GEOPHYSICS. VOL. 48, NO. 5 (MAY 1983); P. 555-568, 17 FIGS

Vertical seismic profiling: Separation of upgoing and downgoing acoustic waves in a stratified medium

B. Seeman* and L. Horowitz*

ABSTRACT

One of the essential steps in the processing of a verti- cal seismic profile is the separation of upgoing and downgoing signals. With this perspective in mind, seis- mic data recorded in a borehole are modeled in terms of these waves and a mathematically optimal “least- squares” technique for extracting them is derived. The method imposes~practically no constraints- on the spac- ing between recording levels and allows almost perfect rejection of a coherent downgoing signal.

The exact formulation of the one-dimensional model requires that acoustic impedance information be includ- ed, but a reasonable and realistic approximation can neglect impedance. We derive frequency-wavenumber response plots for the two limiting cases of even and randomly spaced levels and compare these to the re- sponse of a “conventional” velocity filtering technique. By a careful study of available logs, recording levels can be chosen to optimize geophone coupling rather than insisting on uniform spatial sampling. Data editing, nor- malization, and true amplitude recovery (TAR) are re- quired prior to application of the technique. The TAR correction can be computed from sonic log data, which emphasizes the possibility of more complete synergism between the seismic and logging worlds.

INTRODUCTION

The need to separate upgoing and downgoing signals in order to process vertical seismic profiles (VSPs) has led to a renewal of interest in the techniques of velocity or dip filtering which were extensively discussed in the geophysical literature by Embree et al (1963) and Treitel et al (1967). More recently, frequency-wavenumber techniques have been developed by Christie et al (1980) and Simaan and Jovanovich (1980). The above techniques have generally assumed regular two- dimensional (2-D) data sampling. These methods are widely used to dip filter surface seismic data, but for a VSP the filtering requirements are more restrictive for the following reasons: (1)

the spatial sampling between recording levels is irregular be- cause recording depths are often adjusted to obtain a good acoustic contact with the formation, and (2) the downgoing signal, whose direction is well defined, is an order of magnitude greater than the upgoing signal. Thus filters with efficient rejec- tion capabilities in a narrow band of velocities are needed.

These facts have motivated the development of specialized techniques of velocity filtering particularily adapted to VSP data; however very little, if any, detail exists in the published iiterature on the methods used. The objective of this paper is to present one such technique which, although it does not solve all the problems, is elegant in its simplicity and quite effective in its implementation. We discuss its advantages and outline possible future directions of development. We also stress the need to utilize more fully the available logging information when choosing recording levels and indicate possibilities for using quantitative logging measurements to correct the VSP data further.

DERIVATION OF THE OPTIMUM FILTER

Let us consider a perfectly elastic horizontally stratified medium and the propagation of plane, normally incident upgo- ing and downgoing waves. Neglecting transmission and reflec- tion effects between levels, we have from equation (A-11) in Appendix A the following frequency relationship between the signal S, at level n, the upgoing and downgoing signals U, and D,, at a reference level 0, and a random noise component

R,(f):

S.(f) = u, u,(f) + u. De(f) + R,(f). (I)

The quantity

a, = e i2nScI (2)

and its complex conjugate a are the time shift operators with t, being the vertical traveltime of compressional waves between the depth levels 0 and n (Figure 1). The time-domain repre- sentation of this equation is

s,(t) = uo(t - t,) + tl, (t + t,) + r.(t). (3)

Suppose we have a number N of such measurements at different depth levels, each being contaminated by a random noise com-

Presented at the 51st Annual International SEC Meeting, October 19, 1981, in Los Angeles. Manuscript received by the Editor December 15, 1981; revised manuscript received July 26, 1982. *Etudes et Productions, Schlumberger, 26 Rue de la Cavee, 92140 Clamart, France. 8 1983 Society of Exploration Geophysicists. All rights reserved.

