filtering and stacking

5/24/2018 Filtering and Stacking

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Common Mid Point Stacking

The Common Midpoint StackDefinitions

Seismic lines are shot using the arrangement depicted in Figure 1.

Figure 1

A source-point is recorded by a spread of receivers, which are usually velocity-sensitivegeophones on land, and pressure-sensitive hydrophones at sea. The source-spread system isthen rolled along by a distance equal to the source interval, and the next source-point is recordedat the new receiver positions. At sea, the roll-along is eected by towing the source-spreadsystem behind the ship! on land, it is done electronically, and the planted geophones are notmoved.

"e see romFigure 1that trace # rom record A, trace $ rom record %, and trace & rom record '

sample the same subsurace point! when the relector is parallel to the surace, that subsuracepoint lies directly beneath a surace point around which the three source-receiver pairs are

symmetrically disposed ( Figure # ).
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Figure 2

This surace point is the common mid-point, so called because it is shared by several or manyray-paths, each distinguished by its source-to-receiver oset. *ach raypath corresponds to aseismic trace, but the several or many traces are derived rom dierent source points, and soappear on dierent source records.

The operation o sorting all the traces having a common midpoint is called gathering, and thesuite o traces thus assembled is called a common-midpoint gather, or simply a cmp gather. Thepurpose o the gathering is clear rom Figure #+ because the raypaths impinge on the sameportion o the relector, the traces record substantially the same signal; because the raypaths areotherwise dierent in space and in time, the traces record different ambient noise. Thus, additiono all the traces having a common midpoint ater appropriate corrections enhances thesignal-to-noise ratio. The addition o these traces is called cmp stacking.

"hen the relector has dip, the traces sharing a common mid-point do not share a common depth

point (Figure ).
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Figure 3

The area rom which the relections come in is thereby extended, in the updip direction. Stacingthese traces has a tendency to reduce the amplitude o the staced trace, and thus the generalphase appearance o the relector. The magnitude o this alteration depends on the amount o dipand the steps taen to correct or it. For most commonly encountered dips, however, we have

observed that the stacing process is very tolerant o the dips ( the solution to this problem, bythe way, is dip moveout optimization, or /0- see /eregowsi, 123& ).

%eore we can stac the traces o the gather, we must mae the appropriate corrections wespoe o earlier. These corrections ensure that the signal is substantially the same on all thetraces. This means that+

the traces must sample substantially the same portion o the subsurace!

the signal must have the same time on all the traces! and

the signal must have the same shape on all the traces.

A Review of the Prestack Processes

A relection signal changes rom trace to trace along a common-midpoint gather or severalreasons. The obvious dierences are those o relection time! more subtle dierences occur in theshape or character o the relection.

Time dierences may be static, aecting the entire trace by the same amount, or dynamic,varying with relection time. 4n general, static time dierences arise rom raypath irregularities in


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the near-surace! dynamic time dierences arise rom dierences in the raypath length ( Figure 1

andFigure #).

Figure 2
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Figure 1

The irst o the time corrections occurs early in the processing! this is the 5removal5 o the near-surace eected by the datum corrections. "hen they require no processing analysis, they areoten incorporated with the initial processes.

To mae these corrections, we irst establish a seismic datum, generally below theinhomogeneous near-surace. This datum now serves as our surace o 6ero time, and allsubsequent timing is done rom this origin. "e see rom Figure that the total correction hascontributions rom two parts o the ray-path+ rom source to datum on the way down, and rom

datum to receiver on the way up.
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Figure 3

These are the source static and the receiver static, respectively.

/etermination o both the source static and the receiver static requires some nowledge o thevelocity (or velocities) in the near-surace. (The thickness o the near-surace is, in this context, atrivial problem! it is simply the elevation at the surace minus the elevation o the datum.) 4seismic wor is done with dynamite in drilled holes, the near-surace velocity is inerred rom theuphole time (Figure $ ). The depth o the hole divided by the direct arrival time to an up-holegeophone is the average near-surace velocity. This velocity, in turn, is used to compute the

source and receiver statics.


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Figure 4

4 the seismic source is a surace source such as an air gun or a vibrator, then the near-suracevelocity must be deduced rom previous wor in the area, rom a special velocity survey, or romreraction arrivals. "e must be prepared to repeat the survey down the line i surace conditions

presage a change in near-surace velocity.

nce we have calculated the source and receiver corrections, we have only to apply them! thatis, to subtract these times rom the seismic trace. For each record, we would apply the sourcestatic to all the traces at once! we say the source static is a common-source correction. Then we

would apply the receiver statics as common-receiver corrections ( Figure 7 ).
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Figure 5

The ob8ect o the datum corrections is to simulate the trace that would have been obtained i thesources and receivers had all been on the seismic datum. To the extent this is achieved, theremaining trace-to-trace time variations are such that the relection alignment is essentially a

hyperbola ( Figure & ). The increase in time with oset is, o course, the normal moveout.


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Figure 6

Although we may regard the travel-time pattern as a hyperbola, deined as in Figure &, the realpattern is not so simple. This is because the equation or travel time does not contain only twoterms, but rather an ininite number o them. The so-called higher-order terms can be ignored as

long as the eective spread length (ater the mute) is not much greater than the depth o therelector.

For a given 6ero-oset time, and over a given range o source-to-receiver osets, the travel-timepattern o Figure &is uniquely characteri6ed by a variable that happens to have dimensions olength9time. :eophysicists were thereore quic to call this variable a velocity! because thenormal-moveout correction according to this variable is that used or cmp stacing, it came to benown as a stacing velocity! inally, the determination o this variable came to be called avelocity analysis. This use o the word 5velocity5 is not a good one, but we seem to be stuc withit.

"hen we perorm a velocity analysis, we are searching or hyperbola that best its the travel-timepattern o the relection. The goodness o the it may be determined visually or numerically. Thereis some interpretation involved, o course, because the best it does not always correspond to alegitimate primary relection. 4n Figure ; , or instance, a velocity o 1;


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Figure 7

'learly, the latter two are multiples, so we pic or interpolate a higher velocity or those relectiontimes, even though the velocity analysis gives no indication that the higher velocity correspondsto a primary event.

