Near-Surface Seismology: Wave Propagation

Transcription

1 Chapter 7 Near-Surface Seismology: Wave Propagation John R. Pelton 1 Introduction Near-surface seismology derives much of its identity from the physical characteristics of the near-surface environment. Natural materials encountered at shallow depths possess exceptionally diverse mechanical properties as documented by the classification schemes of soil rock mechanics. Geologic boundaries across which mechanical properties undergo large rapid changes are commonly present, most notably the water table the soil-bedrock interface. Porosity occurs in many forms at a wide variety of scales, tends toward relatively high values because of low confining pressures. Water, air, biogenic gases, fluid contaminants occupy the pore space in spatially varying proportions. Near-surface stress increases very rapidly with depth, but the principal stresses may not align with vertical horizontal directions. At depths where soils rocks are saturated with groundwater, significant pore water pressure acts on the solid frame. All of these physical characteristics combine with the nature of the seismic source to determine the near-surface seismic wavefield. A century of research on the theory of seismic wave propagation has led to many important results of broad applicability. It is now possible to simulate the generation propagation of seismic waves in realistic models of near-surface seismic experiments. Nearly all of this progress has been driven by problems in solid-earth petroleum geophysics where wavelengths, depths, sourcereceiver distances are considerably greater than those encountered in near-surface seismology. Despite these scale differences, seismic wave theory as developed by earthquake exploration seismologists satisfactorily explains common near-surface seismological observations, including the presence of body waves (P-waves S-waves), waves associated with boundaries (reflections, head waves, surface waves), the dissipation of seismic energy. The theory of seismic wave propagation is traditionally developed within the framework of linear elasticity. In 1 Department of Geosciences, Boise State University, Boise, Idaho 83725; jpelton@boisestate.edu. the traditional approach, wave propagation problems are considered for increasingly realistic but linearly elastic earth models. At some point in the development, the earth is more correctly regarded as a dissipative medium, spatial attenuation of plane waves is introduced by allowing the wavenumber to be complex. An alternative approach is to assume at the beginning that the subsurface is composed of linearly viscoelastic materials. Dissipation of energy is encountered as a natural consequence of the viscoelastic assumption but elasticity remains a part of the theory as a special case. Because the viscoelastic constitutive equations involve convolutions, considerable work is done in the frequency domain which is familiar ground for most seismologists. In fact, a useful correspondence exists between the equations of linear elasticity in the time domain the equations of linear viscoelasticity in the frequency domain. Although linear viscoelasticity is an unnecessary complication for many seismological problems, it is especially well suited to the study of harmonic plane displacement waves their spatial attenuation as a result of nonrecoverable energy loss. This chapter serves as an introduction to seismic wave propagation for those waves commonly observed in nearsurface seismology. A reasonably detailed description is provided for linearly elastic linearly viscoelastic constitutive models. Dissipation of energy is systematically developed with careful definition of the relationships between energy loss, complex moduli, the quality factor. The discussion of wave propagation is mostly about plane body waves (viscoelastic case) plane surface waves (elastic case) in homogeneous (or vertically heterogeneous) isotropic media. The discussion also includes a simple model for head waves, the reflection of plane body waves from a plane interface between homogeneous isotropic elastic media, a description of geometric spreading, the radiation pattern, the near field associated with a point source in an unbounded elastic medium. The appendices contains a significant amount of related information including an introduction to linear system theory, Fourier integral transforms the convolution theorem, a discussion of the principles that underlie linear viscoelasticity in the general heterogeneous anistropic case, details for many of the computations. 177

