STATICS CORRECTIONS - A TUTORIALby

Brian H. Russell, Hampson-Russell Software Services Ltd.

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near surface effects, as well as byoffset distance X. Since the seismictravel path is continuous, we cannottheoretically separate the near surfaceeffects from the deeper subsurface ef­fects. However, an approximate solu-

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Fig. J. A hypo/helical 1I0lIZero-ojjsel seismic /0 sho/ S. receiver R, reflec/ioll poill/ P, alld o.f[se/recordillg ShOll'illg a sillgle rej/ee/ioll raypa/h. We X.may slIbdil'ide Ihe raypa/h ill/a cOII/ribwiolls dlle

s~...------x

their elevation depends on topog­raphy. The raypath for a single reflec­tion on a seismic recording is shownin Figure I. From this figure, we cansee that the observed reflection time isinfluenced by both topographic and

Static corrections are important inthe seismic processing flow for a num­ber of reasons:

- They place source and receiverat a constant datum plane.

- They ensure that reflectionevents on intersecting lines willbe at the same time.

- They improve the quality ofother processing steps.

- They ensure the repeatibility ofseismic recording.

Static corrections involve a con­stant time shift of the seismic trace, asopposed to dynamic corrections,which involve a set of time variableshifts. As with most seismic process­ing steps, static corrections representa gross simplification of physicalreality. However, despite the apparentsimplicity of static corrections, theyhave a dramatic effect on the finalquality of the seismic section ifderived and applied correctly.

In this tutorial I shall discuss thetechniques for deriving such optimumstatics values, covering the threemajor approaches to statics computa­tions: field statics, refraction statics,and residual statics. Before discussingthese three computational techniques,let us look at the basic statics model.

INTRODUCTION

THE STATICS/NMOMODEL

Seismic energy travels through theearth as a spherical wave. However,for reasons of simplicity, we will useray theory in the derivations in thispaper. It should be pointed out thatray theory is strictly valid only for ahigh frequency solution and neglectssuch observable features as diffrac­tions. However, we can usually treatthese effects as part of the migrationproblem and thus, in this tutorial, wewill consider only the steps that gointo producing an optimum un­migrated trace.

The basic seismic recording invol­ves a source and receiver which areseparated by a distance called the off­set distance. In marine recording, thesource and receiver are at the samedatum elevation, but in land recording

--------------------16--------------------

tion is to separate the path into fourterms, so that

T = To + Ts + T r + Tx , (1)

where T = Total traveltime,To = Seismic structural time,T 5 = Shot "static" from

datum to surface,T r = Receiver "static" from

datum to surface,and Tx = "Dynamic" time shift

to correct for offset.The term static can be interpreted

as "independent of record time",whereas the term dynamic means "de­pendent on record time". However, ifwe consider two or more reflections,we can see that neither term is precise­ly correct. This is because the "static"is different for each reflection due tochanges in the raypath, and the offset,which defines the "dynamic" correc­tion, depends on the points at whichthe emerging rays intersect the datum.

Nonetheless, these are the assump­tions that are used in the basic seismicprocessing flow, and they can beviewed in idealized form as seen inFigure 2. In this figure, the travel pathof the rays is vertical from surface todatum, and straight between the

datum and the reflecting point. Theseassumptions are reasonable if thevelocities of upper layers are muchlower that those of the deeper layers,which is often the case.

I+~-----X----~.I

s

Ts

p

Fig. 2. Idealized geometry for the statics/ NMOmodel. Stalic corrections are vertical. II'hereasN MO correi'liol1S are horizolllal. The seismic traceis theoretica/~1' correi'led to mid-poilll M.

Using the assumptions shown inFigure 2, the seismic processing flowcan be simplified to two fundamentalsteps. We first derive a velocity versusdepth model, from which static cor­rections are computed and then ap­plied to the data. This is equivalent toplacing the shot and receiver at thedatum. Next, a dynamic, or NMO,correction is appliced to the trace toplace the reflection below midpoint

M, at a time of To. In this simplescheme, the static is a vertical cor­rection, whereas the dynamic is ahorizontal correction. Although thissimple processing scheme is greatlyexpanded in practice, these are theunderlying assumptions of the seismicprocessing method.

FIELD STATICSFrom a knowledge of the topog­

raphy of our seismic line, the sourceand receiver parameters, and thevelocities and thicknesses of the nearsurface layers, a complete statics solu­tion can, in principle, be derived. Thiscomputation is referred to as the "fieldstatic", to differentiate it from othertypes of statics computations. Al­though this static correction shouldtheoretically place the shot andreceiver at the same datum elevation,we shall find that there are many fac­tors that impede our ability to ac­curately determine its value.

