Zero- and first-order approximations for least-squares estimation of seismic signal with coherent and random noise Yuriy Tyapkin1, Bjorn Ursin2, Herve Perroud3, Olena Silinska1 and Olena Tiapkina2

1 Seismic Department, Ukrainian State Geological Prospecting Institute 78 Avtozavodska 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 3 MIGP UMR5212 UPPA-CNRS-TOTAL, Université de Pau et des Pays de l'Adour, BP1155-64013 Pau Cedex, France E-mail: tyapkin@ukrdgri.gov.ua, bjorn.ursin@ntnu.no, herve.perroud@univ-pau.fr, olena@ukrdgri.gov.ua and olena.tiapkina@ntnu.no Abstract

We have proposed a new least-squares method for signal estimation with a complicated and therefore more realistic mathematical model of the multichannel seismic record containing random noise and an arbitrary number of coherent noise wavetrains. The signal and all the coherent noise wavetrains are supposed to bear individual trace-independent waveforms being mutually uncorrelated in time stationary stochastic processes. The amplitudes and arrival times of these record components vary from trace to trace in an arbitrary manner. Random noise is assumed to be a stationary stochastic process uncorrelated with the signal and all the coherent noise wavetrains and from trace to trace as well. Its spectral (autocorrelation) function is trace-independent to within a scale factor, the variance. Under certain conditions, the method may be reduced to two successive stages, namely preliminary subtraction of estimates of all the coherent noise wavetrains and final estimation of the signal from the residual record. At both stages, optimum weighted stacking is used with reference to the variances of random noise and to the amplitudes and arrival times of the corresponding coherent component. A simplified scheme and an advanced scheme for subtracting coherent noise are proposed, which are called the zero-order and first-order approximations, respectively. The first of them can be thought of as the generalization of a conventional approach for subtracting coherent noise to the complicated data model adopted in this paper. The second scheme has an obvious advantage over the first scheme, since it allows the distortions that appear when estimating and subsequently subtracting the coherent noise wavetrains to be compensated. A simulation on synthetic data shows the efficiency of the first-order approximation, and it provides a qualitative and quantitative comparison of those results with the results given by the zero-order approximation. Also, testing the zero-order approximation exploiting the singular value decomposition on synthetic and actual data sets demonstrates the advantage of this method over f-k filtering combined with subsequent straight stacking.

Keywords: least-squares signal estimation, coherent noise, random noise, optimum weighted stacking, complicated mathematical model 1. Introduction When searching for and prospecting of hydrocarbon traps in sedimentary basins, geophysicists are often faced with the problem of recovering signal from multichannel seismic data sets contaminated by both

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spatially coherent and random noise, which is quite a challenge. Coherent noise may be represented by, to name a few, multiples, converted, air, refracted, guided waves, ground roll, etc. Among both kinds of noise, it is coherent noise that is the most persistent problem in seismic imaging because it can obscure, or sometimes be mistaken for, genuine reflections. A number of techniques have been developed to attenuate coherent noise: optimum weighted stacking (Schoenberger 1996), optimum array filtering (Simaan and Love 1984), radial trace filtering (Henley 2003; Zhu et al. 2004), plane-wave prediction filtering (Spitz 1999; Fomel 2002), f-k filtering (Fail and Grau 1963), Radon (τ-p) transform (Hampson 1984; Kelamis and Mitchell 1989; Sacchi and Ulrych 1995), spectral matrix filtering (Mars et al. 1987; Gounon et al. 1998). Furthermore, various modifications of the singular value decomposition (SVD), also known in the literature either as the Karhunen-Loeve or as the principal component transformation, operating in the time or frequency domain (Freire and Ulrych 1988; Trickett 2003; Tyapkin et al. 2004, 2007), are popular tools for removing coherent and random noise. Recently, the SVD-based methods have gained considerable momentum and, in order to take advantage of high order statistics, have been combined with independent component analysis (Vrabie et al. 2004; Bekara and van der Baan 2006). Now, the search for the most convenient and most seismically revealing transformations, which may reach the industrial stage, actively continues. In the 1960s-1970s, in order to solve the problem of retrieving seismic signal imbedded in coherent and random noise, the majority of publications exploited multichannel Wiener filters (Schneider et al. 1965; Meyerhoff 1966; Sengbush and Foster 1968; Galbraith and Wiggins 1968; Cassano and Rocca 1973, 1974) or maximum-likelihood (least-squares) estimators (Nakhamkin 1966), from now on referred to as optimum methods. Then the situation changed and the interest of geophysicists in these methods gradually was lost because of their insufficient effectiveness. Their place was taken by methods implying simplified mathematical models of seismic data, with f-k filtering and Radon transform filtering being best known among them. We refer to them as non-optimum methods. Even though these filters are usually faster and more cost-effective, they are inappropriate to handling the seismic data with a small number of traces, non-uniform spatial sampling, spatial aliasing of noise, and channel inequalities. The most undesirable aspect of them is the mixing of the data, which is usually inherent in these processes, producing a wormy appearance in the output data. This leads to signal distortion and spatial correlation of background noise. The corruption of seismic signal components by these noise attenuation processes is clearly undesirable, and provides us with motivation to seek a new, more effective technique to address the noise attenuation problem. In our opinion, the optimum methods are often less effective because they exploit, first, imperfect mathematical models of the record and, second, an imperfect scheme for subtracting coherent noise. For this reason, in the present paper, we make an attempt to rehabilitate these methods and to reanimate the interest of geophysicists in them. With this purpose in mind, we utilize a more complicated and realistic mathematical model of the record. In this generalized model, the signal component is supposed to be distorted by random noise and an arbitrary number of coherent noise wavetrains. Both noise components are stationary zero-mean Gaussian stochastic processes uncorrelated with each other and with the signal. The signal and all the coherent noise wavetrains are characterized by individual trace-independent waveforms, whereas their amplitudes and arrival times vary across the traces in any manner. In turn, the random noise does not correlate from trace to trace, and its autocorrelation function is assumed to be identical to within a scale factor, the variance, on different channels. The rationale behind this generalized record model, which admits the trace-dependent variability into the signal and noise characteristics, is that it allows us to get rid of or at least to minimize the model assumption errors, which refer to deviations of actual records from the assumed model, and thus to obtain more reliable signal estimates. Tyapkin and Ursin (2005) give a good explanation of such errors in the coherent-noise-free case. With this unconventional mathematical model given, we derive a least-squares optimum method for estimating the underlying signal. The method combines the advantages of the more realistic mathematical data model with those of an advanced scheme for subtracting coherent noise that reduces the corruptive mutual impact of different coherent noise wavetrains. This paper has the following structure. Starting from the generalized mathematical model of the multichannel seismic record, we describe the theory of the proposed least-squares method. In Appendices A-C, we consider the coherent-noise-free case, investigate the signal-to-noise (S/N) ratio at the output from the method and also show that if this S/N ratio is rather high, the multichannel Wiener filter yields almost the same result. Next, we impose two assumptions on the data and demonstrate that under these conditions, the method may be considerably simplified and turns into a two-stage process of successively subtracting all the coherent noise wavetrains and subsequently evaluating the signal on the residual

