Velocity analysis by prestack depth migration: linear theory
, by John T. Etgen

Imaging of seismic reflection data is accomplished by prestack depth migration of individual
shot records followed by stacking the results at common surface locations. If the velocity
model used fo rshot record migration does not correctly model the traveltimes of reflection
events, the images of migrated shot profiles at a constant surface location (CSL) will be
distorted from one another. This distortion can be used to estimate the error in the
velocity model. The distortion among the images of the shot profiles can be removed with
a residual moveout correction. In constant velocity medium the change in depth of images
in a CSL gather due to a change in the velocity model can be found by applying a residual
moveout correction in depth. The moveout is a function of the ratio of the imaging
velocities, the original depth of the image, and the source and receiver locations. For
perturbations to a general interval velocity model, the change in migrated depth of the
migrated events in a CSL gather can be calculated through ray tracing. Applying Fermat's
principle linearizes the relation between changes in the velocity model and changes in
traveltime. Linearized geometrical relations describe how to map the change in traveltime
to a change in migrated depth. Linear least squares theory can then be applied to find a
change to the residual moveout curve that best fits the calculated changes in migrated
depth observed in a CSL gather. The results is a linear operator that relates changes in
interval velocity to changes in the parameters describing residual moveout using the
intermediaries of traveltimes and migrated depth. the change of the stacking semblance
of the images of migrated shot profiles due to a change of interval velocity can be computed
without need to remigrate the data.