Asymptotic pseudounitary stacking operators |

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:

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

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

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,

(36) |

Gritsenko's formula (35) allows us to rewrite equation (34) in the form (Goldin, 1988)

(37) | |||

(38) |

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 ( ).

Asymptotic pseudounitary stacking operators |

2013-03-03