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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 n1 along the
ray , defined by the initial slowness model
n0 . 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.
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Stanford Exploration Project
11/11/1997