1

New multicomponent filters for geophysical dataprocessing

Caroline Paulus∗ and Jérôme I. Mars†

Laboratoire des Images et des Signaux961 rue de la Houille Blanche, BP 46

38 402 Saint Martin d’Hères Cedex, FRANCEFax : 33(0) 474 826 384

∗ Phone : 33(0) 476 827 107, Email: caroline.paulus@lis.inpg.fr† Phone : 33(0) 476 826 253, Email: jerome.mars@lis.inpg.fr

Abstract— As multicomponent processing is a challenge inmany fields such as geophysics, seismology, remote sensing orelectromagnetic, we propose a method which could be used tofilter multispectral, multicomponent or multidimensional data. Inthis paper, the method, called Multicomponent WideBand Spec-tral Matrix Filtering (MC-WBSMF), is applied on geophysicaldata to separate interfering wavefields. The technique is based onthe decomposition of a special multicomponent spectral matrixand could extract a given wavefield from a multicomponentdataset.MC-WBSMF is compared to two classical single-componentseismic methods (the 2DFT or f-k filter and the Radon transform)and to a fully multicomponent method based on singular valuedecomposition (3C-SVD).Finally, MC-WBSMF is tested on a real geophysical subsurfacedataset in order to remove surface waves and to enhancereflection waves.

Index Terms— Geophysical array data, Multicomponent pro-cessing, Wavefield filters

I. I NTRODUCTION

Seismic exploration plays a major role in the search forhydrocarbons and requires three main stages: data acquisition,processing and interpretation. Moreover, in the last decade,the use of multicomponent technology has improvedsignificantly in the field of geophysics but also in seismology,electromagnetic or remote sensing, providing higher fidelityacquisition systems. As a consequence, more advancedanalysis and processing methods for multicomponent recordsare required.

The separation of interfering wavefields is one of the keypoints in geophysical data processing. For example, surfacewaves, also called ground roll, mask the reflections of deepertargets which are of economic interest. As such, the groundroll need to be removed from the data in order to focus onthe underlying reflections.In the single component sensor arrays case, various methodshave already been developed to extract the upgoing waves anddowngoing waves. Wave decomposition is typically carriedout using a 2-D transform techniques such as velocity filter(also called f-k filtering) [1], [2], [3] or Radon transform[4], [5]. In these two techniques, the separation processis accomplished by selecting an appropriate region in the

transformed domain. Nonetheless, for example when P-wavesare interfering with other types of waves, it is difficult toselect the region of the required wave. Moreover, these filtersare deficient in presence of non-plane waves or in case ofshort arrays. In the time-distance domain, median filters havebeen commonly used to separate upgoing and downgoingwaves in VSP and to eliminate spikes [6]. Fuller showed thatsemblance filters also provide efficient algorithm to removenoise and improve velocity analysis [7]. Other separationtechniques based on Singular Value Decomposition [8] arealso used to remove noise. The basic approach to compute theSVD is given in [9] and is based on an eigendecomposition.Nonetheless, in order to be efficient for wavefield separation,these methods require pre-processing such as wave alignment[10].When filtering is performed in the frequency domain, themethod is called spectral matrix filtering. The principle is todecompose an initial dataset into a new space free of noise,with the smallest possible dimension. Signal is separatedfrom noise by finding a subspace defined by the eigenvectorsrelative to the dominant eigenvalues of the spectral matrixissued from the recorded signal. The eigenvalue decompositionallows a separation between signal and noise subspace at eachfrequency bin. The signal is viewed as a juxtaposition of shortband signals since each frequency is treated independently.Nonetheless, as seismic signals are wideband, the previousmethod is not optimal to tackle this problem. Consequently,wideband spectral matrix filtering has been introduced in [11].

Furthermore, since more and more surveys usemulticomponent sensors, specific filtering methods adaptedto multicomponent dataset are required. 3CSVD [12], [13],[14], [15], SVD of quaternion matrices [16], or furthermore3C-SVD compound with partial Independent ComponentAnalysis [17] techniques are efficient to separate wavefield inmulticomponent seismic profile but require wave alignmentas they are based on SVD. To avoid this pre-processing, wepropose to develop a method based on the spectral matrixfiltering adapted to the multicomponent case.

