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Chapter 1
Theory of seismic waves
Theory of elasticity

Seismic waves are stress (mechanical) waves that are generated as a response to
acting on a material by a force.

The force that generates this stress comes from our source of seismic energy such the
Vibroseis or dynamite.

The stress will produce strain (i.e., deformation) in the material. Stress and strain are
related through elasticity theory.

Therefore, we need to study a little bit of elasticity theory in order to better
understand these waves.
Stress

Stress, denoted by , is force per unit area, with units of pressure such as Pascal
(N/m2) or psi (Pounds/in2).

xy denotes a stress produced by a force that is parallel to the x-axis acting upon a
surface which is perpendicular to the y-axis (Figure).

There should be a maximum of 9 stress components associated with every possible
combination of the coordinate system axes (xx, xy, xz, yx, yy, yz, zx, zy, zz).

However, because of equilibrium (i.e., body is not moving but only deformed as a
result of stress application): ij = ji, meaning that xy = yx, yz = zy, and zx = xz.
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
If the force is perpendicular to the surface, we have a normal stress (xx, yy, zz);
while if it’s tangential to the surface, we have a shearing stress (xy, yz, zx).

Therefore, we can define a stress matrix composed of the nine components of the
stress:
 xx  xy

   yx  yy
 zx  zy

 xz 

 yz  .
 zz 
Strain

Strain, denoted by , is the fractional change in a length, area, or volume of a body
due to the application of stress.

For example, if a rod of length L is stretched by an amount L, the strain is L/L.

Evidently, strain is dimensionless.

To extend this analysis to 3-D objects, consider a body with dimensions of X, Y, and
Z along the x-, y-, and z-axes respectively. If this body is subjected to stress, then
generally X will change by an amount of u(x,y,z), Y by an amount of v(x,y,z), and Z
by an amount of w(x,y,z) (Figure).

Again, there are generally 9 strain components corresponding to the 9 stress
components (xx, xy, xz, yx, yy, yz, zx, zy, zz).

However, because of equilibrium: ij = ji, meaning that xy = yx, yz = zy, and zx =
xz.

We can define the following strains:
 Normal strains: xx = u/x, yy = v/y, zz = w/z.
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 Shearing strains: xy = v/x + u/y, yz = w/y + v/z, zx = u/z + w/x.
 The dilatation () is the change in volume (V) per unit volume (V):
  V/V = xx + yy + zz = u/x + v/y + w/z.
 The strain matrix composed of the nine components of strain:
 xx

   yx
 zx

 xy  xz 

 yy  yz 
.

 zy  zz 
Hooke’s Law

It states that, at sufficiently small strains (≤ 10-6), the strain is directly proportional to
the stress producing it.

The strains produced by the passage of seismic waves in earth materials are such that
Hooke’s law is always satisfied.

Mathematically, Hooke’s law can be expressed as:
( 4)
  C
where 
is the stress matrix, 
(1)
( 4)
is the strain matrix, and C is the elastic-constants
tensor, which is a fourth-order tensor consisting of 81 elastic constants (Cxxxx to Czzzz).

Because of the symmetry relations in stress, strain, and strain energy (giving Cijkl =
Cklij), there can only be a maximum of 21 independent elastic constants (IECs) in a
medium. This number reduces as more symmetry relations exist in the medium.

Examples of symmetry relations are crystal systems such as cubic (3 IECs),
hexagonal (5 IECs), triclinic (21 IECs) ... etc.
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
The least number of IECs exists in an isotropic medium, which has only 2 IECs.
These are called Lame’s constants  and .  is also called the rigidity or shear
modulus. Extra material: Forms of Hooke’s law.

A medium with more than 2 IECs is called anisotropic.

Isotropy simply means that a wave property, such as velocity, is independent of wave
propagation direction.

Homogeneity means that a wave property, such as velocity, is independent of
position.

Unless specified otherwise, we will always assume that a single layer is:
 Elastic: meaning that the stress and strain satisfy Hooke’s law.
 Homogeneous: meaning that layer properties (e.g., velocity) are constant across
the whole layer.
 Isotropic: meaning that wave properties (e.g., velocity) are independent of
propagation direction.
Elastic constants in isotropic media

Lame’s constants  and  are defined through the following forms of Hooke’s law
in an isotropic medium: iiii, (i = x,y,z) and ijij, (i ≠ j, i,j = x,y,z).

