Abstract

A reflecting interface with irregular shape is overlain by a material of constant velocity V T . Multifold reflection data are collected on a plane above the reflector and the reflector is imaged by first stacking then migrating the reflection data. There are three velocity functions encountered in this process: the measured stacking velocity V NMO ; the true overburden velocity V T ; and a profile migration velocity V M , which is required by present point-imaging migration programs. Methods of determining V NMO and, subsequently, V T are well-known. The determination of V M from V T , on the other hand, has not been previously discussed. By considering a line-imaging migration process we find that V M depends not only on the true section velocity but also on certain geometrical factors which relate the profile direction to the structure. The relation between V M and V T is similar to, but should not be confused with, the known relation between V NMO and V T . The correct profile migration velocity is always equal to or greater than the true overburden velocity but may be less than, equal to, or greater than the best stacking velocity. When a profile is taken at an angle of (90-theta ) degrees to the trend of a two-dimensional structure, then the appropriate migration velocity is V T /costheta and is independent of the magnitude of any dips present. If, in addition, the two-dimensional structure plunges along the trend at an angle gamma , then the correct migration velocity is given by V T /(1-sin 2 theta cos 2 gamma ) 1 2/. The time axis of the migrated profile for the plunging two-dimensional case must be rescaled by a factor of [(1-sin 2 theta cos 2 gamma )/cos 2 theta cos 2 gamma ] 1 2/, and structures on the rescaled profile must be projected to the surface along diagonal lines to find their true positions. When three-dimensional data are collected and automatic three-dimensional migration is performed, the geometrical factors are inherently incorporated. In that case, the migration velocity is always equal to the true velocity regardless of whether the structure is two-dimensional, plunging two-dimensional, or three-dimensional. Processed model data support these conclusions.The equations given above are intended for use in conventional migration-after-stack. Recently developed schemes combining migration-before-stack with velocity analysis give V M directly. In that case, the above equations provide a method of determining V T from V M .

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