advanced signal processing tools for dispersive waves · sequence used for surface-wave...

Near Surface Geophysics, 2004, 199-210

© 2004 European Association of Geoscientists & Engineers 199

Advanced signal processing tools for dispersive waves

J. I. Mars1, *, F. Glangeaud1 and J. L. Mari2

1 LIS-ENSIEG, Rue de la Houille Blanche Domaine Universitaire, BP 46, F-38402 St.-Martin-d’Heres Cedex, France, 2 Institut Français du Pétrole, ENSPM- IFP School, 238 Avenue Napoléon Bonaparte, 92505, Rueil-Malmaison, France

Received January 2004, revision accepted August 2004

ABSTRACTTwo field examples are presented, showing the advantages of using multicomponent sensors forsurface-wave studies. Multicomponent sensors allow the use of specific signal-processing toolssuch as the multicomponent singular value decomposition filter and the multicomponent polariza-tion filter, which are both very efficient at separating surface waves from the other waves that com-prise a seismic field record.

Firstly, some signal-processing tools for studying surface waves are described. The various fil-ters range from classical to advanced techniques. For processing single-component data, the filtersare the f–k filter and filters based on singular value decomposition and on spectral matrix decom-position. For processing multicomponent data, the filters are the 4C-singular value decompositionfilter and the classical or high-order polarization filter.

Secondly, processing sequences that can be applied to the field data are described and the sin-gle-component processing sequence and the multicomponent processing sequence are compared.Two field examples are presented. The first data set is a land seismic data recording on 2C sensors.The second data set was obtained from a marine acquisition with OBS (4 components). The resultsobtained illustrate the advantages of using multicomponent filters. The efficiency of the 4C-SVDfilter and the high-order statistic polarization filter is demonstrated.

sedimentary layer and of the parameters of dispersion is used todetermine the constraints for modelling the propagation environ-ment (Glangeaud et al. 1999; Nicolas et al. 2003).

The analysis of surface waves and the study of their dispersionallow us to evaluate the shear-wave velocity as a function ofdepth and to determine the shear modulus of the first tens ofmetres below the ground surface. For any recording, we need asource and at least two geophones. The source can be impulsive,i.e. hammer-type, or it can have a single frequency at every shot.Useful frequencies for the determination of the pseudo-Rayleighvelocity vary from 3 to 200 Hz. Surveys are often carried out overthe frequency range from 10 to 100 Hz. Six geophones are gener-ally used, with distances from the source being typically 0.25, 0.5,1, 2, 3 and 4 m for accurate measurement of phase differences.This allows a phase velocity versus wavelength diagram to becomputed. The measurement of phase is carried out in the fre-quency domain after a Fourier transform of traces recorded by thegeophones. When a vibrator is used, one trace is obtained per fre-quency emitted, the frequency being that for which the amplitudespectrum is maximal. With an impulsive source, phases are meas-ured in the frequency zone corresponding to the maximum of theamplitude spectrum and the coherence function. In this approach,we assume that the surface waves are dominant and the effects ofthe other waves are negligible. Once the phase velocity versuswavelength diagram is obtained, an inversion process is used to

INTRODUCTIONIn land surface seismic surveys, the ground roll is composed ofsurface waves, which are pseudo-Rayleigh waves and Lovewaves. They propagate directly from the source to the sensorswithout penetrating deeply into the subsurface. Surface wavesare mainly coherent surface noise with large amplitudes. Theirreflections may create disturbances which may mask (or inter-fere with) the signal of interest. For example, when interest isfocused on reservoir detection, surface waves are considered asunwanted noise and wavefield separation filters are usuallydesigned to attenuate these waves. However, if near-surfacecharacterization is of interest, then surface waves are used toobtain information about the acoustic parameters of the near sur-face, which acts as a seismic waveguide. Others waves, such asreflection arrivals and refracted signals, become the undesirednoise. In marine surveys, especially in the case of shallow water,seismic-wave propagation is affected by frequency dispersionrelated to various waveguides, depending on the depth (Pekeris1948; Jensen et al. 1994; Admundsen and Reitan 1995). Thesewaves are almost unaffected by absorption during their underwa-ter propagation. They propagate at long range and can be used toestimate the geo-acoustical parameters even if the source is farfrom the receivers. The estimation of velocities relative to the

* [email protected]

J. I. Mars, F. Glageaud and J. L. Mari200

© 2004 European Association of Geoscientists & Engineers, Near Surface Geophysics, 2004, 2, 199-210

transform the previous diagram into a graph giving the shearmodulus or stiffness modulus versus depth. The simplest inver-sion is empirical. It consists of assuming a constant ratio betweenwavelength and depth. Generally accepted values are 2 when thestiffness is approximately constant with depth or 4 when the stiff-ness increases significantly with depth. Up to this stage, calcula-tions can be carried out in real time at the acquisition site. Resultscan be then refined using sophisticated inversion techniques,which by iteration adjust a model defined by mechanical param-eters to the field data.

