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## Rays and fronts

Figure 4 depicts a ray moving down into the earth at an angle from the vertical.

 front Figure 4 Downgoing ray and wavefront.

Perpendicular to the ray is a wavefront. By elementary geometry the angle between the wavefront and the earth's surface is also .The ray increases its length at a speed v. The speed that is observable on the earth's surface is the intercept of the wavefront with the earth's surface. This speed, namely , is faster than v. Likewise, the speed of the intercept of the wavefront and the vertical axis is .A mathematical expression for a straight line would be
 (1)
like that shown to be the wavefront in Figure 4.

In this expression z0 is the intercept between the wavefront and the vertical axis. To make the intercept move downward, replace it by the appropriate velocity times time:
 (2)
Solving for time gives
 (3)
Equation (3) tells the time that the wavefront will pass any particular location (x , z). The expression for a shifted waveform of arbitrary shape is .Using (3) to define the time shift t0 gives an expression for a wavefield that is some waveform moving on a ray.
 (4)

Next: Waves in Fourier space Up: PLANE-WAVE SUPERPOSITION Previous: Migration improves horizontal resolution
Stanford Exploration Project
10/31/1997