Accurate estimation of source wavelet is crucial in a successful full-waveform inversion (FWI); however, it cannot be guaranteed in the case of real seismic data. We have developed time-domain source-independent elastic FWI using the convolution-based objective function that was originally developed for acoustic FWI. We have applied a new time window on the reference traces in the objective function to suppress the noises induced by the convolution and crosscorrelation operations. Also, we have adopted -, Huber-, and hybrid-norm objective functions to improve the antinoise ability of our algorithm. We implemented a multiscale inversion strategy to conduct the tests with a quasi-Newton limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method to reduce the sensitivity to initial models and to improve the quality of inversion results. Synthetic tests verified that the new added time window can not only improve the inversion results, but also accelerate the convergence rate. Our method can be implemented successfully without a priori knowledge or accurate estimation of the source wavelet and can be more robust to Gaussian and spike noises, even for a Dirac wavelet. Finally, we applied our method to real seismic data. The similarity between the observed and modeled seismic data, the higher resolution of the migration image, and flatter common image gathers corresponding to the inverted models proved the relevance of our algorithm.