Seismic attenuation factor Q and it's utility.

1. 1
A Report on
“Seismic Attenuation Q & its Utility”
Under the guidance of:
Prof. Dinesh Kumar
Department of Geophysics
Kurukshetra University
Submitted by:
Rashi
Roll No. GP-05
M.Tech. Applied Geophysics, 5th
Semester
Department of Geophysics
Kurukshetra University

2. 2
Table of Contents
1. Introduction…………………………………………………………………………3
2. Mechanism…………………………………………………………………………..6
3. Utility…………………………………………………………………………………18
4. References……………………………………………………………………………20

3. 3
1. Introduction
The seismic attenuation factor Q can be defined as the ratio of 2𝜋 times the
peak energy to the energy dissipated in a cycle; the ratio of 2π times the power
stored to the power dissipated. The seismic Q of rocks is of the order of 50 to
2000.
The further a seismic signal
travels from its source the
weaker it becomes. The
decrease of amplitude with
increasing distance from the
source is referred to as
attenuation. It is partly due to
the geometry of propagation of
seismic waves, and partly due
to anelastic properties of the material through which they travel.
The most important reduction is due to geometric attenuation. Consider the
seismic body waves generated by a seismic source at a point P on the surface of
a uniform half-space (see Fig. 1). If there is no energy loss due to friction, the
energy (Eb) in the wavefront at distance r from its source is distributed over the
surface of a hemisphere with area 2πr2. The intensity (or energy density, Ib) of
the body waves is the energy per unit area of the wavefront, and at distance r
is:
𝐼𝑏( 𝑟) =
𝐸𝑏
2𝜋𝑟2
Fig. 1: Propagation of a seismic
disturbance from a point source P near the
surface of a homogeneous medium. (from
Lowrie, 2007)

4. 4
The surface wave is constricted to spread out laterally. The disturbance affects
not only the free surface but extends downwards into the medium to a depth d,
which we can consider to be constant for a given wave (Fig. 1). When the
wavefront of a surface wave reaches a distance r from the source, the initial
energy (Es) is distributed over a circular cylindrical surface with area 2𝜋𝑟𝑑. At a
distance r from its source the intensity of the surface wave is given by:
𝐼𝑠 ( 𝑟) =
𝐸𝑠
2𝜋𝑟𝑑
These equations show that the decrease in intensity of body waves is
proportional to 1
𝑟2⁄ while the decrease in surface wave intensity is proportional
to 1
𝑟⁄ . Thus, seismic body waves are attenuated more rapidly than surface
waves with increasing distance from the source. This explains why, except for
the records of very deep earthquakes that do not generate strong surface
waves, the surface-wave train on a seismogram is more prominent than that of
the body waves.
Another reason for attenuation is the absorption of energy due to
imperfect elastic properties. If the particles of a medium do not react perfectly
elastically with their neighbours, part of the energy in the wave is lost instead
of being transferred through the medium. This type of attenuation of the
seismic wave is referred to as anelastic damping.
The damping of seismic waves is described by a parameter called the
quality factor Q. It is defined as the fractional loss of energy per cycle
2𝜋
𝑄
= −
∆𝐸
𝐸

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In the expression ∆𝐸 is the energy lost in one cycle and E is the total elastic
energy stored in the wave. If we consider the damping of a seismic wave as a
function of the distance that it travels, a cycle is represented by the wavelength
of the wave. Above equation can be rewritten for this case as:
2𝜋
𝑄
= −
1
𝐸
𝜆
𝑑𝐸
𝑑𝑟
𝑑𝐸
𝐸
= −
2𝜋
𝑄
𝑑𝑟
𝜆
It is conventional to measure damping by its effect on the amplitude of a
seismic signal, because that is what is observed on a seismic record. Thus,
above mentioned equations can be simplified as:
𝐴( 𝑡) = 𝐴0 exp(−
𝜋𝑓𝑡
𝑄
)
This expression gives us the attenuated amplitude at time t. A rock with high Q
value transmits a seismic wave with relatively little energy loss by absorption
and vice-versa. Also, the attenuation of a seismic wave by absorption is
dependent upon the frequency of the signal. High frequencies are attenuated
more rapidly than the lower frequencies. As a result, the frequency spectrum of
a seismic signal changes as it travels through the ground. Thus, the earth acts
as a low pass filter to the seismic signals.

