From kinematics to dynamics

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

$$\frac{\partial^4 P}{\partial t^3 \, \partial \eta} + v^2 \... ...artial \eta} + t v^4 \frac{\partial^4 P}{\partial x^4} = 0 \;,$$ (21)

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

$$P (t, x, \eta) = A (x, \eta) f (t - \tau (x, \eta))$$ (22)

as a trial solution to (21). Here the surface $t = \tau (x, \eta)$ is the anisotropy continuation ``wavefront'' - the image of a reflector for the corresponding value of $\eta$, 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:  

$$\left( 1 + v^2 \tau_x^2 \right) \frac{\partial A}{\partial... ...\eta \tau_{xx} - 6 v^2 \tau \tau_x^2 \tau_{xx} \right) = 0 \;.$$ (23)

We can see that when the reflector is flat ($\tau_x = 0$ and $\tau_{xx}=0$), equation (23) reduces to the equality

$$\frac{\partial A}{\partial \eta} = 0\;,$$

and the amplitude remains unchanged for different $\eta$. 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 $\eta=0$section in the $(t,x,\eta)$ domain. In practice, the initial-value problem can be solved by a finite-difference technique.