"e see, thereore, that the desired nmo correction aligns primary relections, but misalignsmultiple events, thereby attenuating the multiples when we come to stac the traces.

=aving perormed the ield-statics and nmo corrections, we hope that the relection alignment isperect. 4n the real world, o course, it is not, because our simple models do not perectly simulatethe real world. The thicness and velocity o the near-surace are highly variable! these variationsin turn may aect the nmo determination.

This imprecision leaves us with a series o small, unsystematic timing errors rom trace to tracealong the gather, which we reer to as residual static..

4n the standard determination o residual statics the process is called automatic statics, orautostatics the static is considered to comprise our terms+ the residual source static, the

residual receiver static, the residual normal moveout, and the eect o the structure (sometimescalled the dip term). 'onveniently, these our terms correspond to the our directions along whichwe may gather data.

Autostatics programs vary rom one processing house to another, but they all have threeimportant steps in common+

First is an estimate o the trace-to-trace time dierences!


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Then, we decompose those time dierences into the our components!

Finally, we apply the computed source and receiver statics to the traces.

Further, we oten constrain the autostatics solution by invoing the principle o surfaceconsistency. This stipulates that trace-to-trace misalignment can be regarded as a unction o

source and receiver locations at the surface, rather than o raypaths in the subsurace. coursesurace consistency is not always appropriate. "e remember also that time-variant correctionsmay be required.

So, to the degree possible, we have aligned the relections on the cmp gather. There remains thequestion o wave shape or character, because we do not wish to add relections that are out ophase or too unlie in requency content.

The reason we might have a problem is that the traces o a gather all come rom dierent source-points. n land data shot with dynamite, source variables charge si6e, depth, cavity eects,etc. can result in dierent wavelets at each shotpoint. At sea, the ailure o one air gun in thearray can also aect the source signature. *ven when we have a controlled source such asvibrators on land, the pulse-shaping eects o the near-surace can modiy the downgoingwavelets dierently at each source position. Finally, i we are woring in the transition 6one, we

may very well have a combination o sources and receivers ( Figure 3 ).

Figure 8


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To standardi6e all the pulse shapes, we generally require some sort o wavelet processing. Theirst ob8ective is to correct all the source signatures to a uniorm shape! a second ob8ective is toselect a stable and compact shape. "ith marine and land vibroseis data, we now theapproximate source signatures! wavelet processing is thus deterministic, as we preer. "here thesource signature is unnown, as with impulsive land data, we determine the outgoing pulse shapestatistically.

nce we now the source signature, we can calculate its amplitude and phase spectra. Acting onthese with the amplitude-requency and phase-requency responses o the required ilter, weemerge with the desired wavelet. This ilter is then applied to all the traces, so that the relectionsin a cmp gather now align according to both time and shape.

And so it remains to stac the data, and we begin with an analysis o the conventional stac andits beneits.

The Mean-Amplitude Stack

A mean-amplitude stac is a common-midpoint stac in which each staced trace is a simple

summation o the traces in the gather, out to the limits o the mutes. The resulting trace is astatistical average o the constituent traces. 4n essence, we add all the sample values at a giventime, accounting or the ramp in the mute 6one, and divide by the number o traces! thenormalized trace has amplitudes comparable to the input.

'0> stacing has the beneit o improving the signal-to-noise ratio by a actor equal to the squareroot o the number o traces in the stac. "e should qualiy that now with a ew observationsabout both signal and noise.

1. As much recent wor has shown, the relection coeicient is not generally independent

o the oset ( Figure 1 ).


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Figure 1

For some important geologic models, the relection coeicient exhibits a strongdependence on oset or angles o incidence greater than about #


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Figure 2

4n Figure , however, a gassy layer has a much lower >oisson@s ratio.


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Figure 3

As the oset and the incident angle increase, so does the amplitude.

#. Amplitude need not change as only a unction o oset or it to aect the stac.Figure $ reminds us that in the presence o dip, the relection points represented by each

trace do not converge at a point.

Figure 4

The resulting smear is bad enough, but i the physical properties change enough tocause changes in the normal-incidence relection coeicient within the relection area,then we may get spurious amplitudes over one or more traces.

4n general, whenever we see a signiicant amount o dip on the near-trace section or

brute stac, we preer to do aprestack partial migration, or dip-moveout processing, priorto stacing.

. Another simpliying assumption about the signal concerns the correction or thedecrease in amplitude with time. This decrease is due to spherical spreading, additionalspreading rom reraction, and various propagation eects. To compensate or thespherical spreading, which is ully determined, we multiply each sample by its time t. Tocompensate or the remaining amplitude losses, we multiply each sample by anadditional actor ekt, so that the total amplitude ad8ustment is tekt.


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The exponential expansion is intended only as an approximate, temporary correction.Thereore, we should understand now that the amplitude relations among the traces o agather are probably not correct! certainly, now that we have a better nowledge o thevelocity behavior along the line, we can do better.

$. The improvement in the signal-to-noise ratio eected by stacing assumes that

each trace is a composite o relection signal and random noise. *ven i we restrict thediscussion to ambient noise noise which is 8ust 5there5 in the absence o a shot thisassumption applies only to certain types o noise. Thus, the noise generated by towing astreamer may well be random, but common-mode intererence may not be.

Thereore, i we are to spea o such improvements, we must say that we are relating thesignal to incoherent, uncorrelated noise. For most other ambient noises, the signal-to-

noise improvement generally is less than .

7. 'ountering the above is the understanding that, or certain recording geometries and

certain source-generated noise, is the minimum improvement attainable. Thecommon-midpoint gather acts as a very long array, long enough to suppress virtually all

wavelengths o ground roll or water waves (Figure 7 ,

Figure 5

Figure & ,


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Figure 6

Figure ; , andFigure 3 ).


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Figure 7

To ensure continuity o the geophones across the gather, which is the primary criterion othe stack-array, it is more eicient to set up the ield geometry in one o two ways+

Figure 8

At sea, the group length is equal to the group interval, and the source interval is hal the

group interval!

n land, the group length, the group interval, and the source interval are all equal, and

the source is between groups o a split spread.

ther geometries are possible but they require some degree o mixing in creating the

stac-array.