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record. Then, the solution obtained is analyzed. Specifically, a simplified (conventional) scheme and an advanced scheme for subtracting coherent noise are discussed, which are called the zero-order and first-order approximations, respectively. After that, we demonstrate the performance of the first-order approximation on synthetic data sets contaminated by coherent noise of various intensities and compare it with that of the zero-order approximation. Finally, the zero-order approximation using the SVD is tested on simulated and actual data sets and compared with conventional f-k filtering combined with subsequent straight stacking. 2. Mathematical model of the multichannel seismic record Let us first formulate the mathematical model of the record. Suppose that the ith trace of the record that contains N traces may be written as

( ) ( )( ) ( )( ) ( ) Nitntrbtsatu i

L

lilliliii ..., ,1 ,

1rs =+−+−= ∑

=ττ . (1)

Here the signal is described by the first term on the right-hand side of (1) and assumed to have an

identical waveform s(t) on each trace, with arbitrary trace-dependent amplitudes and time delays ia ( )isτ . The amplitudes are permitted to have zero (a signal-free trace) and negative (e.g., due to the AVO-effect) values. The second term represents a superposition of coherent noise wavetrains with individual waveforms . Each of the wavetrains, as well as the signal, bears arbitrary trace-dependent amplitudes b

( ) Lltrl ..., ,1 , =

il and time delays ( )ilrτ . Random noise, in turn, is expressed by the third term and supposed to be a stationary zero-mean Gaussian stochastic process uncorrelated from trace to trace, with identical to within a scale factor, the variance , autocorrelations (power spectra) on different traces. All the coherent noise waveforms and the signal are also assumed to be stationary zero-mean Gaussian stochastic processes uncorrelated with random noise and with each other. To uniquely define the amplitudes of the signal and all the coherent noise wavetrains, we suppose the variances of their forms to be normalized to unity.

2iσ

Due to the above assumptions, the cross-correlation function of the total noise between channels i and j may be expressed as

( ) ( ) ( ) ( )( ) ( )( ) ijiiljll

L

ljlilij RRbbR δτσττττ n

2rrr

1- ++=∑

=, (2)

where ( )τlR )r( and ( )( )τnR are the autocorrelation functions of rl(t) and any ni(t), respectively, at the time lag τ , whereas ijδ signifies the Kronecker delta function.

Fourier transformation of (2) yields the related cross-spectrum:

( ) ( ) ( ) ( ) ( )( )[ ] ( ) ( ) ijiiljll

L

ljlilij RRbbR δωσττωωω n

2rrr

1-iexp += ∑

=, (3)

where ( ) ( )ωlR r and ( )( )ωnR are the power spectra of rl(t) and any ni(t), respectively, at the angular frequency ω . 3. Least-squares signal estimation Given this model, let us state the problem to obtain the best in least-squares sense estimate of the signal shape s(t) in the frequency domain. With this purpose in mind, we represent the signal component as fs, where the column vector f reads ( )( ){ ( )( )}Hs1s1 iexp ..., ,iexp NNaa ωτωτ=f , the scalar s is the Fourier spectrum of s(t), and the superscript H denotes complex conjugate (Hermitian) transpose. For shot, here and in the following the functional dependence on frequency is dropped.

4

The least-squares estimate s) of s minimizes the quadratic form (Helstrom 1968) ( ) ( ufRuf −− − ss ))) 1H (4) and can be represented as

uhH=s) , (5) where

( ) 1H11HH −−−= RffRfh , (6)

the column vector contains the Fourier spectra of all the observed traces { H**1 ,..., NUU=u } iU ( )tui , i = 1,

…, N, R is the matrix whose entries are defined by (3), -1 signifies matrix inverse, and the superscripted asterisk stands for complex conjugation. In order to obtain the sought-for estimate of s, we thus need to invert the composite matrix R. With this aim, let us represent this matrix in the form (Tyapkin et al. 2007)

( )DGBGR nH R+= , (7)

where { } ( )( ){ } , ..., ,1 , ..., ,1 ,iexp r LjNib ijijj ==−== ωτgG is an N by L matrix that consists of the

column vectors ( )( ){ ( )( )}Hr1r1 iexp ..., ,iexp NjNjjjj bb ωτωτ=g , ( ) ( ){ }LRR r1r ..., ,diag=B an L by L diagonal

matrix, and { }221 ..., ,diag Nσσ=D an N by N diagonal matrix.