2 178 Near-Surface Geophysics Part 1: Concepts Fundamentals Constitutive Models for Near-Surface Materials Although most near-surface seismic methods are satisfactorily based on the analysis of seismic wave propagation in a linearly elastic medium, it is well known that the intrinsic attenuation of seismic waves by nonrecoverable energy conversion is not a prediction of the elastic constitutive model. For those applications where observations of intrinsic attenuation are important, the shortcomings of elasticity can be overcome by adopting linear viscoelasticity as the constitutive model for near-surface materials. Linearly elastic linearly viscoelastic constitutive models are introduced in this section by first developing the basic ideas of isotropic linear viscoelasticity, then reducing the viscoelastic constitutive equations to those representing isotropic linear elasticity as a special case. Fundamental results applicable to the intrinsic attenuation General Mathematics d ij d(t) Inverse tangent function (four quadrant) Kronecker delta: d ij = 1 for i = j d ij = 0 otherwise Dirac delta (generalized) function: ˆd Unit vector (unit vectors are indicated by a caret) f(t)*g(t) One-dimensional continuous convolution f(t) f ~ (u) Fourier integral transform pair H(t) Heaviside step function: H(t) = 0 for t < 0 H(t) = 1 for t 0 i = 1 Imaginary unit (when i is not used as a subscript or superscript) u = 2πf, u 0 = 2πf 0 Frequency f in Hz angular frequency u in radian/s t Time x Position vector (x 1, x 2, x 3 ), (x, y, z) Rectangular coordinates of the point indicated by position vector x ˆx 1, ˆx 2, ˆx 3 or î, ĵ, ˆk Unit vectors for the x 1 -axis, x 2 -axis, x 3 -axis, respectively Z R, Z I or Z R, Z I Real imaginary parts of a complex number: Z = Z R + iz I Z Absolute value (modulus) of a complex number: Z = Z 2 R + Z2 I Z Argument of a complex number: Z = arctan(z I /Z R ) Continuum Mechanics D Dt E(x, t), E ij (x, t) e(x, t), e ij (x, t) ε 0 f(x, t), f i (x, t) Material derivative operator Eulerian strain tensor, Cartesian component Deviatoric strain tensor, Cartesian component Amplitude of harmonic uniaxial strain Body force vector (force per unit volume), rectangular component Table of Symbols of seismic waves in a homogeneous isotropic linearly viscoelastic medium are deduced from the general principle of conservation of energy. The equations of motion in linearly elastic linearly viscoelastic media are introduced under the conditions of isotropy homogeneity. Finally, a simple example illustrating the important concepts of this section is provided using a Kelvin-Voigt medium. Basic definitions notation The dynamic process at time t > 0 associated with the passage of a seismic wave through a fixed position x is described by the Cauchy stress tensor T(x, t) the Eulerian strain tensor E(x, t). Both T(x, t) E(x, t) are second-order tensors are viewed as measuring the seismic disturbance relative to background conditions. The Cartesian stress component T ij (x, t) is the j-component of the (x, t), g ij (x,t) Engineering strain tensor, Cartesian component: g ij (x,t) = 2E ij (x,t) γ 0 Amplitude of harmonic engineering strain P ave (x, u 0 ) Potential energy per unit volume averaged over one p (x, t) period 23/u 0 Mean normal pressure: p(x, t) = q(x, t) Q(x, u 0 ), Q(u 0 ) Quality factor p(x, t) Mass density σ(x, t) Mean normal stress: q(x, t) = p(x, t) t Time interval T(x, t), T ij (x, t) Cauchy stress tensor, Cartesian component t(x, t), t ij (x, t) Deviatoric stress tensor, Cartesian component o(x, t) Dilatation: o(x, t) = E kk (x, t) = u(x, t) o 0 Amplitude of harmonic dilatational strain u(x, t), u i (x, t) Displacement vector of particle at x at time t, rectangular component U Internal energy U(x, u 0 ), U(x) Local increase in U over one period 23/u 0 or over an interval t v(x, t), v i (x, t) Velocity vector of particle at x at time t, rectangular component W D Dissipated energy in one cycle (engineering) W S Maximum potential energy during one cycle (engineering) v(u 0 ) Damping ratio (engineering) Linear Viscoelasticity Linear Elasticity G(x, t) G s (x, t), G s (t) G b (x, t), G b (t) G l (x, t), G l (t) G(x, t), G(t) HILE HILV Relaxation tensor Shear relaxation function Bulk relaxation function Combined relaxation function: G l (x, t) G b (x, t) (2/3)G s (x, t) General relaxation function (represents any relaxation function) Abbreviation for homogeneous isotropic linearly elastic Abbreviation for homogeneous isotropic linearly viscoelastic

3 Near-Surface Seismology: Wave Propagation 179 traction acting at x on any smooth surface through x with unit normal ˆx i. The strain component E ij (x, t) is defined by a sum of displacement gradients: (2) where u i (x,t) is the i-component of the Eulerian displacement vector u(x,t). This definition is based on the assumption of infinitesimal deformation so that products of displacement gradients can be neglected in the finite strain version of E ij (x,t). The well-known symmetry E ij (x,t) = E ji (x,t) is clear from definition (1); the analogous symmetry T ij (x,t) = T ji (x,t) holds for the general dynamic case in nonpolar continuum mechanics (Malvern, 1969, ). Some of the results to be presented in this section are given in terms of the deviatoric stress tensor t(x,t) deviatoric strain tensor e(x,t) with components defined by ILE ILV J(x, t) K(x, u 0 ), K(u 0 ) k(x), k L(x, u 0 ), L(u 0 ) l(x), l M(x, u 0 ), M(u 0 ) 2(x), 2 h(x), h X(x, u 0 ), X(u 0 ) Wave Propagation Abbreviation for isotropic linearly elastic Abbreviation for isotropic linearly viscoelastic Creep tensor Complex bulk modulus Elastic bulk modulus Complex Lamé modulus Elastic Lamé modulus (l also used for wavelength) Complex shear modulus Shear modulus or rigidity (elastic or Kelvin-Voight medium) Kelvin-Voigt viscosity General complex modulus (represents any complex modulus) a Angle of incidence of a plane wave on a plane interface a c Critical angle of incidence A P, A S Attenuation vectors (P-wave, S-wave) A General attenuation vector (represents either A P or A S ) A χ, A χ (u 0 ) Attenuation coefficient: A χ A cos χ α, β Complex velocities (P-wave, S-wave) χ Angle between propagation (P) attenuation (A) vectors c Phase velocity for a plane displacement wave c(u 0 ), c(u 0, χ) Phase velocity for harmonic plane displacement wave of frequency u 0 c n (u), c n (k n ) Phase velocity of mode n of a surface wave c R Phase velocity of a Rayleigh wave for a half-space (ct ˆd x) Phase of plane displacement wave (direction of propagation ˆd) f 0 (t) Source-time function γ j Direction cosine of x relative to positive x j -axis k P, k S Complex wave numbers (P-wave, S-wave) κ Wavenumber (magnitude of propagation vector) κ n (u) Wavenumber for mode n of a surface wave: κ n (u) = u/c n (u) λ Wavelength (λ also used for the elastic Lamé modulus) p Horizontal slowness (ray parameter) P(x, t), S(x, t) Lamé potentials (P-wave, S-wave) (1) Table of Symbols where d ij is the Kronecker delta (d ij = 1 when i = j d ij = 0 otherwise). It is also useful to recall the following stard definitions for auxiliary strain stress quantities: p(x, t), s(x, t) Helmholtz potentials for body force f(x, t) in completeness theorem Φ(x, t), (x, t) Helmholtz potentials for U(x) in harmonic displacement U(x)e iu t 0 P P, P S Propagation vector or vector wavenumber (P-wave, S-wave) P General propagation vector (represents either P P or P S ) R p, r p Angles of reflection transmission for a plane P-wave R SH, r SH Angles of reflection transmission for a plane SH-wave R SV, r SV Angles of reflection transmission for a plane SV-wave R, T Plane-wave reflection transmission coefficients, respectively T hw Traveltime of a head wave U n (u), U n (κ n ) Group velocity for mode n of a surface wave u N (x, t), u N i (x, t) Near-field displacement vector, rectangular component u P (x, t), u P i (x, t) P-wave displacement vector, rectangular component u S (x, t), u S i (x, t) S-wave displacement vector, rectangular component V A Apparent velocity of a plane wave along a plane interface V P, V S P-wave S-wave velocities in an HILE medium Additional comments on notation terminology A repeated subscript implies the summation convention. For example, E kk (x, t) refers to a sum over the range of subscript k:, (3), (4), (5), (6) The word complex, when used as an adjective, refers to a quantity (scalar, vector, or tensor) whose component(s) may be purely real or imaginary or may consist of nonzero real imaginary parts.

