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 seis-
mograms. 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 com-
plicated 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. Reflec-
tivity 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 can-
not be used in every case, but when applicable, they often can produce accurate
89
90 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 circum-
stances 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 91
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 de-
pends 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 mod-
eled 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:
yr
d
d
P
planewave dy
dr
The current position of the wavefront is specified as t = 0. Define t0 = d/c
92 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 > t0). 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.
AP (t > t0) = A dS(1/r)(1/dt)
= A2πrc dt(1/r)(1/dt)
= 2πcA (6.2)
Notice that AP has the form of a step function with height 2πcA.
t0
2 cA
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 t0.
A(t0) = f(t) ∗ δ(t0) 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 93
function with height 2πc
A(t0) = f(t) ∗ 2πcH(t0) = f(t) ∗G(t) Huygens result (6.4)
where H is the Heaviside unit step function, and G(t) ≡ 2πcH(t0) 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(t0) = f(t) ∗ (1
2πc)∂
∂tG(t)
=1
2πcf ′(t) ∗G(t) (6.5)
where f ′ = ∂f∂t
and we have used ∂∂t
[f(t) ∗ g(t)] = f ′(t) ∗ g(t) = f(t) ∗ g′(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 ra-
diation from a point source (eqn. 7.12 from the Introduction to Seismology
text)
u(r, t) =(
1
rc
)∂f(t− r/c)
∂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), 2nd edition, section 7.7)
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.
94 CHAPTER 6. SECONDARY SOURCE THEORIES
To model a plane wave, we let r0 → ∞ and assume that the wave has unit
amplitude on S at time t = 0
yr
d
d
P
planewave dy
dr
rd
To obtain an exact solution, let us use ITS 7.66, containing both the near-
field and far-field terms. The 1/r0 geometrical spreading term is not needed
in the case of a plane wave, but the 1/r20 term goes to zero and cos θ0 = −1
since the rays are perpendicular to dS. Thus we have
φP (t) =1
4π
∫S
δ(t− r/c)cos θ
r2dS ∗ f(t)
+1
4π
∫S
δ(t− r/c)1 + cos θ
crdS ∗ f ′(t) (6.7)
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 (6.1) 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
4π
∫S
δ(t− r/c)cos θ
r2dS =
1
4π
d/r
r2(2πrc)H(t− d/c)
6.2. KIRCHHOFF THEORY 95
=dc
2r2H(t− d/c)
=dc
2c2t2H(t− d/c)
=d
2ct2H(t− d/c) (6.8)
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
crdS =
1
4π
1 + d/r
cr(2πrc)H(t− d/c)
=1
2(1 + d/r)H(t− d/c)
=1
2(1 + d/ct)H(t− d/c) (6.9)
Substituting (6.8) and (6.9) into (6.7), we have
φP (t) =d
2ct2H(t− d/c) ∗ f(t) +
1
2
(1 +
d
ct
)H(t− d/c) ∗ f ′(t) (6.10)
Moving the time derivative to the other side of the convolution, we obtain
φP (t) =d
2ct2H(t− d/c) ∗ f(t) +
∂
∂t
[(1
2+
d
2ct
)H(t− d/c)
]∗ f(t)
=d
2ct2H(t− d/c) ∗ f(t) +
[− d
2ct2H(t− d/c) +
(1
2+
d
2ct
)∂
∂tH(t− d/c)
]∗ f(t)
=
(1
2+
d
2ct
)δ(t− d/c) ∗ f(t)
=
(1
2+
d
2cd/c
)δ(t− d/c) ∗ f(t)
= δ(t− d/c) ∗ f(t)
= f(t− d/c) (6.11)
This is what we expected to obtain for the plane wave. The source time func-
tion 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’ prin-
ciple that we showed earlier, the Kirchhoff formula provides an exact solution
without requiring any fudge factors. Note that an approximate solution may
96 CHAPTER 6. SECONDARY SOURCE THEORIES
also be obtained by considering only the far-field term from (6.7)
φP (t) =1
2
(1 +
d
ct
)H(t− d/c) ∗ f ′(t) (6.12)
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 am-
plitude 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, 2nd edition, p. 184–187)
6.3. SCATTERING FROM WEAK HETEROGENEITY—THE BORN APPROXIMATION97
6.2.3 Additional Kirchhoff reading
1. Kampmann, W. and G. Muller, PcP amplitude calculations for a core-mantle 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-Helmholtzintegral 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 multi-
ple 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 me-
dia. 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 equa-
tions 3.1 and 3.6 in the 227a notes).
