Ordinary differential equation representation: anisotropy rays

According to the classic rules of mathematical physics, the solution of the kinematic equations (15) and (16) can be obtained by solving the following system of ordinary differential equations:

$$\begin{eqnarray} \frac{d x}{d m} = -\tau \frac{\partial f_m}{\partial \tau_x} &,... ...\partial m}+\tau_{m} f_m &,& \frac{d \tau_x}{d m} = \tau_x f_m\;.\end{eqnarray}$$ (20)

Here m stands for either v or $\eta$,${\tau }_x$=$\frac{\partial \tau}{\partial x}$, $f=\frac{\partial \tau}{\partial m}$. To trace the v and $\eta$ rays, we must first identify the initial values x₀, $\tau_0$, $\tau_{x0}$, and $\tau_{m0}$ from the boundary conditions. The variables x₀ and $\tau_0$ describe the initial position of a reflector in a time-migrated section, $\tau_{x0}$ describes its migrated slope, and $\tau_{m0}$ is simply $f(m_0,\tau_0,\tau_{x0})$.

Using the exact kinematic expressions for f results in rather complicated representations of the ordinary differential equations. The linearized expressions, on the other hand, are simple and allow for a straightforward analytical solution.