Multicomponent f-x seismic random noise attenuation via vector autoregressive operators

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1 Multicomponent f-x seismic random noise attenuation via vector autoregressive operators Mostafa Naghizadeh and Mauricio Sacchi ABSTRACT We propose an extension of the traditional frequency-space (f-x) random noise attenuation method to 3-component seismic records. For this purpose, we develop a 3-component vector autoregressive (VAR) model in the f-x domain that is applied to the multicomponent spatial samples of each individual temporal frequency. VAR model parameters are estimated using the least-squares minimization of forward and backward prediction errors. VAR modeling effectively identifies the potential coherencies between various components of multicomponent signal. We use the squared coherence spectrum of VAR models as an indicator to determine these coherencies. Synthetic and real data examples are provided to show the effectiveness of the proposed method. INTRODUCTION In exploration seismology we sometimes deal with multicomponent seismic records. For instance, 3-component geophones simultaneously record two horizontal components and one vertical component of the incident wave-field. The common approach in dealing with multicomponent data is to process each component separately. However, vector autoregressive (VAR) modeling presents a promising application for processing multicomponent data. VAR modeling can provide not only a robust analysis of each individual component but also valuable information about the coherency between each component (Pagano, 978; Hrafnkelsson and Newton, ). In this article we will investigate the application of VAR modeling to random noise attenuation. A large class of de-noising methods utilize the Fourier transform. The strategy behind Fourier-based de-noising methods is to retain a few dominant harmonics by preserving a finite number of frequency or wavenumber components in the Fourier domain (Naghizadeh and Sacchi, ). Frequency-space (f-x) domain methods comprise a large group of seismic data interpolation and de-noising methods. For instance, prediction filters are used by Canales (984) and Spitz (99) in the f-x domain for de-noising and data interpolation, respectively. Other methods such as projection filters (Soubaras, 994), noncausal prediction filters (Gulunay, ), Singular Value Decomposition (Trickett, 3), Cadzow de-noising (Cadzow and Ogino, 98; Trickett and Burroughs, 9), and Singular Spectrum Analysis (Oropeza and Sacchi, 9) have also been used for random noise attenuation in the f-x domain. All of the f-x de-noising methods are based on the assumption that the spatial signals at each single frequency are composed of a superposition of a limited number of complex harmonics (Sacchi and Kuehl, ). In this article we introduce a VAR modeling method for multicomponent signal enhancement. The details of computing VAR models and their spectral interpretations are

2 Vector AR presented. The proposed de-noising method is applied to seismic records in the f-x domain. Synthetic and real seismic examples are provided to examine the performance of the proposed VAR de-noising method. We find that VAR filtering is a more powerful noise attenuator than standard AR filtering when components are correlated. THEORY (VAR) operators For a multicomponent signal of length N, we define the M-order forward VAR model as (Leonard and Kennett, 999) g k = M A j g k j, k = M +,..., N. () j= For a 3-component signal the vector autoregressive model is represented by 3 3 matrices of the form a j a j a j3 A j = a j a j a j3, () a j3 a j3 a j33 and g k = (g x, g y, g z ) T k is a 3-component vector at spatial sample k. For instance, expanding equation for order M = we have g x g y g z a = a a 3 a a a 3 a 3 a 3 a 33 g x g y g z +. (3) k k a a a 3 a a a 3 a 3 a 3 a 33 We can also define backward VAR modeling via the following expression g k = M j= g x g y g z k A j gk+j, k =,..., N M, (4) where represents the complex conjugate. The elements of A can be estimated using the least-squares method by simultaneously minimizing the forward and backward prediction errors in equations and 4 (Marple, 987). Spectral analysis of VAR operators The spectral density matrix of a VAR model is defined as (Hrafnkelsson and Newton, ) F(η) = G (η)g H (η),.5 η.5, (5) where M G(η) = I A l e iπlη, (6) l=