The aim of the paper is firstly to present the MC-WBSMF[18] (MultiComponent-WideBand Spectral Matrix Filtering)

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which is more suitable for multicomponent wideband seismicsignal since it takes into account jointly polarization and inter-actions between all the frequencies, sensors and components.The method is used to separate a given wavefield (for example,ground roll) from other events and from noise. Secondly,performances of the MC-WBSMF will be compared with theperformances of classical filtering methods and of 3C-SVD ona synthetic dataset. Finally, results of MC-WBSMF on a realdataset will be shown.

II. M ULTI COMPONENT-WIDEBAND SPECTRAL MATRIX

FILTERING (MC-WBSMF)

A. Model formulation and hypothesisMulticomponent seismic data recordings depend on three

parameters: time (Nt samples), distance (Nx sensors), direc-tions (Nc components) and provide the potential to accesswavefield polarization.The tensor of the data collected duringNt samples on a setof Nx sensors (each composed ofNc = 3 components notedX, Y and Z) is written as:

Tt ε RNx×Nt×Nc . (1)

Applying the Fourier transform, the problem is divided into aset of instantaneous mixtures of signals and we have:

T = TF{Tt} ε CNx×Nf×Nc (2)

with Nf the number of frequency bins.Let us assume that we have a linear array. Using the superpo-sition principle, the signal recorded on a sensor results fromthe linear combination ofNw polarized waves received onthe antenna added with noise. These waves have propagatedthrough a medium which could have attenuated, time delayedor phase shifted them.The tensorT is concatenated into a long-vector notedT l ofsize (NxNfNc) which contains all the frequency bins on allthe sensors for each component:

T l = [X(f1)T , · · · , X(fNf )T , Y (f1)

T , · · · , Y (fNf )T

, Z(f1)T , · · · , Z(fNf )T ]T (3)

whereX(fi), Y (fi) andZ(fi) are vectors of size(Nx) whichcorresponds to theith frequency bin of the signals receivedon each of theNx sensors respectively on components X, Yand Z.The mixture model is defined by:

T l = S · A + B (4)

where:

- S = [S1, S2, · · · , SNw] is a (NxNfNc, Nw) matrixwhose columns are the steering vectors describing thepropagation of each of theNw waves along the receivergroup for all frequencies and all components. This matrixcontains information about the polarization of theNw

waves and information about the waveform emitted bythe sources. All these terms are deterministic;

- A = [a1, a2, · · · , aNw]T is a vector of sizeNw which

contains the random amplitudes of the waves;

- B is a vector of dimension(NxNfNc) which correspondsto the noise component added to each sensor at eachfrequency and on each component. These random noisesare supposed to be additive, temporally and spatiallywhite, uncorrelated with the sources, non polarized andto have identical power spectral densityσ2

b ;

All relationship in frequency between components and sensorsare expressed in the multicomponent wideband spectral matrixdefined by:

Γ = E{T l · THl } (5)

with E the expectation operator andH the transpose conjugateoperation.Γ has dimensions(NxNfNc) by (NxNfNc).

B. Estimation of the spectral matrixAt this step,Γ is non invertible as it has unity rank. To avoid

that fact and to decorrelate sources from noise and sourcesfrom themselves, it is necessary to perform an estimation ofmatrix Γ.Furthermore, the effectiveness of the filtering depends on theestimation stage of the spectral matrixΓ. In practical case ofseismic data processing, the mathematical expectation operatorE is replaced by specific averaging operators, like spatialor frequencial smoothings or both of them [19], [20], [21],[22]. Their purpose is to reduce the influence of the termscorresponding to the interactions of the different sources,making the inversion of the spectral matrix possible. Threepossible averages are:

• spatial smoothing;• frequency smoothing;• ensemble average.