Let us start by assuming that ourproblems in deriving an accurate near­surface model have been overcome.We shall consider the simple case of atwo layer near-surface, consisting of aweathered layer of low velocity un-

_____________________17 _

weathered material, and adding this tothe distance from the base of theweathering to the datum divided bythe subweathering velocity. The totalstatic is then the sum of the shot andreceiver components. In this formula­tion, we have assumed that a buriedsource has been used. For a surfacesource, the source static would bemodified by adding a delay time forthe weathered layer.

In the previous example, we haveassumed that the thickness andvelocity of the weathered layer isknown. In fact, one of the biggestproblems we face is determiningthese values. One technique usedto determine the near surfacevelocities and thicknesses is thevelocity survey, illustrated in Figure 4.Note in the figure that a correction hasbeen done to compensate for non-ver­tical raypaths. One of the mainproblems of this method is that wellvelocity surveys are usually not donefrequently enough along the line todetermine the near surface structure ateach shot point. Another problem isthat there are usually not enough shal­low shots to determine the very near

Fig. 4. The uphole or lI'elf shoo/ing me/hod.II'here (a) shOll'S the slln'e.\' i1self. lI'ith /he recei"erslocated ill/he lI'elf and sho/ is a/ the sllrlace. and (b)sholl'S /he interpreted sllrvey. In this simpleexample, /he. il1lerval velocities have been exactl\,de/ermined.

1'IME (He.) -o 0.02 (). O't 0,06 0.06 (')·1

Total Time llvg. Vel. Int. V"l.(sec) (m/sec) (m/sec)

0.020 500 5000.040 500 5000.050 600 10000.060 667 10000.070 714 10000.075 800 20000.080 875 2000

10

20I\

DEPfH 30I

I n-le..-Va I0

(m)I VelocityI

~o 0

J I\

50 0

\

60 /~Avr;' \

71) '1e/ocHy '.0

0 500 /000 1500 2000

yf.l-DCr-ry (m/~c)'-'

(b)(a)

10 0.02020 0.02030 0.01040 0.01050 0.01060 0.00570 O.OOS;

Depth Con. Int.(m) Time (sec)

datu~ elevation,

depth of shot,

depth of weathering,

receiver static,

shot static,

ES shot elevation,

ER receiver elevation,

dSD = thickness from shot to datum,

dRD = thickness froM receiver to datum,

Vw velocity of weathered layer,

and Vsw velocity of suhweathered layer.

where

DATlM

consolidated material and a sub­weathered layer of more competentlithology. We shall also assume thatthe datum plane down to which wewish to correct the data is in the sub­weathered layer. This situation is il­lustrated in Figure 3. If we know thethickness of the weathered layer, theelevations of the shot and geophone,and the depth of shot, we can thencompute the static corrections. Therewill be two components to the totalstatic: a shot component and' areceiver component.

Although there are many methodsof computing field statics, thesimplest method is shown in Figure 3.The computation of a time delay ineach layer can be found by dividingthe length of the vertical raypath ineach layer by the layer velocity. Forthe two layer case shown in the figure,the shot static is found by dividing thedistance from the base of the shot tothe datum by the subweatheringvelocity. The receiver static is com­puted by dividing the weathering layerthickness by the velocity of the

Fig. J. Field static computations by SIlIII 01lI'eathering alld subll'ea/hering delay.

------------ 18 _

1.D

section, the estimation of static cor­rections can be severely hampered byirregular topography and rapidlyvarying velocity and thickness changesof the weathering and sub-weatheringlayers. One of the best ways to es­timate these changes is to analyze thefirst breaks which result from refrac­tions in the shallow low-velocitylayers. A plot showing such a set ofcomputer picked first breaks is shownin Figure 6.

The key concept in seismic refrac-

TIME(SECI o.s

Fig. 6. A 1'101 of a seislllic profile displayillg alI'eaiherillg tlllolllal.\'. The first arrivals hal'e heellpicked l/sillg a cOlllpl/ter alld are ShOIl'1I as dashes.n1ese .first arril'ols are I'ery "clean". hili otherpro.files are IIIl/ch lIIure di(ficult to pick due 10 thepresence of lIoise. NOle reverse polarit)' trace. /4tracesfrolll the lefl.

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tions are usually made. The first isthat, in general, the top layer of theearth is made of unconsolidatedweathered material of variable thick­ness and low velocity called theweathered layer. As previollsly dis­cussed, the thickness of this layer isoften taken from uphole time meas­urements. This assumes that the shothas been drilled slightly below theweathered layer. Significant devia­tions in this near surface layer thick­ness can be caused by such geologicaleffects as meandering river channels,variations in glacial till thickness, andvariations in the water table.

REFRACTION STATICSAs we discussed in the previous

lOa .. - - - .

200 .", ... -- ..

300

Fig. 5. n,e e/fect of surface lopography onstatics. An earth 1II0dei .sho\l'illg irregular surfacetopography Il'ilh a .flat re.flector lI'il/ he "allti­correlated" \l'ilh the re.flector at depth.