Young’s modulus (E) is defined as: E = xx/xx, (for uniaxial stress along the x-axis
where xx  0, yy = zz = xy = xz = yz = 0.)

Poisson’s ratio () is defined as:  = -yy/xx = -zz/xx. Typically, 0 <  < 0.5. It is
small for hard rocks and large for soft rocks. For a perfect fluid,  = 0,  = 0.5.

The bulk modulus () is defined as:  = /, (for hydrostatic stress: yy=zz=xx=).
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The wave equation

The scalar wave equation of a displacement (u) that depends only on x and t is:
1  2u  2u
( 2) 2  2 ,
V t
x
where V is the wave velocity.

The general solution of this wave equation is a plane wave given by:
u = f(x – Vt)  g(x + Vt),
where f and g are arbitrary functions (e.g., exponential, trigonometric, …etc):
 f(x – Vt) is a wave traveling along the positive x-axis with a velocity V
 g(x + Vt) is a wave traveling along the negative x-axis with a velocity V
 (Figure)

The quantity (x  Vt) is called the phase.

The quantity 1/V is called the slowness.

The surface on which the phase is the same (i.e., have same amplitude) is called the
wavefront.

The normal to the wavefront surface at a point is called ray or propagation direction.

The most commonly used wavefronts in geophysics are the plane and spherical.

Wavefronts are spherical near the source and become planar far from it.
General aspects of seismic waves

A seismic wave consists of a group of sinusoidal waves of different frequencies. The
number of sinusoids with different frequencies forms the frequency band.
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
As the wave’s frequency band increases, its time duration (length) decreases. We
will study this in more details in GEOP320.

Seismic waves are sinusoids that generally have wide frequency bands (2-120 Hz)
and very short time durations (50-100 ms). Such waves are called wavelets.

The wave velocity (V), frequency (f), and wavelength () are related as follows:
V = f.

Typical wave characteristics in petroleum seismic exploration are:
 Most of the reflected energy is contained within a frequency range of 2 – 120 Hz.
 The dominant frequency range is 15 - 50 Hz.
 The dominant wavelength range is 30 – 400 m.

Terminology of waves commonly encountered in seismic exploration include:
 Acoustic wave: wave propagating in a fluid.
 Sonic wave: wave in the hearing frequency range of humans (20 – 20,000 Hz).
 Ultrasonic wave: wave whose frequency is more than 20,000 Hz, commonly used
in acoustic logs and lab experiments.
 Subsonic wave: wave whose frequency is less than 20 Hz, commonly encountered
in earthquake studies.
Huygens’ principle

It states that every point on a wavefront can be regarded as a secondary source that
emits spherical wavefronts. The common tangent to the secondary wavefronts in the
propagation direction defines the new position of the wavefront.

This principle is useful in drawing successive positions of wavefronts. (Figure).
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Fermat’s principle

It states that a wave will take that path which will make its traveltime between the
source and receiver stationary (i. e., maximum or minimum). Mathematically:
dT / dX = 0,
where T: total traveltime from the source to the receiver
X: distance from the source to the point where the wave changes its
direction (e.g., point of reflection or refraction).

In most situations in the earth, the stationary path is the minimum-time path.

This principle is useful in solving problems that require ray-tracing (Figure).
Seismic body waves

They distort the volume element of an elastic medium by traveling inside it.

There are two types of body waves: the primary (P) wave and the secondary (S)
wave.
P-wave

The P-wave has a velocity ():

  2
,

where  is the volume density.

Particle motion is parallel to the wave propagation direction in the form of
compressions and dilatations (expansions).

Figure and movie.
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S-wave

The S-wave has a velocity ():



.

Particle motion is perpendicular to the wave propagation direction. Hence, there are
two S-waves.

To distinguish these two S-waves, we call them the S1- and S2-waves or the SH- and
SV-waves (Link).

Figure and movie.
Notes about P- and S-waves

Since the elastic constants () are always positive (why?):

0



1
2
,
and  ≈ ½ in sedimentary rocks.

