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New multicomponent filters for geophysical data processing Caroline Paulus∗ and Jérôme I. Mars† Laboratoire des Images et des Signaux 961 rue de la Houille Blanche, BP 46 38 402 Saint Martin d’Hères Cedex, FRANCE Fax : 33(0) 474 826 384 ∗ Phone : 33(0) 476 827 107, Email:
[email protected] † Phone : 33(0) 476 826 253, Email:
[email protected]
Abstract— As multicomponent processing is a challenge in many fields such as geophysics, seismology, remote sensing or electromagnetic, we propose a method which could be used to filter multispectral, multicomponent or multidimensional data. In this paper, the method, called Multicomponent WideBand Spectral Matrix Filtering (MC-WBSMF), is applied on geophysical data to separate interfering wavefields. The technique is based on the decomposition of a special multicomponent spectral matrix and could extract a given wavefield from a multicomponent dataset. MC-WBSMF is compared to two classical single-component seismic methods (the 2DFT or f-k filter and the Radon transform) and to a fully multicomponent method based on singular value decomposition (3C-SVD). Finally, MC-WBSMF is tested on a real geophysical subsurface dataset in order to remove surface waves and to enhance reflection waves. Index Terms— Geophysical array data, Multicomponent processing, Wavefield filters
I. I NTRODUCTION Seismic exploration plays a major role in the search for hydrocarbons and requires three main stages: data acquisition, processing and interpretation. Moreover, in the last decade, the use of multicomponent technology has improved significantly in the field of geophysics but also in seismology, electromagnetic or remote sensing, providing higher fidelity acquisition systems. As a consequence, more advanced analysis and processing methods for multicomponent records are required. The separation of interfering wavefields is one of the key points in geophysical data processing. For example, surface waves, also called ground roll, mask the reflections of deeper targets which are of economic interest. As such, the ground roll need to be removed from the data in order to focus on the underlying reflections. In the single component sensor arrays case, various methods have already been developed to extract the upgoing waves and downgoing waves. Wave decomposition is typically carried out 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 process is accomplished by selecting an appropriate region in the
transformed domain. Nonetheless, for example when P-waves are interfering with other types of waves, it is difficult to select the region of the required wave. Moreover, these filters are deficient in presence of non-plane waves or in case of short arrays. In the time-distance domain, median filters have been commonly used to separate upgoing and downgoing waves in VSP and to eliminate spikes [6]. Fuller showed that semblance filters also provide efficient algorithm to remove noise and improve velocity analysis [7]. Other separation techniques based on Singular Value Decomposition [8] are also used to remove noise. The basic approach to compute the SVD 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, the method is called spectral matrix filtering. The principle is to decompose an initial dataset into a new space free of noise, with the smallest possible dimension. Signal is separated from noise by finding a subspace defined by the eigenvectors relative to the dominant eigenvalues of the spectral matrix issued from the recorded signal. The eigenvalue decomposition allows a separation between signal and noise subspace at each frequency bin. The signal is viewed as a juxtaposition of short band signals since each frequency is treated independently. Nonetheless, as seismic signals are wideband, the previous method is not optimal to tackle this problem. Consequently, wideband spectral matrix filtering has been introduced in [11]. Furthermore, since more and more surveys use multicomponent sensors, specific filtering methods adapted to multicomponent dataset are required. 3CSVD [12], [13], [14], [15], SVD of quaternion matrices [16], or furthermore 3C-SVD compound with partial Independent Component Analysis [17] techniques are efficient to separate wavefield in multicomponent seismic profile but require wave alignment as they are based on SVD. To avoid this pre-processing, we propose to develop a method based on the spectral matrix filtering 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 seismic signal since it takes into account jointly polarization and interactions 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 the performances of classical filtering methods and of 3C-SVD on a synthetic dataset. Finally, results of MC-WBSMF on a real dataset will be shown. II. M ULTI C OMPONENT-W IDE BAND S PECTRAL M ATRIX F ILTERING (MC-WBSMF) A. Model formulation and hypothesis Multicomponent seismic data recordings depend on three parameters: time (Nt samples), distance (Nx sensors), directions (Nc components) and provide the potential to access wavefield polarization. The tensor of the data collected during Nt samples on a set of Nx sensors (each composed of Nc = 3 components noted X, Y and Z) is written as: T t ² RNx ×Nt ×Nc .
