Chapter 6. Secondary source theories

Chapter 6 Secondary source theories So far we have concentrated on synthetic seismogram calculations for the case of one-dimensional Earth models in which the velocity varies only as a function of depth. Under this assumption, we have shown how ray theoretical methods such as WKBJ and matrix formulations such as reflectivity can be used to solve the wave equation. Our treatment was based on a flat earth, but can also be used for a radially symmetric earth by applying the earth flattening transformation. To a good first order approximation, the deep earth is close to spherically symmetric so these methods often are adequate for modeling observed seismograms. However, lateral heterogeneity is always present to some degree, particularly in the crust, and is often the target of greatest interest in seismic studies now that the average radial velocity structure has been determined. Computing synthetic seismograms in 3D velocity structures is much more complicated than the 1D calculation. Ray theoretical methods can be generalized to 3D (e.g. Maslov or Gaussian beam methods), but the ray tracing can be tricky and the results still suffer from the limitations of ray theory. Reflectivity methods cannot be generalized to 3D. Finite differences provide exact solutions in 3D, but at great computational cost. Here we present some methods for computing synthetic seismograms that are very useful for certain types of laterally heterogeneous models. They cannot be used in every case, but when applicable, they often can produce accurate 83

84 CHAPTER 6. SECONDARY SOURCE THEORIES synthetics with good computational efficiency. These techniques all involve the concept that each point on the wavefront can be considered in some circumstances to generate a secondary source, and that the response at a receiver can be computing by summing the contributions from the secondary sources. This can be used to generate realistic synthetics for scattering from an irregular interface (Kirchhoff theory) or from 3-D random heterogeneity (the Born approximation). However, to provide some motivation, we begin by considering Huygen s principle. 6.1 Huygens principle This idea was first described by Huygens (c. 1678) and is often called Huygens principle. It is most commonly mentioned in the context of light waves and optical ray theory, but is applicable to any wave propagation problem. If we consider a plane wavefront traveling in a homogeneous medium, we can see how the wavefront can be thought to propagate through the constructive interference of secondary wavelets: t + t t This simple idea provides, at least in a qualitative sense, an explanation for the behavior of waves when they pass through a narrow aperture:

6.1. HUYGENS PRINCIPLE 85 The bending of the ray paths at the edges of the gap is termed diffraction. The degree to which the waves diffract into the shadow of the obstacle depends upon the wavelength of the waves in relation to the size of the opening. At relatively long wavelengths (e.g. ocean waves striking a hole in a jetty), the transmitted waves will spread out almost uniformly over 180. However, at short wavelengths the diffraction from the edges of the slot will produce a much smaller spreading in the wavefield. For light waves, very narrow slits are required to produce noticeable diffraction. These properties can be modeled using Huygens principle by computing the effects of constructive and destructive intererence at different wavelengths. Huygens principle is a useful concept since it provides a simple way to gain an intuitive understanding of many aspects of wave behavior. However, it fails as a quantitative theory in several respects: (1) it says nothing about what amplitude the secondary waves should have, or how their radiation pattern might vary as a function of ray angle, (2) it predicts the wrong phase for the secondary arrivals, (3) it does not explain why the waves should not radiate backwards. 6.1.1 A simple plane wave example To understand this better, let s attempt to use Huygens principle to model plane wave propagation. Consider a receiver P located a distance d in front of a plane wave of amplitude A traveling at velocity c in a homogeneous whole space: dr plane wave y r d dy d P The current position of the wavefront is specified as t = 0. Define t 0 = d/c

86 CHAPTER 6. SECONDARY SOURCE THEORIES as the time that the wavefront will pass by P. Now, let us sum the contributions from the spherical wavefronts generated by each point on the wavefront at time t (t > t 0 ). Waves that arrive within a time interval dt are from a ring on the surface at distance r from P. This ring has a radius y and width dy. The surface area of the ring, ds, may be expressed as ds = 2πy dy = 2π(r sin θ)(dr/ sin θ) = 2πr dr = 2πrc dt (6.1) The expected amplitude at P is then given by multiplying the area ds by the amplitude of the incident plane wave A and the geometrical spreading factor for spherical waves, 1/r, and dividing by the time interval dt. A P (t > t 0 ) = A ds(1/r)(1/dt) = A2πrc dt(1/r)(1/dt) = 2πcA (6.2) Notice that A P has the form of a step function with height 2πcA. 2 ca t 0 Now imagine that the plane wave has a source-time function A = f(t). We would like to form the response at P as a convolution between f(t) and the result of our Huygens calculation. For a plane wave in a homogeneous whole space we already know the answer that we should get a delta function at t 0. A(t 0 ) = f(t) δ(t 0 ) desired result (6.3) The pulse should travel to P unchanged in both amplitude and shape. Yet our calulation predicts that the convolutional operator has the form of a step

