2 depthconversion
TRANSCRIPT
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Creating Depth Images from Seismic Records
The following chapters develop the techniques used to
image the subsurface with seismic reflection data acquired
by a 3D seismic survey. By imaging, we meanprocessing
and displaying the recorded seismic signals on a computer
to facilitate a structural interpretation of the Earthssubsurface.
Here we explain the elementary imaging technique called
depth conversion. Depth conversion allows us to obtain a
simplistic answer to the question, How deep is the
reflector? To produce a depth estimate, geophysicists map
the reflection data, originally recorded in time, into the
depth domain. The interpreter can use this depth
information to deduct the vertical distances to subsurface
horizons and to calculate volumetric estimates of
hydrocarbon reservoirs.
We start with data acquisition geometry in which source
and receiver locations coincide. This situation is referred toas zero-offset acquisition. Second, we explain the simplest
imaging step, the vertical stretching of the observed seismic
time data to depth. Third, we explore the limitations of this
simplistic depth-conversion process as an imaging step and
introduce the seismic migration procedure.
With this foundation, we describe the process of zero-offset
migration in a following chapter and recognize
shortcomings of the zero-offset assumption. Together, this
collection of chapters presents the motivation and basic
principles of depth migrating seismic data while critically
observing the intrinsic limitations of the process. We then
illustrate a pair of universally employed depth migration
algorithms used to create an image of the subsurface usingboth zero and nonzero offsets seismic data.
What ref lect ions can teach u s about
the subsu r face?
Consider the idealized Earth model shown in figure 1. This
Earth model contains only two layers. One of the layer
properties of interest in seismology is the seismic-wave
compressional velocity measured in each layer. This
intervalvelocity in the first layer is called VInt(1) and the
second layer has the interval velocity VInt(2). A plane,
horizontal interface, separates both constant-velocity slabs.
The interval velocity is the key component in the depthconversion and allows converting from time to depth.
Figure 1: A simple 2-layer Earth model. The source is located
at the surface, the receiver on the horizontal interface.
Figure 1 shows an acquisition geometry that makes it feasible to
determinate the acoustic interval velocity VInt(1) in the first layer
With this arrangement, it is straightforward to determine the
velocity: First, place an acoustic source at the surface and an
acoustic receiver at the interface between the two layers, at a
depth z(1). Second, measure the transit time between the
surface source and the buried receiver as t(1). We then define
the interval velocity as
t(i)
z(i)(i)VInt
(1)
where, in this case for the first layer, i = 1.
This interval-velocity-determination procedure uses an acoustic,impulsive surface source. This source, in a constant-velocity
medium, produces a down-going spherical wave front as shown
in figure 2. The figure also shows seismic rays that propagate
perpendicular to the wave fronts1. While nature creates the wave
front, we create the conceptual idealization of the rays. Rays are
a very convenient tool to comprehend the more complex shape
and evolution of wave fronts. Think of rays as the skeleton of the
seismic wave field. Studying the framework of the wave fields
bones allows a more thorough and quick understanding of more
convoluted phenomena of wave propagation.
1 This statement is simplified and only true if the velocity changes
as a function of position in the subsurface and not as a function of
direction. Researchers call this latter situation seismic
anisotropy.
VInt(1)
Surface of Earth
VInt(2)
z
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2 What reflections can teach us about the subsurface?
Figure 2: A down-going wave front from a surface source
shown at a particular instant of time after excitation of
the source. The blue lines denote seismic rays.
In many instances, we are unable to bury the receivers in
the subsurface and are required to determine the depth to
the reflecting interface from surface seismic informationalone. Figure 3 shows the simplest possible arrangement of
source and receiver at the surface. Geophysicists call this
configuration zero-offset geometry, as there is zero
separation (offset) between the surface source and receiver.
Figure 3: Zero-offset geometry. The thick blue ray
illustrates the incident wave. The reflection coefficient
determines the strength of the reflected and transmitted
wave fields.
Figure 3 also illustrates the basic principle of reflection
seismic exploration. At time t= 0, the idealized source
emits an impulsive acoustic wave that travels through the
Earth model. Upon incidence on the interface between the
two layers, the wave splits up into a downward and an
upward traveling part. The downward propagating wave is
the transmittedwave and typically has stronger amplitude
than the upward propagating reflectedwave. The receiver
on the surface records the reflection while the transmitted
part of the wave field continues through the interface into
the deeper layer.
Figure 4 illustrates an idealized seismic trace that records the
reflected impulse as a small red tick at time t0 (the 0 denotes a
zero source-to-receiver separation for the roundtrip travel time).