The eectiveness o the stac-array depends on the constancy o the source-generatednoise rom record to record, and thereore rom trace to trace in the gather. Still, the worstcase is that o random variations in the source-generated noise! then, the near-total

suppression o the noise reduces to statistical, or , suppression.

"e see, thereore, that even straight stacing implies assumptions that occasionally bearexamination. evertheless, the method is powerul, robust, and extremely useul.


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Features of the Conventional Stack

=ere, we summari6e the beneits and limitations o the common-midpoint stac.

bviously, a ma8or beneit o stacing is the improvement in the signal-to-noise ratio.

Stacing also allows us to display the suite o staced traces in a orm that resembles a

6ero-oset, normal-incidence section, such as might be obtained through the method

ideali6ed in Figure 1 and Figure #.

Figure 1

The ield method o multiple coverage allows us to record the required traces more

eiciently, and gives us the signal-to-noise beneits as well.


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Figure 2

"e understand, however, that in the staced trace so obtained, the wavelet and therelection coeicient have been averaged over a range o osets, and sometimes over anenlarged portion o the relector.

For all its appearances, however, the staced section should not be regarded as

equivalent to a geologic cross-section, i only or reasons o dip ( Figure ).


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Figure 3

evertheless, as a irst approximation to the subsurace structure, obtained without anyprior knowledge of that structure, the cmp stac is remarably accurate. aturally, beorewe draw too many inerences about the structure, we must account or whatever verticalexaggeration is present in the display.

%y virtue o the residual normal moveout, the stac is also an excellent attenuator o

certain long-path multiples. The eect is greatest when the velocity increases rapidly withtime, and can be improved (at relection times beyond the end o the mute) by alengthening o the spread. Short-path multiples and water-bottom reverberation are besttreated by deconvolution methods.

The stac also has economic beneits! when we stac -old data, we reduce by a

actor o the number o traces that need subsequent processing. This is no smallconsideration or marine lines that are routinely greater than $3-old.

Stacing is not very useul against noise bac-scattered in the near-surace rom a

scatterer to the side o the spread. 4t can be useul i the scatterer is behind the spread (Figure $ )! where it ails in the latter case, we may require the use o the requency-

wavenumber ilters.


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Figure 4

Irreular !eometr"

=aving said that common depth point and common reflection point are not synonyms or commonmidpoint, we must acnowledge that sometimes even the midpoints are not common. Suraceconditions sometimes prohibit a regular progression o source points, or even a straight line osource points. 4n such cases, the gather may consist o traces whose midpoints lie within a hal

midpoint interval o the nominal midpoint location ( Figure 1 (a), (c)).


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Figure 1

Alternatively, we may gather midpoints that lie within a circle of stack ( Figure 1(c))! this mayinclude some overlap and thereore mild mixing o the staced traces.

The ultimate case, o course, is the crooed line. *ven in level terrain, this ind o geometry may

be dictated by cultural or even political concerns! in mountainous terrain, it is all but unavoidable (Figure # ). bviously, a crooed line cannot be processed in the manner that a straight linewould. 4nstead, we must set up the processing geometry to account or the recording geometry.
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Figure 2

The irst step is to determine the source-to-receiver osets. 4 the source and receiver positionsare nown to good accuracy, this is easily done. 4t is then a simple matter to compute the

midpoint positions ( Figure and Figure $).


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Figure 3

From there,


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Figure 4

the tas is to sort all the scattered midpoints into bins (Figure 7 )! each bin is essentially a gather,and we wish to treat the constituent traces in much the same manner as we treat the traces o a

common-midpoint gather.

Figure 5

Thereore, we adapt the binning methods to the overall curvature o the line, pursuing what isoten called a binning strategy. There are several considerations that dictate the si6e and shapeo each bin.

1. The line connecting the bin centers passes through the greatest density o midpoints,and is oten computed using a locali6ed center-o-gravity scheme. This line issubsequently regarded as the nominal physical location o the staced section. The stack

line need not be straight, o course. 4t may consist o a series o segments, each with adierent a6imuth, as long as the stac line is unbroen.

#. 4 the bin centers satisy the above conditions, then each bin should have a uniormdistribution o osets. A bin with only short osets is useless or multiple attenuation,whereas a bin with only ar osets 8eopardi6es the velocity analysis. course, i a bin isnear a sharp bend in the line, the continuity criterion above may render the desired osetuniormity impossible. 4t is also useul to have a reasonably uniorm number o traceswithin each bin.


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. "e also wish to minimi6e the distribution o source-to-receiver a6imuths within eachbin. This is a more subtle problem, and one that may be expensive to accommodate. 4there is a signiicant amount o dip, traces having the same oset but dierent a6imuthshave dierent relection times or a given relection depth. This dierence in relectiontime alters both the apparent structure and the velocity determination. 4n practice,accommodation o these eects may require iterations through a -/ model! in general,

thereore, we preer to restrict the a6imuth distribution when we select traces or binning (see Barner et al., 12;2 ).

$. 4t is permissible or bins to overlap, as indeed they must at bends in the stac line. 4t isalso permissible or bins to vary in their cross-line dimension, or diameter, as long as thevariation rom one bin to the next is small. Their inline dimension, or bin interval,however, should remain constant.

7. "e are not constrained to put every scattered midpoint into a bin, as long as the osetand a6imuth conditions are met. 4nFigure &, or example, we can probably widen the

bins to include the traces to the northeast, in the manner shown.

Figure 6

"e do this inFigure ;, and then reali6e that little would be gained by widening them

urther, to include the midpoints south o the line.


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Figure 7

The inal bin coniguration ( Figure 3 ) shows that bin diameter and orientation change along theline, to conorm as closely as possible to both the original shot line and the spread o midpoints.


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Figure 8

4n eect, we may regard the binning strategy as a search or a 5best-it5 to the shot line. (And, ocourse, a similar strategy may be used or marine lines plagued by severe streamer eathering.)

