An investigation of the spectral properties of primary reflection coefficients

This paper investigates the form of the nonwhiteness found in the reflection coefficients from a wide variety of rock sequences around the world. In all but one case densities are taken as constant due to the paucity of suitable density data. The reflection sequences are pseudo-white only above a corner frequency, below which their power spectrum falls away according to a power law fβ, where β is between 0.5 and 1.5. This spectrum can be adequately modelled in practice very simply with an ARMA (1, 1) process which acts on a white innovation sequence. The corollary of this is that before wavelet estimation methods are applied (all of which-except those based on synthetic seismograms—presuppose white reflection sequences) or deconvolution filters estimated, seismic traces should be filtered with the inverse of this process. Interestingly, the estimated ARMA processes group themselves into two clearly differentiated categories, having very different indices of predictability (or, strictly, indices of linear determinism). The two categories apparently correlate precisely with two kinds of sedimentation: one which consists largely of sequences of rocks with repeating properties, called “repetitive” in this paper but perhaps loosely describable as “cyclic”, and the other which is randomly bedded with no apparent pattern of components. The former has indices of predictability which are two to four times as great as those of the latter. Another, probably related, property is that β for the repetitive sequences tends to be greater than that for non-repetitive rock columns. The observed power spectra are shown to be consistent with a simple model for the logarithm of acoustic impedance consisting of a mixture of processes where the distribution of (time) scale parameters is reciprocal. Detailed effects of block-averaging and sampling the logs are shown to depend on the type of sequence under examination.