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# Fundamentals of Seismic Wave Propagation

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chapters on ray theory to emphasize the similarities and reuse results. The chapter then develops various solutions of these equations, in homogeneous and inhomogeneous lay- ers, using the propagator and ray expansion formalisms. The WKBJ and Langer asymptotic methods, and the WKBJ iterative solutions, are included. The second chapter of this group, Chapter 8: Inverse transforms for stratified media, then de- scribes the inverse transform methods that can be used with the solutions from the previous chapter. These include the Cagniard method in two and three dimensions, the WKBJ seismogram method and the numerical spectral method. These methods are then used in the final chapter of the group, Chapter 9: Canonical signals, which describes approximations to various signals that occur in many simple problems. These range from direct and turning rays, through partial and total reflections, head waves and interface waves, to caustics and shadows. These results link back to the introductory Chapter 2 where the various singularities, discontinuities and degeneracies of ray results had been emphasized. Particular emphasis is placed on describing the signals using simple, ‘standard’ special functions. The final chapter, Chapter 10: Generalizations of ray theory, describes recent extensions of ray theory which increase the range of application and validity and Introduction 5 include some of the advantages of the transform methods. The methods are Maslov asymptotic ray theory which extends ray theory to caustic regions; quasi-isotropic ray theory which extends ray theory to the near degeneracies that exist in weakly anisotropic media; Born scattering theory which models signals scattered by small perturbations in the model and importantly allows signals due to errors in the ray solution to be included; and the Kirchhoff surface integral method which allows signals and diffractions from non-planar surfaces to be modelled at least approxi- mately. 2 Basic wave propagation This introductory chapter introduces the reader to the concepts of seismic waves – plane waves, point sources, rays and travel times – without math- ematical detail or analysis. Many different types of rays and seismic signals are illustrated, in order to set the scene for the rest of this book. The objective of the book is to provide the mathematical tools to model and understand the signals described in this chapter. In this chapter, we introduce the basic concepts of wave propagation in a simple, stratified medium. None of the sophisticated mathematics needed to solve for the complete wave response to an impulsive point source in an elastic medium is intro- duced nor used. Ideas such as Snell’s law, reflection and transmission, wavefronts and rays, travel-time curves and related properties, are introduced. The concepts are all straightforward and should be intuitively obvious. Nevertheless many sim- ple questions are left unanswered, e.g. how are the amplitudes of waves found, and what happens when wavefronts are non-planar or singular. These questions will be answered in the rest of this book. This chapter sets the scene and motivates the more detailed investigations that follow. 2.1 Plane waves 2.1.1 Plane waves in a homogeneous medium In this section, we introduce the notation and nomenclature that we are going to use to describe waves. We assume that the reader has a basic knowledge of waves and oscillations, and so understands a simple wave equation with the notation of com- plex variables used for a solution. The simplest form of wave equation describing waves in three dimensions is the Helmholtz equation ∇2φ = 1 c2 ∂2φ ∂t2 . (2.1.1) 6 2.1 Plane waves 7 Table 2.1. Symbols, names and units used to describe plane waves. Symbol Name Units φ field variable A (complex) amplitude t time [T] x position vector [L] ω circular frequency [T−1] ν = ω/2π , frequency (Hz) [T−1] k wave vector [L−1] k = |k|, wavenumber [L−1] c = ω/k, wave (phase) velocity [LT−1] T 2π/ω = 1/ν, period [T] λ 2π/k = c/ν, wavelength [L] p = k/ω, slowness vector [L−1T] |p| = 1/c, slowness [L−1T] k · x − ωt phase [0] When the velocity, c, is independent of position, the plane-wave solution of this equation is φ = Aei(k·x−ωt) (2.1.2) = Aeiω(p·x−t). (2.1.3) In Table 2.1 we give the definitions, etc. of the variables used in or related to this equation (see Figure 2.1). Note the signs used in the exponent of the oscillating, travelling wave given by equations (2.1.2) or (2.1.3). The same signs will be used later in Fourier transforms (Chapter 3). Various sign conventions are used in the literature. We prefer this one, a positive sign on the spatial term and a negative sign on the temporal term, as it leads to the simple identification that waves travelling in the positive direction have positive components of slowness. For most purposes, it is convenient to use the slowness vector, p, rather than the wave vector, k, to indicate the wave direction as it is independent of frequency. Surfaces on which k · x is constant, as illustrated in Figure 2.1, are known as wavefronts. 2.1.2 Plane waves at an interface If a plane wave is incident on an interface, we expect reflected and transmitted waves to be generated. Some continuity condition must apply to the field variable, 8 Basic wave propagation λ k x Fig. 2.1. A plane wave in a homogeneous medium. At a fixed time, t , the lines represent wavefronts where k · x is constant. Thus if k · x = 2nπ , the wavefronts are separated by the wavelength, λ. φ, its derivative or combinations thereof. Even without specifying the details of the boundary condition, it is clear that, for a linear wave equation, all waves must have the same frequency ω, and the same wavelength along the interface. Without these conditions, the waves could not match at all times and positions. Let us consider a plane interface at z = 0, separating homogeneous half-spaces with velocities c1 for z > 0, and c2 for z < 0 (Figure 2.2). For simplicity we rotate the coordinate system so the incident slowness vector, pinc, lies in the x–z plane, i.e. py = 0. Then the wavelength along the interface is 2π/kx = 2π/ωpx , so the slowness component px must be the same for all waves. 2.1.2.1 Snell’s law As |p| = 1/c, the incident wave’s slowness vector is pinc = px0 pz = px 0 − ( c−21 − p2x )1/2 = 1 c1 sin θinc0 − cos θinc . (2.1.4) The square root in pz is taken positive, so the minus sign is included as the incident wave is propagating in the negative z direction (Figure 2.2). The angle θinc is the 2.1 Plane waves 9 z prefl pinc ptrans interface preflpinc ptrans normal θinc θrefl θtrans (a) (b) Fig. 2.2. Plane waves incident, reflected and transmitted at a plane interface be- tween homogeneous half-spaces. The situation when c2 > c1 is illustrated. Part (a) shows the wavefronts and (b) the slowness vectors. angle between the slowness vector and the normal to the interface and is taken so 0 ≤ θinc ≤ π/2. The reflected and transmitted waves must have slowness vectors prefl = px 0 + ( c−21 − p2x )1/2 = 1 c1 sin θrefl0 + cos θrefl (2.1.5) ptrans = px 0 − ( c−22 − p2x )1/2 = 1 c2 sin θtrans0 − cos θtrans , (2.1.6) where the angles θrefl and θtrans are similarly defined. Equality of the x slowness component for these vectors immediately leads to the reflection law θinc = θrefl, (2.1.7) and the refraction or Snell’s law sin θinc c1 = sin θtrans c2 . (2.1.8) 10 Basic wave propagation The situation illustrated in Figure 2.2 is when c2 > c1 so θtrans > θinc and the slow- ness vector is refracted away from the normal. The opposite occurs when