seismic refraction statics

8/21/2019 Seismic Refraction Statics

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A unified method for 2-D and 3-D refraction statics

M. Turhan Taner∗, Donald E. Wagner‡, Edip Baysal∗, and Lee Lu∗∗

ABSTRACT

Most of the seismic data processing procedures aredivided into 2-D, 2.5-D, crooked lines or 3-D versionsdictated by the differences in the shot and receiver con-

figurations. In thispaper,we introducea tomographic ap-proach that overcomes these geometrical difficulties andprovides stable statics solutions from picked first-breaktimes. We also show that the first-break picks containboth the short a nd the long wavelength surface statics.The solutions are ob tained by solving a set of general-ized surface-consistent delay-time equations using themethod of weighted least squares and conjugate gradi-ent. While iterating, ea ch fi rst-break pick is evaluatedto ensure its consistency with the least-squa res solution.B ased on consistency, we weight the traveltime picksa nduse them in the next iterat ion. These weights also serveas an indicator of anomalous picks to the user.

We show that long wavelength solutions leave largeresidual errors in the least-squares solutions. We alsouse the expected length of the Fresnel zone to d ifferenti-ate between short a nd long w avelength static solutions.

After removing the influence of long wavelength statics,we apply short wavelength statics to reduce the residualerrors further.

We demonstrate the validity of our unified method byapplying it to actua l dat a examples. The removal of bo thlong and short wavelength statics improves the initialdata set that produces a more consistent set of velocitiesand leaves only the short wavelength residual reflectionstatics, which are generally less than quarter wavelet pe-riod delays. This removes the most probable cause ofthe leg jump contamination and poor velocity estimatesfrom the residual statics computations, especially fromthe 3-D data .

INTRODUCTION

Interpretation of refracted seismic data precedes that of thereflected data inea rly applications. Thornburgh and A nselpub-lished t heir papers in 1930. H agedo orn (1959) ga ve a modifi edgraphical solution of the Ansel and Thornburgh method. Anumber of refraction papers edited by Musgrave (1967) werepublished by SE G . La ter, Pa lmer (1980) published the genera l-ization o f the reciprocal method . Ha mpson and R ussell (1984)and Schneider and K uo (1985) gave solutions ba sed on tra cingrays through refra ctor models. The tomogra phic computa tionswere introduced initially for wellbore to wellbore measure-ments to determine the velocity a nd image structure between

wells. Chun and Jacewitz (1980and 1981) discussed the contentof the first arrivals and presented a surface consistent solutionof refraction statics by the formation of time surfaces. Theyalso indicated that large error distributions often remained af-ter solutions have been obtained. These deviations attra ctedour att ention and led us to classify them a s surface generated

∗R ock Solid I mages, ww w.ro cksolidimages.com, 2600 South G essner Suite 650 H ouston, Texas 77063, - mt.ta ner@rocksolidima ges.com‡AMO CO, B ox 800, Denver, C O 80201. E-mail: dwa [email protected].∗∗Lea ding Seis Inc., 3707 Ingold S treet, H ouston, TX 77085. E-mail: [email protected] 1998 Society of Exploration G eophysicists. All rights reserved.

delays. Farrell and Euwema (1984) gave a least-squares solu-tion tha t computes refractor geo metries from first-break picks.Hampson and Russell (1984) presented a generalized linearinversion scheme to determine the refractor geometry frompicked first breaks. In this least-squa res method, Ha mpson andRussell achieve convergence by minimizing the difference oftraveltimes of rays traced from the computed model versus theactual first-break pick t imes. They constrain t he la teral geo-metrical variation of both the layer thickness and velocities tominimize the effects of the short wavelength undulations inthe picked t imes. The fi rst paper tha t applied t omography torefraction stat ics was given by C hon a nd D illon (1986). Theypresented G auss-Seidel iterative methods for computation of

statics by the delay t ime method. This method wa s describedinitially by Barry (1967), who pointed out that the delay timemethod (G ardner, 1939) can be generalized to arrive at a so-lution for long wavelength 2-D and 3-D refraction statics.