6. 6
Mechanism
After the earthquake arrival, waves travel through the earth and because of the
different properties of the medium that they travel through, there is a decrease
in the amplitude or we say seismic attenuation occurs. This seismic
attenuation is measured by a quantity called as quality factor(Q).
Here , we consider four other processes that can reduce wave amplitude:
1. Geometric spreading
2. Scattering
3. Multipathing
4. Anelasticity(or intrinsic attenuation)
The first three processes are elastic, in which the energy in the propagating
wavefield is conserved. By contrast, anelasticity, sometimes called intrinsic
attenuation, involves conservation of seismic energy in to heat.
As in many seismological applications, it is worth first considering
familiar analogous behaviors for light. As you move away from astreet lamp at
night, the light appears dimmer for several reasons.
The first is Geometrical Spreading : light moves outward from the lamp in
expanding spherical wave fronts. By the conservation of energy , the energy in
a unit area of the growing wavefront deacreases as r-2, where r is the radius of
the sphere or distance from the lamp.
Second is Scattering , the light dims as it is scattered by the air molecules,
dust, and water in the air. As we have discussed , scattering results when the
objects acting as huygens’ sources scatter energy in all the directions. This
effect is dramatic on a foggy night because the scattered light causes a halo
around the lamp.

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Third, the light is focused or defocused by changes in the refractive properties
of the air. This effect is termed multipating in seismology. Focusing and
defocusing can be illustrated by looking at the street light through the
binoculars the usual way, the waves are focused by the lenses, and the lamp
through binoculars. Looking through binoculars makes the lamp appear
further away and dimmer.
Fourth , some of the light energy is absorbed by the air and converted in to
heat. This process differs from the other three in that light energy is actually
lost, not just moved on to a different path.
All the four processes are important for seismic waves. The first three described
by elastic wave theory, and can increase or decrease an arrival’s amplitude by
shifting energy within the wave field. By contrast, anelasticity reduces wave
amplitudes only because energy is lost from the elastic waves. So much of the
seismology is built upon the approximation that the earth responds elastically
during seismic propagation that it is easy to forget that the earth is not
perfectly elastic. However, without anelasticity, seismic waves from every
earthquake that ever occurred would still be reverberating until the
accumulating reverberations shattered the earth. Elasticity is a good
approximation for the earth’s response to seismic waves, but there are many
important implications and applications of anelasticity.
Anelasticity results becase the kinetic energy of elastic wave motion is lost to
heat by permanent deformation of the medium. The large scale, or
macroscopic, term for this process is internal friction. Among the smaller
scale or microscopic , mechanisms that may cause this dissipation are stress
induced migration of defects in the minerals, frictional sliding on the crystal
grain boundaries, vibration of dislocations, and the flow of hydrous fluids or

8. 8
magma through grain boundaries. Theoretical and experimental work is being
carried out to examine the possible mechanism of the seismic attenuation.
The study of anelasticity has lagged behind that of the elastic wave velocities
because of the complexities involved in measuring attenuation and
understanding it’s physical causes. Although measuring seismic wave
amplitudes is straightforward , they depend on both the source, which is not
perfectly known , and all thr elastic and anelastic effects anywhere along the
paths that the seismic energy travelled between the source and receiver. Hence
it can be hard to distinguish the effects of anelasticity from elastic properties.
This inherent uncertainty is somewhat compensated by the fact that variations
in anelasticity are large , as illustrated by comparison of records of an
earthquake in Texas at stations in Nevada and Missouri (Fig. 2). The Nevada
seismogram has much less high-frequency energy, showing that the crust in
the western USA is much more attenuating than that in the Midwest. By
comparison, seismic velocity variations between these areas are generally less
than ±10%. Even so, because of the difficulties in measuring attenuation,
variations in attenuation at both regional and global scales are much less
resolved than similar variations in the velocity .
Geometrical Spreading:
The most obvious effect causing seismic wave amplitudes to vary with distance
is geometric spreading, in which the energy per unit wave front varies as a
wave front expands or contracts. Geometric spreading differs for surface and
body waves. For a homogeneous elastic spherical earth , a surface wave front