Then we can take advantage of the method described by Horn and Johnson (1986) and obtain

( ) ( )( )1H11n

11n

1 −−−−−− −= DGGVIDR RR , (8) where I is an identity matrix,

( ) GDGBV 1H1n

1 −−− += R . (9) To calculate 1−R , it is thus necessary to invert V. Now we can show that under certain conditions, (8) may be simplified for ease of use. To this end, let us first represent the entries of the matrix from (9) as

GDG 1H −

( ) ( )( )[ ]∑=

− −==N

kkjki

k

kjkijiij i

bbc

1rr2

1H exp ττωσ

gDg Lj ..., ,1,, i = , (10)

or as the inner product of the modified vectors : ii gDg 2/1~ −=

jiijc gg ~~ H= . (11)

It is worth noting that measures the spatial interference of coherent noise wavetrains i and j at the current frequency with regard for the presence of random noise.

ijc

Expand V from (9) into three matrices:

210 VVVV ++= , (12)

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where comprises only the diagonal entries of the matrix : 0V ( ) GDG 1H1

n−−R

( ) { LLccR ..., ,diag 111

n0−=V }, (13)

1V contains only the off-diagonal entries of : ( ) GDG 1H1n

−−R

( ) jicRV ijij ≠= − ,1n1 , (14)

whereas

( ) ( ){ }1r

11r

12 ..., ,diag −−− == LRRBV . (15)

Now we can represent V in the form

( )[ ] ( EIVVVVIVV +=++= −021

100 )

)

, (16) where , which permits the first condition to be introduced. Let a matrix norm of E , say

( 211

0 VVVE += −

( )

( ), ,maxmax

r

n jicR

Rcc

eiiiii

ijij ≠+==∞E (17)

satisfy the inequality

( )

( )1max

r

n <+iiiii

ij

cRR

cc

. (18)

Then the inverse matrix ( ) 1

011 −−− += VEIV can be expanded into the infinite series (Horn and

Johnson 1986)

( ) 10

0

1 −∞

=

− ∑ −= VEVm

m . (19)

After restricting this series by the second term, we obtain

( ) ( )[ ] 1021

10

10

1 −−−− +−=−≈ VVVVIVEIV . (20)

Substituting this expression into (8) yields, after some easy algebra,

( ) ( ) ( )[ ]{ }( ) ( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡+∆−−=

+−−=

∑ ∑∑= ≠

−−

=

−−−

−−−−−−−

L

l lk kk

kk

ll

lll

L

l ll

llccc

R

RR

1

1H1H

1

1H11

n

1H1021

10

1n

11n

1

1 DggDggDggID

DGVVVVIGIDR

, (21)

where

6

( )

( ) llll cR

R

r

n=∆ . (22)

In turn, combining (5), (6) and (21) allows the following signal estimate to be obtained:

( ) uDggDggDgg

IDf⎥⎥⎦

⎤

⎢⎢⎣

⎡+∆−−= ∑ ∑∑

= ≠

−−

=

−−−

L

l lk kk

kk

ll

lll

L

l ll

ll

ccccs

1

1H1H

1

1H1H1 1) , (23)

where

∑∑∑= ≠=

+−=L

l lk kkll

klklL

l ll

l

ccccc

cc

cc1

*ss

1

2s

s , (24)

∑=

− ===N

i i

iac1

2

2H1H

s~~

σfffDf , (25)

( ) ( )([∑=

− −===N

kklk

k

klklll

bac1

rs2H1H

s iexp~~ ττωσ

gfgDf )], (26)

fDf 21~ −= . (27)

Now we introduce the second condition. If the record obeys the inequality

1<<∆l (28) for any l, we can neglect the second term in the parentheses on the right-hand side of (23) and thus obtain a somewhat simpler expression:

uDggDggDggIDf ⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∑ ∑∑

= ≠

−−

=

−−−

L

l lk kk

kk

ll

llL

l ll

llccc

cs1

1H1H

1

1H1H1) . (29)

4. Analysis of the solution It might be useful, before illustrating the effectiveness of the described method with synthetic and field data, to dwell a little on its analysis. The structure of (29) permits a straightforward interpretation. Following this equation, we should first estimate and then subtract all the coherent noise wavetrains from the data u. For this aim, the shape of each coherent noise is estimated using optimum weighted stacking represented by the term

. This operation is performed with reference to the variances of random noise and the amplitudes and arrival times of this coherent noise (see Appendix A). Then the result is multiplied by in order to obtain the ultimate estimate of this coherent noise wavetrain, with its amplitudes and arrival times on all the traces. The entire procedure is represented by the second term from the parentheses on the right-hand side of (29).

uDg 1H1 −−lllc

lg

From this equation, it is also apparent that the third term in the parentheses is intended to compensate for the entire distortion caused by the mutual impact of all the coherent noise wavetrains present in the data. Indeed, when estimating the lth wavetrain, all the other wavetrains take part.

In this process, the kth wavetrain, the estimate of which is , participates in the estimation of

the lth wavetrain in accordance with the formula . Summation over all l and allows the entire distortion to be calculated and then compensated.

) ( lk ≠

uDgg 1H1 −−kkkkc

uDggDgg 1H1H11 −−−−kkllkkll cc

lk ≠

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We should point out that the compensator in (29) in not required if the coherent noise wavetrain is single (L = 1). In this special case,

uDggIDf ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−−−

11

1H111H1

ccs) , (30)

where

11

21s

s cc

cc −= . (31)

Also, after representing (29) as

uDggDggIDf ⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∑∑∑

= ≠

−

=

−−−

L

l lk kkll

klklL

l ll

llcc

cc

cs1

1H

1

1H1H1) , (32)

it is clearly seen that there is no need in such a compensator if the signal is corrupted by several (at least two) coherent noise wavetrains and all the corresponding vectors are mutually orthogonal, i.e. if

for all . In this case, (29) turns into lg~

0=lkc lk ≠

uDggIDf ⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑

=

−−−

L

l ll

llc

cs1

1H1H1) , (33)

where

∑=

−=L

l ll

l

cc

cc1

2s

s . (34)

However, we consider this orthogonality assumption unreal because it is too rarely met in practice and may be the case only at separate frequencies. As the vectors depart from being mutually orthogonal, the detrimental effect from the rough estimator of (33) increases and the need of the advanced estimator of (29) grows.

