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

GG 450
April 3, 2008
Refraction, Diffraction, Energy,
Sources, and Sensors
Critical Angle:
Recall:
sin 
 p (Snell's law)
v
When  = 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:
.
1 1
ic
2
v
  sin
v
In some seismic modeling, the critical angle is
important, as we shall see..
c
v1
v2
v1
c  sin
v2
1
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.
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).
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.
Direct Wave: Travels at the
p-wave velocity of the
uppermost layer of the
ground, directly to the
receiver. Direct waves come
in first close to the source,
but often disappear at larger
distances or are lost in
earlier, faster arrivals. The
direct wave arrival time is
usually a straight line or
curved slightly downward.
Reflections:
Reflections arrive from
sharp changes in
velocity (actually
impedance, v) below.
When plotted on a x-t
plot, the first reflection
arrival time is
asymptotic to the direct
arrival. Reflections
bend upwards on x-t
plots.
t
t
t
x
x
x
*
*
Refractions (head waves):
Refractions are arrivals from
faster deeper layers that
arrive first at larger
distances. They are usually
straight or bending slightly
downwards with increasing
distance on x-t plots.
Refractions are often very
small amplitude arrivals, but
often easy to see because
they come in first.
*
Z
direct wave
ref lect io n
ref ract ion
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.
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.
Diffracted arrivals
12
travel time
10
8
depth=1
6
depth=4
4
2
0
0
5
distance from diffractor
10
This plot shows a
PROFILE of the arrival
time of a diffracted
arrival from point
diffractors at depths of
1.5 and 4.5 km below
the surface for a
velocity of 1.5km/s.
seismic profile arrivals
reflections
diffractions
geology
up
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.
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.
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:
I  I 0e
qr
where I is a measure of energy called the
intensity, 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/ (deciBells/ wavelength)
for a particular material.
An attenuation of 0.6 dB/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/ 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.
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.
p1 s1
p2
s2
vp1, vs1
vp2, vs2
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  p1 sin  p 2  sin s1 sin s2 
p



v p1
vp2
v s1
v s2
p1 s1
p2
s2
vp1, vs1
vp2, vs2
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

Ai
Z2  Z1
where Zi  iVi is called the IMPEDANCE.
What happens to the refracted amplitude as 2 approaches
zero ?
Does this happen?
What happens to the energy of the refracted wave as 2
approaches zero ?
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
4Z1Z2

Ii
(Z2  Z1 )2
where Zi  iVi
Notice how the intensity of the refracted wave changes as 2
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.
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.
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.
• 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.
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.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
• Shotgun source. These
sources use 12 gauge
shotgun shells to send a
signal into the earth.
• 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.
• hammer: We will use a
hammer.
The hammer source is
good for small-scale
shallow studies.
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?
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.
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.
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?
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?
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.
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.
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?
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
=slowness
=1/propagation velocity
=1/v0
distance from shot
up
v
0
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:
slope= ray parameter
=slowness
=1/propagation velocity
=1/v1
distance from shot
up
v
v
0
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.
The travel time from source to receiver for a refraction
through a flat 2-layer model is:
2h1 v  v
x
t rfr 
 .
v 2v1
v2
2
2
2
1
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
 ray parameter and a y-intercept equal to the first term.
to the
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:
ti
v2 v1
hi 
, where ti is the y intercept time of the refraction arrival.
1/ 2
2
2
2 v  v 
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.
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.
1) What is the velocity of the upper layer?
2) What is the velocity of the lower layer?
3) What is the depth to the lower layer?
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.
reflection
critical distance
crossover distance
distance
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.
IN CLASS PROBLEM: You observe the following x-t
refraction plot:
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.
1) What is the velocity of the layer observed?
2) 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”?