RETARDED COORDINATES

To examine running horses it may be best to jump on a horse. Likewise, to examine moving waves, it may be better to move along with them. So to describe waves moving downward into the earth we might abandon (x,z)-coordinates in favor of moving (x,z' )-coordinates, where $z' = z\,+\,t\,v$.

An alternative to the moving coordinate system is to define retarded coordinates $\ (x,z,t' ) \$ where $\ t' = t\,-\,z/v$.The classical example of retarded coordinates is solar time. Time seems to stand still on an airplane that moves westward at the speed of the sun.

The migration process resembles the simulation of wave propagation in either a moving coordinate frame or a retarded coordinate frame. Retarded coordinates are much more popular than moving coordinates. Here is the reason: In solid-earth geophysics, velocity may depend on both x and z, but the earth doesn't change with time t during our seismic observations. In a moving coordinate system the velocity could depend on all three variables, thus unnecessarily increasing the complexity of the calculations. Fourier transformation is a popular means of solving the wave equation, but it loses most of its utility when the coefficients are nonconstant.