2 DepthConversion

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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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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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,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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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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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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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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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(3) = 15,000 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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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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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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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


13/15


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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