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

## Constant offset migration

Considering h in equation (6) to be a constant, enables us to write a subroutine for migrating constant-offset sections. Subroutine flathyp() is easily prepared from subroutine kirchfast() by replacing its hyperbola equation with equation (6).

# Flat topped hyperbolas and constant-offset section migration
#
real    t, amp, z,b,            vel(nt), h, t0,dt,dx, modl(nt,nx),data(nt,nx)
do ib= -nx, nx {        b = dx * ib             # b = midpt separation y-y0
do iz= 2, nt {      z = t0 + dt * (iz-1)    # z = zero-offset time
t = .5 * ( sqrt( z**2 +((b-h)*2/vel(iz))**2) +
sqrt( z**2 +((b+h)*2/vel(iz))**2)   )
it = 1.5 + (t - t0) / dt
if( it > nt )                     break
amp = (z/t)/ sqrt(t)
do ix= max0(1, 1-ib),  min0(nx, nx-ib)
data(it,ix+ib)= data(it,ix+ib) + modl(iz,ix   ) * amp
else
modl(iz,ix   )= modl(iz,ix   ) + data(it,ix+ib) * amp
}
}
return; end


The amplitude in subroutine flathyp() should be improved when we have time to do so. Forward and backward responses to impulses of subroutine flathyp() are found in Figures 4 and 5.

 Cos.1 Figure 4 Migrating impulses on a constant-offset section with subroutine flathyp(). Notice that shallow impulses (shallow compared to h) appear ellipsoidal while deep ones appear circular.

 Cos.0 Figure 5 Forward modeling from an earth impulse with subroutine flathyp().

It is not easy to show that equation (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 y equal to the old one shifted by y0. Also make the definitions
 (7)
With these definitions, (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 y and z to the right.
 (8)
Finally, recalling all earlier definitions and replacing y by y-y0, we obtain the canonical form of an ellipse with semi-major axis A and semi-minor axis B:
 (9)
where
 (10) (11)

Fixing t, equation (9) is the equation for a circle with a stretched z-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.

Next: INTRODUCTION TO DIP Up: PRESTACK MIGRATION Previous: Prestack migration ellipse
Stanford Exploration Project
12/26/2000