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The theory outlined in this section closely follows that of Foreman's exact ray theory Foreman (1989), but is summarized here for completeness. Ray theory may be used to compute the characteristics to the time-independent, homogeneous Helmholtz equation,  
\nabla^2\Psi + k^2 \Psi = 0,\end{displaymath} (1)
where $\Psi$ is the desired wavefield solution, and k is the wavenumber. In most ray theoretic developments, the wavefield is represented by an ansatz solution, $\Psi=Ae^{i\phi}$, where $A({\bf r})$ and $\phi({\bf r})$ are the amplitude and phase functions, respectively. Substituting this representation into the Helmholtz equation yields the usual eikonal and transport equations,
K^2= {\bf K} \cdot {\bf K} = \nabla \phi \cdot \nabla \phi & =&...
 ...,\\  2 \nabla A \cdot \nabla \phi+A \nabla^2\phi & =& 0 \nonumber,\end{eqnarray} (2)
where ${\bf K}$ is the phase gradient vector (see figure [*]).

Figure 1
Schematic of phase-front gradient quantities. ${\bf K}$ is the phase gradient vector, $\left\vert K\right\vert{\rm d}s$ is the gradient magnitude along step ${\rm
 d}s$, and $\left\vert K\right\vert{\rm d}x$ and $\left\vert K\right\vert{\rm d}z$ are the projections of $\left\vert K\right\vert{\rm d}s$ along the x and z coordinates, respectively.

Solutions to equations (2) are the ray paths and the amplitudes along these ray paths, respectively. In isotropic media the gradient of the phase function, ${\bf K}=\nabla \phi$, is orthogonal to surfaces of constant phase and represents the instantaneous direction of propagation. Explicitly, this may be written,  
{\bf K}=\nabla \phi = \left\vert K \right\vert \frac{ {\rm d}}{ {\rm d}s}{\bf r},\end{displaymath} (3)
where ${\rm
 d}s$ is an element of length, and ${\rm d}{\bf r}$ is a vector element of the ray-path. A ray-path equation is developed by taking the gradient of the phase gradient magnitude (i.e. $\nabla K$),
K^2 & = & {\bf K}\cdot{\bf K}, \nonumber \\ 2K\nabla K &= & 2 (... \nabla \right) K =\frac{ {\rm d}}{ {\rm d}s}{\bf K}, \nonumber\end{eqnarray}
where $\nabla \times {\bf K}= \nabla \times \nabla \phi =0$ has been employed. Using equation (3) this may be written explicitly,  
\nabla K = \frac{ {\rm d}}{ {\rm d}s}\left(K \frac{ {\rm d}}{ {\rm d}s}{\bf r} \right)\end{displaymath} (5)
Coupling between the eikonal and transport equations is evident through the dependence of the eikonal equation on amplitude function, A. In many cases a high-frequency approximation (i.e. $\frac{\nabla^2
A}{A}\approx 0$) is used to decouple these equations. Use of this approximation yields the usual form of the ray path equation,  
\frac{ {\rm d}}{ {\rm d}s}\left( \frac{1}{c} \frac{ {\rm d}}{ {\rm d}s}{\bf r}\right) = \nabla \left( \frac{1}{c} \right),\end{displaymath} (6)
where c is the velocity. Use of this approximation also eliminates the frequency dependence of ray trajectories (see Appendix A). One manner of reintroducing frequency-dependent ray trajectories is discussed in a latter section.