Transcript No Slide Title

How do waves move from one place to another?
Our initial goal is to develop an understanding
of the different types of events that appear in a
shot record
and to understand how their travel times (t) vary with
source receiver offset (x).
A shot record is a recording of ground movements
produced by a single shot (mechanical disturbance created
at some point on or near the earth’s surface). The recording
is made at several locations usually along a straight line
extending in either or both directions away from the
source.
How will the travel times of the direct arrival
vary with offset? What will a direct arrival look
like in a time distance plot?
Direct Arrival
A shot record
Also need to consider the type of direct arrival ...
Time Distance Plot
The reflection event and its time distance relationships
The direct arrival has the relationship of an asymtote
to the arrival times of the reflection event.
Consider the basic reflection time-distance relationship
t2 
4h12
x2
V12

4h12
V12
2h1
which is the time intercept.
When x = 0, t  2 or t 
V1
V1
2
When x becomes very large with respect to the
thickness of the reflecting layer, the X2/V2 term
becomes much larger than the 4h2/V2 term so
that
x2
x
t 
or t 
V1
V12
2
The single layer refraction time-distance relationship
- but first -
The c’s cancel out and we have ...
One of our assumptions -
Assume V1 < V2 < V3
because sin(/2) = 1
V1
V2
The critical or minimum distance
Basic time-distance relationships for the refracted wave Single horizontal interface
1) straight line
2) Refraction and reflection arrivals coincide at
one offset
3) Refraction arrivals follow a straight line with
4) slope 1/V2, where
5) 1/V2 is less than 1/V1
Diffraction Events
Diffractions, like reflections are hyperbolic in a timedistance plot. They are usually symmetrical about the apex.
Diffractions usually arise from point-like discontinuities and
edges ( for example the truncated edges of stratigraphic
horizons across a fault).
This is a shot record from the Granny Creek oil field in Clay
Co. WV. What kinds of acoustic events can you recognize in
this shot record? Where is the source?
For next time do ray-trace exercises I, II and III
INTRODUCTORY RAY TRACING EXERCISES (I -VI)
GENERAL INSTRUCTIONS
These exercises are designed to illustrates some of the basic
characteristics of wave propagation in a single layer model
use ray-tracing concepts to determine the arrival times of
particular events. These exercises require that you construct
the time-distance plot for the given model (I - VI). In
addition to constructing the time-distance plots,
Be sure to do the following
1) label all plotted curves,
2) label all relevant points, and
3) in a paragraph or so discuss the significance and origins of
the interrelationships portrayed in the resultant time-distance
plots
In exercise II for example, how do you account for the
differences in the two reflection hyperbola? How is their
appearance explained by the equations derived in class for
the reflection time-distance relationship.
In Exercise III, explain the differences observed in the
arrival times of the reflection and diffraction observed in the
shot record.
As noted earlier - accurately portray the arrival
times at different offsets or surface locations.
Your plots should serve as an accurate
representation of the phenomena in question.
If there is time remaining in the class today
we’ll spend it working on Exercises I-III.
For next time -
•Review Chapter 2 and be sure to pay
particular attention to discussions of reflection
and transmission coefficients and the Wave
Attenuation and Amplitude section.
•Work on Exercises I-III and bring
them to class next time along with
any questions you have.
**They are not due next time just
be prepared to ask questions about
them
Discussion of Chapter 2:
What happens to the seismic energy generated by the
source as it propagates through the subsurface?
Basic ConceptsEnergy - The ability to do work. It comes in two forms potential and kinetic
Work expended (W) equals the applied force times the
distance over which an object is moved.
Power is the rate at which work is performed.
As a mechanical disturbance or wavefield propagates
through the subsurface it moves tiny particles back and
forth along it path. Particle displacements are continually
changing. Hence, it is more appropriate for us to consider
the power, or the rate at which energy is being consumed
at any one point along the propagating wavefront.
W (the work)  Fx (force times displaceme nt)
dW
Power 
dt
The rate of change of work in unit time.
dW dFx 
dx