Thisgeneralization becomesa t omography problem inw hichthe refractor velocities and shot and receiver related delay


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Unified Refraction Statics

times are computed from the fi rst-break times with a rbitraryshot-receiver positions, a zimuths, a nd offsets. K irchheimer(1988) an d Tan er et a l. (1988) discuss the long wa velengt h re-fraction stat ics solutions.

Unlike the averaging approach to tomographic computa-tion of sta tics used by C hon a nd D illon (1986) and others, weuse a weighted lea st-squares a pproach in conjunction with theconjugate-grad ient algorithm (Wang and Treitel, 1973; Koehleran d Ta ner, 1985) to solve the large set of eq uat ions ar ising fro mthe tomographic problem. The method is general enough that2-D refraction lines, wide-line profi les, and crooked or slalomlines all become special casesof the 3-D problem.The weightedleast-squa res method is chosen to minimize the conta minatingeffects of unreliable first breaks. A lthough t he computationalprocedure assumes that the input da ta are correct, in reality,we must provide for mispicks and use procedures to minimizetheir effe cts.

SURFACE MODEL

Most papers presenting refraction statics computations as-sume a very simplified model of t he near surface. In a recent

paper (D ocherty, 1992), the layer f rom surface to t he fi rst re-fractor is defined as the weathered layer. This paper consid-ers the variation of thickness and velocity of this near surfacelayer and proposes a tomogra phic solution. D ocherty observesthat in areas of rapid surface elevation changes, the veloci-ties are determined more accurately while in other areas ac-curacy is not maintained. This can be explained easily by theEikonal relationship at the surface. The velocity and arrival-angle ambiguity can be solved only by independent observa-tions of the upward traveling wavefield along x- and z-axes.Since refracted w aves cannot resolve this overburden veloc-ity/propaga tion exit a ngle ambiguity, most papers a ssume tha tthe surface velocity is known or determined from other obser-vations. The near-surface depth models a re computed from

picked first-break times and from these overburden veloci-ties. However, the short wavelength undulations of the picktimes cannot be accurately modeled by ray tracing. Further-more, these short wavelength undulations are not generatedby the refracting interfa ce because their Fresnel zone (Sheriff,1991) a t the surface will be la rge. As we show schematically inFigure 1, waves generated from a smallfea ture on the refractorwill cover a wider area when they a rrive at the surface, hence

FIG. 1. Schematic diagram o f wa ves and Fresnel zone from asmall feature on the refractor.

they will be indistinguishable from longer features. Waves ar-riving from small river channels or washouts, like the one de-picted in Figure 2, will also diffra ct a nd since we pick only thefirst arrivals, some of the short wavelength details will be lost.If w e wish to recover these deta ils, we must use seismic imagingtechniques, rather tha n picked first-break times (Taner et al.,1992).

Our primary objective in this work is to determine thelong wavelength statics caused by the spatial variation of re-fractor geometries and the overburden velocities. Therefore,long wavelength solutions of refractor geometry without theshort wavelength details would be sufficient for our purposes.We cannot determine directly t he interval (overburden) ve-locity of the layer above the refractors from the first-breaktimes. The most convenient procedure is to compute normalmoveout (NMO) velocities from the reflected waves comingfrom the refractors and estimate interval velocities. E stimat-ing overburden velocities from direct arrivals generally isrisky,since most direct a rrivals a re a ctually refracted arrivals. O neof the more convenient but heuristic methods is to try sev-eral velocities a nd generate corresponding stacked sections,then choose the overburden velocity that produces the leastundulating reflectors. This approach was first presented byG eoSource in their Refraction Statics pamphlet in the early1980s.

FIG. 2. Waves arriving from a small river channel on therefractor.


