Elastic full waveform inversion (FWI) attempts to deduce accurate subsurface elastic parameters from multi-component seismic data. However, the inversion accuracy of elastic FWI degrades due to the crosstalk effect caused by coupled elastic wavefields. Decomposing coupled elastic wavefields into pure P- and S-waves is an effective method to suppress the crosstalk noises. Wavefields decomposition method plays a crucial role in wave mode decomposition-based elastic FWI method. Considering trade-off between calculation accuracy and efficiency, we have developed a new elastic FWI method based on decoupled wave equations to improve the inversion accuracy of elastic FWI. New decoupled adjoint state wave equations are proposed to obtain adjoint P- and S-waves. The wave mode decomposition-based gradient equations are further used to update the P- and S-wave velocity models. The synthetic examples reveal that our new elastic FWI method can obtain higher resolution models than conventional elastic FWI method. In addition, our new elastic FWI based on decoupled wave equations does not significantly increase the calculation amount but can speed up the convergence of the elastic FWI.