which is solved by doing NMO as time migration in time and offset coordinates. For the zero-offset case (phh=0), the equation becomes an identity. To verify the partial differential equation, I coded equation (13) for a constant velocity case and for several depth varying velocity functions. Equation (13) can be used to extrapolate the initial wavefield in t0 using a vrms velocity for each zero-offset time value. The results are shown in the following synthetic cases.
Figure shows the velocity and the travel-time map used to model the constant-offset sections. Since for this first case the velocity was constant the information in this figure is trivial, however I kept it just to be consistent with the layout of the other figures. The number of offsets used is 32, with a maximum half-offset h=930 meters. In all cases the structural model was a diffractor at a depth of 1250 m. Figure shows a comparison between the first zero-offset panel obtained from modeling and the output of the MZO algorithm applied to the 32 constant-offset sections. The kinematics of the initial zero-offset section in Figure a coincide with the output of the MZO algorithm in Figure b. The artifacts in Figure b are due to the Fourier domain implementation of the algorithm. The next figures use a depth variable velocity to model the constant-offset sections and the corresponding vrms to migrate the data to zero-offset.
Figure a shows a root-mean-square velocity with a slow increase, corresponding to an interval velocity with several linearly increasing velocity layers. The travel-time map model in Figure b can be used to identify the boundaries of sharper velocity changes. Comparing Figures a and b we notice the excellent kinematic match between the zero-offset model and the output of the migration to zero-offset algorithm using the depth variable velocity.
Figure shows a velocity model with an increasing jump, while in Figure , the velocity has a decreasing jump. Again in both cases, comparing Figures a with b and Figures a with b, the kinematics of the initial zero-offset section and the output of the migration to zero-offset are extremely similar even though the velocities used show quite large variations.