Theory of differential offset continuation

Example 1: plane reflector

The simplest and most important example is the case of a plane dipping reflector. Putting the origin of the $y$ axis at the intersection of the reflector plane with the surface, we can express the reflection traveltime after NMO in the form

$$\tau_n(y,h)=p\,\sqrt{y^2-h^2}\;,$$ (35)

where $p=2\,{ \sin{\alpha} \over v}$, and $\alpha$ is the dip angle. The zero-offset traveltime in this case is a straight line:

$$t_0\left(y_0\right)=p\,y_0\;.$$ (36)

According to equations (28-29), the time rays in this case are defined by

$$y_1\left(t_1\right)={t_1^2 \over {p^2\,y_0}}\;;\; h_1^2\left... ...^2\,y_0^2} \over {p^4\,y_0^2}}\;;\; y_0={{y^2-h^2} \over y}\;.$$ (37)

The geometry of the OC transformation is shown in Figure 3.

ocopln
Figure 3. Transformation of the reflection traveltime curves in the OC process: the case of a plane dipping reflector. Left: Time coordinate before the NMO correction. Right: Time coordinate after NMO. The solid lines indicate traveltime curves at different common-offset sections; the dashed lines indicate time rays.

Theory of differential offset continuation