(1)
Applying the Fourier transform, the problem is divided into a set of instantaneous mixtures of signals and we have: T = T F {T t} ² CNx ×Nf ×Nc
(2)
with Nf the number of frequency bins. Let us assume that we have a linear array. Using the superposition principle, the signal recorded on a sensor results from the linear combination of Nw polarized waves received on the antenna added with noise. These waves have propagated through a medium which could have attenuated, time delayed or phase shifted them. The tensor T is concatenated into a long-vector noted T l of size (Nx Nf Nc ) which contains all the frequency bins on all the sensors for each component: Tl
=
[X(f1 )T , · · · , X(fN f )T , Y (f1 )T , · · · , Y (fN f )T , Z(f1 )T , · · · , Z(fN f )T ]T (3)
where X(fi ), Y (fi ) and Z(fi ) are vectors of size (Nx ) which corresponds to the ith frequency bin of the signals received on each of the Nx sensors respectively on components X, Y and Z. The mixture model is defined by: Tl = S · A + B
(4)
where: - S = [S 1 , S 2 , · · · , S N w ] is a (Nx Nf Nc , Nw ) matrix whose columns are the steering vectors describing the propagation of each of the Nw waves along the receiver group for all frequencies and all components. This matrix contains information about the polarization of the Nw waves and information about the waveform emitted by the sources. All these terms are deterministic; T - A = [a1 , a2 , · · · , aN w ] is a vector of size Nw which contains the random amplitudes of the waves;
- B is a vector of dimension (Nx Nf Nc ) which corresponds to the noise component added to each sensor at each frequency and on each component. These random noises are supposed to be additive, temporally and spatially white, uncorrelated with the sources, non polarized and to have identical power spectral density σb2 ; All relationship in frequency between components and sensors are expressed in the multicomponent wideband spectral matrix defined by: Γ = E{T l · T H (5) l } with E the expectation operator and H the transpose conjugate operation. Γ has dimensions (Nx Nf Nc ) by (Nx Nf Nc ). B. Estimation of the spectral matrix At this step, Γ is non invertible as it has unity rank. To avoid that fact and to decorrelate sources from noise and sources from themselves, it is necessary to perform an estimation of matrix Γ. Furthermore, the effectiveness of the filtering depends on the estimation stage of the spectral matrix Γ. In practical case of seismic data processing, the mathematical expectation operator E is replaced by specific averaging operators, like spatial or frequencial smoothings or both of them [19], [20], [21], [22]. Their purpose is to reduce the influence of the terms corresponding to the interactions of the different sources, making the inversion of the spectral matrix possible. Three possible averages are: • spatial smoothing; • frequency smoothing; • ensemble average. The spatial smoothing could be done by averaging spatial subbands. The uniform linear array with Nx sensors is subdivided into overlapping subarrays in order to have several identical arrays, which will be used to estimate spectral matrices in order to build a smoothed matrix. These subarrays are supposed to be linear and uniform (constant intertrace). If the smoothing order is Ks , the size of each subarray is Nx − 2Ks and the number of subarrays is 2Ks + 1. The spatially smoothed spectral matrix is defined by the average of the spectral matrices obtained for each subarray. Shan and al. [21] have proven that if the number of subarrays is greater than or equal to the number of sources Nw , then the spectral matrix of the sources is non-singular. However, one assumption is that the wave does not vary rapidly over the number of sensors used in the average, in particular, amplitude fluctuations must be smoothed out. To introduce frequency smoothing, two ways can be performed: either by weighting the auto-correlation and crosscorrelation functions (in the time domain) or by averaging frequential subbands (in the frequential domain). These two methods require a velocity correction allowing wave to be shifted in time within the limits of the duration of the weighting function. Therefore, the frequency average requires a rough pre-alignment of the wave (flattening). Considering a frequential smoothing of Kf order, we obtain a number of
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2Kf + 1 frequential subbands of size Nf .
independently on each component.
For a better estimation of the multicomponent wideband spectral matrix, it is suitable to realize jointly spatial and frequential smoothing. An observation generates a set of (2Ks + 1) spatially shifted recurrences. Subsequently, these recurrences can be frequentially smoothed (2Kf + 1) times. Eventually, a set of K = (2Ks + 1)(2Kf + 1) spatio-frequentially shifted b recurrences is obtained and the estimated spectral matrix, Γ, is defined by the average of the spectral matrices obtained for each subarray and each frequency subband. The rank of b is now equal to K which need to be chosen greater than Γ Nw , the number of waves.