The spatial smoothing could be done by averaging spatialsubbands. The uniform linear array withNx sensors issubdivided into overlapping subarrays in order to have severalidentical arrays, which will be used to estimate spectralmatrices in order to build a smoothed matrix. These subarraysare supposed to be linear and uniform (constant intertrace).If the smoothing order isKs, the size of each subarray isNx − 2Ks and the number of subarrays is2Ks + 1. Thespatially smoothed spectral matrix is defined by the averageof the spectral matrices obtained for each subarray. Shanand al. [21] have proven that if the number of subarrays isgreater than or equal to the number of sourcesNw, thenthe spectral matrix of the sources is non-singular. However,one assumption is that the wave does not vary rapidly overthe number of sensors used in the average, in particular,amplitude fluctuations must be smoothed out.

To introduce frequency smoothing, two ways can beperformed: either by weighting the auto-correlation and cross-correlation functions (in the time domain) or by averagingfrequential subbands (in the frequential domain). Thesetwo methods require a velocity correction allowing wave tobe shifted in time within the limits of the duration of theweighting function. Therefore, the frequency average requiresa rough pre-alignment of the wave (flattening). Consideringa frequential smoothing ofKf order, we obtain a number of

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2Kf + 1 frequential subbands of sizeNf .

For a better estimation of the multicomponent widebandspectral matrix, it is suitable to realize jointly spatial andfrequential smoothing.An observation generates a set of(2Ks + 1) spatiallyshifted recurrences. Subsequently, these recurrences can befrequentially smoothed(2Kf + 1) times. Eventually, a setof K = (2Ks + 1)(2Kf + 1) spatio-frequentially shiftedrecurrences is obtained and the estimated spectral matrix,Γ̂,is defined by the average of the spectral matrices obtainedfor each subarray and each frequency subband. The rank ofΓ̂ is now equal toK which need to be chosen greater thanNw, the number of waves.

Finally, the ensemble average would be the most efficientaverage but it requires information redundancy which is rarelyavailable. This average assumes that the signal is invariantfrom one realisation to the others whereas the noise varies.

C. Estimation of signal subspaceOnce the multicomponent wideband spectral matrix has

been estimated, the initial space can be separated into twosubspaces: the signal subspace,Γ̂

s, generated by the first

eigenvectors associated to the highest eigenvalues and itscomplementary, the noise subspace,Γ̂

n. Filtering is then

achieved by projection of the initial data onto the signalsubspace. This procedure is detailed in the following.

1) Structure of the multicomponent wideband spectral ma-trix: Following the assumptions made in part.II-A,Γ̂ can bewritten as:

Γ̂ = S · Γ̂A· SH + σ2

b · I = Γ̂s+ Γ̂

n. (6)

Let Γ̂A

= E[A · AH ] be non-singular (waves partiallycorrelated or uncorrelated) and the columns ofS linearlyindependent, the rank of̂Γ

s= (S · Γ̂

A· SH) is Nw.

Structure ofΓ̂ could be written:

Γ̂ =

Γ̂X,X

Γ̂X,Y

Γ̂X,Z

Γ̂Y,X

Γ̂Y,Y

Γ̂Y,Z

Γ̂Z,X

Γ̂Z,Y

Γ̂Z,Z

(7)

where Γ̂X,X

, Γ̂Y,Y

and Γ̂Z,Z

correspond to the mono-component wideband spectral matrix for respectivelycomponent X, Y and Z. These terms are located on the maindiagonal of Γ̂. The other blocks correspond to the cross-component spectral matrix which contain information relatingto the interaction between the various components. Since thesignals on each component are correlated (because they arepolarized), these extra diagonal terms contain informationrelating to the polarization. Since the multicomponentwideband matrix filtering provides more information on thesignal and especially on polarization, better filtering resultsare obtained rather than applying classical filtering methods

independently on each component.

2) Decomposition of the multicomponent wideband spectralmatrix: By construction, the spectral matrix has a hermitiansymmetry.

