Abstract

Using ray theory and assuming zero offset between source and detector, we have derived an expression for reverberation sequences in two-dimensional models involving a water layer with a dipping bottom. The operators exhibit novel departures from those for the familiar case of a horizontal water bottom. The number of water-layer multiples which contribute to the reverberation operator is limited by the geometry and decreases rapidly with a small increase in dip. Moreover, for a given dip, the time scale of the reverberation operator is compressed linearly as source-receiver position is moved in the up-dip direction. The result is the familiar phenomenon of increasing 'dip' of later multiples on seismic time sections. The introduction of dip causes a splitting of the arrival spikes in the zero-dip operator. Splitting is a simple function of spike number but results in a complicated interfingering of spikes associated with multiples of different orders. The spikes derived from a particular multiple fan out over a time interval which increases with dip and multiple order.Whether the predicted effects cause significant changes in the structure of the zero-dip reverberation operator depends principally on the dip and the impedance contrast. For small dip, significant differences are confined to the later multiples. These multiples have non-negligible amplitudes only for high-impedance contrast situations. As the dip increases, splitting and interfingering occur even in the earlier multiples. Consequently, departures from the zero-dip operator are expected to be significant in areas characterized by small dip and quite persistent reverberation and in areas where the dip is large. Under these conditions, the reverberation operator is not minimum-delay, and standard dereverberation techniques which assume this property are not optimal for resolution improvement. In high-impedance contrast areas, the departure from the zero-dip structure is particularly pronounced for reflections which propagate in the down-dip direction at large angles with respect to the vertical. Narrow-band filtered reverberation operators calculated for high impedance contrasts exhibit irregular beating patterns, late reflection times, phantom reflections, and time-varying frequency content.

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