2) Decomposition of the multicomponent wideband spectral matrix: By construction, the spectral matrix has a hermitian symmetry. bH . b=Γ (8) Γ
Finally, the ensemble average would be the most efficient average but it requires information redundancy which is rarely available. This average assumes that the signal is invariant from one realisation to the others whereas the noise varies. C. Estimation of signal subspace Once the multicomponent wideband spectral matrix has been estimated, the initial space can be separated into two b , generated by the first subspaces: the signal subspace, Γ s eigenvectors associated to the highest eigenvalues and its b . Filtering is then complementary, the noise subspace, Γ n achieved by projection of the initial data onto the signal subspace. This procedure is detailed in the following. 1) Structure of the multicomponent wideband spectral mab can be trix: Following the assumptions made in part.II-A, Γ written as: b · S H + σ2 · I = Γ b +Γ b =S·Γ b . Γ b A s n
(6)
b = E[A · AH ] be non-singular (waves partially Let Γ A correlated or uncorrelated) and the columns of S linearly b = (S · Γ b · S H ) is Nw . independent, the rank of Γ s A b could be written: Structure of Γ b b Γ Γ X,Y X,X b b b= Γ Γ Γ Y,X Y,Y b b Γ Γ Z,X Z,Y
b Γ X,Z b Γ Y,Z b Γ Z,Z
(7)
b b b where Γ , Γ and Γ correspond to the monoX,X Y,Y Z,Z component wideband spectral matrix for respectively component X, Y and Z. These terms are located on the main b The other blocks correspond to the crossdiagonal of Γ. component spectral matrix which contain information relating to the interaction between the various components. Since the signals on each component are correlated (because they are polarized), these extra diagonal terms contain information relating to the polarization. Since the multicomponent wideband matrix filtering provides more information on the signal and especially on polarization, better filtering results are obtained rather than applying classical filtering methods
b can be decomposed in a single way According to Eq. (8), Γ using eigenvalue decomposition (EVD) as: b = U · Λ · UH = Γ
M X
λi ui uH i
(9)
i=1
where M = Nx Nf Nc , Λ = diag(λ1 , · · · , λM ) are the eigenvalues and U is a unitary M by M matrix whose columns b The eigenare the orthonormal eigenvectors u1 , · · · , uM of Γ. values λi correspond to the energy of the data associated with the eigenvector ui . They are arranged as follows: λ1 ≥ λ2 ≥ · · · ≥ λM ≥ 0.
(10)
b has M eigenvalues. Considering Eq. (9), the spectral matrix Γ Nonetheless, the previous rank property (rank(Γs ) = Nw ) implies that: b are equal to σ 2 • the M − Nw minimal eigenvalues of Γ b • the eigenvectors corresponding to these minimal eigenvalues are orthonormal to the columns of the matrix S. Thus, the space generated by the smallest eigenpairs is referred to as the noise subspace and its orthogonal complement as the signal subspace since it is spanned by the steering vectors of the signal. This justifies the benefit of projecting data onto the first eigenvectors to separate signal from noise. b has a Moreover, resulting from smoothing operators, Γ significant rank deficiency since its rank is equal to K and K is much lower than M . Consequently, we have: λi = 0
for K + 1 ≤ i ≤ M.
(11)
Eq. (9) becomes: b= Γ
K X
λi ui uH i .
(12)
i=1
Since the computing time for the diagonalization is proportional to M 3 , it is excessive to diagonalize the whole spectral matrix of size M by M . Only the K first eigenvalues and eigenvectors need to be computed. Methods have been developed to compute only the few dominant eigenpairs (for example the Lanczos algorithm [9]). 3) Filtering by projection onto the signal subspace: The filtering step corresponds to an orthogonal projection of the initial data T l (Eq. (3)) onto the highest eigenvectors corresponding to the signal subspace. As considered previously, the eigenvectors associated with the Nw largest eigenvalues b ), by belong to the same subspace called signal subspace (Γ s b opposition to the noise subspace (Γn ): b +Γ b=Γ b = Γ s n
Nw X i=1
λi ui uH i +
K X
i=Nw +1
λi ui uH i .
