Read Predictive deconvolution in shotreceiver space text version
GEOPHYSICS,
VOL. 48. NO. 5 (MAY
1983); P. 515531, 22 FIGS.
Predictive
deconvolution
in shotreceiver
space
Larry Morley*
and Jon ClaerboutS
spread in seismic data processing that the algorithm may fairly be called the bread and butter of the industry. Previous refinements to predictive deconvolution have centered around the socalled Noah' model which adds the effects s of lowamplitude surface multiples to the waterbottom reverberations (~Riley~ and Claerbout, 1976; Lee and Mcndel, !~981)? Such an approach is limited to zero offset and does not directly deal with the seafloor reflector. This horizon usually has a much stronger influence on the multiples than all other interfaces in the section. Zerooffset models do not address the fact that the water column reverberation response at the shot and at the geophone may be quite different. To deal with the problem of nonzero offset, Estevez (1977) developed a slant frame theory of multiple reflections. His simple theory, without diffractions, included the effects of wide offset and a narrow range of dip. The more general theory, including diffraction terms, allowed for a broader range of dips. An even simpler step beyond the strictly 1D assumptions of predictive deconvolution is a model which relaxes the zerooffset, zerodip requirement for primaries yet assumes vertical incidence propagation of multiple wavefronts in the water. This assumption is poor for water bottom multiples, but it is often surprisingly good for peglegs. This is because the velocity contrast at the seafloor is often so big that a small angle of incidence in the water can imply a large propagation angle in the sediments. Such a model is introduced here as a SplitBackus (SB) model. Backus (1959) developed a theory of multiple reverberation which laid the groundwork for predictive deconvolution. The splitting phenomenon was first noted by Levin and Shah (1977), and refers to the two different periods in the multiple reverberation train resulting from possibly different water depths at shot and receiver locations. Preliminary experience with the SB model showed that there was much to be gained by separate treatment of the water column reverberation times at shot and geophone locations. Observation of field data movies, however, provided the motivation for an extension to the SB approach called "seahoorconsistent multiple suppression." Movie frames of successive commonshot gathers revealed coherent motion of anomalous packets of reflected energy. This motion always stands out on any seismic dataland or marineeven over "pancake" geology. In marine offend shooting geometry, the motion runs from near to far offsets as
ABSTRACT
Standard predictive multiple suppression techniques in marine reflection seismology usually resort to onedimensional assumptions about the underlying earth model. The methods presented here use a multiple model which assumes vertical incidence propagation in the water layer, yet relaxes common assumptions of zero offset and zero dip. In particular, different reflectivities and water depths are assumed at source and receiver locations. One of these methods, seafloorconsistent multiple suppression, models each seismic trace as a convolution of an average frequency response with anomalous shot, geophone, midpoint, and offset responses. In the logfrequency domain, this becomes a separable, additive model which can be solved by linear leastsquares techniques. The anomalous amplitude responses are solved for each frequency in "shotreceiver" space with frequency as the outer loop of the algorithm. Once the seismic data set has been resolved into average and anomalous amplitude responses, the multiple reverberation response is identified with the product of the anomalous shot and geophone responses. Since one can argue on physical grounds that the reverberation response for any particular trace must be minimum phase, it suffices to solve only for amplitude responses and ignore phase contributions. The methods are appliedto adeep water marine seismic line from the Flemish Cap area of the Labrador Sea with extremely encouraging results.
INTRODUCTION
Exploration geophysicists are quite familiar with predictive deconvolution as a tool for marine multiple suppression (Peacock and Treitel, 1969; Robinson and Treitel, 1980). There is a rich literature surrounding this topic which probes many facets of onedimensional (1D) wave propagation theory. Predictive deconvolution uses the WienerLevinson (WL) recursion to solve a system of normal equations derived from a leastsquares predictionerror criterion. Use of the WL recursion is so wide
Manuscriptreceived by the Editor March 22, 1982; revised manuscript received September 7, 1982.
*Geophysics Research and Development, Aramco, Dhahran, Saudi Arabia. SDepartment of Geophysics, Stanford University, Stanford, CA 94305. 80 1983 Socieiy of exploration Geophysicists. All~rights reserved.
