PARAXIAL RAY APPROXIMATION

Ray tracing is a technique for finding the coordinates of a ray. Suppose, by using this technique, that we find the coordinates of the ray. We can then define a new orthogonal coordinate system (s,n), named ray-centered coordinates. As Figure shows, the coordinate s measures the arc-length along the ray, and the coordinate n is the normal distance from the ray at point s on the ray. Let us denote the relations between Cartesian coordinates and ray-centered coordinates as $x=\hat{x}(s,n)$ and $z=\hat{z}(s,n)$. Then, the coordinates of the ray are $(\hat{x}(s,0),\hat{z}(s,0))$. The traveltime on any point of the ray can be computed by integrating the slowness function along the ray, as follows:  

$$\tau(s,0)=\tau(s_0,0)+\int^s_{s_0}m(\xi,0)d\xi,$$ (1)

where m(s,n) is the slowness function in ray-centered coordinates.

 

rccoor Figure 1 Ray-centered coordinates.

The amplitude calculation is more complicated than the traveltime calculation. Cervený et al. (1977) showed that the amplitude function along the ray is related to a Jacobian function determined by  

$$J(s,0)=J(s_0,0)\exp[\int^s_{s_0}{M(\xi,0) \over m(\xi,0)}d\xi],$$ (2)

where M(s,0) is the second-order partial derivative of the traveltime with respect to n along the ray and can be found by solving the dynamic ray-tracing equation  

$${dM \over ds}+{1 \over m}M^2+m^2{\partial^2 \over \partial n^2} \left({1 \over m^2}\right)=0$$ (3)

along the ray.

To compute the traveltimes and amplitudes off but near the ray, we use the paraxial ray approximation. The resulted traveltime is  

$$\tau(s,n)=\tau(s,0)+{1 \over 2}M(s,0)n^2.$$ (4)

The direction of a ray follows the direction of the traveltime gradient that is determined by  

$$\nabla \tau(s,n)= \left[m(s,0)+{\partial m \over \partial n}... ...l M \over \partial s}n^2\right]\vec{\bf s}+ M(s,0)n\vec{\bf n},$$ (5)

where $\vec{\bf s}$ and $\vec{\bf n}$ are unit vectors tangential and normal to the ray at (s,0), respectively. Using the paraxial approximation also yields  

$$M(s,n)=m(s,n){M(s,0) \over m(s,0)}\left[1-{1 \over 2} {M^2(s,0) \over m^2(s,0)}n^2\right]$$ (6)

and  

$$J(s,n)=J(s,0)\left[1+{M^2(s,0) \over 2m^2(s,0)}n^2\right].$$ (7)

The next section explains how these equations are used in the local paraxial ray method.