4 180 Near-Surface Geophysics Part 1: Concepts Fundamentals where g ij (x,t) is the engineering strain, o(x,t) is the dilatation, q(x,t) is the mean normal stress, p (x,t) is the mean normal pressure. A thorough discussion of the basic concepts definitions given above can be found in any of the many excellent reference books on continuum mechanics such as Fung (1965) Malvern (1969). Additional comments on the application of continuum mechanics to near-surface seismology are given in Appendix B. Isotropic linear viscoelasticity The constitutive model for an isotropic linearly viscoelastic medium may be stated in terms of two convolution integrals:, (7) where the functions G s (x,t) G b (x,t) are causal relaxation functions representing shear bulk behaviors, respectively. The convolutions in equation (7) indicate that an isotropic linearly viscoelastic medium can be viewed as a linear shift-invariant system with strain rate as input, stress as output, impulse response given by a relaxation function; see equation (A-13) for clarification of this point. It is possible to combine the two constitutive equations (7) by using the definitions (2) of the deviatoric stress t ij (x,t) deviatoric strain e ij (x,t) followed by consolidation of terms: where, (8) (9) is a causal combined relaxation function introduced for simplicity of notation. The physical dimensions of G s (x,t), G b (x,t), G l (x,t) are those of stress. If the linearly viscoelastic medium is homogeneous as well as isotropic, then the relaxation functions are independent of x are written G s (t), G b (t), G l (t). It is well known that an alternative formulation of linear viscoelasticity can be given in terms of causal creep functions that are convolved with the time derivative of stress to give strain; see equation (B-2). However, the parallels between traditional results for linearly elastic media those for linearly viscoelastic media are more directly developed in terms of relaxation functions, so that approach is followed here. Impulse response of an isotropic linearly viscoelastic medium As mentioned above, the convolution integrals in the constitutive equations (7) indicate that each relaxation function G s (x,t) G b (x,t) may be interpreted as the impulse response of a linear shift-invariant system. To clarify this statement, rewrite (7) using definitions (2) (5) to get the constitutive equations in terms of engineering shear strain dilatation: (10) Now suppose g ij (x,t) o(x,t) each have a time dependency at x given by the Heaviside function H(t) so that the derivatives in the convolution integrals each become a Dirac delta function d(t); see equations (A-1) (A-2) for definitions of H(t) d(t). With these substitutions, the integrations in equation (10) are easily carried out using result (A-7) to give (11) The straightforward interpretation of equation (11) is that G s (x,t) is the shear-stress response to a unit step input of shear strain (i.e., the shear-stress response to a unit impulse of shear-strain rate). Similarly, G b (x,t) is the mean normal stress response to a unit step input of dilatation (i.e., the mean normal stress response to a unit impulse of dilatation rate). Thus, the relaxation functions G s (x,t) G b (x,t) are impulse response functions that define the temporal variation of stress needed to maintain strain at a constant level. Of course, unit strain is inconsistent with the assumption of infinitessimal deformation. The concept of unit strain is a mathematical artifice used here to help clarify the meaning of the relaxation functions. In an actual experiment, the step input of strain would need to be of magnitude 10 6 or less but the interpretation of the relaxation functions would be the same. Characteristics of relaxation functions common relaxation models The desirable time-dependent characteristics of the relaxation functions are determined by considerations such

5 Near-Surface Seismology: Wave Propagation 181 as causality, the fading memory effect, physical requirements at times zero infinity, the results of experiments on actual materials (see Appendix B for additional information). These concerns dictate the nominal shape of a relaxation function G(t) for a solid (Figure 1), a shape that is more or less represented by three common relaxation models (Figure 2). The stard linear solid, (12) meets the nominal shape of Figure 1, but the Kelvin-Voigt model,, (13) has no fading memory because the stress does not continuously relax under constant strain. The Maxwell model,, (14) does not meet the common definition of a solid because shear stress is allowed to relax to zero. The well-known relationships of the stard linear solid, Kelvin-Voigt model, Maxwell model to networks of springs Figure 1. General graphical shape of a relaxation function G(t). Causality requires G(t) to be zero for negative time. The value G(0) of the relaxation function at zero time is greater than zero to represent the instantaneous elastic response of a viscoelastic material. As time becomes large, G(t) approaches G( ), the fully relaxed stress at constant strain, where G( ) < G(0). If the relaxation function is the shear relaxation function G s (t), the medium is a solid, the fully relaxed shear stress G s ( ) must be greater than zero to preclude unrestricted viscous shear flow. dashpots have pedagogical value are discussed in numerous reference books (e.g., Fung, 1965, 20-25). Complex moduli The convolution integrals appearing in constitutive equations (7), (8), (10) may be converted to products in the frequency domain by Fourier integral transformation application of the convolution theorem: a) b) c) 2, (15), (16) Figure 2. Graphs of relaxation functions for (a) stard linear solid, (b) Kelvin-Voigt medium, (c) Maxwell medium. All functions are zero for negative time to meet the causality requirement.