515
516 SPLIT BACKUS MODEL
Morley
and Clambout
=y
FIG. 1. The SplitBackus model. Water depth and seafloor reflectivity are functions of shot and geophone location. Propagation in sea layer is assumed near vertical.
the movie frames run from earlier to later shot gathers. Furthermore, the packet velocities are correlated with the speed of the boat. These observations can be explained by assigning a laterally varying reflectivity to the geology and led to the view that the texture of the seafloor, as well as its depth, is a firstorder effect. This suggested experimenting with a phenomenological model which allowed each part of the seafloor to have its own spectral signature, permitting a statistical exploitation of the spatial organization of the pegleg multiple response. An important byproduct of a seafloorconsistent representation is that the number of free model parameters describing the multiples is vastly reduced in comparison with tracebytrace predictive deconvolution. This means that the primary reflections emerge relatively unscathed after multiple suppression.
NEAR OFFSET SECTION
(iiS = 26M.)
0
SF
PM1
PM2
FIG. 2. Nearoffset section, offshore Labrador. Offset distance is about 9 shotpoints. Shotpoint spacing is 25 m. Labeled events are SFseafloor, BMlfirst bottom multiple, BMZsecond bottom multiple, Pprimary, PMlfirst pegleg multiple, and PM2second pegleg multiple.
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Space PREDICTION OPERATOR
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One of the proven techniques of modeling surface effects in reflection seismology is the "surfaceconsistent statics model" (Taner et al, 1974; Wiggins et al, 1976). Historically this has been used to smooth out tracetotrace static time shifts, permitting a more coherent stack. Recently, Taner and Coburn (1980) introduced the closely related idea of a surfaceconsistent frequency response model to handle the statics problem. They modeled each seismic trace D(o) as a separable product of source S, geophone G, midpoint Y, and offset H responses in the frequency domain. In particular, they assumed that D(w) z S(o)G(w)Y(o)H(w). (1)
SPLITBACKUS
The classical 1D approach to modeling a waterconfined reverberation spike train uses the Backus threepoint operator (Backus, 1959). The assumption is that a reverberation filter with ztransform R(z) =
i=O
f(cz3' +_ =
Modeling the seismic trace in this way allows for removal of nearsurface absorption/transmission anomalies and facilitates conventional surfaceconsistent statics corrections. One might well ask if a similar approach to the multiple suppression problem can be taken. In what sense can one talk about a seafloorconsistent multiple suppression?
operates on the primary response twice prior to observationonce as the source energy passes through the seabottom into the sediments and a second time on return to the hydrophones. The reflection coefficient at the seabottom is c, (I c ) < l), and n is the twoway traveltime of the water column in time samples. The filter which cancels these reverberation poles is the threepoint dereverberation operator A Backus (1 + cz")2 = 1 + 2cz" + CZZ*". =
SF
Bh42
3. Constant offset section (COS) from same line as Figure 2. Offset distance is about 46 shotpoints. Notice that the firstorder pegleg multiple is now split into two distinct arrivals, PMls and PMlg.
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AUTOCORRELATION NEAR OFFSET SECTION
(AS = 26M.)
FIG. 4. Negative autocorrelation of neartrace section. The prominent dipping event gives a good estimate of local seafloor depth.
