Ray tracing 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 (16) to the following form:  

$$F \left(x,y,z,\frac{\partial t}{\partial x},\frac{\partial t}{\partial y} ,\frac{\partial t}{\partial z} \right)=0,$$ (22)

or  

$$F \left(x,y,z,p_x,p_y,p_z \right)=0,$$ (23)

where $p_x=\frac{\partial t}{\partial x}$, $p_y=\frac{\partial t}{\partial y}$, and $p_z=\frac{\partial t}{\partial z}$. According to the classic rules of mathematical physics, the solutions of this kinematic equation satisfy

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

where s is a running parameter along the ray, and it is related to traveltime as follows:

$$\frac{dt}{ds} = p_x \frac{\partial F}{\partial p_x}+p_y \fra... ...artial F}{\partial p_y}+ p_z \frac{\partial F}{\partial p_z}.$$

As a result,

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

Using equation (16) to calculate the various derivates of F, we obtain the following system of ordinary differential equations:

$$\frac{d x}{d s} =\,{v^2}\,{p_x}\, \left( 1 + 2\,\eta \,\left( 1 - {{{p_z}}^2}\,{{{v_v}}^2} \right) \right),$$ (26)

$$\frac{d y}{d s} =\,{v^2}\,{p_y}\,\left( 1 + 2\,\eta \,\left( 1 - {{{p_z}}^2}\,{{{v_v}}^2} \right) \right),$$ (27)

$$\frac{d z}{d s} =\,\left( 1 - 2\,{v^2}\,\eta \,p_r^2 \right) \,{p_z}\, {{{v_v}}^2},$$ (28)

$$\begin{eqnarray} \frac{d p_{x}}{d s} -= -\,v\,\left( 1 + 2\,\eta \right) \,p_r^2... ...{v^2}\,\eta \,p_r^2 \right) \, {{{p_z}}^2}\,{v_v}\,{({{v_v}})_x},\end{eqnarray}$$

$$\begin{eqnarray} \frac{d p_{y}}{d s}-=-\,v\,\left( 1 + 2\,\eta \right) \,p_r^2 \... ...{v^2}\,\eta \,p_r^2 \right) \, {{{p_z}}^2}\,{v_v}\,{({{v_v}})_y},\end{eqnarray}$$

$$\begin{eqnarray} \frac{d p_{z}}{d s} -=-\,v\,\left( 1 + 2\,\eta \right) \,p_r^2 ... ...{v^2}\,\eta \,p_r^2 \right) \, {{{p_z}}^2}\,{v_v}\,{({{v_v}})_z},\end{eqnarray}$$

and

$$\frac{dt}{ds} = \left( {v^2}\,\left( 1 + 2\,\eta \right) \, ... ...eta \,{{{p_y}}^2} \right) \, {{{p_z}}^2}\,{{{v_v}}^2} \right),$$

where $v_x = \frac{\partial v}{\partial x}$, $v_{y} = \frac{\partial v}{\partial y}$, and $v_{z} = \frac{\partial v}{\partial z}$, and the same holds for $\eta$ and (v_v). Recall that p_r²=p_x² + p_y².

 

raysxz
Figure 9 Raypaths (solid curves) and corresponding wavefronts (dashed curves) for an inhomogeneous VTI model with v_v=v=1.5+0.5 z+0.2 x km/s and $\eta=0.1+0.05 z +0.05 x$.The black curves correspond to a ratio of the vertical S-wave to P-wave velocity of one half, and gray curves correspond to a zero vertical S-wave velocity. In this case, the curves practically overlap; they are only barely distinguishable.

Figure 9 shows sixteen rays originating from a source on the surface at the position x=0 for inhomogeneous models all of which have v_v=v=1.5+0.5 z+0.2 x km/s and $\eta=0.1+0.05 z +0.05 x$.These models differ in their shear wave velocities with rays given in the black curves corresponding to the ratio, r, of the vertical S-wave to P-wave velocities of 0.5, and rays given in the gray curves to r=0. The sixteen rays have ray parameters ranging from zero to the maximum value of 1/V_h (V_h is the horizontal velocity), with a fixed ray-parameter spacing of 1/(15 V_h). These rays terminate at the same time of 8 s, and the wavefronts (given by the dashed curves) are plotted at about 1.6-s intervals. The wavefronts corresponding to the different models are practically coincident. This implies that traveltimes extracted from ray tracing are independent of the shear wave velocity, and the acoustic ray equations derived above are almost as accurate as the conventional anisotropic ray tracing equations Cervený and Hron (1980). The distribution of the rays also indicates that the geometrical spreading features of the two models are practically identical.

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