We describe processing methods which can be applied to sur-face waves in order to separate between these waves from otherswaves considered as noise (i.e. reflected, refracted, convertedwaves). After processing, surfaces wave characteristics can bemore clearly identified.

Firstly, we introduce different types of filter, from classical, i.e.the f–k filter (Yilmaz 2002) to advanced filters based on the latestresearch in signal processing, i.e. the high-order polarization filter(Lacoume et al. 1998; Vrabie et al. 2004; Le Bihan and Ginolhac2004). The choice of filter depends mainly on the type of surveyused to record signals (single-component or multicomponent).

Conventionally, seismic data sets are recorded using a single-component sensors array (vertical component). In this case,wavefield separation can be perform by several techniques (Mariet al. 1997), classified in three categories: template methods, suchas f–k or τ–p filters (Yilmaz 2002); inversion methods (Esmersoy1988); and matrix methods, such as the singular value decompo-sition filter (Freire and Ulrych 1988) or filters based on spectralmatrix estimation (Mars et al. 1987, 2004; Spitz 1991; Mari et al.1997). Knowledge of wave polarization requires multicomponentsensor surveys using multicomponent filters, i.e. classical andhigh-order polarization filters (Lacoume et al. 1998) or 4C-singu-lar value decomposition filters (Meunier et al. 2001).

Secondly, we present two field examples. The first one is aland seismic data set recorded on a 2C sensors array. The seconddata set was obtained from a marine acquisition with OBS (4components). Using these examples, we illustrate the processingsequence used for surface-wave characterization and guided-wave extraction.

SOME SIGNAL PROCESSING TOOLS FOR SUR-FACE-WAVE ESTIMATIONFiltering is an essential operation in seismic and near-surfacegeophysics, used to characterize the contribution of each wave-field. We describe various signal processing methods used toseparate each surface wave from the body waves. Our goal is toextract the surface wave in order to characterize its dispersion.Ranging from classical to advanced techniques, we describe thefollowing filters:• the classical f–k filter; • the classical polarization filter and the high-order polarization

filter used respectively on sections recorded by a single-com-ponent sensor array and by a multicomponent sensor array;

• the singular value decomposition filter with its extension tomulticomponent data sets (4C-singular value decomposition);

• the spectral matrix decomposition filter.

ModelHere, we assume that seismic signals are recorded on an array ofscalar sensors (single-component sensors) or vectorial sensors(multicomponent sensors). These signals are generally describedas the summation of several events related to the differentsources (reflected waves, refracted waves, converted waves, sur-face waves) propagating through the media. The observed scalaror vectorial signal, dependant on time t and distance x (a seismicsection), is given by

(1)

where ai(t) is the wavelet from source i, si(t,x) is the propagationvector of the source and * denotes convolution. The signal r(t,x)can be described in dual domains associated with the time anddistance variables as:• Si(f,x) = FTt[si(t,x)], the distance–frequency space representa-

tion;• Si(f,k) = FTx[Si(f,x)], the frequency–wavenumber representa-

tion (2D Fourier transforms on time and distance variables).

FIGURE 1

Non-dispersive plane wave (model). (a) Time–distance

representation; (b) horizontal component.

Advanced signal processing tools for dispersive waves 201


Filtering in the frequency–wavenumber domain on single-component acquisitionsA synthetic example demonstrates the use of the f–k filter on sur-faces waves. A wave propagating in a non-dispersive medium(with constant velocity V) can be expressed in the time–distancedomain as r(t,x) = w(t – x/V) (Fig. 1a). The modulus of the 2DFourier transform of r(t,x) (Bracewell 1986) gives the expressionfor the wave in the f–k plane as

(2)

