Volume 38
`
`August 1973
`
`Number 4
`
`GEOPHYSICS
`
`EFFECT OF NORMAL
`
`MOVEOUT
`
`ON A SEISMIC
`
`PULSE?
`
`J. W. DUNKIN,*
`
`APU‘D
`
`I’. K.
`
`I,EVIN*
`
`Using a synthetic seismogram as input, normal
`moveout correction stretches a reflection pulse
`in such a way that the spectrum of the pulse is a
`linearly compressed version of the uncorrected
`pulse spectrum. The amount of compression de-
`pends on to, the source-detector separation, veloc-
`ity, and the rate at which velocity varies with to.
`
`The amplitude of the spectrum is increased by
`the same factor that expresses the spectral com-
`pression. As a result, the summed pulse from a
`CDP stack is richer in low frequencies than one
`might anticipate and has a smaller signal-to-
`noise ratio than the square root of the number of
`traces in the stack.
`
`INTRODUCTION
`Before seismic traces corresponding to different
`source-detector separations can be stacked, the
`traces must be corrected for normal moveout
`(NMO).
`In a recent paper, Buchholtz (1972) dis-
`cussed the signal distortion
`that results from
`application of NM0 correction. Buchholtz was
`primarily concerned with correction of intersect-
`ing reflections; his discussion was qualitative.
`In
`this paper we shall be concerned with the simpler
`situation of isolated pulses; our work leads to an
`expression describing quantitatively
`the NM0
`stretching of nonintersecting signals. We shall
`also show that the increase in low-frequency con-
`tent known to result from stacking is greater than
`one might suppose.
`
`DERIVATION
`
`OF THE NM0
`
`STRETCHING
`
`EQUATION
`
`Consider a seismic trace from a detector a dis-
`tance X from the source. We’ll assume that the
`interval velocity depends only on depth, or
`equivalently,
`to, where to is the time recorded for
`X = 0. The stretching of a reflected pulse caused
`by application of NM0 correction is illustrated
`
`in Figure 1. In that figure, a pulse centered at LA
`is moved to a time to and stretched. Since NM0
`correction uses the relation
`
`tE = t: + x’/v”(t,),
`
`we know that t is transformed into to in a non-
`linear manner. However, we’ll assume the pulse
`duration is small compared with to or t. Hence,
`we’ll consider a transformation of variables that
`describes a localized stretching of the pulse at la.
`We define variables r=t--ta
`and ro=to-&,a
`(see
`Figure 1) and study the transformation of r
`into 70.
`We expand lo=&,(t) in a Taylor series around,
`t=ta.
`
`tit0
`to = tOA + -dT
`
`(t - fA)
`IA
`
`1 &o’
`,il;i
`
`+r
`
`(t -
`t.4
`
`tA)’
`
`’ “.
`
`(2)
`
`In terms of r and ro, we have
`
`t Manuscript received by the Editor September 22, 1972; revised manuscript received November 28, 1972.
`* Esso Production Research Co., Houston, Texas 77001.
`@ 1973 Society of Exploration Geophysicists. All rights reserved.
`
`635
`
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`
`Dunkin and Levin
`
`/
`
`/
`
`w
`
`t oA
`
`Uncorrected
`
`trace
`
`NM) corrected
`
`trace
`
`* to
`
`_T
`
`-
`
`t
`0
`
`0
`
`-
`
`toA
`
`FIG. 1. Stretching of a pulse by NM0 correction.
`
`3100 F+
`
`1600
`
`4850
`
`7500 wee
`
`8000 ftlrc
`
`9000 wrc
`
`8500 wac
`
`10,000 h/s%
`
`dto
`70 = -
`dt
`
`r+--
`IA
`
`1 &?a
`
`72-j-....
`2 tiv 1*
`
`(3)
`
`We’ll retain only the linear terms in equation (3).
`From equation (i), we find
`
`dto
`=a=-
`_
`dt 1.4
`
`tA
`
`I-_--
`
`X2
`
`t0A
`
`(
`
`vto.4
`
`dV
`
`-l
`
`‘It0
`
`I
`)
`f0A
`
`.
`
`(4)
`
`is, in this approximation, a
`Our transformation
`simple uniform dilatation of time given by
`
`5000
`
`r. = ar.
`
`(9
`
`Equation (5) says that if g(r) is the uncorrected
`pulse and go(ro) is the NM0 corrected pulse, they
`are related by
`
`go(r0) = g(rola).
`
`(6)
`
`if we use a tilde to designate a Fourier
`Further,
`transform, we have
`
`go(f) = S m
`
`--m
`
`go(To)e-2"ifrQh0
`
`=
`
`=
`
`go(f) =
`Equation (7) shows that the spectrum of the
`NMO-corrected pulse is compressed and multi-
`plied by the factor a.
`
`VERIFICATION
`
`OF THE NM0
`
`STRETCHING
`
`EQUATION
`
`To verify
`
`the NM0
`
`stretching predicted by
`
`FIG. 2. The subsurface assumed in the generation
`of the synthetic seismogram.