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556 Seeman and Horowitz

DOWNGOING UPGOING

0 hd I I I I -tN-

FIG. 1. Vertical seismic profile schema illustrating the recording of upgoing and downgoing waves for an irregularly depth sampled profile.

ponent, and we wish to use these measurements to extract the best estimates of v’,(f) and d,(f) in some optimal sense. We choose the L? norm and minimize the following expression with respect to 0, and 6, :

“f(5” - > 2

a,0,-&D”, n-1

This minimization gives us the following two equations for each frequency component:

n=N

X = ND, + 0,x a,” n=O

(5)

and

” = N

Y = 60 c u.’ + ND,, (6) It=*

where X and Y represent the sum of the N traces after they have been time shifted to vertically align the downgoing and upgoing waves, I.e.,

n=N

X= Cans, n=l

(7)

and

n=N

Y = I&“&. (8) “=I

This formulation implies that the best noise rejection is ob- tained by giving equal weights to each level, which is a charac- teristic not generally followed in conventional velocity filtering techniques. We can now solve equations (5) and (6) for 0, and &, to obtain

(9)

n=N

and

NY-Xx$

0, = II=1 n=N “=N

N2 - cti,f 1 a,f PI=1 “=l

“=N NX-Yzai

do = n=l

n=N n=N ’ (10)

N2- x$x ai II=, a=1

Had we used the exact formulation of Appendix A [equation (A-9)] and included the transmission effects defined by A, and its complex conjugate A”, the above equations would have to be modified as follows :

a,+ A,a,,

and

n=N N+ 1 &A,.

n-1

Equations (9) and (10) are optimal, though not robust estimates (Claerbout, 1976, chap. 6) in the_ ptesence of random noise+ ins order to apply them we must have a priori knowledge of the arrival times. These times are routinely obtained from the first breaks of the direct downgoing wave.

No assumptions have been made as to the spatial separation of the recording levels, thus fulfilling our first requirement for the filter.

We now derive the expression for the upgoing signal o0 for

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Vertical Seismic Profiling

DOWNGOING UPGOING DOWNGOING UPGOING

y

-1/z 0 +lli

NORMALIZED WAVENUMBER, K-

FIG. 2. The normalized frequency-wavenumber response of the optimal velocity filter spanning three equispaced recording levels. (Black > 40 dB and white < 6 dB rejection.)

an impulsive plane wave having an arbitrary dip defined by a set of traveltimes ~~ measured relative to the reference level 0 (Figure 1). The appropriate time shift operators are

fl” = e-j2n/r.. (11)

These & are then substituted in place of S,, whose magnitude is set to unity, in equations (7) and (8), and we obtain from equation (9) the expression

“=N “=N n=N N 1 a, P. - c a, 0.1 a.’

o,= n=l n=l II=1 n=N “=N

N* - 1 hi 1 a: n=l n=*

Ctinsider the following two limiting cases.

(12)

(1) For fin = I%, , corresponding to the dip of the downgo- ing wave, we have a response of zero for any number of levels. Since we get perfect rejection of the downgo- ing energy, we fulfill the second requirement of the filter.

(2) For & = a,, corresponding to the dip of the upgoing wave, we have a response of unity at all frequencies except for certain limiting values which will be dis- cussed later.

We should also note that exact filter equations can be derived to model correctly any upgoing event with an arbitrary but known dip. This filter would involve only minor changes in equations (9) and (10).

We now study the significance of the denominator in equa- tion (B-2). This function is plotted in Figures 6 and 7 for the cases of 3 and 11 levels. In fact the denominator, which is real, represents the amplitude distortion of the vertical upgoing signal introduced by the numerator of equation (B-2). By di- vision we are compensating for this distortion and evaluating o0 in its undistorted form. Note the wider and flatter response for the filter spanning 11 levels.

