2. Chapter 2 Elementary Theoretical Recapitulation

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Chapter 2 Elementary Theoretical Recapitulation

“Profound study of nature is the most fertile source of mathematical discoveries.”

— Joseph Fourier (1878, p. 7)

Introduction

Even though the theoretical bases that are needed to design and perform a seismic

experiment (detonating a source and deploying receivers to observe the resulting

motion of the ground) can be restricted to Pythagoras’ theorem and Snell’s law, using

the resulting data to build an image of the earth’s interior requires much broader

knowledge. Understanding the mathematical and physical bases of this broader

knowledge is valuable for acquisition geophysicists who seek to design and perform

better seismic experiments that lead to better images of the earth’s interior. Those bases

will be used widely throughout the remainder of this book.

Mathematics

Perhaps many readers will skip these basic reminders, but others might find it

convenient to refresh their memories. To understand seismic acquisition, there is no

need for deep mathematical knowledge. Beyond elementary calculus, linear algebra, andtrigonometry, I can think of only three domains that the acquisition geophysicist needs

to be familiar with. The first is Fourier analysis, which was not taught in high school

when I was a student. Next, basic sampling theory must be understood. Space must be

sampled because it cannot be measured continuously, and it is convenient to sample

time as well. Finally, because redundancy is one of the best tools to separate signal

from noise, a limited knowledge of statistics is necessary.

Fourier analysis

Fourier analysis can be considered as representing mathematical functions as sums

of sinusoids (Figure 1). The input is a function of time g (t ). Time is the vertical dimen-

sion of the figure. The Fourier transform of this function is noted as G(f ). It is a com-plex function of frequency that can be represented by its modulus, noted as amp(G(f )) in black, and by its phase, written below it as phi(G(f )) in light gray. Frequency is the

horizontal dimension of the figure. A sinusoid Sf (in fact, the cosine is used) of fre-

quency f , amplitude (amp), and phase (phi) is associated with each value of G(f ). The

sum of these sinusoids is equal to the input function g (t ). In this representation, the

Fourier analysis can be considered the decomposition into sinusoids, and the Fourier

synthesis (the inverse transform) is their sum. A more formal representation follows.

Distinguished Instructor Short Course • 15



The continuous Fourier transformThe continuous Fourier transform of a complex but well-behaved function g of the

real variable t is the complex function G of the real variable f given by

G f g t e dt jft ( ) ( ) ,= -

-•

+•

Ú 2p

(1)

where t is time and f is frequency.

The exact inverse transform has a very similar expression:

g t G f e df jft ( ) ( ) .=-•

+•

Ú 2p

(2)

This transform can be generalized to multidimensional functions:

G f k k g f x y e dt dx dy x y

i ft k x k y x y ( , , ) ( , , ) .( )= - + +ÚÚÚ 2p

(3)

Figure 1. Principle of Fourier transform. Here, g (t ) is a function of time represented vertically,and G (f ) is its Fourier transform. Function G (f ) is a function of frequency represented horizon-

tally by its amplitude amp(G (f )) and phase phi(G (f )) components. In addition, S f is a set ofsinusoids of frequency f , amplitude amp(G (f )), and phase phi(G (f )). The sum of thesesinusoids equals g (t ). Courtesy of CGGVeritas. Used by permission.

Seismic Acquisition from Yesterday to Tomorrow

16 • Society of Exploration Geophysicists/European Association of Geoscientists & Engineers



The continuous transform G(f ) can be decomposed into its amplitude amp(f ) and

its phase phi(f ), called amplitude and phase spectra:

G(f ) = amp(f )eiphi(f ). (4)

Three basic properties of the Fourier transform are

k(t ) = ag (t ) + b(h(t )) ⇔ K (f ) = aG(f ) + bH (f ) (linearity), (5)

k(t ) = g (t ) * (h(t )) ⇔ K (f ) = G(f ) H (f ) (convolution), (6)

and

k(t ) = conj( g (t )) ⇔ K (f ) = conj(G(−f )) (conjugate symmetry). (7)

A consequence of the latter property is that the Fourier transform of a real function

is conjugate symmetric, i.e., g t G f G f ( ) ( ) ( )Œ¬ ¤ = -( )conj .

The discrete Fourier transform (DFT)

In practice, sampled functions of finite length L will be used. A finite-length

function can be considered as one period of a periodic infinite function. These

functions can be decomposed into a sum of sinusoids with frequencies that are

multiples of 1/ L.

