is also a controlling factor for transport properties of a rock. Now that S‐wave information can be obtained from the sonic log, it may be used with P‐waves to provide a better understanding of pore structure. A new acoustic velocities‐to‐porosity transform based on an elastic velocity model developed by Kuster and Toksöz is proposed. Employing an approximation to an equivalent pore aspect ratio spectrum, pore structure for reservoir rocks is taken into account, in addition to total pore volume. Equidimensional pores are approximated by spheres and rounded spheroids, while grain boundary pores and flat pores are approximated by low aspect ratio cracks. An equivalent pore aspect ratio spectrum is characterized by a power function which is determined by compressional‐and shear‐wave velocities, as well as by matrix and inclusion properties. As a result of this more sophisticated elastic model of porous rocks and a stricter theory of elastic wave propagation, the new method leads to a more satisfactory interpretation and fuller use of seismic and sonic log data. Calculations using the new transform on data for sedimentary rocks, obtained from published literature and laboratory measurements, are presented and compared at atmospheric pressure with those estimated from the time‐average equation. Results demonstrate that, to compensate for additional complexity, the new method provides more detailed information on pore volume and pore structure of reservoir rocks. Examples are presented using a realistic self‐consistent averaging scheme to consider interactions between pores, and the possibility of extending the method to complex lithologies and shaly rocks is discussed.



ACCURATE FINITE‐DIFFERENCE OPERATORS FOR MODELLING THE ELASTIC WAVE EQUATION^{1}
Authors CORD JASTRAM and ALFRED BEHLEAbstractA method is presented for computing the coefficients of finite‐difference operators for the elastic wave equation. As opposed to other algorithms, in this case the amplitude spectrum of the source function is taken into account. Test calculations show that this approach gives more accurate results than operators based on a Taylor expansion or on minimizing the maximum error of the group velocity in a spatial frequency interval.



RADON TRANSFORM APPLICATION TO THE IMPROVED GRIDDING OF AIRBORNE GEOPHYSICAL SURVEY DATA^{1}
By ZHOU YUNXUANAbstractThe Radon transform is applied to airborne geophysical data, which consist of parallel profiles, analogous to a seismic record. The plane‐wave decomposition (PWD) thus becomes the strike‐direction decomposition (SDD) since the observed spatially distributed information is represented by its strike directions in a domain achieved by the transformation. It is important that, after the SDD, we can identify anomalies and work on them according to their strikes. In particular, for gridding purposes, we may guide the second interpolation of the bi‐directional gridding approach along the strike directions.
In principle, the proposed Radon transform gridding method (RTGM) transforms the observed parallel profiles into a domain where information is mapped as its strike‐direction ‘traces’ against its wavelengths. The number of strike directions into which the data are decomposed is equal to the number of lines to be interpolated. The Fourier spectrum of the grid is reconstructed from the strike‐wavenumber domain by using the projection‐slice theorem and the final square grid is obtained by performing an inverse Fourier transformation on the spectrum.
The SDD is restricted to the Nyquist wavenumber bandwidth imposed by the survey line‐spacing, so that there is no addition of ambiguous short wavelengths in the gridded data. A tapering window is employed to prevent any Gibb's oscillation in the final grid because of the sharp Nyquist cut‐off in the reconstructed spectrum due to the survey line‐spacing.
The RTGM is first tested on a set of synthetic line‐based data. It is also applied to aeromagnetic profile data from northern Botswana as a practical example.



PARABOLIC AND HYPERBOLIC PARAXIAL TWO‐POINT TRAVELTIMES IN 3D MEDIA^{1}
Authors J. SCHLEICHER, M. TYGEL and P. HUBRALAbstractThe 4 × 4 T‐propagator matrix of a 3D central ray determines, among other important seismic quantities, second‐order (parabolic or hyperbolic) two‐point traveltime approximations of certain paraxial rays in the vicinity of the known central ray through a 3D medium consisting of inhomogeneous isotropic velocity layers. These rays result from perturbing the start and endpoints of the central ray on smoothly curved anterior and posterior surfaces. The perturbation of each ray endpoint is described only by a two‐component vector. Here, we provide parabolic and hyperbolic paraxial two‐point traveltime approximations using the T‐propagator to feature a number of useful 3D seismic models, putting particular emphasis on expressing the traveltimes for paraxial primary reflected rays in terms of hyperbolic approximations. These are of use in solving several forward and inverse seismic problems. Our results simplify those in which the perturbation of the ray endpoints upon a curved interface is described by a three‐component vector. In order to emphasize the importance of the hyperbolic expression, we show that the hyperbolic paraxial‐ray traveltime (in terms of four independent variables) is exact for the case of a primary ray reflected from a planar dipping interface below a homogeneous velocity medium.

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