GEOPHYSICS. VOL. 52. NO. 7 (JULY 1987); P. 973-984, 15 FIGS

Wave-equation trace interpolation

Joshua Ronen*

ABSTRACT

Spatial aliasing in multichannel seismic data can be overcome by solving an inversion in which the model is the section that would be recorded in a well sampled zero-offset experiment, and the data are seismic data after normal moveout (NMO). The formulation of the (linear) relation between the data and the model is based on the wave equation and on Fourier analysis of aliasing.

A processing sequence in which one treats missing data as zero data and performs partial migration before stacking is equivalent to application of the transpose of the operator that actually needs to be inverted. The inverse of that operator cannot be uniquely determined, but it can be estimated using spatial spectral balancing in a conjugate-gradient iterative scheme. The first iter- ation is conventional processing (including prestack partial migration). As shown in a field data example in which severe spatial aliasing was simulated, a few more iterations are necessary to achieve significantly better results.

INTRODUCTION

The Nyquist (1928) condition for sampling a wave field is that the sampling interval may not exceed half the smallest wavelength. While this is achievable in two-dimensional (2-D) surveys, practical difficulties arise in three-dimensional (3-D) surveys. The result is a problem of missing data; one does not have enough data to sample the seismic wave field adequately.

On the other hand, it seems that one has too much data in multichannel seismic surveys; there are many traces in every common-midpoint (CMP) gather, and summing (stacking) is used to merge what seems to be redundant information from different channels. The questions are whether the information is redundant and whether summing is the optimal way to merge it.

These questions, in a general context, have been discussed by several authors (Shannon, 1949; Linden, 1959; Papoulis, 1977; Brown, 1981), and in particular they found that if a function is filtered by some independent filters before sam- pling, as ~&own in Figures I, a certairr anmurrt of aiiasing is allowed and one can still recover the original signal.

The same idea can be applied to seismic data (Bolondi et al., 1982; Claerbout, 1984, p. 219). The earth’s image is the signal that is filtered by some different filters before subsampling. The Hj filters, based on the wave equation, in Figure 2 are the operators that produce seismic data for a given earth model and survey geometry. The data they produce may be aliased in each channel. but the combination of all channels can give a high-resolution image.

FORMULATION

Multichannel abased one-dimensional signal

When a model function #z(x) is sampled with a certain sam- pling interval AX, its Fourier transform is replicated: if @I(X) has the Fourier transform m(k), then the Fourier transform of the sampled signal is

FT{sampled[ril(ll]~ = njcm(k - nk,J,

where the sampling wavenumber k, is

k, = 2nlAx.

(1)

Nyquist (1928) showed that if the function m is band-limited,

m(k) = 0 for Ikl 2 W,

then the summation in equation (1) has a finite number of terms N, where N is the smallest integer that is larger than 2 W/k,. In particular, if k, is more than 2 W, then N = 1 and

FT (sampled [K(x)]} = m(k),

for (k ( < W. In this case, m(k) can be easily restored from its samples simply by extracting the spectrum for 1 k 1 < kJ2.

Manuscript received by the Editor March 7, 1986; revised manuscript received September 29, 1986. *Formerly Department of Geophyscs, Stanford University; presently Center for Wave Phenomena, Colorado School of Mines, Golden, CO 80401. al 1987 Society of Exploration Geophysicists. All rights reserved.

973

974 Ronen

Recovering from aliasing

Analog- Sub- Aliased- filters sampling sequences

FIG. 1. Overcomimg aliasing of a one-dimensional signal. The signal at the left is filtered by five filters H,, .., If,, before sampling by the analog-to-digital (A/D) converters. The data are five different aliased sequences. The original signal cannot be recovered from any one sequence alone, but the combination of them may be suffkient.

Multioffset data

Wave propagjation

Constant- offset

sections [aliased 1

FIG. 2. Overcoming aliasing in reflection seismology. The earth reflectivity model is the signal sampled after it has propagated through the filters Hj.

Wave-equaiion Trace interpolation

The point is that aliasing can be overcome if the function 6i(.x) is filtered by some convolution filters hj before sampling. The data in channel j are samples from the function h,(x) * G(x). In the wavenumber domain, one has

d,(k) = C h,(k - nk,)m(k - nk,s), forj=1,2 ,._., J.