The amplitude measured on the seismic trace is proportional to
the reflection coefficient between layers 1 and 2,R(1,2). We
define the reflection coefficient,R, as the ratio of the up-coming
amplitude to the incident amplitude at the interface. The
amplitude at the receiver only relative and not equal to the
reflection coefficient because there are many other phenomenathat also alter the amplitude observed on the surface. A later
chapter presents some of these other wave-propagation
phenomena.
Figure 4: Recorded seismic trace. The impulse recorded as a
function of time is proportional to the reflection coefficient.
The reflection coefficient that describes the bounciness of an
interface depends on the rock properties in layer 1 and 2. One
can derive the specific functional relationship from basic
physical principles, such as requiring the two blocks to stay in
welded contact at the interface and not to start slipping relative
to each other or develop voids in the subsurface. For a
perpendicularly incident wave, the reflection coefficient yields:
))(V))(V
))(V))(V),R(
IntInt
IntInt
1(12(2
1(12(221
(2)
Equation 2 shows that the reflection coefficient depends upon the
velocity of the upper and lower layer, and the densities () of
both layers. More precisely, relative products of interval
velocities and densities govern the reflection coefficient. This
product of the interval velocity and the density is termed
impedance. Zero-offset reflections occur at locations of
differences in impedances, and the reflection strength is
proportional to the relative difference in impedance.
In our desire to determine the depth of the reflecting horizon,
only one portion of the wave field is of interest. That portion
travels from the source to the reflecting horizon and returns to a
coincident receiver. For diagrammatic purposes, figure 3 doesshow a slight separation between the source S, and the receiver
R. Likewise, again for visual clarity, figure 3 shows a slight
separation of the roundtrip ray path. In fact, the downward-going
and upward-going zero-offset ray paths are coincident. A
subsequent section shows that the ray path shown is an
idealization of the real world. For example, we will find that
many locations along the reflecting interface contribute to the
total energy received at the surface receiver, although the path
shown in the figure does represent the path associated with the
dominant portion of the reflected energy.
VInt(1)
VInt(2)
z
z
VInt(1),(1)
VInt(2),
(2)
1 R(1,2)
Amplitude
t
)2,1(R
t0
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Creating Depth Images from Seismic Records 3
From Time Measurements to Depth
EstimatesWe now determine the depth to the seismic reflector.
The general time-distance relationship is:
time).traveldistance/(velocity (3)
Or, in terms of the interval velocity,
me)(Travel Ti
z)(VInt 1
(4)
Solving for the depth in our simplistic zero-offset
experiment with a single horizontal reflector, we have
2
11 0
t)(V)z( Int
(5)
The 2 in the denominator ofequation (5) is required
because the observed time, t0, is the two-way roundtrip
travel time from the surface to the reflector and back to thecoincident receiver.
Example: Constant-Velocity DepthConversion
If we take the case of an interval velocity of 3,000 m/s and a
roundtrip travel time of 2.0 seconds then
.00032
sec2sec0003m,
m/,
zDepth
(6)
Variable-Velocity Depth ConversionClearly, equation (5) is only appropriate for a very
simplistic constant-velocity world. A depth-dependent
velocity represents a much more realistic subsurface
situation.
Figure 5: A zero-offset experiment in a more realistic,
depth-dependent velocity model.
In Figure 5 we see a series of layers, each with a varying
thickness zand an interval velocity ofVInt. In this
configuration, the following equation gives the roundtrip travel
time for the figures reflection ray path.
)(V
)z(
)(V
)z(
)(V
)z()(t
IntIntInt 3
32
2
22
1
1230
(7)
By knowing the round-trip travel times between the surface and
each of the layer interfaces, we can calculate the thickness of
each of the slabs. For example, if we have already determined
the thickness ofz(1) and z(2), then the following equation
gives the thickness ofz(3).
)(V
)z(
)(V
)z()(t
)(V)z(
IntInt
Int
2
22
1
123
2
33 0
(8)
Probing the subsurface with a single source-receiver pair is not
sufficient to determine the lateral continuity of the reflectors. To
overcome this shortcoming, we can expand the experiment
illustrated in figure 5 to that offigure 6. In Figure 6, we haveadded additional coincident source and receiver pairs. Each
source fires independently of the other sources. This experiment
with a series of coincident sources and receivers represents a
multiple-trace, zero-offset acquisition.
Figure 6: A multiple-trace zero-offset acquisition.
From this sequence of zero-offset recordings, we obtain a zero-
offset seismic section as shown in figure 7.
Figure 7: Idealized traces in a zero-offset seismic section with
multiple source-receiver pairs.
