NANJING INSTITUTE OF GEOPHYSICAL PROSPECTING AND INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF GEOPHYSICS AND ENGINEERING
doi:10.1088/1742-2132/2/3/001
J. Geophys. Eng. 2 (2005) 177–187
Optimum stacking of seismic records with irregular noise Yuriy Tyapkin1 and Bjorn Ursin2 1 Seismic Department, Ukrainian State Geological Prospecting Institute (UkrDGRI), 78 Avtozavodskaya St., 04114 Kiev, Ukraine 2 Department of Petroleum Engineering and Applied Geophysics, Norwegian University of Science and Technology (NTNU), SP Andersensvei 15A, N-7491 Trondheim, Norway
E-mail:
[email protected] and
[email protected]
Received 13 April 2005 Accepted for publication 29 June 2005 Published 25 July 2005 Online at stacks.iop.org/JGE/2/177 Abstract Conventional straight stacking has proved to be an effective tool to extract the signal from a multichannel seismic record. However, it maximizes the signal-to-noise ratio only when a rather simple and generally rare model of the record is true. For this reason, some authors try to optimize stacking using more complicated record models. They assume that either the signal amplitudes are allowed to vary from trace to trace in any manner with the noise variances being constant or vice versa. However, in practice, it is commonly the case that these model assumptions are seriously violated. Then, methods based on them become ineffective or even deleterious. We show that these methods produce signal estimates distorted to a considerable extent right up to being absolutely uncorrelated with the sought-for actual signal. This fact motivates our search for new methods for better estimating seismic signals. We therefore introduce a more realistic model that supposes a signal with an identical shape on each trace to be embedded in spatially uncorrelated irregular noise. The signal amplitudes and the noise autocorrelations are allowed to vary across the traces in an arbitrary manner. Given this model, a solution to the maximum likelihood estimation of the signal shape is derived. The effectiveness of the method is highly dependent on the accuracy of determining the signal amplitudes and the noise autocorrelations prior to stacking. We therefore supply the method with estimates of the required parameters. When the noise autocorrelations are trace independent to within a scale factor, the variance, the method becomes much easier to embody and yields the well-known optimum weighted stack (OWS). We compare the performance of the OWS theoretically with that of the straight stack and show that the optimum procedure has obvious advantages over the conventional one. This paper is mainly focused on further developing the OWS. With the complicated record model used, the shortcomings of the above imperfect stacking algorithms can be cancelled. The results of testing our approach to optimum stacking on a variety of field data, some of which are demonstrated, indicate that in many circumstances it can significantly outperform straight stacking and should therefore be prescribed as a better choice than the conventional process. Keywords: optimum stack, maximum likelihood, signal estimation, signal processing, irregular noise
1. Introduction Seismic exploration objectives become more demanding all the time, requiring data with the highest possible signal1742-2132/05/030177+11$30.00
to-noise (S/N) ratios. Limitations to the S/N ratio are common to all signal extraction processes and various methods are employed to bring about improvements. The most widely used and effective general data processing technique
© 2005 Nanjing Institute of Geophysical Prospecting
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for enhancing the S/N ratio of reflection seismograms is stacking (i.e. summing traces sharing a common signal). The most unsophisticated method for stacking is to use the arithmetic mean (conventional straight stacking). Under certain conditions (i.e. equal S/N ratios, irregular stationary noise that does not correlate from trace to trace and a perfectly time-corrected signal), the conventional stack ensures the maximum S/N ratio. In any other case, the straight stack does not maximize the S/N ratio. In general, optimally recovering signals from noisy multichannel seismic data sets is quite a challenge. The problem can be significantly simplified if regular noise is removed perfectly at the beginning of processing. We then have the right to utilize a mathematical model of spatially uncorrelated irregular noise that results in different noise autocorrelations on different traces. Such a model is used in this paper. In turn, the signal component is supposed to have the same shape on each trace, while its amplitudes vary across the traces in any manner permitting even zero and negative values. With this model, the amplitude variation with offset (AVO) effects are taken into account. Given this model, a method for estimating the signal shape, based on the principle of maximum likelihood (ML), is presented. Furthermore, rather