(13)
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If Nw is unknown, various statistical methods can be used to estimate Nw [23]. As soon as Nw is found, T l is projected onto its Nw first eigenvectors, following the expression: T ls =
Nw X
hT l , ui i · ui .
(14)
i=1
The projection onto the noise subspace (T ln ) is obtained by subtraction of T ls from the initial data: T ln = T l − T ls =
K X
hT l , ui i · ui .
(15)
i=Nw +1
Fig. 2.
Hodogram of the initial dataset
Fig. 3.
FK diagram of the initial dataset
The last steps consist of rearranging the long-vectors T ls and T ln in the tensor form (reverse of Eq. (3)) and computing an inverse Fourier transform in order to come back to the timedistance-component domain. III. A PPLICATION ON SYNTHETIC DATASET OF VARIOUS FILTERING METHODS
In this part, the performances of the MC-WBSMF will be tested on a synthetic multicomponent seismic dataset. The goal is to extract the various wavefields from noise and also to separate the waves between themselves. Afterwards, we described briefly various signal processing methods commonly used in seismic processing. We test the following filters: • the velocity filter (2DFT); • the τ -p filter (Radon filter); • the multicomponent singular value decomposition (3CSVD); MC-WBSMF and 3C-SVD methods are fully multicomponent method whereas the classical velocity and τ − p filters work on monocomponent datasets. Consequently, they are applied independently on each component.
non-anysotropic. One wave (referred as "1st wave" in the following) has an infinite velocity, an elliptical polarization and a phase-shift which is varying with offset. The second one (referred as "2nd wave") is a non-plane wave linearly polarized and with a non-infinite velocity. This additive noise is uncorrelated with the sources and temporally and spatially white. 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). Since the initial data are very noisy, one could not distinguished information on wavefields on this hodogram. The f-k diagram of one component corresponding to the initial dataset is shown on Fig. 3. A. MC-WBSMF
Fig. 1.
Initial dataset
The synthetic dataset contains two waves added with noise received on a simulated 3C-sensors array (Fig. 1) composed of 20 sensors. Ricker wavelet has been used as the source wavelet. The signal has propagated through a medium composed of horizontal geological layers that are assumed to be
The results obtained with MC-WBSMF are presented. The spectral matrix is estimated with Ks = Kf = 3 for the spatial and frequential smoothing orders. The amplitude of the ten first eigenvalues resulting from the decomposition of the global spectral matrix is presented on Fig. 4. The first eigenvector allows us to recover the first wave with the infinite velocity and the second eigenvector, the second wave with the non-infinite velocity. Fig. 5 and Fig. 7 show the two extracted waves. One can notice a loss of sensors after filtering which is due to the spatial smoothing. On the hodograms presented on Fig. 6 and Fig. 8, the continous line corresponds to the extracted waves with MC-WBSMF and the dashed line to the true waves (model). One can notice that the ellictic polarization of the first wave and the linear polarization of the second waves were
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relatively well recovered. The amplitudes were also preserved and 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 the 1st extracted wave with MC-WBSMF
Fig. 7.
2nd extracted wave with MC-WBSMF
Fig. 8.
Hodogram of the 2nd extracted wave with MC-WBSMF
Fig. 9.
1st extracted wave with FK filter
B. Velocity filter 1) Method: The velocity filter uses the (f,k) transform which corresponds to the double Fourier transform of a seismic section in time-distance. This is one of the most classical filtering method used for 2D filtering [1], [2], [3]. 2) Application: Results of velocity filtering performed on the synthetic dataset are shown on Fig. 9 for the first wave and Fig. 11 for the second wave. Fig. 13 and Fig. 14 correspond to the f-k diagram and Fig. 10 and Fig. 12 to the hodograms of
the two extracted waves. The results are satisfactory since the two waves have been well separated. The elliptic polarization of the first wave has been well recovered whereas the linear polarization of the second wave has been totally changed into an elliptic polarization. Moreover, f-k filter has major limitations. For instance, it requires a large number of traces and sufficient data sampling to avoid problems related to aliasing whereas that is not the case with MC-WBSMF.
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Fig. 10.
Hodogram of the 1st extracted wave with FK filter Fig. 14.
FK diagram of the 2nd extracted wave with FK filter
slowness and V the velocity. Thus, we obtained a function of time (with null offset); time is then noted τ . The result is presented according to variables τ and p. The method has been extended to the case of non-plane waves. Summation is done along curves instead of lines and the method is called hyperbolic τ -p filter [4]. This is the one we used on this example.
