Systematics of time‐migration errors

Even if the correct velocity is used, time migration mispositions events whenever the velocity changes laterally. These errors increase with lateral velocity variation, depth of burial, and dip angle θ. Our analyses of two model types, one with an implicit gradient and one with an explicit gradient, yield simple “rules of thumb” for these errors to first order in the lateral gradient. The x error is A(1+3tan2θ), and the z error is -2Atan3, where the quantity A = A(x, z) contains the information about depth of burial and magnitude of lateral gradient. These rules can be used to determine when depth migration is needed. Further analysis also shows that the image‐ray correction to time migration is accurate only at small dip. For dipping events, the image‐ray correction must be supplemented by a shift in x of the form 3Atan2θ and a shift in z given by -2Atan3θ. These time‐migration corrections take the same form for both the models we have studied, suggesting a general scheme for correcting time migration, w...