The simplest environment for reflection data is a single horizontal reflection interface, which is shown in Figure 12.

Figure 12

As expected, the zero-offset section mimics the earth model.
The common-midpoint gather is a hyperbola
whose asymptotes are straight lines
with slopes of the inverse of the velocity *v _{1}*.
The most basic data processing is called

The next simplest environment is to have a planar reflector that is oriented vertically rather than horizontally. This is not typical, but is included here because the effect of earth dip is more comprehensible in an extreme case. Now the wave propagation is along the air-earth interface. To avoid confusion the reflector may be inclined at a slight angle from the vertical, as in Figure 13.

Figure 13

Figure 13 shows that the travel time does not change as the offset changes. It may seem paradoxical that the travel time does not increase as the shot and geophone get further apart. The key to the paradox is that midpoint is held constant, not shotpoint. As offset increases, the shot gets further from the reflector while the geophone gets closer. Time lost on one path is gained on the other.

A planar reflector may have any dip between horizontal and vertical.
Then the common-midpoint gather lies between
the common-midpoint gather of Figure 12
and that of Figure 13.
The zero-offset section in Figure 13 is a straight line,
which turns out to be the asymptote of a family of hyperbolas.
The slope of the asymptote is the inverse of the velocity *v _{1}*.

10/31/1997