In this figure, the tick-marks denote each of the roundtrip travel
times. With knowledge of the interval velocities, we can use
VInt(1)
VInt(2)
VInt(3)
VInt(4)
z(1)
z(2)
z(3)
z(4)
z
VInt(1)
VInt(2)
VInt(3)
VInt(4)
z(1)
z(2)
z(3)
z(4)
z
x
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4 What reflections can teach us about the subsurface?
equation (8) (generalized for any number of reflections) to
convert the vertical axis to depth as in figure 8.
Figure 8: Depth-converted zero-offset seismic section.
Figure 8provides an approximate depth-converted image of
the subsurface because it portrays the depths to each of the
reflecting horizons. In a subsequent section, we investigate
the imaging shortcomings of the depth conversion of the
zero-offset shooting.
So, did we solve the depth conversion problem? Clearly, the
investigated Earth models are too simplistic and we need togo beyond constant velocity layers separated by horizontal
interfaces. More importantly still, we did not discuss how to
obtain the crucial interval velocity parameter. While
practitioners know this quantity in many physical contexts
such as medical imaging, the subsurface rock velocity can
change significantly and cannot easily be determined in
situe. Thus, we now turn our attention to the estimation of
the interval velocities that are required as a part of the depth
conversion process.
Figure 9: Laterally interpolated version of previous
figure.
Can we estimate interval velocit ies from
the surface?
While we can use seismic recordings to determine roundtriptravel times, we do not yet know the values of the interval
velocities in the subsurface. We require additional, surface-
acquired information to obtain the interval velocities to
estimate the depths to the reflecting interfaces (i.e., depth
conversion). The primary ingredient for the interval velocity
determination is a set of surface observations obtained at a
variety of offset distances between the sources and
receivers. Figure 10 illustrates such a nonzero-offset surface
geometry. The following demonstrates the basic techniques
of interval velocity estimation from nonzero-offset
geometries.
Figure 10: Simple non-zero-offset acquisition geometry.
Consider two possible ray paths, as shown figure 11. In this
figure, the two legs of the zero-offset raypath, A and A have
equal lengths. Likewise, the path length for B equals that of B.
Figure 11: Zero and far-offset ray paths.
The A - A' ray path is the zero-offset ray path while the B - B' is
a nonzero-offset ray path. In order to calculate the relationship
between the roundtrip travel times observed at zero offset and at
nonzero offset x, we use figure 12, which shows ray path lengths
that are equivalent to the lengths of the paths in figure 11. Figure12 mirrors the situation for the upward-traveling ray paths at the
reflector horizon in order to produce equivalent-length ray paths.
Figure 12: Equivalent ray paths B - B'.
From this new, right-triangle geometry in figure 12, Pythagorean
Theorem provides the following relationship between the ray
path lengths:
x
x
Offset =xOffset
A
B
A
B
VInt
Offset = xOffset
A
B
A
B
A B
2A
VInt
Offset = xOffset
B
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Creating Depth Images from Seismic Records 5
,22 222 OffsetxA)(B)( (9)
where the hypotenuse is 2B and the catheti of the right
triangle are 2A andxOffset.
In order to cast equation (9) in terms of the observed
roundtrip travel times, divide this equation by the interval
velocity to obtain
2
2
2
2
2
2 22
Int
Offset
IntInt V
x
V
A)(
V
B)(
(10)
The first term is the square of roundtrip travel time
observed at offsetxOffset, represented as tx. The second term
is the zero-offset roundtrip travel time, represented as t0.
With this nomenclature, equation (10) becomes
2
2
2
0
2
Int
Offset
xV
xtt
Offset
(11)
Taking the square root of each side provides
2
2
2
0
Int
Offset
xV
xtt
Offset
(12)
This equation is the constant-velocity form of what is
termed the Normal Move-Out equation, also known as the
NMO equation. It describes how the travel time of a seismic
reflection changes if the receiver is moved out away from
the source.
We can solve equation (12) for the interval velocity
2
0
2
2
tt
xV
Offsetx
Offset
Int
(13)
For the following discussions, please keep in mind that this
equation is accurate only under rather restrictive conditions.
In particular, recall that the derivation of this equation
assumed straight rays (i. e., constant velocity) and a
horizontal reflector.
Figure 13: Observed zero and non-zero offset travel times.
Figure 13illustrates the procedure to record seismic reflections
at two offsets. Seismic data processors use the observed travel
time differences to calculate the interval velocity.
We now consider a real-world example. Figure 14 shows twoseismic traces, the first one at zero offset and the second one at
an offset of 2650 feet. These two traces record the roundtrip
travel time for a seismic experiment in a marine setting. The
strongest amplitude reflections are the water bottom reflections.
For our analysis, we pick these arrivals precisely and provide the
measured times in the figure.
Figure 14: Observed water-bottom two-way travel times.