In a matrix form, for five channels (J = 5) and three-fold alias- ing (N = 3),

The solution of this system of equations interpolates on the x-axis because it extrapolates on the k-axis. The data are known up to the wavenumber kJ2, the model is found (in this case) for ) k ) c 3kJ2, because m(k), m(k - k,), and m(k + k,J are found for ) k ) < kJ2.

Multichannel abased seismic data

To apply the idea of equation (2) to reflection seismology, one must first describe the relations between the earth (the model) and the seismic data.

For the purpose of trace interpolation, I chose the model to be the section that would be recorded if a well-sampled, zero- offset survey were performed. The relationship between the zero-offset section and the earth’s reflectivity is approximately linear; the linear operator is poststack migration. Non-zero offset data are also approximately linearly related to the re- flectivity; the linear operator is prestack migration. Both the zero-offset section and the non-zero offset section are linear with reflectivity; therefore, they are linearly related to each other; the linear operator is a prestack-partial migration also known as dip moveout (DMO).

The prestack migration process can be performed in three steps: normal moveout (NMO), DMO, and poststack migra- tion (Figure 3). Choosing the model as the unmigrated zero- offset section and taking the data after NM0 leave only the DMO and the spatial sampling between the data and the model (Figure 4).

To set up the relation between the model and the data, the inverse DMO, and not the DMO itself, is needed. It is shown in the Appendix that the relation between the time-space Fou- rier transform of the zero-offset section m(k, w) and the space Fourier transform of a common-offset section dj(t, k) is

d, (t, k) = dw A ‘~-‘oartn(~, k),

where rl = ,/I + [hj k/w]’ and llj is the half-offset. A discrete form of this integral is

dj(t, k) = 1 A-‘e ~i2noAr~nrtn(w, k), w

(nt is the number of time samples in a trace). This is (for each

k) a matrix-vector multiplication,

d,(k) = Q,? (k)m(k).

The vectors are the common-offset section

975

(3)

dj(k) =

(nt is the number of samples in a trace) and the zero-offset section

m(k) =

(no is the number of frequencies in the Fourier transforin of a trace). The matrix Q,’ is the inverse DMO operator (an nt by nw matrix). The tth row and the wth column element are

A - 1 e - ilnodrlnt

One can now merge sampling theory [equation (i)] and migration [equation (3)]. When spatial aliasing is introduced, equation (3) becomes

d,(k) = c Qf (k - nk,)m(k - nk,). n

(4)

Ptestack Prestack migration partial migration

mid oint

Constant - offset 22

section $J ie .

Model ,$ 5 m

NM0 1

$: B i f

DMO r-i

FIG. 3. Decomposition of prestack full migration into NMO, DMO, and poststack migration. An impulse on a common- offset section can be migrated to the expected ellipse with the shot and the receiver in the foci, either in one process (pte- stack full migration), or in three steps (NMO, DMO, and zero-offset migration. which can be done as a poststack pro- cess).

