While the travel-time curves resulting from a dipping bed are simple, they are not simple to derive. Before the derivation, the result will be stated: for a bed dipping at angle from the horizontal, the travel-time curve is

(3) |

For a common-midpoint gather at in (*h*,*t*)-space,
equation (3) looks
like .Thus the common-midpoint gather contains an
*exact*
hyperbola, regardless of the earth dip angle .The effect of dip is to change the asymptote of the hyperbola,
thus changing the apparent velocity.
The result has great significance in applied work and is
known as Levin's dip correction [1971]:

(4) |

Figure 14 depicts some rays from a common-midpoint gather.

Figure 14

Notice that each ray strikes the dipping bed at a different place.
So a common-*midpoint * gather is not a common-*depth-point * gather.
To realize why the reflection point moves on the reflector,
recall the basic geometrical fact that an
angle bisector in a triangle generally doesn't bisect the opposite side.
The reflection point moves
*up*
dip with increasing offset.

Finally, equation (3) will be proved. Figure 15 shows the basic geometry along with an ``image'' source on another reflector of twice the dip.

Figure 15

For convenience, the bed intercepts the surface at .The length of the line *s*' *g* in Figure 15 is determined by
the trigonometric Law of Cosines to be

Another facet of equation (3) is that it describes the constant-offset section. Surprisingly, the travel time of a dipping planar bed becomes curved at nonzero offset--it too becomes hyperbolic.

10/31/1997