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The kinematic -continuation equation (19) corresponds
to the following linear fourth-order dynamic equation

| |
(21) |

where the *t* coordinate refers to the vertical traveltime , and
is the migrated image, parameterized in the anisotropy
parameter . To find the correspondence between equations
(19) and (21), it is sufficient to apply a
ray-theoretical model of the image
| |
(22) |

as a trial solution to (21). Here the surface is the anisotropy continuation ``wavefront'' - the image of a
reflector for the corresponding value of , and the function *A*
is the amplitude. Substituting the trial solution into the partial
differential equation (21) and considering only the terms
with the highest asymptotic order (those containing the fourth-order
derivative of the wavelet *f*), we arrive at the kinematic equation
(19). The next asymptotic order (the third-order derivatives
of *f*) gives us the linear partial differential equation of the
amplitude transport, as follows:
| |
(23) |

We can see that when the reflector is flat ( and
), equation (23) reduces to the equality
and the amplitude remains unchanged for different . This is of
course a reasonable behavior in the case of a flat reflector. It
doesn't guarantee though that the amplitudes, defined by
(23), behave equally well for dipping and curved
reflectors. The amplitude behavior may be altered by adding low-order
terms to equation (21). According to the ray theory, such
terms can influence the amplitude behavior, but do not change the
kinematics of the wave propagation.
An appropriate initial-value condition for equation (21) is
the result of isotropic migration that corresponds to the section in the domain. In practice, the initial-value
problem can be solved by a finite-difference technique.

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Stanford Exploration Project

11/11/1997