Γ̂ = Γ̂H

. (8)

According to Eq. (8),̂Γ can be decomposed in a single wayusing eigenvalue decomposition (EVD) as:

Γ̂ = U · Λ · UH =M∑

i=1

λiuiuHi (9)

where M = NxNfNc, Λ = diag(λ1, · · · , λM ) are theeigenvalues andU is a unitaryM by M matrix whose columnsare the orthonormal eigenvectorsu1, · · · , uM of Γ̂. The eigen-valuesλi correspond to the energy of the data associated withthe eigenvectorui. They are arranged as follows:

λ1 ≥ λ2 ≥ · · · ≥ λM ≥ 0. (10)

Considering Eq. (9), the spectral matrixΓ̂ hasM eigenvalues.Nonetheless, the previous rank property (rank(Γ

s) = Nw)

implies that:• the M − Nw minimal eigenvalues of̂Γ are equal toσ2

b

• the eigenvectors corresponding to these minimal eigen-values are orthonormal to the columns of the matrixS.

Thus, the space generated by the smallest eigenpairs is referredto as the noise subspace and its orthogonal complement asthe signal subspace since it is spanned by the steering vectorsof the signal. This justifies the benefit of projecting data ontothe first eigenvectors to separate signal from noise.

Moreover, resulting from smoothing operators,Γ̂ has asignificant rank deficiency since its rank is equal toK andK is much lower thanM . Consequently, we have:

λi = 0 for K + 1 ≤ i ≤ M. (11)

Eq. (9) becomes:

Γ̂ =K∑

i=1

λiuiuHi . (12)

Since the computing time for the diagonalization isproportional toM3, it is excessive to diagonalize the wholespectral matrix of sizeM by M . Only theK first eigenvaluesand eigenvectors need to be computed. Methods have beendeveloped to compute only the few dominant eigenpairs (forexample the Lanczos algorithm [9]).

3) Filtering by projection onto the signal subspace:Thefiltering step corresponds to an orthogonal projection of theinitial data T l (Eq. (3)) onto the highest eigenvectors corre-sponding to the signal subspace. As considered previously,the eigenvectors associated with theNw largest eigenvaluesbelong to the same subspace called signal subspace (Γ̂

s), by

opposition to the noise subspace (Γ̂n):

Γ̂ = Γ̂s+ Γ̂

n=

Nw∑

i=1

λiuiuHi +

K∑

i=Nw+1

λiuiuHi . (13)

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If Nw is unknown, various statistical methods can be used toestimateNw [23]. As soon asNw is found, T l is projectedonto itsNw first eigenvectors, following the expression:

T ls =

Nw∑

i=1

〈T l, ui〉 · ui. (14)

The projection onto the noise subspace (Tln) is obtained bysubtraction ofT ls from the initial data:

T ln = T l − T ls =K∑

i=Nw+1

〈T l, ui〉 · ui. (15)

The last steps consist of rearranging the long-vectorsT ls andT ln in the tensor form (reverse of Eq. (3)) and computing aninverse Fourier transform in order to come back to the time-distance-component domain.

III. A PPLICATION ON SYNTHETIC DATASET OF VARIOUS

FILTERING METHODS

In this part, the performances of the MC-WBSMF will betested on a synthetic multicomponent seismic dataset. Thegoal is to extract the various wavefields from noise and alsoto separate the waves between themselves. Afterwards, wedescribed briefly various signal processing methods commonlyused in seismic processing. We test the following filters:

• the velocity filter (2DFT);• the τ -p filter (Radon filter);• the multicomponent singular value decomposition (3C-

SVD);

MC-WBSMF and 3C-SVD methods are fully multicomponentmethod whereas the classical velocity andτ − p filters workon monocomponent datasets. Consequently, they are appliedindependently on each component.