976 Ronen

For J = 5 offsets and an aliasing of N = 3, the system is

~~~=~~~i._~~~]~~~~~.

(5)

This is the same as equation (2), except d, and m are vectors and &),T is a matrix.

m- 4 D; H A/D k dj

- . .

I* DMO - sub- --

DMO sampling

Prestack migration

1

Common-

-__I offset section

b 0

Ii r E 2;

Common

m offset

section I I

FIG. 4. The choice of the model and the data. The model m is the ideal zero-offset section. The data d. are the common- offset sections after NMO. The Q,’ are &verse-DMO oper- ators.

Equation (5) is a system of J x nt equations (one equation for every time sample of every common-offset section), with N x no unknowns for each wavenumber k. The matrix has J x N blocks with each block Q,? (k - nk,) being an nt x nw matrix. Solving this equation extrapolates the data in the offset direction (to zero offset), and also in the wavenumber direction from the low wavenumbers (k 1 < k,/2, on which the aliased data dj are given, to the full range (k 1 < Nk,/2 re- quired to describe the unaliased zero-offset section.

This formulation is similar to the multichannel deconvolu- tion given by Davies and Mercado (1968). Both formulations involve inversion of large block matrices, except that here the

blocks are inverse-DMO operators instead of convolutions.

Wave-equation trace interpolation in two steps

Conventional data processing does not involve inversion of huge matrices, such as the one in equation (5), and missing data are often assumed to be zero. Nevertheless, it works quite well when the data are not severely aliased.

If there is no spatial aliasing, then N = 1 in equation (5), and one has

(6)

If DMO is applied to each common-offset section,

stack = i Qjdj j=l

= CQjLlfm ,: 1

.I

x Cm j=1

= Jm. (7)

The processing in equation (7) is the application of the Hermitian transpose of the matrix in equation (6),

D,(k) Q,(k) Dh(k) Q,(k) 1 (the asterisk denotes taking the Hermitian transpose). This application uses the result

p+ *,(,4) = Q, {k) (8)

given in the Appendix. When the data are not spatially aliased, equation (6) can be

solved by applying the Hermitian transpose matrix. Since this is adequate processing for the nonaliased case, it is interesting to see what the results of applying the Hermitian transpose are in general.

Wave-equation Trace Interpolation 977

N

400 600 800

FIG. 5. Synthetic example. (a) The model: a well-sampled, zero-offset section. (b) The data: four midpoint gathers after NMO. The midpoint interval is 200 m. Each gather has five traces (at the same midpoint but with different offsets). Note that the dipping reflector is overcorrected by the NMO. (c) Transpose processing: the result of DMO and stacking after addition of zero traces. (d) The result of wave-equation trace interpolation.

ORIGINRL OQTFI SRtiPLED DRTG

I I I I I I I r 1 I

I I E

8

1 I 1 I I I I 1 1 I I I 4 9

shot 5oom shot

(4 @I

FIG. 6. (a) Stacking chart of the original data. The original data do not have an aliasing problem: 25 m shot interval, 12.5 m group interval (one offset is missing because of a dead receiver). (b) Stacking chart of the data used for the trace interpolation. Less than 1 percent of the data were used, simulating severe spatial aliasing.

978 Ronen

If equation (5) is written as Synthetic data example

d=Gm, (9) The two-step method was applied to a synthetic data set of

(where G is the J x nt by N x nw matrix), then the vector ti four CMP gathers, 200 m apart. Each gather had five traces

can be defined as with 0 to 800 m offset (Figure 5b).

m = G*d. The earth model includes a 90 degree dipping reflector and

(10) a flat reflector. The ideal solution, the well sampled (25 m

To find 61, consider the example (N = 3 and J = 4). Then midpoint interval) zero-offset section, is shown in Figure 5a.

_+E;/; ;; ;;;;I.

This section is not part of the simulated data but is included for comparison with the results of the trace interpolation.

The results of applying the transpose operator (DMO and stacking) are shown in Figure SC. Poststack correction, using QR factorization to invert G*G + C, (with a diagonal c), generated the section shown in Figure 5d. The improvement

The transpose of G is by the poststack correction is substantial, in particular for the Rat reflector.