6 182 Near-Surface Geophysics Part 1: Concepts Fundamentals. (17) The frequency-domain results given in equations (15) (17) use the notation f(x,t) ~ f (x, u) to indicate a Fourier integral transform pair with angular frequency u (SI unit of radian/s). See equation (A-14) for the definition of the Fourier integral transform, equation (A-17) for a statement of the convolution theorem. The complex shear modulus M(x,u), complex bulk modulus K(x,u), complex Lamé modulus L(x,u) in equations (15) (17) are complex functions of u defined by (18) where i represents the imaginary unit (i = 1). The (iu) factors in definitions (18) arise from the time derivatives of the strain terms in the convolution integrals; see equation (A-16). However, because the (iu) factors are included in the definitions of the complex moduli, the inverse transforms of the complex moduli are the time derivatives of the relaxation functions: (19) Furthermore, the linear shift-invariant systems that correspond to equations (15) (17) have strain as input (as opposed to strain rate), stress as output, derivatives of the relaxation functions as impulse responses. Definitions (9) (18) are used to deduce the following relationships between the complex moduli:. (20) homogeneous as well as isotropic, then the complex moduli are independent of x are written M(u), K(u), L(u). Frequency response of an isotropic linearly viscoelastic medium The frequency-domain products in equations (15) (17) indicate that the stress in an isotropic linearly viscoelastic medium subject to a given strain is determined by the complex moduli acting as frequency response functions; see equations (A-17) (A-18) for a general explanation of the frequency response of a linear shift-invariant system. The role of the complex moduli as frequency response functions may be clarified by computing the stress resulting from steady-state harmonic strain in a simple example. Consider steady-state harmonic engineering shear strain g ij (x,t) dilatation o(x,t) of angular frequency u 0 amplitudes g 0 (x) o 0 (x): (21) As explained in equations (A-19) (A-21), the complex exponentials in equation (21) are eigenfunctions of the linear shift-invariant systems represented by equation (17). Thus, the strains may be directly multiplied by the complex moduli to get the stresses:. (22) The physical stresses are given by the real parts of T ij (x,t) q(x,t): (23) which are obtained from equation (22) by putting the complex moduli into polar form, adding exponents, then extracting the real part on the right-h side of each equation. The stress-strain equations in (23) show that the stress is harmonic of the same frequency as the strain, that the stress amplitude is given by the product of the strain amplitude the absolute value of the complex modulus, where the absolute values are given by The physical dimensions of M(x,u), K(x,u), L(x,u) are those of stress. If the linearly viscoelastic medium is (24)

7 Near-Surface Seismology: Wave Propagation 183 where subscripts R I indicate real imaginary parts, respectively. Furthermore, the stress-strain equations in (23) show that the stress is out of phase with the strain by the argument of the complex modulus, where the arguments are given by (25) The arguments of M(x,u 0 ) K(x,u 0 ) in equation (25) are called loss angles their corresponding tangents are called loss tangents. This terminology stems from the fact that the arguments are related to energy dissipation through the imaginary parts of the complex moduli; see equation (32) for the relationship between energy dissipation the complex moduli. It is important to realize that a complex modulus cannot be viewed as an arbitrarily specified frequency response function for a viscoelastic material. One problem is that inverse Fourier transformation of an arbitrarily specified modulus may lead to an acausal relaxation function. A complex modulus that corresponds to a causal relaxation function has real imaginary parts that satisfy the Kramers-Kronig relations, see equation (B-3). Isotropic linear elasticity The constitutive model for an isotropic linearly elastic medium is obtained as a special case of isotropic linear viscoelasticity by choosing the relaxation functions to be independent of time except for a step discontinuity at t = 0: (26) where 2(x) is the elastic shear modulus (or rigidity), k(x)is the elastic bulk modulus, l(x) is the elastic Lamé modulus. All three elastic moduli have physical dimensions of stress, definition (9) implies the following relationships between them: isotropic, then the elastic moduli are independent of x are written 2, k, l. Notice that the relaxation functions (26) imply no stress relaxation (Figure 3), which is entirely consistent with the basic assumptions that underlie linear elasticity. Computation of the complex moduli using definitions (18) gives M(x) = 2(x), K(x) = k(x), L(x) = l(x), where each modulus is independent of frequency. Also, because the complex moduli are all real, the loss angles are zero, no phase difference exists between stress strain in an isotropic linearly elastic medium. Substitution of the relaxation functions (26) into the constitutive equation (8), followed by straightforward integration, gives the familiar constitutive equation for an isotropic linearly elastic medium (Malvern, 1969, ):, (28) where it is assumed that the strains are zero in the distant past. By applying the definitions (2) for deviatoric stress strain relationships (27), it is easy to show that constitutive equation (28) is equivalent to. (29) Finally, constitutive equation (29) may be expressed in terms a) b) (27) If the linearly elastic medium is homogeneous as well as Figure 3. Relaxation functions for an isotropic linearly elastic medium. (a) Shear relaxation function G S (t). (b) Combined relaxation function G l (t) = G b (t) (2/3)G s (t). Both functions are zero for negative time to meet the causality requirement.