Fig. 1. Initial dataset

The synthetic dataset contains two waves added with noisereceived on a simulated 3C-sensors array (Fig. 1) composedof 20 sensors. Ricker wavelet has been used as the sourcewavelet. The signal has propagated through a medium com-posed of horizontal geological layers that are assumed to be

Fig. 2. Hodogram of the initial dataset

Fig. 3. FK diagram of the initial dataset

non-anysotropic. One wave (referred as "1st wave" in thefollowing) has an infinite velocity, an elliptical polarizationand a phase-shift which is varying with offset. The secondone (referred as "2nd wave") is a non-plane wave linearlypolarized and with a non-infinite velocity. This additive noiseis uncorrelated with the sources and temporally and spatiallywhite. The hodogram of the initial dataset are plotted on Fig. 2.It corresponds to the layout of the amplitude of one component(Z for example) versus the amplitude of an other component(X) for one trace (chosen in the middle of the array). Sincethe initial data are very noisy, one could not distinguishedinformation on wavefields on this hodogram.The f-k diagram of one component corresponding to the initialdataset is shown on Fig. 3.

A. MC-WBSMF

The results obtained with MC-WBSMF are presented. Thespectral matrix is estimated withKs = Kf = 3 for the spatialand frequential smoothing orders. The amplitude of the tenfirst eigenvalues resulting from the decomposition of the globalspectral matrix is presented on Fig. 4. The first eigenvectorallows us to recover the first wave with the infinite velocity andthe second eigenvector, the second wave with the non-infinitevelocity. Fig. 5 and Fig. 7 show the two extracted waves. Onecan notice a loss of sensors after filtering which is due tothe spatial smoothing. On the hodograms presented on Fig. 6and Fig. 8, the continous line corresponds to the extractedwaves with MC-WBSMF and the dashed line to the true waves(model). One can notice that the ellictic polarization of the firstwave and the linear polarization of the second waves were

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relatively well recovered. The amplitudes were also preservedand most of the noise has been removed.

Fig. 4. First ten eigenvalues of the spectral matrix

Fig. 5. 1st extracted wave with MC-WBSMF

Fig. 6. Hodogram of the1st extracted wave with MC-WBSMF

B. Velocity filter1) Method: The velocity filter uses the (f,k) transform

which corresponds to the double Fourier transform of a seismicsection in time-distance. This is one of the most classicalfiltering method used for 2D filtering [1], [2], [3].

2) Application: Results of velocity filtering performed onthe synthetic dataset are shown on Fig. 9 for the first wave andFig. 11 for the second wave. Fig. 13 and Fig. 14 correspond tothe f-k diagram and Fig. 10 and Fig. 12 to the hodograms of

Fig. 7. 2nd extracted wave with MC-WBSMF

Fig. 8. Hodogram of the2nd extracted wave with MC-WBSMF

Fig. 9. 1st extracted wave with FK filter

the two extracted waves. The results are satisfactory since thetwo waves have been well separated. The elliptic polarizationof the first wave has been well recovered whereas the linearpolarization of the second wave has been totally changedinto an elliptic polarization. Moreover, f-k filter has majorlimitations. For instance, it requires a large number of tracesand sufficient data sampling to avoid problems related toaliasing whereas that is not the case with MC-WBSMF.

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Fig. 10. Hodogram of the1st extracted wave with FK filter

Fig. 11. 2nd extracted wave with FK filter

Fig. 12. Hodogram of the2nd extracted wave with FK filter

Fig. 13. FK diagram of the1st extracted wave with FK filter

C. τ -p filterTheτ -p filter consists in making a summation of the seismic

section according to a line of slopep = 1/V with p the

Fig. 14. FK diagram of the2nd extracted wave with FK filter

slowness andV the velocity. Thus, we obtained a functionof time (with null offset); time is then notedτ . The resultis presented according to variablesτ and p. The method hasbeen extended to the case of non-plane waves. Summation isdone along curves instead of lines and the method is calledhyperbolic τ -p filter [4]. This is the one we used on thisexample.

