Zero-offset migration

Dipping-reflector shifts

A little geometry gives simple expressions for the horizontal and vertical position errors on the zero-offset section, which are to be corrected by migration. Figure 5.2 defines the required quantities for a reflection event recorded at $S$ corresponding to the reflectivity at $R$.

reflkine Figure 2. Geometry of the normal ray of length $d$ and the vertical ``shaft'' of length $z$ for a zero-offset experiment above a dipping reflector.

The two-way travel time for the event is related to the length $d$ of the normal ray by

$$t \eq {2 d \over v} ,$$ (1)

where $v$ is the constant propagation velocity. Geometry of the triangle $CRS$ shows that the true depth of the reflector at $R$ is given by

$$z \eq d \cos\theta ,$$ (2)

and the lateral shift between true position $C$ and false position $S$ is given by

$$\Delta x \eq d \sin\theta \eq {v t \over 2} \sin\theta .$$ (3)

It is conventional to rewrite equation (5.2) in terms of two-way vertical traveltime $\tau$:

$$\tau \eq {2 z \over v} \eq t \cos\theta .$$ (4)

Thus both the vertical shift $t - \tau$ and the horizontal shift $\Delta x$ are seen to vanish when the dip angle $\theta$ is zero.

Zero-offset migration