8 184 Near-Surface Geophysics Part 1: Concepts Fundamentals of engineering shear strain dilatation using definitions (2) (5): (30) The identical mathematical forms of equations (15) (29), (16) (28), (17) (30), (20) (27), are examples of the correspondence principle between time-domain equations in linear elasticity frequencydomain equations in linear viscoelasticity (Christensen, 1982, 45 46). Corresponding equations are gathered in Table 1 for ease of comparison. Abbreviations The abbreviations ILE ILV are used for convenience to refer to isotropic linearly elastic media isotropic linearly viscoelastic media, respectively. As shown above, an ILE medium is a special case (no stress relaxation at constant strain) of an ILV medium. If the medium is also homogeneous so that its properties are independent of the location x, the abbreviations ILE ILV become HILE HILV, respectively, where the H refers to homogeneous. Dissipation of energy Because of the relatively short duration of a seismic event, the propagation of seismic waves in the region away from the source is typically viewed as an adiabatic process in a material lacking internal sources of heat. Under these assumptions, the amount of energy lost (dissipated) by a seismic wave in a given time interval is equal to the corresponding gain in internal energy U of the body occupied by the wave disturbance. The discussion given below is concerned with computing the local increase in U per unit volume in an HILV medium for two special cases of strain: (1) harmonic uniaxial strain (2) harmonic plane shear strain. These special cases are chosen because they correspond to the strain generated by harmonic plane displacement P-waves S-waves, respectively, in an HILV medium. The local rate of increase of internal energy U per unit volume for any medium undergoing deformation is given by the energy equation which is based on the principle of conservation of energy (Malvern, 1969, 230). The energy equation may be simplified under the assumptions of seismic wave propagation (infinitessimal deformation, adiabatic conditions, no internal heat sources), then integrated to give Table 1. Examples of the correspondence principle between an ILE medium an ILV medium. Isotropic linear elasticity (ILE) 2(x) shear modulus or rigidity k(x) bulk modulus l(x) Lamé modulus Isotropic linear viscoelasticity (ILV) M(x, ω) complex shear modulus K(x, ω) complex bulk modulus L(x, ω) complex Lamé modulus µ(x) = (3/2)[k(x) l(x)] M(x, ω) = (3/2)[K(x, ω) L(x, ω)] k(x) = l(x) + (2/3)2(x) K(x, ω) = L(x, ω) + (2/3)M(x, ω) l(x) = k(x) (2/3)2(x) L(x, ω) = K(x, ω) (2/3)M(x, ω) (Equation 27) (Equation 20) t ij (x, t) = 22(x)e ij (x, t) ~ tij (x, ω) = 2M(x, ω) ~ e ij (x, ω) ~ T kk (x, t) = 3k(x)E kk (x, t) T kk (x, ω) = 3K(x, ω) E ~ kk (x, ω) (Equation 29) (Equation 15) T ij (x, t) = l(x)e kk (x, t)d ij + 22(x)E ij (x, t) ~ T ij (x, ω) = L(x, ω) E ~ kk (x, ω)d ij + 2M(x, ω) E ~ ij (x, ω) (Equation 28) Equation (16) ~ T ij (x, t) = 2(x)γ ij (x, t) (i j) T ij (x, u) = M(x, ω) g ~ ij (x, u) (i j) q(x, t) = k(x)o(x, t) ~ q(x, u) = k(x, u) (Equation 30) (Equation 17)

9 Near-Surface Seismology: Wave Propagation 185, (31) where U(x) is the increase in internal energy U, over the interval t, of the material occupying an infinitesimal volume dv located at x. Evaluation of energy equation (31) for the special cases of harmonic uniaxial strain harmonic plane shear strain, in an HILV medium deforming at angular frequency u 0, gives the energy dissipated per unit volume in one period (23/u 0 ): (32) where e 0 (x) g 0 (x) are the amplitudes of the uniaxial strain plane shear strain, K 1 (u 0 ) M 1 (u 0 ) are the imaginary components of the complex bulk modulus complex shear modulus, respectively. Because the dissipation of energy in these special cases is directly proportional to the imaginary parts of the complex moduli, the term loss modulus for K 1 (u 0 ) or M 1 (u 0 ) is commonly used, especially in the engineering literature. The computational details leading to result (32) are given by equations (B- 10) (B-19). Quality factor (Q) The quality factor Q is a dimensionless parameter used to measure the tendency of a material to dissipate energy during deformation. Although different fundamental definitions have been proposed for Q, the common idea has been to form a ratio of potential energy to dissipated energy over one period of harmonic deformation. The fundamental definition suggested by O Connell Budiansky (1978) is especially appropriate for linear viscoelasticity:, (33) where P ave (x,u 0 ) is the potential energy per unit volume averaged over one period (23/u 0 ) of harmonic deformation at angular frequency u 0, U(x,u 0 )/dv is the increase in internal energy per unit volume over the same period. If the medium is homogeneous, then the quality factor is independent of x is written Q(u 0 ). It is important to notice that the definition (33) does not specify the type of harmonic deformation that is to be considered in determining the ratio of potential to dissipated energy. This imprecision gives a phenomenological basis to the theory measurement of the quality factor. Formulas for the average potential energy for the special cases of harmonic uniaxial strain harmonic plane shear strain, in a broad class of HILV media, can be deduced by reference to O Connell Budiansky (1978): (34) where e 0 (x) g 0 (x) are as defined for equation (32), K R (u 0 ) M R (u 0 ) are the real components of the complex bulk modulus complex shear modulus, respectively. Equations (32) (34) are used with definition (33) to give Q(u 0 ) for an HILV medium: (35) Notice that if the loss moduli are zero (as for an HILE medium), then equation (32) indicates zero dissipation of energy during harmonic strain, equation (35) indicates an infinite quality factor. On the other h, large loss moduli correspond to substantial energy dissipation a small quality factor. Damping ratio The damping ratio is another dimensionless measure of energy dissipation, it is defined as follows by the geotechnical engineering community (Kramer, 1996, 231):, (36) where the deformation is assumed to be harmonic plane shear strain of angular frequency u 0, W D (u 0 ) is the dissipated energy over one period (23/u 0 ), W S (u 0 ) is the maximum potential energy achieved during that period. If result (32) for the case of harmonic plane shear strain is used for W D (u 0 ), the customary engineering expression