Typical P-wave velocities ():
 In air:
 = 331 m/s (increasing 0.6 m/s per C)
 In water:
 = 1,500 m/s (increasing slightly with salinity)
 In sedimentary rocks:
1,800    6,500 m/s
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 In the weathered layer (soil):
50    1,000 m/s (dry soil)
1,000    2,500 m/s (saturated soil)
Seismic surface waves

They exist due to the presence of a free surface (vacuum over any material) or an
interface that separates two highly-contrasting media.

They are called surface waves because they are tied to the free surface or interface.

Their amplitudes decay exponentially with the distance from the surface.
Rayleigh waves

They propagate along the free surface of a solid (i.e., surface between solid and
vacuum).

The ground surface is considered as a free surface in seismic exploration.

Rayleigh waves are called ground roll in seismic exploration.

The following is true about the relation of Rayleigh wave velocity (VR) to body-wave
velocities in the same material:
VR <  < .

Typically in sedimentary rocks, VR ≈ 0.9

Most of the Rayleigh wave’s energy is confined to 1-2 wavelengths of depth.

Particle motion is largest and elliptical retrograde near the surface and becomes
smaller and elliptical prograde deeper.

Figure, movie, and a summary of waves.
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Tube waves

Tube waves constitute of waves that travel in a borehole parallel to the borehole axis.

They can provide information about the formation surrounding the borehole.

The most common types of tube waves are P-waves propagating in the borehole fluid,
Stoneley, and pseudo-Rayleigh (a.k.a. shear-surface) waves propagating along the
borehole wall.

They can be generated by almost any disturbance of the borehole fluid.

Figure.
Anisotropy

Seismic anisotropy is the variation of a seismic property (e.g., velocity) with the
direction along which it is measured.

The anisotropy type in a medium depends on its symmetry system (e.g., cubic,
hexagonal … etc).

The symmetry system of a medium defines what happens to its properties upon
geometrical manipulations such as inversion and rotation.

Transverse isotropy (TI) is the most common type of anisotropy encountered in
seismic exploration studies.

TI involves a property that is the same within a plane (called the isotropy plane) but
different along an axis (called the symmetry axis), which is perpendicular to the
isotropy plane (Figure).

Two important types of TI are observed in seismic exploration:
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(1) Vertical Transverse Isotropy (VTI) that has a vertical symmetry axis. The main
cause of VTI is the thin layering of shales in the subsurface
(2) Horizontal Transverse Isotropy (HTI) that has a horizontal symmetry axis. The
main cause of HTI is the presence of vertical aligned fractures.

Figure.

There are 5 independent elastic constants for a TI medium.

In TI media, velocity is lowest when measured parallel to the symmetry axis and
highest when measured perpendicular to the symmetry axis.

In TI media, S-wave splits into a fast (S1) wave perpendicular to the symmetry axis
and a slow (S2) wave parallel to the symmetry axis.

Anisotropy () = (Vmax – Vmin)/Vmax.

Most transversely isotropic sedimentary rocks have weak anisotropy (i.e.,  < 0.1).
Medium effects on waves
Geometrical spreading

As the wavefront gets farther from the source, it spreads over a larger surface area
causing the intensity (energy density) to decrease. This is called geometrical
spreading or spherical divergence.

Generally, the intensity (i.e., energy density = energy/wavefront surface area) is
related to distance (r) from source as follows:
I (r )  I 0 r  m ,
where I0 and I(r) are intensities on the wavefront at the source (r = 0) and a distance r
from the source, respectively and:
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o m = 0 for plane waves
o m = 1 for cylindrical waves
o m = 2 for spherical waves.

In correcting for geometrical spreading, spherical wavefronts are assumed (m = 2).

However, since we usually record the amplitude, which is the square root of intensity,
we correct for geometrical spreading using this relation:
A0  A(r ).r
where A0 and A(r) are amplitudes on the wavefront at the source (r = 0) and a distance r
from the source, respectively.

Example.
Absorption

It is the loss of wave amplitude due to the transformation of elastic energy to thermal
energy as the seismic wave passes through the medium.