Inserting the round-trip travel times into equation (13), we have
.ft./,
..
tt
x)(V
Offsetx
Offset
Int
sec1855
008207222650
1
22
2
2
0
2
2
(14)
This value for the water interval velocity is in very good
agreement with the nominal water velocity of 5,000 ft/s.
To summarize, in order to determine the depth of the reflector,
we obtain the interval velocity from equation (13) from
knowledge of the zero-offset round-trip travel time, t0, and a non
zero offset round-trip travel time, tx, along with the offset itself,
xOffset. Then, knowing this estimate of the interval velocity,
Offset = xOffset
VInt
t0 txOffset
0 Feet 2650 Feet
2.008 s 2.072 s
2.0
2.2
Time(s)
xOffset
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6 What reflections can teach us about the subsurface?
substitute it and the value of the zero-offset round trip travel
time t0 into equation (5) to provide an estimate of the
reflector depth.
Can We Go Beyond Simple Earth
models?In the previous sections, we saw that we could use the zero
and nonzero-offset round trip travel times to obtain therequired information for equation (5) in order to determine
the depth of a horizontal reflecting interface. This simple
procedure assumed that the interval velocity is a constant.
In particular, the procedure assumes that the interval
velocity is independent of depth. It also assumes that the
reflector used for the interval velocity calculation is
horizontal. Reaching beyond this simplistic assumption is
the subject of the next section.
Dipping Reflector Interval VelocityThe case of a dipping interface modifies the constant-
velocity NMO equation (equation (12)). To simplify this
analysis, we first alter our simple acquisition set-up infigure 11 to that seen in figure 15. The ray path lengths are
unchanged. However, in figure 15 the zero-offset and the
nonzero-offset ray paths reflect from the same idealized
reflection location. This change of geometry simplifies the
calculation of the interval velocity for the dipping reflector
configuration.
Both source-receiver pairs in figure 15 share a common
midpoint location. Sorting seismic traces into gathers that
share a common midpoint location is advantageous for
several data processing applications. For example, all traces
in a Common Midpoint Gather (or CMP gather) record
signals reflected from the same location on a horizontal
reflector. In other words, the traces in a CMP gather provide
redundant information on the same reflecting segment. This
data redundancy can be exploited for noise reduction or
velocity analysis.
Figure 15: Both zero- and finite-offset rays reflect at the
same subsurface locations.
This coincidence of reflection points is no longer true with
the introduction of a dipping reflector. Zero and far-offset
ray paths no longer reflect from a common subsurface
location. The ray path geometry, for the same source and
receiver locations, becomes more complex as shown in figure 16
In general, as the offset increases, the idealized reflection points
diverge and the finite-offset ray reflection point moves in the up-
dip direction.
Figure 16: Reflections from a dipping interface.
Let us briefly study the geometry of the nonzero offset ray path.
The law of reflections (Snells law) requires the incidence andreflection angles with respect to the interface normal to be equal.
To locate the reflection point on the dipping reflector, we can use a
geometric construction using an image source as shown in Figure
17. First, find the mirror image of S on the opposite site of the
reflector. The line from this mirror image to the receiver location
R intersects the dipping interface at the reflection point.
Keep this concept of image sources in mind, as it is very handy in
comprehending the offset dependence of recorded travel times. For
example, can you tell where you need to place a receiver to record
the minimum travel time of a signal reflected off the dipping
reflector?
Figure 17: Use of an image source to locate the reflection point
The geometry shown in Figure 16 and Figure 17 also answers the
question of how travel times at zero and non-zero offset yield the
interval velocity in the layer above the reflector. After a more
involved trigonometric derivation, the horizontal reflector
relationship of equation (13) generalizes to:
Offset =xOffset
VInt
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Creating Depth Images from Seismic Records 7
2
0
2
2
)cos tt
x
(
V
Offsetx
OffsetInt
(15)
Introducing dip introduces an inverse cosine dependency. If
the dip angle (measured from horizontal) is zero, equation
15 reduces to equation 13. For a finite dip angle, the
apparent velocity described by the square root in equations13, multiplied by the cosine of the dip angle, produces the
physical layer velocity. In other words, neglecting the
interface dip overestimates the layer velocity. This is true
whether the layer dips to the left or to the right.
NMO VelocityHaving covered the complexity introduced by the presence
of dip, we now consider the complexity introduced by the
presence of a velocity gradient. The obstacle is our
ignorance of the values of the depth-dependent interval
velocities. To this point, we have a methodology to estimate
the interval velocity for the constant-velocity world, but not
the world of velocity gradients. To estimate such velocities,we will define a NMO velocity along with an RMS
velocity to achieve the goal.