In the f–k plane, δ(k + f/V) represents the straight-line k + f/V = 0passing through the origin with slope –1/V (Fig. 1b). In a disper-sive medium, the dispersion law is not linear. The wave velocitydepends on frequency. The group velocity and phase velocity ofthe wave can be estimated as vg = df/dk and vφ = f/k, respective-ly. For a seismic section with one dispersive event, the signalr(t,x) is written in the time–distance domain (Fig. 2a) and in thefrequency–wavenumber domain (Fig. 2b), respectively, as

(3)

and

(4)

In the f–k plane, a dispersive wave is characterized by a non-straight line passing through the wavenumber axis at position ko,not equal to zero (Fig. 2b). In order to select dispersive waves bya masking filter in the f–k plane (fan filter, strip filter; Fail andGrau 1963), it is necessary to apply a group-velocity correctionand a phase-shift correction (i.e. translation of the ko value on thewavenumber axis) (Yilmaz 2002).

Filtering by singular value decomposition on single-compo-nent dataAfter propagation through the media, the received signal r

i(t) on

sensor i is the result of the superposition of Ns (number of

sources) waves [a1(t),...,a

Ns(t)] via the transfer functions s

i,p(t),and is given by

(5)

where bi(t) is noise, assumed to be Gaussian, white and centred,

and Nc is the number of sensors. With signals sampled in time,we write the received signals in a data matrix as

(6)

The singular value decomposition of the time–space datamatrix r provides two orthogonal matrices u and v, and one diag-onal matrix ∆∆ made up of singular values (Klema and Laub1980; Golub and Van Loan 1996; Vrabie et al. 2004). The initialdata matrix is expressed as

(7)

• where u = [u1,…,uk,...,uN] is an Nc× N orthogonal matrix madeup of left-hand singular vectors uk giving the amplitude in thereal case (amplitude and phase in the complex case), thereforecalled propagation vectors;

• v = [v1,...,vk,...,vN] is an Nt× N orthogonal matrix made up ofright-hand singular vectors vk giving the time dependence,hence called normalized wavelets;

• D = diag(λ1,...,λk,...,λN) is an N× N diagonal matrix with thediagonal entries ordered λ1 > …> λk >.. > λN > 0.The product ukvk

T is an Nc× Nt unitary rank matrix known asthe kth singular image of data matrix r. Therefore, r is given bythe sum of all the kth singular images multiplied by their corre-sponding kth singular values λ. The rank of the matrix r is thenumber of non-zero singular values in ∆∆. In the noise-free case, ifthe recorded signals are linearly dependent (for example, if theyare equal to within a scale factor, i.e. one wave with an infinitevelocity), the matrix r is of rank one and a perfect reconstructionrequires only the first singular image (Freire and Ulrich 1988;Mars et al. 1999; Vrabie et al. 2004). For one dispersive wave,

FIGURE 2

Dispersive plane wave (model). (a) Time–distance repre-

sentation; (b) horizontal component.



which has an infinite velocity and is noise-free, two singularimages are required. In practice on a real data set, a maximum offive singular values is usually necessary. If the Nc recorded sig-nals are linearly independent, the matrix r is full rank and the per-fect reconstruction requires all singular images. Hence, in prac-tice, before performing the SVD filtering, a velocity correction isapplied to the initial data to obtain an infinite apparent velocityfor the selected wave. This preprocessing enables decompositionwith a smaller space (Mari et al. 1997; Glangeaud et al. 1999).

Using the SVD filter, the separation between the signal andthe noise subspace is given by

(8)

The signal subspace rsig is characterized by the first Ns higher sin-gular images (associated with the first Ns higher singular values).It gives roughly the waveform of the dominant wave, its energyand its amplitude repartition on the sensors. The remainder, sub-space rnoise, contains the waves with a low degree of sensor-to-sen-sor correlation and the noise (Mars et al. 2004; Vrabie et al. 2004).

Filtering by singular value decomposition on multicompo-nent data (4C-SVD)For a multicomponent section (3C or 4C), the objective of the fil-ter is to obtain the polarization state of the waves. In order tocharacterize it, we apply the singular value decomposition filterfor each 4-component trace (4C-SVD) (De Franco andMussachio 2001; Meunier et al. 2001). Each section we consid-er is composed of each individual sensor of the four componentscontained in a matrix r ∈ℜ 4×Nt. As no time delay appearsbetween the components of each sensor, SVD processing isapplied under optimal conditions (no velocity correction isrequired in this case). The 4C-SVD filtering on each sensorenables decomposition with four singular images as

(9)

The singular vector ui describes a wavelet associated with a prop-agation mode. The singular vector vi expresses the amplitude variations of a wavelet (versus distance). The singular value λi

characterizes amplitudes relating to the section of rank i. For eachsensor (4-component vectors, X, Y, Z and hydrophone), we onlykeep the first singular image (equal to λ1.u1.v1

T). At each sensor,the components vx, vy, vz, vhy of the first singular vector v1 are usedto calculate the rotation angles in order to line up the seismic sen-sor. The angle α = atan(vx/vy) is used to realign the sensor in thehorizontal plane to obtain in-line and cross-line sections.