Fig. 15. 1st extracted wave withτ − p filter

Fig. 16. Hodogram of the1st extracted wave withτ − p filter

1) Application: Results ofτ − p filter performed on thesynthetic dataset are shown on Fig. 15 and Fig. 16 for the firstwave and on Fig. 17 and Fig. 18 for the second wave. Theresults are really satisfactory since the two waves have beenwell separated. However, the amplitudes are not preserved and

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Fig. 17. 2nd extracted wave withτ − p filter

Fig. 18. Hodogram of the2nd extracted wave withτ − p filter

the method is really dependant on how the mask in theτ − pdomain is designed.

D. 3C-SVD filtering1) Method: The 3C-SVD is an extension of the classical

SVD method used for monocomponent datasets. It is basedon HO-SVD (Higher Order Singular Value Decomposition).To generalize to the multicomponent case, we present themultilinear algebra notation [12], [13], [14].

We first define the×n operation, called n-mode product,which corresponds to a product of 3D arrays with matrices.The ×n product of a tensor A of size(I1 × I2 × · · · × In ×· · · × IN ) by a matrix B of size(J × In) is a tensor of size(I1 × I2 × · · · × J × · · · × IN ) and follows the equation:

(A ×n B)(i1,i2,··· ,j,··· ,iN ) =∑

in

ai1i2···in···iN.bjin

. (16)

For a 2D seismic sectionX ε RNx×Nt , the Singular Value

Decomposition could be written as:

X = ∆ ×1 Ux×2 U

t(17)

where∆ is a pseudo-diagonal matrix containing the singularvalues of X which define the relative amplitudes of thevarious eigensections.U

tcontains vectorsuti

which havethe dimensions of a trace and corresponds to the normalizedwavelet.U

xcontains vectorsuxi

which convey information

about the amplitude variations of the waveutialong the

seismic section. In the previous equation,∆ is multiplied onits first mode (rows) by the left singular matrixU

xand on its

second mode (columns) by the right singular matrixUt.

Using this notation, the extension of the SVD toTt ε R

Nx×Nt×Nc is obvious:

Tt = D ×1 Ux×2 U

t×3 U

c(18)

whereUt, U

xandU

ccontain the normalized wavelets, their

behaviour along the x-direction and their polarization.D isthe equivalent of the pseudo-diagonal matrix∆ and containsthe singular values. The singular matricesU

x, U

tandU

care

obtained by applying classical SVD to the unfolding versionsof Tt. Three unfolded matrices can be built withTt:

Ttx

ε RNx×NtNc

Tttε R

Nt×NxNc

Ttc

ε RNc×NxNt .

(19)

TheUx, U

tandU

care the left singular matrices coming from

the SVD of the respectively unfolded matricesTtx, Tt

tand

Ttc. D is obtained using the inverse expression of the 3C-

SVD:

D = Tt ×1 UT

x×2 UT

t×3 UT

c. (20)

Tt is characterized by three ranks, one for each mode:rx,rt, rc. These ranks are estimated using the SVD of the threeunfolded matrices.The separation of the seismic events is then realized usinga method derived from the 1C case. The original space isdecomposed into a signal subspace and a noise subspace:

Tt = Tts+ Tt

n(21)

where the signal subspace (Tts) is the (r̃x, r̃t, r̃c) rank

approximation ofTt defined as:

Tts

= Tt ×1 Uxs

UT

xs×2 U

tsUT

ts×3 U

csUT

cs(22)

where the Uxs

, Uts

and Ucs

are obtained by keepingrespectively the first̃rx, r̃t, r̃c vectors ofU

x, U

tandU

c. The

numberr̃t corresponds to the number of normalized waveletsnecessary to describe the signal subspace.r̃x and r̃c describethe behaviour of each selected wavelet along the x-directionand their polarization.

2) Application: Results of 3C-SVD filtering performed onthe synthetic data set are shown on Fig. 19 and Fig. 20 forthe first wave and on Fig. 21 and Fig. 22 for the second wave.Most of the noise has been removed. Although the first waveseems to be relatively well extracted (a very small part of thesecond wave, collinear to the first wave, is present), the ellipticpolarization of this wave was not well estimated (Fig. 20).Moreover, the second wave with a non-infinite velocity hasbeen badly separated.

