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 # subroutine flathyp( adj, add, vel , h, t0,dt,dx, modl,nt,nx, data) integer ix,iz,it,ib, adj, add, nt,nx real t, amp, z,b, vel(nt), h, t0,dt,dx, modl(nt,nx),data(nt,nx) call adjnull( adj, add, 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) if( adj == 0 ) 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 y_{0}. Also make the definitions
(7) | ||
(8) |
(9) |
(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.