This analysis ignores the fact that both water depth and seafloor reflectivity can depend upon shot and geophone locations (Figure 1). The dereverberation operator, Asplitaac~us (1 + c,z"Xl + c,Z' = X (4)
where c, and cg are the seafloor reflectivities at the shot and geophone locations, is obtained by a direct extension of the reverberation model. The symbols z' and zB are the ztransforms of the vertical water column delays at the shot and the geophone. The chief approximation here is that the water reverberation paths are indeed nearly vertical (Figure 1). This is a poor approximation for wideangle water bottom multiples. It is, however, valid for many types of pegleg multiples. The reason the Split Backus (SB) model works where apparently more sophisticated techniques fail is because it recognizes that zs and zg may differ. SPLIT PEGLEGS: A DATA EXAMPLE The next two figures show that split pegleg multiples are an observable phenomenon on real data. Figure 2 is a near trace section from a line of offshore Labrador (Flemish Cap) data' on which two strong pegleg multiples can be seen cutting across the section between 2.5 and 3.5 sec. Figure 3 is a constantoffset section (COS) from the same line for an offset ` ourtesy of Amoco CanadaLtd. C
halfway down the cable (a separation of 45 shotpoints with this geometry). The firstorder pegleg multiple starting at 2.5 set on the left and running across to 3 set on the right is "degenerate" (unsplit) on the neartrace section but is split on the COS due to the seafloor topography. The maximum split is some 200 msec around shotpoints H&200. This occurs, as one might expect, where the seafloor has maximum dip, i.e., where the difference between seafloor depths at the shot and geophone positions is greatest. In order to make an SB prediction, we need to know the water depth at each shot and geophone location along the line. The water depth cannot be obtained from the first break times on the neartrace section since there is a substantial gun delay of 100 msecvarying by as much as 15 msec from shot to shot. The autocorrelation of the near trace section (Figure 4) however, does provide a good estimate of the seafloor reverberation time as a function of shot location. In Figure 4 the seafloor shows up as an event dipping down to the right. The flat events are associated with the shot waveform. The signaltonoise (S/N) ratio of the seafloor event is good enough to track automatically the water depth completely along the line. The prominent dipping event gives a good estimate of local seafloor depth. Figure 5 is a pegleg multiple model of the COS of Figure 3 windowed from 2 to 4 sec. It was obtained by delaying one copy of the COS by the twoway water time estimate at the shot location, delaying another copy of the COS by the estimated reverberation time 45 shotpoints down the line (the hydro
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Space
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In N .
PM
PM
v) 0 .
FIG. 5. SplitBackus multiple model of constant offset section obtained by superposing two independently delayed copies of Figure 3. The delay times are obtained from Figure 4. The predicted firstorder peglegs match the corresponding data peglegs to within onequarter of the dominant period.
phone locations), and superimposing the two data fields. The lowvelocity water bottom multiples were removed by dip filter preprocessing. The arrival times of the predicted firstorder peglegs match the data peglegs to within a quarter wavelength of the dominant seismic period (i.e., within 8 to 10 msec) across the section. The predicted peglegs consistently precede the data peglegs. This is to be expected since the actual reverberation paths for the constant offset section are at a slight angle to the vertical. Now that we' e seen that split pegleg multiples do indeed v exist and can be explained by an SB model, we turn our attention to the problem of suppressing these multiples, i.e., estimating the seafloor reflectivity c. One model which worked well for this purpose was a twochannel shaper filter. The prediction error
TWO CHANNEL
DECONVOLUTION
was minimized by simultaneous variation of & and tg.
FIG. 6. Block diagram for twochannel deconvolution of shot and geophone multiples.
520
PRMWY
m
N
PM19
PMls PRIMARY
In . 0
FIG. 7. Twochannel deconvolution of commonoffset gather of Figure 3. The split pegleg multiple is well suppressed and the primary near 3 set comes out better.
In equation (5) D is the observed seismic trace, M, = Dz" or , D delayed by the estimated water column reverberation time at the shot location, and M, = Dzg. The solution of the normal equations for this model (block diagram, Figure 6) requires inversion of a block Toeplitz matrix. Wiggins and Robinson (1965) and Robinson (1967) gave an efficient algorithm for this. This model ignores the term in ?sscj, that results from a complete expansion of equation (4). Thus it does not precisely account for second and higher order pegleg multiples. It is, however, consistent with a general philosophy of attacking the higher amplitude multiples first. Figure 7 is the result of applying the process depicted in Figure 6 to the same C' S example. The primarytumultipie O ratio is visibly better than the predictive decon result. The split pegleg is well suppressed across the section, and the primary around 3 set stands out better than after predictive deconvolution (Figure 8). This is chiefly because the total number of filter parameters was cut in half. As a bonus, the additional sym
metry of the block Toeplitz matrix reduced the computation time significantly over the time of standard predictive deconvolution. The technique just described requires the user to obtain an estimate of 2 and zg. In cases where the S/N ratio is fairly strong, such estimates can be obtained by automatic picking. If the S/N ratio is weaker, however, an unacceptable amount of human interaction may be required to pick zs and zg and the following method is more appropriate.