ρui = ∂i(λ∂kuk) + ∂j[µ(∂iuj + ∂jui)] (6.13)
where u is the displacement vector, ρ is density, and λ and µ are the Lame
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
98 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 perturba-
tions that make up the heterogeneity. Then the perturbed medium properties
may be expressed as
ρ = ρ0 + δρ
λ = λ0 + δλ (6.14)
µ = µ0 + δµ
where ρ0, λ0 and µ0 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
(ρ0 + δρ)ui = ∂i[(λ0 + δλ)∂kuk] + ∂j[(µ0 + δµ)(∂iuj + ∂jui)] (6.15)
Now separate the homogeneous terms from the perturbed terms, remembering
that the spatial derivatives of ρ0, λ0 and µ0 are zero.
ρ0ui − λ0∂i∂kuk − µ0∂j(∂iuj + ∂jui) = −δρui + ∂i(δλ∂kuk) + ∂j[δµ(∂iuj + ∂jui)]
ρ0ui − (λ0 + µ0)∂i∂kuk − µ0∂j∂jui = −δρui + δλ∂i∂kuk + (∂iδλ)∂kuk + δµ∂j∂iuj
+δµ∂j∂jui + (∂jδµ)(∂iuj + ∂jui)
= −δρui + (δλ + δµ)∂i∂kuk + δµ∂j∂jui
+(∂iδλ)∂kuk + (∂jδµ)(∂iuj + ∂jui) (6.16)
We can also express this as
ρ0ui − (λ0 + µ0)∂i(∇ · u)− µ0∇2ui = −δρui + (δλ + δµ)∂i(∇ · u) + δµ∇2ui
+(∂iδλ)(∇ · u) + (∂jδµ)(∂iuj + ∂jui) (6.17)
where we have used ∂kuk = ∇ · u and ∂j∂j = ∇2. Now let us write the
displacement u as the sum of primary waves u0 and scattered waves u11
u = u0 + u1 (6.18)
6.3. SCATTERING FROM WEAK HETEROGENEITY—THE BORN APPROXIMATION99
u0 is the solution for the unperturbed medium and so satisfies (6.17) with the
r.h.s. set to zero
ρ0u0i − (λ0 + µ0)∂i(∇ · u0)− µ0∇2u0
i = 0 (6.19)
Now substitute (6.18) into (6.17) to obtain
ρ0(u0i + u1
i )− (λ0 + µ0)∂i[∇ · (u0 + u1)]− µ0∇2(u0i + u1
i )
= −δρ(u0i + u1
i ) + (δλ + δµ)∂i[∇ · (u0 + u1)] + δµ∇2(u0i + u1
i )
+(∂iδλ)[∇ · (u0 + u1)] + (∂jδµ)[∂i(u0j + u1
j) + ∂j(u0i + u1
i )] (6.20)
Notice that the u0 terms on the l.h.s. will sum to zero from (6.19). On the
r.h.s. we drop the u1 terms since these are second order terms that involve
products between the scattered waves (assumed small) and the medium per-
turbations (also assumed small). In other words, we consider only single scat-
tering and neglect any higher order scattering. We then have
ρ0u1i − (λ0 + µ0)∂i(∇ · u1)− µ0∇2u1
i = −δρu0i + (δλ + δµ)∂0
i (∇ · u0) + δµ∇2u0i
+(∂iδλ)(∇ · u0) + (∂jδµ)(∂iu0j + ∂ju
0i )(6.21)
Let us identify and define the r.h.s. as the local body force Q so we have
ρ0u1i − (λ0 + µ0)∂i(∇ · u1)− µ0∇2u1
i = Qi (6.22)
where
Qi = −δρu0i + (δλ + δµ)∂0
i (∇ · u0) + δµ∇2u0i
+(∂iδλ)(∇ · u0) + (∂jδµ)(∂iu0j + ∂ju
0i ) (6.23)
(6.22) is the equation of motion for the scattered wavefield u1 in a ho-
mogeneous isotropic medium with body force Q that results from the local
interaction of the heterogeneity with the primary wavefield u0. Let us now see
what form of Q results when P or S plane waves are assumed as the primary
wavefield.
