1. GG 450
April 3, 2008
Refraction, Diffraction, Energy,
Sources, and Sensors

2. Critical Angle:
sinq
Recall: = p(Snell's law)
v
When θ = 90° , the ray travels horizontally through the earth. The
"critical" angle is a special case where the velocities are constant in
the layers. The critical angle is the angle in the upper layer where
the ray becomes horizontal in the layer below:
.
v1
θ ic = sin -1
v2
In some seismic modeling, the critical angle is
important, as we shall see..

3. θχ
v1
v2
v1
θ c = sin
-1
v2

4. Don't confuse the refraction method with the reflection method:
REFLECTION Method: Geometry: A common seismic method
involves the source - usually explosive, being moved along the
surface of the earth at the same speed as the receivers, so that the
distance between the source and receiver remains constant.
This method is termed "profiling" and the resulting records are
called profiles, often plotted as distance along the profile vs. time
after the "bang". Profiles often show a close resemblance to
geological profiles. A marine profile is shown as an example.

6. REFRACTION METHOD: A second type of geometry
has the source remaining at one spot and the receivers
spaced at increasing distances from the source. In this
case, seismic arrivals as they change with distance are
plotted. The resulting plot is an x-t plot, travel time plot,
or "record section".
The source MOVES WITH
the receiver in a PROFILE.
The source stays fixed
(usually at x=0) in an
x-t plot (record section).

7. Seismic Arrivals
When you start a seismic wave at the earth's surface - as
we will with the refraction system - several waves fan out
in more-or-less spherical (waves that go through the earth)
and cylindrical wave fronts.
Air wave: travels through the air at about 330 m/s (1,083
ft/s), only seen close to the source (if at all). Velocity is
constant, so a plot of arrival time of the air wave vs.
distance from the source is a line with a slope of 0.92
ms/foot. This is a SLOW wave, usually mixed in with
surface wave arrivals.
•How far away is lightning.

8. Direct Wave: Travels at the p- Reflections: Reflections Refractions (head waves):
wave velocity of the uppermost arrive from sharp changes Refractions are arrivals from
layer of the ground, directly to in velocity (actually faster deeper layers that arrive
the receiver. Direct waves impedance, ρv) below. first at larger distances. They
come in first close to the When plotted on a x-t are usually straight or bending
source, but often disappear at plot, the first reflection slightly downwards with
larger distances or are lost in arrival time is asymptotic increasing distance on x-t plots.
earlier, faster arrivals. The to the direct arrival. Refractions are often very
direct wave arrival time is Reflections bend upwards small amplitude arrivals, but
usually a straight line or curved on x-t plots. often easy to see because they
slightly downward. come in first.
t t t
x x x
* * *
Z
d i r e c t w a v e r e f l e c t i o n r e f r a c t i o n

9. Surface waves: Ground roll: Ground roll are Rayleigh
waves traveling at the surface of the earth. They are
usually the largest signals on a seismic record, but are
considered NOISE in most studies, because they only
yield information about shallow layers.
SLOWNESS: The slope of the arrival time vs distance
curve – or SLOWNESS - is 1/velocity of the wave at its
deepest point. The slowness is another name for the RAY
PARAMETER.

10. diffractions: when a wave hits a sharp boundary along a profile,
that boundary acts as a wave radiator, and a diffracted arrival is
generated. When diffracted seismic arrivals are plotted as arrival
time vs distance from the diffracting boundary, the arrivals are
hyperbolic in shape. Diffracted arrivals come from boundaries that
are NOT directly below the source and receiver.
This plot shows a
Diffracted arrivals
12
PROFILE of the arrival
10
time of a diffracted arrival
from point diffractors at
8
depth=1
6
travel time
4
depth=4
depths of 1.5 and 4.5 km
2
below the surface for a
0
0 5 10
velocity of 1.5km/s.
distance from diffractor

11. seismic profile arrivals
reflections
diffractions
geology
up

12. Attenuation and amplitude changes with distance.
Spreading: as waves radiate away from the source, the energy
spreads out. The energy of a wave is proportional to the square of
the amplitude, so as the energy spreads out, the amplitude decreases,
although the total energy remains constant.
spherical spreading: For body waves (P and S), the waves spread
out in spherical shells. Since the surface area of a sphere is
proportional to the square of the radius, the energy per unit area
(energy density) decreases as r2 , E=E0/r2 (), and the amplitude
decreases as r, A=A0/r.
cylindrical spreading: Surface waves, like ground roll, are confined
to the surface, so they spread out on a cylindrical shell. The area of
a cylindrical shell is proportional to r, so the amplitude of a surface
wave decreases only as 1/√r, and the energy density decreases as 1/r.

