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Here I show that traveltime data are easily linearized
as a function of the parameters describing the velocity
basis functions.

The functional form of an anisotropic group velocity must
depend on the location and on the direction of a raypath.
The vector points
tangent to a given point on a raypath.
Since is the angle of a raypath from the vertical axis
of symmetry , we can calculate
the angle in the group velocity (4) as
a function of the position along the raypath:

| |
(8) |

For this reason, we can write the parameterized group velocity
in (4)
as a function of the location and a tangent vector (with arbitrary magnitude).
A given raypath , for *r* from 0 to 1, integrates
for the traveltime

| |
(9) |

Because the raypath represents a stationary minimum, tomography recognizes
that perturbation of a valid raypath affects
traveltime only to second order. To perturb traveltimes linearly
with finite perturbations of slowness along a path, we need only
integrate the slowness perturbations along the original path:
| |
(10) |

The perturbation of slowness is given by
the parameterization in (5).
This formulation provides the linearized perturbation
of the traveltime data as a function of perturbed velocity parameters.
The adjoint uses the same weights
for a backprojection of traveltime perturbations upon the
velocity parameters.

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** Up:** Harlan: Flexible tomography
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Stanford Exploration Project

11/12/1997