simple algorithms are derived for preliminarily determining the signal amplitudes and the noise autocorrelation functions (covariance matrices) necessary to put the method into practice. Provided the noise autocorrelations are identical on all traces to within a scale factor, the variance, the method yields the well-known optimum weighted stack (OWS), in which each trace is multiplied by a weighting factor before stacking (Robinson 1970, White 1977, Brown et al 1977, Rietsch 1980, Bykov 1982, 1984, Tyapkin 1991, 1994b). In fact, the weighting factor is proportional to the signal amplitude divided by the noise variance. In this paper, we shall mainly be concerned with the use of the OWS as an alternative stack. Analytical estimates of the S/N ratio improvement through optimum weighted stacking as compared with the conventional straight stacking are demonstrated. Some other properties of the OWS are explored and compared with those of the conventional stack. Various approaches to weighting factor determination have been proposed in the literature and applied in recent years. Most methods calculate the weights using the correlation coefficients of traces (Robinson 1970, White 1977, Rietsch 1980, Bykov 1982, 1984, Tyapkin 1991, 1994b). In other methods (Brown et al 1977, Anderson and McMechan 1989, 1990), to estimate the optimum weights, the noise energies (variances) are measured using the data before the first seismic arrivals or at late record times. The signal amplitudes are derived using the amplitude-decay rates (Anderson and McMechan 1989, 1990) or the difference in the total energy and the noise energy (Brown et al 1977). Some methods that assume the same model and pursue the same objective are iterative (Ursin 1979, Gimlin et al 1982, Tyapkin 1994a). They avoid calculating the cross-correlation matrix of traces and derive the signal directly from a multichannel record. There are authors who exploit simplified versions of the above model assuming either the signal amplitudes (Brennan 178
1955, Meyerhoff 1966, Krey 1969, Gimlin et al 1982, Bykov 1982, 1984, Kirlin 2001) or the noise variances (Ursin 1979, Ursin and Ekren 1995, Bykov 1984) to be constant on different traces. The approaches using the singular value decomposition (SVD), which is also known in the literature either as the Karhunen–Loeve or as the principal component transformation (Hemon and Mace 1978, Jones and Levy 1987, Freire and Ulrych 1988, Bruland 1989, Ulrych et al 1999, Kirlin 2001, Trickett 2003), may be regarded as relating to the constant noise variance class. It is shown in this paper that, in practice, the simplified methods produce errors in signal estimates since actual data very seldom conform to the assumptions of these methods. These errors may distort the signal estimate to a considerable extent right up to being absolutely uncorrelated with the sought-for actual signal, which seems to go unnoticed in exploration seismology. This fact motivates our search for new methods that avoid such pitfalls. As distinguished from the simplified approaches, our method does not suffer from this shortcoming. In such a situation, where the simplified approaches fail because the model assumptions on which they are based are incorrect, our method, due to its more realistic record model, yields more accurate signal estimates, with a superior S/N ratio. The results of its testing on various field data, some of which are demonstrated, indicate that in many circumstances our method can significantly outperform straight stacking and should therefore be prescribed as a better choice than the conventional process.
2. Maximum likelihood signal estimation Let multichannel data, say a common midpoint (CMP) gather after correcting for gain, statics and normal moveout (NMO), be given in matrix notation as X = S + N,
(1)
where S = {sik } and N = {nik } are, respectively, the signal and noise components of the record X = {xik }, i = 1, 2, . . . , L, k = 1, 2, . . . , M; M is the number of traces and L is the number of samples per trace. The signal is assumed to have the same form, s = {s1 , s2 , . . . , sL }T , but not necessarily the same amplitude, a = {a1 , a2 , . . . , aM }T , on different traces: S = saT
with
sik = si ak .
(2)
Here T denotes transposition. To be more specific, suppose that the noise is independent of the signal, stationary and Gaussian with a zero mean and the ML×ML positive definite covariance matrix Φ having elements ij km = E(nik nj m ),
(3)
where E is an expectation operator. We can now state the problem of the ML estimation of the signal form s. Taking into account the normal distribution of the noise, the problem can be formulated as the minimization of the following weighted quadratic form (e.g. Helstrom (1968)): L M i,j =1 k,m=1
−1 ij km (xik − si ak )(xj m − sj am ).
(4)
Optimum stacking of seismic records with irregular noise
Differentiating (4) with respect to si and setting the result to zero yield L M
−1 ij km (xj m − sj am )ak = 0.
