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The expression for the ray velocity in a medium with elliptical velocity dependency is given by the expression

(9) |

When the axis of symmetry is rotated an angle (Figure A-1-b), the expression for the ray velocity becomes (for the same ray direction):

(10) |

Figure 11

where is the angle from the axis of symmetry to the ray (group velocity angle).

If the ray travels a distance *d*
between two points (Figure 1),

(11) |

(12) | ||

To further simplify this equation we need to know the values
of and . In order to do this, we
need to be careful about the sign of (clockwise
or counterclockwise) for the
given ray direction. We also need to be careful about the sign of
and .It turns out that regardless of how the signs of these quantities
are defined (as long as they are consistent)
the final
expression for *t ^{2}* is always, as expected, the same. The result
is

(13) |

In this appendix, I show the expressions for the partial
derivatives of the traveltime *t*_{i}
(equation (5))
with respect to the model parameters *m*_{k}, where *m*_{k}
is a component of the vector ** m**:

First, the derivatives with respect to the interval parameters , and ():

The derivatives of the traveltime with respect to
the boundary parameters (** a_{j}** and

When a ray travels horizontally, only the first
** N** components of the vector of partial derivatives
are non zero. For the forward modeling
this is not problem and it
simply means that the horizontal
component of the velocity is the only parameter that affects the traveltime
of a horizontally traveling ray.
Problems arise, however, when we try to
invert these traveltimes because
there are infinite combinations of the other parameters that
satisfy the data equally well (null space).
This translates into instability
of the inversion procedure. For these reason, this
inversion
does not use rays that travel exactly along the horizontal.

After making the appropriate simplifications in the above equations for the case of isotropic media, it results a set of equations similar to the ones obtained by Lee (1990). Note two misprints in Lee's equations.

11/18/1997