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LONG AND SHORT WAVELE NGTH STATICS

Wave propagation t hrough the ea rth is controlled directlyby the structure of the velocity field. We measure the wavesarriving at the surface by receiver arrays at some predesignedintervals. D iscretely sampled continuous wavefi eld can be re-constructed up to some maximum frequency and minimumwa velength. Therefore we cannot a ccount for the perturbation

of the wavefi eld shorter than some wavelength. Sampling the-ory suggest t hat we need at least four samples per (shortest)wavelength to be able to estimate the amplitude and the phaseof the propagating wavefi eld.Furthermore, our ability to definethe velocity structure of the earth is also limited, on the orderof several hundreds of meters. Therefore, the combination ofthe limited defi nition of the velocities and discrete samplingof the wa vefield will confine us to defi ne the propagation withlonger wavelengths. We will call all of the disturbances below this limit “ the short wavelength statics.” D imensionally, theywill be on the order of fo ur to eight times the receiver interval.All of the longer wavelength near-surface effects will be clas-sified as the long wavelength statics. B ecause of the nature ofthe short wavelength statics, we claim that, they are generated

near the surface, basically effecting only the upward travellingwaves.

PRESENT WORK

In our work we divide the near surface into two zones. Thefirst zone, the w eathered layer at the surface, consists of verylow velocity material. Its surface and base are assumed to haveany type of undulations including the short wavelength undu-lations as shown in Figure 3. Since we measure the surfaceelevations, both its thickness and velocity are unknown. Thelayer below the weathered layer is more consolidated and itcontinues more or less uniformly down to the refracting layer.We a ssume this layer and the top of t he refracting layer varysmoothly in the spatial direction, both in thickness and veloc-

ity. This model fits with our observations and as we will show later it a lso fi ts with our computational model. As depicted inFigure 4, the observed first-break arrival times contain bothshort and long wavelength undulations. Because of the lengthof theFresnelzoneof wavesarriving fromthe refracting surface(as shown on Figure 1) only the longer wavelength undulationsof the fi rst-break times can be att ributed to the refracted fi rstarrivals. Therefore, the short wavelength perturbations mustbe generated in the surface weathered layer. We assign the

FIG. 3. The near-surface model.

long wa velength portion of the fi rst-break pick times to the re-fractor computation and the short wavelength portion to thenear-surface statics computation. The general model will bedefined as

t (n,m) = S (n) + R(m) +

[σ (k ) ·(k )]+ s(n) + r (m),

(1)where

t (n,m) = picked first-break time for n th shot andmth receiver location.

S (n) = d elay time at n th shot location,corresponding to the long wavelengthsource sta tics.

R(m) = d elay time at m th receiver location,corresponding to the long wavelengthreceiver sta tics,

σ (k ) = slowness at k th cell located along thehorizontal path between n th and m th surfacestations,

(k ) = subpath length in the k th cell of the pathbetween n th and m th surface sta tions. The

sum of the product of subpath length andslowness will give the traveltime along therefractor between the shot and thereceiver,

s(n) = short wavelength surface static at t he n thshot position n .

r (m) = short wavelength receiver static at the m threceiver position m .

The components of the tra veltime are shown on Figure 5. Weare also assuming that the picked traveltimes are corrected tothe surface by a dding the uphole time to the picked first-breaktime. In the case of deep hole dynamite, we may not have a

residual surface shot static component. However, experiencehas shown that the uphole times usually contain timing errors.Thus, we leave t he shot residual sta tics as one of t he unknownsto detect or guard against errors in uphole times.

INITIAL CONSIDERATIONS

D ata used for refraction statics computation a re the first-brea k picks. These picks, however, contain a number of errorsoriginating from several factors. One of the most common er-rorsis erroneous pick time. Thiscould occur for several reasons:

1) A simple leg-jump, usually one or two w avelengths aw ayfrom the good pick position,

2) A leg-jump caused by following the wrong refractor a t abranch point,

3) Leg jump caused by the uncertainty of the first break ofpoorly correlated vibratory data ,

4) Picking noise caused by the changing first-break charac-teristics or wavelet shapes,

5) Picking time error due to the errors in uphole times fordynamite data ,

6) Changing wavelet shapes from shot to shot.

