creating depth images from seismic records

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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.

    WT

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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 DepthEstimates

    We 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 velocities fromthe 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

    ,22222

    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

    222

    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 Earthmodels?

    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 and

    reflection 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)

    VInt

    (feet/s)t0

    (s)

    tx

    (s)

    VNMO

    (feet/s)

    VInt

    (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)

    VInt

    (feet/s)t0

    (s)

    tx

    (s)

    VNMO

    (feet/s)

    VInt

    (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.

    InterpretersRole(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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