13. On a sphere, you might expect a surface wave to be just as
large at the antipode (the point directly opposite the source)
as it was at the source. This doesn't happen (although there
are signs of large amplitudes on Mars opposite large impact
craters) because of ATTENUATION and
SCATTERING.
Scattering of waves changes the direction of
propagation of part of the wave when it hits a rough barrier
or irregular surface, or any region where the elastic
constants change over a small area. The ENERGY is still in
the waves, but the DIRECTIONS of energy movement
changes. Scattering is very important in some situations.

14. Absorption: Attenuation is the result of absorption of energy. A
small amount of energy is lost from seismic waves to heat as the
wave moves through a material. Absorption takes the form:
- qr where I is a measure of energy called the intensity,
eI = I 0 e
is the exponential constant, q is the absorbtion coefficient, and r
is distance. q has units of dB/wavelength. So, at a given frequency,
the energy decreases with distance at a certain number of
dB/wavelength. Note that even if q is constant, the energy in a
high frequency wave will decrease faster than the energy in a low
frequency wave.
You often see attenuation in dB/λ (deciBells/ wavelength) for a
particular material.

15. An attenuation of 0.6 dB/λ implies that a signal with a ten km
wavelength will decrease in size by a factor of 2 in 100 km.
Note that if I said the material would lose one tenth of its
amplitude every wavelength, that does NOT mean that it will be
completely gone after ten wavelengths.
Attenuation by a constant dB/λ implies that waves with short
wavelengths will decrease in size faster than those with long
wavelengths.
This means that if we want to use seismic waves to see
deep into the earth, we need to use either long
wavelength waves , or very high amplitude sources.

16. Energy Partitioning
When a seismic wave hits an interface, it splits into different waves,
both reflected and refracted. In the most general case an incident P
wave will split into reflected P and S waves and refracted P and S
waves, although generation of some of these waves may be
forbidden by Snell's Law.
θπ1 θσ1
vp1, vs1
vp2, vs2
θπ2
θσ2

17. Note that the ray parameter is CONSTANT for any
ray -EVEN if the wave changes from a P wave to an
S wave. This means that the angle of reflection of a
p-wave equals the angle of incidence. In general:
sin(q p1 ) sin(q p 2 ) sin(q s1 ) sin(q s2 )
p= = = =
v p1 vp2 v s1 v s2
θπ1 θσ1
vp1, vs1
vp2, vs2
θπ2
θσ2

18. The energy in these waves depends on the densities and velocities of
the two materials, as given by Zoeppritz. For NORMAL incidence
- that is the seismic ray hitting an interface perpendicular to that
interface. The refracted amplitude is given by:
Arfr 2Z1
= where Z i = r iVi is called the IMPEDANCE.
Ai Z2 + Z1
What happens to the refracted amplitude as ρ2 approaches zero ?
Does this happen?
What happens to the energy of the refracted wave as ρ2
approaches zero ?

19. Remember – ENERGY is conserved – not amplitude - and energy
changes as the SQUARE of the amplitude. The formula for
energy (or INTENSITY) of refracted arrivals is:
I rfr 4Z1Z 2
= where Z i = r iVi
Ii (Z2 + Z1 )2
Notice how the intensity of the refracted wave changes as ρ2
approaches zero.
Wave amplitudes in low-density or low-velocity materials become
large – which is why tsunamis get big near shore and why you
shouldn’t build a house on soft fill in a place prone to earthquakes.

20. Careful consideration needs to be given to the problem to be
investigated. Some geological problems just can't be solved with
seismology, while others are best attacked by looking at reflections
from layers and others by looking a refractions.
The deeper into the earth you need to see, the stronger your source
energy and lower the frequency of the source must be. This is the
problem of PENETRATION. The other side of the coin is
RESOLUTION.
If the feature you are trying to study has dimensions of less than
about 1/4 of a wavelength of your seismic signal, then you won't be
able to RESOLVE it in the seismic data.

