3. velocity analysis and statics corrections

3 Velocity Analysis andStatics Corrections

• Introduction • Normal Moveout • NMO for a Flat Reflector • NMO in a Horizontally Stratified Earth • Fourth-Order Moveout • NMO Stretching • NMO for a Dipping Reflector • NMO for Several Layers with Arbitrary Dips •Moveout Velocity versus Stacking Velocity • Velocity Analysis • The Velocity Spectrum • Measure of Coherency• Factors Affecting Velocity Estimates • Interactive Velocity Analysis • Horizon Velocity Analysis • CoherencyAttribute Stacks • Residual Statics Corrections • Residual Statics Estimation by Traveltime Decomposition• Residual Statics Estimation by Stack Power Maximization • Traveltime Decomposition in Practice • MaximumAllowable Shift • Correlation Window • Other Considerations • Stack-Power Maximization in Practice • RefractionStatics Corrections • First Breaks • Field Statics Corrections • Flat Refractor • Dipping Refractor • The Plus-Minus Method • The Generalized Reciprocal Method • The Least-Squares Method • Processing Sequence for StaticsCorrections • Model Experiments • Field Data Examples • Exercises • Appendix C: Topics in Moveout andStatics Corrections • The Shifted Hyperbola • Moveout Stretch • Equations for a Dipping Reflector • TraveltimeDecomposition for Residual Statics Estimation • Depth Estimation from Refracted Arrivals • Equations for a DippingRefractor • The Plus-Minus Times • Generalized Linear Inversion of Refracted Arrivals • Refraction TraveltimeTomography • L1-Norm Refraction Statics • References

3.0 INTRODUCTION

A sonic log represents direct measurement of the ve-locity with which seismic waves travel in the earth asa function of depth. Seismic data, on the other hand,provide an indirect measurement of velocity. Based onthese two types of information, the exploration seismol-ogist derives a large number of different types of velocity— interval, apparent, average, root-mean-square (rms),instantaneous, phase, group, normal moveout (NMO),stacking, and migration velocities. However, the veloc-ity that can be derived reliably from seismic data is thevelocity that yields the best stack.

Assuming a layered media, stacking velocity is re-

lated to normal-moveout velocity. This, in turn, is re-lated to the root-mean-squared (rms) velocity, fromwhich the average and interval velocities are derived.Interval velocity is the average velocity in an intervalbetween two reflectors.

Several factors influence interval velocity within arock unit with a certain lithologic composition:

(a) Pore shape,(b) Pore pressure,(c) Pore fluid saturation,(d) Confining pressure, and(e) Temperature.

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272 Seismic Data Analysis

FIG. 3.0-1. Change of P- and S-wave velocities as afunction of confining pressure observed in dry and water-saturated Bedford limestone samples with pores in the formof microcracks. Fluid volume has been kept constant dur-ing measurements. Here, S = saturated, D = dry, vP =P-wave velocity, and vS = S-wave velocity. (Adapted fromNur, 1981.)

These factors have been investigated extensively underlaboratory conditions. Figure 3.0-1 shows laboratorymeasurements of velocity as a function of the confin-ing pressure in a Bedford limestone sample with poresin the form of microcracks. The experiment was con-ducted using enclosed samples to control the pore fluidpressure independent of the confining pressure.

From Figure 3.0-1, we make the following observa-tions:

(a) Both compressional (P) and shear (S) wave ve-locities increase with increasing confining pres-sure. More specifically, velocity generally increasesrapidly with confining pressure at small confiningpressures, then gradually levels off at high confin-ing pressures. The reason for this is that as theconfining pressure increases, pores close. However,at a high confining pressure, not much deformablepore space is left. Therefore, any further increase inthe confining pressure will not cause a significantincrease in velocity.

(b) Note that, regardless of confining pressure, P-wavevelocity is greater than S-wave velocity. This is truefor any rock type.

(c) The saturated rock sample has a higher P-wavevelocity than the dry sample at low confining pres-sure. At high confining pressures, P-wave velocity

FIG. 3.0-2. Change of P- and S-wave velocities as a func-tion of confining pressure observed in Berea sandstone sam-ples with rounded pores. Fluid volume has been kept con-stant during measurements. Here, vP = P-wave velocity andvS = S-wave velocity. (Adapted from Nur, 1981.)

in the dry sample approaches the magnitude of theP-wave velocity in the saturated sample.

(d) Note also that the P-wave velocity in the saturatedsample does not change as rapidly as in the drysample. This is because the fluid is almost as in-compressible as the rock. Whether the pores arefilled with fluid or not has little effect on S-wave ve-locity, since fluids cannot support shear-wave prop-agation.

We now examine velocity as a function of confiningpressure for an enclosed sample of Berea sandstone withrounded pores (Figure 3.0-2). Again, note the increasein velocity with increasing confining pressure. The im-portant difference between this sample and the one inFigure 3.0-1 is the range of magnitude of the velocity.The rock with microcracks has a higher velocity thanthe rock with rounded pores at any given confining pres-sure. The reason for this is that it is easier to close thepores formed as microcracks than it is to close thosethat are round.

The most prominent factor influencing velocity in arock of given lithology and porosity probably is confin-ing pressure. This type of pressure arises from the over-burden and increases with depth. It is generally truethat velocity increases with depth. However, because offactors such as pore pressure, there may be inversion inthe velocity within a layer.

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Velocity Analysis and Statics Corrections 273

FIG. 3.0-3. Velocity range for rocks of different lithologiccompositions at different depths of burial. (Adapted fromSheriff, 1976; courtesy American Association of PetroleumGeologists.)

Figure 3.0-3 shows the variation of velocity withdepth for various types of lithology. We make the fol-lowing observations:

(a) Tertiary clastics, which usually are less induratedthan other rocks, occupy the low-velocity end ofthe graph. They generally start out with a velocitythat ranges from 1.5 to 2.5 km/s at or near thesurface, then gradually increase to from 4.5 to 5.5km/s at depths greater than 5 km.

(b) Carbonates with high porosity occupy the centralportion of the graph, starting at about 3 km/s andincreasing to nearly 6 km/s.

(c) Carbonates with low porosity, on the other hand,have a smaller range of variation in velocity. If thereis not much pore space to close, then the confiningpressure cannot cause much of an increase in ve-locity.

This chapter discusses ways to estimate veloci-ties from seismic data. Velocity estimation requires thedata recorded at nonzero offsets provided by common-midpoint (CMP) recording. With estimated velocities,we can correct reflection traveltimes for nonzero offsetand compress the recorded data volume (in midpoint-offset-time coordinates) to a stacked section (Figure 1.5-1).

For a single constant-velocity horizontal layer, thereflection traveltime curve as a function of offset is a

hyperbola (Section 3.1). The time difference betweentraveltime at a given offset and at zero offset is callednormal moveout (NMO). The velocity required to cor-rect for normal moveout is called the normal moveoutvelocity. In the case of an earth model with a singlehorizontal reflector, the NMO velocity is equal to thevelocity of the medium above the reflector. In the caseof an earth model with a single dipping reflector, theNMO velocity is equal to the medium velocity dividedby the cosine of the dip angle. When the dipping re-flector is viewed in three dimensions, then the azimuthangle (between the dip direction and the profiling di-rection) becomes an additional factor. Traveltime as afunction of offset for a series of horizontal isovelocitylayers is approximated by a hyperbola. This approxi-mation is better at small offsets than large offsets. Forshort offsets, the NMO velocity for a horizontally lay-ered earth model is equal to the rms velocity down tothe layer boundary under consideration. In a mediumcomposed of layers with arbitrary dips, the traveltimeequation gets complicated. However, in practice, as longas dips are gentle and the spread is small (less than re-flector depth), the hyperbolic assumption still can bemade. For layer boundaries with arbitrary shapes, thehyperbolic assumption breaks down.

There is a difference between the NMO and stack-ing velocities that often is ignored in practice. The NMOvelocity is based on the small-spread hyperbolic travel-time (Taner and Koehler, 1969; Al-Chalabi, 1973), whilestacking velocity is based on the hyperbola that best fitsdata over the entire spread length. Nevertheless, stack-ing velocity and NMO correction velocity generally areconsidered equivalent.

Conventional velocity analysis is based on the hy-perbolic assumption. Various methods for velocity anal-ysis are discussed in Section 3.2. The hyperbolic trav-eltime equation is linear in the t2 − x2 plane, where tis the two-way traveltime and x is the source-receiveroffset. Zero-offset time and stacking velocity for a givenreflector can be estimated from the line that best fits thetraveltime picks plotted on the t2 − x2 plane. Anotherway to estimate the NMO velocity is to apply differ-ent NMO corrections to a CMP gather using a range ofconstant velocity values, then display them side by side.The velocity that best flattens each event as a functionof offset is picked as its NMO velocity. Alternatively, asmall portion of a line can be stacked with a range ofconstant velocity values. These constant-velocity stacks(CVS) can be plotted in the form of a panel. Stackingvelocities that yield the desired stack then can be pickedfrom the CVS panel.

Another commonly used velocity analysis tech-nique is based on computing the velocity spectrum(Taner and Koehler, 1969). The idea is to display some

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measure of signal coherency on a graph of velocity ver-sus two-way zero-offset time. The underlying principleis to compute the signal coherency on the CMP gatherin small time gates that follow a trajectory in offset.Stacking velocities are interpreted from velocity spec-tra by choosing the velocity function that produces thehighest coherency at times with significant event ampli-tudes.

Occasionally the stacking velocity variation needsto be determined in detail along a particular reflector.Horizon-consistent velocity analysis provides the stack-ing velocity variation in the lateral direction along aparticular horizon of interest.

Reflection traveltimes are not always hyperbolic inhorizontally layered media. One reason that traveltimeoften deviates from a perfect hyperbola is the presenceof static time shifts caused by near-surface velocity vari-ations. Statics can severely distort the reflection hyper-bola when there are large surface elevation changes orwhen the weathering layer varies horizontally. Shownin Figure 3.0-4 is a stacked section that exhibits severedistortions of reflection traveltimes in the susburface,which is known to have generally flat layers, caused bythe complexity in the near-surface that is composed ofglacial tills with irregular shapes. The traveltime dis-tortions caused by the near-surface also are observed inthe CMP gather shown in Figure 3.0-5a. The velocityspectrum derived from the CMP gather that has notbeen corrected for the near-surface does not exhibit areliable stacking velocity trend (Figure 3.0-6a). By esti-mating a model for the near-surface and correcting forits effects on the reflection traveltimes in the subsur-face using the refracted arrivals from the near-surface,the resulting CMP gather (Figure 3.0-5b) yields a moreaccurate estimate of stacking velocities (Figure 3.0-6b).The CMP stack derived from the CMP gathers withstatics corrections exhibits reflection traveltimes free ofthe near-surface distortions (Figure 3.0-7). A close-up ofportions of the CMP stack without (Figure 3.0-4) andwith (Figure 3.0-7) statics corrections clearly demon-strates the improvement achieved by the statics correc-tions as shown in Figure 3.0-8.

Residual statics variations usually remain in thedata even after initial corrections for estimated weather-ing layer variations and elevation changes (field statics)(Section 3.4). Corrections for residual statics normallymust be estimated and applied to CMP gathers beforestacking. Estimation is done after a preliminary NMOcorrection using either a regional velocity function orinformation from a series of preliminary velocity analy-ses along the line. Following the residual statics correc-tions, velocity analyses usually are repeated to revisethe velocity picks for stacking. Various aspects of resid-ual statics corrections are discussed in Sections 3.3 and3.4.

As a final note, velocities required by stacking andmigration are not necessarily the same. In fact, for datacollected parallel to the dip direction of a single dip-ping reflector, stacking velocity is the velocity of themedium above the reflector divided by the cosine of thedip angle, while migration velocity is the velocity of themedium itself. In other words, stacking velocity is dip-dependent, while migration velocity is not. Migrationvelocity estimation is discussed in Section 5.4.

3.2 NORMAL MOVEOUT

Consider a reflection event on a CMP gather. The dif-ference between the two-way time at a given offset andthe two-way zero-offset time is called normal moveout(NMO). Reflection traveltimes must be corrected forNMO prior to summing the traces in the CMP gatheralong the offset axis. The normal moveout depends onvelocity above the reflector, offset, two-way zero-offsettime associated with the reflection event, dip of the re-flector, the source-receiver azimuth with respect to thetrue-dip direction, and the degree of complexity of thenear-surface and the medium above the reflector.

NMO for a Flat Reflector

Figure 3.1-1 shows the simple case of a single horizon-tal layer. At a given midpoint location M, we want tocompute the reflection traveltime t along the raypathfrom shot position S to depth point D then back to re-ceiver position G. Using the Pythagorean theorem, thetraveltime equation as a function of offset is

t2 = t02 +

x2

v2, (3 − 1)

where x is the distance (offset) between the source andreceiver positions, v is the velocity of the medium abovethe reflecting interface, and t0 is twice the traveltimealong the vertical path MD. Note that vertical projec-tion of depth point D to the surface, along the normalto the reflector, coincides with midpoint M. This occursonly when the reflector is horizontal.

Equation (3-1) describes a hyperbola in the planeof two-way time versus offset. Figure 3.1-2 is an exam-ple of traces in a common-midpoint (CMP) gather. Thefigure also represents a common-depth-point (CDP)gather, since all the raypaths associated with eachsource-receiver pair reflect from the same subsurfacedepth point D. The offset range in Figure 3.1-2 is 0to 3150 m, with a 50-m trace separation. The mediumvelocity above the reflector is 2264 m/s. All of the traces

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FIG. 3.0-5. A CMP gather associated with the stacked section in Figure 3.0-4 (a) without, and (b) with statics corrections.Dow

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FIG. 3.0-6. Velocity spectra derived from the CMP gather shown in Figure 3.0-5 (a) without, and (b) with statics corrections.

in this CMP gather contain a reflection from the samedepth point.

From equation (3-1), we see that velocity can becomputed when offset x and two-way times t and t0 areknown. Once the NMO velocity is estimated, the trav-eltimes can be corrected to remove the effect of offsetas shown in Figure 3.1-3. Traces in the NMO-correctedgather then are summed to obtain a stack trace at theparticular CMP location.

The numerical procedure involved in hyperbolicmoveout correction is illustrated in Figure 3.1-4. Theidea is to find the amplitude value at A on the NMO-corrected gather from the amplitude value at A on theoriginal CMP gather. Given quantities t0, x, and vNMO,compute t from equation (3-1). Assume that this is 1003ms. If the sampling interval were 4 ms, then this time isequivalent to the 250.25 sample index. The amplitudevalue at this time can be computed using the ampli-

tudes at the neighboring integer sample values, two oneach side — at 248, 249, 251, and 252 sample indexes.This is done by an interpolation scheme that involvesthe four samples.

An alternative numerical method for mappingtrace amplitudes from a nonzero-offset to zero offsetinvolves a nearest-neighbor sample as the output value.Accurate implementation of this method requires, first,oversampling the traces in a CMP gather along the timeaxis. Specifically, for each trace in the CMP gather, per-form 1-D Fourier transform and pad the frequency axiswith zeroes, usually by a factor of eight. Then, inversetransform back to the time domain and obtain a tracewhich has eight times as many samples at a samplinginterval that is one-eighth of the original sampling rate.Now, given quantities t0, x, and vNMO, again, computet from equation (3-1). Assume that this is t = 1003.4ms. If the original sampling interval were 4 ms, after

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(a) (b)

FIG. 3.0-8. A close-up of portions of the CMP stacks shown in (a) Figure 3.0-4 and (b) Figure 3.0-7.

oversampling, the new sampling interval is 0.5 ms. Thenthe amplitude at t = 1003.4 ms can be borrowed fromthe nearest-neighbor sample with an index of 2006 with-out much sacrifice in accuracy.

The NMO correction is given by the difference be-tween t and t0:

∆tNMO = t − t0, (3 − 2a)

or, by way of equation (3-1),

∆tNMO = t0 1 +x

vNMOt0

2

− 1 . (3 − 2b)

Table 3-1 shows the moveout corrections for twodifferent offset values using a realistic velocity functionthat increases with reflector depth.

From Table 3-1, note that the NMO increases withoffset and decreases with zero-offset time, hence, withdepth. The NMO also is smaller for higher velocities,and the combined effect of higher velocities at largerdepths makes it much smaller.

For a flat reflector with an overlying homogeneousmedium, the reflection hyperbola can be corrected foroffset if the correct medium velocity is used in the NMOequation. From Figure 3.1-5, if a velocity higher thanthe actual medium velocity (2264 m/s) is used, then

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Table 3-1. NMO correction as a function of offset x andtwo-way zero-offset time t0 for a given velocity functionvNMO.

∆tNMO, in s ∆tNMO, in st0, s vNMO, m/s x = 1000 m x = 2000 m

0.25 2000 0.309 0.7800.5 2500 0.140 0.4431 3000 0.054 0.2012 3500 0.020 0.0804 4000 0.008 0.031

the hyperbola is not flattened completely. This is calledundercorrection. On the other hand, if a lower velocityis used, then overcorrection results.

Figure 3.1-5 also illustrates the basis of conven-tional velocity analysis. NMO correction is applied tothe input CMP gather using a number of trial con-stant velocity values in equation (3-2b). The velocitythat best flattens the reflection hyperbola is the velocitythat best corrects for NMO before stacking the tracesin the gather. Furthermore, for a simple case of a sin-gle horizontal reflector, this velocity also is equal to thevelocity of the medium above the reflector.

NMO in a Horizontally Stratified Earth

We now consider a medium composed of horizontalisovelocity layers (Figure 3.1-6). Each layer has a cer-tain thickness that can be defined in terms of two-way zero-offset time. The layers have interval velocities(v1, v2, . . . , vN ), where N is the number of layers. Con-sider the raypath from source S to depth point D, backto receiver R, associated with offset x at midpoint loca-tion M. Taner and Koehler (1969) derived the traveltimeequation for this path as

t2 = C0 + C1x2 + C2x

4 + C3x6 + · · · , (3 − 3)

where C0 = t02, C1 = 1/v2

rms, and C2, C3, · · · are com-plicated functions that depend on layer thicknesses andinterval velocities (Section C.1). The rms velocity vrms

down to the reflector on which depth point D is situatedis defined as

v2rms =

1t0

N

i=1

v2i ∆τi, (3 − 4a)

where ∆τi is the vertical two-way time through the ithlayer and t0 = N

i=1 ∆τi. By making the small-spreadapproximation (offset small compared to depth), the se-ries in equation (3-3) can be truncated to obtain the

FIG. 3.1-1. The NMO geometry for a single horizontal re-flector. The traveltime is described by a hyperbola repre-sented by equation (3-1).

familiar hyperbolic form

t2 = t20 +x2

v2rms

. (3 − 4b)

When equations (3-1) and (3-4b) are compared, wesee that the velocity required for NMO correction for ahorizontally stratified medium is equal to the rms veloc-ity, provided the small-spread approximation is made.

How much error is caused by dropping the higherorder terms in equation (3-3)? Figure 3.1-7a shows aCMP gather based on the velocity model in Figure 3.1-8. Traveltimes to all four reflectors were computed bythe raypath integral equations (Grant and West, 1965)that exactly describe wave propagation in a horizon-tally layered earth model with a given interval velocityfunction. We now replace the layers above the secondshallow event at t0 = 0.8 s with a single layer with avelocity equal to the rms velocity down to this reflector— 2264 m/s. The resulting traveltime curve, computedusing equation (3-4b), is shown in Figure 3.1-7b. Thisprocedure is repeated for the deeper events at t0 = 1.2and 1.6 s as shown in Figures 3.1-7c and d. Note thatthe traveltime curves in Figures 3.1-7b, c, and d are per-fect hyperbolas. How different are the traveltime curvesin Figure 3.1-7a from these hyperbolas? After carefulexamination, note that the traveltimes are slightly dif-ferent for the shallow events at t0 = 0.8 and 1.2 s onlyat large offsets, particularly beyond 3 km. By droppingthe higher order terms, we approximate the reflectiontimes in a horizontally layered earth with a small-spreadhyperbola.

Fourth-Order Moveout

A review of the moveout equation (3-3) to attain higheraccuracy at far offsets is given in Section C.1. At first,

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FIG. 3.1-2. Synthetic CMP gather associated with the ge-ometry in Figure 3.1-1. Traveltime curve for a flat reflectoris a hyperbola with its apex at zero-offset trace.

it seems that including the terms up to the fourth-orderin equation (3-3) should achieve this objective:

t2 = t20 +x2

v2rms

+ C2x4. (3 − 5a)

Nevertheless, to compute a velocity spectrum using thisequation requires scanning for two parameters — vrms

and C2; thus, making equation (3-5a) cumbersome touse for velocity analysis. Below, a practical scheme to

FIG. 3.1-3. NMO correction (equation 3-2a) involves map-ping nonzero-offset traveltime t onto zero-offset traveltimet0. (a) Before and (b) after NMO correction.

FIG. 3.1-4. Computational description of NMO correction.For a given integer value for t0, and velocity v and offset x,compute t using equation (3-1). The amplitude at time t,denoted by A, does not necessarily fall onto an input integersample location. By using two samples on each side of t(denoted by solid dots), we can interpolate between the fouramplitude values to compute the amplitude value at t. Thisamplitude value then is mapped onto output integer samplet0 denoted by A at the corresponding offset.

compute a velocity spectrum using equation (3-5a) issuggested:

(a) Drop the fourth-order term to get the small-spreadhyperbolic equation (3-4b). Compute the conven-

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FIG. 3.1-5. (a) CMP gather containing a single event with a moveout velocity of 2264 m/s, (b) NMO-corrected gather usingthe appropriate moveout velocity, (c) overcorrection because too low a velocity (2000 m/s) was used in equation (3-2b), and(d) undercorrection because too high a velocity (2500 m/s) was used in equation (3-2b).

tional velocity spectrum (Section 3.2) by varyingvrms in equation (3-4b), and pick an initial velocityfunction vrms(t0).

(b) Use this picked velocity function in equation (3-5a) to compute a velocity spectrum by varying theparameter C2, and pick a function C2(t0).

(c) Use the picked function C2(t0) in equation (3-5a) torecompute the velocity spectrum by varying vrms.Finally, pick an updated velocity function vrms(t0)from this velocity spectrum.

Castle (1994) shows that a time-shifted hyperbolaof the form

t = t0 1 − 1S

+t0S

2

+x2

Sv2rms

(3 − 5b)

is an exact equivalent of the fourth-order moveout equa-tion (3-5a). Here, S is a constant (Section C.1). ForS = 1, equation (3-5b) reduces to the conventionalsmall-spread moveout equation (3-4b).

As for the fourth-order moveout equation (3-5a),the time-shifted hyperbolic equation (3-5b) can, in prin-ciple, be used to conduct velocity analysis of CMP gath-ers.

FIG. 3.1-6. A horizontally layered earth model geometry.

The traveltime to the deepest reflector is described by equa-

tion (3-3). An approximate form of this equation used in

practice is given by equation (3-4b).

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FIG. 3.1-7. (a) A synthetic CMP gather derived from the velocity function depicted in Figure 3.1-8; (b), (c), and (d) areCMP gathers derived from the rms velocities (indicated at the top of each gather) associated with the second, third, andfourth reflectors from the top. The traveltimes in (a) were derived using the raypath integral equations for a horizontallylayered earth model.

(a) Set S = 1 in equation (3-5b) to get equation (3-4b).Compute the velocity spectrum by varying vrms inequation (3-4b), and pick an initial velocity func-tion vrms(t0).

(b) Use this picked velocity function in equation (3-5b) and compute a velocity spectrum by varyingthe parameter S. Pick a function S(t0), and

(c) use it in equation (3-5b) to recompute the velocityspectrum by varying vrms. Finally, pick an updatedvelocity function vrms(t0) from this velocity spec-trum.

De Bazelaire (1988) offers an alternative moveoutequation to achieve higher-order accuracy at far offsets:

t = (t0 − tp) + t2p +x2

v2s

, (3 − 5c)

where t0 is the two-way zero-offset time, tp is related tothe time at which the asymptotes of the hyperbolic trav-eltime trajectory converge (Section C.1), and vs is thereference velocity assigned to the layer below the record-ing surface (not the near-surface layer). When tp = t0,equation (3-5c) reduces to the small-spread hyperbolicequation (3-4b).

Thore and Kelly (1992) demonstrate the use ofequation (3-5c) to obtain a stacked section with a higher

stack power compared to the conventional stack de-rived from the small-spread moveout equation (3-4b).To use equation (3-5c) for velocity analysis, choose afixed value of reference velocity vs. Then, for each out-put time t0 and for each offset x, apply time shift tp totraces in the CMP gather and compute the input timet for the offset under consideration. Compute a veloc-ity spectrum for a range of tp values. Finally, pick afunction tp(t0) from the velocity spectrum.

NMO Stretching

Figure 3.1-9b shows the CMP gather in Figure 3.1-7a after NMO correction. The rms velocity functionshown in Figure 3.1-8 was used in equation (3-2b)for this correction. As a result of the NMO correc-tion, a frequency distortion occurs, particularly for shal-low events and at large offsets. This is called NMOstretching and is illustrated in Figure 3.1-10. The wave-form with a dominant period T is stretched so thatits period T0, after NMO correction, is greater than T .Stretching is a frequency distortion in which events areshifted to lower frequencies. Stretching is quantified by

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FIG. 3.1-8. A hypothetical velocity function used in gen-erating the synthetic CMP gather in Figure 3.1-7a.

∆f

f=

∆tNMO

t0, (3 − 6)

where f is the dominant frequency, ∆f is the change infrequency, and ∆tNMO is given by equation (3-2b). Thederivation of equation (3-6) is given in Section C.1.

Table 3-2 lists the percent frequency changescaused by the NMO stretching associated with the ve-locity function in Table 3-1. Note that stretching is con-fined mainly to large offsets and shallow times. For ex-ample, a waveform with a 30-Hz dominant frequencyat 2000-m offset and t0 = 0.25 s shifts to nearly 10 Hzafter NMO correction.

Because of the stretched waveform at large off-sets, stacking the NMO-corrected CMP gather (Figure3.1-9b) will severely damage the shallow events. Thisproblem can be circumvented by muting the stretchedzones in the gather. Automatic muting is done by us-ing the quantitative definition of stretching given byequation (3-6). Figures 3.1-9c and d show two versionsof the CMP gather after NMO correction and muting;one version has a stretch limit of 50 percent, while theother has a stretch limit of 100 percent. The 50-percentstretch limit does not show significant frequency distor-tion. However, the stretch limit can be extended to 100percent because we want to include as much of the CMPgather in the stack as possible without degradation.

A trade-off exists between the signal-to-noise ra-tio and mute. In particular, if the signal-to-noise ra-tio is good, then it may be preferable to mute morethan stretch mute requirements to preserve signal band-width. On the other hand, if the signal-to-noise ratio is

Table 3-2. NMO stretching.

%∆f/f for %∆f/f fort0, s vNMO, m/s x = 1000 m x = 2000 m

0.25 2000 123 3120.5 2500 28 891 3000 5 202 3500 1 44 4000 0.2 0.8

poor, it may be necessary to accept a large amount ofstretch to get any events on the stack. A real data ex-ample is provided in Figure 3.1-11. Here, the stretchedzone is seen as the low-frequency zone at the shallowpart of the CMP gathers without mute applied.

Another method for optimum selection of the mutezone is to progressively stack the data. Figure 3.1-12a isan NMO-corrected CMP gather without mute applied.Figure 3.1-12b shows the stack traces derived from theCMP gather (Figure 3.1-12a). The far right trace is thesame as the far right trace in the input CMP gather.The second trace from the right is the sum of the twonear-offset traces, and so on, progressively increasingthe stacking fold. The far left trace is the full-fold stackof the input CMP gather. By following the waveformalong a certain event and observing where changes oc-cur, the mute zone is derived as shown in Figure 3.1-12b.A similar procedure can be followed to determine an in-side mute. This time, the stacking fold is progressivelyincreased in the near-offset direction.

