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1 October 1998 Comparison of propagator decomposition in seismic imaging by wavelets, wavelet packets, and local harmonics
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Kirchhoff migration operator is a highly oscillatory integral operator. In our primary work (Wu and Yang, 1997), it has been shown that the matrix representation of Kirchhoff migration operator for homogeneous background in space-frequency domain is a dense matrix, while the compressed operator in beamlet-frequency domain, which is the wavelet decomposition of the Kirchhoff migration operator, is a highly sparse matrix. Using the compressed matrix for imaging (beamlet migration), we can retain the wide effective aperture of a full-aperture operator, and hence achieve higher resolution and image quality with reduced computational cost. However, as well known, wavelets work best for zero-frequency stationary signals. But, for the Kirchhoff migration operator, it is both space-varying in the near-field region and high-frequency stationary in the far-field zone. Therefore, wavelets are not very efficient for this kind of operator. In this research, we first summarize the results of maximum sparsity adapted wavelet-packet transform (MSAWPT) for the decomposition and compression of Kirchhoff migrator (Wang and Wu, 1998), and then further study the decomposition and compression of Kirchhoff migration operator by local harmonics (i.e., local cosines/sines). It was found in (Wang and Wu, 1998) that the MSAWPT can generate a more efficient representation for the imaging operator than the standard discrete wavelet transform (DWT) and the compression capability of MSAWPT is much greater than that of DWT. In this paper, we also observed that for low frequency operator, the compression capability of uniform local cosine bases is equivalent to that of standard wavelets and is weaker than that of adapted wavelet-packets; while, for high frequency operator, uniform local cosine bases are more powerful than both the standard wavelets and adapted wavelet-packets. Furthermore, for local cosine transform, a good parameter setting (i.e., type and smoothness of the bell function, edge extension, overlapping radius, folding style, and window size) can generate a higher compression ratio.
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Ru-Shan Wu and Yongzhong Wang "Comparison of propagator decomposition in seismic imaging by wavelets, wavelet packets, and local harmonics", Proc. SPIE 3453, Mathematical Methods in Geophysical Imaging V, (1 October 1998);

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