REFERENCES

• Byun, B. S., 1982, Seismic parameters for media with elliptical velocity dependencies: Geophysics, 47, 1621-1626.

• Chapman, C. H. and Pratt, R. G., 1992, Traveltime tomography in anisotropic media: Theory: Geophys. J. Intl., 109, 1-19.

• Dellinger, J. and Muir, F., 1991, The double elliptic approximation in the group and phase domains: SEP-70, 361-366.

• Gill, P. H., Murray, W., and Wright, M. H., 1981, Practical Optimization: Academic Press.

• Harlan, W. S., 1989, Tomographic estimation of seismic velocities from reflected raypaths: presented at the 59th Ann. Internat. Mtg., Soc. Expl. Geophys.

• Harris, J., M., 1988, Cross-well seismic measurements in sedimentary rocks: presented at the 58th Ann. Internat, Mtg., Soc. Expl. Geophys.

• Lazaratos, S. K., Rector, J. W., Harris, J. M. and Van Schaack, M., 1991, High resolution imaging with cross-well reflection data: presented at the 61st Ann. Internat. Mtg., Soc. Expl. Geophys.

• Lee, M.,W.,1990, Traveltime inversion using transmitted waves of offset vertical seismic profiling data: Geophysics, 55, 1089-1097.

• McMechan, G.A., 1983, Seismic tomography in boreholes: Geophys. J. Roy. Astr. Soc., 74, 601-612.

• Michelena, R. J., 1990, Traveltime inversion with constraints: STP-1, paper H.

• Michelena, R. J., 1992, Kinematic ray tracing in anisotropic layered media: SEP-73.

• Michelena, R. J., Muir, F. and Harris, J. M., 1992, Anisotropic traveltime tomography: submitted for publication in Geophysical Prospecting.

• Miller, D. E., and Chapman, C. H., 1991, Incontrovertible evidence of anisotropy in cross-well data: presented at the 61st Ann. Internat. Mtg., Soc. Expl. Geophys.

• Muir, F., 1990, A modified anisotropic system: SEP-67, 41-42.

• Nolet, G., 1987, Seismic wave propagation and seismic tomography, in Nolet, G., Ed., Seismic tomography: D. Reidel Publ. Co., 1-23.

• Onishi, M., and Harris, J., M., 1991, Anisotropy from head waves in cross-well data: presented at the 61st Ann. Internat. Mtg., Soc. Expl. Geophys.

• Pratt, R. G., Chapman, C. H., 1992, Traveltime tomography in anisotropic media: application: Geophys. J. Intl.,109, 20-37.

• Raikes, S., 1991, Shear-wave characterization of the BP Devine test site, Texas: presented at the 61st Ann. Internat. Mtg., Soc. Expl. Geophys.

• Van Trier, J., 1988, Migration velocity analysis using geological constraints: presented at the 58th Ann. Internat. Mtg., Soc. Expl. Geophys.

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

$$\frac{1}{V^2 (\alpha)} \ =\ \frac{\cos^2 \alpha }{V_{\parallel}^{2}} + \frac{\sin^2 \alpha }{V_{\perp}^{2}},$$ (9)

where $\alpha$ is the ray angle measured from the axis of symmetry (positive counterclockwise) and $V_{\parallel}$ and $V_{\perp}$ are the velocities in the directions parallel and perpendicular to the axis of symmetry (Figure A-1-a). When the axis of symmetry is vertical, the angle that measures the direction of propagation of the ray with respect to the vertical is the same as the group velocity angle.

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

$$\frac{1}{V^2 (\alpha)} \ =\ \frac{\cos^2(\alpha - \gamma)}{... ...parallel}^{2}} + \frac{\sin^2(\alpha - \gamma)}{V_{\perp}^{2}}$$ (10)

 

tilted-ellipses
Figure 11 Ray velocity as a function of direction in an elliptically anisotropic medium: (a) Axis of symmetry vertical (ray angle = group velocity angle). (b) Axis of symmetry tilted (ray angle = $\alpha$; group velocity angle = $\phi$).

where $\alpha - \gamma$ 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),

$$d \ =\ \sqrt{\Delta x^2 + \Delta z^2}$$ (11)

the corresponding traveltime t is

$$\begin{eqnarray} t^2 & = & \frac{(d \cos(\alpha - \gamma))^2}{V_{\parallel}^{2}... ...alpha \cos \gamma - d \cos \alpha \sin \gamma )^2} {V_{\perp}^{2}}\end{eqnarray}$$ (12)

To further simplify this equation we need to know the values of $d \cos \alpha$ and $d \sin \alpha$. In order to do this, we need to be careful about the sign of $\alpha$ (clockwise or counterclockwise) for the given ray direction. We also need to be careful about the sign of $\Delta x$ and $\Delta z$.It turns out that regardless of how the signs of these quantities are defined (as long as they are consistent) the final expression for t² is always, as expected, the same. The result is

$$t^2 \ =\ \frac{( -\Delta z \cos \gamma + \Delta x \sin \gam... ...elta x \cos \gamma + \Delta z \sin \gamma )^2} {V_{\perp}^{2}}.$$ (13)

This is the expression for the traveltime of a ray that travels between two points separated by a distance d in an homogeneous elliptically anisotropic medium with axis of symmetry forming an angle $\gamma$ with the vertical. This equation is the heart of the inversion procedure proposed in this paper.

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:

$$\begin{eqnarray} \mbox{\boldmath$m$} & = & (m_1,...,m_N,m_{N+1},...,m_{2N},m_{2N... ...lel_N}, \gamma_1,...,\gamma_N, b_1,...,b_N,a_1,...,a_N). \nonumber\end{eqnarray}$$

First, the derivatives with respect to the interval parameters $S_{\perp_j}$, $S_{\parallel_j}$ and $\gamma_j$ ($1 \le j \le N$):

$$\begin{eqnarray} \frac{ \partial t_{i} }{\partial m_k} & = & \left\{ \begin{arra... ...}} & \mbox{if $2N+1 \leq k \leq 3N$}\end{array} \right. \nonumber\end{eqnarray}$$

Note that for the interval parameters the derivatives with respect to the j^th variable depend only upon the properties of the j^th layer.

The derivatives of the traveltime with respect to the boundary parameters (a_j and b_j), depend on the properties of the medium above and below that boundary, as follows:

$$\begin{eqnarray} \frac{\partial t_i}{\partial m_k} & = & \left\{ \begin{array} ... ... & \mbox{if $4N+2 \leq k \leq 5N$} \\ \end{array}\right. \nonumber\end{eqnarray}$$

Since the traveltime t_i is not affected by the position of the top boundary, $\frac{\partial t_i}{\partial a_1} \ =\ \frac{\partial t_i}{\partial b_1} \ =\ 0$.

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.