Aside from the signal-to-noise ratio, attenuation ofmultiples dictates the choice of a suitable mute pat-tern. Specifically, large offsets often are needed to at-tenuate multiple reflections based on moveout discrimi-nation between primaries and multiples. An inside mutemay be needed in addition to the application of a mul-tiple attenuation technique to alleviate the small move-out discrimination between primaries and multiples atsmall offsets. An inside mute also may be applied to landrecords to suppress ground-roll energy and air waves as-sociated with surface sources.

Muting also is dependent upon the ultimate use ofthe stacked data. If the stacked data are intended asinput to amplitude inversion, you may want to applya harsh mute to minimize the angle-dependency of re-flection amplitudes inferred by the Zoeppritz equations(Section 11.3). If the stacked data are intended as inputto poststack depth migration, then a limited-offset stackmay be needed to minimize the amplitude and travel-time distortions caused by the nonhyperbolic moveout

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FIG. 3.1-9. (a) Same gather as in Figure 3.1-7a, (b) after moveout correction using the rms velocity function depicted inFigure 3.1-8, (c) and (d) after muting using threshold stretch limits of 50 and 100 percent, respectively.

associated with reflections below complex overburdenstructures (Section 8.0).

NMO for a Dipping Reflector

Figure 3.1-13 depicts a medium with a single dipping re-flector. We want to compute the traveltime from sourcelocation S to the reflector at depth point D, then backto receiver location G. For the dipping reflector, mid-point M is no longer a vertical projection of the depthpoint to the surface. The terms CDP gather and CMPgather are equivalent only when the earth is horizontallystratified. When there is subsurface dip or lateral ve-locity variation, the two gathers are different. MidpointM and the normal-incidence reflection point D remaincommon to all of the source-receiver pairs within thegather, regardless of dip. Depth point D, however, isdifferent for each source-receiver pair in a CMP gatherrecorded over a dipping reflector.

Levin (1971), using the geometry of Figure 3.1-13,derived the following two-dimensional (2-D) traveltime

FIG. 3.1-10. A signal (a) with a period of T is stretched toa signal (b) with a period of T0 > T after NMO correction.

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FIG. 3.1-11. NMO correction and muting of a stretched zone on field data, (a) CMP gathers, (b) NMO correction, and (c)mute.

equation for a dipping reflector (Section C.3):

t2 = t20 +x2 sin2 α

v2, (3 − 7)

where the two-way traveltime t is associated with thenonzero-offset raypath SDG from source S to reflectionpoint D to receiver G, the two-way zero-offset time t0is associated with the normal-incidence raypath MDat midpoint M , and α is the angle between the normalto the dipping reflector and the direction of the line ofrecording (Figure 3.1-13). The moveout velocity is thengiven by

vNMO =v

sinα. (3 − 8)

For the 2-D geometry of the dipping reflector shown inFigure 3.1-13, note that

sinα = cos φ, (3 − 9)

where φ is the dip angle of the reflector. Hence, equa-tions (3-7) and (3-8) are written in terms of the reflectordip φ

t2 = t20 +x2 cos2 φ

v2(3 − 10)

and

vNMO =v

cos φ. (3 − 11)

The traveltime equation (3-10) for a dipping reflectorrepresents a hyperbola as for the flat reflector (equa-tion 3-1). However, the NMO velocity now is given bythe medium velocity divided by the cosine of the dipangle as defined by equation (3-11). This equation indi-cates that proper stacking of a dipping event requires avelocity that is greater than the velocity of the mediumabove the reflector.

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Table 3-3. NMO velocity for various earth models.

Model NMO Velocity

Single Horizontal Velocity of the medium aboveLayer the reflecting interface

Horizontally Stratified The rms velocity functionEarth provided the spread is small

Single Dipping Layer Medium velocity divided bycosine of the dip angle

Multilayered Earth The rms velocity functionwith Arbitrary Dips provided the spread is small

and the dips are gentle

FIG. 3.1-12. Optimum mute selection. Starting with theNMO-corrected CMP gather in panel (a), a substack gather(b) is obtained. The far right trace in this gather is the sameas that in the original gather. The second trace from theright is the stack of the two near traces of the original gather.Finally, the far left trace is the full-fold stack obtained fromthe original gather. The area above the dotted line in (b) isthe mute zone.

FIG. 3.1-13. Geometry for NMO of a single dipping reflec-tor. See text for details.

In conclusion, the NMO velocity for a dipping re-flector depends on the dip angle. The larger the dip an-gle, the higher the moveout velocity, hence the smallerthe moveout. There is a 4 percent difference betweenmoveout velocity vNMO and medium velocity v for a 15-degree dip. The difference is 50 percent at a 30-degreedip and rapidly increases at steep dips. An accompany-ing observation is that a horizontal layer with a highvelocity can yield the same moveout as a dipping layerwith a low velocity, as illustrated in Figure 3.1-14.

NMO for Several Layers with Arbitrary Dips

Figure 3.1-15 shows a 2-D subsurface geometry that iscomposed of a number of layers, each with an arbitrarydip. We want to compute the traveltime from sourcelocation S to depth point D, then back to receiver loca-tion G, which is associated with midpoint M. Note thatthe CMP ray from midpoint M hits the dipping inter-face at normal incidence at D , which is not the same asD. The zero-offset time is the two-way time along theraypath from M to D .

Hubral and Krey (1980) derived the expression fortraveltime t along SDG as

t2 = t20 +x2

v2NMO

+ higher order terms, (3 − 12)

where the NMO velocity is given by

v2NMO =

1t0 cos2 β0

N

i=1

v2i ∆ti

i−1

k=1

cos2 αk

cos2 βk. (3 − 13)

The angles α and β are defined in Figure 3.1-15. Fora single dipping layer, equation (3-13) reduces to equa-tion (3-8). Moreover, for a horizontally stratified earth,equation (3-13) reduces to equation (3-4). As long as

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FIG. 3.1-14. Moveout for low-velocity event (a) is largerthan for high-velocity event (b). Moveout for low-velocitydipping event (c) may not be distinguishable from high-velocity horizontal event (b). These observations are directconsequences of equation (3-7).

the dips are gentle and the spread is small, the travel-time equation is approximately represented by a hyper-bola (equation 3-5), and the velocity required for NMOcorrection is approximately the rms velocity function(equation 3-4).

Moveout Velocity versus Stacking Velocity

Table 3-3 summarizes the NMO velocity obtained fromvarious earth models. After making the small-spreadand small-dip approximations, moveout is hyperbolicfor all cases and is given by

t2(x) = t2(0) +x2

v2NMO

. (3 − 14)

The hyperbolic moveout velocity should be distin-guished from the stacking velocity that optimally allowsstacking of traces in a CMP gather. The hyperbolic formis used to define the best stacking path tstk as

t2stk(x) = t2stk(0) +x2

v2stk

, (3 − 15)

where vstk is the velocity that allows the best fit of thetraveltime trajectory on a CMP gather to a hyperbolawithin the spread length.

The optimum stacking hyperbola described byequation (3-15) is not necessarily the small-spread hy-perbola given by equation (3-14). Refer to the travel-times illustrated in Figure 3.1-16 and note the follow-ing:

(a) The observed two-way zero-offset time OC = t(0)in equation (3-14) can be different from the two-way zero-offset time OB = tstk(0) associated withthe best-fit hyperbola (equation 3-15). This occurs,

FIG. 3.1-15. Geometry for the moveout for a dipping in-terface in an earth model with layers of arbitrary dips.(Adapted from Hubral and Krey, 1980.)

for example, if some heterogeneity exists in the ve-locity layers above a reflector under consideration.

(b) The difference between the stacking velocity vstk

and NMO velocity vNMO is called spread-lengthbias (Al-Chalabi, 1973; Hubral and Krey, 1980).From equations (3-14) and (3-15), the smaller thespread length, the closer the optimum stacking hy-perbola to the small-spread hyperbola, hence thesmaller the difference between vstk and vNMO.

In practice, when we refer to stacking velocity andthe zero-offset time associated with the optimum stack-ing hyperbola described by equation (3-15), we almostalways think of the moveout velocity and the zero-offsettime associated with the small-spread hyperbola givenby equation (3-14).

3.2 VELOCITY ANALYSIS

Normal moveout is the basis for determining velocitiesfrom seismic data. Computed velocities can in turn be

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FIG. 3.1-16. The equation for moveout velocity is derivedby assuming a small-spread hyperbola (equation 3-14). Onthe other hand, stacking velocity is derived from the best-fit hyperbola over the entire spread length (equation 3-15).Here, (a) is the actual traveltime, (b) is best-fit hyperbolaover the offset range OA, and (c) is small-spread hyperbola.(Adapted from Hubral and Krey, 1980.)

used to correct for NMO so that reflections are alignedin the traces of a CMP gather before stacking. Fromequation (3-15), we can develop a practical way to de-termine stacking velocity from a CMP gather. Equation(3-14) describes a line on the t2−x2 plane. The slope ofthe line is (1/v2

NMO) and the intercept value at x = 0is t0. The synthetic gather in Figure 3.2-1 was derivedfrom the velocity model in Figure 3.1-8. The far rightframe of Figure 3.2-1 shows the picked traveltimes offour events at a number of offsets plotted on the t2−x2

plane. To find the stacking velocity for a given event,the points corresponding to that event have been con-nected by a straight line. The inverse of the slope of theline is the stacking velocity. (In practice, least-squaresfitting can be used to define the line slopes.) A compar-ison between the computed stacking velocities and theactual rms velocities is made in Table 3-4.

The t2−x2 velocity analysis is a reliable way to es-timate stacking velocities. The accuracy of the methoddepends on the signal-to-noise ratio, which affects thequality of picking. In Figure 3.2-1, results are comparedwith the velocity spectrum (center frame) approach,which is discussed later in the section.

A real data example is shown in Figure 3.2-2.Velocities estimated from the t2 − x2 analysis areshown by triangles on the velocity spectrum. Note that

Table 3-4. Computed stacking and actual rms veloci-ties for the synthetic model in Figure 3.2-1.

Computed Stacking Actual rmst0, s Velocities, m/s Velocities, m/s

0.4 2000 20000.8 2264 22641.2 2519 25331.6 2828 2806

agreement between the t2 − x2 approach and the picksfrom the velocity spectrum are satisfactory.

Claerbout (1978) proposed a way to determine in-terval velocities manually from CMP gathers. The ba-sic idea is illustrated in Figure 3.2-3. First, measure theslope along a slanted path that is tangential to boththe top and bottom reflections of the interval of inter-est (slope 1). Then, connect the two tangential pointsand measure the slope of this line (slope 2). The intervalvelocity then is equal to the square root of the productof the two slope values. The accuracy of this methodprimarily depends on the signal-to-noise ratio.

The method of constant velocity scans of a CMPgather is an alternative technique for velocity anal-ysis. Figure 3.2-4b shows a CMP gather which hasbeen NMO corrected repeatedly using a range of con-stant velocities between 1500 and 4500 m/s. Scan theconstant-velocity moveout-corrected gathers displayedto the right of the original gather (b) starting fromthe low-velocity end and identify flat events. A velocityfunction can be composed by noting the velocity-timepairs that correspond to the flat events (Table 3-5). Byusing this velocity function, the CMP gather (Figure3.2-4b) is moveout corrected for stacking (Figure 3.2-4a).

Accuracy in velocity picking depends on cablelength, the two-way zero-offset time associated with thereflection event, and the velocity itself. The higher thevelocity, the deeper the reflector and the shorter the ca-ble length, the poorer the velocity resolution. The res-olution in velocity picking also depends on the signalbandwidth; the more compact the wavelet is along thereflection traveltime trajectory in the CMP gather, themore accurate is the velocity pick. Prestack deconvo-lution (Section 2.5) prior to velocity analysis aimed atwavelet compression helps to improve velocity resolu-tion.

While Figure 3.2-4 exhibits primary reflectionevents before 3 s, Figure 3.2-5 exhibits primary reflec-tions below 3 s. Follow the NMO for event A. Note

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FIG. 3.2-1. The t2−x2 velocity analysis applied to the synthetic gather derived from the velocity function depicted in Figure3.1-8. The center panel is the velocity spectrum based on equation (3-19b).

that this event is overcorrected at low velocities andundercorrected at high velocities. The event is flat onthe NMO-corrected gather that corresponds to the 2500m/s velocity; thus, this is the optimum stacking veloc-ity for event A. Event B is flat on the NMO-correctedgather that corresponds to the 2800 m/s velocity. Nev-ertheless, there is a range of velocities around 2800 m/swhich exhibits nearly flat character for event B. This re-sults in an uncertainty in making an accurate velocitypick.

The most important reason to obtain a reliable ve-locity function is to get the best quality stack of signal.Therefore, stacking velocities often are estimated fromdata stacked with a range of constant velocities on thebasis of stacked event amplitude and continuity. Figure3.2-6 illustrates this approach. Here, a portion of a linecontaining 100 CMP gathers has been NMO-correctedand stacked with a range of constant velocities. Theresulting constant-velocity CMP stacks then were dis-

played as a panel. Stacking velocities are picked directlyfrom the constant-velocity stack (CVS) panel by choos-ing the velocity that yields the best stack response at aselected event time.

The panel of constant-velocity stacks in Figure 3.2-7 demonstrates how velocity resolution decreases withincreasing depth. The deep event at 3.6 s seems to stackat a wide range of velocity values.

A variation of CVS analysis is a panel of CMPstacks using a family of velocity functions that are afixed percentage higher and lower than a base velocityfunction (Figure 3.2-8). This type of velocity analysispanel usually is used in combination with a velocityspectrum computed at the central CMP location to pickan optimum stacking velocity function.

The constant velocities used in the CVS methoddescribed above should be chosen carefully. There aretwo issues to consider besides the expected range of ac-tual velocities in the subsurface: (a) the range of ve-

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FIG. 3.2-2. The t2 − x2 velocity analysis applied to a CMP gather. The triangles on the velocity spectrum (center panelbased on equation 3-19b) represent velocity values derived from the slopes of the lines shown on the graph at the right.

Table 3-5. Velocity function derived from the constant-velocity panel of Figure 3.2-4.

Two-way Zero-Offset RMS Velocitytime, ms Picked, m/s

0 1500100 1500760 1900

1400 27001800 30002150 36005000 4000

locities needed to stack the data and (b) the spacingbetween trial stacking velocities. In choosing a range,consideration should be given to the fact that dip-ping events and useful out-of-plane reflections may haveanomalously high stacking velocities. In choosing thespacing of constant velocities, keep in mind that it ismoveout, not velocity, that is the basis for velocity esti-mation. Thus, it is better to scan in increments of equal∆tNMO than equal vNMO. This prevents oversamplingof the high-velocity events and undersampling of thelow-velocity events. A good way to choose ∆(∆tNMO)is to pick it so that the moveout difference between adja-cent trial velocities at the maximum offset to be stacked

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FIG. 3.2-3. The interval velocity between two reflectors isequal to the square-root of the products of the slope valuesmeasured as shown above. This is the same gather as inFigure 3.1-7a. Trace spacing is 50 m, slope 1 = 3150/0.43,slope 2 = 550/0.44, and thus, the interval velocity between0.8 and 1.2 s is 3026 m/s.

is approximately 1/3 of the dominant period of the data(S. Doherty, 1986, personal communication). Shallowdata have short maximum offsets because of muting,while deep data have large dominant periods. Thus, thenumber of trial stacking velocities needed to adequatelysample the data can be reduced considerably.

The CVS method is especially useful in areas withcomplex structure (Exercise 3-5). In such areas, this

method allows the interpreter to directly choose thestack with the best possible event continuity. (Oftenthe stacking velocities themselves are of minimal im-portance.) Constant-velocity stacks often contain manyCMP traces and sometimes consist of an entire line.

The velocity spectrum method is described in thenext section. Unlike the CVS method, it is based onthe crosscorrelation of the traces in a CMP gather andnot on lateral continuity of the stacked events. Becauseof this, when compared to the CVS method, it is moresuitable for data heavily contaminated with multiple re-flections and somewhat less suitable for data associatedwith complex structures.

The Velocity Spectrum

The input CMP gather in Figure 3.2-9a contains a singlereflection hyperbola from a flat interface. The mediumvelocity above the reflector is 3000 m/s. Suppose thatthis gather is NMO-corrected and stacked, repeatedly,using a range of constant velocities from 2000 to 4300m/s. Figure 3.2-9b displays the resultant stack tracesfor each velocity side by side on a plane of velocity ver-sus two-way zero-offset time. This is called the velocityspectrum (Taner and Koehler, 1969). We have trans-formed the data from the offset versus two-way timedomain (Figure 3.2-9a) to the stacking velocity versustwo-way zero-offset time domain (Figure 3.2-9b).

The highest stacked amplitude occurs with a veloc-ity of 3000 m/s. This is the velocity that should be usedto stack the event in the input CMP gather. The low-amplitude horizontal streak on the velocity spectrumresults from the contribution of small offsets, while thelarge-amplitude region on the spectrum is due to thecontribution of the full range of offsets (Sherwood andPoe, 1972). Hence, we need long offsets for good res-olution on the velocity spectrum. A way to minimizethe streak effect of finite-cable length on the velocity-spectrum is to transform the CMP gather from offsetto velocity domain by way of discrete Radon transform(Section 6.4).

A CMP gather associated with a layered earthmodel is shown in Figure 3.2-10a. Based on the stackedamplitudes, the following picks for stacking velocityfunction are made from the velocity spectrum (Figure3.2-10b): 2700, 2800, and 3000 m/s. These picks corre-spond to the shallow, middle, and deep events, respec-tively. The velocity spectrum not only can provide thestacking velocity function, but it also allows one to dis-tinguish between primary and multiple reflections.

The quantity displayed on the velocity spectra inFigures 3.2-9b and 3.2-10b is the stacked amplitude.

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FIG. 3.2-5. Constant-velocity moveout corrections applied to a CMP gather using a velocity range of 1500-4200 m/s withan increment of 100 m/s. Events A and B are discussed in the text.

When the signal-to-noise ratio of the input data is poor,then the stacked amplitude may not be the best displayquantity. The aim in velocity analysis is to obtain picksthat correspond to the best coherency of the signal alonga hyperbolic trajectory over the entire spread length ofthe CMP gather. Neidell and Taner (1971) describedvarious types of coherency measures that can be usedas attributes in computing velocity spectra.

Measure of Coherency

Consider the CMP gather with a single reflectionsketched in Figure 3.2-11. Stacked amplitude S at two-

way zero-offset time t0 is defined as

S =M

i=1

fi,t(i), (3 − 16)

where fi,t(i) is the amplitude value on the ith trace attwo-way time t(i), and M is the number of traces in theCMP gather. Two-way time t(i) lies along the stackinghyperbola associated with a trial velocity vstk:

t(i) = t02 +

x2i

v2stk

. (3 − 17)

Normalized stacked amplitude is defined as

NS =Mi=1 fi,t(i)

Mi=1 |fi,t(i)|

, (3 − 18)

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FIG. 3.2-7. Constant-velocity stacks of 24 CMP gathers using a range of velocities 1500-4400 m/s with an increment of 100m/s.

where the range of NS is 0 ≤ NS ≤ 1. As for thestacked amplitude given by equation (3-16), the nor-malized stacked amplitude given by equation (3-18) isdefined at two-way zero-offset time.

Another quantity that is used in velocity spectrumcalculations is the unnormalized crosscorrelation sumwithin a time gate T that follows the path correspond-ing to the trial stacking hyperbola across the CMPgather. The expression for the unnormalized crosscor-relation sum is given by

CC =12

t

M

i=1

fi,t(i)

2

−M

i=1

f2i,t(i) , (3 − 19a)

or, by way of equation (3-16),

CC =12

t

S2t −

M

i=1

f2i,t(i) , (3 − 19b)

where CC can be interpreted as half the difference be-tween the output energy of the stack and the input en-ergy. The outer summation is over the two-way zero-offset time samples t within the correlation gate T .

A normalized form of CC is another attribute thatoften is used in velocity spectrum calculations and isgiven by

NC = MFt

M−1

k=1

M−k

i=1

fi,t(i)fi+k,t(i+k)

t f2i,t(i) t f2

i+k,t(i+k),

(3 − 20)where MF = 2/[M(M − 1)].

Another coherency measure used in computing ve-locity spectrum is the energy-normalized crosscorrela-tion sum

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FIG. 3.2-8. Stacks of 100 CMP gathers using seven velocity functions which are a percent apart from a central function.

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FIG. 3.2-9. Transformation of a synthetic CMP gather con-taining a single reflection event from offset to velocity do-main. Each trace in the velocity-stack gather (b) is a stack ofthe traces in the CMP gather (a) using a constant-velocityNMO correction.

FIG. 3.2-10. Transformation of a synthetic CMP gathercontaining three reflection events from offset to velocity do-main. Each trace in the velocity-stack gather (b) is a stack ofthe traces in the CMP gather (a) using a constant-velocityNMO correction.

FIG. 3.2-11. Stacked amplitude along a hyperbolic trajec-

tory. Amplitudes fi,t(i) along the best-fit hyperbola (equa-

tion 3-17) defined by optimum stacking velocity vstk are

summed to get the stacked amplitude S (equation 3-16).

EC =2

(M − 1)CC

tMi=1 f2

i,t(i)

. (3 − 21)

The range of EC is [−1/(M − 1)] < EC ≤ 1.Finally, semblance, which is the normalized output-

to-input energy ratio, is given by

NE =1M

tMi=1 fi,t(i)

tMi=1 f2

i,t(i)

. (3 − 22a)

The following expression shows the relation of NE toEC:

EC =1

M − 1M × NE − 1 . (3 − 22b)

The range of NE is 0 ≤ NE ≤ 1.Table 3-6 shows the values of the attributes defined

by equation (3-16) and equations (3-18) through (3-22)for the special case of a two-fold CMP gather where thesecond trace is a scaled version of the first as follows:

f1,t = ft, (3 − 23a)

f2,t = a ft. (3 − 23b)

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Table 3-6. Various measures of coherency as appliedto the two-fold case given by equation (3-23).

Attribute a = 0.5 a = −0.5

Stacked Amplitude 1.5f(t) 0.5f(t)S (equation 3-16)

Normalized 1 0.333Stacked AmplitudeNS (equation 3-18)

Unnormalized 0.5 t f2(t) −0.5 t f2(t)Crosscorrelation SumCC (equation 3-19b)

Normalized 1 1Crosscorrelation SumNC (equation 3-20)

Energy-Normalized 0.8 −0.8Crosscorrelation SumEC (equation 3-21)

Semblance 0.9 0.1NE (equation 3-22a)

Several conclusions can be made from the resultsshown in Table 3-6. Note that stacked amplitude is sen-sitive to trace polarity. The unnormalized crosscorrela-tion offers a better standout of the strong reflections onthe velocity spectrum, while the normalized or energy-normalized crosscorrelation brings out weak reflectionson the velocity spectrum. As equation (3-22b) implies,semblance is a biased version of the energy-normalizedcrosscorrelation sum.

The velocity spectrum normally is not displayed asshown in Figures 3.2-9b or 3.2-10b. Instead, two pop-ular types of displays are used to pick velocities in theform of a gated row plot or a contour plot as shownin Figure 3.2-12. Another quantity that helps pickingis the maxima of the coherency values from each timegate displayed as a function of time next to the veloc-ity spectrum, as shown in Figure 3.2-12. Unless other-wise indicated, the unnormalized correlation was usedto construct the velocity spectrum of the synthetic CMPgather (Figure 3.2-12a) that is used in subsequent dis-cussions.

Factors Affecting Velocity Estimates

Velocity estimation from seismic data is limited in ac-curacy and resolution for the following reasons:

(a) Spread length,(b) Stacking fold,(c) signal-to-noise ratio,(d) Muting,(e) Time gate length,(f) Velocity sampling,(g) Choice of coherency measure,(h) True departures from hyperbolic moveout, and(i) Bandwidth of data.

Figure 3.2-13 shows a synthetic CMP gather withvelocity spectra generated by using gradually decreas-ing spread lengths. Lack of large-offset informationmeans lack of the significant moveout required for ve-locity discrimination. Note the loss in sharpness of thepeaks in the velocity spectra computed from the small-spread portion of the CMP gather. Resolution decreasesfirst in the deeper part of the spectrum where there islittle moveout (Table 3-1).

Figure 3.2-14 shows velocity spectra computedfrom a real data set using spread lengths as indicated.The broadened peaks caused by the use of smallerspreads indicate loss of resolution in the velocity spec-trum. This problem may be compounded by the poorsignal-to-noise ratio or residual static shifts. An exam-ple of residual statics effect is shown in Figure 3.2-15.Velocity spectrum computed from a spread length thatincludes small offsets (center panel) infers an incorrectvelocity function. As the spread length is made smaller(right panel), the velocity trend becomes indistinct.

What if only the far offsets are included when com-puting the velocity spectrum? Although far-offset dataare needed to better resolve the velocity picks, there is astretching problem in the far-offset region. Therefore, avelocity spectrum computed on the basis of only the far-offset region of a CMP gather suffers from the effects ofmuting at shallow times. This problem is demonstratedin Figure 3.2-16, where the spread is increasingly con-fined to the far-offset region of the input CMP gather.Note the loss of coherency peaks from the shallow eventsbecause of muting, and the further degradation of thecoherency peaks corresponding to deeper events. Thus,adequate resolution in the velocity spectrum can onlybe obtained with a sufficiently large spread that spansboth near and far offsets. This is analogous to the les-son learned in Section 1.1 on temporal resolution, whichrequires both low and high frequencies.

Stacking fold plays a significant part in the degreeof resolution achieved from velocity spectra. In contem-porary seismic data acquisition, it is common to recorddata with 240 or more channels. For computational sav-ings, high-fold data sometimes are reduced to a low-foldequivalent gather by partial stacking. The idea is to

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FIG. 3.2-12. (a) A CMP gather, and two ways of displaying velocity spectrum computed from this gather: (b) gated rawplot, and (c) contour plot.

FIG. 3.2-13. Effect of spread length on velocity resolution. Lack of long offsets causes loss of resolution, especially at latertimes.

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FIG. 3.2-14. Missing large-offset traces cause loss of resolution in the velocity spectra, especially at later times.

FIG. 3.2-15. Velocity spectra can be distorted severely, particularly in areas with statics problems. When coupled with alack of long-offset traces, the velocity function that is derived can be misleading (center and right).

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FIG. 3.2-16. Lack of short-offset traces can degrade the velocity spectrum. Note the loss of information at shallow timesand poor picks at later times.

FIG. 3.2-17. Partial stacking can reduce computational cost. However, do not use partial stacking if it could degrade thevelocity spectrum. (In this example, 8-fold partial stacking is too much.)

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FIG. 3.2-18. The synthetic CMP gather derived from the velocity function depicted in Figure 3.1-8 and the same gather withnoise added at various strengths. The numbers on top represent the ratio of peak signal amplitude to peak noise amplitude.

stack a number of traces in a CMP gather from adjacentoffsets to produce a CMP gather with lower fold. Forexample, a reduction of fold from 64 to 16 amounts toproducing one output trace for each set of four adjacentinput traces. Partial stacking involves differential NMOapplication to each group of adjacent traces using a rea-sonable, previously estimated velocity function so thatprimaries are aligned before stacking. The CMP gatherin Figure 3.2-17 was partially stacked down to 32-, 16-and 8-fold gathers. Corresponding velocity spectra alsoare shown in Figure 3.2-17. No harm was done by re-ducing the fold to 32. Even the 16-fold data seem toproduce accurate picks. However, use of lower fold sig-nificantly shifts the peaks in the spectrum. Reducingthe fold by partial stacking merely to save computationmust not be done at the expense of accuracy.

Noise in seismic data has a direct effect on the qual-ity of a velocity spectrum. Add band-limited randomnoise to the CMP gather at increasingly higher levelsof amplitude (Figure 3.2-18). The corresponding veloc-ity spectra are shown in gated row plot form in Figure3.2-19 and, for comparison, in contour form in Figure3.2-20. The velocity spectrum distinguishes signal alonghyperbolic paths even with high levels of random noise.(Refer to the velocity spectrum for SNR = 3 in Figure3.2-19.) This is because of the power of crosscorrela-tion in measuring coherency. The accuracy of the veloc-ity spectrum is limited when the signal-to-noise ratiois poor. Refer to SNR = 1 in Figures 3.2-19 or 3.2-20.The event at 0.8 s still can be picked, but the others aredifficult to distinguish.

