In hydrological research, data assimilation (DA) is a powerful tool for integrating observational data with numerical models, significantly enhancing predictive accuracy. However, non-linear groundwater systems often exhibit high-dimensional and non-Gaussian characteristics in observations, parameters, and state variables, posing substantial challenges for traditional DA methods such as Markov chain Monte Carlo and ensemble smoother based on the Kalman update (ES(K)). To address these challenges, we previously introduced ES(DL), which replaces the linear Kalman update with a non-linear deep learning (DL)-based update, enabling improved handling of non-Gaussian issues. Despite its advantages, ES(DL) is constrained by the high computational cost of DL model training and limited utilization of ensemble statistics. In this study, we propose ES(K-DL), a hybrid DA approach that integrates Kalman with DL-based updates to overcome these limitations. Tailored for non-linear and non-Gaussian groundwater systems, ES(K-DL) combines the computational efficiency of ES(K) with the adaptability of ES(DL). To evaluate ES(K-DL), we apply it to a challenging case study involving the joint inversion of eight contaminant source parameters and a 3,321-dimensional non-Gaussian hydraulic conductivity field. Comprehensive numerical experiments are conducted to investigate factors influencing performance, including the number of DL-based updates, the sequencing of Kalman and DL-based updates, and the configuration of error inflation factors. The results demonstrate that the hybrid updating strategy reduces computational costs while maintaining stability and reliability in DA outcomes. The optimal ES(K-DL) variant achieves superior performance compared to ES(K) and ES(DL) individually, highlighting the benefits of this complementary approach.  

To reduce simulation uncertainty and improve process understanding of hydrological systems, integrating models with observations through data assimilation techniques is paramount. Among these, the ensemble smoother (ES) stands out for its ease of implementation and high computational efficiency. When tackling problems involving non-linear processes and non-Gaussian distributions, leveraging deep learning (DL) within ES, termed the ES(DL) method, proves superior to the Kalman-based counterpart. However, the original ES(DL) method is constrained by the traditional paradigm that builds a mapping from the innovation vector (i.e., the difference between observations and model predictions) to the update vector (i.e., the difference between posterior and prior state/parameters). In this study, we introduce a generalized form of ES(DL), where the traditional ES(DL) approach becomes its special case. We explore the optimal implementation of ES(DL) through surface and subsurface scenarios, spanning various dimensions (low to high) and parameter distributions (Gaussian to non-Gaussian). Notably, certain implementations of ES(DL), which diverge significantly from the traditional approach, can yield similar or even better outcomes, especially under the non-Gaussian condition. While this study’s focus lies on the smoothing approach for parameter estimation, the proposed formulation can be extended to filtering problems, facilitating model state updates.