`
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`
`(6), and (7), we generated a
`equations (4), (j),
`simple synthetic seismogram for the section of
`Figure 2. The section was chosen so that the re-
`flections, all primaries, did not intersect for the
`spread length chosen. Our input pulse was a sym-
`metrical Ricker wavelet with a spectrum peaked
`at 30 hz. The resulting seismogram appears as
`Figure 3. After NM0 correction, we recorded the
`seismogram of Figure 4. The NM0 process in-
`volved
`linear
`interpolation on 2-msec sampled
`data.
`We considered two pulses. Both were for the
`largest source-detector separation of 9600 ft. One
`pulse had a to of 0.814 set; the other pulse, a to
`of 1.214 sec. Figures 5 and 6 are the spectra of
`these pulses. Approximating d V/dt,, from
`Figure
`
`2, we used equations (4) and (7) to rescale the
`spectra to the spectrum of our unstretched wave-
`let. Figure 7 is the result. The agreement is re-
`markably good.
`Before proceeding, we must warn the reader
`that our figures illustrate extreme cases. The re-
`flections whose spectra appear in Figures 5 and 6
`are from interfaces at depths of 3100 it and 4850
`ft, as recorded by a geophone 9600 it from the
`source. To avoid excessive pulse distortion, a
`geophysicist would likely kill the part of the trace
`that included these pulses.
`
`EFFECT OF NM0
`
`STRETCHING
`
`ON CDP STACK
`
`When CDP data are stacked, signals with dif-
`ferent source-detector separations are summed
`
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`
`638
`
`Dunkin and Levin
`
`*-
`
`5-
`
`4-
`
`2-
`
`0
`
`0
`
`FIG. 5. Spectrum of an NMO-corrected pulse: ta=0.814 set and X=9600 ft.
`
`that the
`after NM0 correction. We anticipate
`stacked pulses will be broader than pulses uncor-
`rected for NM0 but narrower than the far-trace,
`NM0corrected
`pulses. We might guess also that
`the summed pulses will be stretched versions of
`the original uncorrected pulses, but we have no
`assurance that this will be the case. Figure 8 is a
`suite of 12-fold CDP traces. We’ll examine traces
`23 and 24, which are full stacks. For record times
`greater than 1.5 set, the reflections do resemble
`the symmetric
`input pulse. At shorter record
`times, the pulses are distorted slightly, presum-
`ably because NM0 correction did not line up the
`central peaks of the pulses precisely. The reflec-
`tion between 0.8 and 0.9 set also suffers because
`some of the individual pulses that were summed
`were clipped.
`How much broader, i.e., how much lower in fre-
`quency than the original pulse, are the stacked
`pulses? To answer, we turn once more to the
`spectra. We know that the spectrum of a stacked
`pulse will have its maximum at a frequency lower
`than the 30 hz of the uncorrected pulse. What we
`
`might not anticipate is that the spectrum of the
`stacked pulse is not simply the sum of frequency-
`compressed, normalized spectra, i.e., spectra with
`the same maximum amplitudes, for the pulses
`being stacked. The sum of frequency-compressed,
`normalized spectra, shown in Figure 9, peaks at
`18.5 hz and is broad. The spectrum of the CDP-
`stacked pulse (Figure 10) peaks at 16.5 hz and
`is narrow.
`Why
`is the actual stacked-pulse spectrum
`richer in low frequencies than the spectrum we
`expected? When we stretch a pulse without reduc-
`ing its amplitude, we increase the area included
`within the pulse or, equivalently, add energy to
`the system. This energy appears at low frequen-
`cies. In the case of sampled data, the stretched
`pulse develops gaps that are filled by interpolation
`of extra samples, a process again equivalent to
`adding energy. In equation (7) we show that the
`spectrum of an ru’MO-corrected pulse has its
`amplitude multiplied by the stretch factor a.
`Thus, the pulse in Figure 4, with to=0.8 set and
`geophone distance of 9600 ft, was stretched by a
`
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`
`Normal Moveout
`
`639
`
`c
`
`FIG. 6. Spectrum of an NMO-corrected pulse: to= 1.214 set and X=9600
`
`ft
`
`stacked record of Figure 8. Overall, the agreement
`factor of 2.6. Hence, the amplitude of the spec-
`is excellent. Presumably, the discrepancy at very
`trum was multiplied by 2.6 before adding the
`low frequencies results from the effect of clipping
`spectrum into the stack displayed in Figure 10.
`In Figure 11, we have superimposed on the the- mentioned above.
`That stacking results in pulses richer in low fre-
`oretical spectrum of Figure 10 the values found
`by frequency-analyzing a pulse from the CDP-
`quencies than might be anticipated
`is not gen-
`
`FIG. 7. Resealed values from Figs. 5 and 6 plotted on the spectrum of the input pulse.
`to=0314 set (circles). to= 1.214 set (triangles)
`
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`
`640
`
`Dunkin and Levin
`
`trace spectra that go into the 12.fold pulses with
`FIG. 9. The spectrum formed as the sum of the individual
`to=0.814 set of Figue 8. The individual trace spectra have been normalized before summing.