D,(k - k,) D, (k - k,) !I, (k - k,) D,(k - k,)

G* = [

D,(k) Dz (k) !I, (4 124(k) 1 I

The conjugate gradient method with spectral balancing

@(k + A,, ” &Z(k+!e) &(k+k,) ?,(k+kj 5 “‘J The two-step method is not easily applicable to field data

where equation (8) was used again. KI is therefore because the formation and the inversion of G*G are too com-

D,(k - ks) b(k - k,J P,(k - k,) d,(k)

9, (k) Da(k) P,(k) d, (k)

Dz (k + k,) Dx (k + k,) Ii I D,(k + kJ d,(k) ’

4 (k)

for ) k ) < kJ2. This can be written as

!

i D,(k)dj(k+kJ for - 3k,/2 < k =c - kJ2 j=l

k(k)= i Dj(k)d,(k) for - kJ2 < k < kJ2 (11) j= 1

i Qj(k)dj(k-ks) for k,/2 < k < 3k,/2. j= I

putationally intensive. Fortunately, there are practical itera- tive techniques, such as the conjugate gradient method, that produce good results at a reasonable cost.

Equation (1 I) describes DMO (application of Dj) and stack- ing

Conjugate-gradient inversion is an iterative method in which each iteration involves the application of a forward operator and its transpose (Golub and Van-Loan, 1983; Luen- berger, 1984; Paige and Saunders, 1982). Conjugate-gradient inversion of an M x M matrix converges within M iterations. Often, just a few iterations (much less than M) are actually needed.

using the (unjustified) assumption

To solve equation (9) each conjugate-gradient iteration in- volves DMO stacking (G*) and transpose DMO stacking (G). Therefore, the total cost in four iterations is the cost of eight DMOs. Actually, the cost is less than that because Fourier transforms are not repeated, data are not resorted, and oper- ators are re-used.

d(k - k,) = d(k) = d(k + k,). (12)

This assumption of replicating the Fourier transform is equiv- alent to adding zero traces between data traces in the space domain. An important conclusion is that G* is equivalent to DMO and stacking, treating missing data as zero data.

Applying C* [equation (lo)] can be the first step in solving equation (9). The second step is

[Actually G*G is singular, so the pseudoinverse (Strang, 1980) should be used.]

When applied to field data, straightforward conjugate- gradient inversion converged reasonably quickly, but to a dis- appointing result. The reason for the poor results is that solv- ing equation (9) involves inversion of singular matrices that have no exact inverse. Moreover, even the approximate solu- tion is not unique. There are many models that reasonably fit the data. Those models differ by components of what is called the null space. Null-space components are the eigenvectors that correspond to zero singular values. [See Strang (1980) for a diszussiorr of singular-vaiue~ decomposition and null space.] To achieve a unique solution, the null-space components in the model must be estimated.

To summarize, the two steps are (1) DMO and stacking with zero traces in place of missing traces [equation (lo)], and (2) a poststack process [equation (13)].

A standard estimation procedure for the null space in least- squares problems is damping: no null-space components in the solution. Damping was used in the synthetic example to gcneratc Figue 5d, which is of a reasonable quality. For the

Wave-equation Trace Interpolation 979

(m) 0 150 300 450

Fto. i. Tne data used for the trace interpolation. Every trace is corrected for gain and NMO, and is plotted at its midpoint position. As seen on the stacking chart of Figure 6b, there are four groups of five near traces and four groups of far traces. This figure illustrates what would be obtained by NM0 and stacking.

RAW 1 RAW 2

200 400 200 400

RAW 3 RAW 4 RAW 5 RAW 0 200 400 200 400 200 400 200 400 (m)

field data, one has to be more careful, because in practice the null space is composed of the eigenvectors that correspond to small (not necessarily zero) singular values. How small is small is determined by the noise level. The synthetic example is free of noise, so the null space is small and damping produces a reasonable model. Field data, however, have enough noise to require a more sophisticated estimation of the null space.

The estimation of the null-space components is an opti- mization problem, the objective of which is to fit a priori as- sumptions. A safe a priori assumption is that the spatial spec- trum of the earth model should be balanced.