 F  Fv
dt
dt
dt
This force is the force exerted by the seismic wave at
specific points along the propagating wavefront.
and since force is pressure (p) x area (A), we have
dW
Power 
 Fv  pvA
dt
The power generated by the source is PS. This
power is distributed over the total area of the
wavefront A so that
PS  pvA
We are more interested to find out what is going
on in a local or small part of the wavefield and
so we would like to know * Note we are trying to quantify the effect of wavefront
spreading at this point and are ignoring heat losses.
PS
pv 
A
Over what surface area is the energy
generated by the source distributed?
Area of a hemisphere is 2R2 hence -
pv 
PS
2R 2
This is the power per unit area being dissipated along
the wavefront at a distance R from the source. (Recall
p is pressure, v is the particle velocity, PS magnitude of
the pressure disturbance generated at the source and R
is the radius of the wavefront at any given time.
In the derivation of the acoustic wave equation (we’ll
spare you that) we obtain a quantity Z which is called the
acoustic impedance. Z = V, where  is the density of the
medium and V is the interval velocity or velocity of the
seismic wave in that medium.
Z is a fundamental quantity that describes reflective
properties of the medium.
We also find that the pressure exerted at a point along the
wavefront equals Zv or Vv, where v is the particle
velocity - the velocity that individual particles in the
disturbed medium move back and forth about their
equilibrium position.
pv   Vv  v
pv  Zv 2
Combining some of these
ideas, we find that v, the
particle displacements vary
inversely with the distance
traveled by the wavefield R.
2
Zv 
v
Ps
2R 2
Ps
Z 2R 2
1
Ps
v
R Z 2
•We are interested in the particle velocity variation with
distance since the response of the geophone is
proportional to particle (or in this case) ground
displacement. So we have basically characterized how
the geophone response will vary as a function of distance
from the source.
•The energy created by the source is distributed over an
ever expanding wavefront, so that the amount of energy
available at any one point continually decreases with
distance traveled.
•This effect is referred to as spherical divergence. But in
fact, the divergence is geometrical rather than spherical
since the wavefront will be refracted along its path and its
overall geometry at great distances will not be spherical in
shape.
The effect can differ with wave type. For example, a
refracted wave will be confined largely to a cylindrical
volume as the energy spreads out in all directions
along the interface between two intervals.
z
R
The surface area along the leading edge of the
wavefront is just 2Rz.
Hence (see earlier discussion for the hemisphere)
the rate at which source energy is being expended
(power) per unit area on the wavefront is
Ps
Zv 
A
2
Remember Zv2 is just
Ps
Zv 
2Rz
2
Following similar lines of reasoning as before,
we see that particle velocity
v
Ps
R
v
Ps
R
The dissipation of energy in the wavefront
decreases much less rapidly with distance traveled
than does the hemispherical wavefront.
This effect is relevant to the propagation of waves in
coal seams and other relatively low-velocity intervals
where the waves are trapped or confined. This effect
also helps answer the question of why whales are
able to communicate over such large distances
using trapped waves in the ocean SOFAR channel.
Visit
http://www.beyonddiscovery.org/content/view.page.a
sp?I=224 for some info on the SOFAR channel.
See also -
http://www.womenoceanographers.org/doc/MTolstoy/Les
son/MayaLesson.htm
To orient ourselves, you can think of the particle
velocity as the amplitude of a seismic wave recorded
by the geophone - i.e. the amplitude of one of the
wiggles observed in our seismic records.
The amplitude of the seismic wave will decrease, and unless
we correct for it, it will quickly disappear from our records.
Since amplitude (geophone response) is proportional to
the square root of the pressure, we can rewrite our
divergence expression as
Ps
Ar 
r
As
Ar 
r
Thus the amplitude at distance r from the source (Ar)
equals the amplitude at the source (AS) divided by the
distance travelled (r).
This is not the only process that acts to decrease the
amplitude of the seismic wave.
Absorption
When we set a spring in motion, the spring oscillations
gradually diminish over time and the weight will cease to move.
In the same manner, we expect that as s seismic wave
propagates through the subsurface, energy will be consumed
through the process of friction and there will be conversion of
mechanical energy to heat energy.
We guess the following - there will be a certain loss of amplitude
dA as the wave travels a distance dr and that loss will be
proportional to the initial amplitude A.
i.e.
dA  AS dr
How many of you remember how to solve such
an equation?
dA(r )   AS dr
dA(r )
 dr
AS
 is a constant referred to
as the attenuation factor
In order to solve for A as a function of distance traveled (r)
we will have to integrate this expression In the following
discussion,let
A0  AS
A dA
r
A0 A  0dr
ln A  ln A0  r
ln A  ln A0  r
A
ln
 r
A0
ln
e
A
A0
 e r
A
 e r
A0
A  A0er
Mathematical Relationship
A(r )  A0er
Graphical Representation
A(r )  A0er
 - the attenuation factor is also a function of additional terms  is wavelength, and Q is the absorption constant


Q
1/Q is the amount of energy dissipated in one
wavelength () - that is the amount of mechanical
energy lost to friction or heat.


Q
 is also a function of interval velocity, period and frequency


r
A(r )  A0e Q


A(r )  A0e QV
r
 is just the reciprocal of the frequency so we
can also write this relationship as

f
A(r )  A0e QV
r
Smaller Q translates into higher energy loss or
amplitude decay.
A(r )  A0e

f
QV
r
increase f and decrease A
Higher frequencies are attenuated to a much
greater degree than are lower frequencies.
When we combine divergence and absorption we get the
following amplitude decay relationship
A0 r
A(r ) 
e
r
The combined effect is rapid amplitude decay as the
seismic wavefront propagates into the surrounding
medium.
We begin to appreciate the requirement for high
source amplitude and good source-ground coupling to
successfully image distant reflective intervals.
But we are not through - energy continues to be
dissipated through partitioning - i.e. only some of the
energy (or amplitude) incident on a reflecting surface
will be reflected back to the surface, the rest of it
continues downward is search of other reflectors.
The fraction of the incident amplitude of the
seismic waves that is reflected back to the surface
from any given interface is defined by the reflection
coefficient (R) across the boundary between layers
of differing velocity and density.
Z 2  Z1
R

Ainc Z1  Z 2
Arefl
Z 2  Z1
R

Ainc Z1  Z 2
Arefl
Z1 and Z2 are the impedances of the bounding layers.
 2V2  1V1

1V1   2V2
Z 2  Z1
R
Z1  Z 2
The transmitted wave amplitude T is
T 1 R
Z1  Z 2  Z 2  Z1 

T
 
Z1  Z 2  Z1  Z 2 
2 Z1
T
Z1  Z 2
Hand in your discussion of problems 1 and 2
At a distance of 100 m from a
source, the amplitude of a P-wave is
0.1000 mm, and at a distance of 150
m the amplitude diminishes to
0.0665 mm. What is the absorption
coefficient of the rock through
which the wave is traveling?
(From Robinson and Coruh,
1988)
Any Questions?