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Let x denote a point on the surface at which the propagating wavefield is recorded. Let y denote a point on another surface, to which the wavefield is propagating. Then the summation path of the stacking operator for the forward wavefield continuation is  
\theta(x;t,y) = t - T(x,y)\;,\end{displaymath} (36)
where t is the time recorded at the y-surface, and T(x,y) is the traveltime along the ray connecting x and y. The backward propagation reverses the sign in (36), as follows:  
\widehat{\theta}(y;z,x) = z + T(x,y)\;.\end{displaymath} (37)
Substituting the summation path formulas (36) and (37) into the general weighting function formulas (34) and (35), we immediately obtain  
w^{(+)} = w^{(-)} = {1\over{\left(2\,\pi\right)^{m/2}}} \, 
 ...\partial^2 T}\over{\partial x\,\partial y}}\right\vert^{1/2}\;.\end{displaymath} (38)
Gritsenko's formula Goldin (1986); Gritsenko (1984) states that the second mixed traveltime derivative ${{\partial^2 T}\over{\partial x\,\partial y}}$is connected with the geometric spreading R along the x-y ray by the equality  
R(x,y) = {\sqrt{\cos{\alpha(x)}\,\cos{\alpha(y)}}\over v(x)}...
 ...partial^2 T}\over{\partial x\,\partial y}}\right\vert^{-1/2}\;,\end{displaymath} (39)
where v(x) is the velocity at the point x, and $\alpha(x)$ and $\alpha(y)$ are the angles formed by the ray with the x and y surfaces, respectively. In a constant-velocity medium,
R(x,y) = v^{m-1}\,T(x,y)^{m/2}\;.\end{displaymath} (40)
Gritsenko's formula (39) allows us to rewrite equation (38) in the form Goldin (1988)
w^{(+)}(x;t,y) & = & {1\over{\left(2\,\pi\right)^{m/2}}} \,
{\sqrt{\cos{\alpha(x)}\,\cos{\alpha(y)}}\over {v(y)\,R(y,x)}}\;.\end{eqnarray} (41)

The weighting functions commonly used in Kirchhoff datuming Berryhill (1979); Goldin (1985); Wiggins (1984) are defined as
w(x;t,y) & = & {1\over{\left(2\,\pi\right)^{m/2}}} \,
 ...\,\pi\right)^{m/2}}} \,
{{\cos{\alpha(y)}}\over {v(y)\,R(y,x)}}\;.\end{eqnarray} (43)
These two operators appear to be asymptotically inverse according to formula (9). They coincide with the asymptotic pseudo-unitary operators if the velocity v is constant (v(x)=v(y)), and the two datum surfaces are parallel ($\alpha(x) = \alpha(y)$).

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