Next: Velocity gradients Up: THE PARAXIAL WAVE EQUATION Previous: Time-shifting equations

## Fourier decomposition

Fourier analyzing the function f(x,t,z=0), seen on the earth's surface, requires the Fourier kernel .Moving on the earth's surface at an inverse speed of ,the phase of the Fourier kernel, hence the kernel itself, remains constant. Only those sinusoidal components that move at the same speed as the Snell wave can have a nonzero correlation with it. So if the disturbance is a single Snell wave, then all Fourier components vanish except for those that satisfy .You should memorize these basic relations:

 (51)

In theoretical seismology a square-root function often appears as a result of using (51) to make a cosine.

Utilization of this Fourier domain interpretation of Snell's parameter p enables us to write the square-root equations (48), (49) and (50) in an even more useful form. But first the square-root equation must be reexpressed in the Fourier domain. This is done by replacing the operator in (48), (49) and (50) by .The result is
 (52)
At present it is equivalent to specify either the differential equation (52) or its solution (47) with f as the complex exponential:
 (53)
Later, when we consider lateral velocity variation v(x), the solution (53) becomes wrong, whereas the differential equation (50) is a valid description of any local plane-wave behavior. But before going to lateral velocity gradients we should look more carefully at vertical velocity gradients.

Next: Velocity gradients Up: THE PARAXIAL WAVE EQUATION Previous: Time-shifting equations
Stanford Exploration Project
10/31/1997