A study of spatial abasing in multichannel seismic data (Rocca and Ronen, 1984) predicted that in the presence of spatial aliasing, when performing DMO and stacking there will be resonance wavenumbers for which the spectrum will have high amplitude. An a priori assumption is that these are artifacts.

This a priori assumption can be used in an iterative opti- mization scheme (such as the conjugate gradient method). Each iteration can involve, in addition to the usual conjugate- gradient routine, the following spectral balancing procedure:

(1) Transforming the raw model into the wavenumber-time (k, r) domain.

(2) Calculating the input envelope (a leaky integra- tion of the absolute values will do). The expected envelope is calculated based on the input envelope monotonously decreasing with wavenumber.

(3) Dividing the input by its envelope, and multi- plying by the expected envelope.

Field-data example

A small part of a marine line from the Gulf of Mexico was used as a test case. The original data do not have any aliasing because a shot interval of 25 m and 240 channels, 12.5 m apart, provides a 6.25 m midpoint interval on the 6000 percent stack. Only a small number of these data were used: 150 m

FIG. 8. The models of the first six iterations. Convergence is achieved within four iterations, but the resulting model is disappointing; it is very different from the true zero-offset section of Figure 12.

980 Ronen

RAW 1 RAW 2 RAW 3 RAW 4 RAW 5 RAW 6

0 0.05 0 0.05 0.05 0.05 0.05 0.05 (cycles/m)

N

FIG. 9. The raw (unbalanced) spectra of the first six iterations. As predicted (Rocca and Ronen, 1984), there are some resonance wavenumbers where the amplitude is too high compared to the true spectrum of Figure 13.

BALANCED 1 BALANCED 2 BALANCED 3 BALANCED 4 BALANCED 5 BALANCED 6

200 400 200 400 200 400 200 400 200 400 200 400 0

N

FIG. 10. The models of the first six iterations with spectral balance. Compared to Figure 8, these sections are much more similar to the near-offset section in Figure 12.

BALANCED 1 BALANCED 2 BALANCED 3 BALANCED 4 BALANCED 5 BALANCED 6 0 0.05 0.05 0.05 0.05 0.05 0.05

0

0-n)

(cycles/m)

FIG. 11. Spectra of the first six iterations with balancing. The spectral balancing routine used the knowledge that the raw spectra of Figure 9 may have resonance wavenumbers in which the amplitude is too high.

Wave-equation Trace Interpolation 981

0-W 200

shot interval (killing five shots out of every six), and ten channels per shot (killing 230 channels out of 240) in two groups, using five near traces and five far traces. This particu- lar selection of data simulates the aliasing that would occur in the cross-line direction of data collected in a two-boat survey with a 150 m line interval where the boats alternate shooting and only one boat is recording (Ronen? 1985). A stacking chart of the data is shown in Figure 6. The data that were used in the inversion are shown in Figure 7. As in many 3-D surveys, the coverage is not uniform. Nonuniform coverage is natural for the data; it is the model, not the data, which should be uniformly and adequately sampled.

The models obtained by the first six iterations (Figure 8) are the disappointing results mentioned earlier in this paper. The spatial spectra are clearly unbalanced (Figure 9). The models for the first six iterations, with spectral balancing, are shown in Figure 10, and their spatial spectra are shown in Figure 11. The results with spectral balancing are a much better estimate for the near-offset section (Figure 12) and its spectrum (Figure 13) compared to the results without spectral balancing (Fig- ures S-9).

The final result (Figure 15), although not precisely the near-

0-N 200 400

0

FIG 12. The near-offset section. This section at offsets of about 220 m is considered the “true solution.” Only four traces from this section were included in the data of Figure 7.

TRUE SPECTRUM

0

FIG. 13. The spectrum of the near-offset section of Figure 12. This spectrum was not used in the trace interpolation, but is shown here for comparison with the estimated spectra of Fig- ures 9 and 11.