We begin with the definition of the NMO velocity. Figure
18 shows the case of a vertical velocity gradient, i.e., the
velocity increases with depth. For this case, the rays are no
longer straight, but curve.
Figure 18: Rays in the presence of a velocity gradient.
In the derivation of the constant velocity interval velocity
equation (equation (13)), we had obtained a right triangle by
reflecting the upcoming rays about the reflector itself to
produce figure 12. While we can apply the same technique
to figure 18, the unsatisfactory pseudo right triangle of
figure 19 results. Because the hypotenuse is not a straight
line, we cannot apply the Pythagorean Theorem to create an
equation similar to equation (9).
Figure 19: Attempt at creating a right triangle.
Liking the simplicity of the NMO (Normal MoveOut) equation
(equation (12)), we will keep its form with a new equation. We
define the NMO velocity, VNMO, as the velocity that satisfiesequation (12), even in the presence of a velocity gradient. By
replacing VInt with VNMO we have
.2
2
2
0
NMO
Offset
xV
xtt
(16)
In other words, the NMO velocity explains the delay in the
arrival time of a reflection away from zero-offset, independent of
the underlying physical model of the subsurface. Likewise,
because we have defined the NMO velocity (VNMO) always to
satisfy equation (16), we have the following equation that is true
without any restriction on the nature of the Earths velocities:
.tt
xV
Offsetx
Offset
NMO 2
0
2
2
(17)
We can view equation (17) as the defining equation for the NMO
velocity. Only for the constant-velocity world with a horizontal
reflector will VIntequal VNMO. A subsequent chapter more fully
explains the role of the NMO velocity.
Dix Interval Velocity
We defined the NMO velocity as our first step in developing aprocedure for depth conversion in the presence of a velocity
gradient. The second step in the quest introduces both the Dix
interval velocity and the RMS velocity.
Offset =xOffset
Offset =xOffset
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8 What reflections can teach us about the subsurface?
Figure 20: Interval velocities in horizontal layering.
For situations of a more complex subsurface, the
relationship between the interval velocity and the observed
roundtrip travel times becomes less straightforward. The
dependency of travel time and offset is not a closed-form
expression for a series of horizontal reflectors such as
shown in Figure 6. Instead, a Taylor series expansionapproximates the offset-dependency of travel time. More
precisely, rather than creating an expansion of travel time, it
is more convenient (and, as it turns outmore accurate) to
create an expansion for the squared travel time. As we have
seen above, the squared travel time depends linearly on
squared offset for a single constant velocity layer and
perturbing a linear relationship is far simpler than using a
hyperbolic starting equation. With this in mind, analyze the
first terms of the following series that expresses the squared
travel times as a function of offset:2
.
4
1
6
4
8
2
4
2
2
0
2
2
2
0
2
)f(x
xV
VV
t
V
x
tt
Offset
Offset
RMS
)(RMS
RMS
Offset
X
(18)
(XXX make period into comma in equation above)
where the last term shown in the equation is a function of
the offset raised to the sixth power.
Before defining all parameters, let us first comprehend the
structure of the equation. Why are only even powers ofoffset in the expansion? What terms can we ignore for small
offsets?
Seismic reciprocitythe fact that interchanging source and
receiver does not change the travel time of an event -
dictates that the travel time can only depend on even powers
of offset. Odd powers of offset would introduce a
dependency of travel time on the sign of the source-receiver
2 (Robein, 2003) In addition, you may find a derivation of the
first two terms of the expansion in (Ikelle & Amundsen,
2005)
offset. Therefore, the traveltime equation cannot have odd
powers.
To further elucidate this equation, divide both sides of the
equation by the square of the normal incidence time t0. Now the
expansion is in terms of the unit-less offset-to-depth ratio. If this
ratio is small, terms of higher power of offset-to-depth ratio
become even smaller and one can neglect them. You will
encounter this short-offset (= small offset-to-depth ratio)
assumption in many practical aspects of seismic processing.
After this preamble, we are prepared to study the coefficients in
equation (18) in more detail. The first coefficient is simply the
zero-offset travel time of the event. The second coefficient
(multiplying the offset-squared term) is the inverse of the Root
Mean Square Velocity, VRMS, defined by
i
i
Int
RMSt(i)
itiV
V
)()(2
2
(19)
V2RMSis the time-averaged squared interval velocity of the layerstack traversed by the incident and reflected field. If we denote
the denominator by t(i),when t(i) isthe round-trip time to the
reflector iand t(i) is the round-trip time spent in the i-th layer.
The coefficient multiplying the fourth power of offset includes a
second, velocity-derived quantity of a form very similar to
equation (19), representing an average of the fourth power of
interval velocity
.t(i)
t(i)(i)V
V
i
i
Int
4
4
)4(
(20)
The coefficient multiplying the fourth power of offset depends
on the difference ofVRMSand V(4) and is always negative (higher
powers of large quantities exceed lower powers). Dropping the
fourth-order term implies that the travel time may be slightly
overestimated.