The discrete Fourier transform (DFT) G of the n-term complex series (or an

n-dimensional complex vector) g is given by

G g ek

j k

n

k

n

= +

-

=

-

Â1

2

0

1 p

.

(8)

If Si is the sampling interval of the input series, the sampling interval of G is 1/nSi,

and G is defined from p = 0 to p = n – 1. Like the continuous transform, the DFT formula

can be extended to multiple dimensions, and there is an exact inverse transform. The

basic properties of the continuous transform (linearity, convolution, and conjugate

symmetry) also are true for the DFT.

A few other properties of the DFT are important. First, the DFT of a finite function

is periodic, and its period is the inverse of the function length. Second, the DFT of a

periodic function is discrete, and the sampling interval is the inverse of the period.

An application of the discrete Fourier transform that is encountered often inseismic survey design is the response evaluation of a spatial array. The most common

1D array is the finite comb, which is the basic component of most 2D or 3D source-and-

receiver arrays. Such an array is shown in Figure 2, in which the number of teeth isn

and the tooth interval is e. The response of the comb is a sync function:

comb( )

sin( )sin( )

.kn ke

n ke=

p p

(9)





Its rst zero is found at wavenumber 1/ne, and its rst one is found at wavenumber 1/e.

A remarkable property of all 1D arrays is that the first secondary maximum amplitude is

close to –13 dB, whatever the number of teeth. Those properties also are found in multi-

dimensional arrays. Figure 3 represents an nx- × -n y 2D array. Figure 3a shows that the

element intervals are ex and e y . The 1D array is a 2D sync function: The first zeros arefound at wavenumbers 1/nxex and

1/n y e y and the first ones at wave-

numbers 1/ex and 1/e y .

Sampling theory

The sampling frequency f s is

the inverse of the sampling interval

f s = 1/Si. (10)

The Nyquist frequency Ny is halfthe sampling frequency

Ny = 1/2Si. (11)

The Shannon-Nyquist theorem

states that if the sampling frequency

of a continuous signal is larger than

twice the maximum frequency of

the signal, it is possible to recon-

struct the continuous signal. The

reconstruction uses a sync interpo-lation filter. Figure 4 illustrates this

Figure 2. Response of an n -tooth comb with toothinterval e . The response of a comb is a sync function.

The first zero is found at wavenumber 1/ ne and thefirst one at wavenumber 1/ e . The value of the first

secondary maximum is close to –13 dB, whatever thenumber of teeth. Courtesy of CGGVeritas. Used bypermission.

Figure 3. Response of an n x - × -n y -teeth 2D brush with intervals e y and e y . Figure 3a representsthe brush, and Figure 3b represents its response. This is a 2D sync function. The first zeros arefound at wavenumbers 1/ n x e x and 1/ n y e y and the first ones at wavenumbers 1/ e x and 1/ e y . Courtesy

of CGGVeritas. Used by permission.





theorem. A sinusoid of frequency f 0 = 162 Hz (with a sampling interval of 1 ms corre-

sponding to a Nyquist frequency of 500 Hz, represented in light gray) has been resam-

pled progressively with sampling intervals Si = 2, 4, 8, 16, and 32 ms, with corre-

sponding Nyquist frequencies 250, 125, 62.5, 31.25, and 15.625 Hz, respectively. As

long as the Nyquist frequency remains larger than the original frequency f 0, the recon-structed frequency f r(Si) is unchanged (f r(Si) = f 0). As soon as the Nyquist frequency

becomes lower than the original frequency, the reconstructed frequency is symmetrical

to the original frequency relative to the Nyquist frequency:

f r(Si) = 2Ny(Si) − f 0. (12)

If the data continue to be resampled, the previously reconstructed frequency is

seen as the true frequency, and the same rule applies to the subsequent reconstruction

f r(Si) = 2Ny(Si) − f r(2Si). (13)

In the example in Figure 4, the reconstructed frequency changes at each iteration.

From a mathematical point of view, the frequency of a sampled sinusoid is

unknown. It can take an infinite number of values (f 0, 2Ny − f 0, . . .). In physics and

particularly in geophysics, antialias filters are used to make sure that there is only one

possible frequency value.

