chapter 6 secondary source theoriesshearer/227c/notes_kirchborn.pdf · 2008-11-12 · chapter 6...

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


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.


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


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


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)


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.


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)


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:


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:


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)


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


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).


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


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.


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


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


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)


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


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


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.


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.

chapter 6 secondary source theoriesshearer/227c/notes_kirchborn.pdf · 2008-11-12 · chapter 6...

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