SEAFLOOR CONSISTENT I)ECOMPOSITJON MODEL We adopt a convolutional model very similar to equation (1). The only refinement is the addition of an average response factor A(w). This last term depends mainly on the average shot waveformsomething we will not attempt to deconvolve here. Our goal is to attenuate the water column reverberations,
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521
PREDlCTlvEDECON
(hS =25M)
PRIM UJ . N
PMlg
PMIS PRIM
FIG. 8. Predictive deconvolution of Figure 3. Basic filter parameters same as Figure 7.
leaving as much primary as possible intact. The model, therefore, is 4% Y>w) = S(& W(g, W)Y(Y, WW, c$+). (6)
Intuitively, we expect S to contain shot ghost responses, water reverberation effects characteristic of shot location, and residual shot waveforms. Receiver ghosts as well as water reverberations under the hydrophone locations are embedded in G. Any anomalous response characteristic of midpoint is handled by the Y factor. H is included to model variation of the seafloor reflection coefficient with offset. Taking logarithms on both sides ofequation (6) gives In D sgz In S, + In G, + In yS+y12+ In H~_~,~ + In A, (7)
where s denotes shot index and g receiver index. Equation (7) holds independently for each frequency and is a complexalgebraic equation. The re;rlLpart~contains !ogamplitudeinformation and the imaginary part defines the phase.
Ideally, we would like to get imaginary as well as real solutions to equation (7). This was attempted but frustrated by the issue of phase unwrapping. The phase unwrapping problem (Tribolet, 1979) is encountered whenever complex logarithm processing of seismic data is attempted. There are ambiguities of 2x in a discretely sampled phase spectrum when the slope of the phase curve becomes large. This is commonly the case in any band where the S/N ratio is poor, such as the very lowfrequency band. For deep water, deep enough to allow discrimination between the bubble pulse and the seafloor reverberation response, it turns out that a considerable amount can be done with only a knowledge of the amplitude responsesin equation (7). There are a number of a priori constraints we can place on the phase. In particular, the shot and receiver reverberation effects are known to be causal and have timelimited inverses. As we shall seelater, WienerLevinson spectral whitening techniques seem to handle phase estimation correctly.
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REDUCTION OF FREE PARAMETER COUNT
Morley
and Claerbout
Similarly,
The standard predictive method of multiple suppression involves estimating a WeinerLevinson (WL) inverse filter for each seismic trace. The WL filter is a stationary model fit to the data. If there are N, shotpoints, N, offsets, and N, adjustable WL filter points for each trace, then the total number of free parameters describing the multiples is N, N, N, In all model fitting it is desirable to describe the phenomenon of interest with as few free parameters as possible. In predictive multiple suppression, these parameters are typically the values of a matched filter or some other crosscorrelation filter. There is a temptation when doing tracebytrace deconvolution either to use a large number of filter parameters in a timestationary model or to use a timeadaptive filter with an adaptation time that is small with respect to the multiple reverberation period. In either case we run a very real risk of "throwing the baby out with the bath water" and attenuating the primaries as well as the multiples. The separable model of equation (1) gives a very parsimonious description of the seismic data set. It reduces the number of independent filter parameters from a total of N, N, N, (as in standard tracebytrace predictive deconvolution) to (N, + N, + N, + N&N,. In most cases this is a very substantial reduction. If there are fewer free parameters involved in the multiple suppression, then we are less likely to attenuate the primaries while suppressing multiples. This improves the poststack primarytomultiple ratio. The problem of estimating leastsquares solutions for the logarithms of S, G, Y, and H in equation (7) is completely analogous to the classic residual statics estimation problem. The problem is overdetermined (since there are N, N, equations for only N, + N, + N, + N, unknowns), but it is known to be underconstrained in the longwavelength components of the solution. This second aspect of the problem will become apparent when we examine real data solutions. The complexity of this problem is increased by a factor of nf over conventional residual statics estimation, where nf is the number of frequency planes over which we wish to do the decomposition. The method of solution follows.