(5)
j =1 k,m=1
Hereafter, the noise is assumed to be irregular. For this reason, the noise on each trace is not correlated with the noise on any other trace and has, in general, individual autocorrelation properties. Formally it can be written as ij km = ij k δkm ,
(7)
Combining (5) and (7) yields the set of linear equations, M k=1
ak2
L
ij−1k sj =
j =1
M k=1
ak
L
Fkm (τ ) = E(xik xi−τ,m ) = E(si ak + nik )(si−τ am + ni−τ,m ) = ak am E(si si−τ ) + E(nik ni−τ,m ) + ak E(si ni−τ,m ) + am E(si−τ nik ), (13) where τ is an arbitrary lag. With the signal and the noise supposed to be mutually uncorrelated, the last two terms on the right-hand side of (13) can be ignored and we obtain Fkm (τ ) = ak am F s (τ ) + Fkn (τ )δkm , s
ij−1k xj k ,
(8)
j =1
which can be written in matrix notation as As = b
denote the autocorrelation functions where F (τ ) and of the signal form and the noise, respectively. To uniquely define the amplitude factors, the energy of the signal form should be normalized to unity:
(9)
Fkm = Fkm (0) = ak am + σk2 δkm ,
where is the noise variance on the kth trace. The result of our attempt to estimate the signal amplitudes depends not only on the choice of the assumed model that matches the actual data, but also on the selection of an appropriate error criterion for minimization. Equation (16) enables the following optimality criterion for estimating the signal amplitudes to be formulated: ak
ak2 Ψ−1 k ,
(10)
ak Ψ−1 k xk ,
(11)
k=1
b=
M k=1
xk = {x1k , x2k , . . . , xLk }T .
(12)
In order to obtain the ML estimate of the signal form, information on the signal amplitudes and the noise covariance matrices is thus required. It is worth noting that due to the Toeplitz nature of the matrices Ψk , their inverses can be easily derived through a fast recursive algorithm (Levinson 1947).
3. Necessary parameter determination The effectiveness of the suggested method is highly dependent on the accuracy of the necessary signal and noise parameter determination, which is a key step in the overall methodology. Therefore, for the method to be feasible, appropriate algorithms, which are based on the same model, are presented below.
(16)
σk2
min A=
(15)
Then the cross-correlation function with zero lag takes the form
with M
(14)
Fkn (τ )
F s (0) = 1.
ij−1k δkm .
=
The cross-correlation function between the kth and the mth traces may be expressed as
(6)
where ij k = i−j,k is the covariance matrix of the noise on the kth trace, while δkm signifies the Kronecker delta. In that case, the inverse covariance matrix of the overall multichannel data has elements −1 ij km
3.1. Signal parameters
M
|Fkm − ak am |.
(17)
k,m=1 k=m
In this criterion, the cumulative least absolute deviation technique is applied in order to approximate the off-diagonal elements of the matrix F and minimize the data misfit. The explanation lies in this norm ensuring a more robust procedure than the conventional least-squares criterion when it handles certain types of errors, e.g. erratic data, and noise distribution, e.g. non-Gaussian (Claerbout and Muir 1973). In our case, some deviations from the ideal model of the matrix F, which is described by equation (16), have such a character. Many factors contribute to these deviations. Among them the main factor is residual regular noise, which is neglected in the aboveformulated record model, but distorts the matrix F in a specific manner (Tyapkin 1991, 1994b). Besides, residual statics and NMO, variations in the signal form from trace to trace, etc, are factors that cause both the actual record and the related matrix F to deviate from their assumed models. Differentiating the functional in (17) with respect to ak and setting the result equal to zero permit the following iteration scheme to be obtained (Tyapkin 1991, 1994b): (i) (i)2 Fkm am am (i+1) = (18) ak , (i) (i) Fkm − a (i) am Fkm − a (i) am m=k
k
m=k
k
ak(i)
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In addition to the choice of an optimality criterion, it is usually advisable to employ constraints in order to regularize the solution. Various a priori information can be introduced in the iterative process by using the universal method of convex projections (Youla and Webb 1982) which permits any number of nonlinear restrictive conditions of a certain type to be subsumed automatically. Let us consider two constraints. The solution allows negative values of ak on some traces. On one hand, this broadens the functional potentialities of the developed method in comparison with those not foreseeing such a case (e.g. Rietsch (1980)). On the other hand, it can considerably reduce the stability of the estimates in the presence of factors causing the above errors, residual statics and NMO foremost among them. If any possibility of recording traces with reversed polarity is excluded, it can be taken into account by projecting each approximation in the iterative process of (18) onto the closed convex set of nonnegative functions (Tyapkin 1991, 1994a). We have also found that, in order to improve the performance of optimum stacking and produce a more correct result, it is useful to impose a constraint on the permissible level of the signal amplitudes. This is intended not only to exclude an unnatural case of ak2 > Fkk , or σk2 < 0, which follows from (16), but also to reduce variations in the signal amplitude estimates. Just as the requirement that ak should not be negative, the restriction ak2 qFkk , where 0 < q < 1 is a constant reflecting a priori information on the average S/N ratio of the input traces, forms a closed convex set. Both stabilizing restrictions can be implemented simultaneously using the projection operator onto the intersection of these two sets (Tyapkin 1991, 1994b): ak < 0 0, 0 ak wk P ak = ak , (19) wk , ak > wk 1
with wk = (qFkk ) 2 . The least absolute deviation technique used requires the number of data to exceed or at least to be equal to that of unknowns. Since the cross-correlation matrix F is symmetric, the number of independent equations in (16) with k = m is M(M − 1)/2. The method can therefore be applied, provided that the record consists of at least three traces. Of course, the result can be expected to be better as M increases. 3.2. Noise parameters The noise covariance matrices are also required to implement the suggested method for estimating the signal form. Let us consider one of the various approaches to the determination of these parameters. Firstly, by dividing any off-diagonal element (k = m) of the matrix F in (14) by ak am , which has been defined above, we can calculate the signal autocorrelation function R s (τ ). Note that averaging the result throughout the off-diagonal elements of F raises the accuracy of the estimate. Then the same set of equations (14) allows the noise autocorrelation function on the kth trace to be determined as Fkn (τ ) = Fkk (τ ) − ak2 F s (τ ). 180
(20)
Finally, since the noise is supposed to be stationary, the elements of its covariance matrix are related to its autocorrelation function in the following way: ij k = Fkn (i − j ).