THE EFFECT OF THENEAR SURFACE

surface velocities.

A simpler method makes use of theuphole time, which is the traveltimefrom the shot to the surface asmeasured by a geophone placed closeto each shot. The basic assumption isthat the shot hole is drilled just belowthe weathered layer, and the upholetime will therefore give us the velocityof this layer. If the shot has beendrilled below the weathered layer, thismethod can give erroneous values forthe weathering velocity and it is thusadvisable to check for uphole timesthat deviate significantly from theaverage value. Using the uphole timesimplification, the delay methodequations from Figure 3 can be rewrit­ten so that the receiver static is simplythe sum of the shot static at thereceiver location plus the uphole time.

The effect of near surface geologyplays a major role in the accuracy ofthe field statics computation. For ex­ample, let us consider the effects oftopography. Ideally, we would like toto record data on a perfectly flat sur­face. If we are recording data in thecentral plains of North America, weare close to this ideal, since the effectsof topography are often insignificant.However, if our recording takes placein the overthurst belt of Wyoming, thefoothills of Alberta, a dune environ­ment in Saudi Arabia, or some othercorrespondingly rugged topography,the effects of variations in elevationcan severely distort our data.

It is an interesting exercise to con­sider the effects on the stack if topog­raphy is ignored, or if the elevationsare assigned incorrectly. In the case ofland data, the effect 'is that of a sub­surface structure which is "anticorre­lated" with the elevation profile of thesurface. That is, highs on the surfaceare seen as lows on the reflector, andvice versa. This is illustrated in Figure5 for the simple case of a constantvelocity earth with an irregular topog­raphy and a flat lying reflector. Noticethat the reflector is the inverse of thetopography since the travel time isgreater from the higher points on thesurface.

Next, let us consider the effects ofvelocity and thickness variations inthe near-surface layers. This is theleast well known part of the problemand will have the greatest effect on thestatics solutions. In the processing ofland data, several simplifying assump-

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CANJAY EXPLORATION (1975) LTD.

a situation in which a layer decreasesin velocity, the refraction will betowards the normal, and there will beno refraction from that particularlayer. This is called a low speed layer.A related, and more common, situa­tion is where either a layer is too thinor has too small a velocity contrast tobe resolved by the refraction method.This is called a 'hidden layer'.

Another complication is that ofdipping layers. In general, we can saythat the intercept time will be less atthe updip location than at the downdiplocation, and that the slope of the firstbreaks in the updip direction will beless (and hence the velocity greater)than in the downdip direction. How­ever, the total traveltime is the samein either the updip or downdip direc­tions. This time is often referred to asthe reciprocal time.

REFRACTIONINTERPRETATIONPROCEDURES

Having looked at the basic theoryof seismic refraction, we are now in aposition to look at methods of inter­pretation of refracted arrivals whichare based on this theory. This area ofresearch has been active since the earlydays of seismic prospecting. General­ly, we may classify the approaches

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the second layer. This equation can begeneralized to more layers quite easilyby using the general form of Snell'slaw.

Figure 7 assumed a shot on thesurface. The effect of a buried shot inthe first, or weathered, layer is to adda delay which is equal to the upholetime. In the figure, we also assumedthat the velocities of the near surfacelayers increased with depth. If we have

Xcro.u.

lI'ave ol'ertakes the direct arrival at the distance

V1

REFRACTED WAVE

TIME

Fig. 7. 77,e recording ala refraCled lI'ave along aseismic spread. No refraCled energy reaches thespread before the critical dislOllce Xc. 77,e refracted

tion is that when a seismic ray hits ageological boundary, it is refracted or'bent' by an amount depending on thevelocities of the two geological layers.The amount of the refraction isgoverned by Snell's Law, which statesthat the ratio of the sines of the inci­dent and refracted angles is equal tothe ratio of the velocities of the twolayers. As long as the velocity of thelower layer is greater than the velocityin the upper layer, there comes a pointwhere the refracted angle is equal toninety degrees, and a refracted waveis set up which travels along the inter­face between the layers at the velocityof the lower layer. This is shown inFigure 7, as well as the distance atwhich the first refracted wave willreach the receivers, called the criticaldistance. A theoretical plot of firstbreaks from a surface shot is alsoshown in Figure 7. If we considerthe geometry of the ray from shot S toreceiver R in Figure 7, the total travel­time can be shown to be

2 Zo cos ic XT = Vo + ~, (2)

where ic = critical angle,Vo

and sin ic = -VI

The preceding equation is the equa­tion for a straight line, where the firstterm is the intercept time and thesecond term contains the slope as theinverse of the second layer velocity.The intercept time will reveal thedepth 0 f the first layer, and the slopeof the line will reveal the velocity of

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Once we find the delay times, wemay compute the depths by simplyrearranging equation 3 for the depthterm. The difficult part is of coursefinding the delay times for each in­dividual receiver location. This isdone by finding the average intercepttime from both a forward and reverseprofile, and partitioning the intercepttime into its receiver and shot com­ponents.

