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The eikonal equation in two dimensions relates the two gradient components of
the traveltime field:

| |
(1) |

where *s* is a two-dimensional slowness model and is the traveltime field.
Subscripts *x* and *z* denote partial derivatives with respect to *x* and
*z*, respectively. Using a finite-difference method, one can solve this
equation for . The contour lines of define the trajectories
of wavefronts. It is well known that, in an isotropic medium,
the trajectories of rays are always perpendicular to
the trajectories of wavefronts and that the gradient directions of a
field are always perpendicular to the contour lines of the field.
Therefore, the gradient of is orthogonal to the gradient of
function *p* whose contour lines are the trajectories of rays:
If we assume that the traveltime field is known,
this equation leads to an linear, first-order partial differential equation
for *p*
| |
(2) |

In the following sections, I will explain how to solve this equation for
two specific cases.

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Stanford Exploration Project

12/18/1997