6.2. KIRCHHOFF THEORY 87 function with height 2πc A(t 0 ) = f(t) 2πcH(t 0 ) = f(t) G(t) Huygens result (6.4) where H is the Heaviside unit step function, and G(t) 2πcH(t 0 ) is the result of our Huygens calculation. Since the derivative of H(t) is δ(t) we can fudge our solution into the correct form by taking the derivative and dividing by 2πc A(t 0 ) = f(t) ( 1 2πc ) t G(t) = 1 2πc f (t) G(t) (6.5) where f = f and we have used [f(t) g(t)] = f (t) g(t) = f(t) g (t). t t Is there a simple explanation for where these additional terms might come from? Some insight may be gained from the expression for the far-field radiation from a point source (eqn. 7.12 from the Introduction to Seismology text) ( ) 1 f(t r/c) u(r, t) = rc t (6.6) This provides some rationale for the 1/c factor in (6.5) and for using the time derivative of f(t) as our effective source-time function for the Huygens wavelets if we assume that we are in the far-field. However, we are still left with a factor of 1/2π that is unexplained. Although we can scale the Huygens result to produce the correct answer in this particular case, we have no guarantee that this will work for other situations, nor do we understand where these correction factors come from. 6.2 Kirchhoff theory (Introduction to Seismology (ITS) textbook, p. 133 138, eqn. 7.34 7.66) 6.2.1 The plane wave example revisited Before continuing, let us now try out our new formalism on the simple plane wave example that we earlier attempted to solve using Huygens principle.

88 CHAPTER 6. SECONDARY SOURCE THEORIES To model a plane wave, we let r 0 and assume that the wave has unit amplitude on S at time t = 0 dr plane wave y r d dy rd d P To obtain an exact solution, let us use ITS 7.65, containing both the nearfield and far-field terms. The 1/r 0 geometrical spreading term is not needed in the case of a plane wave, but the 1/r 2 0 term goes to zero and cos θ 0 = 1 since the rays are perpendicular to ds. Thus we have φ P (t) = 1 4π + 1 4π S S δ(t r/c) cos θ ds f(t) r 2 δ(t r/c) 1 + cos θ ds f (t) (6.7) cr Although (6.7) was derived assuming a closed surface around P, it will be sufficient to evaluate the integral only over the plane, since we can imagine the curve being closed far enough away from P that its contributions would arrive later than any time of interest. As before, to evaluate the integral we define a ring with surface area ds P Recall (hk.1) for the area of the ring, ds = 2πrc dt, and note that cos θ = d/r. We thus have for the first term in (6.7) 1 δ(t r/c) cos θ 4π S r ds = 1 2 4π d/r (2πrc)H(t d/c) r2

6.2. KIRCHHOFF THEORY 89 = dc H(t d/c) 2r2 = dc H(t d/c) 2c 2 t2 = d H(t d/c) (6.8) 2ct2 where H(t d/c) is the Heaviside step function and we have used r = ct. Similarly, the second term in (6.7) becomes 1 4π S δ(t r/c) 1 + cos θ ds = 1 cr 4π 1 + d/r (2πrc)H(t d/c) cr = 1 (1 + d/r)h(t d/c) 2 Substituting (6.8) and (6.9) into (6.7), we have φ P (t) = = 1 (1 + d/ct)h(t d/c) (6.9) 2 d 2ct H(t d/c) f(t) + 1 ( 1 + d ) H(t d/c) f (t) (6.10) 2 2 ct Moving the time derivative to the other side of the convolution, we obtain φ P (t) = d 2ct H(t d/c) f(t) + 2 t = d [ H(t d/c) f(t) + 2ct2 ( 1 = 2 + d ) δ(t d/c) f(t) 2ct ( 1 = 2 + d ) δ(t d/c) f(t) 2cd/c = δ(t d/c) f(t) [( 1 2 + d ) 2ct d H(t d/c) + 2ct2 H(t d/c) ] f(t) ( 1 2 + d ) ] H(t d/c) f(t) 2ct t = f(t d/c) (6.11) This is what we expected to obtain for the plane wave. The source time function is delayed by the travel time to the point P but the amplitude and wave shape are unchanged. Unlike the simple calculation based on Huygens principle that we showed earlier, the Kirchhoff formula provides an exact solution without requiring any fudge factors. Note that an approximate solution may