Filtering by spectral matrix filtering on single-component dataOn a single-component section, surface-wave separation and/orenhancement of the signal-to-noise ratio can be performed byapplying an eigendecomposition filter in the frequency domain

(Mars et al. 1987). In the frequency domain, the received signalresulting from the superposition of Ns waves is given by

(10)

where aiis the amplitude (at frequency f) of the wave i, Si is the

vector of the wave i describing the propagation between sensorsand bi(f) is Gaussian, white, centred noise. The spectral matrix ofthe received signal is defined as

(11)

where †

denotes the transpose conjugate operation and E (themathematical expectation) is a smoothing operator applied in thefrequency domain, in the distance domain or by experienceaccording to the experimental context (Shan et al. 1985; Mari etal. 1997). Thus we obtain an Nc× Nc spectral matrix at each fre-quency bin f. Under the assumption of white noise, the spectralmatrix is expressed as

(12)

where γ is the noise power spectral density and

The rank of is equal to Ns (the number of sources).

The eigendecomposition of gives

(13)

where λ1 > λ 2 > …> λ Nc > 0 are the eigenvalues and V1, V2, …,VNc are their corresponding vectors. The number Nc–Ns of mini-mal eigenvalues of equal to λ is γ. The eigenvectors asso-ciated with these minimal eigenvalues are orthonormal to thecolumns of the matrix . The subspace generated by the Nc– Ns eigenvalues could be referred to as the ‘noise space’ andits orthogonal complement as the ‘signal space’. For each fre-quency bin, the subspace decomposition is written as

(14)

In the case of white noise, the number of dominant eigenval-ues gives the number of waves. Projection between the eigenvec-tors associated with the dominant eigenvalues and the initial datagives the signal subspace. In some specific cases, the waves canbe identified through the spectral matrix when • only one source is recorded on the data; the dominant eigen-

vector gives the wave vector of this source;• one dominant and energetic wave is present; eigenvectors

associated with the highest eigenvalues provide a good esti-mator of the dominant wave.



For the case of one dominant surface wave, after eigendecom-position of the estimated spectral matrix , we obtain thedominant eigenvector V1(f) associated with the highest eigenval-ue. By projection of the initial data on to this eigenvector, we candescribe the dominant surface wave:

(15)

Using the inverse Fourier transform of S1(f), we obtain the dom-inant wave in time and distance and the residual waves are givenon

Filtering by polarization filter on multicomponent dataWhen seismic sections are recorded on multicomponent sensors,the objective of the filter is to determine the polarization state ofthe waves (Achenbach 1973; Waters 1978). Polarization filtersenable the separation of two-component signals into a newdomain or into the signal domain, where we find all the waveshaving a given polarization ‘pol’, such that in the complementa-ry domain (the so-called noise domain) none of the waves arepresent. In this complementary domain, all the other waves havean orthogonal polarization. In order to distinguish pseudo-Rayleigh waves from other waves, the polarization filter (Marset al. 1999) is designed by successive operations as follows:• A phase shift is applied to linearize polarization figure

(ellipse). • This operation is followed by a rotation to bring the orienta-

tion of the major axis of the ellipse on to a particular compo-nent (for instance, the vertical component). At this step, theselected wave is now entirely localized on the vertical com-ponent V. The V-component is chosen as the signal domain.Once this operation has been carried out, the second wave isrepartitioned over the two components as its polarization isnot orthogonal to the selected pseudo-Rayleigh wave.

• The phase of the horizontal component H of the second wavemust be changed to obtain a rectilinear polarization (polariza-tion of the pseudo-Rayleigh wave, which has been entirelyprojected on to the vertical component, has not changed).

• The polarizations of the two waves must be made orthogonalby amplifying the horizontal component H. To avoid an infi-nite amplification factor, a rotation of the reference (V, H) iscarried out before calculating the amplification factor. Oncethe amplification factor has been determined, an oppositeangle of rotation must be applied to replace the reference inits initial position.

