THE HOMOGENEOUS INVERSION SCHEME

For elliptically anisotropic media, that is, media in which the group velocity V has an elliptical dependence on the angle of energy propagation $\phi$, such that  

$${(V \cos \phi)^2 \over V_x} + {(V \sin \phi)^2 \over V_z} = 1,$$ (1)

the relation between group and phase velocities and angles is given by a closed analytical form (Levin, 1978):

$$\begin{eqnarray} {v(\theta) \over V(\phi)} & = & \cos(\phi-\theta) \\ \tan(\phi) & = & \left({V_z \over V_x}\right)^2 \tan(\theta)\end{eqnarray}$$ (2) (3)

where v and $\theta$ are the phase velocity and angle, respectively. Equation ( 1) also represents an elliptical dispersion relation, since $V \! \sin\phi = K_x / \! \omega$ and $V \! \cos\phi = K_z / \! \omega$.It turns out that elasticity theory predicts this elliptical dependence only for S_H in a transverse isotropic symmetry, or for all wave types in the trivial case of isotropic material where the ellipse becomes a circle. However, the use of an elliptical approximation for the dispersion relation can considerably simplify the problem of estimating the elastic parameters from the recorded wavefields.

If we consider a source and a receiver separated by a vertical distance z and a horizontal distance x, the traveltime t predicted by equation (1) is  

$$t^2 = M^x \: x^2 + M^z \: z^2.$$ (4)

The relation between the slownesses squared (or sloth) M^x and M^z that best fits our data and the true vertical and horizontal sloths of the medium M^v and M^h respectively will depend on the specific geometry in which the data was collected. For an usual surface-seismic geometry, equation (4) correspond to a paraxial approximation around the vertical axis. In this case, M^x corresponds to the estimated normal-moveout (NMO) sloth, while $M^z \, z^2$ is the vertical traveltime. If z is unknown, then M^z is an arbitrary factor that cannot be estimated (Dellinger and Muir, 1985).

If both distances, x and z, are known it is possible to estimate both sloth parameters by fitting the data to equation (4). For surface seismic surveys the estimated M^z will be close to the vertical sloth M^v, whereas for a cross-well geometry the estimated M^x will be close to the horizontal sloth M^h. Karrenbach (1989) described a scheme to estimate these two sloth parameters using traveltime data from the three wave types, for the case of a homogeneous medium. He also derived the relations between these parameters and the elastic constants corresponding to a transversely isotropic symmetry, for the specific case of a cross-well geometry.