Non-minimum phase wavelet estimation by non-linear optimization of all-pass operators |
| |
| Authors: | Somanath Misra Mauricio D Sacchi |
| |
| Institution: | Department of Physics, University of Alberta, Edmonton, Canada |
| |
| Abstract: | Convolution of a minimum‐phase wavelet with an all‐pass wavelet provides a means of varying the phase of the minimum‐phase wavelet without affecting its amplitude spectrum. This observation leads to a parametrization of a mixed‐phase wavelet being obtained in terms of a minimum‐phase wavelet and an all‐pass operator. The Wiener–Levinson algorithm allows the minimum‐phase wavelet to be estimated from the data. It is known that the fourth‐order cumulant preserves the phase information of the wavelet, provided that the underlying reflectivity sequence is a non‐Gaussian, independent and identically distributed process. This property is used to estimate the all‐pass operator from the data that have been whitened by the deconvolution of the estimated minimum‐phase wavelet. Wavelet estimation based on a cumulant‐matching technique is dependent on the bandwidth‐to‐central‐frequency ratio of the data. For the cumulants to be sensitive to the phase signatures, it is imperative that the ratio of bandwidth to central frequency is at least greater than one, and preferably close to two. Pre‐whitening of the data with the estimated minimum‐phase wavelet helps to increase the bandwidth, resulting in a more favourable bandwidth‐to‐central‐frequency ratio. The proposed technique makes use of this property to estimate the all‐pass wavelet from the prewhitened data. The paper also compares the results obtained from both prewhitened and non‐whitened data. The results show that the use of prewhitened data leads to a significant improvement in the estimation of the mixed‐phase wavelet when the data are severely band‐limited. The proposed algorithm was further tested on real data, followed by a test involving the introduction of a 90°‐phase‐rotated wavelet and then recovery of the wavelet. The test was successful. |
| |
| Keywords: | |
|
|
