Foundations of Anisotropy for Exploration Seismics

Chapter headings and selected contents: Fundamentals. What is homogeneity? What is anisotropy? What is dispersion? What causes anisotropy of wave propagation. Appendix 1A: Analytical derivation of the relation between anisotropy and dispersion. Tools for the Description of Wave Propagation under Piecewise Homogeneous Anisotropic Conditions. Ray velocity and normal velocity. The ray-slowness surface; slowness and wave surface as polar reciprocals. Snell's Law. Appendix 2A: Formal description of the transformations used in this chapter. Analytic expression for inversion (reflection in a circle). Analytic derivation of the tangent curve from the footpoint curve. Analytic description of polar reciprocity. Elasticity. Tensors and vectors. Infinitesimal strain. Basic symmetries of the elastic tensor and the contracted notation. The elastic constants and material symmetry. Appendix 3A: The relation between elastic constants and rotational symmetry. Reduction of an arbitrary rotation to a sequence of rotations about one axis each. Tensors of rank two under rotation of the coordinate system. Appendix 3B: Invariants of the elastic tensor. Contraction of the elastic tensor on itself. Appendix 3C: FORTRAN subroutines for operations on elastic tensors in four- and two- subscript notation. Elastic Waves - The Dispersion Relation and some Generalities about Slowness and Wave Surfaces. The wave equation. Elements of inflection of the slowness surface. Slowness, polarization and symmetry. Singular directions. Appendix 4A: Orthogonality of polarization vectors. Appendix 4B: Explicit versions of the characteristic equation. Appendix 4C: Subroutines for the Kelvin-Christoffel matrix. Stability Constraints. The general stability condition. Stability conditions for isotropic, hexagonal, cubic, strong tetragonal and orthorhombic media. One-Parameter Expressions for the Slowness Surfaces of Transversely Isotropic Media and the Slowness Curves in the Planes of Symmetry of Orthorhombic Media. Decoupling of the across-plane polarization and a parameter expression for the "coupled" slowness curves. The "basic curves", the separating ellipse and the general shape of the "coupled" slowness curves. The associate parameter, the representing point, and a geometric construction for the polarization. A nomogram for the in-plane polarization. Appendix 6A: Closed explicit expressions for the coupled slowness curves in the symmetry planes of orthorhombic media. One-Parameter Expressions for the Wave Curves in the Symmetry Planes of Orthorhombic Media. Expressions for the wave surface in Cartesian and polar coordinates. Cusps. A measure of anisotropy. Squared Slowness Surfaces and Squared Slowness Curves. Phenomenology of slowness surfaces in the squared domain. The "framework" in the planes of symmetry. Deviations from the "framework" in the planes of symmetry. Appendix 8A: Determination of the type of the squared slowness curve. Appendix 8B: Properties of the square transformation. Coordinate grids. Straight lines. Symmetrically centered conics with aligned axes. General centered conics. Appendix 8C: Geometric tools for the conversion of a squared slowness curve to the ordinary domain. Geometric determination of curvature. Causes of Anisotropy: Periodic Fine Layering. Simple quasi-static strain modelling. Constraints on layer-induced anisotropy. Inversion of the compound stiffnesses to constituent stiffnesses. A nomogram for the determination of layer parameters. Appendix 9A: Generalized averages. Anisotropy and Seismic Exploration. Elliptical anisotropy. An equivalence theorem for surface-to-surface seismics. Some aspects of reflection seismics. Eigentensors of the Elastic Tensor and their Relationship with Material Symmetry. Rudimentary definition of a tensor space. Strain tensors and wave propagation. Coordinate-free representation of "wave-compatibility". Eigentensors and symmetry. Eigensystems of specific symmetries. Determination of the symmetry class. Appendix 11A: Construction of media with particular eigentensors. Reconstruction of an elastic tensor from its eigensystem. Construction of an eigensystem with particular eigenvectors. References. Index.

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