** Next:** Migration
** Up:** EXAMPLES
** Previous:** EXAMPLES

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

| |
(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:
| |
(37) |

Substituting the summation path formulas (36) and (37) into
the general weighting function formulas (34) and (35), we
immediately obtain
| |
(38) |

Gritsenko's formula Goldin (1986); Gritsenko (1984) states that the second mixed
traveltime derivative is connected with the geometric spreading *R* along the *x*-*y* ray by
the equality
| |
(39) |

where *v*(*x*) is the velocity at the point *x*, and and
are the angles formed by the ray with the *x* and *y*
surfaces, respectively. In a constant-velocity medium,
| |
(40) |

Gritsenko's formula (39) allows us to
rewrite equation (38) in the form Goldin (1988)
| |
(41) |

| (42) |

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

| |
(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 ().

** Next:** Migration
** Up:** EXAMPLES
** Previous:** EXAMPLES
Stanford Exploration Project

11/12/1997