90 CHAPTER 6. SECONDARY SOURCE THEORIES also be obtained by considering only the far-field term from (6.7) φ P (t) = 1 ( 1 + d ) H(t d/c) f (t) (6.12) 2 ct At t = d/c this function steps from zero to one and then slowly decays as t increases: t = d/c The derivative of this function is δ(t d/c) with a growing negative amplitude tail. The delta function will dominate the response except at relatively long periods, where it becomes necessary to include the near-field term to cancel the effect of this tail. Kirchhoff methods would not be very useful if they were only used to compute simple examples like this where we already know the answer. Their advantages come from the fact that they remain valid when the integration surface ds or the incident wavefield becomes more complicated. In these cases analytical solutions are generally impossible and the integral must be evaluated numerically. 6.2.2 Kirchhoff applications (Introduction to Seismology, p. 138 141)

6.3. SCATTERING FROM WEAK HETEROGENEITY THE BORN APPROXIMATION91 6.2.3 Additional Kirchhoff reading 1. Kampmann, W. and G. Müller, PcP amplitude calculations for a coremantle boundary with topography, Geophys. Res. Lett., 16, 653 656, 1989. 2. Longhust, R.S., Geometrical and Physical Optics, John Wiley and Sons, New York, 1967. 3. Scott, P. and D. Helmberger, Applications of the Kirchhoff-Helmholtz integral to problems in seismology, Geophys. J. Roy. Astron. Soc., 72, 237 254, 1985. 6.3 Scattering from weak heterogeneity the Born approximation We have just seen how the idea of secondary sources as developed in Kirchhoff theory provides a way to obtain solutions for waves interacting with a rough interface. In our Kirchhoff formulas there is no limit regarding the size of the velocity contrast that may be present across the interface; the Kirchhoff approximation is accurate provided the interface is not so rough that multiple scattering becomes important. In the case of weak heterogeneity, there is another equivalent source theory that can be applied. The theory is based on the Born approximation for single scattering in weakly heterogeneous media. In this method, we assume that the wavefield consists of two parts: (1) a primary, background wavefield that is unperturbed by the heterogeneity, and (2) a secondary, scattered wavefield that is generated at sources in the heterogeneities through scattering of the background wavefield. Our discussion will closely follow section 13.2 of Aki and Richards (1980). We begin with the momentum equation for isotropic material (e.g., see equations 3.1 and 3.6 in the 227a notes). ρü i = i (λ k u k ) + j [µ( i u j + j u i )] (6.13) where u is the displacement vector, ρ is density, and λ and µ are the Lamé parameters. At this point we are assuming a general inhomogeneous medium, so the partial derivatives on the r.h.s. will apply to λ and µ as well as to the

92 CHAPTER 6. SECONDARY SOURCE THEORIES displacement. Now assume that the inhomogeneous medium consists of the sum of two parts, an unperturbed homogenous medium and the perturbations that make up the heterogeneity. Then the perturbed medium properties may be expressed as ρ = ρ 0 + δρ λ = λ 0 + δλ (6.14) µ = µ 0 + δµ where ρ 0, λ 0 and µ 0 are for the unperturbed medium and are constant, and where δρ, δλ and δµ are the perturbations (functions of position but assumed to be much smaller than the unperturbed values). Substituting (6.15) into (6.13) we obtain (ρ 0 + δρ)ü i = i [(λ 0 + δλ) k u k ] + j [(µ 0 + δµ)( i u j + j u i )] (6.15) Now separate the homogeneous terms from the perturbed terms, remembering that the spatial derivatives of ρ 0, λ 0 and µ 0 are zero. ρ 0 ü i λ 0 i k u k µ 0 j ( i u j + j u i ) = δρü i + i (δλ k u k ) + j [δµ( i u j + j u i )] ρ 0 ü i (λ 0 + µ 0 ) i k u k µ 0 j j u i = δρü i + δλ i k u k + ( i δλ) k u k + δµ j i u j +δµ j j u i + ( j δµ)( i u j + j u i ) = δρü i + (δλ + δµ) i k u k + δµ j j u i +( i δλ) k u k + ( j δµ)( i u j + j u i ) (6.16) We can also express this as ρ 0 ü i (λ 0 + µ 0 ) i ( u) µ 0 2 u i = δρü i + (δλ + δµ) i ( u) + δµ 2 u i +( i δλ)( u) + ( j δµ)( i u j + j u i ) (6.17) where we have used k u k = u and j j = 2. Now let us write the displacement u as the sum of primary waves u 0 and scattered waves u 1 1 u = u 0 + u 1 (6.18)