Stackin# Alterative Forms

A$T%R&ATI'% F(RMS (F STAC)I&!

There are several alternatives to mean-amplitude common midpoint stacing. To place these inperspective, we must see clearly that stacing accomplishes several ob8ectives.

The enhancement o signal relative to ambient noise. This is particularly useul at late

relection times.

The enhancement o signal relative to coherent noise. This is particularly useul over the

time range at which source-generated noise is received, and is strengthened i the ieldgeometry satisies the stac-array criteria.

The enhancement o primary signal relative to multiples. This is particularly useul or

multiples involving only relectors at medium depths, and or velocities increasingmaredly with depth.

The maintenance o subsurace continuity in those situations where it would otherwise be

impaired by missing source positions, missing receiver positions, local highly absorptiveweathering, and blind spots caused by reraction, aulting, and racturing.

The empirical conclusion rom many years o experience is that the ordinary mean-amplitudestac provides, at minimum cost, an excellent balance among these several ob8ectives. 4n somesituations, however, one o these ob8ectives is much more important than the others. Then, we


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must be prepared to strie a different balance, so that we may strengthen one beneit at theexpense o the others.Sometimes, thereore, we consider alternative types o stac, select one appropriate to theproblem, and as whether its beneit 8ustiies its riss and its costs.

*eihtin +" Mutes

The simplest weights to apply to data are < and 1. "hen the weights are all 1, o course, we havea mean-amplitude stac. "hen some o the weights are


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Figure 2

"e oten call this distortion nmo stretch; in act, however, the nmo correction itsel is a stretch o

the time axis to mae all traces loo lie 6ero-oset traces.


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Figure 3

"e remember the mechanism or the distortion. 4n Figure $we have the 6ero-oset and ar-oset

traces o an uncorrected cmp gather.


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Figure 4

The width o the pulse, which is centered at 1.


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Figure 5

As we can see rom Figures 1-7 , the distortion is worst at ar osets and early times. %ut, as wesee rom Figure 7and the exercise, the amount o the distortion and, by extension, the degreeo spectral change on the staced trace can be calculated in advance. The permissible

spectral change should be time-variant, to accommodate the decreasing expectation o goodbandwidth with time. :iven this speciication, the computer then may calculate an appropriatemute we may call it an 5automute5- on each trace in time. 4n general, this is probably suicientor a irst estimate o the velocity distribution.

"hat the computer cannot do is decide a mute based on considerations other than nmo stretch.4n particular, there may be troublesome source-generated noise, such as a head wave or a shearreraction. 4n this case, the processor or the analyst has to mae a decision based on a visualinspection o the gathers. And, o course, it is best i the mute is selected gather by gather, ratherthan on representative gathers.

So the choice o the mute depends on the stacing velocities and on the source-generated noise,and thereore varies along the line. 4t also depends, to a lesser degree, on the requencies. 4 wewish to preserve the high requencies, then the maximum allowable distortion must be low perhaps #


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The muting o marine data may depend on the water depth. ten, a shallow water bottom maycause relected, reracted, and direct arrivals to appear on the near traces. =ere, thesimultaneous arrival o relections and coherent noise prevents us rom muting only the noise. 4nsuch cases, we preer to ilter the traces in the f-k domain.

To assess the eects o the mute, and to see i the mute is hurting us in any way, it is helpul to

generate a series o stac panels, such as those o Figure & ,

Figure 6

Figure ; ,


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Figure 7

Figure 3 ,


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Figure 8

Figure 2 ,

Figure 9

Figure 1< ,and Figure 11


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Figure 11

.


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Figure 10

=ere, a portion o the line has been staced with dierent mutes, and thereore dierent olds ocoverage at a critical relection time. As we add more and more o the ar osets ( Figures &-3 ),the shallow section beneits. *ventually, however, the mute ails to ill the early waterbornearrival, and the shallow section deteriorates ( Figures 2-11 )

*eihtin +" (ffset

To improve the stac response o primaries relative to multiples, we may weight the traces byoset. *ssentially, we put more emphasis on the ar osets, where the multiples are more out ophase. The scheme may not be elegant, but it is easy and inexpensive, and thereore a goodalternative to straight stacing, especially when the problem is a particular multiple.

For best results, the weights are calculated by the program based on the velocities o the primaryand the multiple. The velocities are determined in the usual way the moveout dierence betweenthe primary and the problematic multiple is then used to calculate the weights. 0ore-advancedschemes may also account or requency content, amplitude ratios between primary and multiple,and perhaps the level o random noise.

Figure 1 , andFigure #describe how oset weighting wors.


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Figure 1

4n Figure 1,


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Figure 2

we see the nmo-corrected travel-time patterns corresponding to a primary and, 8ust below it, anoninterering multiple. The primary is well aligned, and thereore yields a good stac response,

but the misalignment o the multiple is not enough to cause it to stac out. %y giving more weightto the ar traces than to the near traces ( Figure #), we ensure that each in-phase trace is addedto an out-o-phase trace. The multiple is minimi6ed, but the primary is unaected. "e see the

result on real data in Figure .

Figure 3

"e should note that oset-weighted stacing is most eective in those cases where mean-amplitude stacing would wor airly well anyway. 4n general, this means that we need at leastone-hal cycle o residual normal moveout across the gather, and preerably one or more ull

cycles. :iven this level o rnmo, stacing without weights reduces the multiple amplitudemeasurably! oset weighting simply taes it urther. Figure $quantiies the improvement.
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Figure 4

The curves show the approximate ration o multiple amplitude to primary amplitude, as a unctiono stacing velocity! the primary velocity is assumed to be


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A ma8or use o this method is in compensating or large noise bursts in unsummed vibroseis data.4n eect, we weight each trace in inverse proportion to the amount o noise in it+ the greater thenoise, the lower the weight. The method proceeds as ollows+

1. 'alculate the average power within a speciic time window or each trace. "e do thisby calculating the energy and dividing by the time.

#. 'ompute a scaling actor such that the product o actor and average power is aconstant.

