Effect of horizontal velocity gradient

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Problem

In Figure 10.14a, , , the horizon dips , and the diffracting point is at a vertical depth of 2.00 km, the dipping horizon being midway between and (at the surface). Compare the diffraction curve with what would have been observed on a CMP section if .

Figure 10.14a.  Geometry of problem.

Solution

The curve for the diffracting point below the dipping interface can be solved analytically or graphically by ray tracing. The crest of the diffraction is located beneath where the angle of approach to the surface is ; such a raypath is called an image ray. If we denote the angle between a ray at and the vertical as , then Snell’s law at the interface gives for the image ray the equation


The fact that an image ray approaches the surface vertically is employed in depth migration (see problem 10.16) to accommodate horizontal changes in velocity.

The diffraction curve (solid curve, Figure 10.14c) is not symmetrical and the crest is displaced about 500 m updip from . The right limb is nearly flat because the increased traveltimes at the high velocity is almost compensated for shorter travel distance at the low velocity.

Figure 10.14b.  Ray tracing for the diffracting point .
Figure 10.14c.  Diffraction curves. Solid curve is 2-layer case, dashed is constant velocity case.

The diffraction curve for the constant velocity case (dashed curve, Figure 10.14c) can be calculated using the equations

where is the angle of approach at the surface.

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