LEASTSQUARES MODEL DECOMPOSITION
(1 lb)
Making the index transformation
M=2;
s+g
s9
n=2'
Y,H,)'
(12)
we can rewrite equation (8) as E=CC(D,+,.,.S,+,G,.m " (13)
Y, Zeroing aE/L7 and SE/SH, in equation (13) gives Y,=&D,+,,,~.S,,, h n and H, = $1 (D,+,.,,m, Ym  S,+,  G,_,  Y,). (1 Id)  G,m,  H,) (llc)
Equations (1 la))( 1 Id) tell us what we may have intuitively expected : the L, norm solutions to the decomposition problem can be obtained by averaging model residuals over the direction orthogonal to the response component of interest. It is worth noting in passing that L, solutions can be obtained by replacing the averaging operators in equations (11) with median operators. The solution strategy involves cycling through equations (11) for all desired values of k, f, p, and 4 until convergence at the spatial wavelengths of interest is obtained. (This turns out to be equivalent to solving the implied system of equations by GaussSeidel iteration.) The outermost loop of the algorithm is over frequency. An alternative approach to this problem involves working in the midpointwavenumber domain rather than the space domain (see Appendix). This is similar to Marcoux' (1981) s approach to the statics problem. In the statics case, the problem decouples in the wavenumber domain, allowing an explicit solution. Although our problem does not fully decouple with this approach, it does make the computations better organized.
After extracting the average magnitude In A(o) from the s  g plane, our problem is to lind min E(S, G, Y, H), where E = 11 (D,,  S,  G,  yS+g,z Hsg,#.
DECOMPOSITION
OF FLEMISH CAP DATA
s 9
(8)
In equation (8), D, S, G, Y, and H are redefined to be the logmagnitudes of their values in equation (7). The s, g subscripts range over the appropriate regions of the s, g plane. The minimum coincides with the vanishing of the partial derivatives of E with respect to the unknowns S, , G,, Y, , and H, Setting, for example,
_=() 8%
dE (9)
yields 1 (D,,  S,  G,  yk+9,2 H,_,,,) 9 or = 0, (10)
Sk= + 1 U',  G,  Ktg,z  Hlrg/z , 1.
9 9
(lla)
To test the concepts presented to this point, we return to the Flemish Cap data set, partially displayed in Figures 2 and 3. The data are both 48 trace and 48 fold with 25 m shot spacing and 50 m group spacing. The near offset is 273 m, the equivalent of 11 shotpoints. A 2 to 4 set window of two successivecommonshot gathers is displayed in Figure 9. It is more difficult to pick out the split pegleg multiples on gathers than on commonoffset sections. The arrow just above 3 set points to the nearoffset node of a split pegleg event. The steeply dipping events on the far offsets are water bottom multiples. The GaussSeidel algorithm described above was run on the inner 24 offsets of this line. The outer 24 offsets were excluded since they contained water bottom multiples which violated the vertical reverberation path assumption of the model of equation (6). Figures 1OaalOe show the logarithms of the amplitude responses of the various model components. They are plotted as functions of frequency (fraction of Nyquist) and inline coordinate.
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The average response A is a much smoother function of frequency and of higher amplitude than the residual responses. The offset response shows a trend of diminishing amplitude toward larger offset. It also has some color, but this differs markedly from the color of the other three responses, S, G, and Y. The main feature of the S and G plots is a trend toward higher frequencies from left to right. This is because of the increased water depth or reverberation time at the right end of the section (see Figure 2). The S and G plots have much more
lateral continuity and energy than the Y plot. They appear to be more sensitive to the seafloor than the midpoint response. This observation is reinforced by the fact that the best visual correlation between the S and G plots occurs when their two right edges are offset by 11 shotpoints (exactly the distance of the near offset). The useful bandwidth of the data appears to run from 5 to 50 percent of the Nyquist frequency. There is no horizontally coherent pattern in any of the residual model responsesoutside this passband.
0
24
\i
c
24
6
UJ .
N
FIG. 9. Two commonshot gathers from the Flemish Cap line shown in Figures 2 and 3. Group spacing is 50 m. The arrow just above 3 set points to a split pegleg multiple. Only the 24 nearest offsets on each profile were used in the decomposition.
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Morley
and Claerbout
AK?ZRAGE
OFFSET > (AH = 6Om. )
n
0
.
v)
d
FIG. t0ac. Logarith? plqts of (a) average, (b) offset, and (c) shot responses are displayed clockwise from top left. Plots (b) and (c) are dlsplayed after 4 lteratrons of equations (11). Maximum value of (a) is about 6 times the clip level of(b) and (c).