. Taing the scaling actor to apply to the center o its window, interpolate linearlybetween actors, thereby emerging with a gain trace.

$. 'ross-multiply the data trace and the gain trace to yield a scaled trace.

7. Add the scaled traces, add the gain traces, and divide the irst sum by the second.

The process assumes a constant signal amplitude rom trace to trace! by so doing, it ascribes the

trace-to-trace dierences in average power entirely to the dierences in noise power (a goodassumption i the noise amplitude is much higher than the signal amplitude, which was our initialcondition anyway). Furthermore, it does so while preserving the original amplitude variations othe signal in the input records.

Although the diversity stac, or inverse-power-weighted stack, is most common as a vibroseissumming technique, it has uses in the stacing step which mae its inclusion here logical.principally, these uses are in the stacing o marine data contaminated by noise intererence,really rom other seismic vessels.

4n general, we preer to shoot marine data when other crews are inactive. This is because marinewor is essentially a continuous operation, meaning that the recorders are woring almostconstantly. 4n the past, to prevent the recording o noise rom other crews, companies would

speciy a maximum intererence level, beyond which the crew must stop recording. 'rewsgenerally cooperated with each other on a 5time-sharing5 basis, whereby they too turnsrecording.

4n many cases, however, the crew-intererence speciication was too conservative, so that muchtime was lost waiting or the interering crew to inish. "ith diversity stacing, noise that loosvirulent, even hopeless, on a ield record can be suppressed to reasonable, even imperceptible,levels on the inal stac.

Figure 1 shows ive consecutive shot records showing the sort o noise we get rom intereringcrews! here, the pea amplitudes are at least three times greater than many old contract

speciications would allow.


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Figure 1

Figure #, however, shows what these noise patterns loo lie on common-midpoint gathers.


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Figure 2

bviously, even a mean-amplitude stac should go a long way toward attenuating the worst othe noise.

ne o the reasons the crew noise is so disorgani6ed on the gathers (as opposed to itsappearance on the record) is that the source o the intererence is not synchroni6ed with thesource o the survey ship. The noise arrives at dierent times on ad8acent records, so thatgathering urther misaligns it. 0ore important, however, is that the asynchronous noiseguarantees that the noise does not appear on all the traces of the gather. Thus, i is the numbero traces in the gather, 0 is the number o traces on which the noise appears, and the noise is

suiciently incoherent, then the suppression is by a actor o . This becomes i and0 are equal.

"e see 8ust how eective mean-amplitude stacing is in Figure ,

Figure 3

which is a stac o the line rom which the records and gathers o Figure 1, and Figure #are

taen. For comparison, Figure $shows the same line without any interering crew noise.
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Figure 4

*arly in the section, we see that alternative stacing is unnecessary! it is only in the deeper parto the section that the intererence is apparent.

To combat the deeper noise, we resort to the diversity stac, which requires only that the signal

amplitude be approximately uniorm rom trace to trace. Figure 7 ,


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Figure 5

Figure & ,


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Figure 6

andFigure ; ,


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Figure 7

compares the results o the mean-amplitude and diversity stacs to the case o no intererence,

andFigure 3quantiies the improvement or three levels o signal-to-ambient noise.

Figure 8

=ere, the poststac signal-to-burst-noise ratio is plotted as a unction o the prestac ratio or a&


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the amplitude range. Thus, these stacs are sometimes called -trimmed-mean stacs. ("hen

D


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Figure 2

Figure ,


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Figure 3

Figure $ ,

Figure 4

andFigure 7 show the eect o trimming $


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Figure 5

As with the diversity stac, the relative incoherence o the noise on the gather helps the trimmingdo its 8ob.

4n deciding on the level o trimming, two considerations are important. ne is that we preer the

re8ected samples to be distributed symmetrically about the mean. This proviso permits us to bereasonably conident that we do not change the estimate o the mean or a normal distribution oamplitudes.

"e should also avoid discarding too many traces, lest we compromise too severely the beneits

o the subsequent stacing. A trimming o $


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4n constructing a median stac, it is important to exclude rom consideration any 6ero values thatresult rom muting. Further, the median is inappropriate i there are high amplitudes say, roma multiple on more than about a third o the traces.

The median stac combines very low cost with quite remarable eicacy against large noise, andit does not require (as does the diversity stac) that the noise come in bursts. 0edian stacing is

standard in the processing Eertical Seismic >roiling ( ES> )

The loss that we ace, in using both the trimmed-mean and median stacs, is that they do lesswell against multiples, and again are nonlinear! a median stac is sub8ect to abrupt changes inthe amplitude values o consecutive samples o the output. Although the resulting appearance ohigh-requency noise may be removed by iltering, there remains the uncertainty o relativeamplitudes which is always introduced by nonlinearity.

Coherence Stack

There are several variations to the coherence stac, but they are all designed to enhance theamplitudes o coherent signal while suppressing the amplitudes o incoherent noise.

4n essence, we perorm on nmo-corrected gathers the same sort o coherence scan we do in avelocity analysis! indeed, semblance is a common measure o the coherence (basically outputenergy divided by input energy). 4 the semblance window contains a primary event, thesemblance is high, and this high value is assigned to the center o the window. 4 there is noprimary in the window, the semblance is low, and we assign a low value at the center. 4n thismanner, we emerge with a coherence trace, interpolating linearly between window centers. Thecoherence stac is then ormed by scaling the staced trace by the semblance.

The coherence stac (which appears under many trade names) is very eective in improving theratio o properly corrected primaries to both multiples and noise. 4ts weaness springs directlyrom its strength+ it is intolerant o minor errors in the stacing velocity. Thus, each poor velocityanalysis, or inappropriate velocity interpolation, leads to wea 6ones in the aected relections.

The result, which tends to loo geologically unnatural, is oten unacceptable to the interpreter.

A variation to the coherence stac is the correlation stac. =ere, we cross-correlate the stacedtrace with each o its input traces. 4 the correlation coeicient alls below a certain threshold, wescale down that input trace (or ill it entirely i the coeicient is much below the threshold). "emay reine this by maing the process iterative! we restac without the oending trace, andperorm another set o cross-correlations.