10 186 Near-Surface Geophysics Part 1: Concepts Fundamentals (37) is used for W S (u 0 ), then the damping ratio for an HILV medium can be written. (38) As pointed out by O Connell Budiansky (1978), equation (37) does not correctly give the maximum strain energy in a general HILV medium unless M I (u 0 ) is small so that the medium is nearly nondissipative. Thus, rather than base the definition of damping ratio on (36) with the customary use of equation (37) for W S (u 0 ), it seems reasonable to adopt equation (38) as the definition of the damping ratio v(u 0 ) while keeping in mind that equations (36) (37) hold as approximations for small dissipation. Equations of Motion The general equation of motion for any medium can be written in terms of stress, body force, particle velocity (Malvern, 1969, 214):, (39) where p(x,t) is the mass density, f i (x,t) is the i-component of the body force per unit volume f(x,t), v i (x,t) is the i- component of the velocity v(x,t) of the particle occupying position x at time (i.e., v(x,t) = Du(x,t)/Dt). The operator D/Dt is the material derivative of a time-dependent Eulerian variable (Malvern, 1969, ). The general equation of motion (39) is greatly simplified by the following approximations: (40) Approximations (40) are generally assumed to hold under the conditions of infinitesimal deformation. Given approximations (40) assuming zero body force, the equation of motion (39) becomes, (41) which is a useful simplification suitable for many applications. Specialization of the general equation of motion (39) or its simplified version (41) for a particular medium is accomplished by substitution of the appropriate constitutive equations giving stress in terms of a second-order tensor describing deformation. The equation of motion for an HILE medium is derived by substituting the HILE constitutive equation (28) into the equation of motion (39), using approximations (40) to obtain the elastodynamic equation of motion: (42) The elastodynamic equation of motion can also be written in terms of the bulk modulus k by using (27) to substitute for l to get (43) Notice that in the absence of body force, the equations of motion (42) (43) for an HILE medium are linear homogeneous so that any number of solutions for the displacement field may be summed to provide another solution. The derivation of the equation of motion for an HILV medium also uses approximations (40), but is complicated by the fact that the constitutive equation (8) involves convolution integrals. However, the convolutions can be converted to products by Fourier integral transformation to give the following frequency-domain expression of the HILV equation of motion: (44) Equation (44) can also be written in terms of the complex bulk modulus K(u) by using (20) to substitute for L(u) to get (45) In equations (44) (45), ~ u(x,u) ~ f(x,u) are the Fourier integral transforms of the displacement u(x,t) the body force per unit volume f(x,t), respectively. As is the case with an HILE medium, notice that in the absence of body force, the HILV equations of motion (44) (45) are linear homogeneous. Thus, any number of solutions for the displacement field in the frequency domain may be summed to provide another solution (which then can be converted to the time domain by inverse Fourier transformation).

11 Near-Surface Seismology: Wave Propagation 187 Kelvin-Voigt model The Kelvin-Voigt model has been used in near-surface seismology geotechnical engineering to represent the dynamic behavior of subsurface materials under infinitessimal plane shear strain. Because the Kelvin-Voigt model is mathematically simple, it is also a commonly used example of isotropic linear viscoelasticity. The shear relaxation function G s (x,t) for the Kelvin- Voigt model is defined by equation (13) with traditional symbols used for the Kelvin-Voigt shear modulus 2(x) the Kelvin-Voigt viscosity h(x): (46) Comparison of the shear relaxation functions (26) (46) indicates that the only difference between a Kelvin-Voigt medium an ILE medium is the viscosity term. The Kelvin-Voigt complex shear modulus is given according to definition (18) by Fourier integral transformation of G s (x,t) multiplication by (iu) to get. (47) The corresponding loss angle is arctan[uh(x)/2(x)], indicating that the shear stress strain under harmonic conditions are out of phase in a Kelvin-Voigt material except at zero frequency. Substitution of the Kelvin-Voigt shear relaxation function (46) into constitutive equation (10) gives the Kelvin- Voigt constitutive equation for shear stress engineering shear strain:. (48) Comparison of equations (30) (48) again confirms that the viscosity term is the only difference between a Kelvin- Voigt medium an ILE medium. The expression for the Kelvin-Voigt quality factor Q(u) is derived for a homogeneous medium so that 2(x) h(x) are now denoted by 2 h, respectively. Equations (35) (47) lead to the following expression for Q(u) for plane shear strain:. (49) The frequency dependence of Q(u) evident in equation (49) has led some to question the application of the Kelvin- Voigt model in seismic bwidths where experimental evidence shows the quality factor to be approximately independent of frequency (e.g., Kjartansson, 1979). Finally, the Kelvin-Voigt equation of motion is developed for infinitessimal plane shear strain in a homogeneous medium. Consider (without loss of generality) a displacement function that gives plane shear strain in the x 1 - x 2 plane:. (50) The simplified equation of motion (41) for the u 2 -component is, (51) where the spatial derivatives of T 22 T 23 have been set to zero because the problem can only have spatial dependence on x 1. Computation of the displacement gradients shows that the only nonzero strains are. (52) Thus, according to the Kelvin-Voigt constitutive equation (48), the only nonzero shear stresses are T 12 (x 1,t) T 21 (x 1,t) (which are equal because of symmetry of the stress components): (53) Substitution of the stress component (53) into the equation of motion (51), using the traditional notation v for u 2 x for x 1, gives the Kelvin-Voigt equation of motion under the condition of plane shear strain in a homogeneous medium:. (54) Equation (54) is the one-dimensional version of the equation of motion assumed by Ricker (1953) in a classic paper on the propagation of body waves in a dissipative medium. It has also been used as the governing equation of motion for the study of the attenuation velocity dispersion of SH-waves in near-surface materials (e.g., Michaels, 1998). Seismic Body Waves A body wave is a propagating disturbance that can exist in an unbounded medium. Basic results for body waves in linearly elastic linearly viscoelastic media are summarized here but only for homogeneous isotropic media to keep the presentation within reasonable bounds. The emphasis is on plane displacement waves follows the elegant treatment by Borcherdt (1973) who, along with Buchen (1971), considered plane waves in HILE HILV media from a unified viewpoint. The point body force problem in an unbounded HILE medium as formulated by

