Abstract

J. P. Lindsey, (1988) in a clearly written short piece, opens an old question which concerns the analytic properties of seismic wavelets. This well conceived study concludes that most of the roots of a seismic wavelet as expressed by its z transform representation lie on or are very near the unit circle.The present discussion does not seek to characterize the form of all seismic wavelets, but only many if not most of those which have been processed with deconvolutions or 'inversion' type operators to have reduced length, broadened bandwidth, and some desirable phase property. For such wavelets, despite the diversity by which they are obtained, remarkably simple operations having very few parameters can be extremely effective. As a case in point, constant-phase rotations appear to carry such wavelets to zero-phase symmetric form to a very good approximation.I start with empirical attributes which appear to characterize most processed seismic wavelets. Such wavelets tend to be of 40-100 ms duration with a smooth and unimodal amplitude spectrum of 'peak' or 'central' frequency between 15 and 30 Hz. The amplitude spectrum itself is further largely concentrated at frequencies between 5 and 55 Hz.A z transform root structure having essentially all of its roots only on the unit circle and on the real axis seems able to characterize all of the observed attributes rather well. This structure will be termed the band-limiting root approximation (BLRA) and describes the attributes I seek to explain which are not as readily understood from alternative descriptions of the wavelets. Since the class of wavelets we address is obtained by a variety of means, and because the differences in character are at best subtle according to interpretive criteria, my justification is heuristic.The BLRA wavelet structure can be represented with remarkably few parameters (typically fewer than five). Of these few parameters, two relate to the frequency distribution. Such a formalism should be exceptionally useful for designing seismic techniques which seek to extract interpretive information based on properties of the wavelet.

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