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FIG. 3.2-19. Velocity spectra derived from the CMP gathers in Figure 3.2-18. The display mode is gated row.

FIG. 3.2-20. Velocity spectra derived from the CMP gathers in Figure 3.2-18. The display mode is contour. See Figure 3.2-19for the gated row plots of the same spectra for comparison.

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FIG. 3.2-21. Muting effect on correlation values: (a) CMP gather, (b) mute compensated, (c) no compensation.

As a result of moveout correction, the waveformalong a reflection hyperbola is stretched (Section 3.1).Stretching is more severe in the shallow part of themoveout-corrected gather, especially at large offsets.The stretched zone must be muted to prevent degra-dation of the stacked amplitudes associated with shal-low events. However, muting reduces fold in the stack-ing process for shallow data (Figure 3.1-11c). It alsohas an adverse effect on the velocity spectrum, for itcauses weakening of the peak amplitude that falls withinthe mute zone, as demonstrated in Figure 3.2-21. Thesepeaks must be corrected for the weakening effect of themuting process. This is done by multiplying stacked am-plitudes by a scale factor equal to the ratio of the actualmultiplicity to the number of live traces in the mutezone.

The velocity spectrum is computed along hyper-bolic search paths for a range of constant velocity val-ues, or constant ∆tNMO (equation 3-2a). The hyper-bolic path spans a two-way time gate specified at zero-offset. Figure 3.2-22 shows velocity spectra computedwith four different gate lengths. If the gate length cho-sen is too coarse, the spectrum suffers especially fromlack of temporal resolution. This becomes more evidenton the same spectra displayed in contour form in Fig-ure 3.2-23. In practice, the gate length is chosen betweenone-half and one times the dominant period of the sig-nal, typically 20 to 40 ms. Since the dominant periodcan be time-variant (small in early and large in latetimes), the gate length can be specified accordingly.

The velocity range used in the analysis must bechosen carefully; it should span the velocities that cor-respond to those of primary reflections present in the

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FIG. 3.2-22. Too large a correlation gate length can lower resolution.

FIG. 3.2-23. The same velocity spectra as in Figure 3.2-22 displayed using the contour display mode for comparison.

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FIG. 3.2-24. The CMP gathers associated with six neighboring midpoint locations. The reflectors have a gentle downdipfrom left to right.

FIG. 3.2-25. (a) Velocity spectrum derived from a single CMP gather (CMP 1) in Figure 3.2-24, (b) from the sum of sixCMP gathers in Figure 3.2-24, and (c) from the sum of six individual velocity spectra.

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Table 3-7. Acceptable velocity errors (after Schneider,1971).

% Error % ErrorUse of Velocity for rms for Interval

NMO corrections for 2-10 —conventional stack

Structural anomaly detection: 0.5 —30-m anomaly at 3000-m depth

Gross lithologic identification: 0.7 10300-m interval at 3000-m depth

Stratigraphic detailing: 0.1 3150-m interval at 3000-m depth

CMP gather. The velocity increment must not be toocoarse, for it can degrade the resolution, especially forhigh-velocity events.

Several options are considered in constructing thevelocity spectrum. Partial stacking is one option that al-ready was discussed. Band-pass filtering and automaticgain control (AGC) sometimes can improve the cross-correlation process, especially when the input gatherhas poor signal-to-noise ratio.

Another way to improve the quality of a velocityspectrum is to use several neighboring CMP gathers inthe analysis. Figure 3.2-24 shows six neighboring CMPgathers. By using the first CMP gather in the group, weget the velocity spectrum in Figure 3.2-25a. There aretwo ways to analyze these gathers as a group. One wayis to sum the gathers and compute the velocity spec-trum from the sum. This is shown in Figure 3.2-25b.Another way is to compute the velocity spectra fromeach individual gather and sum the spectra as shown inFigure 3.2-25c. Clearly, the former is more cost-effectivethan the latter. In practice, the number of CMP gathersthat may be used must be chosen so that there is negli-gible dip across the gathers under consideration. If thestructural dip is significant, then the number of CMPgathers included in the velocity analysis must be keptsmall. Note that the peak corresponding to the shallowevent in Figure 3.2-25b is smaller than its counterpartin Figure 3.2-25c. Look closely at the CMP gathers inFigure 3.2-24 and note the slight difference in travel-times from gather to gather, especially for the shallowevent. Summing these gathers distorts the hyperbolicpath and causes degradation in the velocity spectrum.

When the input gather has a significant noise level,some smoothing may be done on the velocity spectrum

matrix by averaging over velocity or time gates, or bysome combination of the two. Another way to suppresssmall-amplitude correlation peaks that may be relatedto the ambient noise level in the data is to apply somepercentage of bias to the correlation values. Biasingrefers to subtracting a constant value from the corre-lation values over the entire velocity spectrum. Variouscombinations of averaging and biasing of correlation val-ues also are used in practice. Finally, for computationalefficiency, correlation values may be computed within aspecified velocity corridor. The corridor must be chosenso that it spans the velocity variations vertically andlaterally in the survey area.

Experience in a survey area helps when pickingappropriate stacking velocities for primary reflectionsfrom velocity spectra. Acceptable velocity errors varydepending on use of the estimated velocities (Table 3-7).

Interactive Velocity Analysis

With the availability of powerful workstations, ef-ficiency in seismic data analysis at large has in-creased enormously. Applications that involve numer-ically intensive computations and large input-outputoperations are performed using multiprocessor servers,and results are viewed and evaluated using high-performance graphics workstations. Interactive dataanalysis enables efficient parameter testing that isneeded for many of the steps in a processing se-quence, such as filtering, deconvolution, and gain. Ad-ditionally, interactive analysis provides efficiency inpicking events — first breaks on shot records forrefraction statics, reflection times on migrated sec-tions, and velocity functions from velocity spectra.

Figure 3.2-26 shows a CMP gather and its veloc-ity spectrum displayed in color. Note the distinct ve-locity trend with changes in vertical gradient at 600,1400, and 2000 ms. Color display of velocity spectra isused for interactive picking of semblance peaks, whereasthe contour and gated row plots are used for the tra-ditional paper display of velocity spectra. Figure 3.2-27 shows the same velocity spectrum as in Figure 3.2-26 with the velocity picks associated with primary re-flections. The hyperbolic traveltime trajectories thatcorrespond to the velocity picks are superimposed onthe CMP gather to observe any discrepancy betweenthe modeled and the actual traveltimes, and thus ver-ify the accuracy of the velocity picks. Further verifi-cation of the picks can be made by applying moveoutcorrection and examining the flatness of events on the

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CMP gather as shown in Figure 3.2-27. The undercor-rected event at about 800 ms is the water-bottom mul-tiple; this event is represented in the velocity spectrumby the isolated peak to the left of the velocity trend.

Figure 3.2-28 shows the same gather and veloc-ity spectrum as in Figure 3.2-27 but with a velocityfunction erroneously picked along a high-velocity trendto cause undercorrection of the reflections. Note thediscrepancy between the modeled and actual travel-times on the CMP gather before moveout correction,and the misalignment of the reflections after moveoutcorrection. The case of overcorrection caused by erro-neously low velocities is demonstrated in Figure 3.2-29.

A localized mispick along a velocity trend yieldsa physically implausable interval velocity value as ill-strated in Figures 3.2-30 and 3.2-31. The strategyfor picking velocities from velocity spectra is basedon tracking the velocity trend that coincides withsemblance peaks associated with the primary reflec-tions. Whether these reflections are associated withkey geological markers or not is irrelevant — as manypicks as necessary should be picked so as to honorchanges in vertical velocity gradients and thus obtainthe best stack. However, picks at too close time in-tervals can yield anomalous interval velocities fromDix conversion (Section J.4); therefore, they shouldnot be used for deriving interval velocities. (Chap-ter 9, in addition to Dix conversion, is devoted toseveral techniques for estimating interval velocities.)

To derive plausable interval velocities from thepicked rms velocity functions, first intersect the timehorizons picked from the time-migrated volume ofdata with the velocity functions at analysis loca-tions and extract horizon-consistent rms velocity func-tions. Then, perform spatial interpolation to derivehorizon-consistent rms velocity profiles along line tra-verses or maps over the survey area. Next, per-form Dix conversion of the horizon-consistent rmsvelocities to interval velocities vint (Dix, 1955):

vint =v2

ntn − v2n−1tn−1

tn − tn−1, (3 − 24)

where vn and vn−1 are the rms velocities atthe layer boundaries n and n − 1, respectively;and tn and tn−1 are the horizon times at theselayer boundaries. An alternative method for derivinghorizon-consistent rms velocities is presented next.

Horizon Velocity Analysis

One way to estimate velocities with the accuracy re-quired for detailed structural or stratigraphic studies is

to analyze the particular horizon of interest, continu-ously. Such a detailed velocity estimation is called hori-zon velocity analysis (HVA). Horizon velocity analysisis an efficient way to get velocity information at everyCMP location along selected key horizons, as opposedto the conventional velocity analysis that provides ve-locity information at every time gate at selected CMPlocations. The underlying principle is the same as thatof the velocity spectrum. The output coherency valuesderived from hyperbolic time gates are displayed as afunction of velocity and CMP position. Correlation val-ues are computed from a gate that includes the horizonof interest. Horizon times are digitized and input to thehorizon velocity analysis. Figures 3.2-32 shows a stackedsection and Figure 3.2-33 shows HVA semblance spec-tra over five horizons. Note the short-wavelength vari-ations of stacking velocities along the line traverse —such variations ordinarily are not captured by veloc-ity analyses conducted at sparse CMP intervals. Thesesemblance spectra can be picked to obtain horizon-consistent rms velocities which are then used to de-rive interval velocities (equation 3-24). Similar typesof computational details, such as smoothing and bi-asing, are considered applicable to velocity spectrum.

Whenever there are structural discontinuities on astacked section, HVA is carried out on segments of thehorizon that are separated by faults. Horizon velocityanalysis can improve a stacked section in areas withcomplex overburden structure that may cause nonhy-perbolic moveout. This is somewhat surprising, sinceHVA still is based on hyperbolic moveout. Neverthe-less, in practice, HVA provides the detailed lateral ve-locity variations along a marker horizon, which may bemissed by conventional velocity analysis locations thatare sparsely spaced along the line. Consider horizon Ain Figure 3.2-34, which is below the salt dome S. Thesalt dome behaves as a complex overburden, causingthe raypaths that are associated with the underlyingreflectors to bend. Note the rapid lateral changes invelocity and the improvement in the CMP stack afterusing the HVA picks. The rapid change in stacking ve-locity associated with the base of the salt is typical.Starting on the left, the reflector is deep in time andflat. Then it dips; hence, the higher stacking velocity.Then it gets shallower; hence, the lower stacking veloc-ity. Then it dips again, yielding a higher stacking veloc-ity. Finally, it becomes flat; hence, a decrease in velocity.

A velocity section derived from HVA is structure-consistent, whereas a velocity section derived from ver-tical velocity functions picked at selected analysis lo-cations along a line traverse is, in general, structure-independent. Consider the stacked section in Figure3.2-35 with interpreted time-horizon segments associ-ated with subsurface geological markers. The semblance

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FIG. 3.2-26. A CMP gather and its velocity spectrum.

FIG. 3.2-27. The same CMP gather and its velocity spectrum (left and center panels) as in Figure 3.2-26 with the pickedvelocities denoted by + marks coincident with the semblance peaks. The curve to the right of the semblance peaks is theinterval velocity function derived from the picked rms velocity function. The far right panel shows the CMP gather aftermoveout correction using the picked rms velocity function.

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FIG. 3.2-28. The same CMP gather and its velocity spectrum (left and center panels) as in Figure 3.2-26 with the erroneouslypicked velocities denoted by + marks. The curve to the right of the semblance peaks is the interval velocity function derivedfrom the picked rms velocity function. The right-hand panel shows the CMP gather after moveout correction using the pickedrms velocity function. Note the undercorrection of events caused by the erroneously too high moveout velocities.

FIG. 3.2-29. The same CMP gather and its velocity spectrum (left and center panels) as in Figure 3.2-26 with the erroneouslypicked velocities denoted by + marks. The curve to the right of the semblance peaks is the interval velocity function derivedfrom the picked rms velocity function. The right-hand panel shows the CMP gather after moveout correction using the pickedrms velocity function. Note the overcorrection of events caused by the erroneously too low moveout velocities.

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FIG. 3.2-30. The same CMP gather and its velocity spectrum as in Figure 3.2-26 with an erroneously picked velocity at1875 ms. The curve to the right of the semblance peaks is the interval velocity function derived from the picked rms velocityfunction. Note the anomalous interval velocity derived from the erroneously picked interval velocity.

FIG. 3.2-31. The same CMP gather and its velocity spectrum as in Figure 3.2-26 with an erroneously picked velocity at1875 ms. The curve to the right of the semblance peaks is the interval velocity function derived from the picked rms velocityfunction. Note the anomalous interval velocity derived from the erroneously picked interval velocity.

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FIG. 3.2-32. A stacked section with five marker horizons as indicated.

FIG. 3.2-33. Horizon velocity analyses along five marker horizons indicated in Figure 3.2-32. The vertical and horizontalaxes in each panel are stacking velocity and CMP axis, respectively.

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FIG. 3.2-34. A CMP-stacked section obtained by sparsely spaced conventional velocity analysis (top) and the stacked section(bottom) obtained by using velocities derived from horizon velocity analysis (HVA) (middle). The HVA for horizon A belowthe salt dome S is shown in the center.

spectra associated with the eight horizons picked fromthe stacked section computed by HVA are shown inFigure 3.2-36. The top spectrum corresponds to theshallowest time horizon (dark blue) after the water-bottom horizon. These spectra were picked to obtainthe horizon-consistent rms velocity profiles, which werethen used to derive the rms velocity section shown inFigure 3.2-37 (top). The time horizons picked from thestacked section (Figure 3.2-35) are superimposed on thevelocity section. Compare the velocity sections in Fig-ure 3.2-37 derived from HVA (top) and vertical func-tions (bottom), and note that the HVA-based velocitysection is more consistent with the subsurface struc-ture. Hence, the HVA results are more appropriate touse in Dix conversion to derive structure-consistent in-

terval velocities with meaningful magnitudes (equation3-24).

What about the quality of stacks (Figure 3.2-38)obtained from the velocity sections (Figure 3.2-37) de-rived from HVA and vertical functions? Two sets of de-tails from the stacked sections shown in Figures 3.2-39and 3.2-40 reveal that, indeed, there can be more thanmarginal differences. While the HVA-based stack showsbetter continuity of reflections, it is inferior to the con-ventional stack with regard to diffractions. This shouldbe expected since the HVA analysis is done along re-flection events and not diffractions (Figure 3.2-35).

In conclusion, if the objective is to obtain an op-timum CMP stack with the highest stack power pos-sible, conventional velocity analysis at selected CMP

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FIG. 3.2-35. A CMP-stacked section with interpreted time horizon segments associated with geological markers.

locations along the line traverse or over the 3-D surveyarea yields a robust velocity section. If, on the otherhand, the objective is to derive interval velocities fromDix conversion, then horizon-consistent velocity analy-sis yields structure-consistent results that are geologi-cally plausable.

Coherency Attribute Stacks

The various measures of coherency discussed in this sec-tion to compute velocity spectrum can also be used togenerate coherency attribute stacks. These stacks areobtained as follows:

(a) Choose a specific measure of coherency — stackedamplitude (equation 3-16), normalized stacked am-plitude (equation 3-18), unnormalized crosscorre-lation sum (equation 3-19), normalized crosscor-relation sum (equation 3-20), energy-normalizedcrosscorrelation sum (equation 3-21), or semblance(equation 3-22).

(b) Compute velcoity spectra at selected CMP loca-tions along the line and pick rms velocity functions.

(c) By interpolating between the vertical functions, de-rive an rms velocity section.

(d) Extract vertical rms velocity functions from the ve-locity section at each CMP location along the linetraverse.

(e) Apply moveout correction to CMP gathers usingthe extracted vertical functions.

(f) Now, compute not just the stacked amplitudes(equation 3-16), but also the coherency attributesusing equations (3-18) through (3-22) and thus ob-tain the coherency attribute sections.

Figure 3-2.41 shows portions of coherency attributesections associated with a field data set. These sections,in conjunction with conventional stack, may be usefulin enhancing fault patterns associated with structuralplays and identifying amplitude anomalies associatedwith stratigraphic plays. Note the discriminating powerof semblance for the most coherent reflection events inthe section. Conventional CMP stacking seems to yieldthe most robust section that preserves reflections anddiffrations. The coherency attribute sections based oncrosscorrelation show an apparent higher frequency con-tent compared to the stacked and normalized stackedsections.

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FIG. 3.2-36. Semblance spectra computed along the time horizons in Figure 3.2-35. The top spectrum corresponds to theshallowest time horizon in dark blue.

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FIG. 3.2-37. Top: the velocity section derived from the HVA-based semblance spectra shown in Figure 3.2-36; bottom: thevelocity section derived from the interpolation of the vertical rms velocity functions picked at selected CMP locations alongthe line traverse. Superimposed on these sections are the time horizon segments interpreted from the stacked section in Figure3.2-35.

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FIG. 3.2-38. Top: the CMP-stacked section derived from the HVA-based velocity section shown in Figure 3.2-37 (top); bot-tom: the CMP-stacked section derived from the velocity section shown in Figure 3.2-37 (bottom) derived from the interpolationof the vertical rms velocity functions picked at selected CMP locations along the line traverse.

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FIG. 3.2-39. A detailed portion of the sections in Figure 3.2-38.

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FIG. 3.2-40. A detailed portion of the sections in Figure 3.2-38.

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FIG. 3.2-41. Coherency attribute sections: (a) stack (equation 3-16), (b) normalized stack (equation 3-18), (c) unnormal-ized crosscorrelation (equation 3-19), (d) normalized crosscorrelation (equation 3-20), (e) energy-normalized crosscorrelation(equation 3-21), and (f) semblance (equation 3-22).

3.3 RESIDUAL STATICS CORRECTIONS

Reflection times often are affected by irregularities inthe near-surface. This is best demonstrated by the realdata example in Figure 3.3-1. While the shot gatherson the left contain reflections that exhibit nearly hy-perbolic moveout, those on the right have reflectionsthat significantly depart from hyperbolic moveout. Al-though such distortions can be caused by a structuralcomplexity deeper in the subsurface, more often theyresult from near-surface irregularities.

For land data, reflection traveltimes are reduced toa common datum level, which may be flat or vary (float-ing datum) along the line. Reduction of traveltimes toa datum usually requires correction for the near-surfaceweathering layer in addition to differences in elevation

of source and receiver stations. Estimation and correc-tion for the near-surface effects usually are performedusing refracted arrivals associated with the base of theweathering layer (Section 3.4).

Traveltime corrections to account for the irregu-lar topography and near-surface weathering layer arecommonly known as field statics or refraction staticscorrections (Section 3.4). These corrections remove asignificant part of the traveltime distortions from thedata — specifically, long-wavelength anomalies. Nev-ertheless, these corrections usually do not account forrapid changes in elevation, the base of weathering, andweathering velocity.

Removal of near-surface distortions on reflectiontimes associated with deeper reflectors is routinely doneby lowering the shots and receivers along vertical ray-

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FIG. 3.3-1. Common-shot gathers from a land profile. Note the departures from hyperbolic traveltimes on the gathers atthe right.

paths from the surface to a datum below the weather-ing layer. The positioning of shots and receivers to adatum along vertical raypaths amounts to static timecorrections in a surface-consistent manner (Taner et al.,1974). The term static implies that it is a constant timeshift for an entire trace, and the term surface-consistentimplies that the time correction depends only on thesurface location of the shot and receiver associated withthe trace.

Figures 3.3-2a and 3.3-3a are selected CMP gath-ers (with field statics corrections) which were NMO cor-rected using a set of preliminary velocity picks derivedfrom the velocity analyses in Figure 3.3-4. Deviationsfrom the hyperbolic trends on the CMP gathers sig-nificantly degrade the quality of some of the velocityspectra. For instance, velocity analysis at CMP loca-tion 188 yields relatively poorer quality picks than thosefrom other velocity analyses. The CMP gathers in theneighborhood of CMP location 188 have more travel-time distortions compared to some other CMP gathers(Figure 3.3-2a). The resulting stacked section could bemisleading in that residual statics may cause dim spotsalong the reflection horizons as well as false structures(Figure 3.3-5a), particularly between midpoints 101 to245. False structures also are apparent on the rms AGCgained stack (Figure 3.3-6a) in which dim spots maynot be apparent.

Obviously a more correct picture of the subsur-face should be attained from data corrected for rapidlyvarying near-surface effects. After making these resid-ual statics corrections, the CMP gathers with traveltime

deviations show better alignment of reflections (Fig-ure 3.3-2b), while those that did not require such cor-rections are essentially unchanged (Figure 3.3-3b). Af-ter the residual statics corrections, the ungained (Fig-ure 3.3-5b) and gained stacked sections (Figure 3.3-6b)show improvement in the continuity of reflections aswell as significant elimination of false structures (referto the segment between midpoints 101 to 245).

Following the residual statics corrections, velocityanalyses almost always are repeated to update the ve-locity picks (Figure 3.3-7). Comparison of Figures 3.3-4and 3.3-7 shows that residual statics corrections haveimproved the velocity analysis. The same CMP gath-ers after NMO corrections using the updated velocitypicks are shown in Figures 3.3-8a and 3.3-9a, whilethe same gathers after residual statics corrections areshown in Figures 3.3-8b and 3.3-9b. Comparison of theCMP gathers before and after residual statics correc-tions shows significant elimination of time deviations.The resulting stacked sections using the revised velocityestimates are shown in Figure 3.3-10, while the gainedstacks are shown in Figure 3.3-11.

Compare the stacked sections in Figures 3.3-6a,band 3.3-11a,b, and observe the gradual improvement inthe following order:

(a) CMP stack based on preliminary velocity picks(Figure 3.3-4) but with no residual statics correc-tions applied (Figure 3.3-6a),

(text continues on p. 336)

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FIG. 3.3-2. CMP gathers from a land profile: (a) before residual statics corrections, (b) after residual statics corrections.(Shot gathers are shown in Figure 3.3-1.) NMO correction was applied using preliminary velocity picks derived from thespectra in Figure 3.3-4. The CMP stacks are shown in Figure 3.3-5.

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FIG. 3.3-3. CMP gathers from the same land line as in Figure 3.3-2 (a) before and (b) after residual statics corrections. Thispart of the line does not have as severe a statics problem as that shown in Figure 3.3-2. The NMO correction was appliedusing preliminary velocity picks derived from the spectra in Figure 3.3-4.

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FIG. 3.3-4. Velocity analyses before residual statics corrections along the land line shown in Figure 3.3-5. Figures 3.3-2 and3.3-3 show selected CMP gathers.

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FIG. 3.3-5. CMP stacks derived from gathers in Figures 3.3-2 and 3.3-3. Stack (a) before and (b) after residual staticscorrections. NMO correction was applied using preliminary velocity picks derived from the spectra in Figure 3.3-4.D

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FIG. 3.3-6. The same CMP stacks as in Figure 3.3-5 with rms gain applied. Gained stacks (a) before and (b) after residualstatics corrections.D

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FIG. 3.3-7. Velocity analyses after residual statics corrections along the line shown in Figure 3.3-5. Figures 3.3-8 and 3.3-9show selected CMP gathers.

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FIG. 3.3-8. CMP gathers from the land line shown in Figure 3.3-10. CMP gathers (a) without and (b) with residual staticscorrections. (Figure 3.3-1 shows the shot gathers from the same line.) NMO correction was applied using final velocity picksderived from the spectra in Figure 3.3-7.D

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FIG. 3.3-9. CMP gathers from the same land line as in Figure 3.3-8. CMP gathers (a) without and (b) with residual staticscorrections. This part of the line does not have as severe a statics problem as the part in Figure 3.3-8. NMO correction wasapplied using final velocity picks derived from the spectra in Figure 3.3-7.D

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FIG. 3.3-10. CMP stacks derived from the gathers in Figures 3.3-8 and 3.3-9. Stack (a) before and (b) after residual staticscorrections and using revised velocity estimates from Figure 3.3-7.D

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FIG. 3.3-11. The same CMP stacks as in Figure 3.3-10 with rms gain applied. Gained stacks (a) before and (b) after residualstatics corrections and using revised velocity estimates from Figure 3.3-7.

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FIG. 3.3-12. Processing flowchart with residual statics cor-rections.

(b) stack based on preliminary velocity picks (Figure3.3-4) and with residual statics corrections applied(Figure 3.3-6b),

(c) stack based on final velocity picks (Figure 3.3-7)but with no residual statics corrections applied(Figure 3.3-11a), and

(d) stack based on final velocity picks (Figure 3.3-7)and with residual statics corrections applied (Fig-ure 3.3-11b).

Here, the velocity picks made from CMP gathers withno residual statics corrections applied are referred toas preliminary and those made from CMP gathers withresidual statics corrections applied are referred to asfinal.

Residual statics corrections usually are discussedin terms of applications to land data. However, in cer-tain cases, residual statics corrections have produceddramatic improvement in marine data. Areas with ir-regular water-bottom topography in shallow water (lessthan 25 m), and areas with rapidly varying velocity inthe sediments near the water bottom are places wherestatics corrections have been successful.

Figure 3.3-12 shows a commonly used flowchart forresidual statics corrections and velocity analysis aimedat producing an optimum stacked section. Start withCMP gathers with field statics or refraction staticscorrections applied (Section 3.4), and perform velocityanalysis, usually no more than a few locations along theline. Then, apply NMO correction with the preliminaryvelocity picks and compute residual static shifts. Applythese corrections to the original gathers, and repeat thevelocity analysis — this time at all necessary locationsalong the line. Finally, apply NMO correction and stackthe data. In some cases, there may be more than oneiteration of estimating and applying the residual staticscorrections.

In practice, the flowchart in Figure 3.3-12 usuallyis augmented with additional quality control steps. Itoften is necessary to examine CMP gathers and ve-locity analyses after residual statics corrections. Di-agnostic tools allow determination of the magnitudeof these corrections. For example, common-shot andcommon-receiver gathers indicate relative static shiftsfrom one receiver location to another and from one shotlocation to another (Figures 3.3-13 and 3.3-14, respec-tively). Also, common-shot-point and common-receiver-point stacks can be used in combination with common-receiver and common-shot gathers, respectively. Acommon-shot-point stack (Figure 3.3-15) should indi-cate the range of magnitude of shot static shifts; acommon-receiver-point stack (Figure 3.3-16) should in-dicate the range of magnitude of receiver static shiftsalong the line. These displays enable the determinationof an optimum maximum allowable shift to considerfor picking traveltime deviations from the moveout-corrected CMP data for input to residual statics esti-amtion algorithms. From the example in Figures 3.3-15and 3.3-16, the receiver component of static shifts isgreater than the shot component.

Residual Statics Estimationby Traveltime Decomposition

Static deviations from a hyperbolic traveltime trajec-tory are illustrated in Figure 3.3-17. After NMO cor-rection, the misalignment of the wavelet associated witha reflection time horizon h along the offset axis in theCMP gather will yield a poor stack trace. We want toestimate the time shifts from the time of perfect align-ment and correct for these time shifts. To do this, a

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FIG. 3.3-13. Moveout-corrected common-shot gathers from the same land line as in Figure 3.3-1. Common-shot gathers (a)before and (b) after residual statics corrections.

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FIG. 3.3-14. Moveout-corrected common-receiver gathers from the same land line as in Figure 3.3-13. Common-receivergathers (a) before and (b) after residual statics corrections.