6.3. SCATTERING FROM WEAK HETEROGENEITY THE BORN APPROXIMATION93 u 0 is the solution for the unperturbed medium and so satisfies (6.17) with the r.h.s. set to zero ρ 0 ü 0 i (λ 0 + µ 0 ) i ( u 0 ) µ 0 2 u 0 i = 0 (6.19) Now substitute (6.18) into (6.17) to obtain ρ 0 (ü 0 i + ü 1 i ) (λ 0 + µ 0 ) i [ (u 0 + u 1 )] µ 0 2 (u 0 i + u 1 i ) = δρ(ü 0 i + ü 1 i ) + (δλ + δµ) i [ (u 0 + u 1 )] + δµ 2 (u 0 i + u 1 i ) +( i δλ)[ (u 0 + u 1 )] + ( j δµ)[ i (u 0 j + u 1 j) + j (u 0 i + u 1 i )] (6.20) Notice that the u 0 terms on the l.h.s. will sum to zero from (6.19). On the r.h.s. we drop the u 1 terms since these are second order terms that involve products between the scattered waves (assumed small) and the medium perturbations (also assumed small). In other words, we consider only single scattering and neglect any higher order scattering. We then have ρ 0 ü 1 i (λ 0 + µ 0 ) i ( u 1 ) µ 0 2 u 1 i = δρü 0 i + (δλ + δµ) 0 i ( u 0 ) + δµ 2 u 0 i +( i δλ)( u 0 ) + ( j δµ)( i u 0 j + j u 0 i )(6.21) Let us identify and define the r.h.s. as the local body force Q so we have ρ 0 ü 1 i (λ 0 + µ 0 ) i ( u 1 ) µ 0 2 u 1 i = Q i (6.22) where Q i = δρü 0 i + (δλ + δµ) 0 i ( u 0 ) + δµ 2 u 0 i +( i δλ)( u 0 ) + ( j δµ)( i u 0 j + j u 0 i ) (6.23) (6.22) is the equation of motion for the scattered wavefield u 1 in a homogeneous isotropic medium with body force Q that results from the local interaction of the heterogeneity with the primary wavefield u 0. Let us now see what form of Q results when P or S plane waves are assumed as the primary wavefield.

94 CHAPTER 6. SECONDARY SOURCE THEORIES 6.3.1 Primary plane P-waves Assume the waves are propagating in the x 1 direction. Then the u 2 and u 3 components of displacement are zero and we may write where α 0 = u 0 i = δ 1i e iω(t x 1/α 0 ) (6.24) (λ 0 + 2µ 0 )/ρ 0 is the P velocity in the unperturbed medium. We may then express the temporal and spatial derivatives of u 0 as ü 0 i = δ 1i ω 2 u 0 1 u 0 = 1 u 0 1 = (iω/α 0 )u 0 1 i ( u 0 ) = δ 1i (ω 2 /α 2 0)u 0 1 2 u 0 i = δ 1i k k u 0 1 = δ 1i (ω 2 /α 2 0)u 1 i u 0 j = δ 1i δ 1j (iω/α 0 )u 0 1 (6.25) Substituting into (6.23) we obtain the three components of Q [ Q 1 = δρω 2 (δλ + 2δµ)ω2 + i ω α0 2 1 (δλ) + 2i ω ] 1 (δµ) α 0 α 0 Q 2 = i ω α 0 2 (δλ)e iω(t x 1/α 0 ) e iω(t x 1/α 0 ) Q 3 = i ω α 0 3 (δλ)e iω(t x 1/α 0 ) (6.26) Note that Q 2 and Q 3 are only excited by spatial gradients in λ. The first two terms in the expression for Q 1 may be related to the P velocity perturbation as follows δα = α α 0 λ0 + 2µ 0 + δλ + 2δµ = ρ 0 + δρ For x dx and y dy, we have the approximation x + dx y + dy = x ( 1 + dx y x dy ) y λ0 + 2µ 0 ρ 0 (6.27) (6.28)

6.3. SCATTERING FROM WEAK HETEROGENEITY THE BORN APPROXIMATION95 and thus we can express (6.27) as δα = λ ( 0 + 2µ 0 1 + = ρ 0 [ λ0 + 2µ 0 1 + ρ 0 δλ + 2δµ δρ ) λ0 + 2µ 0 λ 0 + 2µ 0 ρ 0 ρ 0 ] δλ + 2δµ δρ 1 λ 0 + 2µ 0 ρ 0 Next, note that for ɛ 1, we have the approximation (6.29) 1 + ɛ = 1 + ɛ/2 (6.30) Thus, we can express (6.29) as [ λ0 + 2µ 0 δα = 1 + 1 δλ + 2δµ 1 ] δρ 1 ρ 0 2 λ 0 + 2µ 0 2 ρ 0 = α [ 0 δλ + 2δµ δρ ] 2 λ 0 + 2µ 0 ρ 0 2 δα = 1 [ δρ + ρ ] 0(δλ + 2δµ) α 0 ρ 0 λ 0 + 2µ 0 2ρ 0 δα α 0 = δρ δλ + 2δµ α 2 0 (6.31) Note that this is in a form that may be substituted for the first two terms of the expression for Q 1 in (6.26), e.g. δρω 2 (δλ + 2δµ)ω2 α 2 0 = 2ω 2 ρ 0 δα α 0 (6.32) In this way, the dependence on δρ, δλ and δµ may be replaced with dependence on δα, and we are left with only 3 independent parameters that determine the scattering. For the case of an incident P wave, the components of Q are sensitive to perturbations in: (1) P velocity, (2) the gradient of δλ, and (3) the gradient of δµ. Let us now explore what the far-field radiation of the scattered energy will look like in each case, assuming a localized perturbation small enough to be considered a point source. The P velocity perturbation term only enters into the x 1 component of Q and acts as a single force in the x 1 direction. A small element of this source will generate scattered far-field P and S-waves with a radiation pattern that looks like:

96 CHAPTER 6. SECONDARY SOURCE THEORIES Scattered P-waves Scattered S-waves x1 x1 Now consider the (δλ) term. Let us imagine that we have a small blob of λ Cross-section Contour plot The (δλ) vectors point outward in all directions and the far-field radiation pattern will look like: Scattered P-waves Scattered S-waves x1 x1 Note that this term acts like an explosive source, radiating P waves equally in all direction and generating no S waves. Recall the definition of the moment tensor in terms of body force equivalents (e.g., p. 55 of Aki and Richards) Mpq = Z V fp xq dv (x) (6.33) where f is the body force vector and x is the position within V. If δλ is localized in a small region V, we then have Z V xi k (δλ)dv = δik Z δλ dv (6.34) V and we see that the moment tensor is diagonal. Finally, consider the x (δµ) term. This term only affects Q1 and acts as a (1,1) dipole for a localized δµ anomaly, giving a far-field radiation pattern:

6.3. SCATTERING FROM WEAK HETEROGENEITY THE BORN APPROXIMATION97 Scattered P-waves Scattered S-waves x 1 x 1 In this case the moment tensor only has a M 11 element. Note that in all three cases that we have considered, there is no scattered S energy in the exact direction of the plane P wave or back-scattered energy opposite to this direction. Note also that the scattering is frequency dependent with more scattering predicted at larger values of ω. Often we will replace the ω/α 0 factors in (6.26) with the wavenumber (k = ω/α 0 ); thus the δα scattering will scale as k 2 while the (δλ) and (δµ) scattering scale with k. 6.3.2 Primary plane S-waves Now let us consider the case of an incident S plane wave traveling in the x 1 direction with particle motion in the x 2 direction u 0 i = δ 2i e iω(t x 1/β 0 ) (6.35) where β 0 = µ 0 /ρ 0 is the S velocity in the unperturbed medium. The temporal and spatial derivatives of u 0 are ü 0 i = δ 2i ω 2 u 0 2 u 0 = 1 u 0 2 = (iω/β 0 )u 0 2 i ( u 0 ) = δ 1i (ω 2 /β0)u 2 0 2 2 u 0 i = δ 2i 1 1 u 0 2 = δ 2i (ω 2 /β0)u 2 0 2 i u 0 j = δ 1i δ 2j (iω/β 0 )u 0 2 (6.36)

98 CHAPTER 6. SECONDARY SOURCE THEORIES Substituting into (6.22), we obtain Q 1 = i ω β 0 2 (δµ)e iω(t x 1/β 0 ) Q 2 = [ δρω 2 δµ ω2 + i ω ] 1 (δµ) β 0 β 2 0 e iω(t x 1/β 0 ) Q 3 = 0 (6.37) As in the previous section, we can express the first two terms in the equation for Q 2 in terms of the velocity perturbation δβ δρω 2 δµ ω2 β 2 = 2ω2 ρ 0 δβ β 0 (6.38) This may be derived as in (6.32) by substituting the λ + 2µ terms with µ. Thus we see that the scattering from an incident S wave is sensitive to perturbations in β and in the spatial derivatives of µ. No scattering is caused by inhomogeneities in λ or its spatial derivatives. Let us now consider the far-field radiation from small perturbations in β and µ. A localized anomaly in δβ will act as a single force in the x 2 direction and radiate both P and S energy: Scattered P-waves Scattered S-waves x 1 x 1 The terms due to the spatial derivative of δµ correspond to a double couple when δµ is confined to a small region V. The moment tensor has nonvanishing elements M 12 = M 21, proportional to V δµ dv Note that, in this case, there is no scattered P energy in the direction of incident S propagation. 6.3.3 Wave equation solution for the scattered waves In the previous section, we derived expressions for the body force Q that is the effective source for the scattered waves. Now, let us write down solutions

