Statistic inversion of multi-zone transition probability models for aquifer characterization in alluvial fans

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.

💡 Research Summary

The paper presents a comprehensive statistical inversion framework that integrates a multi‑zone transition probability (TP) approach for characterizing the heterogeneous architecture of alluvial fans. Transition probability matrices describe the likelihood of moving from one hydrofacies to another as a function of distance, and the authors first derive an analytical solution that explicitly links these probabilities to three physically meaningful parameters: the mean length of each facies, its integral scale, and its volumetric proportion. This analytical formulation preserves the geological meaning of the parameters while enabling efficient computation.

To estimate these parameters, the authors employ a modified Gauss‑Newton‑Levenberg‑Marquardt (GN‑LM) algorithm. The modification lies in the calculation of the Jacobian matrix via a sensitivity‑equation method rather than finite differences. This yields highly accurate sensitivity information, accelerates convergence, and allows the simultaneous quantification of parameter uncertainties through the covariance matrix derived from the Jacobian.

The methodology is demonstrated on the Chaobai River alluvial fan in the Beijing Plain, China. Based on borehole logs and sedimentological surveys, the fan is divided into three distinct sedimentary zones (upper, middle, lower), each exhibiting markedly different depositional environments. For each zone a separate TP model is constructed, and the GN‑LM inversion is performed independently, producing zone‑specific optimal values of mean facies lengths, integral scales, and volumetric proportions. The results reveal that the upper zone is dominated by short mean lengths and small integral scales, reflecting a tightly interbedded sand‑clay fabric; the middle zone shows larger mean lengths and integral scales, indicating more continuous sand bodies; the lower zone again exhibits reduced mean lengths but higher variability in proportions, consistent with fine‑grained overbank deposits and irregular channel remnants.

Using the optimized TP parameters, the authors conduct sequential indicator simulations. First, the hydrofacies distribution for the upper zone is generated; its realization serves as a boundary condition for the middle‑zone simulation, and finally the lower‑zone simulation is performed. This sequential approach respects the spatial continuity across zone boundaries while preserving each zone’s unique statistical structure. The resulting three‑dimensional realizations capture the multi‑scale heterogeneity of the fan far more realistically than traditional single‑scale Markov chain or Gaussian random field models.

A key contribution is the rigorous uncertainty analysis. By propagating the Jacobian‑derived covariance matrix, confidence intervals for each TP parameter are obtained, and these intervals are subsequently used to assess the reliability of the simulated hydrofacies fields. The authors argue that such quantified uncertainty is essential for downstream groundwater flow and contaminant transport modeling, where predictive error bounds directly influence risk assessments and management decisions.

In summary, the paper advances alluvial‑fan characterization through four major innovations: (1) an analytical TP formulation that ties probabilistic transitions to physically interpretable geological parameters; (2) a robust GN‑LM inversion scheme enhanced by sensitivity‑equation Jacobian computation; (3) a multi‑zone TP framework that accommodates abrupt changes in sedimentary architecture; and (4) a systematic quantification of parameter uncertainty. The approach is not limited to the Chaobai fan; it can be applied to any depositional system with pronounced spatial heterogeneity, providing high‑resolution geological models that are ready for integration into flow and transport simulations.