lg~

Once all the coherent noise wavetrains have been estimated and subtracted, the residual data undergoes optimum weighted stacking described by the term in (29) and intended to obtain the final signal estimate. This process is performed with regard for the variances of random noise and the amplitudes and arrival times of the signal. When coherent noise is absent, the term c in (29) is equal to

(Appendix A). In general case the value of c, represented by (24), accounts for the corruptive impact on the signal of both subtracting coherent noise and compensating for the mutual influence of all the coherent noise wavetrains. This can easily be shown by substituting the signal fs instead of the entire data u in (29). From (24), it is evident that the equality

1H1 −− Dfc

sc

scc = could also be the case with the vector f~ orthogonal to all the vectors , when lg~ 0s =lc is valid for all l. However, this condition, as well as the above assumption of the mutual orthogonality of all , does not comply with reality. lg~

We call the procedure described by (29) the first-order approximation in contrast to the zero-order approximation represented by (33). The use of the first-order, rather than the zero-order approximation has a distinct advantage: the advanced approximation accounts for the mutual impact of all the coherent noise wavetrains, whereas in the rough approximation this effect is neglected. In fact, (33) is the generalization of a conventional unsophisticated scheme for estimating seismic signal contaminated by both coherent and random noise to the complicated data model adopted at the outset. This scheme is proposed for simple data models by many researchers, including Linville and Meek (1995), Chiu and Butler (1997), Al Dossary et al. (2001), and Lu et al. (2003), who follow the principle

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that pre-subtraction of strong coherent noise produces less distortions than does direct estimation of the signal. The result of our study may be considered to be a theoretical background for such a heuristic approach and permits one to realize under what conditions this rough scheme can be justified. For instance, this scheme may be quite effective with a single coherent noise wavetrain [see (30)]. Let us analyze the two constraints imposed on the data in the previous section in order to obtain (29). The first of them is represented by (18). This condition may be open to questions or even inapplicable if at least a pair of coherent noise wavetrains with both a small difference between their arrival times and a large difference between their amplitudes occurs in the data. Indeed, since

( )

( )

( )

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛≤+

ii

ij

iiiii

ij

iii cc

cRR

cc

cRR

maxmaxmaxr

n

r

n , ji ≠ , (35)

(18) is valid if

( )

( )1maxmax

r

n <⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

ii

ij

iii cc

cRR

, (36) ,ji ≠

which is equivalent to

( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−<⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

iiiii

ij

cRR

cc

r

nmax1max , (37) .ji ≠

The weak wavetrain, say i, is characterized by a relatively small value of and thus reduces the right-hand side of (37). At the same time, the strong wavetrain, say j, entails a relatively large value of

iic

ijc and thus raises the left-hand side of (37). For this reason, as the difference between the amplitudes

increases and also the difference between the arrival times decreases, the validity of (18) falls down. If

ijc is so large that satisfies the inequality

( ) ( )n1

r RRcc iiiij−−> , (38)

(18) changes its sign, and therefore expansion into the series of (19) becomes illegal. In this case, in order to avoid a detrimental effect, all the relatively weak constituents of coherent noise not satisfying inequality (18) should simply be ignored at the subtraction stage. It means that this part of coherent noise is supposed to be indistinguishable from random noise and may therefore be suppressed at the final stage of signal estimation. Before analysing the second constraint, let us compare (29) with (33). For the simplified solution described by (33), a priori information on the amplitudes and arrival times of all the coherent components and on the variances of random noise is necessary. The solution of (29) is much more inconvenient to embody for the following two reasons. First, it requires, in addition, a priory information on the power spectra of all the noise components and, second, it cannot be represented in the time domain as a combination of two frequency-independent processes, as in the case of (33). The second constraint is represented by (28), which is equivalent to

( ) ( ) .1r1

n >>−lllcRR (39)

The left-hand side of this inequality characterizes the accuracy of estimating the lth coherent noise waveform. In fact, it is the ratio of the power spectrum of the lth coherent noise wavetrain to that of random noise at the output from optimum weighted stacking performed with regard for the amplitudes

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and arrival times of this wavetrain and for the variances of the random noise (see Appendix A). From (10), it follows that this quantity may be represented as

( ) ( )( )

( )

( )∑∑==

− ==N

k

lk

N

k k

klllll R

R

bRcRR

112

n

2r

r1

n σ, (40)

where is the ratio of the above two spectra on the kth trace. Relation (40) implies, however, that it is

not obligatory for all to be high in order to satisfy this constraint. The condition of (39) can be met

even when the value of is rather small on each trace but the number of traces involved in this process is large enough. As well as in the case of the first constraint, if some coherent noise wavetrain is too weak as compared with random noise and therefore does not satisfy (39), it should simply be ignored at the subtraction stage.

( )lkR

( )lkR

( )lkR

Since one of the main characteristics of any process is the output S/N ratio, it is instructive to derive this characteristic for the signal estimator obtained. This issue is discussed in Appendix B. It is shown that subtracting estimates of coherent noise with the use of the zero-order approximation reduces c as compared with cs and thus reduces the S/N ratio at the output from the least-squares estimator with respect to the maximum value attained in the coherent-noise-free case. This reduction is partially compensated by taking into account the mutual impact of coherent noise wavetrains in the first-order approximation.