6.3. SCATTERING FROM WEAK HETEROGENEITY THE BORN APPROXIMATION99 Scattered P-waves Scattered S-waves x 1 x 1 for the scattered wavefield, borrowing from results that we obtained earlier in the 227a class (i.e. solving the wave equation, seismic sources). Recall (6.22) ρ o ü 1 i (λ 0 + µ 0 ) i ( u 1 ) µ 0 2 u 1 i = Q i (6.39) As we showed in the wave equation derivation in 227a, this can be rewritten in the form ρ 0 ü 1 (λ 0 + 2µ 0 ) ( u 1 ) + µ 0 ( u 1 ) = Q (6.40) By taking the divergence and curl of this equation, we can separate the P and S wave solutions and obtain for the P waves and ü 1 α 0 2 ( u 1 ) = Q/ρ0 (6.41) ü 1 β 2 0 2 ( u 1 ) = Q/ρ 0 (6.42) for the S waves. These have solutions ( u 1 1 1 (x, t) = 4πα0ρ 2 0 V x ξ Q ξ, t ) x ξ dv (ξ) (6.43) α 0 and u 1 (x, t) = 1 4πβ 2 0ρ 0 V ( 1 x ξ Q ξ, t ) x ξ dv (ξ) (6.44) β 0 Notice that 1/ x ξ is our familiar 1/r geometrical spreading factor from the point of scattering to a receiver at x and x ξ /c is simply the propagation time between the scattering point and receiver (where c is the P or S velocity).

100 CHAPTER 6. SECONDARY SOURCE THEORIES 6.3.4 Scattering due to velocity perturbation Now consider the case where velocity varies in all directions with a finite scale length and we have a scalar P wave (Φ = u). Our simplest result will be obtained if we neglect the terms involving the spatial gradients in the medium properties; this approximation is valid if the inhomogeneities are smooth relative to the seismic wavelength. If we assume a plane wave propagating in the x 1 direction, then the primary wave has the form Φ 0 = Ae iω(t x 1/c 0 ) (6.45) where A is amplitude and c 0 is the unperturbed velocity. The solution for the scattered wavefield (6.43) requires Q. We have from (6.26) and (6.32) that Q 1 = 2Aω 2 ρ 0 δc c 0 e iω(t x 1/c 0 ), Q 2 = 0, Q 3 = 0 (6.46) where we have dropped the (δλ) and (δµ) terms. We thus have Q = [ ] 2Aω 2 δc ρ 0 e iω(t x 1/c 0 ) x 1 c 0 2Aω 2 δc ρ 0 i ω e iω(t x 1/c 0 ) c 0 c 0 (6.47) where we again neglect the term involving the gradient of the velocity perturbation. Substituting into (6.43) we obtain Φ 1 (x, t) = u 1 (x, t) ( 1 1 = 4πc 2 0ρ 0 V x ξ Q ξ, t = Aω2 1 2πc 2 0 V r where r = x ξ and V is the region where δc 0. ) x ξ dv (ξ) c 0 δc c 0 e iω(t r/c 0 ξ 1 /c 0 ) dv (ξ) (6.48) This equation could be used in a computer program if one actually knew δc everywhere in the scattering volume of interest. However, normally one has no hope of actually resolving all of the individual scatterers but only some statistical measure of their scale length and strength. A standard way to

6.3. SCATTERING FROM WEAK HETEROGENEITY THE BORN APPROXIMATION101 Incident plane P wave dv(ξ) V (volume with perturbation δc) r Receiver at position x describe the spatial fluctuation of a random field is with the autocorrelation function. Let us define the fractional velocity perturbation as: µ = δc/c 0 (6.49) (do not confuse this parameter with the shear modulus!) where we assume the fluctuation of µ is isotropic and stationary in space. autocorrelation function is N(r) = µ(r )µ(r + r) µ 2 The normalized (6.50) where is a spatial average over many statistically independent samples. Two specific forms for N(r) are often modeled: Gaussian Exponential N(r) = e r /a (exponential model) (6.51) = e r 2 /a 2 (Gaussian model) (6.52) where a is called the correlation distance. Note that the Gaussian model will have blobs of relatively uniform size, whereas the exponential model will have greater heterogeneity structure at both smaller and larger wavelengths.

