## 1. Introduction

[2] Determining field-scale parameters in sufficient detail to capture aquifer heterogeneities is one of the greatest challenges for predicting flow and contaminant transport in large-scale subsurface systems. Uncertainty in the form of conceptual model bias can be introduced in modeling groundwater flow and contaminant transport when aquifer structure is fixed based on sparse or incomplete geophysical data [*Chen and Rubin*, 2003]. Once the structure has been fixed in this way, focus is placed on model calibration of zoned parameters through parameter estimation. This type of inverse method has been applied in many instances to estimate flow and transport parameters at various spatial scales [e.g., *Cooley*, 1983; *Carrera and Neuman*, 1986; *Dai and Samper*, 2006]. Deterministic aquifer structure may introduce larger bias and uncertainty into a model than an inappropriate choice of facies hydraulic parameters [*Ye et al.*, 2004; *Lu and Robinson*, 2006].

[3] Given uncertainty in both aquifer structure and hydrolic parameters *Sun* [2005] suggested “adaptive parameterization” to couple structure identification and parameter estimation in contaminant transport model calibration. This provides a more complete inversion of the model, allowing the optimization to explore combinations of structure geometry and hydraulic parameters. Previous studies usually assumed that the aquifer parameter zonations are randomly distributed. However, recent geological and geostatistical studies indicate that aquifer facies distributions are spatially correlated [*Carle and Fogg*, 1997; *Ritzi et al.*, 2004].

[4] We propose a structure identification method that accounts for spatial correlation by means of a stochastic inversion of a transition probability model, in an analytical framework [*Dai et al.*, 2007], describing facies volume proportions, mean lengths, and juxtapositioning. The transition probability model provides a nonparametric, Markov chain approach to indicator geostatistics that is well suited to applications with sparse information [*Carle and Fogg*, 1997]. The analytical solution allows structure identification to be cast as a conventional inverse modeling problem, using statistical structure parameters (such as facies volume proportions and mean lengths) to iteratively update the transition probability model. The facies proportions and mean lengths define the transition probability matrix. Indicator cokriging simulation produces the aquifer facies distributions, ensuring the statistical properties defined by the transition probability model are maintained. In this way, the aquifer structure is updated in the inversion, while the information provided by the conditional data (the sparse geological and geophysical data used to describe the facies distribution in boreholes) is honored. The optimization of the model inversion is performed using a genetically-adaptive multi-method search algorithm called AMALGAM-SO. This method was chosen as it combines the strengths of several different evolutionary search approaches and has been shown to achieve good efficiency across a range of difficult synthetic benchmark problems [*Vrugt et al.*, 2008]. While other optimization algorithms could potentially be implemented to drive the stochastic inversion, AMALGAM-SO was selected to illustrate the stochastic inversion methodology, without comparing its performance to other optimization algorithms. This decision was partly based on the assumption that, although an analytical solution of the transition probability model is utilized here, gradient-based methods would still have difficulty given the stochastic nature of the structural variables, which serve as inputs to the stochastic simulation. The analytical solution of the transition probability model provides the computational efficiency necessary for the large number of model evaluations required by the stochastic inversion. The combination of the analytical solution of the transition probability model and AMALGAM-SO provides a robust, computationally efficient model inversion with the ability to deal with complex fitness response surfaces. While in the past, the stochastic inversion described here was not possible, due to computational and algorithmic limitations, we propose that through the use of modern computers and analytically derived structure parameters [*Dai et al.*, 2007], this type of inversion can be realized.