2 Seismic Wave Propagation and Earth models

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2. Seismic Wave Propagation and Earth models Young’s modulus, the bulk modulus and the shear modulus all have the same physical units as pressure and stress, namely (in international standard (SI) units): 1 Pa = 1 N m -2 = 1 kg m -1 s -2 (with 1 N = 1 Newton = 1 kg m s -2 ). (2.1) 2.2.2 Stress-strain relationship The most general linear relationship between stress and strain of an elastic medium is governed in the generalized Hook’s law (see Eqation (10) in the IS 3.1) by a fourth order parameter tensor. It contains 21 independent moduli. The properties of such a solid may vary with direction. Then the medium is called anisotropic. Otherwise, if the properties are the same in all directions, a medium is termed isotropic. Although in some parts of the Earth’s interior anisotropy on the order of a few percent exists, isotropy has proven to be a reasonable first-order approximation for the Earth as a whole. The most common models, on which data processing in routine observatory practice is based, assume isotropy and changes of properties only with depth. In the case of isotropy the number of independent parameters in the elastic tensor reduces to just two. They are called after the French physicist Lamé (1795-1870) the Lamé parameters λ and μ. The latter is identical with the shear modulus. λ does not have a straightforward physical explanation but it can be expressed in terms of the above mentioned elastic moduli and Poisson’s ratio, namely σ E λ = κ - 2μ /3 = . (2.2) ( 1 + σ )( 1 − 2σ ) The other elastic parameters can also be expressed as functions of μ, λ and/or κ: and ( 3λ + 2μ) μ E = (2.3) ( λ + μ) λ σ = 2( λ + μ) 3κ − 2μ = . (2.4) 2( 3κ + μ) For a Poisson solid λ = μ and thus, according to (2.4), σ = 0.25. Most crustal rocks have a Poisson’s ratio between about 0.2 and 0.3. But σ may reach values of almost 0.5, e.g., for unconsolidated, water-saturated sediments, and even negative values of σ are possible (see Tab. 2.1). The elastic parameters govern the velocity with which seismic waves propagate. The equation of motion for a continuum can be written as ∂ 2 i ρ 2 ∂t u = ∂ jτ ij + f i , (2.5) 4
5 2.2 Elastic moduli and body waves with ρ - density of the material, ui – displacement, τij – stress tensor and fi – the body force term that generally consists of a gravity term and a source term. The gravity term is important at low frequencies in normal mode seismology (see 2.4), but it can be neglected for calculations of body- and surface-wave propagation at typically observed wavelengths. Solutions of Eq. (2.5) which predict the ground motion at locations some distance away from the source are called synthetic seismograms (see Figs. 2.54 and 2.55). In the case of an inhomogeneous medium, which involves gradients in the Lamé parameters, Eq. (2.5) takes a rather complicated form that is difficult to solve efficiently. Also, in case of strong inhomogeneities, transverse and longitudinal waves (see below) are not decoupled. This results in complicated particle motions. Therefore, most methods for synthetic seismogram computations ignore gradient terms of λ and μ in the equation of motion by modeling the material either as a series of homogeneous layers (which also allows to approximate gradient zones; see reflectivity method by Fuchs and Müller, 1971; Kennett, 1983; Müller, 1985) or by assuming that variations in the Lamé parameters are negligible over a wavelength Λ and thus these terms tend to zero at high frequencies (ray theoretical approach; e.g., Červený et al., 1977; Červený, 2001). In homogeneous media and for small deformations the equation of motion for seismic waves outside the source region (i.e., without the source term fs and neglecting the gravity term fg) takes the following simple form: ρ ü = (λ + 2μ)∇∇⋅u - μ ∇×∇×u (2.6) where u is the displacement vector and ü its second time derivative. Eq. (2.6) provides the basis for most body-wave, synthetic seismogram calculations. Although it describes rather well most basic features in a seismic record we have to be aware that it is an approximation only for an isotropic homogeneous linearly elastic medium. 2.2.3 P- and S-wave velocities, waveforms and polarization The first term on the right side of Eq. (2.6) contains the scalar product ∇⋅u = div u. It describes a volume change (compression and dilatation), which always contains some (rotation free!) shearing too, unless the medium is compressed hydrostatically (as in Fig. 2.2 top). The second term is a vector product (rot u = ∇×u) corresponding to a curl (rotation) and describes a change of shape without volume change (pure shearing). Generally, every vector field, such as the displacement field u, can be decomposed into a rotation-free (u r ) and a divergence-free (u d ) part, i.e., we can write u = u r + u d . Since the divergence of a curl and the rotation of a divergence are zero, we get accordingly two independent solutions for Eq. (2.6) when forming its scalar product ∇⋅u and vector product ∇×u, respectively: and ∂ ∂ 2 2 ( ∇ ⋅u) λ + 2μ 2 r = ∇ ( ∇ ⋅ u ) 2 ∂ t ρ (2.7) ( ∇ × u) μ 2 d = ∇ ( ∇ × u ) . (2.8) 2 ∂ t ρ
Page 1 and 2: CHAPTER 2 Seismic Wave Propagation
Page 3: 3 2.2 Elastic moduli and body waves
Page 7 and 8: 7 2.2 Elastic moduli and body waves
Page 9 and 10: 9 2.2 Elastic moduli and body waves
Page 11 and 12: 11 2.3 Surface waves S waves are al
Page 13 and 14: 2.3.2 Dispersion and polarization 1
Page 15 and 16: 15 2.3 Surface waves Strictly speak
Page 17 and 18: 17 2.3 Surface waves (Z-E) plane, w
Page 19 and 20: 2.3.4 Mantle surface waves 19 2.3 S
Page 21 and 22: 2.4 Normal modes 21 2.4 Normal mode
Page 23 and 24: 23 2.4 Normal modes First observati
Page 25 and 26: 2.5 Seismic rays, travel times, amp
Page 43 and 44: 2.6 Seismic phases and travel times
Page 53 and 54: Overlay to Fig. 2.47 2.6 Seismic ph
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2.6 Seismic phases and travel times
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57 2.7 Global Earth models on verti
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59 2.7 Global Earth models As shown
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61 2.7 Global Earth models The mode
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2.8 Synthetic seismograms and wavef
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