VELOCITY PERTURBATIONS

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 $\partial \vec{ \bf x}(r) / \partial r$ points tangent to a given point on a raypath. Since $\phi$ is the angle of a raypath from the vertical axis of symmetry $\vec{ \bf \hat z}$, we can calculate the angle in the group velocity (4) as a function of the position along the raypath:

$$\phi(r) = \arccos \{ [\partial \vec{ \bf x}(r) / \partial ... ...vec{ \bf x} \Vert _x^2 \equiv \vec{ \bf x} \cdot \vec{ \bf x} .$$ (8)

For this reason, we can write the parameterized group velocity $V [ \vec{ \bf x}(r) , \phi ]$ in (4) as a function of the location and a tangent vector (with arbitrary magnitude).

A given raypath $\vec{ \bf x}(r)$, for r from 0 to 1, integrates for the traveltime  

$$t = \int_0^1 V [ \vec{ \bf x}(r) , \phi(r) ]^{-1} \Vert \partial \vec{ \bf x}(r) / \partial r \Vert _x dr .$$ (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:  

$$\Delta t = \int_0^1 \Delta \{ V [ \vec{ \bf x}(r) , \phi(r)... ...-1} \} \Vert \partial \vec{ \bf x}(r) / \partial r \Vert _x dr.$$ (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.