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FIG. 3.3-15. A diagnostic display for residual statics corrections. Common-shot-point (CSP) stack (a) before and (b) afterresidual statics corrections. Note the missing shots between CMP 151 and 243. The CSP stack can be used to estimate themagnitude and spatial variation of shot statics along a line.D

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FIG. 3.3-16. A diagnostic display for residual statics corrections. Common-receiver-point (CRP) stack (a) before and (b)after residual statics corrections. The CRP stack can be used to estimate the magnitude and spatial variation of receiverstatics along a line.D

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FIG. 3.3-17. Picking traveltime deviations from NMO-corrected gathers.

model is needed for the moveout-corrected traveltimefrom a source location to a depth point on a reflect-ing horizon, then back to a receiver location. Figure3.3-18 shows the geometry and notation that will beused in defining this model. The key assumption in thecommonly used traveltime model discussed here is thatresidual statics are surface-consistent (Hileman et al.,1968; Taner et al., 1974). This means that static shiftsare time delays that depend solely on source or receiverlocations at the surface, not on raypaths in the sub-surface. Such an assumption is valid if all raypaths, re-gardless of source-receiver offset, are vertical within thenear surface layer. The surface-consistent assumptionusually is a good one, because the weathering layer of-ten has very low velocity and a strong refraction at itsbase tends to make travel paths vertical. The assump-tion may not be good for a high-velocity permafrostlayer that causes rays to bend away from vertical.

The picked traveltime tij that corresponds to thejth source station, the ith receiver station, and the kthmidpoint [k = (i + j)/2] along a specific reflection timehorizon h (Figure 3.3-17) can be approximately modeledby the following equation (Taner et al., 1974; Wigginset al., 1976):

tij = sj + ri + Gk + Mkx2ij , (3 − 25)

where sj is the residual static time shift associated withthe jth source station, ri is the residual static time shiftassociated with the ith receiver station, and Gk is thedifference in two-way time at a reference CMP locationand the traveltime at the kth CMP location along thehth horizon. This term refers to structural variationsalong the horizon and is called the structure term. Theterm Mkx2

ij is the residual moveout which is assumedto be parabolic. It accounts for the imperfect moveoutcorrection within the specific time gate that includes

FIG. 3.3-18. Surface-consistent statics model to establishthe traveltime model equation (3-25). Here, T = topographiclayer, B = base of weathering layer, D = datum to whichstatic corrections are made, R = deep reflector, j = shot sta-tion index, i = receiver station index, k = midpoint locationindex, xij = offset between the shot and receiver stations.

the reflection time horizon h. The coefficient Mk has thedimensions of (time / distance2).

To be specific in an analysis of the system of equa-tions implied by equation (3-25), assume ns shot lo-cations, nr receiver locations, and nG CMP locations.Then define the fold as nf . The purpose is to decom-pose the observed traveltimes estimated (picked) fromthe data tij to their individual components as definedon the right side of equation (3-25). The number of timepicks (or individual equations) is equal to nG ×nf . Thenumber of unknowns is ns + nr + nG + nG. Typically,(nG ×nf ) > (ns +nr +nG +nG); hence, there are moreequations than unknowns.

We can formulate a least-squares problem in whichwe must minimize the sum of the least-squares errorenergy between the observed traveltime picks tij andthe modeled traveltimes tij :

E =ij

tij − tij2. (3 − 26)

Residual statics corrections involve three phases:

(1) Picking traveltime deviations tij based on crosscor-relation of traces in a CMP gather with a referenceor pilot trace that needs to be defined in some fash-ion,

(2) Modeling tij by way of equation (3-25) and decom-posing it into its components: source and receiverstatics, structural and residual moveout terms, and

(3) Applying the derived source and receiver termssj and ri, respectively, to traveltimes on the pre-NMO-corrected CMP gathers.

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The picking phase relates to estimating traveltimestij from the data. Several picking schemes are in use inthe industry. The one described here often is known as apilot trace scheme. Starting with the CMP gathers thatare NMO-corrected using a preliminary velocity func-tion(s), trace amplitudes are scaled to a common rmsamplitude in the time gate(s) to be used for picking.For clarity, consider a time gate over the kth midpointgather as illustrated in Figure 3.3-17. (It is preferable tostart with a gather that has a good signal-to-noise ra-tio.) A stack trace then is constructed within the timegate h. Each individual trace in the gather is cross-correlated with the stack trace. Time shifts tij , whichcorrespond to maximum crosscorrelations, are picked.A preliminary pilot trace is constructed by stacking thetime-shifted traces in the gather. This preliminary pilottrace is, in turn, crosscorrelated with the original tracesin the gather and new values for tij are computed. A fi-nal pilot trace is constructed again by stacking the orig-inal traces shifted by the new values for tij . This finalpilot trace is crosscorrelated with the traces of the nextgather to construct the preliminary pilot trace for thatgather. The process is performed this way on all CMPgathers moving to the left and right from the startingpoint. The picked time deviations tij are passed on tothe next phase, which involves decomposing these timepicks into components as defined by equation (3-25).

Several practical issues are involved in the pick-ing phase. Band-pass filtering often helps to estimate areliable time shift that corresponds to the peak cross-correlation value. Selection of the time window used forcrosscorrelation is another important factor. If needed,the window should be allowed to change laterally so thatit follows the marker horizon(s). A maximum thresholdfor the correlation shift can be imposed to prevent unre-alistically large time deviations from being passed on tothe decomposition phase. Any time deviations greaterthan a specified maximum allowable shift can be set tothat shift, or rejected altogether. Alternatively, the re-jected shift can be replaced by a secondary correlationpeak value. Finally, the input CMP gathers must beNMO-corrected by using a regional velocity function orvelocities derived from a preliminary velocity analysis.These parameters are discussed in detail later in thesection.

The next step in residual statics corrections in-volves least-squares decomposition of the time picks tij .For a basic understanding of this step, consider thefollowing special problem. Suppose there are four ob-servations tij measured at receiver locations xi, wherei = 1, 2, 3, 4. We want to fit the observed data into astraight line t = a + bx, which is best in a least-squares

error sense. Start with the following set of equations:

t1 ≈ a + bx1

t2 ≈ a + bx2

t3 ≈ a + bx3

t4 ≈ a + bx4.

(3 − 27)

There are two unknowns, a and b, and four equations.This problem is similar to decomposing the time picksinto various components as described by equation (3-25). We define the error series ei, such that

−t1 + a + bx1 = e1

−t2 + a + bx1 = e2

−t3 + a + bx1 = e3

−t4 + a + bx1 = e4.

(3 − 28)

We want to minimize the energy of the cumulative er-rors as defined by

E =i

e2i . (3 − 29a)

The energy for the ith error is computed by squar-ing both sides of equation (3-28)

e2i = −ti + a + bxi

2. (3 − 29b)

By substituting into equation (3-29a) and summing overi = 1, 2, 3, 4, we obtain

E =i

t2i + 4a2 + b2

i

x2i − 2a

i

ti

− 2bi

xiti + 2abi

xi.(3 − 30)

To find the best line, we want to determine un-knowns a and b so that error energy E is minimal. Thesum E of the squared errors will attain a minimum if aand b are chosen so that

∂E

∂a= 8a − 2

i

ti + 2bi

xi = 0, (3 − 31a)

and

∂E

∂b= 2b

i

x2i − 2

i

xiti + 2ai

xi = 0. (3 − 31b)

We now have two equations with two unknowns thatcan be put into a matrix form

4 i xi

i xi i x2i

a

b

=

i ti

i xiti

. (3 − 32)

Equation (3-32) can be solved for the unknowns a and b.The minimum energy between the estimated model andthe actual data values then can be computed by solvingfor a and b and substituting into equation (3-30).

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Table 3-8. Time picks ti at various xi locations.

i xi ti

1 1 2.42 2 2.93 3 3.64 4 4.1

Consider the time values at various x locationsspecified in Table 3-8. Using these time picks, the el-ements of the coefficient matrix on the left side and thecolumn matrix on the right side of equation (3-32) are

4 1010 30

ab

= 1335.4 . (3 − 33)

The solution for equation (3-33) is a = 1.8 andb = 0.58. Note that we are not constrained by the num-ber of observations when setting up the least-squaresproblem. This problem involves more equations (3-27)than unknowns, a and b.

The least-squares approach has a wide range ofapplications in applied geophysics. We also formulatethe inverse filtering used in deconvolution based on aleast-squares procedure (Section B.5). The Wiener fil-ter itself is based on a least-squares estimation of a fu-ture sample point in a given time series. Here, we dis-cussed another application of the least-squares method— residual statics estimation by traveltime decompo-sition, which involves more equations than unknowns.In Chapter 9, when discussing 2-D surface data pro-cessing, we encounter the problem of fitting irregularlyspaced observations into a uniform grid. This involvesleast-squares fitting to a local plane (Section J.5).

We now return to the problem of residual staticsestimation and refer to equation (3-26). We have theobserved traveltime deviations tij picked from moveout-corrected CMP gathers within a specified time gate thatincludes a strong reflection. We want to estimate theparameters — the surface-consistent source and receiverstatic shifts, sj and ri, respectively. These parametersare related to the modeled traveltime deviations tij byway of the model equation (3-25). If there are 50 000picks, then you have 50 000 model equations.

Substitute for tij from equation (3-25) and mini-mize the error energy E in equation (3-26) by requiring

∂E

∂sj=

∂E

∂ri=

∂E

∂Gk=

∂E

∂Mk= 0, (3 − 34)

which yields ns+nr +nG+nG equations and that manyunknowns.

These equations can be solved for the residual stat-ics associated with ns source locations, nr receiver lo-cations, nG structural terms, and nG residual moveout

terms. For a real data set, the number of these terms canbe large. It is easy to solve the problem with the twoparameters given by equation (3-34). However, whendealing with a large number of linear equations, the so-lution must be done accurately and efficiently. Wigginset al. (1978) used the Gauss-Seidel iterative procedureto solve for equations (3-34) (Section C.4).

The Gauss-Seidel method is best described by re-turning to the earlier line fitting example and solvingfor a and b in equation (3-33). When written as normalequations, we have

4a + 10b = 13 (3 − 35a)

and

10a + 30b = 35.4. (3 − 35b)

These equations are rearranged as

a = 3.25 − 2.5b (3 − 36a)

and

b = 1.18 − 0.333a. (3 − 36b)

Since the Gauss-Seidel technique is iterative, startingvalues are needed. To start the iteration, set a = b = 0.Substituting b = 0 in equation (3-36a), we obtain a =3.25. Putting this updated solution into equation (3-36b) gives b = 0.0977. At the end of the first iteration,a = 3.25 and b = 0.0977. For the second iteration, putb = 0.0977 into equation (3-36a) to get a = 3.0075. Puta = 3.0075 into equation (3-36b) to get b = 0.1785. Thisiterative procedure is continued in Table 3-9.

The solution from the iterative procedure slowlyconverges toward the actual values, a = 1.8, b = 0.58.Convergence is not always guaranteed. Nevertheless,convergence can be attained with the Gauss-Seidelmethod, provided the unknowns are ordered properly;that is, iteration starts with the correct unknown. This

Table 3-9. Gauss-Seidel iteration for solving equation(3-33) for a and b. (Values were rounded off for tabula-tion.)

Actual values: a = 1.8, b = 0.58.Iteration a b

1 3.25 0.09775 2.4918 0.350210 2.0712 0.490215 1.9031 0.546220 1.8358 0.568625 1.8090 0.577929 1.7997 0.5807

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problem is addressed in Exercises 3-11 and 3-12. The ad-vantage of the Gauss-Seidel method is its ability to solvethe large number of simultaneous equations rapidly.

The question of when to terminate the iterationsremains. The rate at which the solution changes canbe examined after each iteration. The computation isstopped when this rate falls below a specified thresholdvalue.

The starting values for solving the normal equa-tions (3-34), can be chosen as sj = ri = Gk = Mk = 0,as in the simple numerical example. To some extent, thebest order of iteration depends on the exact nature ofthe statics problem at hand. One order of iteration thatis commonly in use follows: Compute the structure termG, the residual moveout term M , the static shift associ-ated with source location s, then the static shift associ-ated with receiver location r. The procedure cycles backto G in the next iteration and continues until conver-gence is satisfactory. The order in which the individualterms are computed theoretically can be interchanged.However, the above order forces long-wavelength vari-ations of the time picks to be concentrated mainly inthe structure term. This leads to a lesser number of it-erations (typically 2 or 3) for wavelength componentsof the statics that are less than half the largest dataoffset. A large number of iterations are required to han-dle static variations that are greater than the maximumoffset.

After computing the individual static shifts asso-ciated with each source and receiver location, theseshifts are passed on to the application phase, wherebythe shifts are applied to the pre-NMO-corrected gathertraces (Figure 3.3-13). In areas with extremely poorsignal-to-noise ratio or complicated near-surface varia-tions, multiple passes of residual statics corrections maybe necessary. In other words, output from the first passmay be NMO-corrected, new picks estimated, decom-posed, and applied, and so on.

The question remains as to whether equations (3-25) are independent of each other. This is important,for it tells whether the solution is unique or not. Forsimplicity, consider the case of zero structure and zeroresidual moveout. In that case, equations (3-25) takethe form

tij = sj + ri. (3 − 37a)

Consider only four unknowns: sj , where j = 1, 2, andri, where i = 1, 2, which yield four equations:

t11 = s1 + r1

t12 = s2 + r1

t21 = s1 + r2

t22 = s2 + r2.

(3 − 37b)

However, from a close examination of equations (3-37b),we note that

t11 + t22 = t12 + t21. (3 − 38)

Therefore, one of the four equations (3-37b) is redun-dant, leaving three independent equations for four un-knowns.

This simple exercise shows that the statics solu-tion obtained by solving equations (3-34) suffers froman uncertainty. In particular, the solution obtained by,say, the Gauss-Seidel iteration, is nonunique. It is butone of many possible solutions. In fact, the solution maynot even be physically reasonable. Because of the prob-lem of fewer independent equations than unknowns, itmay be necessary to impose a constraint on the solutionfrom the traveltime decomposition (equation 3-25):

(a) A plausable constraint is that the difference be-tween shot and receiver statics be minimal (Gulu-nay, 1985). It can be argued that this may not bevalid, as is the case with dynamite data in whichshots and receivers do not occupy the same physi-cal locations.

(b) Other possible constraints can be in the form ofrestrictions on the spatial variation of structure,moveout, or statics terms themselves; all are usedin various practical implementations. For instance,one may postulate a model equation for the resid-ual statics problem, where the structure term hasbeen omitted. Then, you have to design your pick-ing strategy accordingly, such that you do not ac-cumulate the traveltimes as you go from one CMPlocation to another. This may be preferable in ar-eas with very low-relief structures.

(c) We may want to impose the surface-consistencyrule in a strict sense, and assign the same staticshift to a shot and receiver if they occupy nearlythe same physical location.

(d) We may also opt to set the residual moveout co-efficient M to a constant across the line if we arefairly confident of the velocity control. This thengives us fewer parameters to estimate, thus mak-ing the residual statics solution more likely to bestable and physically plausable.

Residual Statics Estimationby Stack-Power Maximization

Estimation of traveltime deviations from NMO-corrected CMP gathers may fail with land data whichhave low fold and poor signal-to-noise ratio. As a re-sult, residual statics solution by traveltime decomposi-tion can be erratic and unstable. A more robust alterna-tive for surface-consistent estimates of shot and residual

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static shifts is based on minimizing the difference be-tween modeled and actual traveltime deviations (equa-tion 3-26) associated with a reflection event on moveout-corrected gathers. Specifically, surface-consistent staticshifts also can be determined by maximizing the powerof stacked traces (Ronen and Claerbout, 1985).

The conceptual basis of the method of stack powermaximization is intuitively simple. Consider determin-ing the residual static at a shot station. As in the caseof residual statics estimation by traveltime decomposi-tion, this method also is applied to moveout-correcteddata.

(a) Apply a static shift to all the traces in the common-shot gather associated with the station under con-sideration.

(b) Stack over a time gate the CMP gathers that in-clude traces from that shot gather.

(c) Compute the cumulative energy of the stackedtraces from step (b) by summing the squared am-plitudes.

(d) Repeat steps (a), (b), and (c) for a range of staticshifts.

(e) Choose the static shift that yields the highest stackpower and assign it to the shot location under con-sideration.

(f) Apply the shot residual static shift associated withthe highest stack power to all the traces in the shotgather.

(g) Stack the CMP gathers that include traces fromthis shot gather.

(h) Move to the next shot station and repeat steps (a)through (g).

The process is then repeated for the receiver stationsusing common-receiver gathers.

This formal recipe for stack-power maximizationis intensive both computationally and in terms of datamovement. A practical alternative involves creating twosupertraces — one from the traces of the common-shotor common-receiver gather under consideration, and asecond one from the traces of the stacked traces associ-ated with the common-shot or common-receiver gather(Ronen and Claerbout, 1985). A supertrace is createdby augmenting the individual segments of traces withinthe specified time gate in a gather, one followed by theother with a zone of zero-amplitude samples betweenthem. The subtlety of the method to keep in mind isthat the stack supertrace does not include the contri-bution of the traces from the common-shot or common-receiver gather.

Define the shot and stack supertraces by the timeseries F (t) and G(t), respectively. The stack power de-fined as the power of the sum of these two traces over

the time gate t is

P (∆t) =t

[F (t − ∆t) + G(t)]2, (3 − 39a)

where ∆t is the trial static shift applied to the shotsupertrace F (t). By expanding the squared term, weobtain

P (∆t)=t

F 2(t−∆t)+t

G2(t)+2t

F (t−∆t)G(t).

(3 − 39b)The first two terms are the powers of the two su-pertraces that can be defined by a constant, and thethird term is the crosscorelation of the two supertraces.Therefore, maximizing the stack power is equivalent tomaximizing the crosscorrelation (Ronen and Claerbout,1985).

Now, consider, again, determining the residualstatic at a shot station.

(a) Create the shot supertrace. To circumvent end ef-fects in step (c), place zero-amplitude samples be-tween the trace segments when creating the super-traces.

(b) Create the stack supertrace.(c) Crosscorrelate the two supertraces.(d) Determine the correlation lag associated with the

peak crosscorrelation value — this is the shot resid-ual static shift.

(f) Apply the shot residual static shift associated withthe highest correlation value to all the traces in theshot gather.

(g) Stack the CMP gathers that include traces fromthis shot gather.

(h) Move to the next shot station and repeat steps (a)through (g).

(i) Repeat steps (a) through (h) for all receiver sta-tions.

Steps (a) through (i) usually are applied iteratively toconverge to a solution of shot and residual static shifts.

Traveltime Decomposition in Practice

As stated early in this section, residual statics esti-mation by traveltime decomposition consists of threestages: picking, decomposing, and applying residualstatic shifts. The picking phase, in which traveltime de-viations are derived from trace crosscorrelations, deter-mines the effectiveness of residual statics corrections.We now examine various parameters involved in thepicking phase.

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FIG. 3.3-19. CMP-stacked section associated with a syn-thetic data set. Shot and receiver statics were appliedon moveout-corrected gathers in a surface-consistent man-ner, while the structure term was applied in a subsurface-consistent manner. These terms are plotted above thestacked section. Random noise was added to prestack datawith a spatially varying signal-to-noise ratio.

Maximum Allowable Shift

Consider the CMP-stacked section in Figure 3.3-19 as-sociated with a synthetic data set. This data set wascreated by using the field geometry of a real seismicline. The CMP traces were derived from the first traceof the first CMP location from that real line. This tracefirst was zeroed out within selected time gates, thentreated with the shot and receiver static shifts in Fig-ure 3.3-19 in a surface-consistent manner and with astructure term that only depended on midpoint location(subsurface-consistent). The shot and receiver staticshifts were varied from +32 to −32 ms. Finally, the syn-thetic traces were blended with a band-limited randomnoise, whose strength varied spatially. (The noise levelwas set to zero at both ends of the profile and to max-imum at the center.) The stacked section constructedfrom data before treatment with synthetic shot and re-ceiver static shifts is shown in Figure 3.3-20. Once resid-ual statics corrections are made, the stacked section inFigure 3.3-19 should resemble that in Figure 3.3-20.

FIG. 3.3-20. CMP-stacked section associated with the syn-thetic data set as in Figure 3.3-19, but without syntheticshot or receiver statics applied. Once residual statics correc-tions are applied to the data set associated with the stackedsection in Figure 3.3-19, the resulting stacked section shouldresemble the section shown here.

Note how degrading the effect of the synthetic shot andreceiver static shifts is on the continuity of reflectionsin Figure 3.3-19. The sole effect of random noise is seenin Figure 3.3-20.

Consider three different tests of residual statics cor-rections: one with a small maximum allowable shift (24ms in Figure 3.3-21), one with a moderately sized shift(80 ms in Figure 3.3-22), and one with a fairly largeshift (192 ms in Figure 3.3-23). All three tests had thesame input CMP gathers. All three were run using thesame set of parameters except for the maximum allow-able shift.

The maximum value of combined shot and receiverstatic shifts for any given trace implied by the modelin Figure 3.3-19 is ∓64 ms. When the maximum allow-able shift is insufficient (a value of less than 64 ms),then the derived static shifts (Figure 3.3-21) are signif-icantly smaller than the actual shifts (Figure 3.3-19).Thus, stack quality, although significantly improvedwhen compared to that in Figure 3.3-19, is far fromthe quality of the section shown in Figure 3.3-20.

When the maximum allowable shift is sufficient,then the derived static shifts (Figure 3.3-22) are like theactual shifts imposed on the input model shown on thegraph in Figure 3.3-19. Also, stacking quality (Figure3.3-22) improved so that it now is comparable to theno-static model (Figure 3.3-20).

Reasonable results (Figure 3.3-23) also were ob-tained for the case allowing excessively large maximumallowable shift, up to 192 ms. However, this result doesnot imply that we can be liberal on the upper bound

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FIG. 3.3-21. CMP-stacked section associated with the syn-thetic data set as in Figure 3.3-19 with residual statics cor-rections. The derived shot, receiver, and structure terms areplotted at the top. Compare these estimates with the actualvalues in Figure 3.3-19. The maximum allowable shift is 24ms. Also compare the resulting stacked section with thatshown in Figure 3.3-20.

of maximum shift in real data situations. In the pres-ence of short-period multiple or reverberation energy,or data with a narrow bandwidth or high noise level,crosscorrelation can yield a multiple number of peaksand cause uncertainty in the estimated time shifts (cy-cle skipping). In this case, a large maximum allowableshift could cause anomalously large time shifts to bepicked.

Based on the tests shown in Figures 3.3-21 through3.3-23, the maximum allowable shift used in the pickingphase should be greater than all possible combined shotand receiver static shifts at any given location along theprofile. On the other hand, jumping a leg in correlatingevents from trace to trace in a CMP gather, commonlyknown as cycle skip, especially in poor signal-to-noiseratio conditions, also is more likely to occur if the maxi-mum allowable shift is greater than the dominant periodof the data.

We may argue that the result of cascading a num-ber of small-shift residual statics solutions is as good asa single-step large-shift solution. This approach might

FIG. 3.3-22. CMP-stacked section associated with the syn-

thetic data set as in Figure 3.3-19 with residual statics cor-

rections. The derived shot, receiver, and structure terms are

plotted at the top. Compare these estimates with the actual

values in Figure 3.3-19. The maximum allowable shift is 80

ms. Also compare the resulting stacked section with that

shown in Figure 3.3-20.

have the same effectiveness as the large-shift solution,while avoiding the possibility of cycle skipping. Unfor-tunately, cascading small-shift solutions does not work.Starting with the CMP gathers associated with thestack in Figure 3.3-19, we get the CMP gathers cor-rected for shot and receiver statics based on a 24-msshift (first pass). The stack is shown in Figure 3.3-21.Using these gathers, a new statics solution was derivedand applied to the data (second pass). This processwas repeated for the third and fourth times. The re-sult of this last iteration (Figure 3.3-24) does not havethe quality of the solution derived with the 80-ms shift(Figure 3.3-22).

Now consider maximum allowable shift tests on thefield data as in Figure 3.3-2. Figure 3.3-25 shows CMPgathers from the problem zone of the profile in Figure3.3-5. Refer to the panels for 24- and 40-ms shifts inFigure 3.3-25, and note that an insufficient maximumallowable shift does not completely correct for all static


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FIG. 3.3-23. CMP-stacked section associated with the syn-thetic data set as in Figure 3.3-19 with residual statics cor-rections. The derived shot, receiver, and structure terms areplotted at the top. Compare these estimates with the actualvalues in Figure 3.3-19. The maximum allowable shift is 192ms. Also compare the resulting stacked section with thatshown in Figure 3.3-20.

FIG. 3.3-24. CMP-stacked section associated with the syn-

thetic data set as in Figure 3.3-19 with residual statics cor-

rections. The derived shot, receiver, and structure terms are

plotted at the top. This is the output from the four itera-

tive passes of the statics estimation and application. Com-

pare these estimates with the actual values in Figure 3.3-19.

First-pass results are shown in Figure 3.3-21. The maximum

allowable shift is 24 ms on each pass. Also compare the re-

sulting stacked section with that shown in Figure 3.3-20.

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FIG. 3.3-25. Test of maximum allowable shift. CMP gathers after residual statics corrections using five different maximumallowable shifts. Figure 3.3-28 shows the CMP stacks.

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FIG. 3.3-26. Diagnostics for maximum allowable shift tests (Figure 3.3-25) showing common-shot-point-stacked data.

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FIG. 3.3-27. Diagnostics for maximum allowable shift tests (Figure 3.3-25) showing common-receiver-point-stacked data.

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FIG. 3.3-28. Test of maximum allowable shift: CMP stacks after residual statics corrections using five different maximumallowable shifts. Figure 3.3-25 shows selected CMP gathers.

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FIG. 3.3-29. Test of correlation window: CMP gathers after residual statics corrections using five different correlationwindows. Figure 3.3-32 shows the CMP stacks.

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FIG. 3.3-30. Diagnostics for correlation window tests (Figure 3.3-29) showing common-shot-point-stacked data.

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FIG. 3.3-31. Diagnostics for correlation window tests (Figure 3.3-29) showing common-receiver-point-stacked data.

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FIG. 3.3-32. Test of correlation window: CMP stacks after residual statics corrections using five different correlation windows.CMP gathers are shown in Figure 3.3-29.

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FIG. 3.3-33. Test of correlation window: CMP gathers after residual statics corrections using five different correlationwindows. CMP stacks are shown in Figure 3.3-36.

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FIG. 3.3-34. Diagnostics for correlation window tests (Figure 3.3-33) showing common-shot-point-stacked data.

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FIG. 3.3-35. Diagnostics for correlation window tests (Figure 3.3-33) showing common-receiver-point-stacked data.

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FIG. 3.3-36. Test of correlation window: CMP stacks after residual statics corrections using five different correlation windows.CMP gathers are shown in Figure 3.3-33.

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FIG. 3.3-37. Pilot traces (bottom) and shot and receiver statics solutions (top) for the 80-ms shift and the 400- to 1200-mswindow. CMP stack is shown in Figure 3.3-36.

shifts. Excessively large shifts, however, such as 120- or160-ms, do not seem to harm this particular data set.While common-shot-point (CSP) stacks (Figure 3.3-26)indicate small shot-static shifts, common-receiver-point(CRP) stacks (Figure 3.3-27) indicate a zone of signif-icant receiver-static shifts. Again, small maximum al-lowable shifts have not corrected completely for thesestatics anomalies.

The ultimate judgment is made by examining thestack response and plots of the estimated statics them-selves. From Figure 3.3-28 (ungained stack responses),it is clear that the maximum allowable shift must beadequate to accommodate the combined shot and re-ceiver statics present in the data at any location alongthe profile.

Correlation Window

From the same field data example (Figure 3.3-5), theresults of residual statics corrections (for the right halfof the stacked section) are examined by using differentcorrelation windows while keeping all other parametersconstant. The maximum allowable shift was 80 ms inthese tests. From Figure 3.3-29, note that a correlationwindow confined to the mute zone (400 to 1200 ms)is not desirable. It does not provide sufficient statisticsbecause of the low fold of coverage and the shortness ofthe data window available for crosscorrelation with thepilot traces.

