The traveltime for a ray that travels a distance d in an homogeneous medium with elliptical anisotropy and axis of symmetry forming an angle with respect to the vertical (Figure ) is
If the model is described as a superposition of N homogeneous orthogonal regions, the traveltime for the ith ray traveling across the jth region is
Figure shows the type of model that I will consider in this paper. It consists of homogeneous elliptically anisotropic blocks separated by straight interfaces of variable dip (aj) and intercept (bj). I assume that all the axes of symmetry for the different layers lie in the same plane of the survey geometry. If and are defined as
the expression (2) for the traveltime ti,j of the ith ray in the jth cell becomes
(xi,j, zi,j) is the point of intersection between the ith ray and the jth interface. If the axis of symmetry is vertical (), it follows that and .
Besides the interval parameters previously described, I have added two more parameters to describe how the boundaries that separate different intervals may change their positions. I call these parameters boundary parameters. Figure shows how to count both intervals and boundaries.
The total traveltime for a ray that travels from source to receiver is
Equation (5) is the system of nonlinear equations that relates the model parameters with the measured traveltimes. A linearized version of this equations will be used in the next section to solve the inverse problem. As explained in Figure , b1 and a1 are known. This makes the number of variables in equal to 5N - 2.