( ) ( )s1

ns RRc −

In order to recover the signal, the multichannel Wiener filter can be utilized instead of the least-squares estimator (Tyapkin et al. 2006). In Appendix C, it is shown that if the estimator enables a rather high S/N ratio at the output, the Wiener filter leads to the same solution of the problem. Although, from a theoretical point of view, the advantage of using the more complicated mathematical model looks indisputable, its practical 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 this more realistic model. This conclusion was drawn by White (1977; 1984) for the coherent-noise-free data, when the signal is estimated via optimum weighted stacking. We extend this conclusion to the more complicated data with coherent noise because in this case both stages of the signal estimator also are based on optimum weighted stacking. 5. Synthetic data examples The application of the described method thus requires good a priori information on the variances of random noise and on the amplitudes of both the signal looked for and coherent noise. In general, it is difficult, however, to find a reliable source of this information. For this reason, when testing the first-order approximation, we used synthetic data with these parameters being trace-independent. In this case optimum weighted stacking, which is utilized at both stages, turns into straight stacking. It was also supposed that the time delays of both the signal and coherent noise were known exactly. In our first experiment demonstrated below, emphasis was put on the comparison of the first-order and zero-order approximations. Even though the basic model in that simulation experiment was simplified, we could show both qualitatively and quantitatively the advantage of the advanced scheme for subtracting coherent noise over the conventional scheme. To compare the performances of both schemes, we created a set of synthetic records, each of which was composed of 21 traces with the sampling rate of 1 ms. The signal component was chosen to be identical for all the records and aligned in time. It was contaminated by two overlapping linear coherent noise wavetrains with the dips of 1ms and -1ms per trace and various relative, in regard to the signal, amplitudes ranging from 1 to 16 on different records. The waveforms of the signal and both coherent noise wavetrains were generated by convolution of independent stochastic processes with a 20 Hz Ricker wavelet. Random noise was chosen to be negligible. Figures 1(a) to (d) display the entire record along with its components for the noise-to-signal ratio of 4:1. Here, in order to obtain the same amplitude range, the amplification factor for visualization of both the coherent noise and entire data was chosen 4 times smaller than that for the signal. It is obvious that the

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coherent noise is so strong that before it is removed, there is almost no signal that can be identified in figure 1(d).

Figure 1. (a) Signal, (b) positively and (c) negatively dipping coherent noise and (d) entire data for the noise-to-signal ratio 4:1. Note that the coherent noise is so strong that before it is removed, there is almost no signal that can be identified on panel (d). First, we tested the zero-order approximation. In figure 2, we illustrate that this scheme is incapable of subtracting the coherent noise from the data depicted in figure 1(d). It is clearly seen that the residual record before feeding to the final signal estimation process [figure 2(a)] contains large artefacts not allowing the signal to be identified. As a result, this record differs considerably from the original signal, ten traces of which are put on the right-hand side of each record here and in figure 3 for comparison. This distinction is also confirmed by the difference between the residual data and the signal [figure 2(b)], which has higher amplitudes than the original. For this reason, the subsequent straight stack does a poor job of the ultimate signal approximation, and one can see a contrasting boundary between the signal and its severely damaged estimate in figure 2(c). This is a typical result of delay and sum filter. Thus, we can infer that a small difference between the noise apparent velocities is a strong handicap for this rough process.

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Figure 2. (a) Result of subtracting the coherent noise using the zero-order approximation, (b) difference between the residual data and the signal, (c) final signal estimate. Note that the signal estimate differs drastically from the original, ten traces of which are put on the right-hand side of each panel here for comparison.

A much better signal estimate is obtained using the first-order approximation (figure 3). The first step of this process removes the coherent noise almost perfectly, thereby producing the result that can hardly be distinguished visually from the signal [figure 3(a)]. This is also confirmed in figure 3(b) by much lower amplitudes of the remnants as compared with those shown in figure 2(b). As a result of the more effective noise removal, the final estimate has a striking resemblance to the signal, and one therefore can hardly find the boundary between the signal and its estimate in figure 3(c).

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Figure 3. (a) Result of subtracting the coherent noise using the first-order approximation, (b) difference between the residual data and the signal, (c) final signal estimate. Note that the signal estimate has a striking resemblance to the original, ten traces of which are put on the right-hand side of each panel here for comparison. The advantage of the first-order approximation over the zero-order approximation is demonstrated with the coefficient of correlation between the original and estimated signals as a function of the noise-to-signal ratio (figure 4). We consider this coefficient to be a quantitative measure of the quality of the signal estimate. By examining figure 4, it can be seen that the advanced process always yields superior results with more accurate signal estimates.

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Figure 4. Correlation coefficients between the original signal and its estimates obtained using the first- (solid line) and zero-order (dashed line) approximations as a function of the noise-to-signal ratio. Note that the first-order approximation always yields better results. The encouraging results shown suggest that the first-order approximation is substantially more effective than the zero-order approximation because the advanced procedure takes into account and compensates for the interaction between different constituents of coherent noise.

Before describing the second synthetic experiment, it is worth noting that optimum weighted stacking may be replaced by the SVD if the signal is contaminated only by random noise with variances being quite stable on different traces (Tyapkin and Ursin 2005). In this case the signal component can be estimated optimally as the dominant term of the SVD, with the record treated as a rectangular matrix formed with regard for the corresponding arrival times. We extend this approach to the zero-order approximation to estimate and remove coherent noise and then to estimate the signal (Tyapkin et al. 2007). This technique was used to process simulated data sets demonstrated below. In this experiment, emphasis was put on the comparison of the performance of the SVD-based zero-order approximation with that of f-k filtering combined with subsequent straight stacking.

We created a set of synthetic shot gathers each of which consisted of 44 traces distributed between 50 and 2200m in 50-m increments. The sampling interval is 2 ms. The signal was chosen to be identical for all the gathers. Its waveform was obtained by convolution of an uncorrelated in time stochastic process with a 20 Hz Ricker wavelet. The signal has an infinite apparent velocity and bears root-mean-square (rms) amplitudes decreasing linearly between 1500 and 500 with offset. It is corrupted by a single linear coherent noise wavetrain, whose waveform was obtained by convolution of another stochastic process with the same Ricker wavelet. The coherent noise has various relative, in regard to the signal, amplitudes (0.5, 1, 2, 4, 8 and 16) and dips (1, 2, 4, 8 and 16 sample intervals per trace). The amplitudes of the coherent noise wavetrain are coordinated in space with those of the signal. White noise with an rms amplitude 100 is always superimposed on each trace to simulate additive random noise. Table 1 presents the correlation coefficients between the known ‘pure’ signal waveform and its estimates. By examining this table, we see that the SVD-based method always yields superior results. This advantage over f-k filtering combined with straight stacking is most pronounced when the relative dip of the coherent noise is either too small (1 sample interval per trace) or too large (16 sample intervals per trace), with the relative amplitude being large enough (16).

