  Understanding the heterogeneity arising from the complex architecture of
sedimentary sequences in alluvial fans is challenging. This paper develops a
statistical inverse framework in a multi-zone transition probability approach
for characterizing the heterogeneity in alluvial fans. An analytical solution
of the transition probability matrix is used to define the statistical
relationships among different hydrofacies and their mean lengths, integral
scales, and volumetric proportions. A statistical inversion is conducted to
identify the multi-zone transition probability models and estimate the optimal
statistical parameters using the modified Gauss-Newton-Levenberg-Marquardt
method. The Jacobian matrix is computed by the sensitivity equation method,
which results in an accurate inverse solution with quantification of parameter
uncertainty. We use the Chaobai River alluvial fan in the Beijing Plain, China,
as an example for elucidating the methodology of alluvial fan characterization.
The alluvial fan is divided into three sediment zones. In each zone, the
explicit mathematical formulations of the transition probability models are
constructed with optimized different integral scales and volumetric
proportions. The hydrofacies distributions in the three zones are simulated
sequentially by the multi-zone transition probability-based indicator
simulations. The result of this study provides the heterogeneous structure of
the alluvial fan for further study of flow and transport simulations.