21. SEISMIC SOURCES
We need to put enough energy into the seismic waves to be
sure we can see the necessary signals at our receivers.
• Earthquakes: Signals from earthquakes have large
amplitudes, but the source can be complicated by energy
coming from many places and poorly known origin time.
• Underground Nuclear explosions. Great seismic
sources, but wildly unpopular.
• Conventional explosives: Can be made as large as
desired but somewhat unpredictable amplitudes, expensive,
and dangerous. Permitting and drilling required. Explosives
are used for refraction work and used to be used for
reflection profiling.

22. • Conventional
explosives: Can be
made as large as
desired but have
somewhat unpredictable
amplitudes, expensive,
and dangerous.
Permitting and drilling
required. Explosives
are used for refraction
work and used to be
used for reflection
profiling.

23. Airguns fire a pulse of air
into the water as a marine
seismic source. A problem
is that the bubble of air
oscillates generating a
complex “bubble pulse.”
Many guns are used to kill
the bubble pulse and add
more energy.
QuickTimeª and a
TIFF (Uncompressed) decompressor
are needed to see this picture.

24. • Shotgun source. These
sources use 12 gauge
shotgun shells to send a
signal into the earth.

25. • Vibroseis: vibroseis is the
most commonly used seismic
source on land. Rather than
send an impulse into the
ground, these trucks send a
“chirp” into the ground tha
must be processed to see the
seismogram.
The big advantages are
excellent control, no loud
noise, minimal permitting, and
no drilling.

26. • hammer: We will use a
hammer.
The hammer source is
good for small-scale
shallow studies.

27. SEISMIC INSTRUMENTS
What is a seismometer?
What is meant by "motion of the ground"?
What about tides and gravity?
What does a seismometer measure?
Displacement?
Velocity?
Acceleration?
Stress?
Strain?
Propagation velocity?

28. A TRANSDUCER: changes one type of energy into another
[motors, generators, etc.]. In seismometers we change ground
shaking into electrical signals.
There are several ways to do this. If we use a magnet surrounded by
a coil of wire to generate an electric signal when the magnet and coil
are moved with respect to each other, then the VELOCITY of the
coil relative to the magnet gives us the signal.
This gives a measurement of the velocity of the ground motion IF
the frequency is high - what if it isn't? NOTE: this is NOT the
propagation velocity - it's the PARTICLE velocity!
Nearly all land seismic sensors used for exploration have velocity
transducers.

29. To measure the motion of the earth, we need to be able to measure
the motion of some point connected to the earth RELATIVE to
some point that is NOT moving with the earth.
The simplest way we know of to do this is with mechanical
oscillators, of which there are two basic types: masses attached to
springs (used for detection of VERTICAL motion of the earth), and
pendulums, used to measure HORIZONTAL motion of the earth.

30. While these two instruments look pretty much alike, one measures
horizontal motion in-and-out-of the page, and one measures
vertical motion.
The SIGNAL is the relative motion between the frame fixed to the
earth and the mass. The frame moves with the ground at all
frequencies.
At high frequencies, the mass is stationary, or inertial, so the
relative position of the mass relative to the frame is a measure of
the displacement of the ground.
At very low frequencies, the mass is no longer stationary. Does it
still move relative to the frame? What would cause a change in
the location of the mass relative to the frame at very low
frequencies?

31. A horizontal seismometer like that shown on the previous
pages can be centered easily in the same way that you
would adjust a swinging gate to always remain closed.
How do you adjust a gate to always swing to the “closed”
position?
Which seismometer measures vertical motion and which
horizontal motion?
A gravity meter uses the same design as the vertical
seismometer. In the case of the gravity meter, the
displacement of the mass (stretch of the spring) is a
measure of what parameter?
If the seismometer frame is tilted, what is the effect on the
seismometers?

32. This seismometer is called a GEOPHONE. The motion of
the ground is detected by a coil of wire moving through a
magnet attached to the frame generating a current. If the
coil is not moving relative to the frame, no signal is
generated. Geophones sense the VELOCITY of the ground
at high frequency.

33. What are desirable characteristics of a seismometer?
Fidelity:
A seismometer should yield the motion of the ground with high
"fidelity", where fidelity is a measure of accuracy. It should be
possible to reconstruct the motion of the ground from the recorded
signal. Any distortions should be linear, or at worst, well known.
Noise:
A seismometer should have the lowest possible noise level
Bandwidth: It should have a large frequency band across which it
has a low noise level.
Dynamic Range:
Large dynamic range to record both very large and very small
signals without distortion.

