The advantage of approximating the dispersion relation is that we do not need to use special tools to estimate the parameters, but merely the same velocity estimation and imaging techniques which have proved their usefulness in conventional scalar processing. The four parameters in two dimensions are horizontal and vertical normal moveout and the direct propagation velocities. Using conventional velocity-analysis tools paraxially, NMO velocities can be solved for or direct-wave velocities if the data coverage is sufficient. A surface seismic experiment, a cross-well experiment, and a VSP used together (each having limited aperture) would give the following information:

- 1.
- horizontal moveout velocity (surface seismic).
- 2.
- vertical direct-wave velocity (VSP).
- 3.
- horizontal direct-wave velocity (cross-well).
- 4.
- vertical moveout velocity (cross-well).

It must be kept in mind that there exist scale differences between the surface seismic, cross-well and VSP experiment types. The most obvious difference is in the source frequency content, which can vary as much as several orders of magnitude, depending on whether the source is placed at the surface or in the well. Spatial and temporal sampling varies from experiment to experiment, but typically in surface experiments frequencies are low, in cross-well experiments frequencies are high, and in VSPs frequencies are somewhere in between. Scale differences must be equalized in a ``consistent manner'' before the combined measurement values can be used. After scale equalization, conventional velocity analysis can be applied giving an elliptic parameter estimate. The parameters can be estimated either all together in one pass or separately. Approximation does not make it necessary to estimate the parameters either separately or together; and both ways have their advantages. The elliptic approximation does not require any particular anisotropic symmetry model. It rather serves as a basis for a least-squares estimate of elastic stiffness constants.

12/18/1997