In Appendix B we derive the upgoing response function of The second term in the denominator,

0 +1/z

NORMALIZED WAVENUMBER, K-

FIG. 3. The normalized frequency-wavenumber response of the optimal velocity filter spanning three randomly spaced record- ing levels. (Black > 40 dB and white < 6 dB rejection.)

arbitrary dipping events for the two limiting situations of regu- lar and random positioning of the recording levels. These equa- tions can be represented in the frequency-wavenumber domain and are plotted in Figures 2 to 5 for filters spanning 3 or 11 levels, regularly and randomly spaced. The shading in these figures follows a logarithmic scale where full black 2 40 dB rejection and white 5 6 dB. The elimination of the downgoing events is good irrespective of the number of levels used in the filter design. For upgoing events. the passband becomes pro- gressively narrower as the number of levels N increases. Thus if we suspect that the data contain reflections from dipping hor- izons, there is some justification for sacrificing the noise rejec- tion capabilities of the filter and using fewer levels to better identify various dips.

Note that the response functions are similar in the two limiting cases of regular and random positioning, so that it is not unreasonable to suppose they would apply for any irregular distribution of levels. However, randomness is an idealization, particularly for sparse data sets, and some regularity should be expected so that equation (B-2) is more appropriate than equa- tion (B-4) with the vertical time sampling interval 6t replaced by some average spatial time sampling %.

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558 Seeman and Horowitz

UPGOING DOWNGOING UPGOING DOWNGOING

NORMALIZED WAVENUMBER, K -

FIG. 5. The normalized frequency-wavenumber response of the optimal velocity filter spanning eleven randomly spaced re- cording levels. (Black > 40 dB and white < 6 dB rejection.)

-l/2 0 +1/2

NQRMAJJZED WA!.!EMUMBER,_ K -

FIG. 4. The normalized frequency-wavenumber response of the optimal velocity filter spanning eleven equispaced recording levels. (Black > 40 dB and white < 6 dB rejection.)

(13) Although the paper has not taken this approach, neglecting the term in the denominator of equation (I 3) introduces the

is the only frequency-dependent part, and if we ignore it, the possibility of applying the filter by a relatively efficient nonre- upgoing wave field would be correspondingly distorted. This cursive convolution in the time domain. On the other hand, this distortion in itself may not be too important since in a normal would entail processing at a finer sampling in time than abso- processing sequence a subsequent waveshape deconvolution lutely necessary. can restore the signal to some standard shape. Figures 6 and 7 also indicate that for frequencies around

-

:

J -c

/

0.

: : j : / i N=lI j ~

sin (271 Nf dt) 2 sin (2n f dt) 1

N=3

/dt=8MS : : /

/ i i

/ / / / /

/ dt=8MS :

i

10 15 28 25 30 35 40 45 50 55 60 J

FREQUENCY (HERTZ) 3 Irr is 20 25 38 35 40 45 50

FREQUENCY (HERTZ)

FIG. 6. Plot of the denominator of the response function of the optimal velocity filter spanning three recording levels and pro-

FIG. 7. Plot of the denominator of the response function of the

viding a vertical sampling interval of 8 msec. optimal velocity filter spanning eleven recording levels and providing a vertical sampling interval of 8 msec.

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Vertical Seismic Profiling

DOWNGOING f=& k:O, 1,2...,

559

(14) PGOING

BAND .j / ALIAS

. BAND

0 l/2 NORMALIZED WAVENUMBER, K -

Q

FIG. 8. The normalized frequency-wavenumber response of a conventional velocity filter spanning eleven equispaced record- ing levels after 45 degrees rotation and shift of the axes. (Black > 40 dB and white < 6 dB rejection.)

there will be severe attenuation which cannot be compensated for by using the exact form of the denominator without magni- fying certain low- and high-frequency components of noise. This effect restricts the usable frequency passband to

(15)

The cut-off frequencies SfO and fif; should be chosen to reflect the useful frequency band of the signal.