(21)
4. Optimum weighted stack Let us now consider the case when the noise covariance matrices on all traces are identical to within a normalization factor, the variance: Ψk = σk2 Ψ.
(22)
Substituting in (9) yields the well-known formula for the OWS: s = Xp =
M
pk xk ,
(23)
k=1
where p = {p1 , p2 , . . . , pM }T with M 2 −2 am σm . pk = ak σk−2
(24)
m=1
In this case, the ML estimation of the unknown vector s gets considerably simplified because it requires neither the inversion of the covariance matrices Ψk nor the solution of the set of equations (9) of a large dimension. Given the signal amplitude estimates, the values of σk2 , needed for calculating the optimum weights pk , can be obtained from equation (16) as σk2 = Fkk − ak2 .
(25)
5. Optimum weighted stack versus straight stack Now we implement an analytical comparison of the performance of the OWS with that of the straight stack in the presence of uncorrelated noise. The S/N ratio of the OWS is (e.g. White (1977)) RW =
M
ak2 σk−2 ,
(26)
k=1
while the same characteristic of the straight stack is M 2 M S R = ak σk2 . k=1
In order to show that R W R S , or M 2 M M 2 2 −2 σk ak σk ak , k=1
k=1
(27)
k=1
(28)
k=1
let us take the following two vectors into consideration: f = {f1 , f2 , . . . , fM }T
with
fk = σk
(29)
gk = ak σk−1 .
(30)
and g = {g1 , g2 , . . . , gM }T
with
Optimum stacking of seismic records with irregular noise
Then the proof of (28) can be obtained immediately if we remember the Cauchy–Schwarz inequality (Horn and Johnson 1986): (f T f)(gT g) |gT f|2 .
(31)
Equality in expression (31) arises if and only if the vectors f and g are proportional: g = f · const. Whence ak σk−2 = const.
(32)
That is, the straight stack gives the same result as that of the OWS if and only if the values of ak σk−2 , being proportional to the optimum weights, are identical on all traces. Furthermore, it is obvious that the more the difference between the characteristics ak σk−2 , the more advantages over the conventional stack are gained by the OWS. It may be illustrative to consider the case of identical noise variances. Then, from equations (26) and (27), we get (Bykov 1984) M 2 M W S 2 R /R = M ak ak = 1 + (σa /a )2 (33) k=1
Although the theoretical advantage of the OWS over the straight stack is indisputable, its realization requires rather accurate estimates of the necessary signal and noise parameters. Otherwise, too big errors in the estimates may largely counteract the theoretical benefits of using the OWS. As shown by White (1977, 1984), the OWS ensures the advantage over the straight stack, provided the S/N ratio varies considerably from trace to trace and its average is high enough. The conclusion was drawn after using the multiple coherence function, but in fact it reflects a fundamental property of any estimation procedure. As an argument in favour of this universal conclusion, we consider the results of the testing of various algorithms for optimum stacking that was carried out by Bykov (1984). As the case of (32) is generally rare, optimum weighted stacking should be used as a substitute for conventional straight stacking if the appropriate conditions occur.