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The Reciprocal MethodThe reciprocal method in various

forms has been one of the mostpopular methods of refraction inter­pretation (Hawkins, 1961).

The basis of the method is the time­depth term, which is virtually identicalin definition to the delay time. Thetime-depth term is best understood byreferring to Figure 10.

From this diagram, we can definetime-depth as

ZG cos ic (4)tG =Vo

As you can see, this is identical inform to the delay time. However, thekey difference is in how we find thetime-depth term. To find the time­depth for a particular geophone, con-

(3)Zd cos ic

td =Vo

where td = delay time below shotor receiver,

and 2'd = depth below shot orreceiver.

These definitions can be seen inFigure 9.

The Delay Time Method

The delay time (Barry, 1967) isdefined as the time between the datumand the refractor minus the normalprojection of the raypath on therefractor. Thus, the total travel pathhas two delay times associated with it,for the shot and the receiver, whichare given by the general form:

Fig. 9. The basic principles behind the delay timemethod. Dip is assumed negligible in the vicinity ofthe shot and receiver.

Fig. 8. A hand interpretation of a set of pickedfirst arrivals. Note that small wavelength anomalieshave been smoothed OUI.

that have been taken into the follow­ing headings:

- Slope/Intercept Method

- Delay Time Methods

- Reciprocal Methods

- The Generalized Linear Inverse(GLI) Method

- The Time-Term Method.

Let us briefly consider each of thesemethods.

The Slope/Intercept MethodThis is the simplest method of in­

terpreting first breaks, and followsclosely the theory that was discussedin the previous section. The first stepis to fit slopes to a set of picked arrivaltimes, and thus find the seismicvelocities. Each slope is then extrapo­lated back to the shot location to findthe intercept time, and hence thedepth to a particular layer.

Figure 8 shows a hand interpreta­tion of a set of picked first arrivals,and the resulting geological model.Notice that the method has found onlythe very smoothly-varying componentof the near-surface, and that smallervariations, such as those indicated -atlocations 141 and 161, have not beenaccounted for correctly.

------------------__21 _

sider Figure 10 again. We simply addthe traveltimes from the bracketingshot points, subtract the total timefrom shotpoint to shotpoint (this isdefined as the reciprocal time), andhalve the result. In symbols, we candefine the time-depth as

tG = 1/2 (tSIG + tS2G - ts1S2) (5)

Fig. 10. 7111' basic concept of Ihe reciprocalmelhod. 7111' lime-deplh is defined as Ihe distallcefrom Cia Pin Ihe diagram limes Ihe cosine of fhecrilical all~le ie. divided by Ihe I'elocily of Va.

We can then transform the time­depth into the depth to the refractorby using equation 4.

A more advanced form of thereciprocal method is the generalizedreciprocal method (or GRM),developed by Derecke Palmer (198 I).The method is basically an extensionof the classical reciprocal method, buthas the advantages of good dip han­dling and recognition of very irregular

refractor surfaces. The key differencebetween GRM and the classicalreciprocal method is that Palmer usesa pair of geophones, which areseparated by a variable distance X,rather than a single geophone. If thedistance X is zero, the method is iden­tical to the reciprocal method.

The GLI Method

We have seen that all refractionanalysis methods assume some modelof the near-surface geology, and thatthis model is normally a series oflayers whose thicknesses andvelocities may vary both laterally andvertically. Since first arrival timesdepend on thickness and velocity, thedirect approach is to perform somecalculation using the observed breakswhich yields the parameters themsel­ves, such as the slope/interceptmethod, delay-time method, orreciprocal method. Since thesemethods make restrictive assumptionsabout the model, an alternativemethod is to iteratively build a modelof the subsurface based on the infor­mation provided by the first breaks.This is called the generalized linear

inverse, or GLI, method (Hampsonand Russell, 1984).

MODIFY NEAR·SURFACE MODEL

Fig. I I. FlolI'chart \I·hich sholl's Ihe basic COllceplof refraci iOIl anall'sis by generalized lineari/ll'ersion (G LI) (Hampson alld Russell. 1984).

Figure II illustrates the GLImethod in a non mathematical man­ner. The user inputs an initial guess ofthe near-surface model which tells the

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Fig. /2. A brute stack lI'itll elel'ation and upllolecorre("(ion onl)' (Hampson and Russell, 1984).