With high-fold data, the mute zone problem is han-dled to a degree by limiting the correlation to small

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FIG. 3.3-38. CMP-stacked section associated with the syn-thetic data set as in Figure 3.3-19. In addition to the surface-consistent shot and receiver statics and the subsurface-consistent structure term, residual moveout shifts were in-troduced to the CMP gathers used in Figure 3.3-19.

offsets. In this particular part of the profile, a large win-dow including both the mute zone and deep data (800 to2300 ms), a deep window (1700 to 2300 ms), a deep largewindow (1400 to 2800 ms), or a deep narrow window(1500 to 1700 ms) made no difference. This is probablybecause of the good signal-to-noise ratio in this part ofthe profile.

These observations are verified by the diagnosticsbased on the CSP and CRP stacks in Figures 3.3-30and 3.3-31. Ungained stack responses are shown in Fig-ure 3.3-32. In particular, note the relatively poor stackresponse using a window confined to the mute zone (400to 1200 ms).

The choice of the correlation window is more crit-ical in conditions of poor signal-to-noise ratio. Refer tothe same diagnostics for the left half of the stacked sec-tion in Figure 3.3-5. These diagnostics are shown in Fig-ures 3.3-33 through 3.3-36. Again, a correlation windowconfined to the mute zone not only provides an inade-quate solution, as in the previous case (Figure 3.3-29),but also can be devastating, as shown in Figure 3.3-33.In this case, the CMP gathers with no corrections havebetter signal quality. It now is apparent that a narrowwindow, even if it is outside the mute zone (such as

FIG. 3.3-39. CMP-stacked section associated with the syn-thetic data set as in Figure 3.3-38 after the application ofresidual statics corrections. Compare the results with thosein Figure 3.3-22.

the 1500 to 1700 ms window), may not provide suffi-cient statistics. The CSP stacks (Figure 3.3-34) and theCRP stacks (Figure 3.3-35) show the undesirable as-pects of choosing a window within the mute zone orchoosing a window that is too narrow. The ungainedstacked sections (Figure 3.3-36) clearly demonstrate theadverse effects of an improper choice of correlation win-dow. Moreover, note the poor quality pilot traces foreach CMP gather in Figure 3.3-37 (to the left of mid-point 377). The shot and receiver static solutions shownin the graphs above the pilot traces are totally unreli-able.

These test results suggest choosing a correlationwindow that (a) contains as much signal as possible toimprove correlation values, and (b) is large enough andis outside the mute zone whenever possible.

Other Considerations

The synthetic data set in Figure 3.3-38 is the sameas that in Figure 3.3-19, with the addition of residual

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FIG. 3.3-40. First portion of a land line illustrating the improvement in CMP stacking as a result of residual staticscorrections. Stack A (a) before residual statics corrections and using preliminary velocity picks and (b) after two passes ofresidual statics corrections and using final velocity picks.D

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FIG. 3.3-41. Second portion of the land line shown in Figure 3.3-40 illustrating the improvement in CMP stacking as aresult of residual statics corrections. Stack B (a) before residual statics corrections and using preliminary velocity picks and(b) after two passes of residual statics corrections and using final velocity picks.D

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FIG. 3.3-42. Diagnostics for segment A from the residualstatics corrections applied on the first portion of the landline in Figure 3.3-40.

moveout shifts. Residual moveout was introduced intothe data by inverse NMO correcting the CMP gath-ers associated with the stack in Figure 3.3-19 using avelocity function v1(t), then by NMO correcting usinga velocity function v2(t) = v1(t). The solution (Figure3.3-39) implies some traveltime distortion at the edgesof the stacked section caused by low fold of coverage.(Compare this with the solution in Figure 3.3-22). Oth-erwise, stack response seems to be satisfactory. As longas the residual moveout variations are not large withinthe correlation window, the computed residual staticssolution should be adequate. Use of more than one smallcorrelation window during the picking phase may helpminimize the time-dependent effect of residual moveout.

In some areas, the signal-to-noise ratio is so poorthat a second pass of residual statics corrections mustbe done. The idea is that the first pass of residual staticscorrections improves the signal to such a degree that asecond pass should remove the residuals remaining fromthe first pass. For the second pass, the steps in Figure3.3-12 must be repeated, such that the input are CMPgathers that already were corrected for residual stat-ics. Velocity estimates must be revised between passes.Figures 3.3-40 and 3.3-41 show two different segmentsof a section before and after residual statics correctionsthat were done in two passes. Diagnostic plots of theshot and receiver statics shown in Figures 3.3-42 and3.3-43 indicate that the first pass has taken out a sig-nificant part of the static shifts in the first segment. On

FIG. 3.3-43. Diagnostics for segment B from the residualstatics corrections applied on the second portion of the landprofile in Figure 3.3-41.

the other hand, the second pass was most effective inthe second segment, where the signal-to-noise ratio isrelatively poorer. Repeated estimation and applicationof the residual statics and velocity estimation is com-mon in some processing systems. Use of a large numberof trace correlations and multiple correlation peaks in astatics program tends to minimize the number of passesrequired.

Stack-Power Maximization in Practice

The method of stack-power maximization can yield bet-ter stack compared to the method of traveltime decom-position in areas with poor signal-to-noise ratio. Figure3.3-44 shows a CMP-stacked section along a land pro-file with field statics corrections applied (Figure 3.3-45).While the left-half of the section has a good signal-to-noise ratio, the right-half has a poor signal-to-noise ra-tio resulting from irregular topography and near-surfacecomplexity (Figure 3.3-45). Note, for instance, the lossof continuity along the reflection events at 0.8 and 1.3s at the right-half of the section.

Selected CMP gathers shown in Figure 3.3-46 ver-ify the presence of short-wavelength statics. Followingresidual statics corrections by stack-power maximiza-tion and the subsequent velocity analysis, the same

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FIG. 3.3-45. The elevation profile (top), and the shot (middle) and receiver (bottom) field statics profiles for the stackedsection shown in Figure 3.3-44.

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FIG. 3.3-48. Shot (top) and receiver (bottom) residual static shifts computed by stack-power maximization. The resultingCMP-stacked section is shown in Figure 3.3-47.

gathers indicate that short-wavelength statics have beenlargely resolved (Figure 3.3-46), and thus, the result-ing CMP stack shows significant improvement in thecontinuity of reflections in the right-hand side (Figure3.3-47). The shot and receiver residual static shifts de-rived by using the supertrace scheme described aboveare shown in Figure 3.3-48. Note that mainly large resid-uals are in the right-half of the profile with irregulartopography.

3.4 REFRACTION STATICS CORRECTIONS

An important question in estimating shot and receiverstatics is accuracy of the results as a function of wave-lengths of static anomalies. Figure 3.4-1 is a syntheticdata set that is identical to that in Figure 3.3-19, ex-cept for additional long-wavelength shot and receiverstatic components. (Compare the graphic displays in

Figures 3.3-19 and 3.4-1.) From the solution in Figure3.4-2, note that the long-wavelength components of thestatics were severely underestimated. A significant dif-ference between the stacked sections, in terms of horizontimes, is apparent in Figures 3.3-22 and 3.4-2.

The surface-consistent solution discussed in Sec-tion 3.3 resolves the short-wavelength static shifts (lessthan a spread length), which cause traveltime distor-tions in CMP gathers, and thus yield an improved stackresponse. However, merely improving the stack responseby correcting for short-wavelength statics may notalways be sufficient. The unresolved long-wavelengthcomponents are assigned to the structure term in equa-tion (3-25). If the long-wavelength components arelarge, reflector geometries inferred by the CMP stackcan be distorted significantly. Field statics and refrac-tion statics methods are used to correct for the long-wavelength components.

The statics corrections require knowledge of thenear-surface model. The near-surface often consists of

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FIG. 3.4-1. CMP-stacked section associated with the syn-thetic data set as in Figure 3.3-19 contaminated with long-period statics.

a low-velocity weathering layer. However, there are ex-ceptions to this simplified model for the near-surface.Areas covered with glacial tills, volcanic stringers, andsand dunes often have a near-surface that may consistof more than one layer with different velocities. Layerboundaries can vary significantly from a flat interface toan arbitrarily irregular shape. The single-layer assump-tion for the near-surface also is violated when there is alateral change in rock composition associated with out-crops, pinchouts or a flood plain along a seismic profile.In areas covered with a permafrost layer, which has asignificantly higher velocity than the underlying layer,the surface-consistency assumption for the near-surfacecorrections is not valid. Moreover, the base of the per-mafrost layer does not form a head wave and thereforeis not detectable.

In practice, a single-layer near-surface model oftenis sufficient for resolving long-wavelength statics anoma-lies. Complexities in a single-layer near-surface modelcan be due to one or more of the following:

(a) Rapid variations in shot and receiver station eleva-tions,

(b) Lateral variations in weathering velocity, and

FIG. 3.4-2. CMP-stacked section associated with the syn-thetic data set as in Figure 3.4-1 after the application ofresidual statics corrections. Compare the results with thosein Figure 3.3-22.

(c) Lateral variations in the geometry of the refractor,which, for refraction statics, is defined as the inter-face between the weathering layer above and thebedrock below.

Near-surface velocity-depth models often are esti-mated using refracted arrivals. The refracted energy isassociated with the head wave that travels along the in-terface between the near-surface weathering layer andthe underlying bedrock. If refracted arrivals are observ-able on common-shot gathers, it almost certainly im-plies that the near-surface has a simple geometry. Nev-ertheless, no ray-theoretical method can claim to esti-mate short-wavelength variations in the base of weath-ering that are much smaller than a cable length. Thesevariations are left to be handled by subsequent resid-ual statics corrections using traveltime distortions asso-ciated with reflections on moveout-corrected common-midpoint (CMP) gathers (Taner et al., 1974).

The head wave is distorted in the presence of ir-regularities along the base of the weathering layer, andit turns into a diving wave when there is no sharp ve-locity contrast between the weathering layer and the

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FIG. 3.4-3. A shot record with distinct first breaks.

FIG. 3.4-4. A shot record with a distinct refraction event.

FIG. 3.4-5. A shot record with a shallow and deep refrac-tion event.

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FIG. 3.4-6. A shot record with noise-contaminated firstbreaks.

FIG. 3.4-7. A shot record with a distinct refraction eventalong the right-hand spread.

FIG. 3.4-8. A vibroseis shot record with not-so-distinctfirst breaks.

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FIG. 3.4-9. A shot record in which the onset of the firstarrivals is not clear.

substratum (Hill and Wuenschel, 1985). Such cases, if atall possible, may be handled by wave-theoretical mod-eling and inversion (Hill, 1987) or turning-wave tomog-raphy (Section 9.5).

First Breaks

The refracted energy associated with the base of theweathering layer often constitutes the first arrivals on ashot gather. The onset of these first arrivals is referredto as the first break.

First breaks occur in varying degrees of quality —depending on the source type and the near-surface con-ditions. The common-shot gather shown in Figure 3.4-3has first breaks with clear onset. Deviations from thelinear trend of the first-break times may largely be at-tributed to elevation differences along the shot profile.

FIG. 3.4-10. A near-surface model for statics correctionswhen shots are situated below the weathering layer. Here, S= shot, ES = elevation at the shot station on the ground,R = receiver, ER = elevation at the receiver station on theground, T = surface topography,B = base of weathering, D= datum, ED = datum elevation, vw = weathering velocity,and vb = bedrock velocity.

Figure 3.4-4 shows a record with first breaks associ-ated with a prominent refractor. In Figure 3.4-5, note ashallow and a deep refractor. Figure 3.4-6 shows a shotrecord in which automated procedures would largely failto pick the first breaks. Figure 3.4-7 shows a shot recordwith first breaks that can be detected easily by auto-mated procedures. From the first breaks on the left, onecan infer near-surface irregularity — either in the formof a variable refractor shape or velocity variations inthe near-surface layer. The right-hand side shows thepresence of a distinct refractor. Figure 3.4-8 shows ashot gather recorded with a vibroseis source, which of-ten produces poor first breaks compared to a dynamitesource. A similar situation exists in the record shown inFigure 3.4-9 — it is not simple to detect the first breaks.The remainder of the sidelobes from sweep correlationmasks the onset of the first arrivals.

First-break picking can be done automatically, in-teractively, manually, or as a combination thereof. Tomake reliable picks, first apply linear moveout (LMO)to the data. Once picking is done, the LMO correction isreversed. Note that effectiveness of both reflection- andrefraction-based methods of statics corrections dependson the reliability of the picking process. Apart from thesignal-to-noise ratio, indistinct first breaks (such as invibroseis) sometimes can make picking consistent firstbreaks difficult.

The first-break picks associated with the refractedarrival times are then used in an inversion scheme toestimate the near-surface model parameters. In this sec-tion, we discuss ray-theoretical methods such as plus-minus and its generalized form, the reciprocal method,and the least-squares inversion methods. The basic as-sumption made is that the refractor is flat or nearly

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flat, with a smoothly varying shape along the seis-mic profile. As demonstrated by the field data ex-amples, these methods appear to remove medium- tolong-wavelength statics anomalies associated with var-ious types of near-surface models. Combined with thereflection-based residual statics corrections to resolveany remaining short-wavelength statics variations thataffect the stack quality, we get a final stacked sectionready for poststack processing.

Field Statics Corrections

It is appropriate now to review various methods of fieldstatics corrections. Consider the near-surface model de-picted in Figure 3.4-10. If shots (denoted by S) are lo-cated below the weathering layer, then the total staticcorrection to apply to the trace associated with mid-point M is tD = tS + tR, where tS and tR are the shotand receiver static corrections, respectively, down to aspecified datum D. From the geometry of Figure 3.4-10,the field statics correction ∆τD can be computed by

∆τD = −ES − ED − DS

vb− ER − ED − DR

vb− tUH ,

(3 − 40)

where ED is the datum elevation and ES and ER are thesurface elevations at the shot and receiver stations, re-spectively, DS is the depth of the shot hole beneath theshot station, and DR is the depth of the shot hole nearthe receiver station, tUH is the uphole time measuredat the receiver location (the time associated with thedistance DR in Figure 3.4-3). Finally, vb is the bedrock(subweathering) velocity that may be derived from adeep uphole survey (to a point well below the weather-ing layer) conducted in the area.

An uphole survey involves placing shots down thehole at various depth levels, then recording the arrivalsat the surface near the hole. Alternatively, shots andreceivers can be reciprocated if there is a caving prob-lem down the hole. The hole must be deep enough toreach below the weathering layer. This provides a plotof time versus depth from which the bedrock velocity isobtained.

In land surveys, shots are not always placed in thebedrock for economic reasons, especially in areas with athick weathering layer. Also, impulsive sources are notalways used. Instead, surface sources such as vibroseisoften are used. When surface sources or sources in shal-low holes are used, the refracted arrivals can, at least intheory, be used to compute the static correction ∆τD

down to a specified datum.

Flat Refractor

Consider the refraction wavefront and raypath geome-try in Figure 3.4-11a associated with a single-layer near-surface model. On top, we see a plot of first-breaks. Forsimplicity, consider a flat surface and flat refractor. Forthe head wave to form, and thus the refraction to occur,the requirement is that the overburden velocity vw besmaller than the substratum velocity vb.

The traveltime profile depicts the first breaks seenon the shot record in Figure 3.4-11b. Note that to theleft of the crossover offset xc (also known as critical dis-tance) are the first breaks associated with the the directarrivals. Also note that to the right of offset xc are thefirst breaks associated with the refracted arrivals. Fromthe refraction theory (Dobrin, 1960; Grant and West,1965), the inverse of the slope of the line associatedwith the refracted wave arrivals is equal to the bedrockvelocity vb. Also note that the inverse of the slope of theline associated with the direct wave arrivals is equal tothe velocity of the weathering layer vw.

By picking the first breaks, the weathering andbedrock velocities, vw and vb are estimated. By extend-ing the line associated with the refracted arrivals tozero-offset, intercept time ti, the time at x = 0, is esti-mated. From these three parameters, it is easy to showthat depth to the bedrock zw is given by

zw =vbvwti

2 v2b − v2

w

. (3 − 41a)

We assume that vb > vw. Derivation of this formula isleft to Section C.5.

Alternatively, we can measure the critical distancecorresponding to the change from the direct arrival tothe refracted arrival on the traveltime plot and use it incomputing the depth to the bedrock. Equation (3-41a),in terms of the critical distance xc, takes the form

zw =12

vb − vw

vb + vwxc. (3 − 41b)

It may not be easy to measure the critical distancewhen depth to bedrock is small. In such cases, it is bet-ter to use the intercept time to compute the depth tothe bedrock by way of equation (3-41a).

After computing zw, the total static correction∆τD to the specified datum level can be applied by

∆τD = −2zw

vw+

2(ED − ES + zw)vb

, (3 − 42)

where ES is the surface elevation. If there is a differencebetween the elevations of shot and receiver stations,

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FIG. 3.4-11. (a) Geometry for refracted arrivals. Here, vw = weathering velocity, vb = bedrock velocity, zw = depth to therefractor equivalent to the base of the weathering layer, θc = critical angle, and xc = crossover distance. The direct wavearrival has a slope equal to 1/vw and the refracted wave arrival has a slope equal to 1/vb. (b) A shot record that exhibitsthe direct wave and the refracted wave depicted in (a). (c) Geometry for a dipping refractor with forward traveltime profileassociated with the direct wave and refracted wave arrivals, and (d) with both forward and reverse traveltime profiles. Seetext for details.

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then an additional elevation correction using thebedrock velocity is required. Moreover, if the shots arelocated in boreholes, then the measured uphole timealso must be incorporated into equation (3-42). The es-timated statics correction given by equation (3-42) is anaverage value over a distance that can range from thecritical distance to the spread length, depending on thenumber of traces used in estimating the bedrock veloc-ity. Nevertheless, more than one shot-point is within aspread length. Therefore, an adequate definition of thenear-surface model can be achieved and datum correc-tions can be computed for the entire profile.

Dipping Refractor

When the refractor is dipping, it turns out that the in-verse slope of the refracted arrival is no longer equal tothe bedrock velocity (Figure 3.4-11c). An extra param-eter — the dip of the refractor, needs to be estimated(Section C.6). This requires reverse profiling as illus-trated in Figure 3.4-11d. We have the refracted arrivalin the forward direction and the refracted arrival in thereverse direction obtained by interchanging the shotswith receivers. The traveltimes for the refracted arrivalsof the forward and reverse profiles are expressed as

t− = t−i +x

v−b(3 − 43a)

and

t+ = t+i +x

v+b

. (3 − 43b)

The inverse slopes are given by

v−b =vw

sin (θc + ϕ)(3 − 44a)

and

v+b =

vw

sin (θc − ϕ), (3 − 44b)

where ϕ is the refractor dip and θc is the critical angleof refraction given by

sin θc =vw

vb. (3 − 44c)

Finally, the intercept times are given by the follow-ing relations:

t−i =2zwS cos θc cos ϕ

vw(3 − 45a)

and

t+i =2zwR cos θc cos ϕ

vw. (3 − 45b)

Derivation of the relations (3-44a,b) and (3-45a,b) areleft to Section C.6.

To estimate the thickness of the near-surface layer,first we compute the refractor dip ϕ from the slope mea-surements — vw, v−b , and v+

b . These measurements arethen inserted into the expression

ϕ =12

sin−1 vw

v−b− sin−1 vw

v+b

. (3 − 46a)

Then, we compute the bedrock velocity vb using theexpression

vb =2 cos ϕ

1v−b

+1

v+b

. (3 − 46b)

Finally, we compute the depth to the bedrock atshot/receiver stations

zw =vbvwt−i

2 cos ϕ v2b − v2

w

. (3 − 46c)

Again, equations (3-46a,b,c) are derived in Section C.6.By setting the refractor dip ϕ = 0, equation (3-46c)reduces to equation (3-41a).

Keep in mind that, whether it is the flat refractor(equation 3-41a) or dipping refractor case (equation 3-46c), the depth to bedrock estimation at a shot-receiverstation requires the knowledge of weathering velocity,bedrock velocity and intercept time. In the case of aflat refractor, these can be measured directly from shotprofiles; whereas, in the case of a dipping refractor, theycan be computed by way of equations (3-46a,b,c).

The Plus-Minus Method

It often is difficult to use first breaks to estimate the in-tercept time and velocities for the weathering layer andbedrock. This is primarily because the base of weath-ering typically is an undulating surface, which makestraveltime plots difficult to interpret. Traveltime plotsalso are affected by severe elevation changes. Addition-ally, a typical field cable layout does not provide a suf-ficient number of channels inside the crossover distancexc (Figure 3.4-11a) for a reliable estimate of the weath-ering velocity or thickness. In most cases, vw cannot bemeasured and a reasonable value is assumed for it.

Hagedoorn (1959) formulated a method to indi-rectly estimate intercept time and bedrock velocity. Themethod still requires picking the first breaks. However,it does not require interpreting the traveltime profile(Figure 3.4-11a). (Interpretation means drawing the lin-ear segments for the direct and refracted waves.) Fig-ure 3.4-12a shows three raypaths associated with shot-receiver pairs AD, DG, and AG. The basis of Hage-doorn’s method involves computing two time values, the

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FIG. 3.4-12. (a) Geometry for the plus-minus method. (b) Geometry for the generalized reciprocal method. Here, zw is thedepth to the refractor at the surface station where the plus-minus times as for (a) and intercept times as for (b) are to beestimated, vw is the weathering velocity, and θc is the critical angle of refraction.

plus and minus times given by

t+ = tABCD + tDEFG − tABFG (3 − 47a)

and

t− = tABCD − tDEFG + tABFG. (3 − 47b)

The times given on the right side of these equations arethe measured (picked) values from the first breaks forthe three raypaths shown in Figure 3.4-12a. From theraypath configuration, we find that (Section C.7)

t+ =2zw v2

b − v2w

vbvw. (3 − 48a)

Rewrite equation (3-41a)

ti =2zw v2

b − v2w

vbvw, (3 − 48b)

and note that the plus time t+ in equation (3-48a) isidentical to the intercept time ti in equation (3-48b).Hence, instead of measuring ti directly from the shotrecord, Hagedoorn’s method suggests estimating ti fromthe first break picks given by the right-hand terms inequation (3-47a).

By applying algebra (Section C.7), we find thatminus time t− is related to bedrock velocity vb by

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t− = t+ +2x

vb, (3 − 48c)

where x is the source-receiver separation AD.Thus, Hagedoorn’s plus-minus method involves:

(a) Picking the first breaks,(b) Computing the plus-minus times, t− and t+ (equa-

tions 3-47a and 3-47b),(c) Deriving from the plus-minus times the intercept

time ti (equations 3-48a and 3-48b) and bedrockvelocity vb (equation 3-48c),

(d) Assuming a value for weathering velocity vw,(e) Computing the depth zD to bedrock below station

D (Figure 3.4-12a) from equation (3-41a), and(f) Computing the shot-receiver static shift ∆τD at

that station by

∆τD = −zD

vw+

ED − ES + zD

vb, (3 − 49)

where ES and ED are the surface and datum elevationsat station D (Figure 3.4-12a). If there is a shot at sta-tion D, ∆τD represents the shot static, and if there isa receiver at station D, it represents the receiver static.Again, uphole and elevation corrections are needed be-fore making the plus-minus statics corrections.

The Generalized Reciprocal Method

In practice, raypaths that are suitable for first-breakpicking and are coincident at station D are not alwaysfound. Palmer (1981) generalized Hagedoorn’s methodof using raypaths (Figure 3.4-12b). Palmer’s technique,the generalized reciprocal method (GRM), takes into ac-count offset separation D1D2 when computing the plustime

t+ = tABCD2 + tD1EFG − tABFG − D1D2

vb. (3 − 50a)

The definition of the minus time remains the same as inequation (3-47b), except for accounting for the raypathgeometry in Figure (3.4-12b):

t− = tABCD2 − tD1EFG + tABFG. (3 − 50b)

Note that more than one combination of raypathsassociated with different separations of D1D2 can beused to measure (pick) the traveltimes on the right sidesof equations (3-50a,b). Consequently, there is more thanone estimate of the plus-minus times at a given (shot-receiver) station D. By carefully editing the first breaks,these estimates can be refined and reduced to a singleestimate for each station.

To derive the near-surface model, the generalizedreciprocal method uses the observed traveltimes fromrefracted arrivals that are assumed to be associated withthe base of weathering. A problem arises when a near-surface model with more than one layer needs to be de-fined. This is the case in areas covered with glacial tillsand sand dunes. Several specialized techniques basedon generalized linear inversion (GLI) have been devisedfor these problems (Hampson and Russell, 1984; Schnei-der and Kuo, 1985). The GLI technique is an iterative,model-based approach that provides flexibility in defin-ing a near-surface model consisting of arbitrarily pa-rameterized multilayers. The process begins by comput-ing the refracted arrival times from an assumed initialnear-surface model. These computed traveltimes thenare compared with the actual first-break picks (observedtraveltimes). The procedure tries to minimize the differ-ence between the computed and observed traveltimes byiteratively modifying model parameters for the near sur-face (such as velocities and thicknesses). A GLI methodapplicable to a single-layer near-surface model is pre-sented next.

The Least-Squares Method

We want to estimate the near-surface parameters —weathering and bedrock velocities and thickness of theweathering layer at shot-receiver locations by least-squares inversion of the observed (picked) refractedarrivals. Formulation of this problem using the least-squares inversion leads to an estimate of the near-surface parameters such that the difference between theobserved arrivals and the modeled refracted arrivals isminimum in the least-squares sense. This method is notonly applicable to 2-D line shooting but also to 3-Dswath shooting geometries.

There are several ways to parameterize the near-surface layer. The most general formulation would in-clude varying weathering and bedrock velocities and thethickness of the weathering layer at all shot-receiverstations. This, however, would require linearizing theproblem and iterating over the estimated parameters.The problem also would have to be constrained to sta-bilize the inversion. In a simplified version of this gen-eral formulation, weathering velocity may be fixed andassumed to be known. This leaves the weathering thick-ness and bedrock velocity as spatially varying parame-ters.

As for any inversion problem, we shall need a modelequation that relates the model parameters we want toestimate to the modeled refracted arrival times. Refer tothe sketch of a near-surface model in Figure 3.4-13. If welet the weathering thickness vary as might be the case

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in many field data applications, then we have a problemof not being able to write down an analytic expressionfor the refracted raypath. We would not even begin toanticipate how the head wave would behave or developin case of a laterally varying refractor geometry.

Instead, to be able to make use of the first-breakpicks, we shall take a simpler approach. We want to de-scribe the near-surface with minimal parameterizationand consider the model with a flat refractor. Now, wecan express the modeled traveltime tij for the criticallyrefracted raypath from the source location Sj to thereceiver location Ri (Figure 3.4-13) as

tij =SjB

vw+

DE − DB − CE

vb+

CRi

vw. (3 − 51a)

Keep in mind that we had to make the same as-sumption — that the refractor is flat or nearly-flatwithin the spread length, as in the generalized recip-rocal method. The first and the third terms in equa-tion (3-51a) are associated with the raypaths withinthe weathering layer, and the second term is associatedwith the raypath within the bedrock along the refrac-tor. When the refractor dip is taken into account, theproblem cannot be readily linearized.

By regrouping the terms in equation (3-51a), weget the expression

tij =SjB

vw− DB

vb+

CRi

vw− CE

vb+

DE

vb.