6. Causes of low effectiveness of simplified approaches
k=1
with M 1 a= ak M k=1
(34)
and σa2 =
M 1 (ak − a )2 . M k=1
(35)
Thus, the more the relative amount of variation in the signal amplitude from trace to trace, the more benefits are gained by the OWS in comparison with the straight stack. If one or more traces are reversed in polarity, this has a different impact on the OWS and the straight stack. Optimum stacking identifies such traces and then (if required) takes into account their negative sign automatically. Such traces therefore do not affect the OWS. Straight stacking, as opposed to optimum stacking, is not intended to recognize traces with reversed polarity, and the presence of a great deal of such traces may highly distort the signal estimate right up to producing a signal-free stack. It is clear from (26) that a seismic trace, however poor in signal, always contributes positively to the S/N ratio of the OWS, that is, the performance of the OWS improves monotonically with increasing number of the stacked traces, M. Equation (27) enables us to assert that, in general, the straight stack, as distinguished from the OWS, does not guarantee a monotone dependence of the S/N ratio on M. It follows from the same equation that the Mth trace degrades the performance of the straight stack, providing
M−1 2 2 k=1 ak + aM aM M−1 k=1 σk 2 σM >
M−1 2 k=1 ak
M−1 2 k=1 ak + aM aM = , (36) S RM−1 S where RM−1 denotes the S/N ratio of the straight stack of M − 1 traces.
Because of the pivotal importance of distortionless signal estimates, it might be useful, before illustrating the effectiveness of our method on field data and comparing it with that of straight stacking, to dwell a little on the adverse effect of deviations of actual records from the model assumptions that cause related errors. From now on, they are called the model assumption errors. The developed method is based on the record model that admits arbitrary variations in the signal amplitude and the noise variance across the traces. The use of the complicated model is intended to get rid of or at least to minimize the model assumption errors. Such errors are typical of approaches that are based on simplified versions of the adopted model. In order to demonstrate this and to specify the reasons of failure of the simplified approaches, let us consider algorithms that suppose either ak or σk to be identical on all traces. It is relevant to remark that in such cases explicit expressions for the optimum weights can be obtained. For short, in the following the expectation operator is dropped. 6.1. Unforeseen variations in the signal amplitude When the signal amplitudes are supposed to be constant, the vector of optimum weights can be determined as (Bykov 1982) p = (eT F−1 e)−1 F−1 e
with e = (1, 1, . . . , 1)T .
(37)
We can now easily show that the weighted stack, y = Xp,
(38)
equally correlates with any individual trace: FF−1 e e XT XF−1 e = = T −1 . (39) eT F−1 e eT F−1 e e F e Since the stack equally correlates with different, say the kth and mth, traces, −1 M −1 yT xk = yT xm = (eT F−1 e)−1 = Fkm , (40) XT y =
k,m=1
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a = γ1 u1 .
it is not correlated with their difference, y xk − y xm = y (xk − xm ) = 0. T
T
T
(41)
If the signal amplitudes actually vary across the traces, as is usually the case, (41) takes the form yT [(ak − am )s + (nk − nm )] = (ak − am )yT s + yT (nk − nm ) = 0,
(42)
where nk = {n1k , n2k , . . . , nLk } . When the noise energy is small, as compared with the signal energy, the stack may therefore be almost uncorrelated with the actual signal. The situation becomes most pronounced in the limiting case when the noise vanishes. Then the last equation can be reduced to
Thus, in this case all the signal parameters can be extracted directly from X. Inasmuch as ul are orthonormal, post-multiplication of (45) with u1 yields r Xu1 = γl vl uTl u1 = γ1 v1 = γ1 s, (49)
T
yT s = 0
(43)
because ak = am and therefore ak − am = 0. The output of such methods is thus highly sensitive to the non-anticipated variations in the signal amplitude: the less the relative amount of the noise energy, the more these imperfect methods consider the variations in the signal amplitude to be noise and distort the signal estimate. 6.2. Unforeseen variations in the noise variance As the second illustration, let us now consider simplified approaches that suppose the noise to be white with equal variances on all traces, that is ij km = σ 2 δij δkm . Then, expression (4) gets simplified and the problem can be reduced to the conventional least-squares method that is equivalent to minimizing the square of the Frobenius norm (Horn and Johnson 1986) of the data misfit: X − saT 2F =
L M
(xik − si ak )2 .