Fig. 13. The same line as in Figl/re 12, bur afteral/tomatic G LI refraction statics analysis(Hampson a/l(1 Rllssell, 1984).

program how many layers are ex­pected and their approximatevelocities and thicknesses. The pro­gram then performs a series of itera­tions in which the model breaks are

The Time-Term Method

is reached between observed andmodel first breaks.

Let us look at the application of themethod to a real dataset, chosen be­cause it displayed both long and shortwavelength static anomalies. This isdemonstrated quite clearly in Figure12, which is a brute stack of the linewith elevation and uphole correctionsapplied. The long wavelength staticappears as a slowly varying structuralcomponent with a maximum variationacross the line of about twenty mil­liseconds. The short wavelength com­ponent results in loss of continuity onsome of the weaker reflectors, espe­cially the data above 900 msec, andalso in a vertical swath to the left sideof the low coverage gap.

In Figure 13, we see the same stackafter automatic refraction analysis. Inthis result, both the long and shortwavelength statics solutions have beenimproved. The removal of the longperiod static results in a more struc­turally valid section without having toresort to the artificial solution of flat­tening on a particular event. Also, aswe shall discuss in the next section,residual statics programs based on thecorrelation of reflection zones aretheoretically unable to solve for thislong period static pattern.

The short period statics solutionresults in improved data continuity,both in the events at 900 msec andabove, and also in the vertical swathof data in the central part of the line.

calculated by ray-tracing, comparedwith the measured breaks, the modelis improved, and a new set of breakscalculated. This procedure is repeateduntil some acceptable correspondence

1.0

1.5

2.0

TIME(SEC)

1.0

2.0

1.5

TIME(SEC)

Fig. /4. A seismic se("(ion be/ore and after tileapplicalion 0/ a sllr/ace consistent time termanalysis (Chlln and Jacell'itz, 1981); (a) BrUle slOck

0/ Canadian data and (b) SlIr/ace consislent firstarrival statics-applied slOck.

Until now in our discussion, all ofthe methods that we have consideredinvolved deriving a geological modelof the subsurface and then calculatingstatic corrections from this model.However, it is possible to derive thestatics themselves from the first ar­rivals without the intervening step ofderiving a model. Such methods havebeen referred to as time-term. or leastsquares, methods (Chun andJacewitz, 1981).

To begin with, we may rewrite thebasic travel time equation as

Tij == Si + Rj + Xij/V, (6)

where Si == shot static at shot i,R j == receiver static at

receiver j,Xij == offset from receiver to

shot,and Tij == total travel time from

receiver to shot.

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side by side and their resulting sum orstack. The traces appear identical ex­cept for positive and negative timeshifts. The result of summing thesethree traces is that the individualevents are now 'smeared' in time whencompared to the original traces.

In Figure 16 the result of cross-cor­relating trace A with both trace BandC is shown. These cross-correlationsboth look like auto-correlations ex­cept that the zero lag value has beenshifted, by -10 msec in the first caseand + 10 msec in the second. If theseshifts are applied to the originaltraces, the result is three traces whichare aligned with one another. Theresulting stacked trace shows no'smearing' of the events.

This simple example illustrates thebasic idea behind computing residualstatics. Individual unstacked tracesare corrected for field static andNMO, and are cross-correlated withsome reference trace to compute anoptimum time shift which will removethe static problem completely. Theproblem is in determining the refer­ence trace to use and in deciding how

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Fig. 16. The cross-correlation of traces A and B.and A and C. and Ihe resulling Slack afler Iheindicaled shijis have been applied.

a single CDP gather after NMO cor­rection we can account for subtle er­rors in the deterministic statics solu­tions.

Correlation StaticsFigure 15 shows three seismic traces

Fig. 15. A Ihree Irace Slack. where Ihe /races areout o{ alignl11ell/.

Now the problem can be set up asa series of linear equations and solvedby Gauss-Seidel iteration. The prob­lem can be further simplified by firstremoving a linear moveout (LMO)function from the first breaks. Thethird term in the preceding equationthen becomes a residual moveoutterm. Indeed, if the LMO term repre­sents the true velocity from the secondlayer and we consider only termswhich refract in the second layer,equation 6 can be simplified so thatthe third term is ignored entirely. Fig­ure 14 is an example of a statics solu­tion using a modified time-termmethod.

AUTOMATIC RESIDUALSTATICS

For the above reasons we need amethod for "fine-tuning" the staticssolution. Such a method wasdeveloped in the late sixties and earlyseventies using the technique of reflec­tion correlation. That is, we assumethat by aligning the reflections within

In the previous sections on fieldstatics and refraction statics we haveassumed that the total component ofthe static correction can be derived bymodeling the near-surface layervelocities and thicknesses and com­puting a time shift which will stripaway the effects of the near-surface.However, despite our best efforts, thestatics that are derived using thesemethods do not appear to solve thecomplete problem.