9. 9
would spread as it moved from the source to a distance 90˚ away, refocus as it
approached the antipode on the other side of the earth from the source , and so
on. The amplitudes would be largest at the source and antipode, where all the
energy would be concentrated, and smallest halfway between, 90˚ from the
source. On a homogeneous flat earth, the surface waves would spread out in a
growing ring with circumference 2πr, where r is the distance from the source.
Conservation of energy requires that the energy per unit wave front decrease as
1/r, whereas the amplitudes, which are proportional to the square root of
energy, decreases as 1/√r. However, because the earth is a sphere, the ring
wraps around the globe, making the energy per unit wavefront vary as
1/r= 1/(a sinΔ),
Where Δ is the angular distance from the source. Thus the amplitudes decrease
as (a sinΔ)-1/2, with minimum at Δ=90˚, and maxima at 0˚ and 180˚. Actually,
not all the energy would focus at the antipode and source even if the earth had
no lateral variations in velocity , because some defocusing would result from
the earth’s ellipsoidal shape. Lateral heterogeneity, discussed next , further
distorts the wavefront.

10. 10
For body waves, consider a spherical wavefront moving away from a deep
earthquake. Energy is conserved on the expanding spherical wavefront whose
area is 4πr2, where r is the radius of the wavefront. Thus the energy per unit
wave front decays as 1/r2, and the amplitude decreases as 1/r. In reality,
because body waves travel through an homogeneous earth, their amplitude
depends on the focusing and defocusing of rays by the velocity structure.
Multipathing
As the seismic waves travel downward, there is a variation in the properties of
material and they either get reflected, refracted or transmitted. Similarly, as the
waves travel laterally in the medium , the medium properties change and
similar phenomenon are bound to happen. This phenomena of ray taking
multiple paths is called multipathing.
Figure 3 illustrates this effect for a plane wave passing through a refracting
layer of variable thickness. The ray paths which are normal to the local
wavefront, show how the initially planar wave is refracted . the ray spacing
represents the energy density, so amplitudes are low where the rays are far
apart, and high where they are close together. In some cases the energy
focuses in to caustics, areas of infinitely high energy density, which appear as
solid black regions.

11. 11
Figure 3
figure 4 shows how velocity heterogeneity can cause erroneous of either the
focal mechanism or attenuation. It tells us that rays take the shortest time
taking path to reach the receiver. Hence , multipathing comes in to picture.
figure 4

12. 12
Scattering
A related effect to multipathing is the scattering of seismic waves. Both effects
are complicated and the distinction between them is gradational. So we say, if
the size of the wavelength is similar to velocity heterogeneities then scattering
happens.
Figure 5 demonstrates the scattering for a seismic arrival. The unscattered
wave travels the shortest distance and gives the initial arrival (left). Scattered
energy lost from this arrival that instead arrives later could have been scattered
from an infinite number of locations that would yield the observed travel time.
In a constant-velocity medium, the locus of these possible scatterers form an
ellipsoid with the source and the receiver as foci(center). Larger ellipsoids
define the possible scatterers for energy that arrives later (right). These
ellipsoids are distorted by velocity heterogeneity and are analogous to the
Fresnel volume used when we consider the waves as following distinct ray
paths.

13. 13
figure 5
Scattering is especially noticeable on the moon. Figure 6 contrast seismic
records of an earthquake and impact of a rocket on the moon. Most of the
earthquake’s energy arrives in the main P- and S- wave arrivals. By contrast,
on the moon the energy is intensely scattered , and no main arrivals can be
identified. This is probably because intrinsic attenuation is much larger in the
earth’s crust than on the moon. The movements of interstitial fluids in the
earth’s crust greatly reduce seismic wave amplitudes, whereas energy
scattered by the moon’s highly fractured near surface regolith layer is poorly
absorbed and reverberates. As a result, efforts to identify seismic phases and
use them to study the moon’s internal structure have been generally
unsuccessful.

