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SUMMARY Long-period (T > 10 s) shear wave reflections between the surface and reflecting boundaries below seismic stations are useful for studying phase transitions in the mantle transition zone (MTZ) but shear-velocity heterogeneity and finite-frequency effects complicate the interpretation of waveform stacks. We follow up on a recent study by Shearer & Buehler (hereafter SB19) of the top-side shear wave reflection Ssds as a probe for mapping the depths of the 410-km and 660-km discontinuities beneath the USArray. Like SB19, we observe that the recorded Ss410s-S and Ss660s-S traveltime differences are longer at stations in the western United States than in the central-eastern United States. The 410-km and 660-km discontinuities are about 40–50 km deeper beneath the western United States than the central-eastern United States if Ss410s-S and Ss660s-S traveltime differences are transformed to depth using a common-reflection point (CRP) mapping approach based on a 1-D seismic model (PREM in our case). However, the east-to-west deepening of the MTZ disappears in the CRP image if we account for 3-D shear wave velocity variations in the mantle according to global tomography. In addition, from spectral-element method synthetics, we find that ray theory overpredicts the traveltime delays of the reverberations. Undulations of the 410-km and 660-km discontinuities are underestimated when their wavelengths are smaller than the Fresnel zones of the wave reverberations in the MTZ. Therefore, modelling of layering in the upper mantle must be based on 3-D reference structures and accurate calculations of reverberation traveltimes. 

SUMMARY A number of seismological studies have indicated that the ratio R of S-wave and P-wave velocity perturbations increases to 3–4 in the lower mantle with the highest values in the large low-velocity provinces (LLVPs) beneath Africa and the central Pacific. Traveltime constraints on R are based primarily on ray-theoretical modelling of delay times of P waves (ΔTP) and S waves (ΔTS), even for measurements derived from long-period waveforms and core-diffracted waves for which ray theory (RT) is deemed inaccurate. Along with a published set of traveltime delays, we compare predicted values of ΔTP, ΔTS, and the ΔTS/ΔTP ratio for RT and finite-frequency (FF) theory to determine the resolvability of R in the lower mantle. We determine the FF predictions of ΔTP and ΔTS using cross-correlation methods applied to spectral-element method waveforms, analogous to the analysis of recorded waveforms, and by integration using FF sensitivity kernels. Our calculations indicate that RT and FF predict a similar variation of the ΔTS/ΔTP ratio when R increases linearly with depth in the mantle. However, variations of R in relatively thin layers (< 400 km) are poorly resolved using long-period data (T > 20 s). This is because FF predicts that ΔTP and ΔTS vary smoothly with epicentral distance even when vertical P-wave and S-wave gradients change abruptly. Our waveform simulations also show that the estimate of R for the Pacific LLVP is strongly affected by velocity structure shallower in the mantle. If R increases with depth in the mantle, which appears to be a robust inference, the acceleration of P waves in the lithosphere beneath eastern North America and the high-velocity Farallon anomaly negates the P-wave deceleration in the LLVP. This results in a ΔTP of about 0, whereas ΔTS is positive. Consequently, the recorded high ΔTS/ΔTP for events in the southwest Pacific and stations in North America may be misinterpreted as an anomalously high R for the Pacific LLVP.