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Fig. 19. 1st extracted wave with 3C-SVD

Fig. 20. Hodogram of the1st extracted wave with 3C-SVD

Fig. 21. 2nd extracted wave with 3C-SVD

E. Performances evaluation

Table I quantifies the filtering results obtained on thesynthetic dataset with the previous methods. Mean squareerror (MSE) has been computed to quantify the differencebetween model and filtered wavefield in order to evaluateperformances of the various methods.

In a general way, one can note that for all tested methods,results obtained for the estimation of the first wave are betterthan the one obtained for the second wave. The reason is that

Fig. 22. Hodogram of the2nd extracted wave with 3C-SVD

MSE for 1st wave MSE for 2nd waveMC-WBSMF 8.10

−438.10

−4

Velocity filter 24.01−4

32.10−4

τ -p filter 55.01−4

54.10−4

3C-SVD 35.10−4

122.10−4

TABLE I

MEAN SQUARE ERROR

most methods work better if the wave has an infinite velocity.Moreover, MC-WBSMF and f-k filter are the two methodsproviding the best performances in separating wavefield andenhancing signal-to-noise ratio.

We will now discuss the advantages and disadvantages ofthe tested methods in comparison with MC-WBSMF.

A first type of methods require to work in an associateddomain. The f-k filter andτ -p filter belong to this category.These two methods provide really good filtering results.Nonetheless, the largest disadvantage of these two methodsis that they require user’s intervention in order to realize themask filtering in the associated domain before coming backto the time-distance domain. Moreover, other disadvantagesof f-k filter are that it requires a large number of traces withshort distance sampling and that it is very cost demanding.

A second category which could be referred to as the matrixmethods, includes the Singular Value Decomposition (SVD)[8], [9], the 3DSVD [15], the Karhunen-Loeve method [24],[25], [26], and the various spectral matrix filtering meth-ods: SMF [27], WBSMF [11] and MC-WBSMF [18]. Thesemethods build a matrix (data, covariance or spectral matrix)which is then decomposed according to its eigenvectors. Thesemethods are used to separate waves but also to improvesignal to noise ratio by breaking down the data into a signalsubspace and a noise subspace. SVD-based and Karhunen-Loeve methods require the extracted wave to be flattened(infinite velocity) before separation. In contrast, the spectralmatrix can operate without flattening but the estimation step ofthe spectral matrix determines the effectiveness of the filtering.However, contrary to MC-WBSMF, SMF and WBSMF do nottake into account the wideband characteristics of the seismicsignal and also its polarization.In comparison with all these methods, MC-WBSMF is a low

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cost method which works well without pre-treatment, withouta-priori information and which is not sensitive to aliasing.

IV. A PPLICATION ON REAL NEAR SURFACE DATASET

The MC-WBSMF method has been tested on realmulticomponent seismic datasets and could be used eitherto process seismic reflexion profile, or Vertical SeismicProfile (VSP), or OBS (Ocean Bottom Seismometer) andOBC (Ocean Bottom Cable). In case of seismic reflexion,only reflected waves convey information for imaging thesubsurface. One aim is to separate the reflected waves fromthe other interfering waves (directs arrivals, refracted waves,surface waves, ground roll...). As the algorithm favoursthe waves with the highest energy, the method is used toremove energetic waves (direct waves and ground roll) inorder to enhance reflected waves (which have low energy andwhich are contained in the projection onto the noise subspace).