• A rotation must be carried out to put the first wave back onthe vertical component.Based on the fact that body waves have different states of

polarization from surface waves, several authors have proposeddifferent types of polarization filter to remove the pseudo-Rayleigh wave. To remove ground roll, Becquey and Manin(1996) presented an ellipticity filter using discrete wavelet trans-forms on each component. Becquey and Rodriguez (2000) used

spectral matrix filtering to select linear polarization events.Picheral et al. (2001) proposed a shift invariance method to esti-mate apparent velocity and polarization jointly. Using time–fre-quency tools, Roueff et al. (2002) designed an efficient obliquepolarization filter using a cross scalogram.

Filtering by high-order polarization filter on multicompo-nent dataOver the last 10 years, filter formulations have been extended tothe high-order statistics domain in order to handle the limitationsimposed by order-2 statistics (correlation and spectra). Thepolarization filter has been formulated in the high-order statisticsdomain. For the case of one wave, the analytic signal recordedon a 2-component sensor is given by

(16)

where yv(t) and yh(t) denote, respectively, the signals recorded onthe vertical component V and the horizontal component H, β andα are the polarization parameters (α denotes the phase shiftbetween components and β denotes the amplitude ratio) and z(t)is an analytic wavelet. The normalized polarization vector polmust be identified on each sensor. In the case of two wavesimpinging on a 2-component sensor, we can expressed therecorded signals as

(17)

where ai is a mixing matrix of sensor i describing the polarizationparameters and the wave amplitudes (Lacoume et al. 1998). Thismatrix can be decomposed by singular value decomposition as

(18)

The filtering consists of two steps. The first step, using sec-ond-order statistics, is based on the eigendecomposition of thecovariance matrix at each sensor i as

(19)

As (sources are assumed non-correlated),

(20)

At this step, u and ∆∆ are estimated by second-order processingbut v (see equation (18)) is still missing. Using u and ∆∆, we cancompute the whitened observations as

(21)

where



(22)

The second step of this high-order polarization filter consists ofestimating the rotation matrix v by using, for example, the JADE(jointly approximated diagonalization of eigenelements) algo-rithm, based on the diagonalization of matrices built with 4th-order cross-cumulants (Cardoso and Souloumiac 1993; Le Bihanand Mars 2000).

A LAND EXAMPLEPseudo-Rayleigh waves: 2-component data setSeismic data were obtained using explosive sources and a line of2-component receivers (47 sensors). The distance between adja-cent geophones is 10 m. The vibration axis of the horizontalcomponent H is located in the plane through the source point andthe seismic line (in-line). The second component V is a geophonewith a vertical axis. The offset is 50 m. Data are sampled every16 ms and the recording duration is limited to 4 s. Figure 3 showsthe vertical and horizontal components of the initial data record-ed. On the vertical component, we can identify a refracted wave,a reflected arrival with a strong amplitude associated with a deepreflector, and two dispersive pseudo-Rayleigh waves character-ized by two low apparent velocities and a low-frequency content.Figure 4 shows the 2D amplitude spectrum of the vertical seis-mic recording. The spectrum is presented with (Fig. 4a) andwithout (Fig. 4b) normalization of wavenumber k. Without nor-

malization of k, the two branches associated with the two dis-persed surface waves are clearly visible at low frequencies(under 16 Hz). After normalization in the k-domain, the refract-ed wave appears at frequencies over 17 Hz, and the reflectedwaves are concentrated near the null wavenumber (k=0). Thefirst branch of the pseudo-Rayleigh mode is situated in the fre-quency range 3–10 Hz and the second branch is in the range10–20 Hz. The purpose of the study is to select the slowest pseu-do-Rayleigh wave for analysis. Several types of filter can beapplied to select this slow dispersive pseudo-Rayleigh wave: thef–k filter, the polarization filter, the SVD filter, the SMF filter,and the high-order polarization filter.

f–k filteringA filter designed in the f–k domain allows the extraction of theslowest pseudo-Rayleigh waves after flattening of this waveusing a group-velocity correction of 189 m/s. This filter is equiv-alent to a fan velocity filter. The filtered wave is then subtractedfrom the initial data. The remaining residuals are used to evalu-ate the efficiency of such filtering. Filtered data and the residualsection are respectively presented in the x–t plane in Figs 5(a)and 5(c) and in the f–k plane in Figs 5(b) and 5(d). The 2D ampli-tude spectrum associated with the residual section mainly showsthe fast pseudo-Rayleigh wave (Fig. 5b).