Why is it that we cannot computeour total statics solution in a deter­ministic way? There are a number ofpossible reasons for this. The mostobvious reason is that no matter howwell we think we know the velocitiesand thicknesses of the near-surface,the real earth is actually more complexthan our model tells us. The velocitycan vary both laterally and verticallywithin a single layer due to changinglithology. The thickness of theweathered layer may vary rapidly dueto river deposition or glaciation. Thinlayers with abrupt lateral terminationsmay have been left by terent localizeddeposition patterns, or the watertable may have an effect on thevelocity distribution that is hard topredict. Also, we know that the verti­cal ray approximation used in thestatics model is incorrect. The devia­tions of the raypath from vertical canbe quite significant and will impart anerror into the solution.

____________________24 _

CROSS-eORRELATION STATICS EXAMPLE

0.5

TIME(sec)

1.0

BEFORE AFTER

Fig. 17. A sel 0/600% CDP galhers be/ore andajier Ihe applicalion v/ corre/aliOlI slalics.

BASE WEATHERING

~~-,""",,,-,SURfACE

RECEIVER J'1

Equations like that shown in 7 arereferred to as linear equations. Theseequations consist of a number of ob­servations (the values Tij) from whichthe parameters (S, R. G, and M) mustbe solved. Although we generallythink sets of linear equations in whichthe number of parameters is equal tothe number of observations, ingeophysical problems we usually havethe situation in which the number ofunknowns is not equal to the numberof parameters. Wiggins et al (1976)show that, in the statics problem, thenumber of parameters and observa­tions are always deficient by 13 equa­tions. This means that there is no

=~l'--- __::::REFlECTOR

POSSIBLERNMO

SHOTSTATIC

~HOT I

Fig. 18. A simplified cross-seclion o/Ihe eOrlhsholl'ing sllr/ace and subsurface consis/eJle)'.

--:;L--},:---------\-----......:::"=t==~===-_I_----- DATUM

STRUCTURECHANGE

tent and must not be included in thestatics solution.

To solve for the four componentsshown in Figure 18, we must set up anequation of the form

Tjj = Sj + Rj + Gk + MkXi/, (7)

where Si = shot static for ith shot,Rj = receiver static for jth

receiver,Gk = structure term for kth

COP position,Mk = time averaged RNMO

at kth COP,Xij = offset from shot i to

receiver j,and Tij = total static value.

Linear Surface ConsistentResidual Statics Method

believable each shift is. Figure 17shows a set of 600% COP gathersbefore and after the application ofcorrelation statics. Notice that themethod has done a good job of liningup the events just below 0.7 sec. Thisapproach can be used for computingour final statics, but it will not resolvemajor lateral statics problems on ourline. In the next section we will lookat the more general surface-consistentstatics approach.

We often observe static patternsthat move with the same geophonegroup spacing as the shot rollalong.This is referred to as a surface consis­tent static pattern. Since the samestatic pattern will be spread over dif­ferent CDP gathers, it cannot besolved by simply computing the cross­correlations of traces in a CDP gatherwith the pilot traces derived from theCOP gather itself. Therefore, we mustderive both the shot and receiver com­ponents of the statics in an inde­pendent way. Figure 18 shows thatthere are four possible sources for theobserved static: the shot component,the receiver component, an RNMOcomponent, and a structural com­ponent. The source and receiver termsare referred to as the surface-consis­tent components, and represent thestatics solution. The structure andRNMO terms are subsurface consis-

---------------------25-- _

~ e 7 6 5 A

unique solution to the statics problem.Howcver, by using the followingaveraging technique, we can convergeto a rcasonable solution:

(I) First we sum over the com­mon shot positions to get an es­timate of the shot static.

(2) Next, we sum over the com­mon receiver positions to getan estimate of the receiverstatic.

(3) Next, we sum over the CDP

traces to get the structural com­ponent.

(4) Finally, to get the RNMOvalues we sum over the CDPtraces after weighting themwith a factor equal to theoffset squared.

We then iterate tHrough this cal­culation and measure the error aftereach iteration, hopefully convergingto a solution. In practice it has beenfound that three iterations through theabove proced ure is usually enough to

produce a satisfactory result. Weshould also note that this procedure isoften referred to as Gauss-Seidel itera­tion.

It should be noted that the surfaceconsistent approach, while producingthe best overall statics, does notproduce the best individual staticswithin an individual CDP gather. Forthis reason, it is often advisable to runa final trim or correlation static com­putation on your data prior to finalstack.

Figure 19 shows the effect ofstatics in a particularly bad case.Notice that there is a lack of any eventcontinuity on the line. Figure 20 showsthe result of applying a surface con­sistent statics analysis to the line.

Fig. 19. A II illpur swck hefore swrics applicatioll(Wififiim el a/. /976).