3 3 Vector AR G H is the inverse of the hermitian of G, i =, and I is the identity matrix. The variable η indicates normalized temporal frequency when the original signal evolves in time, whereas it indicates wavenumber for space dependent signals. A set of spectral attributes can be extracted from the spectral density matrix of the VAR models, the main two being the squared coherence and phase coherence spectra of the data. The squared coherence spectrum of the VAR model is Similarly, the phase coherence spectrum is W ij (η) = R {F ij (η)} + I {F ji (η)} F ii (η)f jj (η) Φ ij (η) = arctan. (7) ( ) I{Fji (η)}. (8) R{F ij (η)} where F ji (η) represents the (i, j)th element of F(η). The symbols R and I represent the real and imaginary parts of a complex function, respectively. To gain understanding of VAR modeling we start with a simple D multivariate example. We created three signals of 56 samples each and plotted them in Figures a-c. Figures d-f show the Fourier spectra of the data in Figures a-c, respectively. Each signal contains two harmonics and each signal has one harmonic in common with each of the other two signals. We estimated the VAR model parameters of the 3-component signal using Equations and 4 and then computed the diagonal spectrum, squared coherence spectrum, and phase coherence spectrum of the VAR model. Figures a, b, and c show the spectrum of the diagonal elements of VAR model (F, F, F 33 ) for signals in Figures a-c, respectively. These plots are equivalent to the spectrum of the ordinary autoregressive (AR) modeling applied individually to each signal. Figures 3a, 3b, and 3c show the squared coherence spectra between each pair of signals (W, W 3, W 3 ) shown in Figures a-b, a-c, and b-c, respectively. Notice that the squared coherence spectrum has peaks at the common frequencies of each pair of 3-component data. We have also shown the phase coherence spectra (Φ, Φ 3, Φ 3,) in Figures 4a, 4b, and 4c, respectively. Notice that the ±π discontinuities in the plots are the result of the well-known phase wrapping phenomenon. The spectral estimator derived from VAR models (equation 7) provides a measure to foresee if signal enhancement can be achieved via VAR modeling. The squared coherence amplitude shows how one component of data can benefit from other components. If the squared coherence spectra between two components is close to one, VAR modeling can increase the signal quality. In contrast, if the squared coherence spectra is near zero, VAR modeling will lead to results quite similar to those obtained via classical autoregressive modeling of individual components. Random noise elimination using VAR operators To enhance the signal-to-noise ratio of a multicomponent signal, we first estimate the VAR operator from the noisy data and represent it by Âj. Next, the forward estimate of de-noised data can be obtained using M ĝ f k = Â j g k j, k = M +,..., N. (9) j=

4 4 Vector AR - d) e) f) Sample numbers..4 Figure : A 3-component synthetic signal. a- First, second and third components, respectively. d-f) are the Fourier spectra of data in a-c, respectively Figure : VAR diagonal spectrum F corespondent to the signal in Figure a. VAR diagonal spectrum F corespondent to the signal in Figure b. VAR diagonal spectrum F 33 corespondent to the signal in Figure c.

5 5 Vector AR Figure 3: Squared coherence spectrum W between the signals in Figures a and b. Squared coherence spectrum W 3 between the signals in Figures a and c. Squared coherence spectrum W 3 between the signals in Figures b and c. Phase (rad) Figure 4: Phase coherence spectrum Φ between the signals in Figures a and b. Phase coherence spectrum Φ 3 between the signals in Figures a and c. Phase coherence spectrum Φ 3 between the signals in Figures b and c.

