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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 .