(44)
i=1 k=1
In order to solve this, we can use the SVD that permits any matrix X to be expressed (exactly) as a sum of specific matrices of unitary rank: X=
r
γl vl uTl ,
(45)
l=1
where r = min{L, M} and γl are the rank and the lth singular value of X; vl and ul are the lth eigenvectors of the matrices XXT and XT X, respectively. It can be shown that the singular values γl are the positive square roots of λl , T vl the eigenvalues of XXT and
T X X, while Tthe eigenvectors and ul are orthonormal vl vm = δlm , ul um = δlm . For convenience, the singular values are supposed to be arranged in non-ascending order: γ1 γ2 · · · γr . The theorem of Eckart and Young (1936) asserts that the best (in a least-squares sense) approximation of the matrix X by another one of a lower rank, say r1 < r, is attained when using the first r1 terms in (45), with the others being omitted. Since in our case the approximating matrix saT is of rank 1, the desired solution is the first term of the SVD, saT = γ1 v1 uT1 ,
(46)
whence, in view of the condition of (15), s = v1 , 182
(47)
(48)
l=1
and it follows that s = γ1−1 Xu1 = γ1−2 Xa.
(50)
Taking into account aT a = γ12 uT1 u1 = γ12 ,
(51)
equation (50) can be represented in the form s = (aT a)−1 Xa,
(52)
which is equivalent to (23) when σk = const. It is seen from (47) and (50) that the sought-for solution can be derived either as the first principal component (Bruland 1989) or as the OWS. Both ways lead to the same result. However, since both the eigenvectors v1 and u1 can be determined only to within a polarity, the vector s has the same property. Owing to that, some traces in seismic sections may be reversed in polarity. This effect makes interpretation through correlation of events difficult. Nevertheless, the causeless sign reversals can easily be revealed and then removed by means of analysis of the cross-correlation between the current trace and the previous one, the function being calculated in a wide time interval. If the noise variances actually vary across the traces, which is a common occurrence, the elements of the matrix F = XT X have the form of (16). Hence, with the increase of the relative amount of the noise, the second term on the right-hand side of (16) prevails over the first one and mainly affects the solution. In the limiting signal-free case, (16) becomes Fkm = σk2 δkm .
(53)
Since F is now diagonal, all components of the first eigenvector are equal to zero except for the one which is equal to unity and has the index equal to that of the trace with maximum noise variance. Thus, the simplified methods exclusively choose such a trace, with the others being absolutely ignored. Consequently, as well as in the previous case of methods supposing constancy of ak , this estimate is entirely uncorrelated with the actual signal, which should be reproduced. On the other hand, when σk are assumed to be trace independent, the energy of the OWS is greater than or equal to that of all other stacking techniques and equal to the maximum eigenvalue λ1 of the matrix F (Bruland 1989). In this connection, let us show how the unforeseen variations in the noise variance influence this characteristic. As the matrices in (16) are both Hermitian, it is convenient to make use of inequalities restricting the eigenvalues of a sum of such two matrices (Wilkinson 1965). It yields (Tyapkin 1994a) σ2 T 2 a a + σmax λ1 Tmax 2 (54) a a + σmin ,
Optimum stacking of seismic records with irregular noise
Distance (m) 1080 1440 1800 2160 2520
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2.0
1.5
2.5
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2.0 C1v 2.5 D3 3.0
3.0
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Figure 1. Comparison of straight stacking (top) and optimum stacking (bottom). Note an enhanced continuity of events with neither change in character nor loss of resolution after optimum stacking when compared with straight stacking. Moreover, note the appearance of a reflection from the Devonian formation (D3) on the OWS that is marked with an arrow. The original CMP gathers before stacking were deconvolved and the time sections after stacking were filtered. (A line from the Dnieper-Donets Depression in the Ukraine).
2 2 where σmax and σmin are the maximum and minimum variances, respectively. It follows from (54) that with a decreasing relative amount of signal energy, the OWS becomes more and more dependent on the trace with maximum noise variance. In the limiting case of vanishing signal, the energy of the 2 OWS is equal to σmax because the procedure chooses this sole anomalous trace with the rest being completely ignored.
1440
2.0
C1v
2880
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C1v
720
Time (s)
1440
1.5
Time (s)
Time (s)
1.5
Distance (m) 1080
Time (s)
720
Figure 2. Comparison of straight stacking (left) and optimum stacking (right). To demonstrate exclusively the results of the different approaches to stacking, no filtering was applied to the data before and after stacking. The data are from the left side of the section shown in figure 1.
Furthermore, when the noise variance on all traces is supposed to be constant, the optimum signal estimate maximizes the sum of the squares of the inner products between this estimate and the individual traces (Bruland 1989). The energy in the estimate cannot therefore be less than that in the trace of maximum energy. This also explains why the signal estimate is now most influenced by the traces with highest noise energy. Despite the above fault, the first few terms of the SVD may be successfully used to enhance the S/N ratio of seismic sections (Hemon and Mace 1978, Jones and Levy 1987, Freire and Ulrych 1988) because after stacking or migration the noise energy is much more stable and lower on different traces than before these processes.
