Anisotropy Models of Sedimentary Sections and Characteristics of Wave Propagation
A material is said to be anisotropic if its properties, when measured at the same point, change with direction. Anisotropy is exhibited in its pure form in crystals. A crystal’s properties are governed by the periodic structure of its atoms. The study of
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Anisotropy Models of Sedimentary Sections and Characteristics of Wave Propagation
5.1 Introduction
A material is said to be anisotropic if its properties, when measured at the same point, change with direction. Anisotropy is exhibited in its pure form in crystals. A crystal's properties are governed by the periodic structure of its atoms. The study of anisotropic behavior of crystals has greatly helped understanding seismic anisotropy of sedimentary rocks and fractured zones. With this perspective, some salient points of anisotropy in crystals are introduced first. A collection of points in a periodic arrangement is referred to as a lattice. A lattice could be created by translation of a point. By repeated translation of a point in one direction, two directions and three directions, one-dimensional, two-dimensional and three-dimensional lattices could be produced (Fig. 5.1 a-c). Periodic repetition of points may also be created by rotation about a line and reflection in a plane. Plane and space lattices could be produced by the operation of translation, rotation, reflection and their combinations. Crystal faces are referred to in three or four lines or axes, which pass through a common point, the center of the crystal. These lines are called the crystallographic
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Fig. 5.1 a-c. Generation of: a one-dimensional; b two-dimensional; and c three-dimensional lattice, by translation of a point
S. K. Upadhyay, Seismic Reflection Processing © Springer-Verlag Berlin Heidelberg 2004
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5 Anisotropy Models of Sedimentary Sections and Characteristics of Wave Propagation
axes. The direction of these lines is generally parallel to the edges of a unit cell, which is the same as parallel to the edges of possible faces of the crystal. It may be remarked that the physical properties of an isotropic material can be represented by scalar quantities. In a crystalline solid, the properties of the material depend on the direction. Supposing a force or a field is applied to a crystal. The response of the crystal would depend not only on the magnitude of the force or field, but also on the direction in which it is applied relative to reference directions in the crystal. Since crystals are built on a periodic structure of atoms, they have the same symmetry as this structure. Because of symmetry, sets of all directions break up into subsets. The physical properties represented by scalar quantities in isotropic material are therefore replaced by a finite set of quantities in an anisotropic solid, which are direction-dependent. It is important to note that variation of physical properties with direction in a crystal depends on the symmetry of the crystal. This may be illustrated by considering a unit cell in the form of a parallelogram (Fig. 5.2 a) and square (Fig. 5.2 b) in a plane lattice.
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