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## From kinematics to dynamics

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.

Next: synthetic test Up: Ordinary differential equation representation: Previous: Ordinary differential equation representation:
Stanford Exploration Project
11/11/1997