Fig. 11.
Fig. 12.
Fig. 13.
2nd extracted wave with FK filter
Fig. 15.
1st extracted wave with τ − p filter
Fig. 16.
Hodogram of the 1st extracted wave with τ − p filter
Hodogram of the 2nd extracted wave with FK filter
FK diagram of the 1st extracted wave with FK filter
C. τ -p filter The τ -p filter consists in making a summation of the seismic section according to a line of slope p = 1/V with p the
1) Application: Results of τ − p filter performed on the synthetic dataset are shown on Fig. 15 and Fig. 16 for the first wave and on Fig. 17 and Fig. 18 for the second wave. The results are really satisfactory since the two waves have been well separated. However, the amplitudes are not preserved and
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about the amplitude variations of the wave ut i along the seismic section. In the previous equation, ∆ is multiplied on its first mode (rows) by the left singular matrix U x and on its second mode (columns) by the right singular matrix U t . Using this notation, the extension of the SVD to T t ² RNx ×Nt ×Nc is obvious: T t = D × 1 U x ×2 U t ×3 U c
(18)
where U t , U x and U c contain the normalized wavelets, their behaviour along the x-direction and their polarization. D is the equivalent of the pseudo-diagonal matrix ∆ and contains the singular values. The singular matrices U x , U t and U c are obtained by applying classical SVD to the unfolding versions of T t. Three unfolded matrices can be built with T t: Fig. 17.
2nd extracted wave with τ − p filter
T tx ² RNx ×Nt Nc T tt ² RNt ×Nx Nc T tc ² RNc ×Nx Nt .
(19)
The U x , U t and U c are the left singular matrices coming from the SVD of the respectively unfolded matrices T tx , T tt and T tc . D is obtained using the inverse expression of the 3CSVD: D = T t ×1 U Tx ×2 U Tt ×3 U Tc . Fig. 18.
Hodogram of the 2nd extracted wave with τ − p filter
the method is really dependant on how the mask in the τ − p domain is designed.
(20)
T t is characterized by three ranks, one for each mode: rx , rt , rc . These ranks are estimated using the SVD of the three unfolded matrices. The separation of the seismic events is then realized using a method derived from the 1C case. The original space is decomposed into a signal subspace and a noise subspace:
D. 3C-SVD filtering 1) Method: The 3C-SVD is an extension of the classical SVD method used for monocomponent datasets. It is based on HO-SVD (Higher Order Singular Value Decomposition). To generalize to the multicomponent case, we present the multilinear 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: X (A ×n B)(i1 ,i2 ,··· ,j,··· ,iN ) = (16) ai1 i2 ···in ···iN .bjin . in
For a 2D seismic section X ² RNx ×Nt , the Singular Value Decomposition could be written as: X = ∆ × 1 U x ×2 U t
(17)
where ∆ is a pseudo-diagonal matrix containing the singular values of X which define the relative amplitudes of the various eigensections. U t contains vectors ut i which have the dimensions of a trace and corresponds to the normalized wavelet. U x contains vectors ux i which convey information
Tt = Tt + Tt s
(21)
n
where the signal subspace (T t ) is the (˜ rx , r˜t , r˜c ) rank s approximation of T t defined as: T t = T t ×1 U x U Tx ×2 U t U Tt ×3 U c U Tc s
s
s
s
s
s
s
(22)
where the U x , U t and U c are obtained by keeping s s s respectively the first r˜x , r˜t , r˜c vectors of U x , U t and U c . The number r˜t corresponds to the number of normalized wavelets necessary to describe the signal subspace. r˜x and r˜c describe the behaviour of each selected wavelet along the x-direction and their polarization. 2) Application: Results of 3C-SVD filtering performed on the synthetic data set are shown on Fig. 19 and Fig. 20 for the first wave and on Fig. 21 and Fig. 22 for the second wave. Most of the noise has been removed. Although the first wave seems to be relatively well extracted (a very small part of the second wave, collinear to the first wave, is present), the elliptic polarization of this wave was not well estimated (Fig. 20). Moreover, the second wave with a non-infinite velocity has been badly separated.
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Fig. 22.
Fig. 19.
