Improved PSO Algorithm Wavelet Estimation Via Fourth-order Cumulant Matching

Seismic wavelet extraction is one of the important long-standing research works in seismic data processing. Nowadays, there are two kinds of methods to estimate seismic wavelet. The first have the limit that the reflectivity series must be known. The second has the hypothesis that the reflectivity series is in some certain form. In this paper the second is adopted under the condition that the cumulant of the reflectivity series is a multidimensional spike at zero lag. Matching between the cumulant and the wavelet moment is done in a minimum mean-squared error sense under the assumption of a non-Gaussian, stationary, and statistically independent reflectivity series. This leads to a highly nonlinear optimization problem. An improved particle swarm optimization is applied to solve the optimizationproblem. The new method proposes a linear time-varying acceleration co-efficient and brings in two mutations including differential mutation and random mutation. Also, some betterment is made over the bound constraints which keep the escaped particles diversity. The results show that the improved algorithm is better in finding the global optima not only in velocity but also in precision.During the extraction of the wavelet, fourth-order cumulant is adopted. The second-order cumulant is the covariance. It is a phaseless function. The third-order cumulant preserves phase information, but when the process is symmetrically distributed (such as most reflectivity series), it vanishes for all lags. This is not true for the fourth-order cumulant function, which preserves phase information as well.In order to improve the precision of the extracted wavelet, first the length of the wavelet is computed. Then a multidimensional window is used to smooth the trace cumulant. At last, an initial wavelet is obtained via cumulant matching matrix equation. It is regarded as the initial population of the improved particle swarm algorithm.In the end, some experiments are done to extract the minimum-phase wavelet, mixed-phase wavelet and zero-phase wavelet. Also the trace is contaminated with Gaussian noise. The results show that the method proposed to extract wavelet is possible and effective. The precision of the extracted wavelet is better improved.