Most published shear-wave (VS) velocity models of cratons include a VS increase with depth below the Moho, with a maximum at 100-150 km depth. This feature is seen in regional and global 3D tomography models and in regional 1D VS profiles. Taken at face value, it implies an oscillatory geotherm, with a ubiquitous temperature decrease below the Moho, which is implausible. The VS increase with depth has thus been attributed to strong compositional layering in the lithosphere. One recent model postulated widespread hydration and metasomatism in the uppermost cratonic mantle, decreasing VS just below the Moho. An alternative model suggested a strong enrichment of the lower cratonic lithosphere in eclogite and diamond, increasing VS but implying an unusual lithospheric composition. Here, we assemble a representative dataset of phase-velocity curves of Rayleigh and Love surface waves for cratons globally, including the all-craton averages, averages over regions in southern Africa, and interstation measurements elsewhere. We perform both thermodynamic and purely seismic inversions and show that the sub-Moho VS increase is not required by the data. Models with equilibrium, conductive lithospheric geotherms and ordinary, depleted-peridotite compositions fit the surface-wave data fully. A model-space mapping quantifies the strong trade-off between seismic velocities just below the Moho and at 100-150 km depth, which is the cause of the ambiguity. The reason why most seismic models contain a VS increase with depth below the Moho is regularization that penalizes deviations from global average reference models, which are much slower than cratonic VS profiles.

Surface waves propagating from earthquakes, active sources or within the ambient noise wavefield are widely used to image Earth structure at various scales, from centimeters to hundreds of kilometers. The accuracy of surface-wave, phase-velocity measurements is essential for the accuracy of the Earth models they constrain. Here, we identify a finite-frequency phase shift in the phase travel time that causes systematic errors in time-domain, phase-velocity measurements. The phase shift arises from the approximation of monochromatic surface waves with narrow-band filtered surface waves. We derive an explicit formula of the finite-frequency phase shift and present a numerical method for its evaluation and for the correction of the measurements. Applications to high-frequency and long-period examples show that the phase shift is typically around π/60-π/16 for the common settings of ambient-noise imaging studies, which translates to 0.2-0.8% phase-velocity measurement errors. The finite-frequency phase shift depends on the (1) second derivative of the wavenumber with respect to frequency; (2) width of the narrow-band filter; (3) epicentral or interstation distance; (4) center frequency of the filter. In conversion to phase velocity, the last two factors cancel out. Frequency-domain methods for phase-velocity measurements have the advantage of not producing the finite-frequency phase shift. Both time- and frequency-domain measurements, however, can be impacted by a break-down of the far-field approximation (near-field phase shift), which our calculations also show. Our method offers an effective means of improving the accuracy of the widely used time-domain, phase-velocity measurements via the evaluation of and corrections for the finite-frequency phase shift.