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GEOPHONE> (AG =
26m.)
MiLlPOINT> (AY= 25m.l
FIG. 10de. Logarithm plots of(d) geophone, and (e) midpoint amplitude responses are displayed top to bottom. Clip level is same as in plots lob and 10~.
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Morley
and Claerbout
SHOTERROR
FIG. 11. Estimate of the error in shot response.Most of the error energy is in the long spatial wavelength components, i.e., wavelengths greaterthan a cable length.
iter=l
ifer=3
tier=6
tier=7
FIG. 12. Logamplitude plots of a commonshotgather after correction for geophone,midpoint, and offseteffects.The quantity in expression (15) is displayedafter oddnumberediterations of equations(11). These tracesare summedto give the S estimatefor this particular shotpoint.
Predictive Deconvolution in ShotReceiver Space
CONVERGENCE PROPERTIES PROCESSING ALGORITHM
527
Since the previously posed model decomposition problem is underconstrained in the very longwavelength solutions, its solutions are only conditionally stable. Although lowwavelength instability was never observed to be a problem in practice, the potential for it is always there. It is possible, for example, for a large, positive zerowavenumber solution in S to be cancelled by an equally large but negative dc component in G, without changing the value of E in equation (8). The program implementing equations (11) for the Flemish Cap data resulted in arithmetic overflow when allowed to iterate several (more than 20) times. The algorithm can be bullet proofed by redefining the error functional to be E' = E + h
1 S,;+
c C; + c
9 P
Y; +
c Hi
Y /
(14)
with li > 0. This change adds a term of h to each of the denominators of equations (11). The plots of Figure 10 were made after four iterations of equations (11) in the order d, a, b, c. To probe the errors involved in this solution, we iterated four times, again from the same initial data, but in the order b, a, c, d. The difference between the two final S amplitude responses is displayed in Figure 11 at the same clip level as the plotsof Figure 10. This plot shows that the main errors are confined to long wavelengths. The corresponding difference plots for the other model components (not displayed) support this contention. We can therefore be confident that the decomposition is unique at wavelengths less than a cable length or, in this case, about fifty traces. Figure 12 gives some idea of the speed of convergence of equations (11). In this figure, logamplitude versus frequency for the traces of a particular commonshot gather (SP # 100) is displayed after oddnumbered iterations of the GaussSeidel algorithm. The displayed quantity is
D kegGg Y,,+g,, Hk,gn (15)
FIG. 13. Procedure for processing the seismic trace d, recorded at sth shotpoint and gth geophone location.
for one fixed shotpoint k,. The display changes with each iteration as G, Y, and H are updated. Convergence is obtained in about four iterations. The current estimate of the S response, obtained by averaging over the g coordinate, is displayed to the right of each gather. An obvious difference between the first and third iterations is the power level on the near (leftmost) offsets. The slanted events, which stack out in the sum trace, represent the water column reverberations at the geophone locations. They diverge with increasing offset, since for a fixed shotpoint the water depth increases with offset.
SUPPRESSING THE MULTIPLES
The plots in Figure 10 represent the anomalous spectra, characteristic of shot, geophone, and midpoint locations. Our goal is to remove anomalous color and power associated with seafloor location. The algorithm used for this is shown in Figure 13. We start with a data base of logamplitude spectra S, and G, obtained from the seismic data by equations (11). For each trace d,, we form the sum 5, + G,, multiply by 2, and exponentiate to obtain a model power spectrum. It is essential to pay careful attention to scale factors in the definition of S and G
up to the exponentiation stage since these have a nonlinear effect in further processing. Factors of 2 and log, 10 are most likely to cause trouble. Having obtained a power spectrum, we inverse Fourier transform to obtain an autocorrelation. We then use a standard predictive deconvolution (WeinerLevinson) algorithm to obtain a causal inverse filter asgto the constrained, anomalous power spectrum. The multiplesuppressed trace is the convolution ofa,, with d,, An important issue in the design of asg is the selection of prediction distance and filter length. For the Flemish Cap data the prediction distance in the WL procedure was chosen to be slightly less than the minimum seafloor time across the section. This was done so as to leave the phase of the bubble pulse unchanged. The filter length was chosen just long enough to include the maximum seafloor time across the section. The choice of prediction distance may become critical in shallow water, a case yet to be tested with real data. This point is discussedfurther in a later section. The next three figures demonstrate the final results of applying the above algorithm to the Flemish Cap data. Figure 14 is a 2400 percent brute CDP stack of these data. Figure 15 shows the result of the processing sequence of Figure 13. The primary near 3 set now stands out very clearly, and the multiples are virtually eliminated. A weaker primary some 200 msec below the strongest primary is also visible. The bubble pulse remains in the data, since it is contained in the average response which is not backed out. For comparison, Figure 16 shows a stack of the data after standard predictive deconvolution processing before stack. A gap of 200 msec was used with an operator of 120 msec. Although this conventional processing was successfulin removing the secondorder pegleg from the final result, it failed to do a good job on the firstorder pegleg. Figures 14 through 16 show a definite continuum of quality
PRlM 1ARr  Y) . N
0
mb
9 `
PRIM ARY
a.
