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The DSR equation in midpoint-offset space

By converting the DSR equation to midpoint-offset space we will be able to identify the familiar zero-offset migration part along with corrections for offset. The transformation between (g,s) recording parameters and (y,h) interpretation parameters is
 (37) (38)
Travel time t may be parameterized in (g,s)-space or (y,h)-space. Differential relations for this conversion are given by the chain rule for derivatives:
 (39) (40)

Having seen how stepouts transform from shot-geophone space to midpoint-offset space, let us next see that spatial frequencies transform in much the same way. Clearly, data could be transformed from (s,g)-space to (y,h)-space with (37) and (38) and then Fourier transformed to ( ky , kh )-space. The question is then, what form would the double-square-root equation (35) take in terms of the spatial frequencies ( ky , kh )? Define the seismic data field in either coordinate system as
 (41)
This introduces a new mathematical function U' with the same physical meaning as U but, like a computer subroutine or function call, with a different subscript look-up procedure for (y,h) than for (s,g). Applying the chain rule for partial differentiation to (41) gives
 (42) (43)
and utilizing (37) and (38) gives
 (44) (45)
In Fourier transform space where transforms to i kx, equations (44) and (45), when i and U = U' are cancelled, become
 (46) (47)
Equations (46) and (47) are Fourier representations of (44) and (45). Substituting (46) and (47) into (35) achieves the main purpose of this section, which is to get the double-square-root migration equation into midpoint-offset coordinates:
 (48)

Equation (48) is the takeoff point for many kinds of common-midpoint seismogram analyses. Some convenient definitions that simplify its appearance are
 (49) (50) (51) (52)
Chapter showed that the quantity can be interpreted as the angle of a wave. Thus the new definitions S and G are the sines of the takeoff angle and of the arrival angle of a ray. When these sines are at their limits of they refer to the steepest possible slopes in (s,t)- or (g,t)-space. Likewise, Y may be interpreted as the dip of the data as seen on a seismic section. The quantity H refers to stepout observed on a common-midpoint gather. With these definitions (48) becomes slightly less cluttered:
 (53)

Most present-day before-stack migration procedures can be interpreted through equation (53). Further analysis of it will explain the limitations of conventional processing procedures as well as suggest improvements in the procedures.

EXERCISES:

1. Adapt equation (48) to allow for a difference in velocity between the shot and the geophone.
2. Adapt equation (48) to allow for downgoing pressure waves and upcoming shear waves.

Next: THE MEANING OF THE Up: SURVEY SINKING WITH THE Previous: The DSR equation in
Stanford Exploration Project
10/31/1997