(3 − 51b)Finally, rewriting in terms of the model parameters thatwe want to estimate — vw, vb, and zw, we obtain theexpression for the model equation for the refracted ar-rivals:

tij =zj v2

b − v2w

vbvw+

zi v2b − v2

w

vbvw+

xij

vb. (3 − 51c)

In addition to assuming a flat refractor, we fix thebedrock velocity vb but retain it as a parameter to beestimated. We also assume that vw is known. Underthese assumptions, the model equation (3-51c) for therefracted arrivals can be written in the form (Farrelland Euwema, 1984)

tij = Tj + Ti + sbxij , (3 − 52)

where sb = 1/vb is the bedrock slowness, and

Tj =zj v2

b − v2w

vbvw, (3 − 53a)

Ti =zi v2

b − v2w

vbvw. (3 − 53b)

By comparing equations (3-53a,b) with equation(3-48b), note that Tj and Ti actually are (half) intercept

FIG. 3.4-13. Geometry of refracted arrival used in deriv-ing the least-squares solution for intercept times. Here, Sj

and Ri are source and receiver stations, respectively; θc isthe critical angle of refraction, zj and zi are depths to thebedrock at source and receiver locations, and vw and vb areweathering and bedrock velocities, respectively.

time values at the shot and receiver locations. Hence, forn shot-receiver stations, the parameter vector contains(T1, T2, . . . , Tn; sb).

Equation (3-52), which models refracted arrivals,is solved for the parameter vector in the same mannerusing the generalized linear inversion (GLI) theory (Sec-tion C.8) as for equation (3-25), which models travel-time deviations associated with residual statics (SectionC.4). The generalized linear inversion solution is basedon the objective of minimizing the least-squares differ-ence between the observed refracted arrival times tijand the modeled times tij defined by equation (3-52).Alternatively, the refraction statics solution can be ob-tained by using the minimization criterion that is basedon the L1 norm (Section C.10).

An extension of equation (3-52) to represent the re-fraction statics associated with each shot and receiverlocation in two parts — long- and short-wavelengthcomponents, is proposed by Taner et al. (1998). Thisis done by including two more terms in equation (3-52) to represent the shot and receiver short-wavelengthvariations in refraction statics caused by rapid changesin elevation and near-surface layer geometry. The con-jecture for such an extension is to stabilize the long-wavelength solution derived from equation (3-52). Al-beit the extension includes short-wavelength terms forshots and receivers, you still would need to estimateresidual statics and apply them to your data (equation3-25).

Equation (3-52) describes a variable-thicknessscheme since the weathering velocity is assumed to be

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constant. Note that the traveltime model equation (3-52) suffers from uncertainty in the value for the weath-ering layer velocity, as was the case for the generalizedreciprocal method. The estimated refractor shape usingthe variable-thickness scheme (whether it is based onthe the generalized reciprocal or least-squares method)does not yield the true refractor shape. Instead, the un-certainty in weathering velocity significantly influencesthe implied refractor shape. Perhaps, uphole informa-tion can be used to calibrate the estimated thicknessesand thus an acceptable weathering velocity can be cho-sen accordingly.

Once the parameter vector is estimated, then thethickness of the weathering layer below shot and re-ceiver locations can be computed using the expressionsfor the intercept time values (equations 3-53a,b).

Because equation (3-52) does not contain a struc-ture term, any long-wavelength static anomaly is par-titioned between the other terms. This is not the casefor the reflection-based residual statics model that isbased on equation (3-25). Thus, following the field stat-ics corrections (to account for elevation changes), staticscorrections are estimated and applied in two stages:

(a) Refraction-based statics corrections to removelong-wavelength anomalies, and

(b) Reflection-based residual statics corrections to re-move any remaining short-wavelength static shifts.

Both the generalized reciprocal and least-squaresmethods are based on computing intercept time anoma-lies at shot-receiver stations. The generalized recipro-cal method yields multi-valued intercept times for eachshot-receiver station, which need to be reduced to asingle-valued intercept time profile along the line be-fore computing the shot-receiver statics. On the otherhand, the least-squares method yields a unique inter-cept time value for each shot-receiver station based onthe least-squares minimization.

The generalized reciprocal method requires a spe-cial combination of raypaths associated with the trav-eltime picks that correspond to the terms in equations(3-50a,b) in estimating intercept times. These raypathsare not always attainable by 3-D swath shooting, andtherefore, the generalized reciprocal method is mostsuitable for 2-D seismic profiles. On the other hand,the least-squares method yields intercept times for ar-bitrary shot-receiver locations associated with 2-D or3-D recording geometries. The important point to keepin mind, however, is that neither method yields the truephysical parameters for the near-surface — the variable-thickness solution depends on the assumed value for theweathering velocity. However, with extra informationsuch as from upholes, these solutions may be calibrated.

Processing Sequence for Statics Corrections

It is important that we revisit the processing sequencein Figure 3.3-12 and the near-surface model depictedin Figure 3.4-10 for a rigorous description of moveoutand statics corrections. Starting with unprocessed fieldrecords, a detailed version of the processing sequence inFigure 3.3-12 is described below:

(a) Pick and edit first breaks from unprocessed fieldrecords.

(b) Assume or derive from uphole information a valuefor weathering velocity.

(c) For a downhole source, apply the uphole correction.(d) Compute the bedrock velocity and intercept times

at all shot and receiver stations using a refractionstatics method, such as the generalized reciprocalor the least-squares technique.

(e) By using the weathering velocity, bedrock velocityand intercept times, compute the depth to bedrockat shot-receiver stations (equations 3-53a,b).

(f) Apply the shot and receiver statics to replace theweathering layer with the bedrock while placingthe shot and receivers on a floating datum thatcorresponds to a smoothed form of the topographicsurface. The static time shift ∆τij to apply for agiven source-receiver pair is (Figure 3.4-10)

∆τij = (zj + zi)1vb

− 1vw

− 1vb

ETj − EFDj + ETi − EFDi ,

(3 − 54a)where zj and zi are the thickness of the weatheringlayer at shot and receiver stations, ETj and ETi

are the true shot and receiver elevations referencedto the topography, and EFDj and EFDi are theshot and receiver elevations referenced to the float-ing datum, respectively. The reason for moving theshots and receivers to a floating datum close to thesurface topography, rather than to a flat datum, isto be able to preserve the hyperbolicity of reflec-tion times while placing the shot and receiver pairsassociated with a CMP gather over the local datumlevel that is nearly flat within the spread length.

(g) Apply geometric spreading correction and decon-volution to shot records and sort to CMP gathers.

(h) Perform preliminary velocity analysis and applymoveout corrections.

(i) Apply datum corrections to move the shots andreceivers from the floating datum as specified instep (f) to a flat datum ED to which the CMP stack

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is referenced. Refer to Figure 3.4-10 and note thatthe datum correction ∆τij for a source-receiver pairis given by

∆τij =2ED − (EFDj + EFDi)

vb, (3 − 54b)

where EFDj and EFDi are the shot and receiverelevations with respect to the floating datum spec-ified in step (f).

(j) Estimate surface-consistent shot and residual staticshifts using methods described in Section 3.3.

(k) Apply residual statics corrections to CMP gathersfrom step (i).

(l) Apply the inverse of step (i) to move the shots andreceivers from the flat reference datum back to thefloating datum.

(m) Apply inverse moveout correction using velocitiesfrom step (h).

(n) Perform velocity analysis and apply moveout cor-rection.

(o) Apply datum corrections to move the shots andreceivers from the floating datum to the referenceflat datum as in step (i).

(p) Apply mute and stack the data. The stacked sec-tion is referenced to the flat datum level ED spec-ified in step (i).

Model Experiments

We shall analyze the problem of near-surface modelestimation using two sets of synthetic data. Thenear-surface model for the first data set comprisesa single layer with an undulating refractor and aflat topography. The near-surface model for the sec-ond data set comprises multiple layers below an ir-regular topography. We shall examine the extent ofresolving the long- and short-wavelength anomaliesby way of the refraction and residual statics cor-rection methods described in Sections 3.3 and 3.4.

Figure 3.4-14a shows an earth model that com-prises a simple subsurface velocity-depth model anda near-surface model that comprises a single layerwith an undulating refractor but flat surface topog-raphy. The refractor geometry has wavelength vari-ations that range from short wavelengths that areless than a cable length to long wavelengths greaterthan a cable length. The weathering velocity is1200 m/s and the refractor velocity is 2000 m/s.

Figure 3.4-14b shows the zero-offset section derivedfrom the earth model in Figure 3.4-14a. Note the travel-time distortions associated with the flat reflectors. Also,

in this section we see the multiples from the refrac-tor and the peglegs from reflector 1. The zero-offsetsection is appropriately aligned with respect to thevelocity-depth model in the lateral direction. The objec-tive in this model experiment is, following the applica-tion of refraction and residual statics corrections, to ob-tain a stack response similar to this zero-offset section.

A two-way acoustic wave equation was used tomodel a total of 154 shot records along the line. Bothshot and receiver group intervals are 50 m, and the num-ber of channels is 97, including the zero-offset trace. Thesplit-spread recording geometry has a maximum offsetof 2350 m. Selected shot records shown in Figure 3.4-15from the synthetic data set associated with the earthmodel exhibit traveltime distortions on the reflectionevents caused by the undulating refractor. Note also thedistinct refracted arrivals and the ground-roll energy.

Figure 3.4-16a shows the CMP-stacked section withno statics corrections. Compare with the zero-offset sec-tion (Figure 3.4-14b) and note some differences. As aresult of velocity discrimination, multiples have beenattenuated to some extent by CMP stacking. Both theCMP-stacked and zero-offset sections have the imprintof the near-surface effects on reflection times. Note thatCMP stacking has given rise to the spurious structuraldiscontinuities on reflections below 1 s between CMP100-200. Following residual statics corrections (Figure3.4-16b), these short-wavelength anomalies appear tohave been resolved. The moderate-to-long wavelengthanomalies expressed by the reflection traveltime undu-lations, however, have remained in the section. Theseanomalies have been resolved by refraction statics cor-rections as shown in Figure 3.4-17a using, in this case,the generalized linear inverse (GLI) method to solveequation (3-52). Nevertheless, some residual anomaliesstill remain. After residual statics corrections based onthe solution to equation (3-25), the remaining long-wavelength anomalies are untouched, while the resid-ual short-wavelength anomalies have been further re-solved. Unfortunately, not all of the traveltime distor-tions due to the near-surface layer (the undulating re-fractor R in Figure 3.4-14a) have been eliminated. Note,for instance, the slight undulations in Figure 3.4-17bon events at 0.5 and 1 s which correspond to horizons1 and 2 in Figure 3.4-14a. Note also in Figure 3.4-17bthe distorted structural high represented by the reflec-tion between 1-1.5 s, which corresponds to horizon 3in Figure 3.4-14a, and the sagging reflection below 1.5s, which corresponds to horizon 4 in Figure 3.4-14a.

The results of the GLI statics estimates are sum-marized in Figure 3.4-18. For the variable-thicknessestimate (equation 3-52), the weathering velocity was

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FIG. 3.4-14. (a) A velocity-depth model with a near-surface refractor (R) and simple subsurface structure; (b) the corre-sponding zero-offset section with trace spacing of 50 m.D

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FIG. 3.4-15. Selected shot records from the synthetic data set associated with the earth model in Figure 3.4-14a. Numberson top of each record indicate the CMP location in the vicinity of the shot.

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FIG. 3.4-16. CMP stack associated with the model data in Figure 3.4-14: (a) with no statics corrections, (b) section as in(a) after residual statics corrections. Datum level is 0 m in both sections.

FIG. 3.4-17. CMP stack associated with the model data in Figure 3.4-14 — (a) with refraction statics corrections using theGLI solution, (b) section as in (a) after residual statics corrections. Datum level is 0 m in both sections, and the weatheringvelocity is assumed to be 1200 m/s.

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FIG. 3.4-18. Summary of the variable-thickness GLI solution for refraction statics associated with the CMP stacked sectionin Figure 3.4-17a. See text for details. Plot direction is the same as that in Figure 3.4-17. Except in frame 1, shot attributesare denoted with × and receiver attributes are denoted with vertical bars. Estimated parameters from equation (C-54) areplotted in frame 1 with no distinction made between shot and receiver locations.

set to 1200 m/s — the correct velocity for the near-surface layer. Frame 1 shows the estimated GLI pa-rameters — the intercept time anomalies (equations 3-53a,b), as a function of the shot-receiver station num-ber. Frame 2 shows the pick fold, namely the number ofpicks in each shot (denoted by ×) and receiver (denotedby the vertical bars) gather. Note the tapering of thepick fold at both ends of the line.

A quantitative measure of the accuracy of the GLIsolution to refraction statics is the sum of the differencesbetween the observed picks tij and the modeled trav-eltimes tij (equation 3-52) over each shot and receivergather. These residual time differences are plotted inframe 3 of Figure 3.4-18. Large residuals often are re-lated to bad picks. Nevertheless, even with good picks,there may be large residuals attributable to inappropri-ateness of the model assumed for the near-surface.

Figure 3.4-18 also shows the estimated weatheringthicknesses at all shot-receiver stations (frame 4). Fi-

nally, the computed statics and the near-surface modelare shown in frames 5 and 6, respectively.

Uncertainty in the assumed value for weatheringvelocity is an important practical consideration in re-fraction statics. Figure 3.4-19 shows results of GLI stat-ics solution using two different weathering velocities.Compare with the result using the correct weatheringvelocity (Figure 3.4-17a) and note that the GLI solutiontolerates reasonable departures from the correct weath-ering velocity.

The ability of statics solutions to resolve the effectof the near-surface layer with the undulating refractorshown in Figure 3.4-14a is further tested by applyingthe generalized reciprocal method (GRM) described byequations (3-50a,b). Compare the results of refractionstatics corrections using the GLI method (Figure 3.4-17) and the GRM method (Figure 3.4-20), and notethat, in this case, the differences are marginal. Never-theless, it appears that neither of the statics solutionsappear to have resolved the time anomalies caused by

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FIG. 3.4-19. CMP stack associated with the model data in Figure 3.4-14 with refraction statics corrections using weatheringvelocity (a) 1000 m/s and (b) 1400 m/s. Datum level is 0 m in both sections. Compare with the section in Figure 3.4-17a.

FIG. 3.4-20. CMP stack associated with the model data in Figure 3.4-14 — (a) with refraction statics corrections using thegeneralized reciprocal method (GRM), (b) section as in (a) after residual statics corrections. Compare with the GLI resultsin Figure 3.4-17.

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the undulating near-surface layer, completely. Assump-tions made about the near-surface model always limitthe resolving power of all the statics corrections meth-ods.

Although in most cases the near-surface often con-sists of a low-velocity weathering layer, in some explo-ration basins, a single-layer near-surface model may notbe adequate. We shall examine the response of staticssolutions to a multilayer near-surface model shown inFigure 3.4-21a. The zero-offset section at z = 0 is free ofthe near-surface effects (Figure 3.4-21b) since the near-surface anomalies are above z = 0 in the model (Fig-ure 3.4-21a). Following the application of refraction andresidual statics corrections, ideally, we should like to ob-tain a stack response similar to this zero-offset section.

The near-surface consists of layers with irregu-lar geometry and an irregular topography. A two-wayacoustic wave equation was used to model a total of 154shot records along the line. Both shot and receiver groupintervals are 50 m, and the number of channels is 97,including that associated with the zero-offset trace. Thesplit spread recording geometry has a maximum offsetof 2350 m. Selected shot records from the synthetic dataset associated with the earth model (Figure 3.4-22) ex-hibit traveltime distortions on reflection events result-ing from the complex near-surface model. Note also thedistinct refracted arrivals and the ground-roll energy.

Figure 3.4-23a shows the CMP-stacked section withno statics corrections. Compare with the zero-offsetsection (Figure 3.4-21b) and note the significant dif-ferences. Note the severe time anomalies caused bythe complexity of the near-surface layer. Some short-wavelength anomalies have been resolved by residualstatics corrections (Figure 3.4-23b). Nevertheless, thisstacked section is far from implying the simple subsur-face structure (Figure 3.4-21a).

By using the near-surface model (Figure 3.4-21a),statics at all shot-receiver stations were calculated byhand and corrections were applied to the data. The re-sulting stacked section is shown in Figure 3.4-24a. Notethat most of the long-wavelength anomalies have beenremoved. Remaining disortions on reflection times, par-ticularly betwen CMP 300-400, imply that correctingfor the near-surface effects by statics shifts applied toCMP traces is a simplistic approach given the complex-ity of the near-surface model. Residual statics correc-tions do not help in removing the time anomalies thatare beyond the limit of statics corrections (Figure 3.4-24b).

Now assume a near-surface model that comprises asingle layer with constant velocity (1400 m/s). The topand base of this layer are defined by the elevation curveand the flat datum at z = 0, respectively. Then, com-pute the elevation statics at each shot-receiver stationusing the thickness of the constant-velocity layer, and

apply them to the CMP traces. Figure 3.4-25 showsthe CMP stack with elevation statics and the subse-quent residual statics corrections. Compare with Fig-ure 3.4-23 and note that much of the time anoma-lies are due to elevation differences along the line.

Elevation statics corrections are basically a sim-ple alternative to statics corrections based on an esti-mate of a near-surface model. Using the GLI solutionbased on equation (3-52), inversion of the refracted ar-rivals yields the stacked section in Figure 3.4-26a. Here,the near-surface model was assumed to consist of asingle layer, as in the case of elevation statics correc-tions, but with varying refractor geometry. Note theimprovement on reflection times on the stack associ-ated with the GLI solution. Clearly, a more compli-cated, multi-layered near-surface model can, in princi-ple, be estimated from inversion of refracted arrivals.However, the more complicated the model, the moreparameters need to be specified. This in turn will re-quire a more complicated inversion scheme. Generally,in practice, one should model the near-surface simply.If traveltime distortions are not resolved adequately bya simple near-surface model, it often means that theproblem is not solvable by statics methods. Specifi-cally, the near-surface corrections should not be doneusing vertical time shifts applied to CMP traces. Un-der those circumstances, very little can be achieved byresidual statics corrections (Figure 3.4-26b). Instead,the problem should be characterized as dynamic andbe solved by earth modeling in depth (Chapter 9).

The results of the GLI statics estimates are sum-marized in Figure 3.4-27. For the variable-thicknesssingle-layer near-surface, the weathering velocity wasassumed to be 1400 m/s. Frame 1 shows the estimatedGLI parameters — the intercept time anomalies (equa-tion 3-53a,b), as a function of the shot-receiver sta-tion number. Frame 2 shows the pick fold, namely thenumber of picks in each shot (denoted by ×) and re-ceiver (denoted by the vertical bars) gather. Note thetapering of the pick fold at both ends of the line. Thesum of the differences between the observed picks tijand the modeled traveltimes tij (equation 3-52) overeach shot and receiver gather is shown in frame 3.Large residuals, in this case, are attributable to theinappropriateness of the model assumed for the near-surface. Figure 3.4-27 also shows the estimated weath-ering thicknesses at all shot-receiver stations (frame4). Finally, the computed statics and the near-surfacemodel are shown in frames 5 and 6, respectively.

For comparison, the GRM statics solution for themultilayered near-surface model of Figure 3.4-21a isshown in Figure 3.4-28. While both the GLI (Figure3.4-26a) and GRM (Figure 3.4-28a) solutions are com-parable, it appears that neither of the statics solutions


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FIG. 3.4-21. (a) A velocity-depth model that comprises a multilayer near-surface with a strong refractor (R) and simplesubsurface structure; (b) the corresponding zero-offset section with trace spacing of 50 m.

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FIG. 3.4-22. Selected shot records from the synthetic data set associated with the earth model in Figure 3.4-21a. Numberson top of each record indicate the CMP location in the vicinity of the shot.

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FIG. 3.4-23. CMP stack associated with the model data in Figure 3.4-21 — (a) with no statics corrections, (b) section as in(a) after residual statics corrections. Datum level is 0 m in both sections. Elevation curve is plotted on top of each section.

FIG. 3.4-24. CMP stack associated with the model data in Figure 3.4-21 — (a) with statics corrections using hand-calculatedstatics, (b) section as in (a) after residual statics corrections. Datum level is 0 m in both sections. Elevation curve is plottedon top of each section.

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FIG. 3.4-25. CMP stack associated with the model data in Figure 3.4-21 — (a) with elevation statics corrections, (b) sectionas in (a) after residual statics corrections. Datum level is 0 m in both sections. Elevation curve is plotted on top of eachsection.

FIG. 3.4-26. CMP stack associated with the model data in Figure 3.4-21 — (a) with refraction statics corrections using theGLI solution, (b) section as in (a) after residual statics corrections. Datum level is 0 m in both sections, and the weatheringvelocity is assumed to be 1400 m/s.

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FIG. 3.4-27. Summary of the variable-thickness GLI solution for refraction statics associated with the CMP stacked sectionin Figure 3.4-26a. Plot direction is the same as that in Figure 3.4-26. Except in frame 1, shot attributes are denoted with× and receiver attributes are denoted with vertical bars. Estimated parameters from equation (C-54) are plotted in frame 1with no distinction made between shot and receiver locations.

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FIG. 3.4-28. CMP stack associated with the model data in Figure 3.4-21 — (a) with refraction statics corrections using thegeneralized reciprocal method (GRM), (b) section as in (a) after residual statics corrections. Compare with the GLI resultsin Figure 3.4-26.

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appears to have resolved the time anomalies caused bythe complex near-surface layer, completely. Again, as-sumptions made about the near-surface model alwayslimit the resolving power of all the statics methods.

Field Data Examples

We shall analyze field data with three different near-surface characteristics. Specifically, near-surface modelswith combinations of irregular topography and refractorgeometry are examined. Refraction statics solutions arebased on the variable-thickness scheme based on equa-tion (3-52) and residual statics solutions are based onequation (3-25), both solved by the generalized linearinversion schemes (Sections C.4 and C.8).

The first field data example is from an area withnearly flat topography and presumably irregular base ofweathering. Shown in Figure 3.4-29a is a CMP-stackedsection based on elevation statics corrections that in-volved a flat datum and constant weathering velocity.Note the presence of traveltime distortions along themajor reflections down to 2 s caused by the unresolvedlong-wavelength statics anomalies. We also note veryshort-wavelength traveltime distortions, much less thana cable length. This latter component of the statics canbe resolved by surface-consistent residual statics correc-tions as shown in Figure 3.4-29b. Although the CMPstacking quality has been improved after the residualstatics corrections, the long-wavelength statics anoma-lies remain unresolved.

Figure 3.4-30 shows plots of the first-break picksfrom the far-offset arrivals associated with the refractedenergy. While most of the first-break picks consistentlyfollow a linear moveout from shot to shot, note thatthere are some local deviations that indicate a mod-erate degree of complexity in the near surface. Figure3.4-31 shows the CMP-stacked section after the applica-tion of refraction statics using the generalized recipro-cal method. Compare with Figure 3.4-29a and note thesignificant elimination of long-wavelength statics. Alsoplotted are the intercept time anomalies at all shot-receiver stations. Recall that equations (3-50a,b) yieldmultiple values of intercept time estimates at each sta-tion. These multiple values need to be reduced to uniqueintercept time values at each station so as to be able toestimate the thickness of the weathering layer at eachstation, uniquely. The statics solution at all shot andreceiver stations shows that the generalized reciprocalmethod can correct for all wavelengths of statics causedby undulations along the base of the weathering layer.Any remaining (residual) very short-wavelength stat-ics should be corrected for by using a reflection-basedmethod (Section 3.3).

Figure 3.4-32a shows the CMP-stacked section af-ter the application of refraction statics corrections basedon the variable-thickness, least-squares scheme (equa-tion 3-52a). Compare this result with Figure 3.4-29aand note that the long-wavelength statics anomalieshave been removed. Also, note that both the general-ized reciprocal method (Figure 3.4-31a) and the least-squares method (Figure 3.4-32a) yield comparable re-sults. The section in Figure 3.4-32a can further be im-proved by applying residual statics corrections to re-move the short-wavelength statics components (Figure3.4-32b).

The results of the least-squares statics estimatesare summarized in Figure 3.4-33. The weathering veloc-ity was assumed to be 450 m/s. Frame 1 shows the esti-mated intercept times as a function of the shot/receiverstation number. Frame 2 shows the pick fold, namelythe number of picks in each shot and receiver gather.Note the tapering of the pick fold at both ends of theline.

A quantitative measure of the accuracy of the least-squares solution is the sum of the differences betweenthe observed picks tij and the modeled traveltimes tij(equation 3-52a) over each shot and receiver gather.These cumulative residual time differences over eachshot and receiver are plotted in frame 3 of Figure 3.4-33.Large residuals often are related to bad picks. Neverthe-less, even with good picks, there may be large residualsattributable to the inappropriateness of the model as-sumed for the near-surface.

Figure 3.4-33 also shows the estimated thickness ofthe weathering layer at all shot-receiver stations (frame4). Finally, the computed statics and the near-surfacemodel are shown in frames 5 and 6, respectively.

The next field data example is from an area withirregular topography associated with a sand dune andpresumably a near-flat base of weathering. Figure 3.4-34a shows the CMP-stacked section with elevation stat-ics corrections. Note the severe distortions of the geome-try of shallow reflections and a very poor signal-to-noiseratio in the central part of the section. Residual staticscorrections (Figure 3.4-34b) cannot improve the inter-pretation, especially in the center of the line where thefirst breaks show significant departures from a consis-tent linear moveout (Figure 3.4-35).

Figure 3.4-36 shows the CMP-stacked section withthe application of refraction statics corrections usingthe generalized reciprocal method. Again, note themultiple-valued intercept time values at shot/receiverstations. The statics solution based on the reducedintercept times shows a significant medium- to long-wavelength variations. After the application of these


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FIG. 3.4-29. (a) A CMP stack with elevation statics corrections, (b) same as in (a) with residual statics corrections. (Datacourtesy Nederlandse Aardolie Maatschappij B.V.)

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FIG. 3.4-30 First-break picks associated with the data shown in Figure 3.4-29. The top and the bottom plots correspondto the left- and right-hand of the split-spread geometry. The results of the refraction statics solution based on these picks areshown in Figures 3.4-31 to 3.4-33.D

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FIG.3.4-31. (a) A CMP stack with refraction statics applied using the generalized reciprocal method (compare with Figures3.4-29a and Figure 3.4-32a), (b) estimated multiple intercept times at each shot-receiver station and the computed shot (opensquares) and receiver (solid dots) statics.

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FIG. 3.4-32. (a) A CMP stack with refraction statics applied using the least-squares method (compare with Figures 3.4-29aand Figure 3.4-31a), (b) same as in (a) with residual statics corrections.

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FIG. 3.4-33. Results of the least-squares refraction statics estimated from and applied to the data in Figure 3.4-32. In eachframe, shot attribute is denoted by × and receiver attribute is denoted by the vertical bars. R is the refractor that representsthe base of the weathering layer. See text for details.D

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FIG. 3.4-34. (a) A CMP stack with elevation statics corrections, (b) same as in (a) with residual statics corrections.

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FIG. 3.4-35. First-break picks associated with the data shown in Figure 3.4-34. The top and the bottom plots correspondto the left- and right-hand of the split-spread geometry. The results of the refraction statics solution based on these picks areshown in Figures 3.4-36 to 3.4-38.

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FIG. 3.4-36. (a) A CMP stack with refraction statics applied using the generalized reciprocal method (compare with Figures3.4-34a and Figure 3.4-37a), (b) estimated multiple intercept times at each shot-receiver station and the computed shot (opensquares) and receiver (solid dots) statics.D

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statics corrections, the near-surface effects on the re-flector geometries have been largely removed (comparewith Figure 3.4-34a).

Figure 3.4-37a shows the CMP-stacked section af-ter the application of refraction statics corrections basedon the least-squares method. Compare with Figure 3.4-34a and note the significant improvement in the centerof the line. This section can be improved further byapplying residual statics corrections and thus removingthe short-wavelength statics components (Figure 3.4-37b).

The results of the least-squares statics estimatesare summarized in Figure 3.4-38. The weathering ve-locity was assumed to be 800 m/s. (The description ofthe frames in Figure 3.4-38 is the same as that of Figure3.4-33.)

The third field data example is from an areawith an abrupt change in topography and presumablysurface-following the base of weathering. The CMP-stacked section with elevation statics corrections isshown in Figure 3.4-39a. Residual statics correctionssignificantly improve the stacking quality (Figure 3.4-39b); but the long-wavelength statics anomalies remainon the section and appear as spurious structural anoma-lies.

