Backus-Gilbert inversions of a new differential PcP data set result in estimates of the core radius, 3484-3486 km, compatible with those obtained from the inversion of large sets of normal mode data. For that part of the core sampled by the data, undulations in the core-mantle boundary can be of order no more than 5-10 km, in the worst possible case.Signal misidentification and lateral inhomogeneities in the upper mantle must be accounted for in observed short-period differential PcP travel times. In the case of Amchitka data, seismic ray tracing is used to describe the effects of a dipping lithospheric plate beneath the Island.