We analyze the sensitivity of cross correlations to the anisotropy of the incident field in the context of seismic ambient noise monitoring of small velocity changes. Numerical simulations of elastic waves are performed in a 2D scattering plate with a focus on the comparative character of the direct and coda waves in the cross-correlation. We show that coda waves reconstructed from cross-correlations are far more robust than direct waves in the presence of azimuthal anisotropy of the incident field. We observe similar behavior with real data recorded on Erebus volcano, where a database of impulsive icequakes is used to simulate an anisotropic source field. We propose a simplified approach to evaluate the sensitivity of scattered waves to the anisotropy of the incoming noise field. We rely on previous results obtained for direct waves and on intrinsic properties of scattered waves to predict the errors produced by strong source anisotropy with numerical experiments. These results also yield realistic values for monitoring the accuracy to be expected with real data at crustal scales. Our analysis shows that high-precision noise-based monitoring could be performed with coda waves in the correlation functions, even in the presence of variations in the azimuthal distribution of the ambient noise field.