6 6 Vector AR Similarly the backward estimate of de-noised data is given by M ĝk b =  jg k+j, k =,..., N M, () j= where  j represents the complex conjugate of the VAR operator. The final estimate of the de-noised data is given by averaging forward and backward estimators ĝk t = ĝf k + ĝb k. () In seismic data, the noise attenuation is performed in the f-x domain for the spatial samples of each individual frequency. Clearly, the proposed method extends Canales f-x random noise attenuation technique (Canales, 984) to the multivariate case. Notice that Canales noise attenuation method, also called f-x deconvolution, uses the scalar version of equations and 4 for the estimation of a scalar prediction filter. It also assumes the scalar version of equations 9- for noise attenuation. In addition, the complex conjugate constraint between the forward and backward prediction filter in Canales s methods imposes a constraint on the lateral gain variations that may exist in the data. Such f-x filters are unable to predict lateral changes in amplitudes. One may have to design non-casual f-x prediction filters for such cases (Gulunay, ). EXAMPLES D synthetic example To examine the performance of VAR de-noising we started with the 3-component signal. We added random noise to the original data in Figures a-c to produce the noisy data in Figures 5a-c. The signal to noise ratio (SNR) for all 3 components is equal to.75. Figures 6a-c show the de-noised data using the VAR operator. The very high amplitude random noise has been successfully eliminated from data in Figures 5a-c. Synthetic seismic example We test the VAR de-noising method for a synthetic multicomponent seismic record composed of the three linear events shown in Figures 7a-c. Notice that the linear events in each seismic component have different amplitudes and polarities but they have the same dip structure. Figures 7d-f show the f-k spectra of the data in Figures 7a-c, respectively. Each seismic section in Figure 7 is contaminated with random noise of SNR =. to obtain the noisy seismic sections in Figure 8. Figures 8d-f show the f-k spectra of the noisy seismic sections in Figures 8a-c, respectively. Figures 9a-c show the results of VAR de-noising of the seismic sections in Figures 8a-c. We used 3-component VAR operators of order M = 4 to de-noise the data. Figures 9d-f show the f-k spectra of the VAR de-noised seismic sections in Figures 9a-c, respectively. Figures a-c show the results of Canales f-x de-noising method applied individually to the seismic sections in Figures 8a-c. Figures d-f show the f-k spectra of the seismic sections in Figures a-c, respectively. A detailed comparison

7 7 Vector AR Sample number Figure 5: - are the noisy data created from data in Figures a-c by adding noise with SNR =.75.

8 8 Vector AR Sample number Figure 6: - are the de-noised data from Figures 5a-c using VAR modeling, respectively.

9 9 Vector AR d) e) f) Figure 7: - are the three components of a synthetic linear seismic data. d)-f) are the f-k spectra of a-c, respectively. of Figures 9 and reveals that VAR de-noising method performs better for de-noising multicomponent data since it simultaneously uses information from all of the components. Application of the VAR de-noising method to seismic sections with curved events requires spatial windowing in order to satisfy the linear seismic events assumption (Naghizadeh and Sacchi, 9). The presence of linear events in the t-x domain implies having a few dominant harmonics at a given frequency in the f-x domain. Therefore, VAR modeling can be used to identify dominant harmonics present in the multicomponent signal. Figures a-c show a simulated 3-component seismic record with one vertical and two horizontal components. The data are contaminated by random noise with SN R =.. For illustration purposes we plotted only every 4 traces of the original data in each section. Figures d-f depict the f-k spectra of the data in Figures a-c, respectively. Figures a-c show the de-noised data using VAR modeling. Figures d-f show the f-k spectra of the data in Figures a-c, respectively.

10 Vector AR d) e) f) Figure 8: - are the noisy data created form the data in Figures 7a-c with SNR =.. d)-f) are the f-k spectra of a-c, respectively.

11 Vector AR d) e) f) Figure 9: - are the de-noised data using VAR modeling for the data in Figures 8a-c. d)-f) are the f-k spectra of a-c, respectively.

12 Vector AR d) e) f) Figure : - are the de-noised data using Canales f-x denoising method for the data in Figures 8a-c. d)-f) are the f-k spectra of a-c, respectively.

13 3 Vector AR d) e) f) Figure : - are the three components of simulated multicomponent seismic data with SN R =.. d)-f) are the f-k spectra of a-c, respectively.

14 4 Vector AR d) e) f) Figure : - are the de-noised data using VAR modeling for the data in Figures a-c. d)-f) are the f-k spectra of a-c, respectively.