Of the data problems, the uphole times and leg jumps arethe lea ding timing errors.


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NATURE OF FIRST-BREA K WAVELETS

We assume that first breaks usually are the arrivals fromshallow refractors. The refracted arrivals caused by their an-gle of emergence will have a phase rotation difference fromthe reflected waves beyond the critical angle. They a lso ha vea dispersive nature, therefore the rotation angle changes overdistance. For accurate picking, it would be ideal if the picks

could be mad e over shorter offset ra nges. Their shapes also de-pend on the source wa velet shape, which changes from shot t oshot on land records.This change leads to shot to shot pickt imeerrors. To a lleviat e this, first-break w avelet shaping, similar t othe seismic signature estimation and shaping, isperformed (SeeR obinsona nd Osman, 1996). Thisforces first-break wavelets tohavea uniform shape,hence the picks will occura t the samepo-sition of the wavelets. Vibratory data generally have the poor-est fi rst breaks. Wavelet shaping the fi rst brea ks to a minimumphase wavelet with a bandwidth providing the least precursorreverberation is a good procedure for successful automat ic andinteractive picking.

UNIFIED METHOD FOR STATICSCOMPUTATION FROM

FIRST-BREA K PICK TIMES

We start with the classical dela y time eq uation (Thornburgh,1930; Ansel, 1930; H ag edo orn, 1959) and de velop a n algo rithmfor estimating delay times for refractors, residual statics fromnear-surface anomalies a nd refractor velocities from pickedfirst-arrival times. From these estimates of residual statics, de-lay times, and refractor velocities, we compute a set of surfaceconsistent static corrections for each source a nd receiver lo-cation. Care is taken to cast the problem in a X ,Y , Z coordi-nat e system for applicat ion to 3-D da ta , wide-line profi les, andcrooked lines, as w ell as conventiona l 2-D linear profi les.

FIG. 4. A seismic record depicting the first-break times with bot h short and long w avelength undulat ions.

COMPUTATIONAL STEPS

To ensure the consistency of pick times, we shape the fi rst-brea k zones to a uniform wa velet shape. The optimum shape isa minimum-phase wavelet with the same bandwidth as the first-brea k wa velet. This is very importa nt particularly for vibratoryrecords with reverberant precursors. U niformly shaped fi rst-break wavelets produce lesspicking noise.D etails of automatic

picking using artificial intelligence techniques are given else-whe re (Veezhina tha n, 1990; Veezhinat ha n et a l., 1991; Tan er,1988). Here we assume the first breaks of shot records arepicked properly. By this we mean that the picked times areconsistent and have a minimum number of errors. These er-rors are assumed to be scattered throughout the data and a renot congested in a particular area. We will address the pickerrors further in our computation. The refraction statics com-putational steps include both the long and short wavelengthsolutions as fo llows.

FIG. 5. The travel pat h in the near surface model.


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LONG WAVELE NGTH SOLUTION

If the source is dynamite, then we modify the traveltime, inequation (1), by adding the uphole times u p(n) to the pickedfirst-break time,

T (n,m) = t (n,m) + up(n). (2)

Since near-surface elevat ion and the surface sta tics aff ect therefracted arrival times, we compute a smooth (consistent withthe wavelength separation of the short and long wavelengthstatics) near-surface elevation ˜ z(n) and an elevation correc-tion between the smooth surface and the actual surface ele-vation z(n). The pick times to the smooth surface are thencorrected,

t̃ (n,m) = T (n,m) − elcor (n) − elcor (m), (3)

where t̃ (n,m) is the modified pick times and elcor (n) is theelevation correction at n th surfa ce position. The elevat ion cor-rection at a location (n) is defined by

elcor (n) = ( z(n) − ˜ z(n))/v(weathering ). (4)

The more a ccurate surface wea thering velocity v(weathering )may be determined experimentally by applying elevation cor-rections with different velocities and choosing the one produ-cing the smoothest first-break arrival times.