c1
v)
.
m
FIG. 14. Flemish Cap 2400 percent brute CDP stack.
PRIMA!? Y ul hl PRINAR Y

m .
m
FIG. 15. Flemish Cap: CDP stack of processed data.
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from a worst case situation (Figure 14), to a better one (Figure 16), to a best result (Figure 15). MOVIES OF INVERSE OPERATORS A better understanding of these methods can be obtained from movies of the inverse operator as9 (Figure 17). In each frame of Figure 17 the vertical axis is time and the horizontal axis is shotpoint location. Each frame is displayed at a constant but successively larger offset and may be described by the heuristic formula Frame,_,(s, t) = Levinson (FFTl{e*[Ss(w)+G~(w' )) l (16)
the upper part since the geophone locations move into shallower water as offset increases for fixed shotpoints. It is not significant that there is motion in this movie since the underlying mathematical construction calls for two images moving with respect to each other. Even if random numbers were substituted for S, and G,, this movie would still give the effect of motion. What is significant, however, is the spatial coherence of the two seafloor branches and the fact that they coincide at zero offset. These results are so convincing that we no longer need to restrict ourselves to the best available data. THE SHALLOW WATER PROBLEM The Flemish Cap data are an example of a deep water multiple problem. By deep water we mean that the decay time of the bubble pulse z0 is less than the minimum water column reverberation time ~~(i.e., Q, < r,). A great deal of seismic data are recorded over shallow bottom areas such as the Gulf Coast, where q, > TV. Here also the multiples are usually more subtle than those of the Flemish Cap since the bottom is not as hard. Is this method of an:L use in these areas?We be!ieve that the answer is yes, but only if the bubble pulse is constant from trace to trace. In general, the anomalous shot response S will be composed
which summarizes the right half of Figure 13. It is difficult to appreciate all of the subtleties in the movie from a series of still frames; nevertheless, the main phenomena can still be seen. In viewing the movie, the eye can pick out two distinct components in the total image. One component is motionless and is associated with the shotpoint reverberations. (The shots are fixed in this frame of reference.) The other component appears to slide over the top tithe background~and~is associated with the geophone reverberations. The overall moving image looks like the pincers of a crab opening and closing as offset increases and decreases.The moving part of the pincer is
FIG. 16. Standard multiple suppression processing. Tracebytrace predictive decon was run before stack with a 200 msec gap and a 120 msec filter. The secondorder pegleg multiple was attenuated, but the firstorder pegleg remains quite strong.
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and Claerbout
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in ShotReceiver
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of two factors: an anomalous bubble response S,, and an anomalous water column reverberation response S, at the shot location. In the same spirit, we can model the anomalous geophone response G as a product of G, and G, If the survey is carefully controlled, then both S, and G, should be small with respect to A. The remaining factors S, and G, are both minimum phase on physical grounds. We are therefore justified in backing them out with WienerLevinson techniques. If this is done, we will be left with a data set whose color cannot be attributed to any seafloorconsistent response. This should aid further deconvolution by leaving an apparent bubble which is more consistent from trace to trace.