To evaluate the benefits of the filtering processing, we com-pute the phase velocities before and after processing. Figure 6(a)shows the initial data of the vertical component windowed in

FIGURE 3

Land seismic example: raw data

showing vertical component (V),

horizontal component (H).

FIGURE 4

Land seismic example: 2D ampli-

tude spectrum of vertical compo-

nent. (a) Without wavenumber

amplitude normalization; (b) with

wavenumber amplitude normal-

ization.

time and after a rough group-velocity correction. We can see thatthe pseudo-Rayleigh wave interferes with other waves along theoffset axis. Interference is especially strong in the short offset(between 50 and 200 m). After filtering by an f–k filter, the pseu-do-Rayleigh wave is well isolated without any interference (Fig. 6b), and at this step, we can obtain the geophysical param-eters of this pseudo-Rayleigh wave. We observe two branchesalong the distance axis. The first one, the ‘short offset’, presentsa well-aligned pseudo-Rayleigh wave (correct velocity andphase-shift corrections). The second one, the ‘long offset’, showsa pseudo-Rayleigh wave with a non-infinite group velocity.Computing the phase velocity for each branch (Fig. 6c) enablesus to characterize a change in behaviour in the propagation ver-sus distance plot (probably due to a change of petrophysicalparameters). These phase velocities are frequency dependent.Computation of the phase velocity using the initial wave (withinterference) gives incorrect information on the propagation.

Polarization filteringThe polarization parameters of the slow pseudo-Rayleigh waveare the phase shift and the rotation, which will make the polar-ization linear and vertical. The rotation and the phase shift val-ues have to be refined so that the slow pseudo-Rayleigh wave isonly present on one component. Phase shift (–85°) and rotation(–45°) parameters are estimated from sensor #23 where the pseu-do-Rayleigh is fairly visible from 1.5 s onwards. In the field casestudied, as the parameters of the polarization of the wave are sta-



FIGURE 5

Land seismic example: f–k filter-

ing on vertical component.

Extracted surface wave after fil-

tering (a) in the time–distance

domain and (b) in the f–k domain.

Residual section (c) in the

time–distance domain and (d) in

the f–k domain.

FIGURE 6

Land seismic example: phase velocity before and after filtering. (a)

Vertical component in the time–distance domain between 0 and 0.8 s

before processing; (b) after f–k processing: pseudo-Rayleigh wave

extraction; (c) phase velocities from short and long offsets.



tionary in distance, this wave was well isolated. After the filter-ing process, the extracted wave and the residual wavefield areshown in Figs 7(a) and 7(b).

SVD filteringSingular value decomposition (SVD) is used to separate waves.Applying a group-velocity correction (189 m/s) and a phase-shiftcorrection (22°) reduces the dispersion of the slow pseudo-Rayleigh wave, which has a main frequency of 7 Hz. After thesecorrections, this wave is flattened and rendered non-dispersive.The pseudo-Rayleigh wave can be characterized by SVD filter-ing (five singular images are selected to estimate this wave).After filtering, the slow pseudo-Rayleigh wave is returned to its

initial position in the x–t plane. The extracted pseudo-Rayleighwave and the residual section are shown in Figs 8(a) and 8(b). Itcan be seen that the filtering is effective due to the fact that pseu-do-Rayleigh wave is flattened prior to the filtering.

SVD and SMF filteringConsidering the vertical component section, flattened by correc-tion of the group velocity and dispersion (phase shift) of the slowpseudo-Rayleigh wave, we want to estimate it by using the SMFfilter. As the main frequency of the slow pseudo-Rayleigh waveis 7 Hz, we can estimate the spectral matrix at this specific fre-quency. Eigendecomposition of this matrix allows us to recoverthe pseudo-Rayleigh wave characterized by the first eigenvector

FIGURE 7

Land seismic example: after

polarization filter on vertical

component. (a) Extracted surface

wave; (b) residual section.

FIGURE 8

Land seismic example: after sin-

gular value decomposition filter

on vertical component. (a)

Extracted surface wave estimated

by the stack of the fifth greatest

singular sections; (b) residual

section.

FIGURE 9

Land seismic example: spectral

matrix filtering on vertical com-

ponent. (a) Extracted surface

wave computed from eigenvec-

tors estimated at frequency 7 Hz;

(b) residual section.