Distance (m) 4035
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5475
6195
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0.8 D2 C 1.5
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D2 C 1.5
Figure 3. Comparison of straight stacking (top) and optimum stacking (bottom). Note a change in the character of the reflections from the Cambrian formation (C −): their prominent continuity on the OWS allows us to correlate events more confidently and accurately. Moreover, after optimum stacking a fault in this formation (marked with an arrow) becomes much more pronounced. (A line from the East European Platform in the Western Ukraine).
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7. Field data examples
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In our experiments, optimum weighted stacking of CMP gathers was carried out using equation (18) to estimate the signal amplitudes and equation (25) to estimate the noise variances, with the restrictions of (19). The effectiveness of the method is demonstrated with various field data sets. Emphasis is put on the comparison of its performance with that of the conventional straight stacking. Figure 1 presents time sections obtained from straight and optimum stacking. A close look at the sections demonstrates an amelioration of the coherence by optimum stacking. Moreover, the OWS permits a weak reflection from the Devonian deposits to be identified. The straight stack does not indicate this reflection because of a relatively low S/N ratio and the screening effect of the overlying rock mass. For a more detailed analysis, the left-hand side of this section is shown in figure 2 on a larger scale. Figure 3 demonstrates how conventional stacking fails to correctly localize a fault in Cambrian sediments, while optimum stacking is a success in solving this problem. It is seen that optimum stacking yields the clearest seismic section, with the best definition and correct position of the fault. Figure 4 is a good example of the capabilities of optimum stacking. It can be clearly seen here that some parts of the time section obtained using straight stacking are of inferior quality because of distortion by appreciable irregular noise. With optimum stacking the severe noise is greatly diminished and the refinement of the data is significant, which favours a more confident correlation of reflections. The OWS technology for noise attenuation has been found to work well for both land and marine data. Results of processing a real data set from an offshore seismic field experiment are demonstrated in figure 5. They show that after optimum stacking, the seismic section is more regular and the spatially coherent noise in the right-hand side of the section is
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Figure 4. Comparison of straight stacking (top) and optimum stacking (bottom). Note the apparent suppression of irregular noise by optimum stacking producing a stratigraphically viable and interpretable section. (A line from the Timan-Pechora Oil and Gas Province in Russia).
Thus, as with the methods supposing constancy of ak , the methods supposing constancy of σk cannot resist the model assumption errors. For this reason, both classes of imperfect approaches must be avoided in case of suspected deviations of actual records from the assumed oversimplified model. Because of these shortcomings, such methods become ineffective or deleterious and are not widely used in practice. The results of their testing on field data have even made some geophysicists question the optimization of stacking. A situation like this may be regarded as a challenge to devise methods that ensure a more reliable estimation of the signal, being based on more realistic record models. The results of testing our approach on field data, some of which are given below, allow us to look optimistically at the prospects of such development. These results also confirm that with the more complicated mathematical model adopted in the present Distance (m) 11980
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Figure 5. Comparison of straight stacking (left) and optimum stacking (right). Note especially the apparent suppression of regular noise in the right side of the section after optimum stacking and a more continuous basement of a hypothetical reef in deposits of the Paleogene period (P −) that is marked with an arrow. (A line from the North-Western continental shelf of the Black Sea in the Southern Ukraine).
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Figure 6. Comparison of straight stacking (left) and optimum stacking (right). Note the more detailed and clear character of weak reflections from mainly terrigenous sediments of the Carboniferous period (C) and, in particular, evidence of cross-bedding in some stratigraphic intervals after optimum stacking (one example is marked with an arrow). Moreover, note an apparent structural unconformity between Turnaisian terrigenous sediments and overlying Turnaisian–Visean (C1t-v) carbonate interval. (A line from the Dnieper-Donets Depression in the Ukraine). Distance (m) 13200
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in some stratigraphic intervals becomes more apparent. Furthermore, reflections from the Turnaisian terrigenous sediments gain a less inherited character and look more unconformable to those from the overlying Turnaisian and Visean carbonates. It may be interpreted as a result of a more effective multiple removal. These waves are related to the carbonate interval which is notable for its sharply heightened acoustic impedance, relative lateral uniformity and considerable thickness. The interval therefore originates a high level of multiples masking weak reflections from underlying boundaries, which we are trying to enhance. Some advantages of optimum stacking over conventional straight stacking are demonstrated in figure 7. It is seen that there is a significant improvement in the interpretability of the data after applying the optimum procedure, specifically within the time interval 0.5–0.7 s, due to a more effective suppression of regular and irregular noise. The real data examples presented so far indicate that in many cases optimum stacking may be preferred to straight stacking and more successfully applied to the S/N ratio enhancement in order to provide favourable conditions for confident and detailed interpretation of seismic data. They also provide verification for the complicated record model that is the basis for optimum stacking.
