S U M M A R YCurrent models of upper-mantle discontinuity topography derived from traveltime variations of long-period underside reflections (PdP or S d S waves) may contain gross errors due to the use of geometrical optics in relating observed traveltimes to depth variations of the discontinuity near the reflection point. The use of geometrical optics is not valid if variations in the depth of the discontinuities exist on a lateral scale smaller than the size of the Fresnel zone in the data. Geometrical optics does not take into account the large size and the complex structure of the traveltime surface of underside reflections and may introduce spurious structure when used in inversions. Examples of synthetic long-period PdP waveforms for reflections near subducting lithosphere show that scattering from expected small-scale depth variations of the discontinuities inside slabs causes complex waveform variations. For a 15-20 s PdP wave, these small-scale depth variations (with a scale-length of 4") are smaller than the Fresnel zone (roughly 20") but larger than the P d P wavelength at 420 km depth (about 150-200 km). The synthetic waveforms are processed to obtain PdP traveltimes, which are in turn converted to apparent discontinuity depth at the reflection point using geometrical optics. The apparent discontinuity-depth variations are not only of the same order of magnitude as those obtained from real data but also exhibit artefacts that closely resemble features observed in current models. It is shown that large-scale (wavelengths of the order of 1000 km) downwarping of the 670-km discontinuity near subduction zones derived from underside reflection data can be explained as the longwavelength manifestation of a single, small-scale (wavelengths of several hundreds of kilometres) deflection of the discontinuity inside and near subducting slabs.