Seismic Attribute Analysis Using Higher Order Statistics

Abstract

Seismic data processing depends on mathematical and statistical tools such as convolution, crosscorrelation and stack that employ second-order statistics (SOS). Seismic signals are non-Gaussian and therefore contain information beyond SOS. One of the modern challenges of seismic data processing is reformulating algorithms e.g. migration, to utilize the extra higher order statistics (HOS) information in seismic data. The migration algorithm has two key components: the moveout correction, which corresponds to the crosscorrelation of the migration operator with the data at zero lag and the stack of the moveout-corrected data. This study reformulated the standard migration algorithm to handle the HOS information by improving the stack component, having assumed that the moveout correction is accurate. The reformulated migration algorithm outputs not only the standard form of stack, but also the variance, skewness and kurtosis of moveout-corrected data. The mean (stack) of the moveout-corrected data in this new concept is equivalent to the migration currently performed in industry. The variance of moveout-corrected data is one of the new outputs obtained from the reformulation. Though it characterizes SOS information, it is not one of the outputs of standard migration. In cases where the seismic amplitude variation with offset (AVO) response is linear, a single algorithm that outputs mean (stack) and variance combines both the standard AVO analysis and migration, thereby significantly improving the cost of seismic data processing. Furthermore, this single algorithm improves the resolution of seismic imaging, since it does not require an explicit knowledge of reflection angles to retrieve AVO information. In the reformulation, HOS information is captured by the skewness and kurtosis of moveout-corrected data. These two outputs characterize nonlinear AVO response and non-Gaussian noise (symmetric and nonsymmetric) that may be contained in the data. Skewness characterizes nonsymmetric, non-Gaussian noise, whereas kurtosis characterizes symmetric, non-Gaussian noise. These outputs also characterize any errors associated with moveout corrections. While classical seismic data processing provides a single output, HOS-related processing outputs three extra parameters i.e. the variance, skewness, and kurtosis. These parameters can better characterize geological formations and improve the accuracy of the seismic data processing performed before the application of the reformulated migration algorithm.