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# DMO IN THE PROCESSING FLOW

Instead of implementing equation (28) in one step we can split it into two steps. The first step converts raw data at time th to NMOed data at time tn.

 (29)

The second step is the DMO step which like Kirchhoff migration itself is a convolution over the x-axis (or b-axis) with
 (30)
and it converts time tn to time t0. Substituting (29) into (30) leads back to (28). As equation (30) clearly states, the DMO step itself is essentially velocity independent, but the NMO step naturally is not.

Now the program. Backsolving equation (30) for tn gives

 (31)

Like subroutine flathyp() , our DMO subroutine dmokirch() is based on subroutine kirchfast() . It is just the same, except where kirchfast() has a hyperbola we put equation (31). In the program, the variable t0 is called z and the variable th is called t. Note, that the velocity velhalf does exclusively occur in the break condition (which we have failed to derive, but which is where the circle and ellipse touch at z=0).

subroutine dmokirch( adj, add, velhalf, h, t0,dt,dx, modl,nt,nx, data)
real    amp,t,z,b,             velhalf, h, t0,dt,dx, modl(nt,nx),data(nt,nx)
if( h == 0)     call erexit('h=0')
do ib= -nx, nx {        b = dx * ib             # b = midpt separation
do iz= 2, nt {      z = t0 + dt * (iz-1)    # z = zero-offset time

if( h**2 <= b**2 )                  next
t= sqrt(  z**2 / (1-b**2/h**2) )
amp= sqrt(t) * dx/h
if( velhalf*abs(b) * t*t > h**2*z)  break
it = 1.5 + (t - t0) / dt
if( it > nt )                       break
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


Figures  24 and  25 were made with subroutine dmokirch() . Notice the big noise reduction over Figure 18.

 dmatt Figure 24 Impulse response of DMO and NMO

 coffs Figure 25 Synthetic Cheop's pyramid

Next: Residual NMO Up: Dip and offset together Previous: Restatement of ellipse equations
Stanford Exploration Project
12/26/2000