Common causes of absorption in seismic exploration are:
o Friction along fracture and sediment-grain boundaries
o Differential pore-fluid movements.

Absorption follows an exponential relation:
A(r )  A0 e  .r ,
where A0 and A(r) are amplitudes of a plane wavefront at two points a distance r
apart,
: Absorption coefficient ( = 10-5 - 10-3 m-1 in sedimentary rocks).
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
Therefore, to correct amplitudes for absorption effects, we use the following relation:
A0  A( r )e .r .

Geometrical spreading dominates at low frequencies and short distances from the
source, while absorption dominates at high frequencies and greater distances from the
source.

The distances and frequencies involved in seismic exploration are such that
geometrical spreading is far more effective than absorption. Hence, we usually
correct for geometrical spreading and neglect absorption (Figure).

Summary:
Dispersion

It is the dependence of seismic velocity on its frequency.

Dispersion is negligible for body waves but considerable for surface waves (Figure).
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Interface-related effects

When a wave encounters an interface (i.e., an abrupt change in the elastic
properties), some of the energy is reflected back to the incident medium and the rest
is refracted (transmitted) into the other medium.

Snell’s law governs reflection and refraction angles:
Sin1 Sin 2

 p,
V1
V2
where 1: angle of incidence,
2: angle of refraction,
V1: velocity of the incident medium,
V2: velocity of the refraction medium,
p: ray parameter, which is constant for the same ray.

Snell’s law applies even when the wave mode (P- or S-wave) differs.

The critical angle (c) takes place when 2 = 90:
 V1 
 .
V
 2
 c  Sin 1 

When 1 = c, head waves are generated which travel along the interface in the
refraction medium with a velocity V2 (why?).

Note that c will not exist when V2 < V1 (why?).

For 1 > c, total internal reflection takes place. That is, no energy will be transmitted
to the refraction medium for rays that have 1 > c.

Diffraction takes place when the wave encounters an abrupt lateral change in
lithology (e.g., fault, wedge … etc.).
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
Snell’s law does not apply for diffractions and Huygens’s principle is used instead.

(Summary).

Link (Note that n2/n1 = v1/v2).
Amplitude partitioning at an interface

A P or SV wave incident on an interface between two solids will generally generate:
o A reflected P-wave
o A reflected SV-wave
o A refracted P-wave
o A refracted SV-wave.

On the other hand, a SH-wave incident on an interface between two solids will
generate only:
o A reflected SH-wave
o A refracted SH-wave.

What happens in the case fluid/solid and fluid/fluid?

At the interface, the following boundary conditions must be satisfied:
o Normal stresses must be continuous (why?).
o Tangential stresses must be continuous (why?).
o Normal displacements must be continuous (why?).
o Tangential displacements must be continuous (why?).

The amplitudes of reflected and refracted waves are found by applying the boundary
conditions at the interface and solving the resultant Zoeppritz equations.
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
At nonnormal incidence (i  0), the exact reflection and refraction coefficients we
get from Zoeppritz equations are very algebraically complicated functions of the Pand S-wave velocities and densities in the two media as well as the angles of
reflection and refraction of the P- and S-waves.

At normal incidence (i = 0), the reflection (R) and transmission (T) coefficients
reduce to the following simple forms:
R
Z 2  Z1
Z 2  Z1
T  1 R 
2 Z1
,
Z 2  Z1
where Zi = ii is the acoustic impedance of a medium (why is it called acoustic?).

The normal incidence formulas can be used for slight deviation from the normal (i 
15) without introducing considerable error.

Approximations of Zoeppritz equations (e.g., Shuey, 1985) can be used up to i  30
to calculate R. These approximations are commonly used in amplitude-variationwith-offset (AVO) studies.

A reflection coefficient of -0.3 means that 30% of the seismic energy will be reflected
to the incident medium after amplitude polarity reversal. The remaining 70% will be
transmitted into the refraction medium also with no amplitude polarity reversal.

A reflection coefficient of +0.3means that 30% of the seismic energy will be reflected
to the incident medium with no amplitude polarity reversal. The remaining 70% will
be transmitted into the refraction medium also with no amplitude polarity reversal.

Figure.

Link.

Exact expression of the P-P reflection coefficient.