Figure 3.4-40 shows the first-break picks from thefar-offset arrivals associated with the refracted energy.Figure 3.4-41 shows the CMP-stacked section with therefraction statics applied using the generalized recipro-cal method and the first-break picks in Figure 3.4-40.Compare with Figure 3.4-39b and note the removal ofthe spurious structural discontinuity along the strongreflection just above 2 s on the left half of the section.

By using the first-break picks shown in Figure 3.4-40, the variable-thickness least-squares parameters forthe near-surface were computed (equation 3-52a). Thecorresponding CMP-stacked section is shown in Figure3-4.27a. Note the elimination of the spurious structuraldiscontinuities seen in Figure 3.4-39b between 1 and 2s. The CMP stacked section can be improved further byapplying residual statics corrections (Figure 3.4-42b).

The results of the least-squares statics estimatesare summarized in Figure 3.4-43. For the variable-thickness estimate, the weathering velocity was as-sumed to be 900 m/s. (The description of the framesin Figure 3.4-43 is the same as that of Figure 3.4-33.)

Figure 3.4-44 is a stacked section with only thefield statics applied. The pull-up at midpoint loca-tion A probably is caused by a long-wavelength stat-ics anomaly. Start with CMP gathers (Figure 3.4-45a)and apply linear-moveout (LMO) correction (Figure3.4-45b). Assuming that the first breaks correspond toa near-surface refractor, we use the estimated velocity

from the first breaks (usually from a portion of the ca-ble) to apply the LMO correction. The CMP-refractionstack of the shallow part of the data after the LMOcorrection is shown in Figure 3.4-45c. This section isthe equivalent of the pilot trace section that is associ-ated with the reflection-based statics corrections. (Anexample of this is shown in Figure 3.3-37.)

Traveltime deviations are estimated from theLMO-corrected gathers (Figure 3.4-45b) and are de-composed into shot and receiver intercept time compo-nents based on equation (3-52a). These intercept timesare used to compute shot and receiver static shifts,which are then applied to the CMP gathers shown inFigure 3.4-45a. A comparison of the CMP-refractionstack section with (Figure 3.4-45d) and without (Fig-ure 3.4-45c) refraction statics corrections clearly indi-cates removal of the significant long-wavelength staticsanomaly centered at midpoint location A (Figure 3.4-44). The CMP-stacked section after the refraction stat-ics corrections shown in Figure 3.4-46 no longer containsthe false structure (compare with Figure 3.4-44). Thislong-wavelength anomaly cannot be removed by reflec-tion statics corrections alone (Figure 3.4-47). Neverthe-less, the residual statics corrections resolved the short-wavelength statics components that were present in thedata. By cascading the two corrections — refraction andresidual statics, we get the improved section in Figure3.4-48.

The last field data example for refraction and resid-ual statics corrections is from an overthurst belt with ir-regular topography and large elevation differences alongthe line traverse. Figures 3.4-49 and 3.4-50 show se-lected shot records. Note that the first breaks are verydistinct, and the first arrivals do not manifest significantdepartures from linear moveout. Nevertheless, there aresignificant distortions along the reflection traveltimetrajectories; these are largely attributed to the subsur-face complexity associated with the overthrust tecton-ism in the area.

Figure 3.4-51 shows selected CMP gathers with el-evation corrections applied and the data referenced toa flat datum of 1800 m above the topographic profile ofthe line. Following the normal-moveout correction (Fig-ure 3.4-52), note that the CMP gathers exhibit short-wavelength deviations less than a cable length along thereflection traveltime trajectories. Velocity analysis andmoveout correction were performed from a floating da-tum — a smoothed version of the topographic profile.The CMP stack with elevation corrections is shown inFigure 3.4-53.

The same CMP gathers as in Figures 3.4-51 and3.4-52 with refraction statics applied are shown in Fig-ures 3.4-54 and 3.4-55. A comparison of these sets of


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FIG. 3.4-38. Results of the least-squares refraction statics estimated from and applied to the data in Figure 3.4-37. In eachframe, shot attribute is denoted by × and receiver attribute is denoted by vertical bar. R is the refractor that represents thebase of the weathering layer. See text for details.D

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FIG. 3.4-39. (a) A CMP stack with elevation statics corrections, (b) same as in (a) with residual statics corrections.

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FIG. 3.4-40. First-break picks associated with the data shown in Figure 3.4-39. The top and the bottom plots correspondto the left- and right-hand of the split-spread geometry. The results of the refraction statics solution based on these picks areshown in Figures 3.4-41 to 3.4-43.

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FIG. 3.4-41. (a) A CMP stack with refraction statics applied using the generalized reciprocal method (compare with Figures3.4-39a and Figure 3.4-42a), (b) estimated intercept times and the computed shot (open squares) and receiver (solid dots)statics.

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FIG. 3.4-43. Results of the least-squares refraction statics estimated from and applied to the data in Figure 3.4-42. In eachframe, shot attribute is denoted by × and receiver attribute is denoted by vertical bar. R is the refractor that represents thebase of the weathering layer. See text for details.

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FIG. 3.4-44. A CMP stack with field statics applied. Note the unresolved long-wavelength statics manifested by the spuriosstructural high below midpoint A.

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FIG. 3.4-45. (a) Selected CMP gathers from the section in Figure 3.4-44, (b) CMP gathers after linear-moveout (LMO)correction, (c) stack of the LMO-corrected gathers as shown in (b), and (d) stack of the LMO-corrected gathers after long-period statics were removed; compare with (c).

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FIG. 3.4-46. The CMP stack associated with the data in Figure 3.4-45 after refraction statics corrections. Compare withFigure 3.4-44.

gathers indicates that the statics problem is primar-ily of residual nature — differences between refractionand elevation statics are not significant. In other words,long-wavelength statics, in this case, are associated forthe most part with irregular topography. Differences be-tween the CMP stack with refraction statics (Figure 3.4-56) and the CMP stack with elevation statics (Figure3.4-53) are marginal.

Short-wavelength traveltime deviations observedon the CMP gathers in Figures 3.4-54 and 3.4-55 havebeen resolved by residual statics corrections as shown inFigures 3.4-57 and 3.4-58. Reflection traveltimes in Fig-ure 3.4-57 are much like hyperbolic and those in Figure3.4-58 are reasonably flat after moveout correction. Thecorresponding CMP stack shown in Figure 3.4-59, whencompared with Figure 3.4-56, clearly demonstrates theimprovement attained by residual statics corrections.

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FIG. 3.4-47. The CMP stack associated with the data in Figure 3.4-45 after field statics and residual statics corrections.Compare with Figure 3.4-44 and 3.4-46.

In areas with severely irregular topography andlarge elevation changes along line traverses, one mayconsider extrapolating the recorded data from the to-pographic surface to a flat datum above the topographyby using the wave-equation datuming technique (Sec-tion 4.6). Bevc (1997) applied this technique to the dataas in Figures 3.4-49 and 3.4-50. You still will need toapply residual statics corrections to account for short-

wavelength statics not associated with topography, butrelated to the near-surface layer geometry.

Finally, in the presence of a permafrost layer ora series of lava flows at the near-surface, the prob-lem inherently is dynamic in nature. Specifically, undersuch circumstances, rays through the near-surface donot follow near-vertical paths, and thus the near-surfaceproblem cannot be posed as a statics problem. Instead,D

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FIG. 3.4-48. The CMP stack associated with the data in Figure 3.4-45 after refraction and residual statics corrections.Compare with Figure 3.4-44 and 3.4-46.

one needs to estimate accurately a velocity-depth modelthat accounts for the near-surface complexity so as tohonor ray bending through the near-surface layer.

Figure 3.4-60 shows a CMP-stacked section from anarea with a permafrost layer at the near-surface. Notethat refraction statics followed by residual statics cor-rections (Figure 3.4-61) yield a section with improvedevent continuity. Nevertheless, there still exist a num-ber of spurious structural features that have to be ac-counted for. Figure 3.4-62 shows a CMP-stacked sectionfrom an area with lava flows at the near-surface. Al-

though residual statics corrections have improved eventcontinuity, spurious faults are troublesome (Figure 3.4-63). The traveltime distortions on the stacked sectionsin Figures 3.4-61 and 3.4-63 strongly suggest that theycannot be resolved by statics corrections alone. Addi-tional work, such as velocity-depth modeling (Chapter9) and imaging in depth (Chapter 8), is required toaccount for lateral velocity variations associated withnear-surface complexities that result from lava flows anda permafrost layer.


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EXERCISES

Exercise 3-1. Why does salt have anomalouslyhigh velocity (4.5 to 5.5 km/s)?

Exercise 3-2. Measure the traveltimes corre-sponding to offset values of 1 and 3 km in Figure 3.1-2.Then compute the velocity above the reflector and ver-ify that it is 2264 m/s. Note that the zero-offset traceis not recorded; therefore, t0 normally is not a knownquantity.

Exercise 3-3. The CMP gather in Figure 3.1-5acontains a hyperbola. Following NMO corrections andusing 2000 (Figure 3.1-5c) and 2500 m/s (Figure 3.1-5d)velocities, are the traveltime trajectories hyperbolic?

Exercise 3-4. Make velocity picks from the veloc-ity panels in Figures 3.2-4, 3.2-5, 3.2-6, and 3.2-7.

Exercise 3-5. Make velocity picks from the CVSpanel in Figure 3.E-1.

Exercise 3-6. Consider two intersecting lines.Would you expect that velocity analyses at the inter-section point yield the same velocity function?

Exercise 3-7. Which is correct: velocity analysisfrom datum or velocity analysis from surface?

Exercise 3-8. Fill the missing elements in thefollowing table. Average velocity vavg, which relatesvertical traveltime to depth in a horizontally layeredmedium, is defined as

vavg =Ni=1 vi∆tiNi=1 ∆ti

,

where ∆ti = ∆zi / vi, ∆zi = layer thickness and vi =interval velocity. The rms velocity is given by equation(3-4).

Layer Interval RMS AverageThickness, Velocity, Velocity, Velocity,

m m/s m/s m/s

200 2,000300 3,000400 4,000350 3,500500 5,000

Exercise 3-9. Explain why the velocity for horizonA in Figure 3.2-34 behaves as shown in the HVA displaybelow the salt dome S.

Exercise 3-10. Suppose you want to fit a set ofobserved traveltimes to a parabola of the form t = a +bx + cx2. The tabulated input values are given below.

Observedi xi ti

1 0 0.42 1 1.13 2 3.54 3 7.95 4 14.4

Set up the least-squares problem and solve for a, b, andc. You will have five equations and three unknowns.

Exercise 3-11. Solve the system

x1 − 2x2 = 1x1 + 4x2 = 4

by the Gauss-Seidel iterative method. Verify the resultsby solving these equations by the method of substitu-tion to obtain the correct solution: x1 = 2 and x2 = 0.5.

Exercise 3-12. Solve the system

x1 + 4x2 = 4x1 − 2x2 = 1

by the Gauss-Seidel iterative method. Note that this isthe same problem as in Exercise 3-11, except that theorder of the equations is reversed. The solution shouldbe the same. You will find that the solution cannot beobtained because the process of iteration will not con-verge. This demonstrates the importance of ordering theequations when solving by the Gauss-Seidel method.

Exercise 3-13. Write equation (3-37a) for i =1, 2, 3, and j = 1, 2, 3. You will find that there are sixunknowns, but five independent equations.

Exercise 3-14. Can you use refraction-based stat-ics techniques in permafrost areas?

Exercise 3-15. Which of the following adverselyaffects the quality of velocity spectrum — long-wavelength or short-wavelength (less than a cablelength) statics anomalies?

Exercise 3-16. In what way does inside muting ofa CMP gather affect the velocity spectrum?

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FIG. 3.E-1. Part 1: CVS panels for a line from an area with complex structure associated with overthrust tectonics.

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Appendix CTOPICS IN MOVEOUT AND STATICS CORRECTIONS

C.1 The Shifted Hyperbola

The objective in this section is to review the higher-order accuracy in normal moveout for ahorizontally layered earth model. Refer to the traveltime equation (3-3) by Taner and Koehler(1969):

t2 = C0 + C1x2 + C2x

4 + C3x6 + · · · , (C − 1)

where x is the offset, and

C0 = t20, (C − 2a)

where t0 is the two-way zero-offset traveltime

C1 =1µ2

, (C − 2b)

C2 =14

µ22 − µ4

t20µ42

, (C − 2c)

and

C3 =2µ2

4 − µ2µ6 − µ22µ4

t40µ72

, (C − 2d)

with the additional definitions of the terms (Castle, 1994)

µj =1t0

N

i=1

vji ∆τi. (C − 3a)

Note that

µ2 = v2rms (C − 3b)

since

v2rms =

1t0

N

i=1

v2i ∆τi, (C − 3c)

where ∆τi is the vertical two-way traveltime through the i-th layer, v is the velocity, and t0 =Ni=1 ∆τi. Derivation of equation (C-1) is based on the parametric equations for traveltime and

offset for a horizontally layered earth model (Slotnick, 1959; Grant and West, 1965) and is givenby Castle (1994).

Drop higher-order terms in equation (C-1) to obtain the fourth-order moveout equation

t2 = C0 + C1x2 + C2x

4. (C − 4)

Now, substitute for the coefficients from equations (C-2a), (C-2b) and (C-3a) to obtain

t2 = t20 +x2

v2rms

+ C2x4. (C − 5).

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This equation can in principle be used to compute a velocity spectrum (Section 3.2). First, dropthe fourth-order term to get the small-spread hyperbolic equation

t2 = t20 +x2

v2rms

. (C − 6)

Compute the velocity spectrum using equation (C-6), and pick an initial velocity functionvrms(t0). Then, use this picked velocity function in equation (C-5) and compute a velocityspectrum for the parameter C2. Pick a function C2(t0) and use in equation (C-5) to recomputethe velocity spectrum. Finally, pick an updated velocity function vrms(t0) from this velocityspectrum.

The fourth-order moveout equation (C-4) can be expressed exactly by a time-shifted hy-perbolic traveltime equation of the form (Castle, 1994)

t = t0 1 − 1S

+t0S

2

+x2

Sv2rms

, (C − 7)

where S is a constant of the form

S =µ4

µ22

. (C − 8)

For S = 1, equation (C-7) reduces to the conventional moveout equation (C-6).Figure C-1 shows the traveltime trajectories of the hyperbolic moveout equation (C-6) and

the time-shifted hyperbolic equation (C-7) where τs = t0(1− 1/S). Note that the shifted hyper-bola is a better match to the true traveltime trajectory at far offsets. The latter was computedfor a horizontally layered earth model using the exact parametric equations for traveltime andoffset (Slotnick, 1959; Grant and West, 1965).

The shifted hyperbola equation satisfies the requirements for a moveout equation:

(a) The traveltime trajectory is symmetric with respect to the time axis.(b) The instantaneous velocity dx/dt is never zero.(c) For constant velocity, it reduces to the exact hyperbolic moveout equation.

As for the fourth-order moveout equation (C-5), this equation can, in principle, be used toconduct the velocity analysis of CMP gathers. First, set S = 1 in equation (C-7) to get equation(C-6). Compute the velocity spectrum using equation (C-6), and pick an initial velocity functionvrms(t0). Then, use this picked velocity function in equation (C-7) and compute a velocityspectrum for the parameter S. Pick a function S(t0) and use in equation (C-7) to recomputethe velocity spectrum. Finally, pick an updated velocity function vrms(t0) from this velocityspectrum.

De Bazelaire (1988) offers an alternative moveout equation to achieve higher-order accuracyat far offsets:

t = (t0 − tp) + t2p +x2

v2s

, (C − 9)

where t0 is the two-way zero-offset time, tp is related to the time at which the asymptotes of thehyperbolic traveltime trajectory converge (Figure C-2), and vs is the reference velocity assignedto the layer below the recording surface (not the near-surface layer). When tp = t0, equation(C-9) reduces to the small-spread hyperbolic equation (C-6).

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FIG. C-1. Traveltime trajectories based on (top) the hyperbolic equation (C-23) and (bottom) thetime-shifted hyperbolic equation (C-14). Compare with the true traveltime trajectory associated witha layered model (Castle, 1994).

C.2 Moveout Stretch

Refer to Figure 3.1-10 for a sketch of a wavelet before and after moveout correction. The moveoutequation is

t2 = t20 +x2

v2, (C − 10)

where t is the two-way traveltime associated with a source-receiver separation x, t0 is the two-wayvertical traveltime — the time after moveout correction, and v is the moveout velocity.

Consider a reflection event represented by a wavelet of dominant period T with an arrivaltime t at offset x. After normal-moveout correction, the dominant period becomes T0 = T +∆T .The moveout equation (C-10) is associated with the onset of the wavelet. Similarly, the moveout

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FIG. C-2. The traveltime trajectory associated with the moveout equation (3-5c) (De Bazelaire, 1988).

equation with the termination of the wavelet is expressed by

(t + T )2 = (t0 + T + ∆T )2 +x2

v2. (C11)

Expand the terms on both sides:

t2 + 2tT + T 2 = t20 + 2t0(T + ∆T ) + (T + ∆T )2 +x2

v2. (C − 12a)

By making the substitution from equation (C-10), we obtain

2tT + T 2 = 2t0(T + ∆T ) + (T + ∆T )2. (C − 12b)

Simplify and rearrange the terms

2(t − t0)T = 2(t0 + T )∆T + ∆T 2. (C − 12c)

Now, ignore the second term on the right-hand side of the equation and observe that ∆tNMO =t − t0 to obtain

∆tNMOT = (t0 + T )∆T . (C − 12d)

Assume that t0 >> T and rearrange the terms to obtain a relationship for change in the periodof the wavelet as a result of moveout correction:

∆T

T=

∆tNMO

t0. (C − 13)

Now, we want to express equation (C-13) in terms of the dominant frequency f of thewavelet. Start with the relation

T =1f

, (C − 14)

and obtain

∆T = − 1f2

∆f. (C − 15)

Finally, combine equations (C-13) and (C-15) to obtain the equation for the absolute value offrequency stretching:

∆f

f=

∆tNMO

t0. (C − 16)

This is the same as equation (3-6) in the main text.

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C.3 Equations for a Dipping Reflector

We want to derive equation (3-7) of the main text using the geometry of a dipping reflectoras shown in Figure C-3. (The derivation provided here is courtesy Zhiming Li, 1999). Thedistance from source location S to reflection point R back to receiver location G is given bySR + RG = FR + RG = FG = vt, where point F is the mirror image of the source location S,v is the velocity of the medium above the dipping reflector, and t is the traveltime from F to G.

To compute the traveltime t associated with the distance FG, we shall need to computethe coordinates of F : (xF , zF ) and G : (xG, zG), so that

(FG)2 = xF − xG2 + zF − zG

2. (C − 17)

From the geometry of Figure C-3, the coordinates of F : (xF , zF ) are

xF = OF cos 2φ

and

zF = OF sin 2φ,

where φ is the dip angle of the reflector. Substitute the relation OF = OS = OM + x/2 intothe above equations to obtain

xF = OM +x

2cos 2φ (C − 18a)

and

zF = OM +x

2sin 2φ. (C − 18b)

Again, from the geometry of Figure C-3, the coordinates of G : (xG, zG) are

xG = OM − x

2, (C − 19a)

and

zG = 0. (C − 19b)

Substitute equations (C-18a,b) and (C-19a,b) into equation (C-17)

(FG)2 = OM +x

2cos 2φ − OM − x

2

2

+ OM +x

2sin 2φ

2

, (C − 20a)

then use the trigonometric relation cos 2φ = 2 cos2 φ − 1 and apply some algebra to obtain

(FG)2 = OM +x

2

2

+ OM − x

2

2

− 2 (OM)2 − x2

42 cos2 φ − 1 . (C − 20b)

Further algebraic manipulation yields

(FG)2 = 4(OM)2 sin2 φ + x2 cos2 φ. (C − 21)

Finally, note from the geometry of Figure C-3 the relation

MN = OM sinφ. (C − 22)

Subsitute equation (C-22) into equation (C-21), and use the relations 2MN = vt0 and FG = vtto obtain the desired expression for the reflection traveltime associated with a dipping reflector

t2 = t20 +x2 cos2 φ

v2, (C − 23a)

and the moveout velocity

vNMO =v

cos φ. (C − 23b)

Equations (C-23a) and (C-23b) are the same as equations (3-10) and (3-11) of the main text.

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FIG. C-3. Geometry of a dipping reflector used in deriving the equations in Section C.3.

C.4 Traveltime Decomposition for Residual Statics Estimation

We want to model traveltime deviations tij associated with a reflection event on moveout-corrected CMP gathers by the following equation (Taner et al., 1974; Wiggins et al., 1976):

tij = sj + ri + Gk + Mkx2ij , (C − 24a)

where sj is the residual statics shift at the jth source location, ri is the residual statics shift atthe ith receiver location, Gk is the structure term at the kth midpoint location, [k = (i + j)/2],Mkx2

ij is the residual moveout at the kth midpoint location.For m picks of tij , and ns shot locations, nr receiver locations, nG midpoint locations, we

have the following set of equations:

...

...

...tij.........

=

· · · 1 · · · 1 · · · 1 · · ·x2ij · · ·

...sj

...ri...

Gk...

Mk...

. (C − 24b)

A more parsimonious parameterization can be achieved by an appropriate traveltime pickingstrategy such that the structure term can be set to zero. Additionally, the residual moveout termcoefficient M may be set to a constant. The model equation (C-24a) can be rewritten as

tij = sj + ri + Mx2ij . (C − 25a)

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Equation (C-24b) is modified, accordingly

...

...

...tij.........

=

· · · 1 · · · 1 · · · x2

ij

...sj

...si...

M

. (C − 25b)

Whichever the preferred modeled traveltimes, write the respective equations (C-24b) or(C-25b) in matrix notation as

t = Lp, (C − 26)

where t is the column vector of m-length (number of traveltime picks) in equation (C-24b) or(C-25b), L is the sparse matrix in the same equations with dimensions m× (ns +nr +nG +nG)in case of equation (C-24b) and m × (ns + nr + 1) in case of equation (C-25b), and p is thecolumn vector of (ns +nr +nG +nG)-length in case of equation (C-24b) and (ns +nr +1)-lengthin case of equation (C-25b) on the right-hand side of these equations. Except the three elementsin each row, the L matrix contains zeros.

The generalized least-squares solution to equation (C-26) satisfies the requirement that theenergy of the error vector

e = t − t (C − 27a)

is minimum (Lines and Treitel, 1984). The cumulative error energy is

E = eTe. (C − 27b)

Substitute equation (C-27a) into equation (C-27b) and use equation (C-26) to obtain

E = (t − Lp)T (t − Lp). (C − 27c)

Minimization of E with respect to p requires that

∂E

∂sj=

∂E

∂ri=

∂E

∂Gk=

∂E

∂Mk= 0, (C − 27d)

which, in the case of equation (C-24b), yields ns + nr + nG + nG equations and that manyunknowns. These equations can be solved for the residual statics associated with ns sourcelocations, nr receiver locations, nG structural terms, and nG residual moveout terms.

Finally, the solution that satisfies the minimization requirement given by equation (C-27d)follows (Lines and Treitel, 1984)

p = (LTL)−1LTt, (C − 28)

where t denotes the column vector of m-length that represents the traveltime deviations pickedfrom moveout-corrected CMP gathers, and T denotes the matrix transposition.

A practical scheme for solving equation (C-24a) is based on the Gauss-Seidel method. Inthis scheme, each term on the right-hand side of equation (C-24a) is computed by the set ofrecursive equations

smj =

1nr

nr

i

{tij − Gm−1k − Mm−1

k x2ij − rm−1

i }, (C − 29a)

rmi =

1ns

ns

j

, {tij − Gm−1k − Mm−1

k x2ij − sm−1

j }, (C − 29b)

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Gmk =

1nh

nh

l

, {tij − sm−1j − Mm−1

k x2ij − rm−1

i }, (C − 29c)

and

Mmk =

1nG

nG

k

1x2

ij

{tij − sm−1j − Gm−1

k − rm−1i }, (C − 29d)

where i and j are the receiver and source indexes, respectively, l = |i − j| is the offset index,k = (i+ j)/2 is the midpoint index, m is the iteration index, and nh is the fold of coverage. Thesolutions in equations (C-29) are based on the orthogonality of the shot and receiver axes, andthe orthogonality of the midpoint and offset axes. Equations (C-29) can be modified as

smj =

1nr

nr

i

{tij} − 1nr

nr

i

{Gm−1k − Mm−1

k x2ij − rm−1

i }, (C − 30a)

rmi =

1ns

ns

j

{tij} − 1ns

ns

j

{Gm−1k − Mm−1

k x2ij − sm−1

j }, (C − 30b)

Gmk =

1nh

nh

l

{tij} − 1nh

nh

l

{sm−1j − Mm−1

k x2ij − rm−1

i }, (C − 30c)

and

Mmk =

1nG

nG

k

1x2

ij

{tij} − 1nG

nG

k

1x2

ij

{sm−1j − Gm−1

k − rm−1i }. (C − 30d)

This modification enables us to compute and store the sum of the traveltime deviations tij ,thus circumventing the need for storing the individual traveltime deviations. The process isiterated until an index m that yields the least-squares solution such that the differences betweenthe solutions for sj , ri, Gk, and Mk from the mth and (m−1)st iteration are smaller than somespecified values.

C.5 Depth Estimation from Refracted Arrivals

Refer to Figure 3.4-11a for a sketch of refracted arrivals associated with a flat refractor. We wantto estimate the depth to bedrock zw (thickness of the weathering layer) at some shot-receiverstation. The traveltime t for the critically refracted arrival associated with a shot S and receiverR has three parts

t =SB

vw+

BC

vb+

CR

vw. (C − 31a)

The terms in this equation can be expressed in terms of the near-surface model parameters

t =zw

vw cos θc+

x − 2zw tan θc

vb+

zw

vw cos θc, (C − 31b)

where x is the shot-receiver separation, zw is the depth to bedrock, and vw and vb are weatheringand bedrock velocities, respectively. The critical angle of refraction θc is related to the velocitiesby the relation

sin θc =vw

vb. (C − 32)

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Combine the first and third terms on the right, and split the second term in equation (C-31b)to get

t =2zw

vw cos θc− 2zw sin θc

vb cos θc+

x

vb. (C − 33a)

Substitute equation (C-32) into equation (C-33a) and apply some algebra to obtain

t =2zw v2

b − v2w

vbvw+

x

vb. (C − 33b)

Note that this is the equation of a line

t = ti +x

vb, (C − 34)

with its slope given by 1/vb — inverse of the bedrock velocity, and the intercept time ti givenby

ti =2zw v2

b − v2w

vbvw. (C − 35)

By measuring the slope of the line associated with refracted arrivals in Figure 3-4.11a, weestimate the bedrock velocity vb. We also estimate the intercept time ti by extending the lineto x = 0. Finally, we assume a value for the weathering velocity vw. Then, equation (C-35) canbe rearranged to compute the depth to bedrock zw as

zw =vbvwti

2 v2b − v2

w

. (C − 36)

This is equation (3-41a) in the text.The depth to bedrock can also be computed from the crossover distance xc where the

refracted arrival and the direct arrival coincide (Figure 3-4.11a). The equation for the directarrivals is

t =x

vw, (C − 37)

whereas equation (C-34) describes the refracted arrivals. At the crossover distance xc, we equatethe arrival times in equations (C-34) and (C-37) as

ti +xc

vb=

xc

vw. (C − 38a)

By substituting equation (C-35), we obtain

zw =vbvwxc

2 v2b − v2

w

1vw

− 1vb

. (C − 38b)

By simplifying, we obtain the expression for depth to bedrock in terms of the crossover distance:

zw =12

vb − vw

vb + vwxc. (C − 39)

This is equation (3-41b) in the text.

C.6 Equations for a Dipping Refractor

Consider the dipping refractor depicted in Figure 3.4-11c. The traveltime for the refracted ar-rivals from downdip shooting is given by

t− =SA

vw+

AB

vb+

BR

vw. (C − 40a)

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From the geometry and the variables of Figure 3.4-11c, the terms in equation (C-40a) areexplicitly written as

t− =zwS

vw cos θc

+x cos ϕ − zwS tan θc − (zwS + x sinϕ) tan θc

vb

+zwS + x sinϕ

vw cos θc.