Statistics

Seismic acquisition and processing make extensive use of summation, which

might be the best tool to separate signal from noise. In the field, waves received by

Figure 4. Aliasing of a progressively resampled sinusoid: Light gray indicates the original sinusoid(frequency 162 Hz). Dark gray represents the resampled sinusoid. As long as Ny > f 0, f r = f 0. WhenNy < f r(k − 1), f r(k ) = 2 * Ny (k ) − f r(k − 1). Black dots are the samples of the original sinusoid kept in

the aliased sinusoid. Courtesy of CGGVeritas. Used by permission.





individual geophones belonging to the same receiver station are summed into a single

seismogram. In the processing center, seismograms belonging to a same common

midpoint are summed into a single stack trace. In both cases, the sum is expected to

attenuate noise. Chapter 6 will show how to design arrays to optimize that process in

the case of source-generated noise. The other type of noise — ambient noise, or noisewhich exists in the absence of a seismic source — also is attenuated by those sums,

albeit in a different way. Depending on noise properties, it is possible to predict how

much a sum will attenuate ambient noise.

Figure 5 represents the result of an experiment in which 12 sums are formed from

a set of Gaussian random-noise traces with order varying from two to 4096. The bottom

panels represent the rms amplitude of the sum before (Figure 5b) and after (Figure 5c)

division by the square root of the number of terms. That result can be predicted by the

following simple calculation:

Let Rij be a matrix representing a set of n noise realizations by m-point vectors (or n

traces of m time samples), with i being the trace index from one to n and j the sample

index from one to m.

Figure 5. Sum of Gaussian random noise of unit rms amplitude. (a) Model noise. (b) The rms

amplitude of 12 sums of the above traces of order 21, 22, 23, ... 212. (c) The ms amplitude of thesame 12 sums divided by the square root of the number of terms. Courtesy of CGGVeritas. Usedby permission.





If two assumptions are met, first, that noise is uncorrelated,

R Rik jk

k

m

==

Â 01

,

(14)

and second, that noise power is constant,

1 2

1m

P i i nik

k

m

R = " = º=

Â , , , , ,for all 1 2

(15)

then the power of the sum is

P m hk

h

n

k

m

0

11

2

1=

Ê

Ë Á

ˆ

¯ ˜

==ÂÂ R .

(16)

It can be decomposed into

P m mhk

h

n

k

m

ik

i jk

m

jk02

11 1

1 2= +

== π=ÂÂ ÂÂR R R .

(17)

According to assumptions 14 and 15,

P m

nP hk

h

n

k

m

02

11

1= =

==ÂÂ R ,

(18)

and the rms of the sum is

P n P 0 = .

(19)

The rms amplitude of the sum is proportional to the square root of the number of

terms. Strictly speaking, this property is true when noise is uncorrelated and of constant

power. However, it is approximately true in many other cases, e.g., for the sum of sinu-

soids of identical amplitudes and frequency but of random phases.

PhysicsSnell’s law

Snell’s law can describe most of what will be discussed in later chapters. The law, as

published by Descartes (1637), describes reflection and refraction. To prove it, Descartes

uses the motion of a ball. He also mentions reciprocity as a remarkable property of the

reflection and refraction law.





Reflection

The ray reflected at an interface is in

the plane of the incident ray and of the

normal at the interface. The angle of reflec-

tion equals the angle of incidence.

Refraction

When a ray passes through an inter-

face between two isotropic layers of veloc-

ity V 1 and V 2, it is deviated (refracted) so

that the refracted ray is in the plane of the

incident ray and of the normal at the

interface (Figure 6).

The relationship between the inci-

dent angle θ 1 and the refracted angle θ 2 isgiven by

sin sin.

q q 1

1

2

2V V =

(20)

This law is a consequence of the Fermat

principle (discovered later) that a ray follows the path to travel between two points in

the least time.

Huygens’ principle

In 1690, Christiaan Huygens wrote a treatise on light in which he showed that the

wave theory of light could explain Snell’s law on reflection and refraction. For Huygens’

demonstration, he introduced the notion of secondary waves (Huygens, 1690, p. 19):

“There is the further consideration in the emanation of these waves, that each particle

of matter in which a wave spreads, ought not to communicate its motion only to the

next particle which is in the straight line drawn from the luminous point, but that it

also imparts some of it necessarily to all the others which touch it and which oppose

themselves to its movement. So it arises that around each particle there is made a wave

of which that particle is the centre.” In modern words, he would have said that given an

original wavefront at time t , the new wavefront at time t + dt is the envelope of the

fronts of secondary waves located on the original front.More than a century later, Fresnel (1818) rephrased this principle to account for

diffraction, stating that the amplitude of the wave at any given point equals the super-

position of the amplitudes of all the secondary waves at that point.