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Table 1. Correlation coefficients between the original signal and its estimates obtained using the SVD-based zero-order approximation (in numerators) and using f-k filtering combined with straight stacking (in denominators).

Dips of coherent noise (in sample intervals per trace) Relative

amplitudes of coherent

noise 1 2 4 8 16 0.5 0.981 / 0.960 0.994 / 0.972 0.999 / 0.987 1 / 0.990 0.999 / 0.993 1 0.995 / 0.964 0.998 / 0.972 0.999 / 0.990 1 / 0.993 0.999 / 0.996 2 0.996 / 0.950 0.998 / 0.970 0.999 / 0.990 1 / 0.993 0.999 / 0.990 4 0.996 / 0.903 0.998 / 0.962 0.999 / 0.989 1 / 0.995 0.999 / 0.965 8 0.996 / 0.790 0.998 / 0.927 0.999 / 0.980 1 / 0.995 0.999 / 0.880

16 0.996 / 0.615 0.998 / 0.830 0.999 / 0.941 1 / 0.987 0.999 / 0.678 The first case, when f-k filtering is incapable of separating the signal and coherent noise, is demonstrated in figure 5. Here and also in figure 7, in order to obtain the same amplitude range, the amplification factor for visualization of both the coherent noise and entire data was chosen 16 times smaller than that for the signal. It is seen that almost no signal can be identified on the paned depicting entire data [figure 5(c)] because of too severe coherent noise. The SVD-based method removes the coherent noise almost perfectly, leaving only remnants of an incoherent appearance [figure 5(e]. Then this random noise is successfully suppressed in the second step [figure 5(f)], which produces a signal estimate that can hardly be distinguished visually from the original [figure 5(d)].

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Figure 5. Synthetic data example. (a), (d), (g) ‘Pure’ signal. (b) Coherent noise with a relative amplitude 16 and dip 1 sample interval per trace. (c) Shot gather containing signal, coherent noise and random noise. Result obtained after (e) subtracting coherent noise by SVD, (f) finally approximating signal by SVD, (h) subtracting coherent noise by f-k filtering, and (i) straight stacking of (h) multiplied for a better visual analysis. Note how the SVD-based method subtracts the coherent noise and approximates the signal almost perfectly, whereas f-k filtering and straight stacking fail in both steps. As expected, f-k filtering does a poor job of removing the coherent noise. After this process, there are large artefacts not allowing the signal to be identified [figure 5(h)]. For this reason, the subsequent straight stack [figure 5(i)] differs considerably from the signal [figure 5(g)]. For a better visual analysis, the signal waveform and both its estimates are represented in figure 6 on a large scale.

Figure 6. Signal waveform (black) and its estimates by the SVD-based method (blue) and by f-k filtering combined with straight stacking (red) from figure 5. Note that the SVD-based estimate coincides with the original almost perfectly, whereas the result of the combined method differs considerably from the original.

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The second case, when f-k filtering can not cope with an aliased coherent noise, is exhibited in figure 7. As in the previous situation, the SVD-based method is very effective in removing the coherent noise [figure 7(e)] and yields an estimate [figure 7(f)] that has a strong resemblance to the signal [figure 7(d)]. As opposed to this method, f-k filtering along with straight stacking produces a severely damaged signal estimate [figure 7(i)], which is also confirmed by figure 8.

Figure 7. Synthetic data example. (a), (d), (g) ‘Pure’ signal. (b) Coherent noise with a relative amplitude 16 and dip 16 sample intervals per trace. (c) Shot gather containing signal, coherent noise and random noise. Result obtained after (e) subtracting coherent noise by SVD, (f) finally approximating signal by SVD, (h) subtracting coherent noise by f-k filtering, and (i) straight stacking of (h) multiplied for a better visual analysis. Note how the SVD-based method subtracts the coherent noise and approximates the signal almost perfectly, whereas f-k filtering and straight stacking fail in both steps.

Figure 8. Signal waveform (black) and its estimates by the SVD-based method (blue) and by f-k filtering combined with straight stacking (red) from figure 7. Note that the SVD-based estimate coincides with the original almost perfectly, whereas the result of the combined method differs considerably from the original.

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From the second synthetic experiment we thus can draw a conclusion that the SVD-based zero-

order approximation is successful in estimating the signal without producing any noticeable lateral mixing in seismic data. Unfortunately, this happens to be the case only when we deal with a single coherent noise wavetrain. The situation is much worse if the signal is contaminated by at least two relatively strong noise wavetrains with apparent velocities that differ insignificantly from that of the signal, as in our first synthetic experiment. In this case the SVD-based approach may yield highly unsatisfactory results because of the corruptive mutual impact of all the coherent noise wavetrains. 6. Field data examples Though in the method tested on the synthetic data in the previous section, the waveform of coherent noise is supposed to be quite stable in space, it is not usually the case in practice. It is such a dispersive noise as ground roll that is mostly characterized by significant waveform variations across channels. In order for the method to better approximate and subtract this kind of noise, in the actual data examples that follow and are from Ukraine (seismic data courtesy of the State Geological Enterprise Ukrgeofizyka), we used a few dominant SVD terms instead of only one. This approach has several features that produce better results, as observed in the data examples.

The left-hand panel in figure 9 presents a 96-trace shot gather acquired with a vibratory source and no geophone grouping in the field. The receiver spacing was 10m. In this record, the reflections are obscured by severe coherent source-generated noise including ground roll, air, refracted and guided waves up to the first breaks. Besides, on the right top part of the panel, one can see a technogenic event with an infinite apparent velocity. By examining figure 9, it is obvious that all the noise components are several times stronger that the signal. For this reason, before removing the noise, there is almost no reflection in the original data that can be identified.