8. Discussion and conclusions
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Figure 7. Comparison of optimum stacking (top) and straight stacking (bottom). Note the superior data quality after optimum stacking that favours a more confident correlation of reflections. (A line from the Dnieper-Donets Depression in the Ukraine).
partially suppressed. Besides, on the OWS, the basement of a hypothetical reef of Paleogene age becomes more pronounced and continuous. Figure 6 exhibits much the same results with land data. It illustrates how the section quality improvement gained due to optimum stacking ensures a more detailed analysis of weak reflections from mainly terrigenous sediments of Carboniferous age. Specifically, cross-bedded structure
Stacking is the industry’s most effective and widely used seismic data processing technique based on the S/N ratio enhancement. Therefore, any attempt to improve its performance makes sense. This paper has focused on a solution to the problem. Generalized mathematical models for the signal and irregular noise of multichannel records have been adopted at the outset. The signal component is assumed to have an identical shape on each trace, while its amplitude is allowed to vary across the traces in an arbitrary manner permitting, in general, zero and negative values. The noise on any trace is correlated with neither the signal nor the noise on any other trace and has an arbitrary autocorrelation. Given this model, a method for the ML estimation of the signal shape has been derived. Without providing the method with accurate estimates of the necessary signal and noise parameters, its practical value is open to questions. Therefore, in order to ensure the feasibility of the method, a robust algorithm has been developed for evaluating the parameters needed. It is based on the same record model. Importantly, this rather simple iterative algorithm lends itself to the incorporation of various a priori information via the universal technique of convex projections which permits any number of nonlinear constraints of a certain type to be subsumed automatically if such information is available. For example, it has been pointed out that forcing the signal amplitude estimates to be non-negative and bounded in level proves useful in many cases. The accuracy of the required signal and noise parameter estimates as well as the effectiveness of the overall methodology depends, of course, on the degree to which the 185
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assumption that the signal and the noise on different traces are mutually uncorrelated is actually met. Prior to stacking, spatially coherent noise should therefore be removed as well as possible. Clearly, this is a vital pre-processing step. When the noise autocorrelations on different traces are identical to within a scale factor, the variance, the method reduces to optimum weighted stacking. This procedure is much simpler in realization since it obviates determining the noise covariance matrices, their inversion, and solving a set of equations of large dimension as well. Optimum stacking, as compared with the conventional straight stacking, yields better signal estimates. Also, it is insensitive to polarity reversal of some traces and its performance improves monotonically with the increase of the number of stacked traces. Unfortunately, this is not always the case for the conventional straight stack. With rather convincing examples taken from a variety of land and marine data sets, we have the right to look optimistically at the prospects of stacking optimization and to revise the somewhat pessimistic conclusions of White (1977) and Anderson and McMechan (1990) about the limitations of optimum stacking. The encouraging results of testing the method demonstrate that there are many circumstances in which optimum stacking is justified and preferable in spite of its somewhat higher computational cost than that of straight stacking. They also show the reliability of the complicated record model adopted in the present paper which is at the heart of the overall signal estimation philosophy. We have adopted a complicated record model in order to get rid of or at least to minimize the model assumption errors. These errors refer to deviations of actual records from the assumed model. It turns out that such errors are typical of the methods that are based on simplified versions of the adopted model, e.g. supposing either the signal amplitudes or the noise variances to be identical on all traces. In these cases, signal estimates in reality may be greatly affected by such errors right up to being absolutely uncorrelated with the actual signal. That is likely why these simplified approaches are not widely used in practice. From this, the rationale of using more realistic models becomes clear. Our method was developed with that purpose in mind. Being based on a more complicated record model, our derivation avoids the assumptions of the simplified methods. This may be regarded as a challenge to further develop methods that ensure even more accurate signal estimates, being based on even more realistic record models. Finally, our method is appropriate for both CMP and vertical stacking. It is expected to provide a basis for more effective fulfilment of some seismic data processes, such as residual static and NMO correction, deconvolution, AVO analysis, etc.
Acknowledgments We wish to thank the management of the state geological enterprises Ukrgeofizika (Ukraine) and Pechorageofizika (Russia) for their kind permission to publish the field data examples shown. 186
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