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# Fundamentals of Seismic Wave Propagation

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from the top downwards. The �-th layer is between the �-th and (� + 1)-th interfaces, i.e. z� ≥ z ≥ z�+1 with the �-th interface at the top of the �-th layer. Normally, the first and n-th layers are half-spaces and the source lies at xvi Preliminaries z z1 z2 zS z� z�+1 zn �-th layer �-th interface Fig. 0.1. The interface/layer arrangement for a flat, layered model illustrating the free surface at z1, source at zS, the �-th interface at z�, the �-th layer between z� and z�+1 and the final layer or half-space below the interface zn . z = zS. When needed, the free surface of the Earth is at z = z1. This is illustrated in Figure 0.1. 0.1.7 Acronyms For brevity, several acronyms are used throughout the text. Most of these are widely used but for completeness we list them here: ART asymptotic ray theory is the mathematical theory used to describe seismic rays (see Chapter 5). The solution is normally expressed as an asymptotic series in inverse powers of frequency. AVO amplitude versus offset describes the behaviour of the amplitude of a re- flected signal versus the offset from source to receiver. Normally factors of ge- ometrical spreading and attenuation are removed so the amplitude changes are described by the reflection coefficient (see Chapter 6). 0.1 Nomenclature xvii FFT fast Fourier transform is the efficient numerical algorithm used to evaluate the discrete Fourier transform, which is usually used as an approximation for a Fourier integral (see Chapter 3). GRA geometrical ray approximation is the approximation usually used to describe seismic rays (see Chapter 5). Usually it is the zeroth-order term in the ART ansatz, i.e. the amplitude is independent of frequency and the phase depends linearly on frequency. KMAH the Keller (1958), Maslov (1965, 1972), Arnol’d (1973) and Ho¨rmander (1971) index counts the caustics along a ray (see Chapter 5). In isotropic, elastic media it increments by one at each line caustic and two at each point caustic, but in anisotropic media and for some other types of waves, it may decrement. NMO normal moveout describes the approximate behaviour of a reflected arrival near zero offset (see Section 2.5.1). The travel time is approximately a parabolic function of offset. QI quasi-isotropic ray theory is a variant of ART used to describe the coupling of quasi-shear waves that occurs in heterogeneous, anisotropic media when the shear wave velocities are similar (see Section 10.2). SOFAR a minimum of the acoustic velocity at about 1.5 km depth in the deep ocean (above the velocity increases because of the temperature increase, and below it increases because of the increased hydrostatic pressure) forms the deep ocean sound (or SOFAR) channel. Energy can be trapped in the SOFAR chan- nel, with rays turning above and below the minimum, and sound propagates to far distances (see Section 2.3). TI a transversely isotropic medium is an anisotropic medium with an axis of sym- metry, i.e. the elastic properties only varying as a function of angle from the symmetry axis and are axially symmetric (see Section 4.4.4). It is also known as hexagonal or polar anisotropy. WKB the Wentzel (1926), Kramers (1926), Brillouin (1926) asymptotic solution is an approximate solution of differential equations. It is widely used for wave equations at high frequencies when the phase varies as the integral of the lo- cal wavenumber and the amplitude varies to conserve energy flux. The WKB solution is very useful but, however many terms are taken in the asymptotic solution, it does not describe reflected signals from a smooth but rapid change in properties – the so-called WKB paradox (see Section 7.2.5). WKBJ a variant of the WKB acronym used by geophysicists to honour Jeffreys’ (1924) contribution (another variant is JWKB). The WKBJ asymptotic expan- sion (Section 7.2.5), WKBJ iterative solution (Section 7.2.6) and WKBJ seismo- gram (Section 8.4.1 – so-called as it only depends on the WKBJ approximation) are important solutions for studying seismic waves in stratified media. xviii Preliminaries 0.2 Symbols As the symbols and notation used in seismology have never been successfully standardized, we tabulate in Table 0.1 many of the symbols used in this book, their units, description and reference to an equation where they are first used. The list is not exhaustive and all variants are not included, e.g. forms with alternative subscripts and arguments are not listed. Symbols that are only used locally are not included. Table 0.1. Symbols, units, description and first equations used in the text (continued on the following pages). Symbols Units Description Equation x, r [L] position vector (0.1.1) q, Q, qν [L] wavefront coordinates (5.1.21) ıˆ, jˆ, kˆ [0] unit coordinate vectors (2.5.26) nˆ [0] unit surface normal (4.1.4) R [L] radial length (2.2.2) d [L] layer thickness (8.1.3) k, k [L−1] wavenumber (2.1.2) λ [L] wavelength s, L [L] ray length (5.1.12) � [0] dimension (5.4.28) R(�) [L] effective ray length (5.4.25) t [T] time (2.1.1) ω [T−1] circular frequency (2.1.2) ν [T−1] frequency c, α, β [LT−1] wave (phase) velocity (2.1.1) V, V [LT−1] ray (group) velocity (5.1.14) p, p, q [L−1T] slowness (2.1.3) p [L−1T] sub-space slowness (3.2.13) θ [0] phase-normal angle (2.1.4) φ [0] phase-group angle (5.3.32) X , X, X [L] range function (2.3.4) T [T] travel-time function (2.2.8) τ [T] intercept-time function (2.3.15) T , t [T] reduced travel time (2.5.48) T˜ [T] generalized time function (8.1.2) 0.2 Symbols xix Table 0.1. continued. Note the units of the Green functions are for the three-dimensional case. An extra unit of [L] exists in two dimensions. The units of the field variables, e.g. u, are in the temporal and spatial domain. An extra unit of [T] exists in the spectral domain. Symbols Units Description Equation t [ML−1T−2] traction (4.1.1) σ [ML−1T−2] stress (4.1.5) P [ML−1T−2] pressure (4.4.1) u [L] particle displacement (4.2.1) v [LT−1] particle velocity (4.1.19) e [0] strain (4.2.2) θ [0] dilatation (4.2.8) ρ [ML−3] density (4.0.1) κ [ML−1T−2] bulk modulus (4.4.3) k [M−1LT2] compressibility (4.4.4) ci jkl , Ci j , ci j [ML−1T−2] elastic stiffnesses (4.4.5) ai jkl , Ai j [L2T−2] squared-velocity parameters (5.7.19) si jkl , Si j , si j [M−1LT2] elastic compliances (4.4.40) λ, µ [ML−1T−2] Lame´ elastic parameters (4.4.49) f [ML−2T−2] body force per unit volume (4.1.7) fS [MLT−1] point impulse (4.5.76) M, MS [ML2T−2] moment tensor (4.6.5) u [M−1T] Green particle displacement (4.5.17) v [M−1] Green particle velocity (4.5.20) P [L−2T−1] Green pressure (4.5.20) xx Preliminaries Table 0.1. continued. Note � is the dimensionality of the solution (2 or 3). P(�)(t) has an extra unit [T] compare with P(�)(ω). The Green amplitude coefficients v(0), P(0) and t(0)j are for three-dimensions. They have extra units of [LT−1/2] in two dimensions. Symbols Units Description Equation y – phase space vector (5.1.28) v(m), v(0) – velocity amplitude coefficients (5.1.1) P(m) – pressure amplitude coefficients (5.1.1) t(m)j – traction amplitude coefficients (5.3.1) gˆ [0] normalized polarization (5.2.4) g [M−1/2LT1/2] energy flux normalized polarization (5.4.33) GGG [M−1L2T] polarization dyadic (8.0.8) H(x, p) [0] Hamiltonian (5.1.18) Z , Zi [ML−2T−1] impedance (5.2.8) N [MT−3] energy flux vector (5.2.11) σ [0] KMAH index (5.2.70) LLL,MMM,NNN – anisotropic ART operators (5.3.5) D – dynamic differential system (5.2.19) J – ray tube cross-section (5.2.12) D [0] Jacobian volume mapping (5.2.14) P, P – dynamic propagator matrix (5.2.23) J – dynamic fundamental matrix (5.2.29) M, M [L−2T] wavefront curvature matrix (5.2.46) S(�) [L2(�−1)T1−�] ray spreading function (5.2.67) T (�) [L1−�T(�−1)/2] ray scalar amplitude (5.4.34) P(�)(ω) [L1−�T(�−1)/2] ray propagation term (5.4.36)