A “seismic” picture of Huygens’ principle is proposed in Figures 7 and 8. Figure 7

represents an expanding spherical wavefront in a homogeneous medium. At time

t = 96 ms, the wavefront has reached the position represented by the black semicircle

shown in Figure 8. If secondary (Huygens’) sources were located on this wavefront and

were emitting simultaneously, the envelope of the corresponding wavefront would be the

Figure 6. Snell’s law: Ray behavior at an

interface. The reflected and refracted rays arein the plane formed by the incident ray and thenormal at the interface. The angle of reflection

equals the angle of incidence θ 1. The angle ofrefraction θ 2 is given by V 1 sin θ 2 = V 2 sin θ 1.Courtesy of CGGVeritas. Used by permission.





Figure 7. Expanding spherical wave in a constant-velocity medium. The wavefront at time 96 msis used as the origin in Figure 8. Courtesy of CGGVeritas. Used by permission.

Figure 8. Combination of Huygens’ secondary sources (black stars) on an original wavefrontrepresented by the black line. If the number of sources is adequate, the envelope of their wave-fronts equals the original wavefront at a later time. Courtesy of CGGVeritas. Used by permission.





same as the original wavefront at any later time. The progressive buildup of the enve-

lope can be seen in Figure 8, where the number of Huygens’ secondary sources increases

until the envelope becomes continuous.

In practice, to reconstruct the wavefront properly, only the forward-moving front

should be considered (as in Figure 8). That restriction constitutes the major criticism toHuygens’ theory.

Newton’s laws of motion

In Philosophiæ Naturalis Principia Mathematica (Newton, 1687; trans. Motte, 1729,

p. 19–20), Sir Isaac Newton states the basic laws of motion:

Law 1. Every body perseveres in its state of rest, or of uniform motion in a

right line, unless it is compelled to change that state by forces impress’d

thereon....

Law 2. The alteration of motion is ever proportional to the motive force

impress’d; and is made in the direction of the right line in which that

force is impress’d. . . .

Law 3. To every action there is always opposed an equal reaction: or the mutual

actions of two bodies upon each other are always equal, and directed to

contrary parts. . . .

Hooke’s law

Hooke’s law states that the extension of a spring is in direct proportion with the

force applied to it. Mathematically, Hooke’s law is written

F = −kx, (21)

where F is the force applied to a spring, x is the displacement of the end of the springfrom its equilibrium position, and k is the spring constant.

In elastic materials, Hooke’s law gives the relationship between stresses (in pressure

units) and (dimensionless) strains and can be written in its most general tensor form

σij = Cijkl ⋅ εkl, (22)

where σij is the stress tensor, εkl is the strain tensor, and Cijkl is the stiffness tensor. It is

expressed in pressure units. In the general case, it contains 81 terms, not all of which

are independent.

Symmetry of stress and strain tensors (σij = σ ji and εij = ε ji) allows the use of a simpli-

fied notation:

Posing

s

s

s

s

s

s

s

s

s

s

s

s

11

22

33

23

13

12

1

2

3

4

5

6

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

=

È

ÎÎ

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙

and

e

e

e

e

e

e

11

22

33

23

13

12

˙̇˙˙

=

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

e

e

e

e

e

e

1

2

3

4

5

6

.

(23)





Hooke’s law becomes

s

s

s s

s

s

1

2

3

4

5

6

11 12 13 14 15 16

12 22

È

Î

ÍÍ

ÍÍÍÍÍÍ

˘

˚

˙˙

˙̇˙˙˙˙

=

c c c c c c

c c c 223 24 25 26

13 23 33 34 35 36

14 24 34 44 45 46

15 25 35

c c c

c c c c c c c c c c c c

c c c c 445 55 56

16 26 36 46 56 66

1

2

3

4

5c c

c c c c c c

È

Î

ÍÍ

ÍÍÍÍÍÍ

˘

˚

˙˙

˙̇˙˙˙˙

e

e

e e

e

e 66

È

Î

ÍÍ

ÍÍÍÍÍÍ

˘

˚

˙˙

˙̇˙˙˙˙

.

(24)

The simplified stiffness tensor turns out to present a symmetry that leaves only 21

independent coefficients in the general case.

In isotropic materials, only two independent terms remain, and they can be cho-

sen between various pairs. Lamé coefficients constitute the most common pair. Using

those coefficients and the simplified notation, Hooke’s law for isotropic materials can be

written

s

s

s

s

s

s

l m l l

l l m l

l l l m

1

2

3

4

5

6

2 0 0 0

2 0 0 0

2 0

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

=

++

+ 00 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1

2

3

4

5

6

m

m

m

e

e

e

e

e

e

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

È

Î

ÍÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

,

(25)

where λ and μ are the Lamé parameters. They are homogeneous to a pressure and are

expressed in pascals. Other pairs are the bulk modulus and shear modulus ( K and G) or

the Young modulus and Poisson’s ratio ( E and ν ).The relationship between elastic parameters is

E

K E

G E

= +

+

= +

= - = +

= + - =

m l m l m

n l

l m

n l m

n n n

m

( )

( )

( )

( )( )

.