(because a phase and group velocity correction has beenapplied). Figure 9(a) shows the extracted pseudo-Rayleigh waveafter such processing. The residual section is shown in Fig. 9(b).On this section, it is possible to identify the refracted wave, areflected wave and an aerial wave preceding the fast pseudo-Rayleigh wave (Fig. 9b).

High-order polarization filtering.Figures 10(a) and (b) show respectively the vertical and horizon-tal components, windowed in time and flattened with a constant-velocity correction. The two waves are clearly mixed on the twosections. The objective now is to extract the slow pseudo-Rayleigh wave using a high-order polarization filter. The result



FIGURE 10

Land seismic example: seismic sections

windowed and flattened with constant

velocity correction. (a) Vertical component;

(b) horizontal component.

FIGURE 11

Land seismic example: high-order polariza-

tion filter results. (a) Extracted surface

waves on the vertical component (left) and

the horizontal component (right); (b) rota-

tion angle and phase parameter for the sur-

face-wave estimation; (c) residual sections

on the vertical component (left) and the

horizontal component (right).



shows the estimation of the slow pseudo-Rayleigh wave on eachcomponent (Fig. 11a), and the residual section on the vertical andhorizontal components (Fig. 11c). The polarization parameterswere estimated independently for each trace. For the pseudo-Rayleigh wave, the layout of these parameters (rotation angleand phase shift) shows that the polarization varies gradually (Fig. 11b). The fluctuations that are seen are attributed to theinaccuracy of the measurements. Parameters associated with theresidual section are not meaningful, due to the interferencebetween several waves. For small source–sensor offsets, thisprocessing is more efficient than the other filtering describedabove (polarization filter, f–k filter, SVD and SMF). This is dueto the fact that this processing does not require previous knowledgeas polarization parameters do not change with distance.

A MARINE EXAMPLEMarine guided waves: four-component data set (OBS-4C)A seismic marine survey was provided by the CompagnieGénérale de Géophysique (CGG). Data were recorded on a 4-component sensor (one hydrophone and three geophones: OBS-4C) laid on the sea-floor (14 m). We show common-receiver sec-tions with the following parameters: maximal offset 900 m; lis-tening time 1s; time sampling 2 ms; spatial sampling 50 m. Figure12 shows the initial 4-component data set. On the hydrophone-,X- and Y-components, respectively shown in Figs 12(a), (b) and(c), guided waves are dominant. The Z-component is quite differ-ent (Fig. 12d) due to the fact that the guided wave is stronglyaliased. Considering the hydrophone-component, the guidedwave is dominant (Fig. 13a). In the f–k domain, the 2D amplitudespectrum, presented without normalization in the wavenumber

domain, shows the guided wave with its energy concentrated inthe 60–80Hz frequency bandwidth (Fig. 13c). With normalizationin the wavenumber domain (Fig. 13e), we see that its bandwidthis much wider (20–220 Hz). After the velocity correction has beenapplied in the x–t domain (Fig. 13b), we see that the energy asso-ciated with the guided wave is not centred on the null wavenum-ber in the f–k plane (Figs 13d and f) (Mars et al. 1999). This indi-cates that the guided wave is dispersive.

4C-SVD filtering.In order to characterize multicomponent relationships, the SVDmethod is applied to each 4C trace (4C-SVD). The seismic sec-tion we consider is composed of each trace of the four compo-nents. In order to apply SVD processing trace-by-trace undergood conditions, we checked that the guided wave is in phase onthe four components (after velocity correction). The 4C-SVD fil-tering of each trace (one trace is a 4-component recording) allows

FIGURE 12

Marine seismic data set: 4-component raw data. (a) Hydrophone compo-

nent section; (b) horizontal component section X; (c) horizontal compo-

nent section Y; (d) vertical component section Z.

FIGURE 13

Marine seismic data set: hydrophone component studies. (a) Hydrophone

component section in time–distance domain (raw data); (b) hydrophone

component section in time–distance domain after velocity correction; (c)

2D amplitude spectrum of (a) without wavenumber amplitude normal-

ization; (d) 2D amplitude spectrum of (b) without wavenumber ampli-

tude normalization; (e) 2D amplitude spectrum of (a) with wavenumber

amplitude normalization; (f) 2D amplitude spectrum of (b) with

wavenumber amplitude normalization.



decomposition into four singular sections. For each trace, wekeep only the first singular section. Figure 14 shows, from left toright, according to the source-sensor distance: vector u1: waveleton windowed time (0–0.25 s, Fig. 14a); components vx, vy, vz, vhy

of the first singular vector v1 (Fig. 14b); the rotation angle in thehorizontal plane defined by α = atan(vx/vy) (Fig. 14c).

