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Improv ements on seismic data compression and migration using compressed data with the

exible segmentation sc hemefor local cosine transform

Y ongzhong Wang and Ru-Shan Wu Institute of Tectonics, University of California, Santa Cruz, CA 95064

Summary Best-basis searc hing algorithm based on binary (in general, M-ary) segmentation was constructed by Coifman and Wickerhauser (1992) and widely used for signal processing. How ever, there are several problems with the binary scheme. First, the binary segmentation is in exible in grouping signals along the axis. Secondly, the binarybased segmentation method is very sensitiv e to time/space shifts of the original signal, such that the resulted best-basis will change a great deal if the signal is shifted by some samples. Thirdly, the reconstruction distortion after compression is relatively strong. Wu and Wang (1999) have designed a new exible segmentation algorithm with arbitrary time/space segmentation which addresses the above-mentioned problems caused by the binary segmentation scheme. In that paper, the adv an tages of the new exible segmen tation technique over the binary scheme are demonstrated by sho wing the removal of the constraint of dyadic segmentation, reduction of time/space-shift sensitivity and reconstruction distortions, and superior performance in seismic data compression. We apply our exible segmentation scheme to real 2-D seismic data compression, and study the eects of data compression by the new method on imaging. F rom the comparison with the con ventional binary scheme, we see that the decompressed data has less distortion and the migrated image using the decompressed data has better quality.

Introduction Orthogonal transforms pla y a key role in data compression schemes, suc h as discrete cosine transform, discrete wavelet transform and adapted w avelet-packet transform. Since seismic signals are always nonstationary in time and/or space, compression schemes with exible segmentations in time/space to match the characteristics of nonstationary signals are highly desirable. Much

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progress has been achiev edin signal processing and seismic data compression in the applications of Adapted Local Cosine Transform (ALCT) (Coifman and Meyer, 1991 Coifman and Wickerhauser, 1992 Pascal, Guido and Wickerhauser, 1992 Wesfreid and Wickerhauser, 1993 Wickerhauser, 1994 Fang and Sere, 1994 Bernardini and V etterli, 1998 Wang and Wu, 1999). How ever, the existing methods in the literature are all based on dy adic decomposition trees in selecting bestbasis. There are sev eral problems for the binary scheme. First, the binary time/space segmentation has no exibility at all. F or example, if the peak of a pulse just rides on the midpoint of the whole signal, the binary segmentation scheme will result in either separating the pulse into tw o halves from the peak, or keeping the whole signal as only one segment. Undoubtedly, neither segmenting is satisfactory. Secondly, the binary-based segmentation method is very sensitive to time/space shifts of the original signal, such that the resulted best-basis will change a great deal if the signal is shifted by some samples. Due to these reasons, the reconstruction distortion after compression is relatively strong. In this extended abstract, w e further apply the new exible segmentation algorithm (Wu and Wang, 1999) which addresses the abo ve-mentioned problems caused by the binary segmentation scheme to the real 2-D seismic data compression, then study the eects of data compression by the new method on migration (imaging). F rom our numerical tests, the advan tages of this new exible segmentation technique over the binary searching scheme can be easily seen.

Flexible segmentation scheme Here, we assume that readers are familiar with the adapted binary local cosine basis (if not, please see Wang and Wu, 1999). To overcome the binary segmentation constraint, w edesigned a exible segmentation algorithm (Wu and Wang, 1999). It can be described schematically as follows.

