Decomposition and Compression of Kirchhoo Migration Operator by Adapted Wavelet Packet Transform

Kirchhoo migration operator is a highly oscillatory integral operator. In our previous work (see \Seismic Imaging in Wavelet Domain", Wu and Yang 1], 1997), we have shown that the matrix representation of Kirchhoo migration operator for homogeneous background in space-frequency domain is a dense matrix, while the compressed beamlet-operator, which is the wavelet decomposition of the Kirchhoo migration operator in beamlet-frequency (space-scale-frequency) domain, is a highly sparse matrix. Using the compressed matrix for imaging, we can obtain high quality images with high eeciency. We found that the compression ratio of the migration operator is very diierent for diierent wavelet basis. In the present work, we study the decomposition and compression of Kirchhoo migration operator by adapted wavelet packet transform, and compare with the standard discrete wavelet transform (DWT). We proposed a new maximum sparsity adapted wavelet packet transform (MSAWPT), which diiers from the well-known Coifman-Wickerhauser's best basis algorithm, to implement the decomposition of Kirchhoo operator to achieve the maximum possible sparsity. From the numerical tests, it is found that the MSAWPT can generate a more eecient matrix representation of Kirchhoo migration operator than DWT and the compression capability of MSAWPT is much greater than that of DWT.