We a ssume this corrected t ime now principally conta ins thesum of the delay timesa t shot and receiver positionsa nd the tra-veltime along the refra ctor in betw een. We compute the refra c-tor slownessesa nd thedelay timesby minimizing the traveltimeerror equations

FIG. 6. The generalized tra vel path across a rea l grid for a 3-D geometry.

ε =n

m

w(n,m) ·

t̃ (n,m) − S (n) − R(m)

−k

(σ (k ) ·(k ))

2, (5)

where w (n,m) are the weights of each pick time. These are

initially set equal to 1. In a computational procedure, we stopthe computation after a preset number of iterations (say 15to 20) and check the error for each pick. We determine themean error and the standard deviation. Thisgivesa n idea of theamo unt of errors in the pick times. Ba sed on the error sta tisticswe assign weights to each pickproportional to its error range.I feach error is given by “ ε(n,m),” and the threshold error valueis given as some percentage of the standard deviation “ ε̄,” thenwe compute the weights by the sigmoidal function,

w(n,m) = 1.0/[1.0+ (ε(n,m)/ε̄) p)], (6)

where p isa n even constant. Thisw ill range from 2to 8.Sma llervalues will be more tolerant than the larger values. Weightsgiven by the above expression vary between 0. and 1.0. It will

be equal to 1/2 for errors equal to the chosen threshold. In thefollowing iterations, we locate the picks with large errors andsmaller weights to isolate and reduce their effects on the fi nalsolution.

For crooked lines or for 3-D geometry, we modify equa -tion (1) to handle t ravelpaths across an areal grid. This 3-Dgeneralization becomes a tomography problem in which re-fractor velocities are computed from first-arrival tra veltimesreceived at arbitrary shot-receiver azimuths and offsets. Thegeneralized tra veltime/path model fo r 3-D geometry is shownin Figure 6. In this 3-D case, we subdivide the refractor surfaceinto regular recta ngular bins (cells), with a bin size of X a nd


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Y in the X and Y d irections, respectively. To each bin, w eassign a velocity a nd denote Ea st-West bin indices by n a ndnorth-south b in indices by m . Then, the slowness of t he refra c-tor in bin n,m is given by σ nm or (1/V nm ). The path from sourcelocation i to receiver location j is to be divided into subpathsd i j . Ea ch subpat h represents the individual path a cross a singlebin. Using superscripts to represent the source and receiverpositions and subscripts to represent the bin coordinates of thecomponent segments, the delay time equa tion for the 3-D casebecomes

T i j = S i + R j +

ji

d i jnm · σ nm . (7)

Our objective is to solve equation (7) for S , R , and σ . Wepropose an iterative approach that breaks the problem intotwo subproblems: one solves for the slowness σ , and the othersolves for the delay t imes S a nd R, as shown in the two stepsbelow.

1) Assume S i a nd Ri a re known and solve for σ values.2) Substitute values of σ into eq uation (7) and solve for S i

and Ri .

To solve f or slow ness σ in step (1), modify t he picked timesto

T i j = T i j − S i − R j . (8)

Eq uation (8) multiplied by the corresponding weightsis writ-teninthematr ixformas Dσ = T . To minimize the least-squa reerrors, in the conventional wa y, we determine a nd solve the fol-lowing equation

Dt Dσ = Dt T , (9)

where D t is the transpose of matrix D . The d imensions o f Dconsist of the number of picked times by t he number of binsin the area . The conventional solution to this very large mat rixequa tion is impractical. We solve equation (9) by the conjugategra dient m etho d (Wang a nd Treitel, 1973; Koehler a nd Tan er,1985). The advantage of this method is its robustness, speed ofconvergence, and its immunity to va rious insta bilities inherentin surface consistent eq uations.