100 CHAPTER 6. SECONDARY SOURCE THEORIES
6.3.1 Primary plane P-waves
Assume the waves are propagating in the x1 direction. Then the u2 and u3
components of displacement are zero and we may write
u0i = δ1ie
−iω(t−x1/α0) (6.24)
where α0 =√
(λ0 + 2µ0)/ρ0 is the P velocity in the unperturbed medium. We
may then express the temporal and spatial derivatives of u0 as
u0i = −δ1iω
2u01
∇ · u0 = ∂1u01
= (iω/α0)u01
∂i(∇ · u0) = −δ1i(ω2/α2
0)u01
∇2u0i = δ1i∂k∂ku
01
= −δ1i(ω2/α2
0)u1
∂iu0j = δ1iδ1j(iω/α0)u
01 (6.25)
Substituting into (6.23) we obtain the three components of Q
Q1 =
[δρω2 − (δλ + 2δµ)ω2
α20
+ iω
α0
∂1(δλ) + 2iω
α0
∂1(δµ)
]e−iω(t−x1/α0)
Q2 = iω
α0
∂2(δλ)e−iω(t−x1/α0)
Q3 = iω
α0
∂3(δλ)e−iω(t−x1/α0) (6.26)
Note that Q2 and Q3 are only excited by spatial gradients in λ. The
first two terms in the expression for Q1 may be related to the P velocity
perturbation as follows
δα = α− α0
=
√λ0 + 2µ0 + δλ + 2δµ
ρ0 + δρ−√
λ0 + 2µ0
ρ0
(6.27)
For x � dx and y � dy, we have the approximation
x + dx
y + dy=
x
y
(1 +
dx
x− dy
y
)(6.28)
6.3. SCATTERING FROM WEAK HETEROGENEITY—THE BORN APPROXIMATION101
and thus we can express (6.27) as
δα =
√√√√λ0 + 2µ0
ρ0
(1 +
δλ + 2δµ
λ0 + 2µ0
− δρ
ρ0
)−√
λ0 + 2µ0
ρ0
=
√λ0 + 2µ0
ρ0
[√1 +
δλ + 2δµ
λ0 + 2µ0
− δρ
ρ0
− 1
](6.29)
Next, note that for ε � 1, we have the approximation
√1 + ε = 1 + ε/2 (6.30)
Thus, we can express (6.29) as
δα =
√λ0 + 2µ0
ρ0
[1 +
1
2
δλ + 2δµ
λ0 + 2µ0
− 1
2
δρ
ρ0
− 1
]
=α0
2
[δλ + 2δµ
λ0 + 2µ0
− δρ
ρ0
]
2δα
α0
=1
ρ0
[−δρ +
ρ0(δλ + 2δµ)
λ0 + 2µ0
]
−2ρ0δα
α0
= δρ− δλ + 2δµ
α20
(6.31)
Note that this is in a form that may be substituted for the first two terms of
the expression for Q1 in (6.26), e.g.
