Using the high-frequency approximation, wavefronts in a 3-D model can be described by the eikonal equation,
(1) |
Numerically solving the eikonal equation is probably the most efficient method of obtaining wavefront traveltimes in arbitrary velocity models. One reason for the efficiency is that we can conveniently solve the eikonal equation over a regular grid, which eliminates the need for the interpolation commonly used with other methods, such as ray tracing.
A major drawback of using conventional methods to solve the eikonal equation numerically, is that we only evaluate the fastest arrival solution, not necessarily the most energetic Vidale (1990). This results in less than acceptable traveltime computation for imaging in complex media Geoltrain and Brac (1993). Another drawback of the eikonal solvers is that the conventional solvers are generally unstable in complex media Popovici (1991). Stability usually came at higher price (a finer grid) to the implementation. In addition, conventional eikonal solvers cannot treat turning waves. Using polar coordinates, though it does not eliminate the turning wave problem, allows for wavefronts overturning at a considerable angle, on the condition that they do not overturn on the radial axis.
polar-dif
Figure 1 A spherical coordinate system given by r, , and . The source, s(x_{0},y_{0},z_{0}), is at the origin of the spherical coordinates where r=0. The parameter r (=) is the distance from the source to the point of interest along the wavefront, which is also the magnitude of the vector. The parameter is the angle between the x-axis and the projection of the vector onto the x-y plane. The parameter is the angle between the z-axis and the vector along a vertical plane. |
The eikonal equation can also be described in spherical coordinates (r, , and ; see Figure 1), as follows:
(2) |
In polar coordinates, we eliminate the terms so that the eikonal equation, described by r and , is given by
(3) |