Anisotropy means that the physical properties of a solid medium vary with direction.For example, the speed of travel of a wave in the vertical direction may be different from its speed in the horizontal direction. The real subsurface of the Earth is anisotropic in certain regions. Media containing fractures can be effectively modeled by replacing them with a single anisotropic medium. This chapter introduces the reader to the mathematical theory of how seismic waves propagate in anisotropic media.Isotropic media can be described by two physical parameters, whereas anisotropic media require more. The relatively simple but important case of a transversely isotropic medium, which requires five physical parameters or elastic constants for its description, is covered in relative detail.Transversely isotropic media that are weakly anisotropic are also discussed, as well as special cases, such as elliptical anisotropy. Reflection and transmission are briefly discussed. Slowness surfaces, which are helpful in understanding the nature of anisotropy in a medium, are discussed. The interesting effects that occur in a tilted anisotropic medium are also briefly covered.

This chapter begins with coverage of the quantitative concepts used to describe the deformation of solids by seismic waves, namely the concepts of stress, strain, and dilatation. This is followed by the derivation of equations for describing seismic wave motion in the subsurface, namely, the equation of motion, conservation of energy, kinetic and strain-energy density, intensity or energy flux, the stress–strain relation, isotropy, hydrostatic stress, elastic constants (which are related to the nature of the medium in which waves travel), the wave equations, compressional and shear waves, plane harmonic waves, displacement potentials, Helmholtz equations, near-field and far-field waves, mean values, and the acoustic wave equation. The chapter ends with examples that discuss seismic waves produced by a buried explosive charge and by a directed point force, and discussions of the moment tensor and apparent velocities.

This chapter covers the mathematical theory of how plane waves are reflected by boundaries or interfaces separating solid layers in the subsurface and how they are transmitted through them. Reflections off the Earth's surface are also discussed. The equations describing the physical boundary conditions that incident, reflected, and transmitted waves must satisfy are derived. The calculation of reflection and transmission coefficients, which give the amplitudes of reflected and transmitted waves, is covered.Polarity reversals and phase changes are discussed. Critical angles, which are related to the total internal reflection of incident waves, are studied. This is followed by coverage of the calculation of the amount of seismic wave energy that is reflected and transmitted, reflection and transmission of waves from liquid–liquid, liquid–solid, and rigid boundaries, and approximate formulas for reflection and transmission coefficients.

This chapter looks at how seismic wave theory relates to transforming seismic wave travel-time data into different representations such as the frequency domain (achieved with a 1D Fourier transform), the frequency-wavenumber domain (achieved with a 2D Fourier transform), and the tau-p domain (or intercept time–ray parameter domain). The reason for transforming seismic data into different domains is that the data may be easier to analyze and interpret in other domains. Furthermore, 1D and 2D filtering can be done often more conveniently in the frequency and frequency-wavenumber domains. Also covered are topics related to the tau-p domain, namely, slant-stacking, plane wave decomposition, and the Hilbert and Radon transforms.

For simplicity, calculations in seismic wave theory often assume that the Earth is perfectly elastic. But the real Earth is anelastic, meaning that wave energy is absorbed by internal friction effects. This chapter gives an introductory account of the mathematical theory of seismicwave propagation in anelastic media. The important concepts of the quality factor (Q), the loss factor (1/Q) ,and the complex modulus are introduced. The necessity for including dispersion in computations of the shapes of waveforms in anelastic media is demonstrated. Spring-dashpot models for describing anelastic media, and the more general linear theory of viscoelasticity, are introduced. The nature of Q in the Earth is discussed. The 1D and 3D equations of motion in anelastic media, and their plane wave solutions, are derived. The concept of general plane waves (including homogeneous and inhomogeneous plane waves) is introduced.Reflection and transmission of plane waves, as well as particle motion, in anelastic media are briefly discussed. The theory of exactly constant Q is briefly covered.

Surface waves are waves that are essentially confined to the surface of the Earth, in that their amplitudes decrease with depth in some way. This chapter covers the basic theory of such waves, as well as the theory of normal modes, which are waves confined to a surface layer, and are similar to waves in an organ pipe or the motions of a vibrating string fixed at one end or both ends, for instance. Coverage includes Rayleigh waves, which are a combination of compressional and shear waves and produce ground motions parallel to the vertical plane in which the wave is traveling, and are a dominant type of earthquake wave; and Love waves, which are shear waves that produce ground motions in the horizontal plane, and head waves, which are upgoing waves produced by critically refracted transmitted waves. Waves along an interface between two solids are also discussed. Also discussed are how the different frequencies in the wave pulse travel at different wave speeds, and the corresponding concepts of phase and group velocity.In addition, other types of normal modes are covered, as well as an interesting wave phenomenon known as the Airy phase.

This chapter is a review of much of the mathematical knowledge required for the basic seismic wave theory covered in the book. The topics covered are vector algebra, vector calculus, vector identities used in seismic wave theory, curvilinear coordinates, rotation of coordinates, tensor analysis, Fourier transforms, and convolution.

This chapter covers the computation of synthetic seismograms, or theoretical seismograms. This involves predicting, via computation, what seismic traces might look like for a given subsurface medium model. The relatively simple case of vertically traveling waves in a sequence of flat horizontal layers is discussed in relative detail, including how to compute wave amplitude losses due to reflection, transmission, geometrical spreading of wavefronts, and absorption. The generally more complicated case of nonvertically traveling waves is also briefly summarized. More complete methods such as the finite difference and finite element methods are briefly mentioned. Also covered are the reflectivity function and the interference effects that occur for waves with nearly equal arrival times, such as the tuning effect. The chapter ends with an appendix showing examples of synthetic seismograms computed with the finite difference method.

This chapter shows an example of how seismic wave theory can be used to improve cross-sectional images of subsurface zones with complex geological structure prepared from seismic data. In CMP stack sections, which are basic images of the subsurface constructed from seismograms, the reflectors and layer interfaces are generally not in their correct spatial positions. Seismic migration is a process that attempts to move them to their correct spatial positions. This chapter shows how seismic wave theory can be used to develop the methods of seismic wave equation migration. As an introduction, the chapter begins with basic methods for migrating point and dipping reflectors and describes the relatively simple methods for diffraction-summation and wavefront migration.This is followed by coverage of the basic wave equation migration methods, namely phase-shift migration, frequency-wavenumber migration, finite difference migration, and Kirchoff migration. A brief explanation of the need for depth migration, which is an improvement over the wave equation migration methods, is also provided.

The subsurface of the Earth consists typically of media that are not homogeneous, but rather heterogeneous (or inhomogeneous), such as sequences of homogeneous layers where each layer has a different density and wave velocity, or zones where the wave velocity and medium parameters vary smoothly with position. This chapter looks at the theory of how seismic waves propagate in such heterogeneous media. In addition, coverage includes applications of the theory to well surveys and logs (tools used in exploration seismology), the Wiechert–Herglotz method for computing how the seismic wave velocity varies with position from measurements of seismic wave travel times, the eikonal equation (an equation for computing wave travel times), zero-offset ray tracing (a relatively simple technique for roughly estimating the nature of the seismic data that might be recorded in regions of complex subsurface structure), diffracted seismic waves, acoustic waves in heterogeneous materials, and ray equations (for computing the paths of seismic waves in heterogeneous media).