FIG. 14. The model obtained by the first iteration, without spectral balancing. This would be the result of adding zero traces in place of missing data, then performing DMO and stacking.

982 R0n0ll

offset section (Figure 12), is significantly better than what is achieved by DMO stacking (Figure 14), and tremendously better than the result of NM0 and stacking (Figure 7).

CONCLUSIONS

The trace-interpolation method is based on the wave equa- tion and on an a priori assumption of a smooth spatial spec- trum.

Two alternative methods of trace interpolation presented are a poststack process (on synthetic data) and an iterative, conjugate-gradient scheme (on field data). The poststack pro- cess involves large matrix multiplication and inversion and is too costly to be applied to field data. The iterative method, a combination of conjugate-gradient and spectral balancing, was successfully applied to field data. In both methods, the first step, or first iteration, is DMO and stacking.

The method involves an iterative inversion of the linear relations d = Gm, where the model vector m is the ideal zero- offset section, and the data vector d is the seismic data after NMO.

Treating missing data as zero data and performing DMO and stacking are equivalent to application of the transpose operator I?I = G*d. This processing is adequate only in the absence of spatial aliasing.

0-n)

200 400

FIG. 15. The model obtained at the sixth conjugate-gradient iteration with spectral balancing. This is the estimation for the zero-offset section (Figure 12) that the wave-equation trace interpolation produces. Flat as well as dipping reflectors are well-interpolated.

ACKNOWLEDGMENTS

This work would not have been possible without the guid- ance and support of the staff and sponsors of the Stanford Exploration Project. Special thanks are due to Fabio Rocca who started me on this project; to Jon Claerbout who taught me how to make it work; to Einar Kjartansson, Stew Levin, Rick Ottolini, Chuck Sword, and John Toldi with whom I often discussed raw ideas, and whose software was very useful; to Paige and Saunders for writing LSQR; and to Chevron who donated the field data. Thanks are also due to Jim Black and Dave Hale who carefully reviewed the manuscript.

REFERENCES

Bolondi, G.. Loinger, E., and Rocca, F., 1982, Offset continuation of seismic sections: Geophys. Prosp., 30, 813-828.

Brown, J. L., 1981, Multichannel sampling of lowpass signals: Inst. Electr. Electron. Eng., Trans. Circuit Theory, CAS 28, 101-106.

Claerbout, J. F., 1984. Imaging the Earth’s interior: Blackwell Scien- tific Publ. Ltd.

Davies E. B., and Mercado, E. J., 1968, Multichannel deconvolution filtering of field recorded seismic data: Geophysics, 33,711-722.

Dereeowski. S. M.. and Rocca, F.. 1981, Geometrical optics and wave theory of constant offset sections in layered media: Geophys. Prosp., 29, 384-406.

Golub, G. H., and Van-Loan, C. F., 1983, Matrix computations: The Johns Hopkins Univ. Press.

Hale, D., 1983, Dip moveout by Fourier transform: Ph.D. thesis, Stanford Univ.

Judson, D. R., Schultz, P. S.. and Sherwood, J. W. C., 1978, Equal- izing the stacking velocities of dipping events via Devilish: Present- ed at the 48th Ann. Internat. Mtg., Sot. Explor. Geophys., San Francisco.

Linden, D. A., 1959, A discussion of sampling theorems: Proc., I.R.E., 47, 1219.

Luenberger, D. G., 1984, Linear and non linear programming: Addison-Wes!w.

Nyquist, H., 192& Certain topics in telegraph transmission theory: AIEE Trans., 47,617-644.

Ottolini, R. A., 1982, Migration of reflection seismic data in angle- midpoint coordinates: Ph.D. thesis, Stanford Univ.

Paige, C. C., and Saunders, M. A., 1982, LSQR: An algorithm for sparse linear equations and sparse least squares: Assn. Comp. Math., Trans. Math. Soft., 8, 195-209.

Papoulis, A., 1977, Generalized sampling expansion: Inst. Electr. Electron. Eng., Trans. Circuit Syst., CAS 24, 652-654.

Rocca, F.. and Ronen, J., 1984, Improving resolution by dip moveout: Presented at the 54th Ann. Internat. Mtg., Sot. Explor. Geophys., Atlanta.

Ronen, J., 1985, Multichannel inversion: Ph.D. thesis, Stanford Univ. Shannon, C. E., 1949, Communication in the presence of noise: Proc.

I.R.E., 37, l&21. Strang, G., 1980, Linear algebra and its applications: Academic Press,

