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 t_h to NMOed data at time t_n.

 

$$t_n^2 \eq t_h^2 - {h^2 \over{v_{\rm half}^2}}$$ (29)

The second step is the DMO step which like Kirchhoff migration itself is a convolution over the x-axis (or b-axis) with  

$$t_0^2 \eq t_n^2 \left( 1 - {b^2 \over h^2} \right)$$ (30)

and it converts time t_n 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 t_n gives

 

$$t_n^2 \eq {t_0^2 \over 1-b^2/h^2 } .$$ (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 t_h 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)
integer ix,iz,it,ib, adj, add,                            nt,nx
real    amp,t,z,b,             velhalf, h, t0,dt,dx, modl(nt,nx),data(nt,nx)
call adjnull(        adj, add,                       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)
                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

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