3 2

2

3 1 223

1 1 2

(26)

It must be remembered that Hooke’s law is valid until the limit of elasticity is

reached. At that point, permanent deformations or damages occur, and the relationship

between stress and strain is no longer linear and reversible. Such is the case in the

proximity of most seismic sources. Moreover, the pure elastic model assumes no perma-

nent energy transfer to the medium. This is an approximation that cannot be made

always, especially in materials with high absorption such as that found in the shallow-

est layers of the earth (the so-called weathered zone).





The wave equation

The wave equation results from application of Newton’s law of motion to an ele-

mentary volume of elastic material. If u (u1, u2, u3) is the displacement and σ is the

density (mass per unit of volume), Newton’s law of motion can be written as

∂∂

+ ∂

∂ +

∂∂

= ∂

∂∂∂ +

∂∂ +

∂∂ =

∂

s s s r

s s s r

11

1

12

2

13

3

21

2

21

1

22

2

23

3

x x xu

t

x x x

222

2

31

1

32

2

33

3

23

2

u

t

x x x

u

t

∂∂∂

+ ∂

∂ +

∂∂

= ∂

∂

.

s s s r

(27)

Note that the strain tensor can be expressed as a function of u:

e

e

e

e

e

e

e

e

e

e

e

e

e

=

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

=

È

Î

ÍÍÍ

11

22

33

23

13

12

1

2

3

4

5

6

ÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

=

∂∂∂∂∂∂

∂∂ +

∂∂

Ê Ë Á

ˆ ¯ ˜

U x

U x

U x

U x

U x

11

2

2

3

3

3

2

2

3

12

12

∂∂∂ +

∂∂

Ê Ë Á

ˆ ¯ ˜

∂

∂ +

∂

∂

Ê

Ë Á

ˆ

¯ ˜

È

Î

ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ

U x

U x

U

x

U

x

3

1

1

3

2

1

1

2

1

2ÍÍ

˘

˚

˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙

.

(28)

The wave equation is obtained by substituting the strain ε values given by equations 28

into Hooke’s law in equation 24 and the resulting stress σ in Newton’s law of motion

(equation 27).

In an isotropic medium of infinite dimension, the expressions are simplified.

Moreover, axes can be chosen in such a way that the above system becomes

( )

.

22

1

12

21

2

2

2

12

2

22

m l r

m r

+ ∂

∂ =

∂∂

∂∂ = ∂∂

u

x

u

t

ux ut

(29)

The first equation, which is the P-wave equation, has a pure longitudinal solution.

The P-wave velocity is

V p =

+2m l r

.

(30)





The second equation, which is the S-wave equation, has a pure transverse solution.

The S-wave velocity is

V s = m

r

.

(31)

Wave types

The solution of the wave equation in an infinite isotropic medium predicts one P

body wave and one S body wave. In anisotropic material, the solution is more complex,

and a second S-wave is predicted and observed. In a semi-infinite medium, surface

waves propagating along the surface are predicted and observed. Essentially, there are

two types of surface waves, Rayleigh and Love waves.

Rayleigh waves

Rayleigh waves are solutions of the wave equation propagating on the free surface ofan infinite half-space. The corresponding particle motion is in the plane orthogonal to

the free surface (vertical). On the surface, the motion is elliptical and retrograde. When

depth increases, the motion amplitude decreases progressively, and its ellipticity changes

with depth in such a way that it becomes prograde and eventually fades at depths on the

order of one wavelength. The velocity of Rayleigh waves does not depend on frequency;

it depends on the Poisson’s ratio. The velocity varies from about 93% of the S-wave veloc-

ity for a Poisson’s ratio of zero to 97% for a Poisson’s ratio of 0.5. Rayleigh waves are the

main component of ground roll observed on P-wave seismograms.

Love waves

Love waves are dispersive surface waves guided by a low-velocity layer on top of ahalf-space. The particle motion is purely transverse and is parallel to the free surface.

The motion amplitude decreases with depth. Love waves are observed on horizontally

polarized (or SH) S-wave seismograms.

Partition of energy at interfaces

Behavior of elastic waves at an interface is more complex than predicted by Snell’s

law. An incident P-wave is reflected, transmitted, and converted into an S-wave (Figure 9).

