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The traveltime for a ray that travels a distance d in an homogeneous medium with elliptical anisotropy and axis of symmetry forming an angle $\gamma$with respect to the vertical (Figure [*]) is
t \ =\ \sqrt{
\frac{(\Delta x \cos \gamma + \Delta z \sin \g...
 ... \cos \gamma + \Delta x \sin \gamma )^2}
},\end{displaymath} (1)
where $\sqrt{\Delta x^2 + \Delta z^2} \ =\ d$ and $V_{\parallel}$ and $V_{\perp}$ are the velocities in the directions parallel and perpendicular respectively to the axis of symmetry. In Appendix A, I explain how to derive this expression.

Figure 1
Ray traveling a distance d in a medium with tilted axis of symmetry. $\alpha$ and $\gamma$ are the angles of the ray and the axis of symmetry respectively with respect to the vertical.

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
t_{i,j} \ =\ \sqrt{
\frac{(\Delta x_{i,j} \cos \gamma_j +
 ...mma_j + 
\Delta x_{i,j} \sin \gamma_j)^2}{V_{\parallel_j}^2}.
}\end{displaymath} (2)
Note that each homogeneous region is characterized by three parameters: two velocities and the angle of the axis of symmetry with respect to the vertical. From now on I will refer to this parameters as interval parameters. In the previous equation, $\sqrt {\Delta x_{i,j}^2 + \Delta z_{i,j}^2}$ is the distance traveled by the ith ray in the jth cell. The sum of expressions like equation (2) can be used to compute the traveltime from source to receiver for a ray that travels in an heterogeneous media, assuming the ray path is known. Byun (1982) and Michelena (1992) explain how to do the ray tracing.

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 $\Delta x_{i,j}$ and $\Delta z_{i,j}$ are defined as
\Delta x_{i,j} & = & x_{i,j+1} - x_{i,j} \\  
\Delta z_{i,j} & = & z_{i,j+1} - z_{i,j} \end{eqnarray} (1)

the expression (2) for the traveltime ti,j of the ith ray in the jth cell becomes
t_{i,j} \ =\ \sqrt{ {\Delta X_{i,j}}^2 S_{\perp_j}^2 + 
{\Delta Z_{i,j}}^2 S_{\parallel_j}},\end{displaymath} (4)
where $S_{\perp_j}$, $S_{\parallel_j}$, $\Delta X_{i,j}$ and $\Delta Z_{i,j}$ are equal to
S_{\perp_j} & = & \frac{1}{V_{\perp_j}}, \nonumber \\ S_{\parallel_j} & = & \frac{1}{V_{\parallel_j}}, \nonumber\end{eqnarray}

\Delta X_{i,j} \ =\ \Delta x_{i,j} \cos \gamma_j + 
(a_{j+1} x_{i,j+1} + b_{j+1} - a_{j} x_{i,j} - b_{j}) \sin \gamma_j,\end{displaymath}

\Delta Z_{i,j} \ =\ 
- (a_{j+1} x_{i,j+1} + b_{j+1} - a_{j} x_{i,j} - b_{j}) \cos \gamma_j +
\Delta x_{i,j} \sin \gamma_j.\end{displaymath}

(xi,j, zi,j) is the point of intersection between the ith ray and the jth interface. If the axis of symmetry is vertical ($\gamma_j = 0$), it follows that $\Delta X_{i,j} = \Delta x_{i,j}$ and $\Delta Z_{i,j} = \Delta z_{i,j}$.

Figure 2
Model of velocities and heterogeneities. The top and bottom interfaces are horizontal (a1 = aN+1 = 0) and located at known depths.

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
t_i( \mbox{\boldmath$m$}) \ =\ \sum_{j=1}^{N} t_{i,j} ( \mbox{\boldmath$m$})
\hspace{.5in} i=1,...,M,\end{displaymath} (5)
where $\mbox{\boldmath$m$}$ is the vector of model parameters of 5N elements:
\mbox{\boldmath$m$} & = & (m_1,...,m_N,m_{N+1},...,m_{2N},m_{2N...
and M is the total number of traveltimes.

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 $\mbox{\boldmath$m$}$ equal to 5N - 2.

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Next: INVERSE MODELING Up: Michelena: Anisotropic tomography Previous: Introduction
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