TESTS WITH SYNTHETIC DATA

In Figure 8 we plot the frequency-wavenumber response of a conventional wide-band velocity tilter with an 11 trace input. The axes have been rotated by 45 degrees and shifted to bring out a certain number of points more clearly. The upgoing and downgoing events are indicated, and we see that although a wide band of dipping events is passed, the following incon- veniences are evident: (1) the average rejection of the downgo- ing signal is of the order of only 30 dB and would be even less with fewer levels; (2) rejection of high frequencies in the downgoing signal is limited by the alias passband; (3) there is more distortion at low frequencies because of finite length elects; and (4) the levels should in theory be equally spaced in time

0 time (MS) - 1000

FIG. 9. Synthetic VSP data having irregular spatial sampling, a lo-45 Hz zero-phase signal, and 4 msec sampling. The acoustic impedance log (at left) generates the upgoing reflections.

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Seeman and Horowitz

ACOUSTIC IMPEDANCE

0 -~ time (MS) ------+ 1000

FIG. 10. The upgoing wave field obtained from the synthetic VSP in Figure 9 by applying a conventional velocity filter spanning eleven depth levels.

ACOUSTIC IMPEDANCE

I I I I _ _

I I LI 0 time (MS) - 100 ‘0

FIG. 11. The upgoing wave field obtained from the synthetic VSP in Figure 9 by applying the optimal velocity filter spanning eleven depth levels and neglecting the reflection and transmission effects of the variations in acoustic in~~~anrp Pa& llnonino pvpn+ appears to originate 5 or 6 depth levels below its corresponding impedance contrast.

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Vertical Seismic Profiling 561

ACOUSTIC IMPEDANCE

FIG. 12. The upgoing wave field obtained from the synthetic VSP in Figure 9 by applying the optimal velocity filter spanning eleven depth levels and including the reflection and transmission effects of the variations in acoustic impedance. Each upgoing reflection now originates at the same depth as its corresponding impedance contrast.

Figure 9 shows a synthetic one-dimensional (1-D) VSP with irregularly spaced levels in which the data have been sampled at 4 msec and the signal has a frequency band of 10 to 45 Hz. The estimates of the upgoing wave fields obtained through (1) the conventional velocity filter, (2) the “optimal” filter ignoring reflection and transmission effects, (3) the “optimal” filter in- cluding reflection and transmission effects are plotted in Fig- ures 10, 11, and 12. As expected the conventional filter does a poor job of removing the downgoing signal. The optimal filter does an excellent job, and for a synthetic data set the result is perfect if the effects of the interposed reflectors are included through the complex transmission terms A, described in Ap- pendix A.

TESTS WITH REAL DATA

Before we can apply these filters to real data recorded in a borehole, the quality of the real data must be carefully con- trolled, certain field conditions must be imposed, and correc- tions must be made to make the data closely approximate the 1-D model we have specified.

During data acquisition the available logs (particularly cali- per and sonic) should be carefully studied to choose recording levels which avoid caved zones and low-velocity formations where geophone coupling is likely to be poor. Figure 13 indi-

FIG. 13. Locating VSP recording levels (1, 2, and 3) by studying open-hole logs to avoid a zone of potentially bad coupling (e.g., overpressured shale) and respecting the required vertical sepa- ration (z 8 msec).

shales

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Vertical Seismic Profiling

co t - W

LL

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TWO WAY TIME(MS)

A. B C D 2000 3000

6000

8000 F

FIG. 16. Upgoing wave field obtained from test well VSP by applying the optimal velocity filter spanning eleven depth levels.

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Vertical Seismic Profiling

0 m a

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566 Seeman and Horowitz

cates how these considerations are used to locate recording levels and to ensure the desired average spatial sampling dicta- ted by the frequency band of the seismic signal. The average vertical transit time % of 8 msec would correspond to a high cut below 60 Hz.

During preprocessing of the data, careful editing is required to remove spurious noise spikes which would not be effectively eliminated by a filter based on a nonrobust I? norm.

Prior to velocity filtering we must compensate as far as possible for the effects of transmission losses, spherical diver- gence, refraction due to nonnormal incidence, inelastic attenu- ation, and variability in coupling between levels. The first effect requires the exact formulation derived in Appendix A together with an acoustic impedance log. The second and third effects were discussed by Newman (1973) and are summarized by the divergence factor

i=n CtiV

D i=l one-way time=-

VI ’ (16)

where ti and v are the one-way traveltimes and velocities in the layers through which the waves travel. These values can be derived from the sonic log.