Velocity Filtering

'%$(CIT, FI$T%RI&!

4n ield wor one o the ma8or unctions o the spread is to allow us to recogni6e dierent types oseismic waves by their characteristic velocities across the spread. 4n particular, we can separatedierent arrivals on the basis o these characteristic velocities by using two-dimensional ilters.

Two-dimensional ilters are useul in seismic processing because o their ability to re8ect eventshaving a particular velocity across the spread, or across the section. Such ilters are thereorecalled velocity filters; because they are invariably applied in the requency domain ( as well as inthe slant-stac domain ), they are also nown as fre!uency-wavenumber (f-k filters.


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The Two-Dimensional Fourier Transform

The seismic ield record, which represents signal amplitudes as a unction o time and o oset, isoten reerred to as a time-domain representation. To get to its fre!uency-domain counterpart, weremember the concept o the one-dimensional Fourier transorm+ we correlate the trace with asinusoid, which gives a measure o the amplitude o that requency in the trace. /oing this or all

requencies, we get the spectral density and phase o the trace. Filtering the trace is then amatter o scaling the spectrum as a unction o requency, using a scale actor o one orrequencies we wish to pass, 6ero or those we wish to re8ect, and a suitable ramp in between.eassembly o the spectral components yields the iltered trace.

Seismic data also have a spatial requency. "e see this inFigure 1!at the let is a sinusoidalwaveorm in time, and at the right is the waveorm as it is recorded at a spread o our receivers.

Figure 1

The arriving wave ront is represented by the alignment o the open circles. 4 the receiver spacing

is small enough to prevent aliasing (as it is here), then the 5trace5 connecting the receiveramplitudes may be regarded as a waveorm in space.

Gust as the temporal requency o a waveorm is the reciprocal o its temporal period, so is thespatial requency o a waveorm the reciprocal o its spatial period. Spatial period is simply the

apparent wavelength , so spatial requency is simply . "e call this quantity the

wavenumber, and it has units o inverse length! oten, we speciy the wavenumber in terms osome unit length, such as so many cycles per receiver interval, or per ilometer, or per mile.


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4n Figure 1, the waves are coming up to the surace rom the let. Further, because they impingeon the receivers one at a time, they are seen as having a hori6ontal velocity. This is the apparentvelocity Eap, because it is not a true measure o the speed o the waveront. "e quantiy it as the

dierence in receiver coordinates divided by the dierence in arrival times, or "#t.

"e remember that the apparent velocity is equal to the true velocity divided by the cosine o the

angle o wave propagation. *quivalently, it is equal to the true velocity divided by the sine o theangle o dip. Thus, we may say that the waves inFigure 1impinge on the receivers one at a timeby reason o dip, and that the dip may be quantiied as the dierence in arrival times divided by

the dierence in receiver coordinates, or "#t.

4n Figure #, the waves are coming up rom directly below, rom a lat relector.

Figure 2

%ecause all the receivers sample the same phase at the same time, the apparent wavelength andthe apparent velocity are ininite, and the wavenumber is thereore 6ero. Finally, in Figure , the

waves are coming up rom the right.
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Figure 3

The apparent wavelength is the same as it is in Figure 1, so the wavenumber and the dip havethe same magnitude! however, they must have the opposite sign. 4n this situation, we may alsodeine a negative apparent velocity, meaning that the wave is traveling right to let.

"avenumbers and dips, unlie requencies, are both positive and negative. The sign conventionis this+ A wave front that is down to the right on a section, or down from near offset to far offset ona record, has a positive wavenumber, and the reflector from which it comes has a positive dip. Awave front that is down to the left on a section, or down from far offset to near offset on a record,has a negative wavenumber, and the reflector from which it comes has a negative dip.

4t is possible or a waveront with a positive wavenumber to appear as though it has a negativewavenumber. "e see this in Figure $, where the waveront can be interpreted (wrongly) as

coming up rom the right.


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Figure 4

course, it is also interpreted as having a negative apparent velocity and a negative dip.

Figure $is an example o spatial aliasing, and it occurs because the receivers are too widelyspaced to prevent the ambiguity. To avoid spatial aliasing. we require the receiver spacing to be

smaller than Eap9#f$ , where f$is the yquist requency. The quantity E9f$is the inverse o theyquist wavenumber k$.

eturning now to the waveorms o Figure 1, reproduced in gray in Figure 7, we see the eect o

the same apparent velocity (or the same amount o dip) on a waveorm o higher requency.


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Figure 5

bviously, the higher requency results in a higher wavenumber. Furthermore, when we graphrequency and wavenumber or the two waveorms ( Figure & ), we see that the relation is simpleand direct+ %f two events have the same apparent velocity, their respective fre!uency-

wavenumber pairs lie in a radial line in the f-k plot.


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Figure 6

A urther result apparent in Figure &is that the slope o the radial line is simply f#k; because D

l9a, the slope a is the velocity o the wavetrain. Thus, dierent velocities have dierent

slopes on the f-k plot, even i the events cross or overlap partially on the time-distance plot.

"e can extend the results oFigure &by adding the requency-wavenumber pairs rom Figure #,andFigure . This we do in Figure ; , which represents a generali6ed f-k plot. The vertical axis

represents, again, the temporal requency, and ranges rom 6ero to the yquist requency.
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Figure 7

The hori6ontal axis represents the wavenumber, and ranges between the negative and positiveyquist wavenumbers.

The Two-Dimensional Filter

The concept o a #-/ ilter is much the same as that o a 1-/ ilter. This time, however, wecorrelate an n-trace record with an n-trace sinusoid having a certain velocity, such as Figure 1.


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Figure 1

The irst correlation gives us a value or one - pair. Heeping the requency the same and

changing the velocity ( Figure # ), we correlate again to get a value or a second - pair.


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Figure 2

=aving done this or the ull range o velocities to the limit o aliasing ( Figure ), we then go to

the next requency component, and begin again.


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Figure 3

"hen we have reached the alias requency, we have the complete - density. Then, the velocityilter is a matter o scaling the components as a unction o requency and wavenumber.eassembly o the components yields the velocity-iltered record.