Figure 9. Common-shot gather before (left) and after (right) subtracting severe coherent source-generated noise. The boundaries of four sectors selected for processing are marked with different colours. Note that after using the SVD-based method, the noise was removed considerably and much of the underlying reflection was revealed.

To subtract the noise, which has a divergent, fan-like character, we apply the method suggested in (Tyapkin et al. 2004). It requires specifying lines of demarcation, much like a surgical mute, around the entire noise or each component (wavetrain), by picking segmented straight lines. Ones the lines of demarcation have been determined, each sector is mapped by shifting and stretching along the time axis into a new domain where the segmented straight lines become horizontal. This procedure is intended to align the coherent noise events horizontally in order to favour the subsequent SVD. After that the

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coherent noise may successfully be approximated by a few dominant terms of the SVD and then subtracted from the record.

This can also be considered a process to minimize the misfit between any real coherent noise and its idealized mathematical model adopted in the present paper.

On the left-hand panel in figure 9, four sectors are shown within which the noise was subtracted successively. The top and bottom boundaries of each sector are marked by the same colour. The result of subtracting the noise is depicted on the right-hand panel in figure 9. Comparing the result of processing and raw record, one can see that the severe noise is greatly diminished and the refinement of the data is significant, which favours a more confident identification and correlation of reflection events.

Figure 10 exhibits two common midpoint time sections obtained after application of f-k filtering combined with straight stacking (top) and after application of the SVD-based zero-order approximation (bottom). Note the remnants of coherent noise after applying the f-k filter, specifically on the central and right-hand side of the top panel. This result can be attributed to spatial aliasing, which is a common problem with the performance of f-k filters. From figure 10, it is evident that after applying the SVD-based technique, the section has a better S/N ratio and trace-to-trace continuity of reflected signals. Furthermore, a close examination shows that the SVD-based technique yields a better vertical resolution than f-k filtering along with straight stacking. The improved vertical resolution is most obvious for the events marked with arrows. These improvements are due to the fact that our approach allows the coherent noise to be suppressed more effectively, with negligible effect on the signal.

Figure 10. Stacks after application of f-k filtering combined with straight stacking (top) and after application of the SVD-based zero-order approximation (bottom). The range of time dips or apparent velocities for applying both the f-k filter and the SVD-based technique was chosen to be the same. Except for these two processes, the same processing sequence with identical parameters was used for both sections. Note how the SVD-based approach successfully reduces the remnants of coherent noise, specifically on the central and right-hand sides of the section, revealing the underlying reflections.

The field data example shown indicates that the SVD-based method does not produce the synthetic appearance common in many mixing schemes.

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7. Discussion and conclusions In the 1980s, when estimating seismic signal corrupted by both coherent and random noise, non-optimum methods, such as now industry-standard f-k filtering and τ-p transform filtering, began to take the place of their optimum precursors, such as Wiener filters and least-squares estimators. This tendency was outlined first of all for the simple reason that the non-optimum approaches are usually faster and more cost-effective. However, being based on imperfect mathematical models of the data, they often result in the mixing of the data, producing seismic sections of a wormy appearance. Nevertheless, in many circumstances, the non-optimum methods outperform the optimum methods. We explain this lack of effectiveness of the optimum methods basically by their exploiting, first, simplified and therefore imperfect mathematical models of the record and, second, an imperfect scheme for subtracting coherent noise. From this, the rationale of using more realistic models and more sophisticated schemes for subtracting coherent noise becomes clear.

With this purpose in mind, we have started with adopting a more complicated and adequate mathematical model of the record in order to rehabilitate the optimum methods and to re-attract the attention of geophysicists to them. With this model and the least-squares criterion, the strict solution of the problem to estimate the seismic signal has been derived. We have shown that under certain conditions, this solution reduces to a two-stage process that comprises estimating and subsequently subtracting all the coherent noise wavetrains and then estimating the signal from the residual record. A simplified scheme and an advanced scheme for subtracting coherent noise have been discussed, which are called zero-order and first-order approximations, respectively. The zero-order approximation may be considered to be the generalization of a conventional approach to the more realistic data model adopted in the present paper. This rough scheme may be valid only if either coherent noise is formed by a single wavetrain or the interaction between different wavetrains is negligible. Since any actual seismic data that comprise more than one coherent noise wavetrain and also satisfy the latter condition can hardly ever be met in practice, the signal estimates obtained using this simplified scheme may contain significant distortions caused by deviations of the data from the model assumption. To overcome this drawback, we propose to use the first-order rather than the zero-order approximation because it sufficiently reduces the distortions entailed by the mutual impact of all components of coherent noise when estimating and subtracting them. The novelty of our approach is thus in integration of the advantages of the more realistic mathematical data model with those of the advanced scheme for subtracting coherent noise. We have presented applications to simulated data to compare the behaviour of the first-order approximation with that of the zero-order approximation. Also, the SVD-based zero-order approximation has been compared with a conventional combination of f-k filtering and straight stacking on synthetic and field data. The effectiveness of both approximations relies heavily on the accuracy of the necessary signal and noise parameter determination, which is a key step in the overall technology. At present, to our knowledge, only the coherent-noise-free case is supplied with such parameters (Tyapkin and Ursin 2005). For this reason, in order to ensure the feasibility of the methods and to make them data-adaptive, the aim of our future work is to develop robust algorithms for obtaining good a priori information on the parameters needed. These algorithms should be based on the same generalized mathematical model of the data. The developed methods may be a valuable tool in processing not only surface but also borehole seismic data. There is no conceptual difficulty in extending the model adopted in this paper for vector-sensor signal processing. Acknowledgments We wish to thank the management of the State Geological Enterprise Ukrgeofizyka for their kind permission to publish the field data examples shown. Part of this work was achieved while Yuriy Tyapkin was an invited professor at the Department of Earth Sciences, Université de Pau et des Pays de l'Adour, France. Bjorn Ursin thanks VISTA and the Norwegian Research Council (through the ROSE project) for financial support. We gratefully acknowledge two anonymous reviewers who have given their efforts in order to make this publication more understandable.

