** Next:** About this document ...
** Up:** Fomel: Linearized Eikonal
** Previous:** A SIMPLE derivation of

The eikonal equation (1) can be rewritten in the form

| |
(12) |

where is the unit vector, pointing in the traveltime
gradient direction. The integral solution of equation (12)
takes the form
| |
(13) |

which states that *the traveltime can be computed by
integrating the slowness **n* along the ray ,tangent at every point to the gradient direction .
Similarly, we can rewrite the linearized eikonal equation (5)
in the form

| |
(14) |

where is the unit vector, pointing in gradient
direction for the initial traveltime . The integral solution
of equation (14) takes the form
| |
(15) |

which states that *the traveltime perturbation can be
computed by integrating the slowness perturbation **n*_{1} along the
ray , defined by the initial slowness model
*n*_{0} . This is exactly the basic principle of traveltime
tomography.
I have borrowed this proof from Lavrentiev et al. (1970), who used linearization
of the eikonal equation as the theoretical basis for traveltime inversion.

** Next:** About this document ...
** Up:** Fomel: Linearized Eikonal
** Previous:** A SIMPLE derivation of
Stanford Exploration Project

11/11/1997