Elastic reflection waveform inversion (ERWI) utilizes reflections to update the low and intermediate wavenumbers in the deeper part of elastic models and can provide good initial models for elastic full waveform inversion (EFWI). Although ERWI aims to mitigate the nonlinearity of inversion when starting from a poor initial model, it suffers from the cycle-skipping problem due to the objective function of waveform fitting. Building initial P- and S-wave velocity models for EFWI through elastic wave-equation reflection traveltime inversion (ERTI) would be effective and robust since traveltime information relates to the background model more linearly. However, the current implementations of acoustic traveltime inversion is not straightforward in elastic media due to the existence of S-wavefields. Wave mode decomposition, both on the recording surface and in the extrapolated wavefields, is important for ERTI. First, for seismic data with P-wave sources, the P/S separation of multicomponent seismograms isolates the PP and PS reflection events and thus make it possible to extract the event-to-event time-shifts of these isolated reflections through dynamic image warping (DIW). Then, we can use the traveltime residuals of PP and PS reflections to build the objective function for ERTI. Second, based on the investigation of the complicated reflection kernels in an elastic medium, we demonstrate the necessity of wave mode decomposition applied on the extrapolated elastic wavefields, to suppress the artefacts induced by the undesirable cross-correlations of the components in forward and back-propagated wavefields. Therefore, the decomposition of surface recording data and extrapolated wavefields guarantees the dominate contribution of the traveltime is included during the ERTI. Accordingly, we propose a two-stage method to first build the P-wave background velocity using the separated PP reflections and then build the S-wave background velocity using the separated PS reflections based on the well-recovered P-wave velocity model. A numerical example of the Sigsbee2A model shows the effectiveness of the proposed ERTI approach.