34. Another type of seismic transducer consists of two plates, one
connected to the seismic mass and one to the frame.
As the distance between the plates changes the capacitance
changes, and we get a measure of the DISPLACEMENT of the
mass relative to the frame of the instrument.
How does this relate to the particle motion?

35. REFRACTION SEISMOLOGY METHODS
The DIRECT WAVE travels straight from the seismic source
to the seismometer. The x-t plot for the direct arrival looks
like:
slope= ray parameter
time
=slowness
=1/propagation velocity
=1/v0
distance from shot
up
v
0

36. As soon as we let velocity change with depth in a flat model, the
x-t graph will no longer be a straight line, as the ray path between
any two points will no longer be a straight line, in general.
If the earth is made up of constant-velocity layers, the x-t plot will
be made up of a sequence of straight lines, one for each layer IF
the velocity always increases with depth.
When we have a single horizontal interface separating two layers
that have constant velocities, it's relatively simple to describe the
resulting refracted arrival.
The ray that will arrive at the geophones along the critical path
(horizontal in the lower layer) hitting the lower layer at the critical
angle, thus:

37. slope= ray parameter
=slowness
time =1/propagation velocity
=1/v1
distance from shot
up v
0
v
1
The critical angle is important here. It allows us to determine
the travel time of the refracted arrival, and from there to
calculate the depth to the 2nd layer.

38. The travel time from source to receiver for a refraction through a
flat 2-layer model is:
2 2
2h1 v - v
2 1x
t rfr = + .
v 2v1 v2
The trfr equation is much simpler than it looks, since x only appears
in the 2nd term, it is a straight line with slope equal to the ray
parameter and a y-intercept equal to the first term.
Since we can measure the y-intercept of the refraction (called the
intercept time), and the two velocities can be measured from the
slopes of the direct and refracted arrival, we can solve the above
equation for h1, and obtain the depth to the layer:

39. ti v2 v1
hi = 1/ 2
, where ti is the y intercept time of the refraction arrival.
2 (v - v )
2 2
2 1
Evaluation of refraction data using these formulae, and their
expansion to multiple layers, has been used extensively - so much
that many people have been given the impression that the earth is
made up of constant velocity layers! While the models often fit the
data quite well, so do models with gradients and low velocity
zones.
Great care must be taken in over interpreting model results.

40. A refracted arrival has a slope of 5 ms/10 m. The direct layer
arrival has a slope of 2 ms/m. The intercept time of the
refracted arrival is 20 ms.
• What is the velocity of the upper layer?
• What is the velocity of the lower layer?
• What is the depth to the lower layer?

41. HOMEWORK FOR Tuesday, Apr 8, 2008:
Find the appropriate formulas in text books or on the Web to
determine the critical distance, crossover distance, and intercept
time for a single-layer model with zero dip where the upper
layer has a velocity of 500 m/s and the depth to the lower layer is
5 m. Construct a graph of critical and crossover distance and
intercept time vs. velocity of the lower layer between 510 m/s
and 2000 m/s. What do these results imply in terms of geophone
spacing and detection of the two layers?
Hint: Find the formulas and enter them into a spread sheet or
Matlab, then use the formulas to make the graphs. Come see me if
you have problems.

42. reflection
critical distance
crossover distance
distance
ξ χ ριτ
δ ε πτη θχ
v1
v2
The critical distance is the distance from the source where the refraction can
first be observed. Notice that at the critical distance the reflection from the layer
and the refraction have the same travel time AND the ray parameter (slope) of
both the reflection and the refraction are the same. The cross-over distance is
the distance where the direct arrival and the refracted arrival come in at the same
time.

45. IN CLASS PROBLEM: You observe the following x-t
.
refraction plot:
50
40
30
t rav el t im e, m s
20
10
0
0 5 10 15 20
dist ance f rom shot , m

46. The closest geophone is 5 m from the shot point. Note that when
you extrapolate the first arrival time back to zero distance it
doesn't go through the origin.
3) What is the velocity of the layer observed?
5) Is this the velocity of the surface layer?
3) What limits can you place on the thickness and velocity of the
surface layer?
4) How could you prevent this problem and get the surface layer
model parameters “exactly”?