Numerous sources of evidence point to the fact that heterogeneity
within the Earth's deep crystalline crust is complex and hence may
be best described through stochastic rather than deterministic approaches.
As seismic reflection imaging arguably offers the best means of sampling
deep crustal rocks in situ, much interest has been expressed in using
such data to characterize the stochastic nature of crustal heterogeneity.
Previous work on this problem has shown that the spatial statistics
of seismic reflection data are indeed related to those of the underlying
heterogeneous seismic velocity distribution. As of yet, however,
the nature of this relationship has remained elusive due to the fact
that most of the work was either strictly empirical or based on incorrect
methodological approaches. Here, we introduce a conceptual model,
based on the assumption of weak scattering, that allows us to quantitatively
link the second-order statistics of a 2-D seismic velocity distribution
with those of the corresponding processed and depth-migrated seismic
reflection image. We then perform a sensitivity study in order to
investigate what information regarding the stochastic model parameters
describing crustal velocity heterogeneity might potentially be recovered
from the statistics of a seismic reflection image using this model.
Finally, we present a Monte Carlo inversion strategy to estimate
these parameters and we show examples of its application at two different
source frequencies and using two different sets of prior information.
Our results indicate that the inverse problem is inherently non-unique
and that many different combinations of the vertical and lateral
correlation lengths describing the velocity heterogeneity can yield
seismic images with the same 2-D autocorrelation structure. The ratio
of all of these possible combinations of vertical and lateral correlation
lengths, however, remains roughly constant which indicates that,
without additional prior information, the aspect ratio is the only
parameter describing the stochastic seismic velocity structure that
can be reliably recovered.