Nonlinear Surface ConsistentResidual Statics

An interesting approach to the solu­tion of large statics anomalies hasbeen proposed by Daniel Rothman(1985, 1986). The algorithm which heproposed, called simulated annealing,uses a Monte Carlo optimization tech­nique which solves the statics problemin a way that is similar to suddencrystallization in a melt. Keeping up

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2000

1835 It.eraLion.

100o

1500

Power

(a). T1Iis lI'as ajier 1000 iterations of Ihe melhod.(Rolhman. 1985)0/111 (c) Final Slack flfler Ihe 1835ilerations of Ihe silllulaled annealing algorill/ln(Rothman. 1985).

1000 It.eraLion.

100

1000i terot ion

o

500

InpuL St.ack

100

o

o

LJ)

•...Stack

"-CD....oQ.

Fig. 20. 7711' sallie s/llck as in Fif;ure 19. hUI afierIhe applicOlion ofsurface colISislent residual Slalics(11 'iggillS 1'1 01, 1976).

Fig, 22. A pial oflhe slOck po\l'erfor rhe results ofFigure 21

Fig. 21, Results 0/ the sillll/lated annealingalgorill/ln: (a) An inpm slOck lI'ith 110 staticsapplied. (Rolhman. 1985): (b) The result of Ihesilllulaled annealing lIIethod applied 10 the Slack of

INTERPRETATIVECONSIDERATIONS

Now that we have looked at the nutsand bolts of static corrections, weshall consider how to make sure thata good statics solution has been ob­tained on the final stack. There arethree reasons why we would like agood static solution. These are:

eto obtain a correct structuralinterpretation,

eto obtain a good stratigraphicinterpretation, and

eto optimize the overall qualityof the seismic reflections.

It is important to realize that anyone of the above criteria can be metwithout satisfying the other two. Thatis. we may have obtained a very goodlong period static solution without theequivalent improvement in the shortperiod static solution. Or we may haveobtained the short period static solu­tion without the long period solution.Or, finally, we may have obtainedgood alignment within each CDP

this physical analogy, the statics arelike molecules, and the algorithm al­lows the "temperature" of the systemto change while keeping the"molecules" in thermal equilibrium.Although this method is nonlinear, itpreserves the surface consistent ap­proach of the other methods.

Rothman also uses the concept ofstack power, introduced by Ronenand Claerbout (1985), to measure theeffectiveness of his method. Stackpower is a po\\'(~r estimate based on thesum of the squared amplitudes of theCMP section. The best estimate of thestatics is thus the solution which max­imizes stack power. This method util­izes the stack power concept by firstrandomly generating sets of statics.After the random statics have beengenerated, they are applied to thedata. and a stack is generated. If stackpower is plotted as a function of itera­tion, a point is reached at which thetrue statics suddenly crystallize. Thisis best illustrated by an example. Theexample is taken from the overthrustbelt of Wyoming. Figure 21 (a) showsthe input stack with no statics applied.Figure 21 (b) shows the result after1000 iterations of the method, and theresult is actually worse than the firststack. However, Figure 21 (c) shows thefinal result after 1835 iterations, and is adramatic improvement. The "crystaliza­tion" of the solution happened at about1200 iterations, as can be seen by thestack power in Figure 22.

---------- 27 _

SUMMARY ANDCONCLUSIONS

(c)

51

(b)

5

are not too large and may have to usesome datum other than a perfectly flatone, such as a floating datum, in com­plex areas. In extremely complexareas, the approximations inherent inthe vertical static correction may betoo great, and we may have to use awave-equation datuming method(Berryhill, 1979, 1984, 1986). Thismethod, which was not discussed inthis paper, is commonly used in thecase of a complex marine water bot­tom.

A second key point is that no onemethod that we have discussed cansolve the complete statics problem. Ifwe look at this from an historicalperspective, there have been differentperiods of time in which geophysiciststhought the problem was solved. Forexample, in predigital days, fieldstatics and refraction statics were con­sidered to be the complete solution.Then, in the wake of the success thatresidual statics programs had whenthey were first developed (mid tolate 1970's), it was felt that the statis­tical methods alone were the answer.

However, the consensus now is that

(repealed six tillles). alld (c) is tire stack with 110

statics (repeated six tillles). Notice the phasediSTOrtion 011 tire stacked traces ill (b).

15

(a)

10o

100 -

300 -

4-00

-By looking at the stacked data.Structural and stratigraphicvalidity should be upheld.

In this paper we have had a brieflook at the subject of statics correc­tions in seismic processing. We havereviewed the assumptions behind thestatics correction, the methods used tocompute these corrections, theproblems encountered with our as­sumptions and procedures, and theeffectiveness of statics corrections inthe overall processing flow. Let ussummarize the key points.