Co nventionally used G a uss/Seidel iterat ive method has sev-eral disadvantages. Since the matrices involved in the solutionof statics equations have linear indeterminancies, direct solu-tion w ith G auss/Seide l metho d will not co nverge. To elimina tethis instability the main diagonal of the matrix must be in-creased by a small percentage. This, consequently, will act asa low-pass filter on the computed results, losing fine detailsof short wave-length results. Another problem is, that the re-sults are dependent on the initiation point of the computation.Computations starting with the source statics will differ fromthe ones starting with the receiver statics. Conjugate gradientmethod is a recursive solution of the whole set of equation,thus it will not be affected by the starting points. It is also im-mune from the insta bilities of the equat ions, and the results areminimum distance to t he starting values of the solutions, whichare usually taken as a set of zero values.

A similar procedure using the least-mean-square approachwith the conjugate gradient algorithm can be used to computethe source and receiver delays.

We have illustrated above how the 3-D refraction staticsproblem reduces to a formalism similar to the 2-D refractionstatics solution. This formalism can be considered a unifiedmethod for solutions of 2-D and 3-D refraction statics.

SHORT WAVELENGTH SOLUTION

After the delay times and the slownessesa re computed, their

effects are removed from the original uphole corrected picktimes. This will remove the relative eleva tion correction termsfrom the pick time values written as

t (n,m) = T (n,m) − S (n) − R(m) −k

[σ (k ) ·(k )].

(10)

Therefore t (n,m) now containsonly the nearsurface weath-ering la yer differential statics, which a lso contains the eleva-tion statics in a residual manner, and possible uphole time er-rors. It is interesting to note that t (n,m) is basically the errorterm from the long wavelength solution. Since the long wave-length stat ics were computed from only the long w avelengthsmoothed arrival times and by setting the delay times equalfor upward traveling (receiver) and the downward traveling(shot) wa ves, thus the resulting sta tics will be long wavelengthstatics. As above equa tion indicates that w e subtract these longwavelength delay times (statics) from t he original pick timesto ob tain the residual traveltimes with short wavelength distur-bances. We decompose this traveltime into shot and receiversurface (or short wavelength) statics. Note t hat we now sepa-rate the shot and the receiver statics. They are computed in asurface consistent manner by minimizing the equation

ε =n

m

[w(n,m) · [t (n,m) − s(n) − r (m)]]2. (11)

Here, aga in, we use weights in the same ma nner as the re-fraction statics computation to minimize the effects of erro-neous picks. A fter a ll computations a re completed, based onthe overburden and refractor velocities, we compute the timeit takes to go from the surface down to the refractor with givena overburden velocity and then back up to the datum levelwith the velocity of the consolidated deeper layers (Musgrave,1967). Applying these corrections to data will bring the shotsand receivers to the da tum surface, remove the elevation, nea rsurface weathered layer effects, and the effects of the slowlyvarying layer above the refracting layer.

2-D AND 3-D DATA EXAMPLES

Example1

Our first example is a deep hole dyna mite 2-D line. In thisex-ample, a tot al of 31 shots of 48-channel data w ith a length of 3 sand 2 ms sampling is used. The group interval is 220 ft (67 m),and the data is gathered at 6-fold. The raw stack with 110 ftcommon depth (CD P) trace interval without any statics ap-plication is shown in Figure 7a. We have used a weatheringvelocity of 3500 ft/s (1066 m/s) a nd subwea thering velocity of5500 ft/s (1670m/s). The stack with conventional elevation stat -icsis shown in Figure7b.The first arrivalsof the data was pickedusing a neural net work fi rst-brea k picking technique (Taner etal., 1988) a nd input to the refra ction sta tics computation. Thecomputed long and short period refractionsta ticswere then ap-plied to the d at a. Figures 7c and 7d show sta cks with the short


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or long period sta tics corrections, respectively. Figure 7e showsthe stack with both the short and long period statics correc-tions. To demonstra te the effect iveness of the refra ction stat -ics, examples of shot records with and without refraction stat icscorrections are displayed in Figures 8a and 8b, respectively.