As previously described, assuming small values of the offset, we
approximate equation (18) as
2
2
2
0
RMS
Offset
xV
x
tt
(21)
The similarity of equation (21) to equation (16) supports the
observation that the RMS velocity is approximately equal to the
NMO velocity for this horizontal, layered Earth model. We will
make use of that observation in estimating the interval velocity.
Solving equation (19) for the interval velocity, we have
VInt(1)
VInt(2)
VInt(3)
VInt(4)
Offset =xOffset
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Creating Depth Images from Seismic Records 9
t(i))t(i
(i) t(i)V))t(i(iV(i)V RMSRMSInt
1
11 222
(22)
Making use of the observation that VNMO is approximately
equal to the VRMS
, equation (22) becomes:
.1
11 222
t(i))t(i
(i) t(i)V))t(i(iV(i)V NMONMOInt
(23)
Equation (23) is theDixequation used to calculate the
interval velocity within a series of flat, parallel layers. It
converts VNMO velocities to reflectors above and below the
layer into the layers interval velocity. Geophysicists
routinely apply the Dix equation in practice, but it requires
careful analysis of the intrinsic assumptions, in particular
the horizontal layer supposition. The numeric properties ofthe equation also may introduce problems, especially in the
presence of measurement uncertainties. Please note that
equation (23) implies a differentiation-process and is
inherently unstable as compared to integration processes
(such as the determination of VRMS in equation (19)).
Finally note that casual use of the Dix equation may
introduce a negative radian in the square root.
We now consider a model case to estimate the accuracy of
the interval velocities obtained from equation (23). The
following figure shows a simple test model. The inspiration
for this model is a marine salt sheet with the 15,000-ft/s slab
representing the salt.
Ray tracing provided the roundtrip travel times for both zero-
offset and the far-offset traces. Given the roundtrip travel times,
equation (17) provides the NMO velocities. We then assume that
the NMO velocities are approximately equal to the RMS
velocities and can use equation (23) to calculate the interval
velocity.
Table 1 shows these ray-tracing travel times and subsequently
derived results.
Figure 21: A layered Earth test model.
Table 1: 2,500 foot-offset model and results
Observe that the computed interval velocities are not exactly equal to the original interval velocities in our model. That is because
equation (23) is an approximation. This equation assumed that the NMO velocities were equal to the RMS velocities.
In addition, as can be seen in this table, the value ofVNMO is equal to exactly VInt only for the shallowest, constant velocity, block. In
a constant-velocity world, VNMO equals VInt, which in turnequals VRMS. For deeper blocks, the ray paths traverse more than one
interval velocity. In this situation, the value ofVNMOis not equal to VIntbecause the velocity discontinuities bend the raypaths. The
error in VIntalso produces an error in the derived depths. We cannot correct this error with a constant correction factor.
VInt(1) = 5,000 feet/s
VInt(2) = 6,000 feet/s
VInt(3) = 15,000 feet/s
VInt(4) = 7,000 feet/s
2,000 ft.
4,000 ft.
6,000 ft.
8,000 ft.
Offset =xOffset
Input Values Model Values, 2500-foot Offset
ModeledObservations
Inverted Medium Parameters
Depth(feet)
VIn t
(feet/s)t0
(s)
tx
(s)
VNMO
(feet/s)
VIn t
(feet/s)Depth(feet)
0 0.00 0.00 0
5000 5000
2000 0.800 0.943 5000 2000
6000 6000
4000 1.466 1.535 5478 4000
15000 15090
6000 1.733 1.763 7775 6013
7000 69638000 2.304 2.327 7582 8003
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10 What reflections can teach us about the subsurface?
Overall, though, the accuracy of the predicted depth is amazingly good. Remember that these are synthetic data and that the ray
tracing produces high-precision estimates of the round-trip travel times as input for these calculations. Real data will not afford such
high precision. Measurement uncertainties can easily exceed any imprecision introduced by the assumptions going into the Dix
equation. A later chapter further investigates the introduction of error in the interval velocity determination with real data of finite
frequency bandwidth.
The following table shows the result of increasing the offset of the observations.
Table 2: 5,000 ft-offset model and results
From a comparison of the results ofTable 1 and Table 2, we
can see that an increase in the offset enlarges the estimated
depth error. The error in the assumption that VRMSequals
VNMO increases with an increase in offset because of stronger
ray bending at the velocity discontinuities.
