Seismic noise cross correlations are used to image crustal structure and
heterogeneity. Typically, seismic networks are only anisotropically illuminated
by seismic noise, a consequence of the non-uniform distribution of sources.
Here, we study the sensitivity of such a seismic network to structural
heterogeneity in a 2-D setting. We compute finite-frequency cross-correlation
sensitivity kernels for travel-time, waveform-energy and waveform-difference
measurements. In line with expectation, wavespeed anomalies are best imaged
using travel times and the source distribution using cross-correlation
energies. Perturbations in attenuation and impedance are very difficult to
image and reliable inferences require a high degree of certainty in the
knowledge of the source distribution and wavespeed model (at least in the case
of transmission tomography studied here). We perform single-step Gauss-Newton
inversions for the source distribution and the wavespeed, in that order, and
quantify the associated Cram\'{e}r-Rao lower bound. The inversion and
uncertainty estimate are robust to errors in the source model but are sensitive
to the theory used to interpret of measurements. We find that when classical
source-receiver kernels are used instead of cross-correlation kernels, errors
appear in the both the inversion and uncertainty estimate, systematically
biasing the results. We outline a computationally tractable algorithm to
account for distant sources when performing inversions.Comment: 19 pages, 12 figures, Geophysical Journal Internationa