N. Kazemi* (University of Alberta), A. Gholami (University of Tehran) & M. D. Sacchi (University of Alberta)

Transcription

1 3 May 2 June 216 Reed Messe Wien We STZ1 9 Modified Sparse Multichannel Blind Deconvolution N. Kazemi* (University of Alberta), A. Gholami (University of Tehran) & M. D. Sacchi (University of Alberta) SUMMARY Euclid deconvolution is a multichannel algorithm that leads to the estimation of the multichannel seismic reflectivity via the solution of homogeneous system of equations. In the ideal case, the eigenvector associated to the minimum nonzero eigenvalue of the homogenous system of equations is an estimator of the multichannel reflectivity. However, small amounts of noise impinge on the identification of the eigenvector associated to the impulse response. Recently, we proposed a method called SMBD that solves the homogeneous system of equations arising in Euclid deconvolution by imposing sparsity on the unknown multichannel impulse response. The method can accurately estimate the seismic reflectivity and wavelet in the presence of a moderate amount of noise. However, it does not model the noise properly and there is no automatic way for defining the regularization parameter. In this abstract, we tried to improve the SMBD algorithm by including an extra term to handle additive noise. Moreover, in our new algorithm the regularization parameters can be automatically estimated via line search and cross validation procedures. The method is successfully tested on a realistic synthetic example and on marine and land real datasets. 78 th EAGE Conference & Exhibition 216 Vienna, Austria, 3 May 2 June 216

2 3 May 2 June 216 Reed Messe Wien Introduction Euclid deconvolution is a member of a large group of methods that have been proposed for blind deconvolution of seismic data. The method is discussed and used by the seismic data processing as well as by the signal and image processing literature (Rietsch, 1997; Xu et al., 1995; Sroubek and Milanfar, 212; Kazemi and Sacchi, 214). The idea can be summarized as finding the common factors of the z-transform of the source function embedded in a group of seismograms with different reflectivity sequences. The problem leads to the estimation of the multichannel seismic reflectivity via the solution of an homogeneous system of equations. In the ideal case, the eigenvector associated to the minimum nonzero eigenvalue of the homogenous system of equations is an estimator of the multichannel reflectivity. However, small amounts of noise impinge on the identification of the eigenvector associated to the impulse response. Kazemi and Sacchi (214) proposed an improvement to Euclid deconvolution where the homogeneous equation is satisfied by a sparse solution (sparse impulse response). The problem leads to a non-quadratic minimization technique where the solution must be constrained to be on the unit sphere. The method permits to obtain accurate estimates of the seismic reflectivity and wavelet in the presence of a moderate amount of noise. The method has been recently extended to the surface consistent case (Kazemi et al., in press). A major shortcoming of the method is the automatic selection of its regularization parameter and its stability in the presence of noise. In this paper, we improve our original algorithm by including a term to handle additive noise. We also discuss the estimation of regularization parameters and show that they can be automatically estimated via line search and cross validation procedures. Modified Sparse Multichannel Blind Deconvolution The input-output relationship for earth system, assuming a stationary source wavelet, can be written as d j = w r j + n j, j = 1...J (1) where w is source wavelet, stands for convolution and d j, r j and n j are data, reflectivity and noise time series of channel j, respectively. We also remind the readers that convolution can be represented via the Z-transform as follows and by virtue of equation (2), it is easy to show that D j (z) = W(z)R j (z) + N j (z), j = 1,...,J (2) D p (z) R q (z) D q (z) R p (z) = N p (z) R q (z) N q (z) R p (z), The latter can be rewritten in matrix-vector form as 78 th EAGE Conference & Exhibition 216 Vienna, Austria, 3 May 2 June 216 p,q. D p r q D q r p = N p r q N q r p, (3) where D p and N p in equation (3) represent the convolution matrices of channel p of data and noise terms, respectively. The combination of all possible equations leads to the following system of equations where A = D 2 D 1 D 3 D 1 D 4 D 1 Ax = Ex, (4). D 3 D 2 D 4 D D J D J D J 2 D J 1 (5)

3 3 May 2 June 216 Reed Messe Wien (d) True wavelet Figure 1 Performance of the modified SMBD method using synthetic data with SNR=3.5. a) True synthetic reflectivity sequences. b) Seismic traces with SNR=3.5. b) Estimated sparse reflectivity sequences. d) True and estimated wavelets. and x = [r 1,r 2,r 3,...,r J ] T. (6) Matrix E has the same structure as matrix A but constructed from the noise time series only. We will assume that noise term is white and Gaussian, the reflectivity is sparse and the source wavelet is a smooth function. Therefore, we propose to find the signals x and w that minimize the cost function {ˆx,ŵ} = argmin x,w Ax λ x x 1 + λ n Wx d λ w w 2 2 (7) where W is a block diagonal matrix made of the convolutional matrices of the stationary source wavelet. The cost function is non-linear and we will solve it by an alternating minimization technique. By fixing the source wavelet the problem can be solved for the reflectivity using any L 2 L 1 solvers. By fixing the reflectivity, using the updated version of it, the estimation of the source wavelet can be cast as an L 2 L 2 problem which has a closed form solution. We repeat the alternating process till convergence. There are three regularization parameters one needs to choose. The optimum λ n is the one that satisfies Ax(λ n ) 2 2 = Ex(λ n) 2 2 for any non-zero sparse x where the noise term is calculated via N(λ n) = 78 th EAGE Conference & Exhibition 216 Vienna, Austria, 3 May 2 June 216