4n practice, this sort o sample-by-sample, trace-by-trace correlation would be slow andexpensive. So we do it in the - domain, harmonic by harmonic (so we can incorporate requencyiltering at the same time) and wavenumber by wavenumber. %ut the principle is the same+ weuse other traces, and an alignment across them, to optimi6e a stepout and subtract it.

As an example, we see in Figure $a 5relection5 (event 1), crossed by 5ground roll5 (event #).


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Figure 4

%ecause their spectra overlap, requency iltering would be ineective. 4n the - plot, however,

there is no overlap ( Figure 7 ).


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Figure 5

So, we designate the re8ect band by a wedge emanating rom the origin. As with one-dimensionaliltering, we apply a taper between the re8ect 6one and the pass 6one, steep enough to preventthe attenuation o - components we need, but gentle enough to prevent ringing. The re8ection isbest i the event is straight, and i the event amplitudes on all traces are equal. (The latter

condition is essentially a requirement that noise amplitudes are equal on all traces.) The ilteredrecord is shown in Figure &! we see that the ground roll has been much reduced.


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Figure 6

educed, but not eliminated. "e see why in Figure ; , which extends the - plot oFigure 7.


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Figure 7

The trace spacing o the ground roll, which has a low velocity on the record, is too large toprevent aliasing o the higher requencies. This aliasing shows up in the negative-wavenumberside o Figure ;, as though event # were actually a series o events traveling rom the ar tracesto the near traces! the 5velocity5 varies with requency. %ecause the re8ect region in the - plot

must emanate rom the origin, a substantial portion o the - representation o event # isuntouched. The high-requency remnant o Figure &is characteristic o uniltered aliases.

"e prevent spatial aliasing in the ield by using a iner group interval. 4n the processing, wesimulate the iner interval by trace interpolation. Typically, the procedure begins with trace-to-trace correlations within a sliding time gate to determine the dominant coherence direction. Then,a simple halway-point interpolation between the time samples along that direction gives the tracevalue in the center o the gate ( this trace interpolation procedure is also eective in the slantstac domain).

(n real data, the interpolation scheme must be sensitive to the presence o incoherent noise!otherwise, we run the ris o converting that noise into spurious, laterally coherent signal.

"e also note that trace interpolation cannot replace proper sampling! we cannot expect it toprovide geologic resolution greater than that actually recorded in the ield.)

Figure 3 ,


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Figure 8

Figure 2 ,


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Figure 9

andFigure 1< show the eect o the combination o trace interpolation ( Figure 3)

Figure 10

and f-k iltering (Figure 2) on the record o Figure $. "ith the interpolated traces removed, sothat we restore the original trace spacing ( Figure 1restac iltering is an expensive process, and we mustweigh its cost against its beneit.
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ne inal use o prestac f-k iltering is or the removal o multiples (but, again, only where nothingelse is suicient). 4n a sense, this is a time-variant iltering that requires good nowledge omultiple velocities.

The irst step is to correct the cmp gathers according to the velocity o the multiple system, so that

the primaries are overcorrected ( Figure 1 ).

Figure 1

The f-k plot o this gather then has the multiples aligned on the requency axis, and the primaries

entirely on one side o it (Figure # ).


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Figure 2

To the extent that elimination o the multiples can be eected, the resulting gather has primaries

only ( Figure ).


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Figure 3

The eectiveness o the process can be 8udged by comparing velocity analyses beore and ater

the process (Figure $ ).


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Figure 4

n real data, o course, there may be many multiple systems, each with its own 5velocity.5 So thechoice o one multiple-velocity distribution is equivalent to the choice o one multiple-generatingsystem. This means that f-k iltering o multiples gives improved results only in highly speciic

cases. 4n general then, we use f-k iltering to remove multiples only i there is one multiplesystem. *ven then, the eect is greatest or intermediate or deep hori6ons, where the velocitycontrast with the primaries is greatest. 4 there are several multiple systems, f-k iltering is not aneective option.

Poststack Filterin

The apparent velocity o a (hyperbolic) relection varies over a wide range, rom slightly greaterthan its true velocity to ininity. The low end o this range, which occurs at early times and longosets, may overlap the velocity o some coherent noise. "orse, that coherent noise can be in adifferent part o the record. Thereore, prestac f-k iltering oten needs a wide pass-band, andmay result in indierent noise attenuation.

4t may also be that the coherent noise train is not visible on the ield record. 4nFigure 1(a), or

example, the section is plagued by noise that stacs well.


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Figure 1

Any f-k iltering we wish to apply must be poststac. 4n the present context, what we hope toaccomplish is more appropriately called dip filtering.

The principle is the same, o course, except that alignment o data on an f-k plot has signiicance

as a dip rather than as a velocity. Thus, a hori6ontal event on a seismic section shows up as avertical alignment in f-k space. 0ost events o seismic interest lie within a small wedge straddlingthe requency axis! most coherent-noise trains, on the other hand, exhibit a much steeper dip.Thus, the section o Figure 1(a) is an ideal candidate or dip iltering, because the signal and thenoise should be easily separable on an f-k plot. Ater dip iltering, the relections are much clearer( Figure 1(b)).

n the staced section, o course, both positive and negative dips are equally liely. Thus, itmaes sense to plot both positive and negative wavenumbers. As we might expect, however, areversal o dip may be construed as spatial aliasing. ur rule here is airly simple+ i an alignmentdoes not pass through the origin (that is, D D


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The Choice of 'aria+les

The processing required beore we do an f-k analysis and iltering depends on the reason or theanalysis. 4 we are studying a noise spread or determination o optimum source or receiverarrays, then we probably want to do no processing that aects statics or moveout. As regardsmoveout, the ground roll and reraction that we wish to analy6e do not display any hyperbolic

curvature. Further, i we do mae some nmo correction, then we alter the apparent velocities othe ground roll and the reraction. Inless all the stepouts are still within the re8ect 6one, the nmocorrection could render the f-k analysis useless.

