Elastic full-waveform inversion serves as a fundamental technique for recovering subsurface elastic parameters. However, its application is hindered by mode coupling between P- and S-waves, multi-parameter crosstalk, and a strong dependence on the initial model, all of which degrade the accuracy of velocity model building at depth. The conventional inversion methods employing P-wave sources rely on PP and PS reflections, yet the PS converted wave typically exhibits low energy, making it difficult to jointly construct accurate initial P- and S-wave velocity models. To address this limitation, the present work introduces a traveltime-based inversion framework for P- and S-wave velocities built upon the S-wave source wave equation. First, a curl source function is designed to simulate pure shear-wave source excitation. Second, high-fidelity P- and S-wave mode separation is accomplished through vector decomposition combined with a wavenumber-domain correction for staggered-grid spatial migration. Subsequently, a traveltime minimization objective functional for SS and SP reflected waves is formulated, and the adjoint equations together with the corresponding velocity gradient expressions for P- and S-wave velocities are derived using the Born approximation and the adjoint-state method. The traveltime sensitivity kernels of SS and SP reflections are then extracted via wavefield separation, and the cross-correlation interference among multiple wave modes is decomposed. A stepwise inversion strategy is employed: S-wave velocity is first inverted from SS reflections, after which P-wave velocity is inverted from SP reflections; finally, the two velocity models are jointly refined to mitigate parameter crosstalk. Synthetic tests demonstrate that the proposed method effectively separates the SS and SP traveltime sensitivity kernels, which exhibit smooth shapes and favorable symmetry. Unlike conventional full-waveform inversion, which is prone to being trapped in local minima, this approach can accurately retrieve the long-wavelength components of both P- and S-wave velocities, thereby providing a high-quality initial model for subsequent full-waveform inversion and significantly improving the migration imaging accuracy of multicomponent seismic data.