Imaging in shot-geophone space

Zero-dip stacking (Y = 0)

When dealing with the offset $h$ it is common to assume that the earth is horizontally layered so that experimental results will be independent of the midpoint $y$. With such an earth the Fourier transform of all data over $y$ will vanish except for $k_y = 0$, or, in other words, for $Y = 0$. The two square roots in (9.14) again become identical, and the resulting equation is once more the paraxial equation:

$${dU \over dz } \eq { - i \omega } {2 \over v } \sqrt { 1 -\ { v^2 k_h^2 \over 4 \omega^2 } } U$$ (33)

Using this equation to downward continue hyperboloids from the earth's surface, we find the hyperboloids shrinking with depth, until the correct depth where best focus occurs is reached. This is shown in Figure 9.10.

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Figure 10. With an earth model of three layers, the common-midpoint gathers are three hyperboloids. Successive frames show downward continuation to successive depths where best focus occurs.

The waves focus best at zero offset. The focus represents a downward-continued experiment, in which the downward continuation has gone just to a reflector. The reflection is strongest at zero travel time for a coincident source-receiver pair just above the reflector. Extracting the zero-offset value at $t = 0$ and abandoning the other offsets amounts to the conventional procedure of summation along a hyperbolic trajectory on the original data. Naturally the summation can be expected to be best when the velocity used for downward continuation comes closest to the velocity of the earth.

Actually, the seismic energy will not all go precisely to zero offset; it goes to a focal region near zero offset. A further analysis (not begun here) can analyze the focal region to upgrade the velocity estimation. Dissection of this focal region can also provide information about reflection strength versus angle.

Imaging in shot-geophone space