raytracing equations

Using the method of characteristics, we can derive a system of ordinary differential equations that define the ray trajectories. To do so, we need to transform equation (9) to the following form:  

$$F \left(x,\tau,\frac{\partial t}{\partial x},\frac{\partial t}{\partial \tau} \right)=0,$$ (14)

or  

$$F \left(x,\tau,p_x,p_{\tau} \right)=0,$$ (15)

where $p_x=\frac{\partial t}{\partial x}$ and $p_{\tau}=\frac{\partial t}{\partial \tau}$. According to the classic rules of mathematical physics (Courant, 1966), the solutions of this kinematic equation can be obtained from the system of ordinary differential equations

$$\begin{eqnarray} \frac{d x}{d s} = \frac{1}{2} \frac{\partial F}{\partial p_x} &... ...d p_{\tau}}{d s} = - \frac{1}{2} \frac{\partial F}{\partial \tau},\end{eqnarray}$$

where s is a running parameter along the rays, related to the traveltime t as follows:

$$\frac{d t}{d s} = \frac{1}{2} p_{\tau} \frac{\partial F}{\partial p_{\tau}}+ p_x \frac{\partial F}{\partial p_x},$$

with

$$\begin{eqnarray} \frac{d x}{d t} = \frac{d x}{d s} \left/ \frac{d t}{d s}\right.... ...\tau}}{d t} = \frac{d p_{\tau}}{d s} \left/\frac{d t}{d s}\right..\end{eqnarray}$$

Using equation (9), we obtain  

$$\frac{d x}{d s} = a\,{v^2}\,\left( 1 + 2\,\eta \,\left( 1 - 4\,{{{p_{\tau }}}^2} \right) \right),$$ (18)

$$\frac{d \tau}{d s} = 4\,{p_{\tau }} - a\,{v^2}\,\left( -\sig... ...\,{p_{\tau }} + 8\,\sigma \,{{{p_{\tau }}}^2} \right) \right),$$ (19)

$$\begin{eqnarray} \frac{d p_{x}}{d s} = -a^2\,v\,\left( 1 + 2\,\eta \,\left( 1 - ... ...a - 8\,\eta \,{{{p_{\tau }}}^2} \right) \, {{\sigma }_x} \right),\end{eqnarray}$$

   $$\begin{eqnarray} \frac{d p_{\tau}}{d s} = -v\,a^2\,\left( 1 + 2\,\eta \,\left( 1... ...\,\eta \,{{{p_{\tau }}}^2} \right) \,{{\sigma }_{\tau }} \right),\end{eqnarray}$$

and

$$\frac{d t}{d s} = 4\,{{{p_{\tau }}}^2} + {a^2}\,{v^2}\, \le... ... + 2\,\eta \,\left( 1 - 8\,{{{p_{\tau }}}^2} \right) \right),$$

where

$$a={p_x} + \sigma \,{p_{\tau }},$$

and $v_x = \frac{\partial v}{\partial x}$ and $v_{\tau} = \frac{\partial v}{\partial \tau}$, and the same holds for $\eta$ and $\sigma$.

To trace rays, we must first identify the initial values x₀, $\tau_0$, p_x0, and $p_{\tau 0}$. The variables x₀ and $\tau_0$ describe the source position, and p_x0 and $p_{\tau 0}$ are extracted from the initial angle of propagation. Note that, from equation (9),

$$p_{\tau 0}=1- \frac{v^2 p_{x0}^2}{1-2 \eta v^2 p_{x0}^2},$$

because $\sigma$=0 at the source position (z=0).

The raytracing system of equations (18-21) describes the ray-theoretical aspect of wave propagation in the $(x-\tau)$-domain, and can be used as an alternative to the eikonal equation. Numerical solutions of the raytracing equations, as opposed to the eikonal equation, provide multi-arrival traveltimes and amplitudes. In the numerical examples, we use raytracing to highlight some of the features of the $(x-\tau)$-domain coordinate system.