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Assume that a ray is specified by the parametric equation
 
(2) 
where is the traveltime along the ray, with at the location of
the source. The independent variable is not necessarily . It
can be any other variable which increases monotonously along the ray, e.g.,
arclength is a common choice in the ray tracing algorithm. Here we choose
traveltime because seismic data are sampled along time axis. Therefore, it
is more convenient to compare the ray tracing results with seismic data.
The three coordinates are the solution of geometric ray tracing system
 
(3) 
where s is the slowness along the ray. The reason we use slowness, not
velocity, is to follow the convention of tomography. Here, are the
Cartesian components of the slowness vector and
 
(4) 
Equation (3) is firstorder ordinary differential
equations. Usually, we cannot find an analytical solution for a general
velocity model. It is more common to use some numerical algorithms, e.g., the
RungeKutta method. In this paper, we use a Fortran90 version RungeKutta
solver. In order to fit our expression to the
convention of rksuite90 package, the geometric ray tracing system is
expressed as
 
(5) 
The initial condition is
 
(6) 
where and are the two initial takeoff angles,
as shown in Figure 1.
takeoff
Figure 1 The initial condition of each central ray. (x, y, z) forms a Cartesian coordinate system. O is the source location.

 
Next: 3D dynamic ray tracing
Up: THEORY OF DYNAMIC RAY
Previous: THEORY OF DYNAMIC RAY
Stanford Exploration Project
11/11/1997