δρω2 − (δλ + 2δµ)ω2
α20
= −2ω2ρ0δα
α0
(6.32)
In this way, the dependence on δρ, δλ and δµ may be replaced with de-
pendence on δα, and we are left with only 3 independent parameters that
determine the scattering. For the case of an incident P wave, the compo-
nents 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 radi-
ation 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 x1 component of Q
and acts as a single force in the x1 direction. A small element of this source
will generate scattered far-field P and S-waves with a radiation pattern that
looks like:
102 CHAPTER 6. SECONDARY SOURCE THEORIES
x1
Scattered P-waves ScatteredS-waves
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:
x1
Scattered P-waves ScatteredS-waves
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 =∫
Vfpxq 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∫V
xi∂k(δλ)dV = −δik
∫V
δλ dV (6.34)
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 APPROXIMATION103
x1
Scattered P-wavesScatteredS-waves
x1
In this case the moment tensor only has a M11 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/α0 factors in (6.26) with the wavenumber (k = ω/α0); thus the δα scattering
will scale as k2 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 x1
direction with particle motion in the x2 direction
u0i = δ2ie
−iω(t−x1/β0) (6.35)
where β0 =√
µ0/ρ0 is the S velocity in the unperturbed medium. The tem-
poral and spatial derivatives of u0 are
u0i = −δ2iω
2u02
∇ · u0 = ∂1u02
= (iω/β0)u02
∂i(∇ · u0) = −δ1i(ω2/β2
0)u02
∇2u0i = δ2i∂1∂1u
02
= −δ2i(ω2/β2
0)u02
∂iu0j = δ1iδ2j(iω/β0)u
02 (6.36)
104 CHAPTER 6. SECONDARY SOURCE THEORIES
Substituting into (6.22), we obtain
Q1 = iω
β0
∂2(δµ)e−iω(t−x1/β0)
Q2 =
[δρω2 − δµ
ω2
β20
+ iω
β0
∂1(δµ)
]e−iω(t−x1/β0)
Q3 = 0 (6.37)
As in the previous section, we can express the first two terms in the equation
for Q2 in terms of the velocity perturbation δβ
δρω2 − δµω2
β2= −2ω2ρ0
δβ
β0
(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 per-
turbations 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 x2 direction
and radiate both P and S energy:
x1
ScatteredP-waves
Scattered S-waves
x1
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 M12 = M21, 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 APPROXIMATION105
x1
ScatteredP-waves
ScatteredS-waves
x1
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)
ρou1i − (λ0 + µ0)∂i(∇ · u1)− µ0∇2u1
i = Qi (6.39)
As we showed in the wave equation derivation in 227a, this can be rewritten
in the form
ρ0u1 − (λ0 + 2µ0)∇(∇ · u1) + µ0∇× (∇× u1) = Q (6.40)
By taking the divergence and curl of this equation, we can separate the P and
S wave solutions and obtain
∇ · u1 − α0∇2(∇ · u1) = ∇ ·Q/ρ0 (6.41)
for the P waves and
∇× u1 − β20∇2(∇× u1) = ∇×Q/ρ0 (6.42)
for the S waves. These have solutions
∇ · u1(x, t) =1
4πα20ρ0
∫V
1
|x− ξξξ|∇ ·Q
(ξξξ, t− |x− ξξξ|
α0
)dV (ξξξ) (6.43)
and
∇× u1(x, t) =1
4πβ20ρ0
∫V
1
|x− ξξξ|∇ ×Q
(ξξξ, t− |x− ξξξ|
β0
)dV (ξξξ) (6.44)
Notice that 1/|x− ξξξ| is our familiar 1/r geometrical spreading factor from the
point of scattering to a receiver at x and |x − ξξξ|/c is simply the propagation
time between the scattering point and receiver (where c is the P or S velocity).