Flexible segmentation scheme Let stand for the time/space segmentation resolution, i.e., the length of the nest segment (cell) we want to have, for a 1-D signal with length , w e always assume that is a multiple of , say, = , where is the total number of nest segments. In our segmentation algorithm, w echoose the Shannon Entropy as the cost-functional. Starting with the uniform nest segmentation, w eadopt a left-to-right merging process to optimize the segmentation which doesn't suer from the binary tree restriction. F or eac h possible merge, we compare the cost of the merged entity (segment) with the total cost, i.e., the sum of costs of the t w o separate entities (segments). If the cost of the merged segment is smaller than the total cost of tw oseparate segments, the merge is approved, and the merged segment will be treated as one entity in the next possible merge. Otherwise, if the opposite is true, the merge will be abandoned. Since the merge is only possible between neighboring segments, the segment on the left after disapproval of a merging will be dropped from the list of merging candidates. The new merging process will start from the segment on the right and will never look bac k to the left. In other w ords, we will put a termination node at the right endpoint of the left segment, signaling the termination of an old merging process and the beginning of a new one. A tthe end of the process, w eobtain an optimized segmentation by retrieving all the termination nodes. Note that, in this segmentation, the minimum length of segments is the preset resolution, i.e., the cell length while the maximum length of segments has no limit, and can be the whole length of signal in the extreme case. The binary segmentation process starts from the longest segment, i.e., the whole signal, and proceeds by dividing the segment in the middle, while our segmentation starts from the nest segments (cells) and achiev es the exible segmentation by a neighbor merging process. Many drawbacks ofthe binary segmentation process ha vebeen avoided by the new method. For the 2-D case, based on the BINARY 2-D semi-ALCT scheme proposed by Wang and Wu (1999), w e can parallelly construct the FLEXIBLE 2-D semi-ALCT algorithm. It includes tw o semi-adaptive schemes, i.e., FLEXIBLE 2D time-ALCT and FLEXIBLE 2-D space-ALCT. L

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F or the FLEXIBLE 2-D time-ALCT, it is uniform in space direction, and adaptive in time direction within eac h xed strip along the time direction. Its adaptability is accomplished by the abo ve-mentioned 1-D new exible segmentation algorithm. F or the FLEXIBLE 2-D space-ALCT, the segmentation is uniform in time direction, but adaptive in space direction.

Application to seismic data compression and migration on decompressed data In this section, compared with the traditional BINARY segmentation algorithm, w e will demonstrate the superior performance of the new FLEXIBLE segmentation scheme in seismic data compression and in the eects of data compression on imaging. In our past work (Wang and Wu, 1999), we have proposed tw o 2-D compression schemes based on the BINARY tree algorithm, i.e., 2-D time-ALCT and 2-D space-ALCT, and applied them to the compression of a subset of the SEG-EAEG salt data set, a synthetic zero-oset data from the salt model using a nite-dierence exploding re ector modeling algorithm, generated at AMOCO. We concluded that (32 32) or (32 64) minimum (time, space) window size can generate the best compression result for the data set. In this extended abstract, using the new FLEXIBLE segmentation scheme, w ealso tested the tw o2-D compression schemes, i.e., FLEXIBLE 2-D time-ALCT and FLEXIBLE 2-D space-ALCT (see Section II). Fig. 1 is the compression performance comparison betw een BINARY 2-D space-ALCT and FLEXIBLE 2-D space-ALCT for the salt data. We can see that (30 30) (time, space) segmentation resolution for the new FLEXIBLE algorithm can even provide better compression performance than the (32 32) BINARY 2-D space-ALCT, which is already the best result in binary schemes for the data set. Here, the SNR is dened as follows,











SN R

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= 10 log10 (

2

k

jck j =

k

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j"k j

)

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where k is the coecient above the threshold in the absolute value sense and thus being retained, and k is the coecient discarded. For more detailed comparison, Fig.2a shows the original seismic data from SEG-EAEG salt model, Fig.2b gives the reconstructed data from c

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Flexible segmentation scheme Performance comparison between 2D binary space−ALCT and 2D flexible space−ALCT 35

