Which is the axis of symmetry?

Every ellipse has two axes of symmetry. The inversion proposed in this paper can estimate the inclination with respect to the vertical of either one of them depending on which axis is closer to the initial model. In either case, the estimated parameters will always describe the same elliptical velocity function.

Another way to understand why the inversion can estimate the inclination of either axis of symmetry is by examining the traveltime equation (1). An identical equation can be obtained if we switch in (1) $V_{\parallel}$ and $V_{\perp}$ changing at the same time $\gamma$ by $\gamma \pm \pi / 2$. This means that the traveltimes are affected only by the elliptical function of velocities regardless of how such a function is described. Figure  shows how a given ellipse can be described with two different sets of parameters.

If the inversion procedure is forced to estimate the same axis of the ellipse (either the major or the minor) for every layer, it can be more difficult (or impossible) to get a reliable estimate of the real elliptical velocity function of the medium, especially in cases when the axis of symmetry in the initial model is far from the true answer.

 

two-sets
Figure 3 The same ellipse represented in two different ways by interchanging $V_{\perp}$ and $V_{\parallel}$ and rotating the axis of symmetry $-\pi/2$. Traveltimes are affected by the velocity function (elliptical in this case) regardless of how that function is described.