Dip and offset together

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## Constant offset migration

Considering in equation (8.6) to be a constant, enables us to write a subroutine for migrating constant-offset sections. Forward and backward responses to impulses are found in Figures 8.4 and 8.5.

Cos1
Figure 4.
Migrating impulses on a constant-offset section. Notice that shallow impulses (shallow compared to ) appear ellipsoidal while deep ones appear circular.

Cos0
Figure 5.
Forward modeling from an earth impulse.

It is not easy to show that equation (8.5) can be cast in the standard mathematical form of an ellipse, namely, a stretched circle. But the result is a simple one, and an important one for later analysis. Feel free to skip forward over the following verification of this ancient wisdom. To help reduce algebraic verbosity, define a new equal to the old one shifted by . Also make the definitions

 (7)

With these definitions, (8.5) becomes

Square to get a new equation with only one square root.

Square again to eliminate the square root.

Introduce definitions of and .

Bring and to the right.
 (8)

Finally, recalling all earlier definitions and replacing by , we obtain the canonical form of an ellipse with semi-major axis and semi-minor axis :
 (9)

where
 (10) (11)

Fixing , equation (8.9) is the equation for a circle with a stretched -axis. The above algebra confirms that the string and tack'' definition of an ellipse matches the stretched circle'' definition. An ellipse in earth model space corresponds to an impulse on a constant-offset section.

 Dip and offset together

Next: INTRODUCTION TO DIP Up: PRESTACK MIGRATION Previous: Prestack migration ellipse

2009-03-16