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
#
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 y₀. Also make the definitions

$$\begin{eqnarray} t\,v \ \ \ \ &=&\ \ \ \ 2\ A\ \\ \alpha\ \ \ \ &=&\ \ \ \ z^2 \... ...number \\ \alpha\ \ -\ \ \beta\ \ \ \ &=&\ \ \ \ 4\ y\ h \nonumber\end{eqnarray}$$ (7)

With these definitions, (5) becomes

$$2\ A\ \eq \ \sqrt \alpha \ \ +\ \ \sqrt \beta$$

Square to get a new equation with only one square root.

$$4\ A^2 \ \ -\ \ (\alpha\ +\ \beta) \ \eq \ 2\ \sqrt{ \alpha \beta }$$

Square again to eliminate the square root. Introduce definitions of $\alpha$ and $\beta$.

$$16\ A^4 \ \ -\ \ 8\ A^2 \ [\,2\,z^2 \ +\ 2\,y^2 \ +\ 2\,h^2 ] \ \ +\ \ 16\ y^2 \, h^2 \ \eq \ 0$$

Bring y and z to the right.

$$\begin{eqnarray} A^4 \ \ -\ \ A^2 \, h^2 \ \ \ \ &=&\ \ \ \ \nonumber A^2 \, ( z... ... \ \ \ \ &=&\ \ \ \ {z^2 \over 1 \ -\ {h^2 \over A^2}}\ \ +\ \ y^2\end{eqnarray}$$ (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:  

$${(y\ -\ y_0)^2 \over A^2} \ +\ {z^2 \over B^2} \eq 1 \ \ \ ,$$ (9)

where

$$\begin{eqnarray} A &\eq& {v\ t \over 2} \\ B &\eq& \sqrt{A^2\ -\ h^2}\end{eqnarray}$$ (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.