The outermost part of the inner core is isotropic, though the thickness of this layer varies longitudinally, commonly interpreted to be a zone where compaction of the crystallites randomly adhering to the inner core boundary eventually aligns them. We explore the possibility that the surface layer is also anisotropic, but of a form that thwarts detection by travel time observation. If the surface material is hexagonally anisotropic with a vertical symmetry axis (vertically oriented transverse isotropy), P wave travel times through the shallow inner core will not depend on their angle to the Earth's rotation axis. To detect any anisotropy in this layer, we explore PKiKP amplitudes versus distance to determine whether there is a range trend diagnostic of anisotropy. P wave anisotropy levels of 2–10% can increase reflection amplitudes by a factor of 10. The trend of PKiKP amplitude with range may be fit with vertically oriented transverse isotropy anisotropy levels of 5–10%.