Elastic frequency-domain full-waveform inversion is here applied to a realistic synthetic offshore model to study the behavior of the least-squares and least-absolute-value norms, in the presence of noisy multi-component seis- mic data. As expected, the inversion results show that the L2 norm is highly sensitive to non-Gaussian errors in the data and gives rise to high amplitude artifacts in the reconstructed models. The L1 norm shows more robust be- havior, whatever the noise characteristics, and allows convergence towards acceptable models, for both the compressional-wave and shear-wave velocity models. This study highlights the sensitivity of the full-waveform inversion results for the chosen norm in the case of noisy data. The marginally used L1 norm reveals a robust alternative to the L2 norm for frequency-domain full-waveform inversion of weakly redundant data.