We remind you that this synthetic example has the
advantage of very accurate determinations of the values oft0and tx; a ray-traced model determined them. With real data,
we cannot determine these round-trip travel times with
similar precision. The following, real-world example (figure
22) reveals this limitation.
0 ft 2650 ft
1.430 s
1.828 s
2.395 s
1.520 s
1.893 s
2.413 s
Water Bottom
Salt Top
Salt Base
1.4
2.0
Time(s)
Figure 22: Seismic observations of water-bottom and salt.
In order to use (23) to provide an approximation to the
interval velocity, we must first estimate the values ofVNMO
for the top and bottom of salt. By using the values shown in
figure 22 with equation (17), we have VNMO,1 = 5,388 ft/s and
VNMO,2 = 9,007 ft/s. Using those values in equation (23) we
obtain
.ft./,
..
).()().()(
t(i))t(i
(i)t(i)V))t(i(iV
(i)V
NMONMO
Int
sec78215
82813952
8281538839529007
1
11
22
22
(24)
This result is a reasonable value for the interval velocity of salt.
However, our ability to estimate accurately the top and bottom of
salt reflection times from Figure 22 determines the accuracy of
the interval velocity estimate.
Figure 23: Ray paths for series of dipping horizons.
Input Values Model Values, 5000-foot OffsetModeled
ObservationsInverted Medium Parameters
Depth(feet)
VIn t
(feet/s)t0
(s)
tx
(s)
VNMO
(feet/s)
VIn t
(feet/s)Depth(feet)
0 0.00 0.00 0
5000 5000
2000 0.800 1.281 5000 2000
6000 6009
4000 1.466 1.727 5482 4003
15000 15600
6000 1.733 1.844 7929 60837000 6774
8000 2.304 2.395 7659 8018
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Creating Depth Images from Seismic Records 11
In addition to interval-velocity estimation errors introduced by
the assumption that VRMSequals VNMO and/or the errors
introduced by inaccuracies in estimating the round-trip travel
times, violation of the assumption of horizontal reflectors also
introduces errors. Only in the simplest case, if all of the
reflectors have the same dip, as illustrated in Figure 23, there
is a simple modification of the relationship in equation (15) asshown in the following equation:
2
0
2
2
)cos tt
x
(
V
Offsetx
OffsetRMS
(25)
Even in this simple case of dipping, but parallel reflectors,
equation (25) indicates that we must also know the
geological dip of these parallel reflectors in order to more
accurately determine VRMSas input for the Dix interval-
velocity determination formula.
Subsequent chapters provide additional information about
velocities and their estimation.
Limitations of Vertical Depth Conversionfor Imaging
The use of vertical depth conversion of zero-offset seismic
data as an imaging process is accurate for only the case of
horizontal reflectors in the presence of horizontal layered
velocities such as shown in figure 20. Although horizontal
layering is often encountered in the subsurface and vertical
depth conversion can often be applied successfully, other
geologic features need special attention. Simply applying a
similar depth conversion approach in these more complex
scenarios has serious shortcomings.
The following illustrates this significant limitation.
Figure 24: Observations of a point reflector.
Figure 25 shows the recording of the zero-offset roundtrip
travel times for the physical situation offigure 24, the
reflections from a small, spherical reflector. Thex-axis is the
lateral position of the three, coincident sources and receivers
and they-axis is the roundtrip travel time.
Figure 25: Hyperbolic round-trip travel time recording for a
point reflector.
(XXX make true hyperbola)
The black curve in Figure 25 interpolates the 3 sampled
observations for continuous surface locations. The shape of this
zero-offset travel time curve of an idealized point reflection is
termed diffraction. It describes a hyperbolic shape with its apex
at the x-coordinate of the point diffractor. Of course, if thesubsurface reflector really were a point, then it would not reflect
any amplitude. Therefore, to be more precise, the reflector is, as
shown in the previous illustration, a small sphere; it is small in
comparison to the thickness of the downgoing wave front.
Diffractions are a very common occurrence in seismic data.
Faults, horizon edges, pinch-outs, and rough horizon topology or
karst-type geology all generate diffractions. Clearly, as with the
synthetic point diffractor in figure 25, simple depth conversion
will not produce the image that we see in figure 24. This is the
first example illustrating the failure of depth conversion as an
imaging step.
The second example considers a flat, dipping reflector. Here, thedifference between the true depth and our image of that depth
is subtler than for the point reflector.
Figure 26: Observations of a dipping reflector.
Figure 27 shows the arrival times at continuous surface
locations.
1 2 3
x 1 2 3
VInt
x
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12 What reflections can teach us about the subsurface?
Figure 27: Zero-offset section recorded from the dipping
reflector.