106 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 rela-
tive to the seismic wavelength. If we assume a plane wave propagating in the
x1 direction, then the primary wave has the form
Φ0 = Ae−iω(t−x1/c0) (6.45)
where A is amplitude and c0 is the unperturbed velocity. The solution for the
scattered wavefield (6.43) requires ∇·Q. We have from (6.26) and (6.32) that
Q1 = −2Aω2ρ0δc
c0
e−iω(t−x1/c0), Q2 = 0, Q3 = 0 (6.46)
where we have dropped the ∂(δλ) and ∂(δµ) terms. We thus have
∇ ·Q =∂
∂x1
[−2Aω2ρ0
δc
c0
e−iω(t−x1/c0)
]
≈ −2Aω2ρ0δc
c0
iω
c0
e−iω(t−x1/c0) (6.47)
where we again neglect the term involving the gradient of the velocity pertur-
bation. Substituting into (6.43) we obtain
Φ1(x, t) = ∇ · u1(x, t)
=1
4πc20ρ0
∫V
1
|x− ξξξ|∇ ·Q
(ξ, t− |x− ξξξ|
c0
)dV (ξξξ)
=Aω2
2πc20
∫V−1
r
δc
c0
e−iω(t−r/c0−ξ1/c0)dV (ξξξ) (6.48)
where r = |x− ξξξ| and V is the region where δc 6= 0.
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 APPROXIMATION107
IncidentplaneP wave
V (volume with perturbation δc)dV(ξ)
Receiver at position x
r
describe the spatial fluctuation of a random field is with the autocorrelation
function. Let us define the fractional velocity perturbation as:
µ = −δc/c0 (6.49)
(do not confuse this parameter with the shear modulus!) where we assume
the fluctuation of µ is isotropic and stationary in space. The normalized
autocorrelation function is
N(r) =〈µ(r′)µ(r′ + r)〉
〈µ2〉(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/a2
(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.
108 CHAPTER 6. SECONDARY SOURCE THEORIES
a a
Gaussian Exponential
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.
V
dV
Receiver
L
xr
ξO
In order to evaluate the integral in (6.48), we approximate the scatter-to-
receiver distance as:
r = (|x|2 + |ξξξ|2 − 2x · ξξξ)1/2 (6.53)
≈ |x| − n · ξξξ (6.54)
where n 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:
kL2
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 APPROXIMATION109
wavelength, this condition is equivalent to
L2
Λ|x|� 1
2(6.56)
and we see this is a far-field approximation that is valid provided the wave-
length and distance are large enough compared to the size of the volume het-
erogeneity.
Putting (6.54) into (6.48), replacing 1/r with 1/|x|, using µ = δc/c0, and
setting k = ω/c0, we have
Φ1(x, t) =Aω2
2πc20
∫V−1
r
δc
c0
e−iω(t−r/c0−ξ1/c0)dV (ξξξ)
=Ak2
2π|x|e−i(ωt−k|x|)
∫V
µ(ξξξ)eik(ξ1−n·ξξξ)dV (ξξξ) (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 |Φ1|2. Since |Φ1|2 is equal to the product of Φ1 and its complex
conjugate, we have
|Φ1|2 =A2k4
4π2|x|2∫
V
∫V
µ(ξ′)µ(ξ)eik[ξ1−ξ′1−n·(ξξξ−ξξξ
′)]dV (ξξξ)dV (ξξξ′) (6.58)
Now define e1 as the unit vector in the ξ1 direction (the direction of the
incident wave) and θ as the scattering angle (the angle between the incident
wave direction, e1, and the scattered wave direction, n).
θ
n
e1
K
to receiver
incident wave direction
110 CHAPTER 6. SECONDARY SOURCE THEORIES
We define K = e1− n. From the isosceles triangle in this figure, it is easily
seen that sin(θ/2) = |K|/2, since |n| = |e1| = 1, and hence |K| = 2 sin(θ/2).