Zoeppritz et al. (1912) describe the elastic reflection. Various authors give approxima-

tions of those equations. Hilterman (2001) discusses some of the approximations. These

approximations essentially differ in the emphasis they give to rock moduli, P- and

S-wave velocity and density variations, or P-wave velocity and Poisson’s ratio, as pro-

posed by Shuey (1985).

For an interface between two layers of density ρ1 and ρ2 and P-wave velocity V 1 and

V 2, the P-wave reflection coefficient at incidence angle θ is given by

R R A RV

V ( )

( )sin (tan sin ).q

n

n q q q = + +

-È

ÎÍ

˘

˚˙ + -0 0 0 2

2 2 2

1

12

D D p

p

(32)





The first term gives the amplitude

at normal incidence (θ = 0), the

second term characterizes R(θ ) at

intermediate angles, and the third

term describes the approach tocritical angle. The average P-wave

velocity at the reflecting interface

is V P = (V P1 + V P2)/2. The P-wave

velocity variation at the reflect-

ing interface is ΔV P = (V P2 − V P1).

The average density at the reflect-

ing interface is ρ = (ρ1 + ρ2)/2,

and Δρ = (ρ2 − ρ1) is the density

variation at the reflecting

interface.

RV V V V

V

V 02 2 1 1

2 2 1

12

= -

+ ª +Ê

Ë Áˆ

¯ ˜ r r

r r r

r D D p

p

(33)

is the normal-incidence reflection

coefficient. The term A0 is a con-

stant specifying the normal,

gradual decrease of amplitude with

offset. The average Poisson’s ratio

at the reflecting interface is

ν = (ν 1 + ν

2)/2, and Δν = (ν

2 − ν

1) is

the variation of the Poisson’s ratio

at the reflecting interface.

Transmission losses

Spherical divergence

Spherical divergence is one expression of the law of conservation of energy. In a

homogeneous medium, wavefronts are spherical. Therefore, energy-density decay is

proportional to the square of the radius of the wavefront, and amplitude decays as

the radius of the wavefront. In a horizontally layered medium, refraction will bendthe rays and will increase the surface of the wavefronts as velocity increases.

Figure 10 represents that effect. The black rays propagate linearly in a medium of

constant velocity, and the light gray rays have been refracted at an interface corre-

sponding to a velocity increase. After having propagated the same distance, the

refracted wavefront is larger than the constant-velocity wavefront. Processing geo-

physicists generally take that effect into account when compensating for spherical

divergence.

Figure 9. Energy partition at an interface. An incident

P-wave that makes an angle i i with the normal at theinterface generates four new waves: a reflected

P-wave that makes an angle i rp with the normal at theinterface; a reflected S-wave that makes an angle i rs with the normal at the interface; a transmitted P-wave

that makes an angle i tp with the normal at the inter-face; and a transmitted S-wave that makes an angle

i ts with the normal at the interface. Courtesy ofCGGVeritas. Used by permission.





Absorption

Absorption is the transformation of wave

energy into heat as a consequence of nonelasticity

of the medium. Often it is assumed that absorption

is the same for each wavelength. Usually it is

represented by quality factor Q . Note that this

factor is not the same for P-waves and S-waves.

Absorption is given by

A t f

f t Q t

( , ) exp( )

,= -Ê Ë Á

ˆ ¯ ˜

p

(34)

where t is time and f is frequency.

Because it depends on time and frequency,

absorption is not an easy parameter to control. Itoften is assumed and sometimes is observed that

absorption is associated with velocity dispersion.

Figures 11 and 12 show the effect of attenuation.

Figure 11 represents the attenuated wavelets, and

Figure 12 illustrates their amplitude spectra. The

columns are associated with six values of Q cover-

ing the range of expected absorptions in rocks, and

the rows correspond to five depths from 1 to 5 km.

The input wavelet is the correlation of the far-field

velocity signal received from a vibrator generating a

linear sweep by the pilot sweep in phase with theforce and therefore with far-field displacement. Its

amplitude spectrum rises 6 dB per octave, and its

phase is 90°. The represented effect is highly unre-

alistic because it assumes a constant Q from surface to a depth of 5 km. Figure 12 shows

that the effect can be significant. This is the case in layers with low compaction. The

shallow, weathered layers can have a very important and variable absorption.

Transmission losses at interfaces

Energy conservation requires that the energy of an incident wave be converted

into reflected and transmitted energy (if conversion is ignored). As a result, the normal-incidence transmission coefficient across an interface (ρ1V P1, ρ2V P2) is given by

T

V V V V

= - -

+1 2 2 1 1

2 2 1 1

r r

r r .