12 188 Near-Surface Geophysics Part 1: Concepts Fundamentals Aki Richards (2002) is also reviewed to identify the characteristics of wave propagation that are associated with a finite seismic source. The goal throughout is a detailed but reasonably economical introduction to the primary features of body-wave propagation. Perhaps the most important topic that is not addressed here is body-wave propagation in smoothly varying heterogeneous isotropic linearly elastic media. However, this topic is described in many papers in stard texts such as Hudson (1980) Aki Richards (2002). Readers interested in body-wave propagation in anisotropic linearly elastic materials will find Synge (1957), Musgrave (1961), Kraut (1963), Crampin (1981) to be helpful, useful velocity formulas for different types of anisotropy are given by Mavko et al. (1998). Although solutions of the classical homogeneous inhomogeneous wave equations their interpretations are also not discussed here, many excellent general texts on the subject are available such as Coulson Jeffrey (1977) Barton (1989). Completeness theorem for displacement fields in an HILE medium A fundamental theorem of linear elasticity is that any displacement field in an HILE medium can be represented as the gradient of a real function plus the curl of a vector function, where these two functions satisfy classical wave equations. More precisely, if u(x,t) is any suitably differentiable solution of the elastodynamic equation of motion (43), then u(x,t) can be represented by, (55) where the functions P(x,t) S(x,t) are known as the Lamé potentials. The Lamé potentials satisfy inhomogeneous wave equations: where S(x,t) also satisfies a divergence condition:, (56). (57) The functions p(x,t) s(x,t) in equation (56) are the Helmholtz potentials of the body force f(x,t):. (58) The use of Helmholtz potentials to represent vector functions is described in Appendix C; see equations (C-1) (C-2). The constants V 2 P V2 S are defined by (59) where k is the bulk modulus, 2 is the shear modulus or rigidity, p is the density. The constant V 2 P is also commonly written in terms of the Lamé modulus l by using relations (27) to get. (59) Notice that V P > V S because k, l, 2 are all positive real constants. As is shown later in this section, V P V S are associated with irrotational equivoluminal displacement body waves known as P-waves S-waves, respectively. Thus, V P is known as the P-wave velocity V S is known as the S-wave velocity. The theorem stated above is valid for bounded unbounded bodies (Sternberg, 1960), is called the completeness theorem by Achenbach (1973, 85-88) Lamé s theorem by Aki Richards (2002, 67-69). The converse of the completeness theorem also holds which is easily shown by substituting the displacement (55) into the elastodynamic equation of motion (43), making use of equations (56) (59) to produce an identity. P-waves S-waves in an HILE medium An important consequence of the completeness theorem is that under very general conditions, the displacement field u(x,t) in an HILE material can be expressed as the sum of irrotational body waves equivoluminal body waves. To see this result clearly, assume that u(x,t) is a solution of the elastodynamic equation of motion (43) so that equations (55) (59) hold for Lamé potentials P(x,t) S(x,t). Let the first term P(x,t) in the displacement (55) be denoted by u P (x,t), let the second term S(x,t) be denoted by u S (x,t), so that equation (55) can be rewritten as. (61) Basic vector calculus identities imply that u P (x,t) is irrotational (i.e., the curl is zero) u S (x,t) is equivoluminal (i.e., the divergence is zero):. (62)

13 Near-Surface Seismology: Wave Propagation 189 Furthermore, the gradient of the first equation in (56), the curl of the second equation in (56), show that u P (x,t) u S (x,t) satisfy inhomogeneous wave equations:. (63) The conclusions from (62) (63) are that u P (x,t) u S (x,t) propagate in an HILE medium as irrotational equivoluminal waves, respectively. Furthermore, these waves may properly be classified as body waves because the analysis is based on the completeness theorem, which is valid for both bounded unbounded bodies. Body waves u P (x,t) u S (x,t) are the well known P- wave S-wave of seismology. The important result of this section is that any displacement field in an HILE medium can be viewed as a sum of P-waves S-waves, because, by the completeness theorem, any displacement field u(x,t) satisfying the elastodynamic equation of motion (43) admits the representation (61), where u P (x,t) u S (x,t) are the gradient curl of the Lamé potentials P(x,t) S(x,t), respectively. It is also worth noting that the defining difference between a P-wave an S-wave in an HILE medium is that a P-wave is irrotational an S- wave is equivoluminal. This point is lost easily by the association of these waves with other properties physical quantities, so that P-waves are also called primary, longitudinal, compressional, dilatational waves; S-waves are commonly referred to as secondary, transverse, shear, rotational waves. Plane displacement waves A sufficiently general expression for a plane displacement wave is as follows:, (64) where U i (x) is a real function of position x, F i (x) is a real function of a real variable x, ˆd is a real unit vector, c is a positive real constant called the phase velocity. The properties terminology associated with plane displacement waves are fundamental in seismology are reviewed in this section. The phase of the plane displacement wave (64) is the expression (ct ˆd x) which gives the dependence of each function F i (x) on spatial position time. Setting the phase equal to a real constant C defines a plane of uniform phase called the phase plane that occupies a time-dependent position in space has unit normal ˆd:. (65) The phase plane has important physical meaning if each function U i (x) is a real constant U i. In that case, the phase plane is also a plane of uniform displacement as can be seen by substitution of equation (65) into the plane wave (64) to get a constant displacement vector. The phase velocity c is the speed at which the phase plane advances through space in the direction of its unit normal. This concept is readily grasped by recognizing that ˆd x is the distance D of the phase plane from the origin as measured along the line D ˆd (Figure 4). Thus, equation (65) can be written (66) which shows that the speed dd/dt of the phase plane is indeed the phase velocity c. A special case of the plane displacement wave (64) is the simple plane displacement wave obtained when each function U i (x) is a real constant U i the functions F i (x) are all the same function F(x):, (67) where U = U 1ˆx 1 + U 2ˆx 2 + U 3ˆx 3 is a real vector constant. Harmonic plane displacement waves Consider the following displacement field with harmonic dependence on time distance: where U(x) is a real function of position x, the rectangular components C i of vector C are complex constants, angular frequency u is a positive real constant, P is a real vector constant called the propagation vector or vector wavenumber, Re indicates the operation of taking the real part of the expression in braces. A displacement field of the form of equation (68) is called a harmonic plane displaceu, (68) Figure 4. Parameters describing the geometry of the phase plane of a plane wave. The phase plane is shown by a heavy line propagates in the direction of the unit normal vector ˆd. The distance D = ˆd x measures the distance of the phase plane from the origin O in the direction of propagation. C

