We propose a new method for estimating the seismic wavelet. Suppose a seismic wavelet can be modeled by the formula with three free parameters (scale, frequency and phase). The phase of the wavelet is estimated by constant-phase rotation to the seismic signal, other two parameters are obtained by the higher-order Statistics (HOS) (fourth-order cumulant) matching method. In order to derive the estimator of the higher-order Statistics (HOS), multivariate scale mixture of Gaussians (MSMG) model is applied to formulating the multivariate joint probability density function (PDF) of the seismic signal. By this way, we can represent HOS as a polynomial function of second-order statistics to improve the anti-noise performance and accuracy. In addition, the proposed method can work well for short time series.
The objective of blind deconvolution is to retrieve the reflectivity series without knowledge of the amplitude or phase spectrum of the wavelet. Many deconvolution techniques, under various assumptions about the seismic wavelet and reflectivity series, have been developed with respect to different criteria thus far, and these have been reviewed and described in detail in literature. The brief comments on various seismic deconvolution approaches by Baan [1] and Nsiri et al. [2] and by Larue et al. [3] are worth mentioning. Conclusion on cumulant matching (CM) method drawn from their comments indicates that the reliability of the higher-order Statistics (HOS) matching method depends strongly on amount of data available. Thus, we are interested in the problem of how we can estimate the HOS from the field recordings accurately and efficiently without a large amount of data available. A computationally efficient method is proposed for computing the HOS (fourth-order and sixth-order cumulants) based on joint probability distribution function (PDF) of the output seismic signal which is described as multivariate scale mixture of Gaussians (MSMG) model. Using this PDF model, we can represent HOS as a polynomial function of second-order Statistics (SOS); hence we work just with SOS. Since SOS is phaseless, we apply constant-phase rotations technique to estimating the wavelet phase.
In many real world data sets involving multivariate observations, the data have an empirical distribution which is highly peaked at zero (or the mean vector), and asymptotically falls off more slowly than the Gaussian distribution as the distance from zero increases. We deal these distributions with sparse distributions. Multivariate observations, which are mutually correlated and have higher-order dependencies, have frequently been represented using mixture of Gaussians models. In the above blind deconvolution context, the reflectivity sequence is said a white sparse distribution. As a convolution of a wavelet with the reflectivity series, the clean signal presents sparse and correlated nature, and can also be represented using mixture of Gaussians models. A random vector
with zero mean called Multivariate scale mixture of Gaussians distribution can be described as follows [7].
In this paper, we have proposed a new approach based on MSMG model for estimating the seismic wavelet.