(35)

Each interface above a reflector is crossed twice, by the downgoing and upgoing rays.

Therefore, the contribution of one interface to the total transmission loss is T 2; i.e.,

Figure 10. Section of a normal-inci-

dence wavefront. Black indicates theconstant velocity model. Light gray

marks the two-layer velocity model. If

V 2 > V 1, the wavefront section isenlarged. Courtesy of CGGVeritas.

Used by permission.





Figure 12. Absorption. Amplitude spectrum of observed wavelets at depths of 1, 2, 3, 4, and 5 kmfor Q factors 50, 100, 150, 200, 250, and 300. The input wavelet has an amplitude spectrum with a

gain of 6 dB/octave and a phase spectrum of 90°. Courtesy of CGGVeritas. Used by permission.

Figure 11. Absorption. Observed wavelets at depths of 1, 2, 3, 4, and 5 km for Q factors 50, 100,150, 200, 250, and 300. The input wavelet has an amplitude spectrum with a gain of 6 dB/octave

and a phase spectrum of 90°. Courtesy of CGGVeritas. Used by permission.





it is very close to one. However, the number of interfaces could be very large. O’Doherty

and Anstey (1971) launch an exploration of that effect that continues to be debated

today.

Reciprocity The reciprocity theorem states that if a wave propagating from point A to point B

can be represented by the seismogram

D(f , A, B) = S(f , A)G(f , A, B) R(f , B), (36)

a wave traveling from point B to point A can be represented by

D(f , B, A) = S(f , B)G(f , B, A) R(f , A), (37)

where S and R are the source and receiver functions and G is the propagation function.

The propagation functions are identical:

G(f , B, A) = G(f , A, B). (38)

This theorem results from the symmetry properties of the wave equation, which

remains the same if t is changed into −t . It is illustrated in Figure 13a. The theorem

applies to any raypath, whatever its complexity. It must be noted that it does not apply

to wave generation or wave reception. In other words, because a vibrator is different

from a geophone, the signal emitted by a vibrator at point A and recorded by a geo-

phone at point B is identical to the signal emitted by a vibrator at B and recorded by a

geophone at A only when the

coupling conditions (whichare part of functions S and R)

for both vibrators and geo-

phones are identical at A and

B. For that reason, some small

differences will be observed

on reciprocal body waves,

most often onshore. Offshore,

if the source and receiver

positions are identical, iden-

tity of reciprocal raypaths will

be honored better.An apparent exception

to reciprocity is the case of

converted waves (illustrated

in Figure 13b). In fact, it is not

the reciprocity of propagation

that fails (the reciprocal of

P-to-S conversion is S-to-P

Figure 13. Reciprocity theorem. (a) Reflection and refrac-

tion: The reciprocity theorem applies to the symmetrical ray-path AM1B reciprocal of BM1A as well as to the nonsymmet-rical raypath AM2B reciprocal of BM2A. (b) Conversion at

interface: P-to-S converted raypath DC2E is different fromP-to-S converted raypath EC1D. The reciprocal of a P-to-S

conversion is an S-to-P conversion, which usually is not ob-served because it is not convenient to generate and recordP-to-S and S-to-P conversions at the same time. Courtesy of

CGGVeritas. Used by permission.





conversion). It is the differ-

ence between sources and

receivers that cannot be

resolved easily.

Data domains

The natural way to ex-

haustively sort seismic data

is to use one common prop-

erty of the raypaths. Because

of spatial aliasing, the result-

ing domains are not necessar-

ily adequate for imaging.

Therefore, it is interesting to

construct more sophisticateddomains that are adequate for

imaging. If spatial sampling

intervals (source and receiver

intervals) honor the Shannon-

Nyquist criterion, those do-

mains will provide spatially

unaliased data.

For the orthogonal

geometry depicted in Figure

14, five natural domains and

two unaliased domains willbe presented. An orthogonal

geometry is a 3D geometry

in which source lines and

receiver lines are orthogonal.

It is the most common land

3D geometry.

In Figure 15 through 21,

four attributes are repre-

sented: the source (in red),

the receiver (in blue), the

center of the source-receiver segment or CMP (in green), and the source-to-receiver

raypath (in black).

Natural domains

The source domain

The source domain also is known as the shotpoint domain, noted SP or S (Figure

15). Data originate from the same source point. In this domain, the CMP interval is half

Figure 14. 3D orthogonal geometry. Source lines are orthogo-nal to receiver lines. The most common land-acquisition

geometry, 3D orthogonal geometry provides a CMP grid withintervals equal to half the source and receiver intervals.