The other effects, which can be nonnegligible, are more or less unknown. Thus at present a practical and straightforward way to deal with all of them and to approximate the 1-D model is to equalize the energy in the predominant direct arrival and then apply some empirical true amplitude recovery (TAR) func- tion to correct for the remainder of each trace. This procedure is a way of compensating for all five of the effects mentioned above, and it allows us to apply equations (9) and (10) to extract the upgoing and downgoing wave fields.

In Figure 14 we plot VSP data recorded using a Vibrosei@ source offset 200 ft from the wellhead. Apart from the direct signal, tube waves and their reflections at the bottom of the well are the most prominent features. These phenomena were dis- cussed by Hardage (1981). We also observe a strong resonance especially at the deeper levels. There is little visual evidence of upgoing primary reflections. This data set is thus a good exam- ple to use in evaluating velocity filtering techniques.

The high-frequency tube wave was removed by a 40 Hz low-pass filter, and the data were then processed by a conven- tional velocity filter spanning 11 levels and the optimal velocity filter spanning 11 and 3 levels. The results are plotted in Figures 15, 16, and 17, respectively.

As expected, the conventional velocity filter does a poor job of rejecting the downgoing signal.

The rejection of the direct downgoing signal is excellent for both the 11 and 3 level optimum filters (Figures 16 and 17), and good rejection is also observed for the downgoing tube wave in the case of 11 levels. We are now able to identify several shallow and deep reflectors (A, B, C, D, E, and F in Figure 16), the deepest two reflectors E and F being below the bottom of the well. The resonant energy masks any reflectors below 2.4 sec.

The three level optimal filter has the potential of both ef- fectively removing the direct downgoing signal and more accu- rately preserving the whole upgoing wave field (dips, diffrac- tions, etc). Unfortunately in this example the quality of the upgoing wave field is deteriorated by the tube waves, which emphasizes the need to attack this problem by an appropriate field procedure during data acquisition (Hardage, 1981).

@Trademark of Conoco Inc.

CONCLUSIONS

The “optimal” filter described here is very well adapted to the VSP situation. Not only dots it allow a choice of recording level locations and facilitate the acquisition of better quality data, but its rejection characteristics for coherent downgoing energy are excellent.

Furthermore, we have indicated the benefits to be derived from a synergistic approach to the use of seismic and logging data; more developments in this area will no doubt soon appear.

Nevertheless before the full potential of borehole seismic measurements can be exploited, progress must be made in the following areas: (1) development of improved downhole tools with multiaxial sensors, (2) a better understanding of the effects of tool-to-borehole coupling, (3) acquisition methods to im- prove resolution and suppress unwanted noise, and (4) devel- opment of robustness in the data reduction algorithms.

One aim in vertical seismic profiling should be for more accurate and complete preservation of the scattered wave field, the inversion of which would help to delineate subsurface struc- tures better.

REFERENCES

Christie, P. A. F., Hughes, V. J., and Kennet, B. L. N., 1980, Velocity filtering of seismic reflection data: Presented at the 50th Annual International SEC meeting November 5, in Houston.

Claerbout, J. F., 1976, Fundamentals of data processing: New York, McGraw-Hill Book Co., Inc.

Embree, P., Burg, J. P., and Backus. M. M., 1963, Wide band velocity filtering-The pie slice process: Geophysics, v. 28, p. 948-974.

Hardage, B. A., 1981, An examination of tube wave noise in vertical seismic profiling: Geophysics, v. 46, p. 892-903.

Newman, P., 1973, Divergence effects in a layered earth: Geophysics, v. 38, p. 481488.

Simaan, M., and Jovanovich, D., 1980, Optimum filters for vertical array seismic data processing: Presented at the 50th Annual Interna- tional SEG Meeting, November 7, in Houston.

