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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

$\displaystyle \theta(x;t,y) = t - T(x,y)\;,$ (32)

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 (32), as follows:

$\displaystyle \widehat{\theta}(y;z,x) = z + T(x,y)\;.$ (33)

Substituting the summation path formulas (32) and (33) into the general weighting function formulas (28) and (29), we immediately obtain

$\displaystyle w^{(+)} = w^{(-)} = {1\over{\left(2 \pi\right)^{m/2}}}   \left\vert{{\partial^2 T}\over{\partial x \partial y}}\right\vert^{1/2}\;.$ (34)

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

$\displaystyle R(x,y) = {\sqrt{\cos{\alpha(x)} \cos{\alpha(y)}}\over v(x)}  \left\vert{{\partial^2 T}\over{\partial x \partial y}}\right\vert^{-1/2}\;,$ (35)

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,

$\displaystyle R(x,y) = v^{m-1} T(x,y)^{m/2}\;.$ (36)

Gritsenko's formula (35) allows us to rewrite equation (34) in the form (Goldin, 1988)
$\displaystyle w^{(+)}(x;t,y)$ $\displaystyle =$ $\displaystyle {1\over{\left(2 \pi\right)^{m/2}}}  
{\sqrt{\cos{\alpha(x)} \cos{\alpha(y)}}\over {v(x) R(x,y)}}\;,$ (37)
$\displaystyle w^{(-)}(y;z,x)$ $\displaystyle =$ $\displaystyle {1\over{\left(2 \pi\right)^{m/2}}}  
{\sqrt{\cos{\alpha(x)} \cos{\alpha(y)}}\over {v(y) R(y,x)}}\;.$ (38)

The weighting functions commonly used in Kirchhoff datuming (Wiggins, 1984; Berryhill, 1979; Goldin, 1985) are defined as

$\displaystyle w(x;t,y)$ $\displaystyle =$ $\displaystyle {1\over{\left(2 \pi\right)^{m/2}}}  
{{\cos{\alpha(x)}}\over {v(x) R(x,y)}}\;,$ (39)
$\displaystyle \widehat{w}(y;z,x)$ $\displaystyle =$ $\displaystyle {1\over{\left(2 \pi\right)^{m/2}}}  
{{\cos{\alpha(y)}}\over {v(y) R(y,x)}}\;.$ (40)

These two operators appear to be asymptotically inverse according to formula (10). 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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