Courtesy of CGGVeritas. Used by permission.

Figure 15. Common source-point gather — an ensemble ofseismograms that share the same source. The CMP interval

is half the receiver interval in the inline direction and halfthe receiver-line interval in the crossline direction. Courtesyof CGGVeritas. Used by permission.





the receiver interval in the

receiver-line (inline) direction

and half the receiver-line

interval in the source-line

(crossline) direction.

The receiver domain

The receiver domain is

noted as RP or R (Figure 16).

Data are recorded through the

same receiver. In this domain,

the CMP interval is half the

source-line interval in the

inline direction and half the

source interval in the crossline

direction.

The midpoint domain

The midpoint domain is

known also as the common-

midpoint (CMP) domain. See

Figure 17 for an illustration

of a common-midpoint

gather. Data have an identi-

cal center along the source-

receiver segment.

The constant-offset domain

A constant-offset gather

is illustrated in Figure 18.

Data have the same source-

receiver offset. Note that the

term offset is taken as the

length of the source-receiver

vector.

The constant-azimuth domain

A constant-azimuth gather is illustrated in Figure 19. Data have the same source-

receiver azimuth.

Note that if the receiver-line interval is reduced adequately, the shotpoint domain

becomes unaliased, and reciprocally, if the source-line interval is reduced adequately,

the receiver domain becomes unaliased. Strictly speaking, constant-offset and constant-

azimuth domains are not easy to work with. It is more convenient to work with offset

Figure 16. Common-receiver gather — an ensemble ofseismograms that share the same receiver. The CMP inter-val is half the source-line interval in the inline direction and

half the source interval in the crossline direction. Courtesyof CGGVeritas. Used by permission.

Figure 17. Common-midpoint gather — an ensemble of seis-

mograms that share the same CMP (midpoint between thesource and the receiver). Courtesy of CGGVeritas. Used by

permission.





and azimuth “classes,” which

are ensembles of traces with

offset or azimuth belonging

to a given range (a few hun-

dred meters or a few de-grees).

Unaliased domains

The cross-spread domain

Figure 20 shows a

cross-spread gather. Data are

recorded with their source

on a given source line and

their receiver on a given

receiver line. The CMPcoverage of a cross-spread is a

rectangle sampled with half

the source-point interval in

the source-line direction and

half the receiver interval in

the receiver-line direction.

The size of the rectangle

depends on the recording

strategy (which receiver is

alive when a given SP is

recorded), but it is almostalways a fraction of the

survey size.

The offset vector tile

Offset vector tiles

(OVTs) are elements of

cross-spreads that can be

assembled to construct seamless single-fold data sets that cover the entire survey. As

shown in Figure 21, each tile is a subset of cross-spreads in which sources are selected on

a segment twice the length of the receiver-line interval, and receivers are selected on asegment twice the length of the source-line interval. Relative positions of the source and

receiver segments are identical for all cross-spreads. It is convenient but not necessary to

use segments centered on a line intersection. This domain is remarkable because it covers

the full survey area and because differences in offset and azimuth between components

of a given OVT are relatively small. Offset vector tiles were introduced independently by

Gijs Vermeer in 1998 under the name offset/azimuth slot and by Peter Cary in 1999

under the name common-offset vector (COV) gathers (Vermeer, 1998; Cary, 1999).

Figure 18. Common-offset gather — an ensemble of seismo-

grams that share the same (absolute value of) the source-

receiver offset. Courtesy of CGGVeritas. Used by permission.

Figure 19. Common-azimuth gather — an ensemble of seis-

mograms that share the same source-to-receiver azimuth.Courtesy of CGGVeritas. Used by permission.





Conclusion

Mechanics and optics are the fundamental bases of the relatively young seismic

technique. Even though modern processing, particularly imaging routines, uses sophis-

ticated mathematics, we should not forget that those routines are constructed on these

relatively simple bases.

Figure 20. Cross-spread gather — an ensemble of seismograms whose source points belong tothe same source line and whose receivers belong to the same receiver line. Courtesy of

CGGVeritas. Used by permission.

Figure 21. Offset vector tile — a subensemble of a cross-spread restricted by the condition thatthe source must belong to a segment twice as long as the receiver-line interval and the receiver

must belong to a segment twice as long as the source-line interval. Courtesy of CGGVeritas. Used

by permission.





2. Chapter 2 Elementary Theoretical Recapitulation

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