Isotropic tomography fits circles (t2 = (x2 + z2) S2) to the data. Anisotropic tomography fits ellipses (t2 = x2 Sx2 + z2 Sz2). Depending on the range of ray angles available (or the geometry used) and the wave type under consideration, the estimated slownesses Sx and Sz may or may not correspond to the real slownesses of the medium.
Horizontal or near horizontal rays are typical of a cross-well geometry. These rays sample a portion of the slowness surface close to the horizontal and for this reason the estimated Sx corresponds to the real horizontal slowness. On the contrary, Sz is not sampled by cross-well geometries. The inversion gives the vertical slowness of the best fitting ellipse (Sznmo) which coincides with the real vertical slowness only if the wave type considered is SH. However, we can generally expect Sznmo to be closer than S to the real vertical slowness. This will be illustrated later with field data by comparing S, Sx and Sznmo with sonic logs.
Vertical or near vertical rays are typical of a VSP-like geometry. From this type of geometry we can get Sxnmo and Sz. These two slownesses plus Sx and Sznmo obtained from cross-well geometries can be used to estimate the real slowness surface of the medium using the following approximate expression (Muir, 1990):
This expression is called the double elliptic approximation. Dellinger and Muir (1991) demonstrate that this approximation accurately fits general transversely isotropic media.
In the cross-well geometry examples that follow, the estimated vertical slowness will be referred to as Sz rather than Sznmo.