Fig. 23. Initial dataset

Fig. 24. Estimated slowest pseudo-Rayleigh wavefield

Fig. 25. FK diagram of the initial dataset (up) and the slowest pseudo-Rayleigh wavefield (down)

Fig. 26. First residual section

Fig. 27. Fast pseudo-Rayleigh wavefield

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Fig. 28. Second residual section

Fig. 29. First and second reflected waves

The proposed dataset has been obtained by using explosivesources and a line of 47 2C-receivers in the valley of Chan-tourne near Grenoble-France. The distance between adjacentgeophones is 10 m. The vibration axis of the horizontalcomponent is located in the plane going through the sourcepoint and the seismic line (inline). The second component isa geophone with a vertical axis. The offset is 50 m. Dataare sampled every 16 ms and the recording’s duration islimited to 4 seconds. Fig. 23 shows the initial data recordedon the vertical and horizontal component. On the verticalcomponent, we can identify a refracted wave, a reflectedarrival with a strong amplitude associated to a deep reflector,and two dispersive pseudo-Rayleigh waves characterized bylow apparent velocities and a low frequency content. On thehorizontal component, the two pseudo-Rayleigh waves are alsopresent and a reflected waves (around 1,2 ms) drowned into theothers waves. The aim of the study is to separate the differentwavefields to enhance the reflections.MC-WBSMF has been applied three times. The first time was

on the initial data to isolate the slowest pseudo-Rayleigh wave(Fig. 24). Fig. 25 shows the f-k diagram of the initial datasetand of the slowest pseudo-Rayleigh wave. We clearly seethat the MC-WBSMF filtering is not dependent on the spatialsampling of the data since it does not suffer from aliasing.The impact of this feature is that the survey design may beless constrained by spatial sampling requirement.Difference between initial dataset and estimated slowestpseudo-Rayleigh waves is given on Fig. 26. MC-WBSMF wasapplied a second time on this residual section to isolate thefastest pseudo-Rayleigh wave presented on Fig. 27. Finally,processing has been applied on the second residual sectionbetween 0 and 1.4s (Fig. 28) since we applied a mute between1.4s and 4s. Two reflected waves has been well isolated ataround 0.6s and 1.2s (Fig. 29).Results obtained with MC-WBSMF could be compared withthe ones obtained with f-k filtering, polarization filtering, SVDand SMF since the previous methods have been tested on thesame near surface dataset in [28].

V. CONCLUSION

Since multicomponent geophones are now largely used forseismic acquisitions, specific tools for these kind of dataare required. This paper presents a new method for wave-field filtering of multicomponent data called multicompo-nent wideband spectral matrix filtering (MC-WBSMF). Theperformances of the method have been compared with theones of the classical filtering methods (velocity filter,τ -pfilter) and with multicomponent SVD; and has also beentested on real multicomponent datasets. It has been shownthat MC-WBSMF provides good results in terms of signal-to-noise ratio enhancement and waves separation comparingwith conventional methods. Moreover, the method is relativelylow cost and works without any pre-treatment and a-prioriinformation. The main reason of this achievement comesfrom the consideration of the wideband characteristics and thepolarization (multicomponent aspect) of the seismic signals.

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Caroline Paulus studied Electrical Engineering atthe Telecom departement of the Institut NationalPolytechnique de Grenoble, France, where she re-ceived the Electrical Engineering degree in 2002,and the M.S. degree in Signal, Image, Speech andTelecommunications from the Institut National Poly-technique de Grenoble, France, in 2003. She is cur-rently working toward the Ph.D. degree at the Lab-oratoire des Images et Signaux, Grenoble, France.Her research interests are in signal processing, moreprecisely multidimensional filtering for geophysical

applications.

Jérôme I. Mars received a Ms (1986) in Geophysicsfrom Joseph Fourier University of Grenoble andPhD in Signal Processing (1988) from the InstitutNational Polytechnique of Grenoble. From 1989-1992, he was a postdoctoral research at the Centredes Phénomènes Aléatoires et Geophysiques, Greno-ble. From 1992-1995, he has been visiting lecturerand scientist at the Materials Sciences and MineralEngineering Dept at University of California, Berke-ley. He is currently Assistant Professor in SignalProcessing for the Laboratoire des Images et des

Signaux at the Institut National Polytechnique de Grenoble He is leader ofgeophysical signal processing team. His research interests include seismicand acoustic signal processing, wavefield separation methods, time frequencytime-scale characterization, and applied geophysics. He is member of SEGand EAGE.