NORMAL‐MOVEOUT AND VELOCITY RELATIONS FOR FLAT AND DIPPING BEDS AND FOR LONG OFFSETS

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Abstract
Accurate relations between NMO and velocity are needed in modern exploration seismology, especially in long‐offset CDP work, where accurate NMO corrections must be made for stacking, and where several types of velocity averages may be computed with accuracy from NMO data. The velocity average associated with NMO is the time‐rms velocity [Formula: see text]. Even for long offsets the straight‐ray computation using [Formula: see text] is usually adequate, but a closer approximation for horizontal reflectors is obtained by reducing the NMO calculated from [Formula: see text] or reducing the value of [Formula: see text] calculated from NMO by the factor [Formula: see text], where σ is the rms deviation of the velocity from its mean, T is zero‐offset traveltime, and ΔT the NMO. The difference between time‐average and time‐rms velocities is often several percent. For the velocity function [Formula: see text] and for reflectors of arbitrary dip and strike, the NMO is shown to be [Formula: see text] where X is offset, α is emergence angle, and ψ is the angle between the offset direction and the reflector dip direction. The terms that contain angles can be used as a correction ΔΔT to the NMO value computed as if the seismic energy were reflected from a horizontal reflector, even for offset greater than those for which an NMO expression quadratic in offset is accurate. A further approximation gives [Formula: see text], where δT is dip moveout over a spread of length L, and [Formula: see text] is the angle between the receiver line and the dip direction, differing from ψ only if there is substantial perpendicular offset of the source point. An expression for the degradation of the stacked signals in CDP stacks because of NMO errors is given. It is shown that the criterion that the signal‐to‐random‐noise ratio could not be improved by dropping the longest‐offset trace(s) requires that the NMO error be not much larger than one‐quarter of a dominant period.

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