Inc. Yilmaz, O., 1979, Prestack partial migration: Ph.D. thesis, Stanford

Univ.

Wave-equation Trace Interpolation

APPENDIX

INVERSE DMO

983

Different common-offset sections are not independent be- cause they are all reflected from the same reflectors. Many authors (Judson et al., 1978; Yilmaz, 1979; Deregowski and Rocca, 1981; Bolondi et al., 1982; Ottolini, 1982; Hale, 1983) have shown how to get the zero-offset section from common- offset sections. For the trace interpolation, the inverse oper- ator is needed and it is derived in this Appendix.

Prestack full and partial migration

The presence of a single spike in the data set, at midpoint (x, ,3 = (0, 0), and traveltime t,, with a shot at (x, y, z) = (h, 0. 0) and a receiver at (-h, 0, 0), means there is an ellip- soidal reflector such that the distance from any point (x. I’. z) on the ellipsoid to the source plus the distance from that point to the receiver equals the trayeltirne multiplied by the velocity,

~~(x-~,2+~~+Z*+~~~++)~+~~+z~=uth. (A-l)

Points (x, y, Z) in equation (A-l) are on an ellipsoid which can be written as

a, can be found by setting 2 = 0 and r‘ = 0 in the last two equations, i.e.,

Similarly,

a, = a, = V/(ut,/2)2 - h2 = ut,/2,

where t, is the normal moveout (NMO) time

r; = t,: - (2h/U)Z. (A-2)

Therefore, prestack migration of the impulse produces the ellipsoid

or (A-3)

x2

1 + (2/l,/nr,)Z t 4.2 f 22 = vt,/2

L 1 a,

The change of variables

x2 = x2

1 + (2h,ILQ

compresses the ellipsoid (A-3) to the sphere

X2 + $ + z2 = (ot,/2)2,

(A-4)

which is the zero-offset migration of a spike at the NM0 time

t”

Zero-offset migration is described by the dispersion relation

(A-5)

(k, , li,, kZ , and WI, are the wavenumbers that correspond to X, I’, Z, and t,,) Inserting the Fourier transform of the change of variables equation (A-4)

J

into equation (A-5) gives

which is the dispersion relation for prestack, post-NMO, full

migration. Both t, and w, appear in equation (A-6) because the operator is time-variable.

Prestack full migration can be done in three steps (Figure 3):

(1) NMO: Perform the well-known trace-by-trace mapping,

t; = th’ - (2h/a)Z. (A-2)

(2) DMO: Define

(A-7)

co0 is the frequency that corresponds to the time on the zero-offset section.

(3) Migration: Using

[gyk:+k;+k:]=lu:. (A-8)

obtain the migrated section.

Inverse DMO

Equation (A-7) can be used to derive the relation between the zero-offset section m and the common-offset section d.

Starting from the inverse Fourier transform

d(t,, k,, h) = 1 done-‘“+@,, k,, h) J

and using equation (A-7) to change variables o,, to o,, ,

984 Ronen

Defining with

one obtains

and

A = ,/l + (hkJw, t,)‘,

W” = Ao, (A-10)

a0 4l do, - a,

= A-‘, (A-11)

Substituting equations (A-10) and (A-11) into equation (A-9),

d(t,, k,, h) = s

dw, A-le-‘“OA’nm(oo, k,), (A-12)

which implies a method of generating common-offset sections d(t,, x, h) from the zero-offset m(t, , x).

Inverse DMO in three dimensions

In equation (A-l), I made the assumption that the shot and receiver are on the x-axis. In general they are not, and the half-offset vector has two components:

h=

Let 0 be the angle of rotation such that

One can rotate the (x, y) axis to (x’, y’) by

XI [I [ = Y’ -sin0 cost3

so that the shot and receiver are on the x’-axis, and one can make the same development of equations (A-l) through (A- 12) in the (x’, y’) coordinates, obtaining

d(t,, k,. , k,, , 11) = s

dw, A- ‘e-iooALm(wO k,, , k,.), (A-13)

A= Jl + (hk,./o, t,)‘. (A-14)

To return to the original (x, y) coordinates, recall that a rota- tion in the space domain corresponds to a rotation in the same direction in the wavenumber domain:

k X’

(1[

cos 9 sin 0 k, z-z

k Y’ -sin0 cos8 I I k,

Therefore,

hk,, = h(cos Ok, + sin Ok,)

= (h cos O)k, + (h sin O)k,

= h,k, + h,ky

=h.k.

Equations (A-13) and (A-14) become

d(t,, k,, k,, I?) = s

do, A-le-‘“UA”m(oo, k,, k,,)

and

A = \/l + (h - k/w,t,)‘,

where h. k is the linear product of the offset and the wave- number vectors.

Summary

The inverse DMO operator is given by

d(r, k, h) = dw A- ‘e-iWA’m(w, k).

This is the Hermitian transpose to the DMO operator,

m(w, k) = s

dt Am’e-‘OA’d(t, k, h),

that was derived by Hale (1983).