First of all, the assumption that thestatic correction is a vertical shift ofthe seismic trace is an approximationto reality. This is equally true of theNMO or dynamic correction, which isthe companion to the static correc­tion. Despite the approximate natureof the static correction, however, itworks quite well. We must be carefulto pick a datum that is close to thesurface so that the statics corrections

Fig. 23. n,l' effect ofhighji'equellcy slatics Oil/hestacked trace. where (a) is a twe/l'e trace gatlrerafterNMO. hili wi/Ir sla/ics. (b) is tire stack of (a)

gather without taking proper care tomake sure that each stacked trace hasthe proper physical relationship withthe other stacked traces.

Let us first look at the structuralproblem. The fact that near surfaceeffects can cause problems in the longperiod statics solution suggests thatthese long period anomalies can bepropagated through the seismic sec­tion. In most cases, the fact that thesame structure is visible on allhorizons will indicate to the inter­preter that the structure is due to astatic. However, the structure couldbe the result of a very late intrusion,and therefore it would be preferable toremove the apparent structure if it isindeed due to statics.

As we have seen, the best way toremove such a structure is to performa refraction statics analysis, sincereflection statics can not resolve anystatic with a period longer than a cablelength. Figures 12 and 13 showed theremoval of an apparent structureusing such a refraction statics method.Notice that the seismic events are nowreasonably flat. While this is quiteuninteresting, it is highly preferable todrilling an incorrect structure.

Next, consider the stratigraphiccase. Stratigraphic interpretation in­volves the interpretation of subtlechanges in reflection character. Weoften consider that the main process­ing consideration for a stratigraphi­cally correct interpretation is thewavelet processing technique used.Equally important is the high frequen­cy statics solution. If this solution isnot correct, there will be a misalign­ment of the reflections across eachCDP gather, resulting in the destruc­tive interference of reflection charac­ter. A poor alignment of the traceswithin a CDP gather can result in lossof reflection continuity, and thereforea loss in interpretibility of the section.Figure 23 shows a simple example ofthis loss of reflection continuity.

Finally, how do we check to see ifour statics solutions is correct? Thereare three main ways in which to checkthe validity of the solution:

- By looking at the statics them­selves. The receiver and shotstatics should contain the sameinformation at the same surfacelocations. That is, the solutionshould be surface-consistent.

-By looking at the prestack data.Events should be flat prior tostack.

___________________29 _

each method has its own place in ad­ding to the complete statics solution.Field and refraction statics supply thelong wavelength component, whereasresidual methods supply the shorterwavelength part of the solution.

In summary, no single techniquewill completely solve the statics prob­lem. It is only by iterating through anumber of methods, and by carefullyinterpreting the solutions, that ageologically valid seismic section canbe produced which will aid the inter­preter in his or her quest for thegeophysical holy grail, finding oil!

Rothman, D.H., 1985. Nonlinear in­version, statistical mechanics, andresidual statics estimation:Geophysics, V 50, p. 2784-2796

__~1986. Automatic estimationof large residual statics correc-

tions: Geophysics, V 51, p. 332­346.

Wiggins, R.A., Lamer, K.L., andWisecup, R.D., 1976. Residualstatics analysis as a general linearproblem: Geophysics, V 41, p.922-938.

TE ORKANDTECHNOLOGY

COMBINING TO PRODUCE

STATE OF THE ART/EISMIC DATA PROCESSING

/,,

______---!.1986. Submarinecanyons: Velocity replacement bywave-equation datuming beforestack: Geophysics, V 51, p. 1572­1579.

Chun, J.H., and Jacewitz, CA.,1981. The first arrival time sur­face and estimation of statics:Presented at the 51st AnnualMeeting of the Society of Explora­tion Geophysicists, Los Angeles.

1984. Wave-equationdatuming before stack:Geophysics, V 49, p. 2064-2066

Barry, K. M., 1967. Delay timeand its application to refractionprofile intelpretation, in SeismicRefraction Prospecting: A. W.Musgrave, Ed., S.E. G., Tulsa, p.348-361.

Bel'lyhi//, J.R., 1979. Wave-equationdatuming: Geophysics. V 44, p.1329-1344.

REFERENCES

Hampson, D., and Russell, B., 1984.First-break intelpretation usinggeneralized lineal' inversion: Jour­nal of the C.S.E.G., V 20, p. 45­54.

Hawkins, L. V, 1961. The reciprocalmethod of routine shallow seismicrefraction investigations:Geophysics, V 26, p. 806-819.

-INVEST-

-SIERRA WORKSTATION-

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-GREEN MOUNTAIN REFRACTION STATICS·

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Palmer, D., 1981. The generalizedreciprocal method of seismicrefraction intelpretation: S.£. G.,Tulsa. .~~~

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Ronen, J., and Claerbout, J.F., 1985.Swface-consistent residual staticsestimation by stack-power maxi­mization: Geophysics, V 50, p.2759-2767.

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