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Appendix A. Coherent-noise-free case It may be illustrative to consider the least-squares estimator when coherent noise is negligible and the seismic record is thus supposed to be corrupted by random noise only. From (7), it follows that in this case the matrix R reduces to , and (5) takes the form ( )DR nR=

( ) uDfuDffDf 1H1s

1H11H€ −−−−− == cs . (A.1)

This is the spectral counterpart of optimum weighted stacking, which is performed with regard for the amplitudes and arrival times of the signal and the variances of random noise. We refer the reader to (Tyapkin and Ursin 2005) for a more detailed description of this process. It is easy to show that the ratio of the power spectrum of the signal to that of random noise at the output from this delay and weighted sum process is , where is the power spectrum of the signal shape . ( ) ( )s

1ns RRc −

( )sR ( )ts Appendix B. Signal-to-noise ratio at the estimator output At the output from the least-squares estimator described by (5) and (6), the signal power spectrum may be given as

( ) ( )s2H

souts RRR == fh , (B.1)

whereas the power spectrum of the combination of coherent and random noise may be defined as

( 11HHoutn

−−== fRfRhhR ) . (B.2) When 1−R is described by (21) with 0=∆l , the noise power spectrum takes the form

( )n1out

n RcR −= (B.3) and, consequently, the S/N ratio is

( )

( )n

soutn

outs

RR

cRR

= . (B.4)

Here the term c is represented by (24) and attains the maximum value cs in the coherent-noise-free case (Appendix A). Simple subtraction of coherent noise estimates using the zero-order approximation reduces c as compared with cs, thus reducing the S/N ratio at the output from the least-squares estimator with respect to the maximum value . This reduction is partially compensated by exploiting the first-order approximation, which takes into account the mutual impact of coherent noise wavetrains. For this reason, the least-squares method based on the first-order approximation enables more accurate signal estimates, with a superior S/N ratio.

( ) ( )s1

ns RRc −

Appendix C. Signal estimation with the multichannel Wiener filter Let us consider the optimum multichannel Wiener filter intended to obtain the closest in least-squares sense estimate of the signal form ( )ts by filtering each trace described by (1) with its individual filter along with subsequently summing the outcomes. The spectral characteristic of this multichannel filter may be defined as the column vector (Tyapkin et al. 2006)

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( )

( ) fRffR

h 1Hs

1s

1 −

−

+=

RR

, (C.1)

whence

( )

( ) fRfRf

h 1Hs

1HsH

1 −

−

+=

RR

. (C.2)

The structure of this equation can readily be realized if we remember the following fact, enabling us to establish a connection between the Wiener filter and the least-squares estimator developed in this paper. Any signal-extracting multichannel Wiener filter is equal to the combination of the related multichannel maximum-likelihood signal estimator and the single-channel Wiener filter applied to the output from the estimator (Green et al. 1966). In our case, inasmuch as all the noise components are supposed to be Gaussian stochastic processes, this estimator is the least-squares estimator described by (5) and (6). The signal and noise power spectra at the output from the estimator are represented by (B.1) and (B.2) in Appendix B. For this reason, the final single-channel filter should have the spectral characteristic

( )

( ) ( ) 11Hs

soutn

outs

outs

−−+=

+ fRfR

RRR

R . (C.3)

Combining (6) and (C.3) yields (C.2). When the value of , which is equal to the spectral S/N ratio at the output from the least-squares estimator (see Appendix B), satisfies the inequality

( ) fRf 1Hs

−R

( ) 11Hs >>− fRfR (C.4)

at each frequency, the multichannel Wiener filter of (C.2) reduces to the least-squares estimator of (6). References Al Dossary S, Maddison B, Al Buali A, Luo Y, Al Faraj M and Li Q 2001 Linear adaptive noise attenuation 71st Ann. Int. Mtg. Soc. Expl. Geophys. Expanded Abstracts pp 1993-6 Bekara M and van der Baan M 2006 Local SVD/ICA for signal enhancement of pre-stack seismic data 68th Ann. Int. Conf. and Exh. Eur. Assoc. Geosc. Engin. Extended Abstracts D038. Cassano E and Rocca F 1973 Multichannel linear filters for optimal rejection of multiple reflections Geophysics 38 1053-61 Cassano E and Rocca F 1974 Multichannel filters without mixing effects Geophys. Prospect. 22 330-44 Chiu S K and Butler P 1997 2D/3D coherent noise attenuation by locally adaptive modeling and removal on prestack data 67th Ann. Int. Mtg. Soc. Expl. Geophys. Expanded Abstracts pp 1309-11 Fail J P and Grau G 1963 Les filtres en eventail Geophys. Prospect. 11 131-63 Fomel S 2002 Application of plane-wave destruction filters Geophysics 67 1946-60 Freire S L M and Ulrych T J 1988 Application of singular value decomposition to vertical seismic profiling Geophysics 53 778-85 Galbraith J N and Wiggins R A 1968 Characteristics of optimum multichannel stacking filters Geophysics 33 36-48 Gounon P Y, Mars J I and Goncalves D 1998 Wideband spectral matrix filtering 68th Ann. Int. Mtg. Soc. Expl. Geophys. Expanded Abstracts 1150-3 Green P E, Kelly E J and Levin M J 1966 A comparison of seismic array processing methods Geophys. J. R. Astron. Soc. 11 67-84 Hampson D 1986 Inverse velocity stacking for multiple elimination Can. J. Expl. Geophys. 22 (1) 44-55 Helstrom C W 1968 Statistical Theory of Signal Detection (Oxford: Pergamon)

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