  We use analytical examples and asymptotic forms to examine the mathematical
structure and physical meaning of the seismic cross correlation measurement. We
show that in general, cross correlations are not Green's functions of medium,
and may be very different depending on the source distribution. The modeling of
noise sources using spatial distributions as opposed to discrete collections of
sources is emphasized. When stations are illuminated by spatially complex
source distributions, cross correlations show arrivals at a variety of time
lags, from zero to the maximum surface-wave arrival time. Here, we demonstrate
the possibility of inverting for the source distribution using the energy of
the full cross-correlation waveform. The interplay between the source
distribution and wave attenuation in determining the functional dependence of
cross correlation energies on station-pair distance is quantified. Without
question, energies contain information about wave attenuation. However, the
accurate interpretation of such measurements is tightly connected to the
knowledge of the source distribution.