Statics are a dierent matter. "e have said that coherence on an f-k plot is a unction o apparentvelocity, or apparent dip. 4t is reasonable to expect, thereore, that severe statics will hamper theanalysis. n the other hand, we recogni6e that ground roll, traveling as it does along the surace,is a wea unction o the topographic irregularity. Furthermore, static corrections that wouldoptimi6e relection data may not necessarily do the same or reraction data. n balance,thereore, it is usually best to have neither nmo nor statics corrected beore f-k analysis o ielddata. The rule o thumb is that, to attenuate an unwanted wave train, we wish to eep it straight.The same is not true, o course, or any poststac analysis.

The Fourier transorm o a short time waveorm may produce ambiguities in the requencydomain. The longer the time waveorm, the better the resolution o the component requencies. Asimilar result arises when we consider Fourier transorms to yield wavenumber densities+ thegreater the number o traces considered, the better resolved the wavenumber distribution. Thus,i we have a seismic record in which all the events have comparable moveouts, or apparentvelocities, then the input to the f-k analysis should be the entire record (or, at least, as much o itas contains signal).

4n reality, a record usually has a wide range o moveouts. "hat happens on the f-k plot then isthat the ma8or lobe broadens, encompassing the larger range o wavenumbers and requencies.4n general, this is not a problem as long as the noise is well deined! ater all, the f-k analysis hereis to be used as a re&ect ilter. And we reali6e that because there are so many more time samplesthan there are space samples, the temporal requencies are always better resolved than the

spatial requencies.

nce we have the f-k plot, the tas is to design the ilter, which may be a wedge emanating romthe origin ( it may also be a polygon ). The width o the ilter depends on the range o velocitieswe wish to ilter. As with requency ilters, we also need to speciy slopes, because a sharp cutoo the wedge causes a ringy impulse response.

4ndeed, Figure 1 , andFigure #show that a poor choice o slopes puts us at ris o 5creating5

events.


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Figure 1

The slopes o Figure


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Figure 3

, and Figure $do not have this problem.


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Figure 2

(And slopes need not be straight, but may have curved edges.) 4n some circumstances, however,we are orced to steep slopes or narrow pass-bands, in which case the consequent ringing

imparts a 5wormy5 appearance to the section.


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Figure 4

Then, the standard processing remedy is to add bac some o the uniltered section to the ilteredsection, in ratios o 1+1 up to 1+.

Finally, we must understand that f-k iltering has penalties attached to its use. "e have said that

noise re8ection by f-k iltering wors best i the traces have the same noise amplitude. 4n general,this requires amplitude manipulation that also aects the signal. "e may equali6e the traces aterthe iltering, but this sacriices our conidence in the staced amplitudes, and prevents any avoanalysis we might wish to do.

As an alternative however, some programs store the gain in the header. This gain can then beremoved ater - iltering, which thus prevents this compromising o amplitudes.

Final Filtering

andpass Filterin

%y the time we get to stac, we now the usable signal bandwidth o the data. 4n general, then,our next step is to design a bandpass filter with an output9input response o one over thatbandwidth, and a decreasing response on both sides o it ( Figure 1 ). 4n the requency domain,we multiply the amplitude spectrum o the seismic trace by the response o the ilter. 4n the time

domain, we convolve the input time series with the ilter operator.


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Figure 1

4n Figure 1, we see that the nominal cutoff fre!uencies f'and fdo not actually have ullresponse. ather, they represent - d% points, a remnant o the bandpass ilter@s electrical-engineering heritage! engineers regard the hal-power (- d%) point as the edge o the pass-band.For our purposes, this is acceptable, because a -d% loss o the cuto requencies is not liely tobe noticeable on the iltered trace.

A bandpass ilter is also characteri6ed by its slopes, which tell us how much a requency outsidethe pass-band is reduced by the ilter. ne way to deine the slopes is to speciy amplitude values

or our dierent requencies! the inner two are the cuto requencies, and the outer two are thenull requencies. 4n a design sense, the null requencies have a response o 6ero, but with inite-length ilters, this may not be reali6ed.

Another way to speciy a slope is as a taper, that is, so many d% per octave. Thus, i a ilter isdeined as 1< to &< =6 with slopes o 1# d%9oct and & d%9oct, we would expect a 7-=6component to have one-ourth the amplitude o a 1


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The choice o slopes represents a compromise. The ininite slope o a boxcar ilter cannot beachieved with operators o inite length, so the 5ideal5 bandpass ilter would probably have an

amplitude spectrum lie that o Figure #(a).

Figure 2

Inortunately, the impulse response o this ilter 5rings5 ( Figure #(b))! the ringing is asuperposition o the cuto requencies.

4n Figure , we see the eect o using a gentler slope.


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Figure 3

The time-domain operator is shorter (that is, it has ewer non6ero components), which permitseicient and economic time-domain iltering.

4n bothFigure #and Figure , the slopes are not linear tapers. ather, they are scaled by the

irst quarter-cycle o a cos# unction, which is general practice. This permits the amplitudespectrum to have rounded corners at the cuto and null requencies. 4n general, however, even alinear taper is eective at reducing the ringing.

Although the ilter o Figure has a short, nonringy operator, we see rom its response that it isslow to re8ect requencies above the cuto. To establish the correct operator length, thereore, weremember that the eective length o the operator is inversely proportional to the requencybandwidth. For most applications, an operator length between #


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#. "e ilter the entire trace three times, once by each ilter selected ( Figure 1 ,

Figure 1

Figure # , andFigure ).


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Figure 3


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Figure 2

. "e scale each iltered trace to have ull amplitude within the speciied times, anddecreasing amplitude beyond them.

Thus, or the irst trace o Figure $ ,

Figure 4

Figure 7 ,


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Figure 5

andFigure &, we eep ull amplitude or the irst 1.3 s, steadily decrease the amplitude

rom 1.3 to #.# s, then 6ero the trace beyond #.# s.


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Figure 6

Similarly, the second trace is 6eroed to 1.3 s, ramped up to #.# s, ull amplitude to .< s,ramped down to .7 s, then 6eroed beyond .7 s. Finally, the third trace is 6eroed to .

filtering and stacking

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