102 CHAPTER 6. SECONDARY SOURCE THEORIES Gaussian Exponential a a Now let us consider the scattered waves at a distance far away from an inhomogeneous region confined in a small volume V with linear dimension L. Receiver V O ξ dv x r L In order to evaluate the integral in (6.48), we approximate the scatter-toreceiver distance as: r = ( x 2 + ξ 2 2x ξ) 1/2 (6.53) x ˆn ξ (6.54) where ˆn is the unit vector in the direction of x. Note that the first (exact) expression follows from the law of cosines and the dot product definition. This approximation is valid provided: kl 2 2 x π 2 (6.55) where k is the wavenumber. Recalling that k = ω/c = 2π/Λ where Λ is the

6.3. SCATTERING FROM WEAK HETEROGENEITY THE BORN APPROXIMATION103 wavelength, this condition is equivalent to L 2 Λ x 1 2 (6.56) and we see this is a far-field approximation that is valid provided the wavelength and distance are large enough compared to the size of the volume heterogeneity. Putting (6.54) into (6.48), replacing 1/r with 1/ x, using µ = δc/c 0, and setting k = ω/c 0, we have Φ 1 (x, t) = Aω2 1 δc e iω(t r/c 0 ξ 1 /c 0 ) dv (ξ) 2πc 2 0 V r c 0 = Ak2 2π x e i(ωt k x ) V µ(ξ)e ik(ξ 1 ˆn ξ) dv (ξ) (6.57) If we know only the statistical properties of µ(x) rather than its exact from, we cannot expect to evaluate this expression and obtain individual wiggles on a synthetic seismogram. Fortunately, however, a solution is possible if we consider only the power carried by the scattered waves. The power is proportional to Φ 1 2. Since Φ 1 2 is equal to the product of Φ 1 and its complex conjugate, we have Φ 1 2 = A2 k 4 µ(ξ )µ(ξ)e ik[ξ 1 ξ 4π 2 x 2 1 ˆn (ξ ξ )] dv (ξ)dv (ξ ) (6.58) V V Now define ê 1 as the unit vector in the ξ 1 direction (the direction of the incident wave) and θ as the scattering angle (the angle between the incident wave direction, ê 1, and the scattered wave direction, ˆn). to receiver n K θ incident wave direction e 1

104 CHAPTER 6. SECONDARY SOURCE THEORIES We define K = ê 1 ˆn. From the isosceles triangle in this figure, it is easily seen that sin(θ/2) = K /2, since ˆn = ê 1 = 1, and hence K = 2 sin(θ/2). Note that this definition of K can be used to simplify part of (6.58): ξ 1 ξ 1 ˆn (ξ ξ ) = ê 1 ξ ˆn ξ + ê 1 ξ ˆn ξ = (ê 1 ˆn) ξ (ê 1 ˆn) ξ = K (ξ ξ ) (6.59) Provided our integration volume is large enough to fully sample the heterogeneity, taking the statistical average of (6.58) using (6.50) we have Φ 1 2 = A2 k 4 µ 2 4π 2 x 2 V V N(ξ ξ )e ikk (ξ ξ ) dv (ξ) dv (ξ ) (6.60) To evaluate this integral we change the variables ξ and ξ to the relative coordinate ˆξ = ξ ξ and the center-of-mass coordinate ξ = (ξ + ξ )/2 and obtain where V = V dv (ξ). Φ 1 2 = A2 k 4 µ 2 4π 2 x 2 = A2 k 4 µ 2 V 4π 2 x 2 = A2 k 4 µ 2 V 4π 2 x 2 V V V V N(ˆξ)e ikk ˆξ dv (ˆξ) dv (ξ) N(ˆξ)e ikk ˆξ dv (ˆξ) N(ˆξ)e ikk ˆξ d ˆξ1 d ˆξ 2 d ˆξ 3 (6.61) Next, we change from (ˆξ 1, ˆξ 2, ˆξ 3 ) to the spherical coordinates (r, θ, φ ), with K as the polar axis, obtaining: We then obtain V N(ˆξ)e ikk ˆξ dˆξ1 dˆξ 2 dˆξ 3 = r = ˆξ K ˆξ = K r cos θ dˆξ 1 dˆξ 2 dˆξ 3 = r 2 dr sin θ dθ dφ (6.62) = 4π N(r )e ik K r cos θ r 2 dr sin θ dθ dφ V 0 N(r ) sin(k K r ) k K r dr (6.63)