Example2

The data in this 2-D example was acquired with a dynamitesource in an end -on shooting pat tern. It is a 48-channel record-ing, 3 s in length sampled at 4 ms. The group interval is 50m andit isga thered at 24-fold. The wea thering velocity is1500 m/s and

FIG. 7. (a) The raw stack witho ut stat ics correction. (b) The stack section with elevat ion stat ics correction.

the subw ea thering ve locity is 2300 m/s. The brut e sta ck sectionwith elevation sta tics correction generated a t 25 m CD P inter-vals is shown in Figure 9a. The events in the zone of interest ataround 2 s are d iscontinuous and apparently distorted by thestatics problems. After the refraction statics corrections, theevents are improved as show n in Figure 9b.

Example3This example shows a comparison of a 3-D data set with

and w ithout refra ction statics application. Weat hering velocityof 1300 m/s and a subweat hering velocity of 2900 m/s is


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used. The stacked trace interval is 50 m. The conventionalbrute stack section with elevation correction is shown inFigure 10a. The application of both statics (surface andrefraction) (Figure 10b) successfully has removed the longand short wavelength statics in the data. The effectiveness canalso be illustrated in the comparison of time slices at 940 and944 ms show n in Figures 11a a nd 11b, respect ively. To show effectiveness of convergence by this method, we computedthe error statics associated with the long and short wavelengthstatics computation. Figure 12a shows the error distributionof long wavelength refraction statics and Figure 12b shows

FIG. 7. (c) The stack section with long wa velength refraction statics correction. (d) The stack section with short wavelength refractionstatics correction.

the error distribution of short wavelength surface statics forthis example. The concentrat ion of errors a t the sma ller sta ticsmeans that we have obtained a stable solution for both thelong and short w avelength statics, and thus our model fits thedata within the error ranges indicated on the graph.

CONCLUSION

We developed a near-surface model that is more suitablefor surface and ref raction sta tics computat ion. We have shownthat the picked fi rst-break tra veltimes contain both refractionand near-surface w eathered la yer elevation a nd transmission


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FIG. 7. (e) The stack section with both short a nd long wa velength refraction sta tics correction.

stat ics. This was supported by b oth t he convergence of mat he-matical solutions and the a ppearance of the fi nal stack sections.The ability to determine and apply static corrections for thenear-surface weathering layer as well as the relative elevationstat ics reduces the residual reflectio n stat ics to residual levels,on the order of 1/4 of the wavelet width. With small resid-ual statics, initial velocity estimates are more accurate, henceelimina ting the velocity /sta tics interdepe ndency dilemma . Themethod outlined in this paper is equally applicable to 2-D,3-D and crooked or slalom lines. We have a lso shown tha t the

method is robust and tolerates a certain amount of mispickswithout ill-effects. We used the conjugate gradient solutionwithout making any provisions for instabilities inherent in thestatics equations.

ACKNOWLEDGMENTS

We wish to thank Duane Dopkin and Chung-Chi Shih ofCogniSeis for fi rst-break picking and statics computation onthe FO CU S system. We would a lso like to tha nk John Taner ofSeismic Research Corporation for preparing this manuscript.

REFERENCES

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FIG. 8. (a) The raw shot records. (b) The shot records afte r refraction stat ics correction.


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FIG. 9. (a) The brute sta ck section. (b) The section with refra ction statics correction.


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FIG. 10. (a) A n inline section of a 3-D brute sta ck. (b) The inline section af ter refra ction stat ics correction.


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FIG. 11. (a) Time slices of the 3-D brute sta ck. (b) The time slices after refra ction stat ics correction.


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FIG. 12. (a) Error distribution for long wavelength refraction statics computation. (b) Error distribution for short-wavelengthrefraction statics computation.


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FIG. 12. (Continued.)

seismic refraction statics

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