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Let
denote a point on the surface at which the propagating
wavefield is recorded. Let
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
![$\displaystyle \theta(x;t,y) = t - T(x,y)\;,$](img78.png) |
(32) |
where
is the time recorded at the
-surface, and
is the
traveltime along the ray connecting
and
. The backward
propagation reverses the sign in (32), as follows:
![$\displaystyle \widehat{\theta}(y;z,x) = z + T(x,y)\;.$](img80.png) |
(33) |
Substituting the summation path formulas (32) and (33) into
the general weighting function formulas (28) and (29), we
immediately obtain
![$\displaystyle w^{(+)} = w^{(-)} = {1\over{\left(2 \pi\right)^{m/2}}} \left\vert{{\partial^2 T}\over{\partial x \partial y}}\right\vert^{1/2}\;.$](img81.png) |
(34) |
Gritsenko's formula (Goldin, 1986; Gritsenko, 1984) states that the second mixed
traveltime derivative
is connected with the geometric spreading
along the
-
ray by
the equality
![$\displaystyle R(x,y) = {\sqrt{\cos{\alpha(x)} \cos{\alpha(y)}}\over v(x)} \left\vert{{\partial^2 T}\over{\partial x \partial y}}\right\vert^{-1/2}\;,$](img84.png) |
(35) |
where
is the velocity at the point
, and
and
are the angles formed by the ray with the
and
surfaces, respectively. In a constant-velocity medium,
![$\displaystyle R(x,y) = v^{m-1} T(x,y)^{m/2}\;.$](img88.png) |
(36) |
Gritsenko's formula (35) allows us to
rewrite equation (34) in the form (Goldin, 1988)
The weighting functions commonly used in Kirchhoff datuming
(Wiggins, 1984; Berryhill, 1979; Goldin, 1985) are defined as
These two operators appear to be asymptotically inverse according to
formula (10). They coincide with the asymptotic pseudo-unitary
operators if the velocity
is constant (
), and the two
datum surfaces are parallel (
).
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2013-03-03