Maximum Kurtosis Phase Correction

Abstract

Seismic reflectivity sequences have amplitude distributions that are leptokurtic, that is, they are more heavy-tailed than a Gaussian distribution. This property forms the basis of a method of phase adjustment for seismic reflection data, called maximum kurtosis phase correction (MKPC), that maximizes the ‘stand-out’ of events on seismic sections. the method is normally applied after standard deconvolution and stacking and, unlike methods of minimum entropy deconvolution (MED) which attempt to employ the same property of seismic reflectivities, it is a thoroughly predictable and testable process. the requirements for estimating stable phase corrections can be stated explicitly and, moreover, these show exactly why MED has never been a practical success. The method simply determines, in an efficient way, the frequency-independent phase shift that maximizes the kurtosis of the seismic data. In so doing it realizes the true potential advantage of MED-type methods, the ability to tune the phase, without attempting to control signal whitening and noise amplification, where MED methods offer no advantage over the near-optimum balance already achievable from the usual well-tried deconvolution tests. A constant phase shift is sufficient correction for most post-stack processing and, although stable estimates sometimes demand surprisingly large amounts of data, the selection of the phase correction that maximizes kurtosis is very straightforward and easily tested. the paper gives the theory of the method and describes its data requirements by means of expressions that relate the accuracy of the phase estimates to the amount, bandwidth, and kurtosis of the seismic data. Equations are also given to show how noise and filtering affect the kurtosis of a seismic trace. By analysing the operation of a band-limited phase-shifter, it is shown that the bandwidth of the data must exceed the centre frequency for a successful determination of a phase shift from maximum kurtosis, and the magnitudes of the variations in kurtosis with phase angle for bandwidths above this threshold are derived. the phase correction itself can be determined without applying a whole series of phase shifts to the data. the performance of the method on seismic data is fully consistent with the theory presented here and has been reported in separate studies (Levy & Oldenburg 1987; Longbottom, Walden & White 1988; Hosken et al. 1987).