Analytical Inversion of a Stack of Weakly Anisotropic Layers
Céline de Bazelaire, Eric de Bazelaire, Hervé Perroud, 2001. "Analytical Inversion of a Stack of Weakly Anisotropic Layers", Anisotropy 2000: Fractures, Converted Waves, and Case Studies, L. Ikelle, A. Gangi
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Using the seismic-reflection method, we carry out a 3D study of a stack of horizontal layers characterized by contrasted anisotropic properties (from TI with a vertical axis of symmetry to monoclinic). The inversion proposed here requires two mathematical descriptions of the common mid-point reflection time-distance curve of the anisotropic medium. The first one, called the exact equation, was obtained using the P-wave phase velocity expression for weakly anisotropic media of arbitrary symmetry proposed by Mensch and Rasolofosaon (1997). This equation contains the following unknown parameters: vertical velocity, thickness and anisotropy coefficients. The second equation, also describing the P-wave travel time in the medium, is called the approximate equation. It is a four-parameter mathematical approximation whose deviation from the exact equation is smaller than the seismic time-sampling interval. It contains the measurable parameters needed for the inversion. This equation corresponds to the combination of two hyperbolae tangent at a point (x,t). The physical meaning is that the wavefronts can be described by two tangent circles in a given azimuthal plane. The common mid-point time-distance curve is thus defined by four independent parameters for each azimuth. Thirteen independent values (instead of 16) can be obtained from measurements in four azimuthal orientations (every 45 degrees), since one of the parameters, the zero-offset traveltime, is azimuthally invariant. The inverse problem consists of evaluating the 10 unknown parameters (monoclinic case) of the exact equation using the 13 independent values defined by the four approximate equations. Modeling of synthetic seismic traces in anisotropic media is achieved using the program ANRAY (Gajewski and Psencik 1987) to produce a synthetic data set. On this, the parameter measurements are made using the PSCAN theory (de Bazelaire 1988). The results from the inversion depict a good agreement with true model parameters. Finally, this technique can be applied to a stack of anisotropic media by processing the inversion individually on each of them. To achieve this, we need to find geophysical relationships which can transmit the measurements of the current layer through the upper multilayered stack using Dix-type.
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“This volume contains 25 papers that represent most of the best work in seismic anisotropy in 1998 and 1999. Fracture characterizations and processing of converted waves are the two main topics covered in this volume. They are addressed from both theoretical and practical viewpoints. Also included are papers describing the historical roots of seismic anisotropy.”