15 5 Vector AR Real data example In order to examine the performance of VAR de-noising on real seismic data we use a shot record from an ocean bottom cable (OBC) survey. The data are composed of two horizontal and one vertical component records. Figures 3a-c show the vertical and the two horizontal components of the OBC shot record, respectively. The vertical component of data (Figure 3 contains a noticeable amount of random noise. Figures 3d-f show the result of VAR de-noising method applied to the data in Figures 3a-c, respectively. The order of VAR operator used to de-noise the data was M = 3. The results show that random noise was eliminated from the vertical component (Figure 3d). To further analyze the performance of VAR de-noising, we de-noised the vertical component of the data (Figure 3 using Canales f-x de-noising method. We used the same spatial window size and operator order for both Canales and VAR denoising. Figures 4a-4c show the results of VAR de-noising and Canales f-x de-noising for the vertical component and their difference, respectively. While both methods has been successful in eliminating the random noise, the VAR de-noising method has preserved the signal better. To examine the effectiveness of VAR de-noising in the presence of noise in all three components we added random noise to the original field data to obtain the noise-contaminated data in Figure 5a-c. Figures 5d-f show the results of VAR de-noising applied to the data in Figures5a-c, respectively. Here one can clearly see that the random noise has been effectively removed from all three components of the data. DISCUSSION In this article we used VAR modeling for analysis of multicomponent signals. Special emphasis is placed on the noise elimination of multicomponent noisy data. The VAR modeling preforms better in comparison to ordinary AR modeling, the basis of f-x random noise reduction, when there are common harmonics present in multicomponent data. If the squared coherency term between all data components is zero over all harmonic range, then the VAR de-noising does not provide extra benefits. In addition, our experiments show that for low SNR, higher orders of VAR models improve the quality of de-noising process. We use the conjugate gradient (CG) algorithm to estimate the VAR operators. This way we can control the trade-off between misfit and model norm by adjusting the number of iterations for the CG algorithm. This is important because for noisy data over-fitting the data may result in ineffective VAR parameter estimation for de-noising purposes. Our experiments show that an optimal estimate of VAR de-noising operators can be achieved with a small number of CG iterations ( ). VAR modeling can be extended effectively to the multidimensional case. Possible applications are de-noising and interpolation of multicomponent seismic records with multiple spatial dimensions. Another interesting application for VAR modeling is the band-width extension of seismic records. VAR operators can be used to extract information from wideband components in order to extend the band-width of narrow-band components.

16 6 Vector AR d) e) f) g) h) i) Figure 3: Vertical component of OBC data. and are the horizontal components of OBC survey. d)-f) show the data in a-c after applying the VAR de-noising method, respectively. g)-i) show the diffence sections between a-d, b-e, and c-f, respectively.

17 7 Vector AR Figure 4: The vertical component of OBC data de-noised using VAR modeling. The vertical component of OBC data de-noised using Canales f-x method. The difference between a and b. CONCLUSIONS We introduced and investigated the performance of VAR modeling for seismic data denoising purposes by extending f-x random noise attenuation to the multivariate case. We used the least-squares method to estimate the optimal VAR operators and also described the spectral interpretation of the VAR model. The performance of VAR modeling was analyzed for de-noising of D multicomponent noisy data. We showed that VAR de-noising algorithm improves the noise elimination if common harmonics are present in different components of the signal. The synthetic and real seismic data examples showed the effectiveness of the proposed VAR de-noising algorithm. ACKNOWLEDGMENTS We acknowledge financial support by the sponsors of the Signal Analysis and Imaging Group at the University of Alberta. We also thank Dr. Keith Louden and Mr. Omid Aghaei from Dalhousie University for kindly providing OBC gathers utilized for our real data tests. REFERENCES Cadzow, J. A. and K. Ogino, 98, Two-dimensional spectral estimation: IEEE Transactions on Acoustics, Speech, and Signal processing, 9, Canales, L. L., 984, Random noise reduction: 54th Annual International Meeting, SEG, Expanded Abstracts, Session:S.. Gulunay, N.,, Noncausal spatial prediction filtering for random noise reduction on 3-d poststack data: Geophysics, 65, Hrafnkelsson, B. and H. J. Newton,, Asymptotic simultaneous confidence bands for vector autoregressive spectra: Biometrika, 87, 73 8.

18 8 Vector AR d) e) f) Figure 5: - are the vertical and two horizontal components, respectively, of an OBC survey after adding random noise with SNR =. d)-f) show a-c after applying VAR de-noising.

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