Although figure 27 is similar in appearance to the dipping
reflector seen in figure 26, the straightforward depth
conversion offigure 27provides incorrect depths and dip of
the reflector. To see this, we investigate a concrete example.
Figure 28 illustrates a 30-dipping reflector.
Figure 28: A 30 dipping bed.
Figure 29 shows the zero-offset section for this particular
model.
Figure 29: Zero-offset section.
As our imaging step, convert the zero-offset seismic
section in figure 29 to depth. Using (5), taking the two-way
travel time observation at the 1000-foot lateral location, we
obtain a depth of
.2
00010sec10500 s
ft,.
ft
(26)
Figure 30 shows the full depth conversion offigure 29. This
result is not equivalent to figure 28. The dip in figure 30 is
less than that in figure 28because the 500-foot ray path in
figure 28 is oblique, while the depth conversion of that same
500-feet used in figure 29 is vertical. The greater the
reflectors dip in the initial model, the greater the error in the
depth conversion.
Figure 30: Depth conversion of zero-offset section.
The following generalizes the dip error in depth conversion. is
the true dip of the reflector. The path length of the normal ray to
the surface is zDiag, the diagonally measured depth. In depth
conversion, we assign the diagonally measured depth to a
vertical depth, zVert. Note that zDiagequals zVert. However, the
first is diagonal and the second is vertical. Also, is the apparent
depth from the depth conversion.
Figure 31: True () and apparent () dips.
This geometry provides,
zDiag/x = zVert/x= sin() = tan(). (27)
The sine tangent relationship on the right-hand side of
equation (27) relates the apparent angle, , of depth-converted
data to the true angle in the ground, .
For the previous example, this formula relates the true dip, 30to the apparent dip of 26.56.
sin (30) = tan(26.56). (28)
We can express zDiag in terms of the observed, two-way travel
time, tObs as
zDiag= VInttObs/2. (29)
where VIntis the interval velocity. Combining equations (27) and
(29) and solving for the dip in the ground, we have
= Sin-
(VInttObs/(2 X)). ( 30)
The final example reveals how dramatically different the
reflection in the time section can be from the subsurface
geometry in the originating Earth model. In this case, the
reflector is a syncline whose shape is that of a hemisphere.
Assume that the interior of the hemisphere is a constant velocity
medium.
x
VInt(1) =
10,000 feet/s
30
x
1000 feet
500feet
1000 feet
x
26.56
x
zVert
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Creating Depth Images from Seismic Records 13
Figure 32: Hemisphere reflector.
Only at the surface position that is at the center of the
hemisphere do we record large amplitude from this
subsurface reflector3. To understand special nature of this
location, follow the wave created by the source at the
hemispheres center. The firing of the source creates a
downward-traveling spherical wave front. This wave front
strikes the entire hemispherical reflector at the same instant.
The wave front travels back to the coincident receiver as a
collapsing hemisphere. Thus, the sources amplitude returnsto the coincident receiver. To review, we have a syncline
(more precisely, a hemisphere) in the Earth model. This
hemisphere appears as an isolated point in the zero-offset
time section. Figure 33 shows the zero-offset section
obtained from figure 32.
Figure 33: Zero-offset section of generated for the
hemisphere reflector.
Clearly, depth conversion offigure 33 will not image the
Earth model as seen in figure 32. The preceding series of
examples demonstrates that the depth-converted, roundtrip
travel-time surface observations do not, in general, correctly
image the subsurface. In fact, the only successful example
was the depth conversion of data observed from a horizontal
reflector.
Interpreters Role(Many of the chapters end with this Interpreters Rolesection. This section highlights the conclusions that are of
particular interest to interpreters.)
Understanding and interpreting the subsurface geology in the
depth domain is the ultimate goal of seismic processing and
imaging. It is the bread-and-butter of seismic interpreter.
The link tying data measured in time to images in depth is
3 The other surface locations will also "see" the reflector.
However, at the other locations the returned amplitude is
much smaller than the amplitude indicated in Figure 33.
the seismic velocity. The importance of velocities cannot be
over-emphasized. Here are some points to remember:
Producing a depth image from seismic observations is anon-trivial undertaking that requires the estimation of
the seismic velocities in the subsurface rocks.
Always be aware of the origin of seismic-derived
velocities. Some velocities are reliable; others aresensitive to measurement uncertainties due to
differentiation.
Be aware that dips contaminate the velocity.Velocities appear faster in the presence of dip, no
matter in which direction the horizon may slope.
Depth conversion by simple vertical stretching hasserious limitations. Horizon dip angles are
underestimated and diffractions do not collapse into
their point of origin.
Relating travel times to velocities often involves ashort-offset assumption. Violating this assumption
enlarges the error in computed velocity estimates.
x
VInt(1)
x
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