Treitel, S., Shanks, J. L., and Frarier, C. W., 1967, Some aspects of fan filtering: Geophysics, v. 32,789 X00.

APPENDIX A EXTRAPOLATION OF SEISMIC WAVES

ACROSS A STRATIFIED MEDlUM

We assume plane waves normally incident on a perfectly elastic horizontally stratified medium and thin equal time slices. In terms of z-transforms the relationship between the upgoing and downgoing wave fields at a level k and those at a reference level 0 is given by Claerbout (lY76, chap. 8)

where F, and G, are given by

F,(z) = F,- 1(z) + CkZG,k_,)(Z),

and

(A-2)

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Vertical Seismic Profiling 567

G,(z) = c,F,_ ,(z) + zG,,_ ,\(z).

with F,(z) = 1 and G,(z) = 0.

(A-3)

The physical quantities measured, velocity or pressure, are related to the sum and difference of U, and D, (Claerbout, 1976, chap. 9). If we define two new variables by ”

F, + G, A, = 7

n [I ,= I

and

where A, and B, are given by

A, = $[(I + --)A,_, + (1 ~ z)B, ,].

(A-4)

(A-5)

(A-6)

(A-7)

and

B

k = :(I - (.k)

2 (I+(.,) [(I - dA,- , + (1 + dB,- ,I. (A-8)

with A, = 1 and B, = I. If the sum term in equation (A-6) represents the quantity Sk

we measure, we have

APPENDIX B FREQUENCY-WAVENUMBER RESPONSE

OF OPTIMUM FILTER

Equal time spacing of levels

In this case, supposing 1 to be the reference level, the timeshift terms become

and

!3, = e-- ,277,” 1 b/h

where 6t and ST are, respectively. the vertical and dipping traveltimes between adjacent levels. p, can also be expressed in terms of vertical frequency k by

j2nCn - lbk6r p.=p- (

because we can define a dimensionless velocity ~3 to be given by either

f 6r L’=L k

or (‘z- 67’

The summation terms in equation (9) can be evaluated as geometric means. For example,

n=Y n = N

sin GJ7WW e _ j2ntN ,,f6r

= sin (2xf6f) (B-1)

and we finally obtain for the F-K response to an arbitrarily dipping event

N sin [rrN(f- k)&)] sin (2xNf6t) sin [r‘Y(J+ k)6t]

U,(.l; k) = sin [x(f- k)i%] - sin (2Q6t) sin Cn(f + kPt1

N2 _ sin2 (27tiV,/Ff) e,n,K , ,,/ k,6r

sin’ (2rtffSt)

(B-2)

The term A, is a transmissivity factor representing the effects of all reflectors traversed. If we neglect interbed multiples and suppose that level-to-level signal normalization can be effected independently, the above equation becomes

Sk = $%I, + \:>D,,, (A- IO)

which involves only pure time shifts equal to the one-way traveltime tk between levels 0 and k. Transforming to the fre- quency domain. we obtain

S,(f) = nk u,(f) + a,,Do(l),

where the shift operator ak is given by

a, = e-i2nJ:r,

(A-l 1)

The normalized F-K response is then obtained by setting 6t = I in equation (B-2).

Random time spacing of levels

In this case we replace the geometric series by expected values having a normal distribution that allows all positions to be equally possible. and we integrate the series term by term to obtain

=

N sin (24) e- j2nl,

2rrji (B-3)

where I is the vertical traveltime between levels 1 and N Assuming an average spatial sampling of

St= ’ (N ~ 1)’

we obtain the following expression for the amplitude part of the normalized F-K response

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588 Seeman and Horowitz

(N - 1)2 sin [x(N - I)(+ k)] sin [x(N - l)f] sin [h(N - INS+ k)] _

U”(J k) = C4.f - @I (WI Cn(f+ k)fl

(N _ 1)2 _ sin’ I270 - 1)Sl

(2?fY

The phase delay is the same as for regular sampling, being zero for the vertically upgoing signal (f= k). Otherwise, it is linear with a shift that depends upon the reference level chosen.

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