1st extracted wave with 3C-SVD
Hodogram of the 2nd extracted wave with 3C-SVD
MC-WBSMF Velocity filter τ -p filter 3C-SVD
MSE for 1st wave 8.10−4 24.01−4 55.01−4 35.10−4
MSE for 2nd wave 38.10−4 32.10−4 54.10−4 122.10−4
TABLE I M EAN SQUARE ERROR
most methods work better if the wave has an infinite velocity. Moreover, MC-WBSMF and f-k filter are the two methods providing the best performances in separating wavefield and enhancing signal-to-noise ratio. Fig. 20.
Hodogram of the 1st extracted wave with 3C-SVD
We will now discuss the advantages and disadvantages of the tested methods in comparison with MC-WBSMF. A first type of methods require to work in an associated domain. The f-k filter and τ -p filter belong to this category. These two methods provide really good filtering results. Nonetheless, the largest disadvantage of these two methods is that they require user’s intervention in order to realize the mask filtering in the associated domain before coming back to the time-distance domain. Moreover, other disadvantages of f-k filter are that it requires a large number of traces with short distance sampling and that it is very cost demanding.
Fig. 21.
2nd extracted wave with 3C-SVD
E. Performances evaluation Table I quantifies the filtering results obtained on the synthetic dataset with the previous methods. Mean square error (MSE) has been computed to quantify the difference between model and filtered wavefield in order to evaluate performances 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 better than the one obtained for the second wave. The reason is that
A second category which could be referred to as the matrix methods, includes the Singular Value Decomposition (SVD) [8], [9], the 3DSVD [15], the Karhunen-Loeve method [24], [25], [26], and the various spectral matrix filtering methods: SMF [27], WBSMF [11] and MC-WBSMF [18]. These methods build a matrix (data, covariance or spectral matrix) which is then decomposed according to its eigenvectors. These methods are used to separate waves but also to improve signal to noise ratio by breaking down the data into a signal subspace and a noise subspace. SVD-based and KarhunenLoeve methods require the extracted wave to be flattened (infinite velocity) before separation. In contrast, the spectral matrix can operate without flattening but the estimation step of the spectral matrix determines the effectiveness of the filtering. However, contrary to MC-WBSMF, SMF and WBSMF do not take into account the wideband characteristics of the seismic signal 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, without a-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 real multicomponent seismic datasets and could be used either to process seismic reflexion profile, or Vertical Seismic Profile (VSP), or OBS (Ocean Bottom Seismometer) and OBC (Ocean Bottom Cable). In case of seismic reflexion, only reflected waves convey information for imaging the subsurface. One aim is to separate the reflected waves from the other interfering waves (directs arrivals, refracted waves, surface waves, ground roll...). As the algorithm favours the waves with the highest energy, the method is used to remove energetic waves (direct waves and ground roll) in order to enhance reflected waves (which have low energy and which are contained in the projection onto the noise subspace).
Fig. 23.
Fig. 24.
Initial dataset
Estimated slowest pseudo-Rayleigh wavefield
Fig. 25. FK diagram of the initial dataset (up) and the slowest pseudoRayleigh wavefield (down)
Fig. 26.
First residual section
Fig. 27.
Fast pseudo-Rayleigh wavefield
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Fig. 28.
Second residual section
on the initial data to isolate the slowest pseudo-Rayleigh wave (Fig. 24). Fig. 25 shows the f-k diagram of the initial dataset and of the slowest pseudo-Rayleigh wave. We clearly see that the MC-WBSMF filtering is not dependent on the spatial sampling of the data since it does not suffer from aliasing. The impact of this feature is that the survey design may be less constrained by spatial sampling requirement. Difference between initial dataset and estimated slowest pseudo-Rayleigh waves is given on Fig. 26. MC-WBSMF was applied a second time on this residual section to isolate the fastest pseudo-Rayleigh wave presented on Fig. 27. Finally, processing has been applied on the second residual section between 0 and 1.4s (Fig. 28) since we applied a mute between 1.4s and 4s. Two reflected waves has been well isolated at around 0.6s and 1.2s (Fig. 29). Results obtained with MC-WBSMF could be compared with the ones obtained with f-k filtering, polarization filtering, SVD and SMF since the previous methods have been tested on the same near surface dataset in [28]. V. C ONCLUSION
Fig. 29.