For each trace, the components vx, vy, vz, vhy of the first singu-lar vector v1 are used to calculate the rotation angles to realign thesensor. The guided wave is entirely contained inside the first sin-gular section associated with each component. The other wavescontained in the data are in the complementary space consistingof the sum of the three other sections (from rank 2 to rank 4).

Figure 15 shows the results of the filtering in the 0–1s initialinterval on the different components in the time–distancedomain. A correction of inverse velocity has been applied to thedata in order to place the waves back in their original positions.For the hydrophone (Fig. 15a top), the filtered section is verynear the initial section. On the residual hydrophone component(Fig. 15b top), we can successively observe a refracted wave, ahigh-frequency guided mode with fast apparent velocity and a 10Hz dominant frequency mode with slow apparent velocity. Forthe Z-component, the section filtered by 4C-SVD (Fig. 15a bot-tom) is of very weak amplitude and does not resemble the initialsection because the guided wave is not very energetic. The resid-ual section, on the other hand, is therefore very close to the ini-tial section (Fig. 15b bottom).

The ratio of the guided wave, selected by 4C-SVD, to resid-ual waves, evaluated on the amplitude spectrum, is 20 dB for thehy-, Y- and X-components and –6 dB for the Z-component. Wesee that the refracted wave is dispersive for this shallow depthand the velocity analysis of the lower layer requires the velocitylaw to be designed according to the frequency. This multicompo-nent filtering is limited. We assume that the wave polarization isconstant with frequency. In the present case, for dominant fre-quencies, this proves to be true. On the other hand, it is not trueat the limits of the bandwidth. Indeed, for high frequencies, weobserve the presence of a fast wave for the filtered hydrophone-

component, which precedes the extracted guided wave. For lowfrequencies and angles close to the critical angle, the energy ishigh and the wave polarization varies rapidly. In addition, weobserve the presence of a second residual wave, slower than theselected guided wave.

CONCLUSIONSeismic data acquisition is currently carried out using single-component sensors. To extract reflected waves from the elemen-tary field records, geophysicists use conventional apparent-velocity filters in order to reject or extract dispersive surfacewaves. In near-surface geophysics, these dispersive waves arevery important because they convey information on the subsur-face media. The f–k filter is currently used, but it is not efficientat low frequencies and requires a large number of traces to givegood results. If the number of seismic traces is limited, the sin-gular value decomposition filter (SVD) is a very efficient meansof characterizing the dominant wave, which is usually the sur-

FIGURE 14

Marine seismic section: 4C singular value decomposition. (a) Wavelet u

from hydrophone section; (b) amplitude variation versus offset for all

component sections; (c) rotation angle versus offset for the hydrophone-

component section.

FIGURE 15

Marine seismic section: results after 4C singular value decomposition.

(a) Filtered section (from top to bottom: hydrophone-, X-, Y- and Z-sec-

tions); (b) residual sections (from top to bottom: hydrophone-, X-, Y- and

Z-sections).



face wave associated with ground roll. But the SVD filterrequires preprocessing (group-velocity and phase-shift correc-tions) to flatten the dominant wave before extraction. The twocorrections must be accurate. The preprocessing can also beapplied to the data in order to select the dispersive wave, using amasking filter in the f–k plane. In the same way, the spectralmatrix decomposition filter (SMF) can also be used. However, inthis case, only a coarse group-velocity correction is required. Multicomponent data sets allow the use of specific filters forstudying and processing dispersive waves observed both on landand marine field data. It has been shown that the SVD filter canbe extended to a multicomponent SVD filter (4C-SVD filter). Inthis case, preprocessing is not required and the results are betterthan those obtained with the conventional SVD filter.Multicomponent data sets allow the use of polarization filters. Ithas been demonstrated that the efficiency of conventional polar-ization filters can be increased by criteria based on high-orderstatistics (HOS). The results obtained illustrate the advantages ofusing multicomponent data. The efficiency of the 4C-SVD filterand the high-order statistic polarization filter has been demon-strated. It has been also demonstrated that surface-wave charac-teristics can be better identified after wave separation.

ACKNOWLEDGEMENTSWe acknowledge CGG for providing the data.

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