On the nonuniqueness of traveltime inversion in elliptically anisotropic media

Elliptical velocity dependence on the direction of wave propagation is the simplest type of anisotropy that might be encountered in the subsurface. When the symmetry axis is vertical, describing elliptical anisotropy (EA) requires only one parameter — the ellipticity coefficient — in addition to the conventional isotropic velocity. It is tempting, therefore, to use elliptical anisotropy as a testing ground for various anisotropic parameter-estimation techniques. This apparently straightforward thinking, however, fails to deliver satisfactory results because traveltime inversion in certain EA media is known to be nonunique. Here, I prove this nonuniqueness to be a general property of all EA media and explicitly show that a constant depth stretch generates a family of kinematically equivalent EA models for any given spatial distribution of the ellipticity coefficient and the velocity.