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Figure 1: Compression performance comparison between BINARY 2-D spac e-ALCT and FLEXIBLE 2-D space-ALCT. Obviously, better compression performance is achieved by the new FLEXIBLE segmentation method. the compressed ALCT coecients with Compression Ratio (CR) 67:1 by the BINARY 2-D space-ALCT scheme, while Fig.3c is the reconstructed data from the compressed ALCT coecien ts with CR=69:1 by the corresponding FLEXIBLE scheme. As can be seen, the FLEXIBLE scheme gives better reconstructed data even in the case of higher CR than the BINARY algorithm. Fig.3 is the comparison of migrated images using decompressed data from dierent schemes: Fig.3a is a reference image which is obtained by hybrid pseudo-screen migration (Jin, Wu and Peng, 1998) on the original data (Fig.2a), Fig.3b is the image by the same migration method on the reconstructed data with CR=67:1 (Fig.2b), and Fig.3c is the image by the same migration method on the reconstructed data with CR=69:1 (Fig.2c). F rom Fig.3, we see that a better image including subsalt structures can be obtained ev en from the higher compression ratio data by the new FLEXIBLE scheme.

Conclusions In this extended abstract, w e applied the new FLEXIBLE segmentation scheme (Wu and Wang, 1999) to the SEG-EAEG salt data compression and investigated the eects of data compression on imaging, compared with the corresponding BINARY algorithm. These numerical results demonstrate the superior performance of the new FLEXIBLE scheme for seismic data compression and imaging.

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Acknowledgments This research is supported by the WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) project at the University of California, Santa Cruz. We are grateful to the sponsors. The facilities have been supported in part by the W. M. Keck Foundation.

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References Coifman, R. R. and Meyer, Y., 1991, Remarques sur l'analyse de F ourierafen^etre, serie I, C. R. Acad. Sci. P aris 312, 259-261. Coifman, R. R. and Wickerhauser, M. V., 1992, Entropybased algorithms for best basis selection, IEEE Trans. on Information Theory 38, 713-718. P ascal, A., Guido, W. and Wickerhauser, M. V., 1992, Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets, Wavelets: A Tutorial in Theory and Applications (ed. Charles K. Chui), A cademic Press, Inc., 237-256. Wesfreid, E., and Wickerhauser, M. V., 1993, Adapted local trigonometric transforms and speech processing, IEEE T rans. on Signal Processing, 41, 3596-3600. Wickerhauser, M. V., 1994, Adapted wavelet analysis from theory to soft ware, A. K. Peters, Wellesley, Mass, 103152. Fang, X., and Sere, E., 1994, Adapted multiple folding local trigonometric transforms and wavelet packets, Applied and Computational Harmonic Analysis, 1, 169-179. Bernardini, R., and V etterli, M., 1998, Discrete- and continuous- time local cosine bases with multiple overlapping, IEEE Trans. on Signal Processing, 46, 31663180. Jin, S., Wu, R.S. and Peng, C., 1998, Prestack depth migration using a hybrid pseudo-screen propagator, 68th SEG meeting, Exp anded Abstracts, 1819-1822. Wang, Y. and Wu, R.S., 1999, 2-D semi-adapted local cosine/sine transform applied to seismic data compresssion and its eects on migration, 69th SEG me eting, Exp anded Abstracts, 1918-1921. Wang, Y. and Wu, R.S., 1999, Seismic data compression by adaptiv elocal cosine/sine transform and its eects on migration, Ge ophysical Pr ospecting (accepted). Wu, R.S. and Wang, Y., 1999, New exible segmentation technique in seismic data compression using local cosine transform, Wavelet applications in signal and image processing VII, Proc. SPIE, 3813, 784-794.

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tween the BINARY 2-D space-ALCT scheme and the FLEXIBLE 2-D space-ALCT scheme:

(a) Syn theticzero-oset data from the SEG-EAEG salt model (b) Reconstructed data from the compressed ALCT coecients with compression ratio (CR) 67:1 by the BINARY scheme (c) Reconstructed data from the compressed ALCT coecients with CR=69:1 by the FLEXIBLE scheme.

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Hybrid pseudo-screen migration on the original synthetic data (Fig.2a) (b) Hybrid pseudo-screen migration on the reconstructed data with CR=67:1 from BINARY scheme (Fig.2b) (c) Hybrid pseudo-screen migration on the reconstructed data with CR=69:1 from FLEXIBLE scheme (Fig.2c).