First and second reflected waves
The proposed dataset has been obtained by using explosive sources and a line of 47 2C-receivers in the valley of Chantourne near Grenoble-France. The distance between adjacent geophones is 10 m. The vibration axis of the horizontal component is located in the plane going through the source point and the seismic line (inline). The second component is a geophone with a vertical axis. The offset is 50 m. Data are sampled every 16 ms and the recording’s duration is limited to 4 seconds. Fig. 23 shows the initial data recorded on the vertical and horizontal component. On the vertical component, we can identify a refracted wave, a reflected arrival with a strong amplitude associated to a deep reflector, and two dispersive pseudo-Rayleigh waves characterized by low apparent velocities and a low frequency content. On the horizontal component, the two pseudo-Rayleigh waves are also present and a reflected waves (around 1,2 ms) drowned into the others waves. The aim of the study is to separate the different wavefields to enhance the reflections. MC-WBSMF has been applied three times. The first time was
Since multicomponent geophones are now largely used for seismic acquisitions, specific tools for these kind of data are required. This paper presents a new method for wavefield filtering of multicomponent data called multicomponent wideband spectral matrix filtering (MC-WBSMF). The performances of the method have been compared with the ones of the classical filtering methods (velocity filter, τ -p filter) and with multicomponent SVD; and has also been tested on real multicomponent datasets. It has been shown that MC-WBSMF provides good results in terms of signalto-noise ratio enhancement and waves separation comparing with conventional methods. Moreover, the method is relatively low cost and works without any pre-treatment and a-priori information. The main reason of this achievement comes from the consideration of the wideband characteristics and the polarization (multicomponent aspect) of the seismic signals. R EFERENCES [1] P. Embree, J. P. Burg, and M. M. Backus, “Wide band velocity filtering - the pie-slice process,” Geophysics, vol. 28, pp. 948–974, 1963. [2] C.-M. Chen and M. Simaan, “Velocity filters for multiple interferences in two dimensional geophysical data,” IEEE Trans. Geoscience and Remote Sensing, vol. 29, no. 4, pp. 563–570, July 1991. [3] M. Hanna, “Velocity filters for multiple interference attenuation in geophysical array data,” IEEE Trans. Geoscience and Remote Sensing, vol. 26, no. 6, pp. 741–748, November 1988. [4] O. Yilmaz, Seismic Data Processing. Society of Exploration Geophysicists, 2002, vol. 1, ch. 6, pp. 938–960. [5] S. Greenhalgh, I. Mason, E. Lucas, D. Pant, and R. Eames, “Controlled direction reception filtering of P- and S- waves in τ -p space,” Geophys. J. Int., vol. 100, pp. 221–234, 1990. [6] G. Duncan and G. Beresford, “Median filter behaviour with seismic data,” Geophysical Prospecting, vol. 43, pp. 329–345, 1995. [7] B. Fuller and R. Lynn Kirlin, “Weighted correlation pairs for improved velocity analysis,” in Meeting of Society of Exploration Geophysicists, Expanded Abstract, 1992, pp. 1221–1222. [8] S. L. M. Freire and T. J. Ulrych, “Application of singular value decomposition to vertical seismic profiling,” Geophysics, vol. 53, no. 6, pp. 778–785, June 1988. [9] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, London: The John Hopkins University Press, 1989.
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Caroline Paulus studied Electrical Engineering at the Telecom departement of the Institut National Polytechnique de Grenoble, France, where she received the Electrical Engineering degree in 2002, and the M.S. degree in Signal, Image, Speech and Telecommunications from the Institut National Polytechnique de Grenoble, France, in 2003. She is currently working toward the Ph.D. degree at the Laboratoire des Images et Signaux, Grenoble, France. Her research interests are in signal processing, more precisely multidimensional filtering for geophysical applications.
Jérôme I. Mars received a Ms (1986) in Geophysics from Joseph Fourier University of Grenoble and PhD in Signal Processing (1988) from the Institut National Polytechnique of Grenoble. From 19891992, he was a postdoctoral research at the Centre des Phénomènes Aléatoires et Geophysiques, Grenoble. From 1992-1995, he has been visiting lecturer and scientist at the Materials Sciences and Mineral Engineering Dept at University of California, Berkeley. He is currently Assistant Professor in Signal Processing for the Laboratoire des Images et des Signaux at the Institut National Polytechnique de Grenoble He is leader of geophysical signal processing team. His research interests include seismic and acoustic signal processing, wavefield separation methods, time frequency time-scale characterization, and applied geophysics. He is member of SEG and EAGE.
