The relationship of pre-stack apparent velocity filtering to the symmetry of the CMP stack response

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Wearing my processing geophysicist's hat, I, like others, often ask questions to which there appear to be no answers readily available. One such question that had re-occurred to me was 'what is the relationship between pre-stack apparent velocity filtering and the common midpoint (CMP) stack response?' The elements of any answer were roughly -potential reduction in spatial resolution; -reduction in coherent noise; -apparent velocity in common source or geophone space is twice that observed on the CMP stack. During the summer of 1983 Walt Lynn presented some material at Western Geophysical's London office (as part of a week long 'Wave Equation Update School' lecture course given by Jon F. C1aerbout) entitled 'Dip filtering for noise suppression'. Part of this graphical material demonstrated how apparent velocity filtering could be viewed in source-geophone space as removing ciearly definable parts of a pre-stack diffraction. Upon this basis the following text is an attempt to answer my question by doing three things: • Setting down the necessary equations. • Showing how certain symmetry conditions affect the CMP stack. • Computing synthetic examples. The text will consider a point scatterer in the subsurface, how it responds to the CMP method and what happens when pre-stack apparent velocity filtering is included. The most important conclusion will be that certain apparent velocity filter symmetry conditions are necessary if the CMP stack is to respond similarly to positive and negative dips. The analysis will commence by describing the prestack traveltime surface that is associated with the point scatterer, which is sometimes called Cheop's Pyramid. Equations for apparent velocity on the commonly used axes of Cheop's Pyramid will be derived which show how certain areas of the surface are rejected by apparent velocity discrimination. The idea of symmetry will be described and a set of rules will be presented in order to test whether the pre-stack apparent velocity filter will prejudice dip direction on the CMP stack. Some computed examples will be presented along with a brief description of the computational method used. Finally a potential problem with 'marine' three-dimensional (3D) surveys that can be caused by pre-stack apparent velocity filtering will be briefly discussed.