How are vertical shear wave splitting measurements affected by variations in the orientation of azimuthal anisotropy with depth?

Splitting measurements of teleseismic shear waves, such as SKS, have been used to estimate the amount and direction of upper mantle anisotropy worldwide. These measurements are usually made by approximating the anisotropic regions as a single, homogeneous layer and searching for an apparent fast direction (<(phi)over tilde>) and an apparent splitting time (Delta (t) over tilde) by minimizing the energy on the transverse component of the back-projected seismogram. In this paper, we examine the validity of this assumption. In particular, we use synthetic seismograms to explore how a vertically varying anisotropic medium affects shear wave splitting measurements. We find that weak heterogeneity causes observable effects, such as frequency dependence of the apparent splitting parameters. These variations can be used, in principle, to map out the vertical variations in anisotropy with depth through the use of Frechet kernels, which we derive using perturbation theory. In addition, we find that measurements made in typical frequency bands produce an apparent orientation direction that is consistently different from the average of the medium and weighted towards the orientation of the anisotropy in the upper portions of the model. This tendency of the measurements to mimic the anisotropy at the top part of the medium may explain why shear wave splitting measurements tend to be correlated with surface geology. When the heterogeneity becomes stronger, multiple scattering reduces the amplitude of the tangential-component seismogram and the associated splitting time, so that a null result may be obtained despite the fact that the waves have travelled through a strongly anisotropic medium. Regardless of the amount of vertical heterogeneity, we find that there is very little dependence on backazimuth for the measured fast-axis direction or splitting time if the top and bottom halves of the medium average to similar fast-axis directions. If, however, the average fast-axis direction in the top half of the model differs from that in the bottom half, splitting-time measurements will show a significant dependence on backazimuth, but fast-axis direction measurements will remain relatively constant.