14 190 Near-Surface Geophysics Part 1: Concepts Fundamentals ment wave serves as the basis for describing planewave propagation in HILE HILV media. The mathematical form of the harmonic plane displacement wave (68) is understood best in terms of its components:, (69) where C i C i are the absolute value argument of C i, respectively. Extracting the real part of equation (69) factoring the cosine argument gives, (70) where P is called the wavenumber ˆP = P/ P is the unit vector in the direction of P. Summing equation (70) on index i with appropriate unit vectors gives an equation of the form of equation (64):, (71) thereby showing that equation (68) is indeed a plane displacement wave. The wave propagates in the direction of propagation vector P with phase velocity c given by, (72) which is the key relationship between phase velocity, angular frequency, wavenumber in the theory of harmonic plane displacement waves. It will be useful to describe one additional type of plane displacement wave. A simple harmonic plane displacement wave is obtained from the general form of equation (68) by setting U(x) = 1 C = Ue ij where U is a real vector constant j is a real constant:. (73) Factoring the argument of the cosine function as before gives, (74) where the phase velocity c is given by equation (72). It is clear that the harmonic plane displacement wave (73) or equivalently (74) is also a simple plane displacement wave of the form of equation (67). Because the various mathematical representations of plane displacement waves will be used repeatedly in the remainder of this section, a summary of the representations is provided in Table 2 for convenience. Frequency, period, wavelength A few special parameters applicable to harmonic plane displacement waves are reviewed here. The frequency is denoted by f is defined in terms of the angular frequency u by. (75) Although the term frequency is commonly used to refer to either f (SI unit is s 1 or Hz) or u (SI unit is rad/s), the context of a statement can usually be used to determine the intended meaning. The frequencies recorded in surfacebased experiments in near-surface seismology generally lie well within the Hz range. The frequency bwidth preserved on any given seismogram depends on the frequency response of the recording system, the wave-propagation characteristics of near-surface materials, the presence of noise. The period T wavelength l of a harmonic plane displacement wave are defined with respect to the real part of the complex exponential in general form (68):. (76) If the spatial position x is fixed, the cosine function in (76) is plotted against variable time t, then the temporal interval between peaks is defined to be the period T is related to u f as follows:. (77) On the other h, if time t is fixed the cosine function is plotted against the distance ˆP x, then the spatial interval between peaks is defined to be the wavelength l is related to P as follows:. (78) Based on the expressions for phase velocity (72), frequency (75), wavelength (78), the phase velocity can be written as the product of wavelength frequency:. (79) As an example, a 100-Hz harmonic plane displacement wave with a wavelength of 15 m propagates with phase velocity 1500 m/s. Simple plane displacement waves in an HILE medium Simple plane displacement waves in an HILE medium are either P-waves with longitudinal displacement or S-

15 Near-Surface Seismology: Wave Propagation 191 waves with transverse displacement. This result was shown by Achenbach (1973, ) who substituted the simple plane displacement wave [equation (67)] into the elastodynamic equation of motion (43) to get an equation that constrains the displacement parameters:. (80) The interpretation of equation (80) is facilitated by investigating two mutually exclusive possibilities: either U ˆd are parallel or U ˆd are not parallel. If U ˆd are parallel, then the wave must be longitudinal (particle displacement is parallel to the direction of propagation) U may be written. (81) Substitution of equation (81) into constraint (80) solving for c shows that the phase velocity of the longitudinal wave is the P-wave velocity:. (82) Table 2. Summary of mathematical representations of plane displacement waves. Type Representation for displacement u(x,t) Notes General Simple Harmonic Simple harmonic Equation (64) Equation (67) C Equation (68) Equation (71) Equation (73) However, if U ˆd are not parallel, then constraint (80) requires that both of the following equations hold:. (83) The first equation in (83) shows that the wave is transverse (particle displacement is normal to the direction of propagation), whereas the second equation shows that the phase velocity of the transverse wave is the S-wave velocity:. (84) Thus, using the notation u P (x,t) u S (x,t) introduced earlier for P-waves S-waves in an HILE medium, the two types of simple plane displacement waves that can propagate in an HILE medium are (85) Phase velocity c is a positive real constant. Direction of propagation is given by real unit vector ˆd. Function U i (x) is a real function of position vector x. Functions F i (x) F(x) are real functions of real variable x. Unit vector ˆx i is in the direction of the positive x i - axis. Vector U is a real vector constant. Angular frequency u is a positive real constant. Constant j is a real number. Phase velocity is the real constant c = u/ P where P is the propagation vector. Direction of propagation is given by the unit vector ˆP= P/ P. Function U(x) is a real function of position vector x. Vector U unit vector ˆx i same as above. Vector C has rectangular components C i that are complex constants. Equation (74)