Datuming

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  

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

$$\widehat{\theta}(y;z,x) = z + T(x,y)\;.$$ (37)

Substituting the summation path formulas (36) and (37) into the general weighting function formulas (34) and (35), we immediately obtain  

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

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

$$R(x,y) = v^{m-1}\,T(x,y)^{m/2}\;.$$ (40)

Gritsenko's formula (39) allows us to rewrite equation (38) in the form Goldin (1988)

$$\begin{eqnarray} w^{(+)}(x;t,y) & = & {1\over{\left(2\,\pi\right)^{m/2}}} \, {\s... ..., {\sqrt{\cos{\alpha(x)}\,\cos{\alpha(y)}}\over {v(y)\,R(y,x)}}\;.\end{eqnarray}$$ (41) (42)

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

$$\begin{eqnarray} w(x;t,y) & = & {1\over{\left(2\,\pi\right)^{m/2}}} \, {{\cos{\a... ...\,\pi\right)^{m/2}}} \, {{\cos{\alpha(y)}}\over {v(y)\,R(y,x)}}\;.\end{eqnarray}$$ (43) (44)

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 ($\alpha(x) = \alpha(y)$).