Velocity Analysis by Nonlinear Optimization of Phase-Contoured Shot Profiles
, by Jos van Trier

The curvature of reflections from planar
beds in migrated shot profiles depends on the migration velocity that is
used. Overmigrated reflections curve downwards, while undermigrated
ones curve upwards. Thus, curvature can be used as an objective
function in an optimization scheme to determine the correct migration
velocity.
Curvature is estimated from lines of equal instantaneous phase in a
shot profile by taking the second derivative of these phase contours.
Phase contouring can also be used to calculate local dip and find
static corrections.
In the optimization, the velocity model is parametrized into basisfunctions,
so that the number of unknowns is reduced considerably.
The optimization method is a non-derivative method, i.e., no gradient of the objective function needs to be calculated. The optimization is also nonlinear: no linear approximations are made to relate model and data. Each iteration
in the optimization consists of a shot-profile migration with the current velocity model. The estimation of the curvature is independent of the type of migration used in the optimization. So, by using a depth migration, the model
can be expressed directly in interval velocities.
For general geology (i.e., curved reflectors), several shot profiles
are needed to determine the velocity model. The curvature analysis in
each shot profile (along the geophone direction) then has to be combined
with a curvature estimation in the shot direction.