Note that this definition of K can be used to simplify part of (6.58):
ξ1 − ξ′1 − n · (ξξξ − ξξξ′) = e1 · ξξξ − n · ξξξ + e1 · ξξξ′ − n · ξξξ′
= (e1 − n) · ξξξ − (e1 − n) · ξξξ′
= K · (ξξξ − ξξξ′) (6.59)
Provided our integration volume is large enough to fully sample the hetero-
geneity, taking the statistical average of (6.58) using (6.50) we have
〈|Φ1|2〉 =A2k4〈µ2〉4π2|x|2
∫V
∫V
N(ξξξ − ξξξ′)eikK·(ξξξ−ξξξ′)dV (ξξξ) dV (ξξξ′) (6.60)
To evaluate this integral we change the variables ξξξ and ξξξ′ to the relative co-
ordinate ξξξ = ξξξ − ξξξ′ and the center-of-mass coordinate ξξξ = (ξξξ + ξξξ′)/2 and
obtain
〈|Φ1|2〉 =A2k4〈µ2〉4π2|x|2
∫V
∫V
N(ξξξ)eikK·ˆξξξ dV (ξξξ) dV (ξξξ)
=A2k4〈µ2〉V
4π2|x|2∫
VN(ξξξ)eikK·ˆξξξ dV (ξξξ)
=A2k4〈µ2〉V
4π2|x|2∫
VN(ξξξ)eikK·ˆξξξ dξ1 dξ2 dξ3 (6.61)
where V =∫V dV (ξξξ).
Next, we change from (ξ1, ξ2, ξ3) to the spherical coordinates (r′, θ′, φ′),
with K as the polar axis, obtaining:
r′ = |ξξξ|
K · ξξξ = |K|r′ cos θ′
dξ1 dξ2 dξ3 = r′2 dr′ sin θ′ dθ′ dφ′ (6.62)
We then obtain∫V
N(ξξξ)eikK·ˆξξξ dξ1 dξ2 dξ3 =∫
VN(r′)eik|K|·r′ cos θ′
r′2 dr′ sin θ′ dθ′ dφ′
= 4π∫ ∞
0N(r′)
sin(k|K|r′)k|K|
r′dr′ (6.63)
6.3. SCATTERING FROM WEAK HETEROGENEITY—THE BORN APPROXIMATION111
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/a2(Gaussian), and the result put into (6.61):
〈|Φ1|2〉 =2A2k4〈µ2〉a3V
π|x|21(
1 + 4k2a2 sin2 θ2
)2 for N(r) = e−r/a (6.64)
and
〈|Φ1|2〉 =A2k4〈µ2〉a3V
4√
π|x|2e−k2a2 sin2 θ
2 for N(r) = e−r2/a2
(6.65)
In both cases the power of the scattered waves is proportional to k4 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 pertur-
bations 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 di-
rected 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:
〈|Φ1|2〉 =2A2k4a3〈µ2〉V
πr2
14
(cos θ + 1
3+ 2
3cos2 θ
)2
(1 + 4k2a2 sin2 θ
2
)2 (6.66)
where r is the scattering receiver distance. Note that 〈µ2〉 is simply the square
of the RMS velocity perturbation (µ = δc/c0) of the random medium. This
112 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 APPROXIMATION113
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/r2 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 scat-
tered power that will arrive at the receiver. Using the total source-
to-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.
114 CHAPTER 6. SECONDARY SOURCE THEORIES
3. Pekeris, C.L., Notes on the scattering of radiation in an inhomogeneousmedium, Physical Review, 71, 268, 1947.
4. Sato, H. and M.C. Fehler, Seismic wave propagation and scattering inthe heterogeneous Earth, Springer-Verlag, New York, 1998.
5. Wu, R. and K. Aki, Scattering characteristics of elastic waves by anelastic heterogeneity, Geophysics, 50, 582–595, 1985.
6. Wu, R.S. and K. Aki, Elastic wave scattering by a random medium andthe small-scale inhomogeneities in the lithosphere, J. Geophys. Res., 90,10,261–10,273, 1985.
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