4 3 May 2 June 216 Reed Messe Wien Figure 2 Performance of the modified SMBD method using Gulf of Mexico dataset. a) Near offset section of data set from the Gulf of Mexico. b) Estimated sparse reflectivity. c) Estimated source wavelet. Wx d. Optimum values for λx and λw can be chosen based on L-curve methods or the generalized cross validation approach. In next section we will show the efficiency of the method on synthetic and two real data examples. Comparing to our previously proposed method, SMBD, the current algorithm is more suitable to process noisy data, hence we will call this technique as modified SMBD. In essence, in this paper, we have incorporated the noise term λn Wx d 22 that serves to control the fitting of noisy data. In the absence of this term, the method in only reliable for blind deconvolution scenarios with a high SNR. Examples In following the examples, we use a delta time series as the initial estimate of source function and the data as an initial estimate of reflectivity series. Choosing the data as an initial solution for reflectivity series accelerates the convergence of our our improved SMBD method. To test the algorithm, we generated a synthetic example with realistic signal to noise ratio (SNR = 3.5). The data and reflectivity series are shown in Figures 1a and b. The estimated reflectivity series is represented in Figure 1c. The estimated source wavelet is compared with the true one in Figure 1d. The algorithm is ran for 9 alternating passes with 1 iterations for the reflectivity estimation part. The results are in good accordance with the true reflectivity and source wavelet time series. The quality of estimations are better than our previously proposed SMBD method. Next, we applied our method to a Gulf of Mexico near offset section. We ran the algorithm for 7 passes with 1 iterations for the reflectivity estimation part. Figure 2 shows the results of Gulf of Mexico dataset. As it is clear from the figure, the estimated reflectivity series looks coherent in the offset direction thank to the application of multichannel algorithm which takes advantage of the statistical properties of the signal of interest. The estimated source function is also interesting. We were able to estimate the main phase and the bubble effect of airgun source function. The deconvolved data show higher resolution that the original data set. We also applied our method on a 2D stack section of the Alaska North slope dataset (line 31-81). The data are already processed, so we expect to see a little bit of improvement in the resolution of the dataset and to estimate a zero phase residual wavelet as the common source function after application of the improved SMBD. Figure 3 shows the results for this 78th EAGE Conference & Exhibition 216 Vienna, Austria, 3 May 2 June 216

5 3 May 2 June 216 Reed Messe Wien Figure 3 Performance of the modified SMBD method using Alaska North slope stack section. a) 2D stack section of Alaska North slope dataset. b) Estimated sparse reflectivity. c) Estimated source wavelet. land data example. The estimated reflectivity and source function series are in good accordance with our expectations. Conclusions We have presented an improved version of the previously proposed SMBD method. The improved SMBD algorithm alleviates some of the problems encountered in SMBD. For instance, the new SMBD models the noise, hence it is more applicable to field data. Moreover, the regularization parameters can be automatically defined. We used alternating minimization technique to solve for both reflectivity and source function time series. Starting with constant time series as an initial guess for source function and data as an initial estimate for reflectivity series always results in satisfactory performance of the technique. The method is successfully tested on a realistic synthetic example and on marine and land real datasets. Acknowledgment We thank the sponsors of the Signal Analysis and Imaging Group (SAIG) at the University of Alberta. We also thank WesternGeco for providing the Gulf of Mexico, Mississippi Canyon dataset and the USGS for the Alaska North slope dataset and the SEG for facilitating access to the data via References Kazemi, N., Bongajum, E. and Sacchi, M. [in press] Surface-Consistent Sparse Multichannel Blind Deconvolution of Seismic Signals. IEEE Trans. Geoscience and Remote Sensing. Kazemi, N. and Sacchi, M. [214] Sparse multichannel blind deconvolution. Geophysics, 79(5), V143 V152. Rietsch, E. [1997] Euclid and the art of wavelet estimation, Part II: Robust algorithm and field-data examples. Geophysics, 62(6), Sroubek, F. and Milanfar, P. [212] Robust Multichannel Blind Deconvolution via Fast Alternating Minimization. IEEE Trans. Image Processing, 21, Xu, G., Liu, H., Tong, L. and Kailath, T. [1995] A least-squares approach to blind channel identification. IEEE Trans. Signal Processing, 43(12), th EAGE Conference & Exhibition 216 Vienna, Austria, 3 May 2 June 216