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figure 6
Intrinsic Attenuation
As the waves travel, they attenuate with distance due to different properties of
the medium. This attenuation is quantized by a term known as Quality
factor(Q). To define this attenuation , we study the analogous case of a spring
in a damped oscillatory motion. Equation of motion of this spring is written as

15. 15
    02
2
 tku
dt
tud
m
(1)
Once set in motion by an impulse , this frictionless system has a purely elastic
response described by a perpetual harmonic oscillation.
tiw
o
tiw
o
00
eB+eA=u(t) (2)
Where A and B are constants , and the mass moves back and forth with a
natural frequency
1/2
o (k/m)=w
One example of this general solution is
t)cos(wA=u(t) 0o
However, this is no longer the case if the system contains a dashpot , or
damping term, the damping term is proportional to the velocity of the mass and
opposes its motion. Hence the equation of motion becomes
    0)(2
2
 tku
dt
tdu
m
dt
tud
m 
Where ϒ is the damping factor. To simplify this, we define the quality factor
Q=wo/ϒ,
And rewrite equation… as
    0)(2
2
2
 tkuw
dt
tdu
Q
w
dt
tud
o
o

16. 16
This differential equation , which describes the damped harmonic oscillator,
can be solved assuming that the displacement is the real part of a complex
exponential
ipt
oeA=u(t)
Where p is a complex number. Substituting the equation … in to equation ,,..
Yields
0)/(-p )(22
 pt
ooo eAwQpw 

For this to be satisfied for all values of t,
0/p- 22
 oo wQpw
Because p is a complex , we break it in to real and imaginary parts,
p=a+ib, p2=a2 +2iab-b2,
so equation gives
0//2p- 222
 ooo wQibwQiawbiab
which can be split in to equations for the real and imaginary parts and solved
separately :
Real: 0/- 222
 oo wQbwba
Imaginary: 0/2  Qawab o
Solving the imaginary part for b gives
Qwb o 2/
And putting this in to the equation for the real part gives
)/11(4/ 22
0
22
0
2
0
2
QwQwwa 

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Thus we define
2/12
)4/11( Qwa o 
And rewrite equation with separate real and imaginary parts,
iwtbtibtwti
o eeAeAtu 
 0
)(
)(
The real part is the solution for the damped harmonic displacement,
)cos()( 2/
0 wteAtu Qtwo

This solution shows how the damped oscillator responds to an impulse at time
zero. It is no longer a simple harmonic oscillation because it differs in two ways
from the undamped solution. The exponential term expresses the decay of the
signal’s envelope, or overall amplitude,
figure

18. 18
Utility of Q-value:
So far we have seen the waves attenuate as
Qtwo
eAtA 2/
0)( 

To determine how far in depth the wave has to travel before its amplitude
reduces to, say, one-tenth of its amplitude at the surface z = 0, equation is
written as follows:
z =2.3Qv/πf
Note that the smaller the Q factor, the lower the velocity and the higher the
frequency, the shallower the depth at which the wave amplitude decays to a
fraction of the wave amplitude at z = 0. Table 1 lists the z values for a
frequency of 30 Hz and a velocity of 3000m/s for a range of Q values. Depth at
which wave amplitude drops to one-tenth of its original at the surface for a
range of Q values ,v = 3000 m/s and f = 30 Hz
Q Factor Depth in m
25 1830
50 3660
100 7325
250 18,312
500 36,625
Table 1.
Note that the smaller the Q factor the shallower the depth at which the
amplitude drops to the specified value of one-tenth of the original value at the
surface z = 0. For very large Q values, that is, for small attenuation, the
amplitude reduction to the specified value does not take place until the wave
reaches very large depths beyond the exploration objectives.
So , to compensate for the loss we design the inverse Q-filter as follows:

19. 19
A(t)=A0e(-wt/2Q)
Now I have to find the A0 . so re-arranginf the equation we get
A0=A(t)e(w t/2Q)
Here the exponential term is called as inverse Q-filter. The effects of inverse
Q-filter is visible in the following picture figure 8.
Figure 8

20. 20
References
1. Stein, S., & Wysession, M. (2003). An Introduction To
Seismology, Earthquakes, And Earth Structure. UK: Blackwell
publishing.
2. Yilmaz, O. (2001). Seismic Data Analysis: Processing, Inversion
and Interpretation (2nd ed., Vol. 1). USA: Society of Exploration
Geophysicists
3. Lowrie,W. (2007). Fundamentals of Geophysics. Cambridge
University Press.