6.3. SCATTERING FROM WEAK HETEROGENEITY THE BORN APPROXIMATION105 where the integration limit for r is extended to infinity, assuming that the correlation distance a is much smaller than the linear dimension of V. This integral can be evaluated for the cases N(r) = e r/a (exponential) and N(r) = e r2 /a 2 (Gaussian), and the result put into (6.61): and Φ 1 2 = 2A2 k 4 µ 2 a 3 V π x 2 1 ( 1 + 4k2 a 2 sin 2 θ 2) 2 for N(r) = e r/a (6.64) Φ 1 2 = A2 k 4 µ 2 a 3 V 4 π x 2 e k2 a2 sin2 θ 2 for N(r) = e r2 /a 2 (6.65) In both cases the power of the scattered waves is proportional to k 4 when ka 1. This is termed Rayleigh scattering. If ka is small, the scattered power does not depend upon the scattering angle θ. Thus velocity perturbations with scale length much smaller than a wavelength produce isotropic scattering. However, when ka is small, the gradients of velocity and elasticity perturbation (neglected so far in our analysis) become important and their effects are directional. When ka is large the scattering due to velocity perturbation is mostly directed forward and the scattered power is concentrated within an angle (ka) 1 around the direction of primary wave propagation (θ = 0). Back-scattered power (θ = π) becomes very small, particularly for the Gaussian model. A more complete scattering model can be derived by taking into account the gradients in elastic properties. If we assume that the medium behaves like an Poisson solid (this provides the scaling between the P and S-wave velocity perturbations), then for the exponential autocorrelation model, one can show that the average scattered power is given by: Φ 1 2 = 2A2 k 4 a 3 µ 2 V πr 2 ( 1 4 cos θ + 1 + 2 3 3 cos2 θ ) 2 ( 2 (6.66) 1 + 4k2 a 2 sin 2) 2 θ where r is the scattering receiver distance. Note that µ 2 is simply the square of the RMS velocity perturbation (µ = δc/c 0 ) of the random medium. This

106 CHAPTER 6. SECONDARY SOURCE THEORIES equation does, however, neglect the effect of density perturbations. Density perturbations tend to increase the amount of backscattered energy. If these are important, then still more complete equations need to be used. In addition, in some cases P-to-S and S-to-P scattering should also be included. A good general reference for scattering theories that includes all of these complications is the textbook by Sato and Fehler (1998). 6.3.5 How To Write a Born Scattering Program Most scattering programs are based on ray theory so you will need to be able to trace rays through your model and to compute travel time and geometrical spreading factors. 1. Define the background velocity vs. depth model, the source and receiver locations, and the ray paths to be modeled. 2. Decide on what type of random media (e.g.., exponential, Gaussian, etc.) and what scattering equation you will use (e.g., 6.64, 6.66, etc.). This will determine what parameters you will need to specify the scattering part of the model. Determine the frequency (ω) at which you will model the scattering. 3. Determine where the scattering volume is in the Earth that you will use to model your observations. Specify the heterogeneity parameters that you will need, such as the scale length, the RMS velocity heterogeneity, the P-to-S scaling, etc. 4. Divide the scattering volume into cells that you will use to numerically integrate the scattered power. 5. For each source-receiver pair, initialize a time series to zero values. 6. For each cell in your scattering volume, compute the source-to-cell travel time and amplitude, A, of the incident wave. Compute the scattering angle, θ, the difference between the incident ray direction and the takeoff

6.3. SCATTERING FROM WEAK HETEROGENEITY THE BORN APPROXIMATION107 direction of the scattered ray that will land at the receiver (this is one of the trickier parts so be sure to thoroughly test this part of the code!). Compute the geometrical spreading factor for the scattered ray (this will replace the 1/r 2 factor (6.65), etc.). Compute the local wavenumber k from ω and the average velocity in the cell. 7. Use your preferred scattering equation to compute the amount of scattered power that will arrive at the receiver. Using the total sourceto-scatterer-to-receiver travel time, add this contribution to your time series. 8. Repeat (6) and (7) for all the cells in your scattering volume. 9. Repeat (5)-(8) for all of your source-receiver pairs. 10. Your synthetics will give power as a function of time. If they are noisy looking, try using a longer sample interval dt for your time series or convolve the result with a realistic source-time function (in energy, not amplitude!). 11. Take the square root if you want the amplitude envelopes. 12. Often you will want to compare the scattered power to that in the direct arrival. To do so, simply compute the ray theoretical amplitude for each source-to-receiver ray path. 13. You can add in the effect of Q along the ray paths and reflection and transmission coefficients where the rays cross boundaries if you want to include these effects. 6.3.6 Born scattering references 1. Aki, K. and P.G. Richards, Quantitative seismology: theory and methods (volume 2), W.H. Freeman, San Francisco, 1980. 2. Chernov, L.A., Wave propagation in a random medium, McGraw-Hill, New York, 1960.

108 CHAPTER 6. SECONDARY SOURCE THEORIES 3. Pekeris, C.L., Notes on the scattering of radiation in an inhomogeneous medium, Physical Review, 71, 268, 1947. 4. Sato, H. and M.C. Fehler, Seismic wave propagation and scattering in the heterogeneous Earth, Springer-Verlag, New York, 1998. 5. Wu, R. and K. Aki, Scattering characteristics of elastic waves by an elastic heterogeneity, Geophysics, 50, 582 595, 1985. 6. Wu, R.S. and K. Aki, Elastic wave scattering by a random medium and the small-scale inhomogeneities in the lithosphere, J. Geophys. Res., 90, 10,261 10,273, 1985.