(C − 40b)

Apply some algebra to obtain

t− =2zwS cos θc cos ϕ

vw+

x sin(θc + ϕ)vw

. (C − 41a)

This is the equation of a line

t− = t−i +x

v−b, (C − 41b)

with its inverse slope defined as

v−b =vw

sin(θc + ϕ)(C − 41c)

and the intercept time given by

t−i =2zwS cos θc cos ϕ

vw. (C − 41d)

Equation (C-41a) can be rewritten for the refracted arrivals from updip shooting (Figure3.4-11d) as

t+ =2zwR cos θc cos ϕ

vw+

x sin(θc − ϕ)vw

. (C − 42a)

Again, this is the equation of a line

t+ = t+i +x

v+b

, (C − 42b)

with its inverse slope defined as

v+b =

vw

sin(θc − ϕ), (C − 42c)

and the intercept time given by

t+i =2zwR cos θc cos ϕ

vw. (C − 42d)

To compute the refractor dip and bedrock velocity, first, rewrite equations (C-41c) and(C-42c)

θc + ϕ = sin−1 vw

v−b(C − 43a)

and

θc − ϕ = sin−1 vw

v+b

. (C − 43b)

By subtracting equation (C-43b) from (C-43a), we obtain an expression for the refractor dip ϕ:

ϕ =12

sin−1 vw

v−b− sin−1 vw

v+b

. (C − 44)

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This is equation (3-46a) of the text. Measure the slopes of the downdip and updip refractedarrivals in Figure 3.4-11d — v−b and v+

b , respectively. Then, assume a value for the weatheringvelocity vw. By direct substitution into equation (C-44), compute the refractor dip ϕ.

To obtain the bedrock velocity vb, first, rewrite slopes from equations (C-41c) and (C-42c)as

1v−b

=sin θc cos ϕ + sinϕ cos θc

vw(C − 45a)

and1

v+b

=sin θc cos ϕ − sinϕ cos θc

vw. (C − 45b)

Add equations (C-45a) and (C-45b) and take the inverse. Then use the relation given by equation(C-32) to obtain

vb =2 cos ϕ

1v−b

+1

v+b

. (C − 46)

This is equation (3-46b) of the text. Finally, we compute the depth to the bedrock at shot-receiver stations by substituting the estimates for refractor dip ϕ from equation (C-44) and thebedrock velocity vb from equation (C-46), and the measurements for the intercept times t−i andt+i from the arrival times in Figure 3-4.11d into equations (C-41d) and (C-42d) to obtain

zwS =vbvwt−i

2 cos ϕ v2b − v2

w

(C − 47a)

and

zwR =vbvwt+i

2 cos ϕ v2b − v2

w

. (C − 47b)

These equations are equivalent to equation (3-46c) of the text.

C.7 The Plus-Minus Times

Consider the three raypaths in Figure 3.4-12a associated with shot-receiver pairs AD, DG, andAG. The plus and minus times are defined as

t+ = tABCD + tDEFG − tABFG (C − 48a)

and

t− = tABCD − tDEFG + tABFG. (C − 48b)

The times given on the right side of these equations are the picked values from the first breaksfor the three raypaths shown in Figure 3.4-12a. From the raypath configuration, we have therelation

t+ = 2CD

vw− CH

vb. (C − 49a)

By using the geometric relations from Figure (3.4-12a), equation (C-49a) can be expressed interms of the critical angle of refraction θ and depth to bedrock zw as

t+ = 2zw

vw cos θ− zw tan θ

vb. (C − 49b)

Finally, by using the relation (C-32), we have the relation for the plus time t+ in terms of thenear-surface parameters — depth to bedrock zw, weathering velocity vw, and bedrock velocityvb:

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t+ =2zw v2

b − v2w

vwvb. (C − 48c)

This is equation (3-48a) of the text.Now, consider the minus time t− defined by equation (C-48b). By using the raypath con-

figuration depicted in Figure 3.4-12b, we have

t− =2CD

vw+

2BC

vb+

CE

vb. (C − 49a)

Make substitutions in terms of the near-surface parameters to get

t− =2zw

vw cos θ+

2BC

vb+

2zw tan θ

vb. (C − 49b)

By some algebraic manipulation, we get

t− =2zw

vw cos θ− 2zw tan θ

vb+

2x

vb, (C − 49c)

where x is the source-receiver separation AD. Compare the first two terms on the right withequation (C-48b), and rewrite equation (C-49c) to obtain

t− = t+ +2x

vb. (C − 49d)

This is equation (3-48c) of the main text.

C.8 Generalized Linear Inversion of Refracted Arrivals

For an arbitrary source-receiver geometry, given a set of observed traveltimes associated withthe refracted arrivals, we can estimate the parameters associated with a single-layer near-surfacemodel using the generalized linear inversion (GLI). The parameters consist of the refractorvelocity, and the velocity and thickness of the near-surface layer at all shot/receiver locations.The GLI solution for the parameters satisfies the requirement that the difference between theobserved (picked) refracted arrival times and the estimated (modeled) times is minimum in theleast-squares sense. The modeled times are computed using the traveltime equation for refractedarrivals for a flat refractor considered as the base of a weathering layer.

The GLI schemes that allow velocity and thickness of a near-surface layer to vary spatiallyrequire iterative strategies (Hampson and Russell, 1984; Schneider and Kuo, 1985; De Amorimet al., 1987). Starting with initial estimates for the near-surface layer velocity and thickness,and an initial estimate for the bedrock velocity, these parameters are changed such that thedifference between the observed (picked) refracted arrival times and the estimated (modeled)times is minimum in the least-squares sense. The GLI method is not only applicable to 2-D linerecording but also to 3-D swath recording geometries (Baixas and Du Pont, 1988; Kircheimer,1988). It is important to parameterize the near-surface layer parsimoniously while conformingwith the basic assumptions required for the use of refracted arrivals in estimating the near-surfacemodel.

There are several ways to parameterize the near-surface layer. The most general formu-lation would include varying the weathering and the bedrock velocities and the thickness ofthe weathering layer at all shot/receiver locations. This, however, would require linearizing theproblem and iterating over the estimated parameters (Hampson and Russell, 1984; Schneiderand Kuo, 1985; De Amorim et al., 1987). In a simplified version of this general formulation,the weathering velocity may be fixed and is assumed to be known. This leaves the weatheringthickness and bedrock velocity as spatially varying parameters. A further simplification maybe made by defining a polynomial as a function of the space variable for the bedrock velocity(Farrell and Euwema, 1984). In practice, we often find that iterative GLI schemes suffer from

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stability problems. When an inversion scheme leads to unstable solutions, the most likely reasonis the ill-posed nature of the problem.

In this section, two robust parameterizations of the near-surface layer are presented andthe GLI method is used to estimate the associated parameters. In each, we assume that thenear-surface is made up of a single weathering layer with a significant velocity contrast at itsbase. The variable-thickness scheme allows the thickness of the weathering layer to vary spatially,while assuming a fixed known value for the weathering velocity. The bedrock velocity, however,is included in the parameterization and is assumed to be constant. The variable-velocity schemeallows the weathering velocity to vary spatially, while fixing the refractor position at a specifieddepth. The bedrock velocity, again, is treated as a parameter to be estimated and is asssumedto be constant. We find that in many field data applications, these two inversion schemes areable to remove long-wavelength statics variations from the data.

We want to describe the near-surface with minimal parameterization and consider the modelshown in Figure 3.4-13. The traveltime tij for the refracted raypath from the shot location Sj

to the receiver location Ri is given by

tij = tSjB + tBC + tCRi. (C − 50)

The first and the third terms are associated with the raypaths within the weathering layer andthe second term is associated with the raypath within the bedrock along the refractor. In Figure3.4-13, θc is the critical angle of refraction which is expressed in terms of the weathering andbedrock velocities by the relation θc = sin−1(vw/vb). Also, as depicted in Figure 3.4-13, weassume a flat refractor. When refractor dip is taken into consideration, the problem cannot bereadily linearized.

By rewriting equation (C-50), for a flat or near-flat refractor, we obtain

tij =SjB

vw+

DE − DB − CE

vb+

CRi

vw. (C − 51)

By regrouping the terms, we get

tij =SjB

vw− DB

vb+

CRi

vw− CE

vb+

DE

vb. (C − 52)

Finally, by rewriting equation (C-52) in terms of the near-surface parameters, we obtain themodel equation for the refracted arrivals:

tij =zj v2

b − v2w

vbvw+

zi v2b − v2

w

vbvw+

xij

vb. (C − 53)

In additon to asumming a flat refractor, we fix the bedrock velocity but retain it as aparameter to be estimated. Under these assumptions, equation (C-52) can be rewritten in thefollowing form:

tij = Tj + Ti + sbxij , (C − 54)

where

Tj =zj v2

b − v2w

vbvw, (C − 55)

Ti =zi v2

b − v2w

vbvw, (C − 56)

and

sb = 1/vb. (C − 57)

Note that Tj and Ti are the intercept time anomalies at shot and receiver locations, respec-tively, and sb is the bedrock slowness. Hence, for n shot/receiver stations the parameter vector isp : (T1, T2, . . . , Tn; sb). We refer to the scheme based on equation (C-54) as the variable-thickness

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GLI solution. Once the parameter vector p : (T1, T2, . . . , Tn; sb) is estimated, then the thicknessof the weathering layer below shot and receiver locations can be computed using equations (C-55) and (C-56), respectively. For m picks of tij , and n+1 parameters p : (T1, T2, . . . , Tn; sb), wehave the following set of equations:

...

...

...tij.........

=

· · · 1 · · · 1 · · · xij

...Tj

...Ti...sb

. (C − 58)

By writing these equations in matrix notation, we have

t = Lp, (C − 59)

where t is the column vector of m-length on the left-hand side of equation (C-58), L is thesparse matrix in the same equation with dimensions m× (n + 1), and p is the column vector of(n + 1)-length on the right-hand side of equation (C-58). Except for the three elements in eachrow, the L matrix contains zeros. The GLI solution to equation (C-59) satisfies the requirementthat the energy of the error vector

e = t − t (C − 60)

is minimum and is given by:

p = (LTL)−1LTt, (C − 61)

where t denotes the column vector of m-length that represents the observed (picked) refractedarrival times, and T denotes matrix transposition.

In summary, the variable-thickness scheme for refraction statics is as follows:

(1) Assume a value for the weathering velocity vw. This can be varied spatially based onavailable uphole information.

(2) Estimate the parameter vector p : (T1, T2, . . . , Tn; sb), hence compute the intercept timeanomalies at shot/receiver locations and the bedrock slowness by solving equation (C-61).

(3) Solve equations (C-55) and (C-56) for the weathering layer thickness at shot and receiverstations, respectively.

The shape of the weathering layer derived from the variable-thickness scheme strictly de-pends on the assumed value for the weathering velocity. To demonstrate this important aspect ofthe variable-thickness scheme, consider the near-surface model in Figure C-4a with a flat refrac-tor R1, and constant weathering and bedrock velocities. We use equation (C-53) and computethe refracted arrival times associated with shot/receiver locations as indicated in Figure C-4a.Then we use these arrival times in equation (C-61) and assume a value for the weathering velocitydifferent from its true value to estimate the thickness of the weathering layer at all shot/receiverstations. Figures C-4b,c show the estimated refractor shapes R2, R3 for two different weatheringvelocities. Note the significant departure from the true refractor shape R1. Results of FigureC-4 clearly demonstrate that the estimated refractor shape using the variable-thickness scheme(whether it is based on GRM or GLI) does not yield the true refractor shape. Instead, theuncertainty in weathering velocity significantly influences the implied refractor shape.

We now consider an alternative parameterization for the near-surface model. We assume aflat base of weathering as in Figure 3.4-13. This assumption makes the weathering thickness aknown quantity in equation (C-53), and leaves the weathering and bedrock velocities as unknownparameters to be estimated. We take the similar view as in the variable-thickness scheme, and

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FIG. C-4. (a) A single-layered near-surface model with a flat refractor R1; vw = 3000 f/s, vb = 9000f/s. Shot locations are denoted by × and receiver locations are denoted by the vertical bars. Traveltimescomputed from the model in (a) and equation (C-53) are put into the variable-thickness GLI inversionscheme (equation 2-5) to obtain the results shown in (b) where the weathering velocity is assumed tobe 3500 f/s, and (c) where the weathering velocity is assumed to be 2500 f/s. The estimated refractorshapes are denoted by R2 and R3. Note the departure from the true location of the flat refractor R1due to the uncertainty in the weathering velocity.

FIG. C-5. (a) Same model as in Figure C-4a. (b) Results of the variable-velocity GLI inversion scheme(equation 2-13) using the traveltimes computed from equation (C-53) and the model in (a) with anassumed refractor position R2 that is different from the true position R1 of the refractor. The trueweathering and bedrock velocities, vw and vb, are 3000 f/s and 9000 f/s, respectively. Shot locations aredenoted by × and receiver locations are denoted by vertical bars. While the GLI inversion yields thetrue refractor velocity, note the spatially varying adjustment of the weathering velocity to the changeof the refractor position from R1 to R2 to compensate for the thickness R1-R2.

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further assume the bedrock velocity to be a constant parameter. Under these implicit constraints,equation (C-53) takes the form

tij = αjzj + αizi + sbxij , (C − 62)

where

αj = s2wj − s2

b , (C − 63)

αi = s2wi − s2

b , (C − 64)

sw = 1/vw (C − 65)

and

sb = 1/vb. (C − 66)

Hence, for n shot/receiver stations, the parameter vector is p : (α1, α2, . . . , αn; sb). Werefer to the scheme based on equation (C-62) as the variable-velocity GLI solution. Once theparameter vector p : (α1, α2, . . . , αn; sb) is estimated, then the weathering velocity below shotand receiver locations can be computed using equations (C-63) and (C-64), respectively. For mpicks of tij , and n+1 parameters p : (α1, α2, . . . , αn; sb), we have the following set of equations:

...

...

...tij.........

=

. . . zj . . . zi . . . xij

...αj

...αi...sb

. (C − 67)

We write these equations in matrix notation as in equation (C-59), where t is the columnvector of m-length on the left-hand side of equation (C-67), L is the sparse matrix in the sameequation with dimensions m × (n + 1), and p is the column vector of (n + 1)-length on theright-hand side of equation (C-67). Except for the three elements in each row, the L matrixcontains zeros. The GLI solution is given by equation (C-61) where the L matrix is defined asin equation (C-67).

In summary, the variable-velocity scheme for refraction statics is as follows:

(1) Specify a flat datum to which shot and receivers are to be lowered along vertical raypaths.(2) Estimate the parameters: p : (α1, α2, . . . , αn; sb) using the GLI solution given by equation

(C-61) where the L matrix is defined as in equation (C-67).(3) Solve equations (C-63) and (C-64) for the weathering velocity at shot and receiver stations,

respectively.

The estimated weathering velocity from the variable-velocity scheme depends on the as-sumed refractor position. Consider the near-surface model in Figure C-5a, with a flat refractorR1, and constant weathering and bedrock velocities. We use equation (C-53) and compute therefracted arrival times associated with shot/receiver locations as indicated in Figure C-5a. Wethen use these arrival times in equation (C-61) and assume a refractor position R2 to estimatethe weathering velocity at all shot/receiver stations. Figure C-5b shows the estimated weath-ering velocity which departs from the true value vw. The difference arises from compensatingfor the difference between the true refractor position R1 and the assumed refractor position R2(Figure C-5a).

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In equations (C-58) and (C-67), we made no distinction between a shot and a receiverif they occupy the same location on the surface. This strictly surface-consistent solution is ofcourse not valid for a dynamite source. However, shots can be brought up to surface with anuphole correction prior to setting up equations (C-58) and (C-67).

For most field data applications, both the variable-thickness and the variable-velocityschemes yield comparable statics solutions. The solution from the variable-thickness schemeis sensitive to the uncertainty in weathering velocity, and the solution from the variable-velocityscheme is influenced by the assumed depth of the flat refractor.

C.9 Refraction Traveltime Tomography

Return to equation (C-52) and re-express it in the following manner (De Amorim et al., 1987):

tij = swjZj + swiZi + sbXij , (C − 68)

where

Zj =zj

cos θj, (C − 69a)

Zi =zi

cos θi, (C − 69b)

Xij = xij − zj tan θj − zi tan θi, (C − 69c)

sw =1vw

, (C − 69d)

and

sb =1vb

. (C − 69e)

In matrix form, equation (C-68) now is written as follows:

...

...

...tij.........

=

· · · Zj · · · Zi · · · Xij

...swj

...swi...sb

. (C − 70)

Consider an initial estimate of the parameter vector p : (· · · , swj , · · · , swi, · · · , sb). We wantto minimize the difference between the observed and the modeled times by iteratively perturbingthe initial estimate of the parameter vector. A change ∆p in the paramater vector will changethe modeled times as follows:

tijmodeled

= tijinitial

+∂tij∂p

modeled

∆p. (C − 71)

The error in modeling the traveltimes is given by

eij = tijobserved

− tijmodeled.

(C − 72)

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Substitute equation (C-71) into equation (C-72) to obtain

eij = tijobserved

− tijinitial

− ∂tij∂p

modeled

∆p. (C − 73)

Now define the difference ∆tij between the observed traveltimes tij and the initial estimate ofthe modeled traveltimes tij , and rewrite equation (C-73) to get

eij = ∆tij −∂tij∂p

∆p. (C − 74)

The second term on the right is the amount of change in ∆tij as a result of the change in theparameter ∆p. Define this term as ∆tij , and rewrite equation (C-74) once more to obtain

eij = ∆tij − ∆tij , (C − 75)

where

∆tij =∂tij∂p

∆p. (C − 76)

The derivatives ∂tij/∂p in equation (C-76) can be computed by differentiating the modelequation (C-68) with respect to each of the parameters:

∂tij∂swj

≡ Zj , (C − 77a)

∂tij∂swi

≡ Zi, (C − 77b)

and

∂tij∂sb

≡ Xij . (C − 77c)

These then are substituted back into the right-hand side of equation (C-76) to get

∆tij = Zj∆swj + Zi∆swi + Xij∆sb. (C − 78)

Examine the structure of equation (C-78) and note that, instead of modeling refractiontraveltimes by way of equation (C-68), we can model the change in traveltimes by way ofequation (C-78), and thus estimate the near-surface parameters. Hence, for n shot/receiverstations, the parameter vector is ∆p : (∆sw1,∆sw2, · · · , ∆swn,∆sb). We refer to the schemebased on equation (C-78) as the iterative GLI solution. For m picks of tij , and n+1 parameters∆p : (∆sw1,∆sw2, · · · , ∆swn,∆sb), we have the following set of equations:

...

...

...∆tij

...

...

...

=

· · · Zj · · · Zi · · · Xij

...∆swj

...∆swi

...∆sb

. (C − 79)

We write these equations in matrix notation as in equation (C-59) to obtain

∆t =∂t∂p

∆p (C − 80)

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or

∆t = L∆p, (C − 81)

where ∆t is the column vector of m-length on the left-hand side of equation (C-79), L is thesparse matrix in the same equation with nonzero elements as the partial derivatives ∂t /∂p andwith dimensions m × (n + 1), and p is the column vector of (n + 1)-length on the right-handside of equation (C-79). Except for the three elements in each row, the L matrix contains zeros.

The GLI solution to equation (C-81) satisfies the requirement that the energy of the errorvector

e = ∆t − ∆t (C − 82)

is minimum and is given by

∆p = (LTL)−1LT ∆t, (C − 83)

where ∆t denotes the column vector of m-length that represents the difference between theobserved (picked) refracted arrival times and the initial estimate of the modeled times, and Tdenotes matrix transposition.

In summary, the iterative scheme for refraction statics is as follows:

(a) Specify a flat datum to which shot and receivers are to be lowered along vertical raypaths.(b) Also specify a set of initial model parameters p : (sw1, sw2, · · · , swn, sb).(c) Compute ∆tij , the time difference between the picked (observed) times tij and the initial

modeled times tij .(d) Estimate the change in parameters: ∆p : (∆sw1,∆sw2, · · · , ∆swn,∆sb), by way of the GLI

solution given by equation (C-83).(e) Update the paramater vector p + ∆p, and compute new modeled times tij .(f) Iterate steps (c), (d), and (e) to get a final estimate of the parameter vector p.

The parameter vector for the iterative scheme described above comprises laterally varyingweathering velocity and a constant bedrock velocity. Depth to the refractor is assumed to beknown. An alternative parameterization of the near-surface model may involve estimating alaterally varying weathering velocity and depth to the refractor while assuming a known valuefor the bedrock velocity.

Rearrange the terms in equation (C-62):

tij = tij − sbxij , (C − 84)

so that

tij = Tj + Ti. (C − 85)

Equation (C-84) implies that, in this scheme, we deal with picked times that have been linear-moveout corrected using the assumed value for the bedrock velocity. By way of equations (C-55)and (C-56), equation (C-85) is expressed as follows:

tij = zj s2wj − s2

b + zi s2wi − s2

b . (C − 86)

Compute the partial derivatives of the parameters to be estimated — depth to refractor andweathering velocity at shot and receiver locations:

∂tij∂zj

= s2wj − s2

b ≡ Zj , (C − 87a)

∂tij∂zi

= s2wi − s2

b ≡ Zi, (C − 87b)

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∂tij∂swj

=zjswj

s2wj − s2

b

≡ Swj , (C − 87c)

and∂tij∂swi

=zjswi

s2wi − s2

b

≡ Swi. (C − 87d)

Now rewrite equation (C-76) in terms of the new variable t to obtain

∆t =∂t

∂p∆p, (C − 88)

and substitute equations (2-49a, b, c, d) to get the model equation

∆t = Zj∆zj + Zi∆zi + Swj∆swj + Swi∆swi. (C − 89)

Finally, similar to equation (C-79), we have the following set of equations:

...

...

...∆tij

...

...

...

=

· · · Zj · · · Zi · · · Swj · · · Swi · · ·

...∆zj

...∆zi

...∆swj

...∆swi

...

. (C − 90)

Follow the steps which involve equations (C-80) through (C-83) to estimate the parametervector p : (· · · , zj , · · · , zi, · · · , swj , · · · , swi).

C.10 L1-Norm Refraction Statics

The generalized linear inversion method applied to residual and refraction statics is based onminimization of the quantity

E =ij

tij − tij2, (C − 91a)

where tij are the actual traveltime picks and tij are the modeled traveltimes defined by equation(C-24a) for residual statics and (C-54) for refraction statics.

The minimization norm defined by equation (C-91) is formally referred to as the L2 norm.For statics applications, it may be desirable to use the L1 minimization norm defined by

E =ij

tij − tij . (C − 91b)

Outliers in picked times can cause biased results from the L2-norm, whereas, they may bebetter handled by the L1-norm schemes. The L2-norm solutions to equations (C-24a) and (C-54)are expressed by the generalized linear inversion formula (equations C-28 or C-61). Whereas,the L1-norm solution is based on linear programming techniques (Press et al., 1987) and willnot be dealt with here. Instead, we shall refer to model experiments and a field data examplefor the L1-norm refraction statics corrections.

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FIG. C-6. CMP stack associated with the model data in Figure 3.4-14 — (a) with refraction staticscorrections using the L1-norm, (b) section as in (a) after residual statics corrections using the L1-norm.Compare with the L2-norm results in Figure 3.4-17.

FIG. C-7. Summary of the L1-norm solutionfor refraction statics associated with the CMPstacked section in Figure 3.4-15a. Plot direc-tion is the same as that in Figure 3.4-15. Ex-cept in frame 1, shot attributes are denotedwith × and receiver attributes are denotedwith vertical bars. Estimated parameters areintercept time anomalies and are plotted inframe 1 with no distinction made between shotand receiver locations.

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FIG. C-8. CMP stack associated with the model data in Figure 3.4-21 — (a) with refraction staticscorrections using the L1-norm, (b) section as in (a) after residual statics corrections using the L1-norm.Compare with the L2-norm results in Figure 3.4-26.

FIG. C-9. Summary of the L1-norm solution for refrac-tion statics associated with the CMP stacked sectionin Figure 3.4-26a. Plot direction is the same as that inFigure 3.4-26. Except in frame 1, shot attributes are de-noted with × and receiver attributes are denoted withvertical bars. Estimated parameters are intercept timeanomalies and are plotted in frame 1 with no distinctionmade between shot and receiver locations.

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FIG. C-10. The CMP stacked section of the data in Figure 3.4-29 (a) after refraction statics using anL1-norm scheme for refraction statics, and (b) followed by residual statics corrections. Compare withFigure 3.4-29 and note the significant improvement of CMP stacking as a result of resolving both long-and short-wavelength statics anomalies by refraction and residual statics corrections, respectively. Also,compare with Figure 3.4-32 and note that the results of L1- and L2-norm statics solutions, in this case,yield very similar results.

Consider the single-layered near-surface model shown in Figure 3.4-14. Figure C-6 showsresults of refraction and residual statics corrections using the L1-norm. Note that these resultsare comparable to those obtained from L2-norm minimization (Figure 3.4-17).

The results of the L1-norm statics estimates are summarized in Figure C-7. Compare withthose of the L2-norm solution in Figure 3.4-18 and note that the statics solutions exhibit minordifferences, although the resulting stacked sections are almost identical. Frame 1 shows theestimated intercept time anomalies (equations 3-53a,b), as a function of the shot-receiver stationnumber. Frame 2 shows the pick fold, namely the number of picks in each shot (denoted by ×)and receiver (denoted by the vertical bars) gather.

A quantitative measure of the accuracy of the L1-norm solution to refraction statics is thesum of the differences between the observed picks tij and the modeled traveltimes tij (equationC-91b) over each shot and receiver gather. These residual time differences are plotted in frame3 of Figure C-7. Large residuals often are related to bad picks. Nevertheless, even with goodpicks, there may be large residuals attributable to inappropriateness of the model assumed forthe near-surface.

Figure C-7 also shows the estimated weathering thicknesses at all shot-receiver stations(frame 4). Finally, the computed statics and the near-surface model are shown in frames 5 and6, respectively.

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Now consider the multilayered near-surface model shown in Figure 3.4-21. Figure C-8 showsresults of refraction and residual statics corrections using the L1-norm solution. Again, resultsare comparable to those obtained from L2-norm minimization (Figure 3.4-26).

Results of the L1-norm refraction statics solution are summarized in Figure C-9. Comparewith those of the L2-norm solution in Figure 3.4-27 and note that the statics solutions and theresulting stacked sections are virtually identical. Although the actual near-surface model consistsof several layers (Figure 3.4-21a), the L1-norm solution is based on a single layer with a constantvelocity of 1400 m/s as for the L2-norm solution (Figure 3.4-27). In Figure C-9, Frame 1 showsthe estimated intercept time anomalies (equation 3-53a,b), as a function of the shot-receiverstation number. Frame 2 shows the pick fold, namely the number of picks in each shot (denotedby ×) and receiver (denoted by the vertical bars) gather. The sum of the differences between theobserved picks tij and the modeled traveltimes tij (equation C-91b) over each shot and receivergather is shown in frame 3. Large residuals, in this case, are attributable to inappropriatenessof the model assumed for the near-surface. Figure C-9 also shows the estimated weatheringthicknesses at all shot-receiver stations (frame 4). Finally, the computed statics and the near-surface model are shown in frames 5 and 6, respectively. Whatever the inversion norm, theimportant point is how the near-surface model is parameterized and how close it is to the realsituation.

Figure C-10 shows the results of refraction and residual statics corrections applied to fielddata as in Figure 3.4-29 using the L1-norm solution. The results are comparable to those obtainedfrom L2-norm minimization (Figure 3.4-32).

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3. velocity analysis and statics corrections

Documents

average velocity

normalmoveout velocity

swave velocity

pwave velocity

squared rms velocity

moveout andstatics corrections

order moveout nmo

dipping reector nmo