`
`FIG. 10. The spectrum formed as the sum of the individual trace spectra that go into the 12.fold pulse with to=0.814
`set of Figure 8. The individual spectra have been multiplied by the proper stretch factors before summing.
`
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`
`Normal Moveout
`
`641
`
`08 -
`
`06 -
`
`f
`
`i 0. -
`
`FIG. 11. Values found by frequency analysis of the pulse with to=0.814 SK in
`Figure 8, superimposed on the spectrum of Figure 10.
`
`erally realized. As long as the CDP method is
`used to increase the signal-to-noise (S/N)
`ratio
`in exploration for large structures, the unexpected
`shift toward low frequencies is not serious. When
`the goal is detailed delineation of the subsurface
`for stratigraphic purposes, the decreased resolu-
`tion that goes hand-in-hand with a narrow, low-
`frequency spectrum can be deleterious.
`
`pulse; the second had a spectrum that was the
`same for all frequencies (white noise).
`For both models, we assumed that the signals
`added directly, while the total noise was the
`square root of the sum of the power density
`spectra of the noise on each trace. If we write the
`spectrum of a pulse plus noise before iYMO cor-
`rection as
`
`S/N RATIO
`
`FOR CDP STACKING
`
`The same calculations that let us draw Figure
`10 permitted us to compute the signal-to-noise
`(S/N) ratio for a 12.fold stack. We’ll look at the
`reflection with to slightly greater than 0.8 sec.
`We considered two noise models: The first had a
`spectrum identical to the spectrum of the input
`
`the summed signal after NM0
`stacking becomes
`
`correction and
`
`(9)
`
`I’IG. 12. Theoretical signal-to-noise ratio for a 12-fold stacked pulse, to=0.814 set,
`when the noise spectrum is identical to that of the reflection pulse.
`
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`
`642
`
`Dunkin
`
`and
`
`Levin
`
`The corresponding noise after KM0 correction is
`
`In equations (9) and (lo), k-pulses are being
`summed. In our examples, k = 12.
`For white noise, a(i)
`is a constant independent
`of frequency and the S/N
`ratio for the stacked
`pulse is simply a scaled version of Figure 10. For
`this reason, we show in Figure 13 the S/N
`im-
`provement,
`i.e., the ratio of (S/fl)/(g/fi).
`The
`improvement greater than 412 at low frequencies
`results from signal energy being transferred into
`the low-frequency end of the spectrum, a region
`where little signal existed before NM0 correction
`and stacking. When the signal and noise spectra
`are identical, the stacked S/N
`ratio .?(f)/N(,f)
`has the form plotted as Figure 12. If all 12 pulses
`were the same, s(f)/m(f) would be 412. Because
`of NM0 stretching, no two pulses are identical
`and
`.?(f)/~V(f)
`is smaller than the dl2,
`ap-
`proaching the 412 only in the neighborhood of
`the peak in the summed pulse spectrum.
`Before leaving this subject, we must emphasize
`once more that in this paper our illustrations are
`limited to extreme cases. For Figures 12 and 13,
`for example, the reflector is at 3100 ft and the
`source-geophone separation is 9600 ft. Rarely will
`an interpreter be concerned with reflections from
`interfaces this shallow and recorded with spreads
`this long. As Figure 1 shows, the amount of
`stretching produced by NM0 correction is small
`at depths of exploration interest; the amount is
`given by equation (A). For this paper, we have
`selected to times and spread lengths that illustrate
`the phenomena we are investigating, but the con-
`ditions chosen should not be taken as typical.
`
`DISCUSSION
`The pulse stretching or, equivalently, frequency
`scaling produced by NM0 correction is linear;
`however, it cannot be expressed as a convolution
`
`FIG. 13. Theoretical signal-to-noise improvement for
`a 12.fold stacked pulse, 10=0.814 set, when the noise is
`white.
`
`operation. Although the result of the process is
`deceptively simple [equations (6) and (7)], subse-
`quent processes, such as stacking, are not. To gen-
`erate the spectrum of Figure 11, for example, we
`had to compress and amplify
`the spectrum of
`each pulse to be stacked before we summed. The
`neatness and insight provided by the spectra mul-
`tiplication of linear filtering was missing.
`To the writers, the range over which equation
`(1) was valid was unexpectedly great. We’d as-
`sumed the approximation would describe stretch-
`ing of, perhaps, 10 percent. rlctually, the equation
`was still adequate when the factor was greater
`than 2.5. Of course, we restricted ourselves to the
`simple case of isolated pulses. The complications
`resulting from intersecting reflections that were
`discussed by Buchholtz were not considered here.
`An investigation that combined and extended the
`results of Buchholtz and of our work should cast
`further
`light on the important subject of NM0
`correction.
`
`REFERENCE
`Buchholtz, H., 1972, A note on signal distortion due to
`dynamic (NMO) corrections: Geophys. Prosp., v. 20,
`p. 395-402.
`
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