CONCLUSIONS
Taner, M. T., Koehler, F., and Alhilali, K. A., 1974, Estimation and correction of nearsurface time anomalies: Geophysics, v. 39, p. 441463. Tribolet. J. M., 1979, Seismic applications of homomorphic signal processing: New York, PrenticeHall (signal processing series). Wiggins, R. A., and Robinson, E. A.. 1965, Recursive solution to the multichannel filtering problem: J. Geophys. Res., v. 70, p. 18851891. Wiggins, R. A., Larner, K. L., and Wisecup, R. D., 1976, Residual statics analysis as a general linear inverse problem: Geophysics, v. 41, p. 922938.
APPENDIX MODEL DECOMPOSITION MIDPOINTWAVENUMBER
IN SPACE
This method is a generalization of predictive deconvolution which accounts for different seafloor reflectivities and depths at the source and receiver locations under the assumption of nearvertical incidence propagation in the water. Water bottom multiples which do not obey this assumption must be eliminated by preprocessing. Because of the method' similarity to the s surfaceconsistent statics problem, we call it a seafloorconsistent pegleg attenuation. Highlights of this method are listed below. (1) The SplitBackus reverberation response is automatically estimated, i.e., separate reverberation times are estimated for both shot and geophone locations. (2) The number of free model parameters is greatly reduced in comparison with tracebytrace predictive deconvolution. Thus, primary events are not as severely attacked. (3) The method is relatively easy to implement in typical industrial processing environments. (4) The method is robust in the sense that it does not alter the data if the model assumptions are invalid. The technique may also be useful with shallow marine data, although further development is needed to confirm this.
REFERENCES Backus, M. M., 1959, Water reverberationsTheir nature and elimination : Geophysics, v. 24, p. 23326 I. Claerbout, J. F., 1976, Fundamentals of geophysical data processing: New York. McGrawHill. Estevez, R. J., and Claerbout, J. F., 1982, Wideangle diffracted multiple reflections: Geophysics, v. 47, p. 12551272. Lee, J; S., and Mendel, J. M., 1981, Application of a surface multiple suppression filter to noisy reflection data: Paper presented at 51st Annual International SEC Meeting, October 19, Los Angeles. Levin, F. K., and Shah, P. M., 1977, Pegleg multiples and dipping reflectors: geophysics Y. 42. D. 957. Marcoux, M. 0.: f981, On &e resolution of statics, structure, and residual normal moveout: Geophysics, v. 46, p. 984993. Peacock, K. L., and Treitel, S., 1969, Predictive deconvolution: Theory and practice: Geophysics, v. 34., p. 155. Riley, D. C., and Claerbout, J. F., 1976, 2D multiple reflections: Geophysics, v. 41, p. 593. Robinson, E. A., 1967, Multichannel time series analysis: San Francisco, HoldenDay. Robinson, E. A., and Treitel, S., 1980, Geophysical signal analysis: New York. PrenticeHall. Taner, Mu.T., and Coburn, K. W., 1980, Surface consistent Estimation of source and receiver response functions: Presented at the 50th Annual International SEG Meeting, November 5, in Houston.
The advantage of working in midpointwavenumber (k,) space is that the S, G, and Y solution components decouple for each k,. Using the same symbols as before, we write the residual model error [equation (13)] as E = 11 [D(y, h)  S(y + h)  G(y  h)  Y(y)  H(h)12. h P (Al) Fourier transforming over the midpoint axis and using Parseval' theorem gives s
E =
C 1 11 @k, , h) ~~eikyhS"(k,)
h k,
 eikvh~(k,)  y(k,)  I
Now, for each k, we set
JIz.
(A2)
?fi!LLO
%*(k P)
giving
C h ,ik"h[fi _ eik,h$ _ eik,h c B H(h)] =
'
(A3)
0,
(A4)
or
g= + C {[&
h h
p _ H(h)]eW
 &
*%h}, (A5)
Similarly, (A6)
7=
G1
h h
[B _ eVS
_
e

WG _
I]
(A7)
and
H(h,) = t _
1 Re [@k,, h,)  eikyhS"
Y4
e 
[email protected] _ 81,
I <rPi
N,.
(A8)
Note that the unknowns s' G, and p are now (complex) scalars , instead of vectors, as they were in equations (11).
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