The dipping bed

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 $\alpha$ from the horizontal, the travel-time curve is  

$$t^2 \, v^2 \eq 4 \, (y - y_0 )^2 \, \sin^2 \alpha \ +\ 4 \, h^2 \, \cos^2 \alpha$$ (3)

For $\alpha \,=\,$ 45$^\circ$,equation (3) is the familiar Pythagoras cone--it is just like $t^2 = z^2 \,+\, x^2$.For other values of $\alpha$, the equation is still a cone, but a less familiar one because of the stretched axes.

For a common-midpoint gather at $y\,=\, y_1$ in (h,t)-space, equation (3) looks like $t^2 \,=\, t_0^2 \,+$$4h^2 / v_{\rm apparent}^2$.Thus the common-midpoint gather contains an exact hyperbola, regardless of the earth dip angle $\alpha$.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]:  

$$v_{\rm apparent} \eq {v_{\rm earth} \over \cos ( \alpha ) }$$ (4)

(See also Slotnick [1959]). In summary, dip increases the stacking velocity.

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

 

dipray
Figure 14 Rays from a common-midpoint gather.

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.

 

lawcos
Figure 15 Travel time from image source at s' to g may be expressed by the law of cosines.

For convenience, the bed intercepts the surface at $y_0 \,=\, 0$.The length of the line s' g in Figure 15 is determined by the trigonometric Law of Cosines to be which is equation (3).

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.