Phase estimation in bispectral domain based on conformal mapping and applications in seismic wavelet estimation
Yu Yong-Cai1, Wang Shang-Xu1, Yuan San-Yi1, and Qi Peng-Fei2
1. State Key Laboratory of Petroleum Resource and Prospecting, CNPC Key Laboratory of Geophysical Exploration, China University of Petroleum, Beijing 102249, China.
2. China University of Geosciences (Beijing), Beijing 100083, China.
Abstract Seismic wavelet estimation is an important part of seismic data processing and interpretation, whose preciseness is directly related to the results of deconvolution and inversion. Wavelet estimation based on higher-order spectra is an important new method.However, the higher-order spectra often have phase wrapping problems, which lead to wavelet phase spectrum deviations and thereby affect mixed-phase wavelet estimation. To solve this problem, we propose a new phase spectral method based on conformal mapping in the bispectral domain. The method avoids the phase wrapping problems by narrowing the scope of the Fourier phase spectrum to eliminate the bispectral phase wrapping infl uence in the original phase spectral estimation. The method constitutes least-squares wavelet phase spectrum estimation based on conformal mapping which is applied to mixed-phase wavelet estimation with the least-squares wavelet amplitude spectrum estimation. Theoretical model and actual seismic data verify the validity of this method. We also extend the idea of conformal mapping in the bispectral wavelet phase spectrum estimation to trispectral wavelet phase spectrum estimation.
This research was fi nancially supported by National 973 Program (No. 2007CB209600).
About author: Yu Yong-Cai received his Bachelor’s degree of Information and Calculation Science at China University of Petroleum (Beijing) in 2007. He is now a PhD candidate at China University of Petroleum (Beijing). His research interests mainly include seismic wavelet estimation, seismic data processing, and inversion.
Cite this article:
YU Yong-Cai,WANG Shang-Xu,YUAN San-Yi et al. Phase estimation in bispectral domain based on conformal mapping and applications in seismic wavelet estimation[J]. APPLIED GEOPHYSICS, 2011, 8(1): 36-47.
[1]
Bartelt, H., Lohmann, A. W., and Wirnitzer, B., 1984, Phase and amplitude recovery from bispectra: Applied Optics, 23(18), 3121 - 3129.
[2]
Brillinger, D. R., 1977, The identification of a particular nonlinear time series system: Biometrika, 64(3), 509 - 515.
[3]
程乾生，2003, 数字信号处理：北京大学出版社, 172 - 175.
[4]
Edgar, J., and Van der Baan, M., 2009, How reliable is statistical wavelet estimation? SEG Houston 2009 International Exposition and Annual Meeting, 3233 - 3237.
Kang, M. G., Lay, K. T., and Katsaggelos, A. K., 1991, Phase estimation using the bispectrum and its application to image restoration: Optical Engineering, 30(7), 976 - 985.
[7]
Lazear, G. D., 1993, Mixed-phase wavelet estimation using fourth-order cumulants: Geophysics, 58(7), 1042 - 1051.
[8]
梁昆淼，刘法，缪国庆，1995, 数学物理方法：高等教育出版社, 423 - 458.
[9]
Lii, K. S., and Rosenblatt, M., 1982, Deconvolution and estimation of transfer function phase and coefficients for non-Gaussian linear processes: The Annals Statistics, 10(4), 1195 - 1208.
[10]
Lu, W. K., 2005, Blind channel estimation using zero-lag slice of third-order moment: IEEE Signal Processing Letters, 12(10), 725 - 727
[11]
Lu, W. K., Zhang, Y. S., Zhang, S. W., and Xiao, H. Q., 2007, Blind wavelet estimation using a zero-lag slice of the fourth-order statistics: Journal of Geophysics and Engineering, 4(1),24 - 30
[12]
Marron, J. C., Sanchez, P. P., and Sullivan, R. C., 1990, Unwrapping algorithm for least-squares phase recovery from the modulo 2 bispectrum phase: Journal of the Optical Society of America A: Optics, Image Science, and Vision, 7(1), 14 - 20.
[13]
Matsuoka, T., and Ulrych, T. J., 1984, Phase estimation using the bispectrum: Proceedings of the IEEE, 72(10), 1403 - 1411.
[14]
Mendel, J. M., 1991, Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications: Proceedings of the IEEE, 79(3), 278 - 305.
[15]
Nikias, C. L., and Petropulu, A. P., 1993, Higher-order spectra analysis: A nonlinear signal processing framework: Prentice Hall.
[16]
Pan, R., and Nikias, C. L., 1987, Phase reconstruction in the trispectrum domain : IEEE Transactions on Acoustics, Speech, and Signal Processing, 35(6), 895 - 897.
[17]
Petropulu, A. P., and Pozidis, H., 1998, Phase reconstruction from bispectrum slices: IEEE Transactions on Signal Processing, 46(2), 527 - 530.
[18]
Robinson, E. A., 1967, Predictive decomposition of time series with application to seismic exploration: Geophisics, 32(3), 418 - 484.
[19]
Sacchi, M. D., and Ulrych, T. J., 2000, Nonminimum-phase wavelet estimation using higher order statistics: The Leading Edge, 19(1), 80 - 83.
[20]
Sundaramoorthy, G., Raghuveer, M. R., and Dianat, S. A., 1990, Bispectral reconstruction of signals in noise: amplitude reconstruction issues: IEEE Transactions on Acoustics. Speech. and Signal Processing, 38(7), 1297 - 1306.
[21]
Tekalp, A. M., and Erdem, A. T., 1989, Higher order spectrum factorization in one and two dimensions with applications in signal modeling and nonminimum phase system identification: IEEE Transactions on Acoustics. Speech. and Signal Processing, 37(10), 1537 - 1549.
[22]
西安交通大学高等数学系, 1996, 复变函数（第四版）：高等教育出版社, 186 - 191.
[23]
Yuan, S. Y., Wang, S. X., and Tian, N., 2009, Swarm intelligence optimization and its application in geophysical data inversion: Applied Geophysics, 6(2), 166 - 174.
[24]
Zhang, F., Wang, Y. H., and Li, X. Y., 2008, Mixed-phase wavelet estimation using unwrapped phase of bispectrum: 70th EAGE Conference and Exhibition.
[25]
Zhang, F., Wang, Y. H., and Li, X. Y., 2009, Mixed-phase wavelet estimation using unwrapped phase of bispectrum: CPS/SEG Beijing International Geophysical Conference and Exposition.
2011, Vol. 8 
