In various research fields such as hydrogeology, environmental science and energy engineering, geological formations with fractures are frequently encountered. Accurately characterizing these fractured media is of paramount importance when it comes to tasks that demand precise predictions of liquid flow and the transport of solute and energy within them. Since directly measuring fractured media poses inherent challenges, data assimilation (DA) techniques are typically employed to derive inverse estimates of media properties using observed state variables like hydraulic head, concentration, and temperature. Nonetheless, the considerable difficulties arising from the strong heterogeneity and non-Gaussian nature of fractured media have diminished the effectiveness of existing DA methods. In this study, we formulate a novel DA approach known as PEDL (parameter estimator with deep learning) that harnesses the capabilities of DL to capture nonlinear relationships and extract non-Gaussian features. To evaluate PEDL’s performance, we conduct two numerical case studies with increasing complexity. Our results unequivocally demonstrate that PEDL outperforms three popular DA methods: ensemble smoother with multiple DA (ESMDA), iterative local updating ES (ILUES), and ES with DL-based update (ESDL). Sensitivity analyses confirm PEDL’s validity and adaptability across various ensemble sizes and DL model architectures. Moreover, even in scenarios where structural difference exists between the accurate reference model and the simplified forecast model, PEDL adeptly identifies the primary characteristics of fracture networks.

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.  

Seawater intrusion (SI) poses a substantial threat to water security in coastal regions, where numerical models play a pivotal role in supporting groundwater management and protection. However, the inherent heterogeneity of coastal aquifers introduces significant uncertainties into SI predictions, potentially diminishing their effectiveness in management decisions. Data assimilation (DA) offers a solution by integrating various types of observational data with the model to characterize heterogeneous coastal aquifers. Traditional DA techniques, like ensemble smoother using the Kalman formula (ESK) and Markov chain Monte Carlo, face challenges when confronted with the non-linearity, non-Gaussianity, and high-dimensionality issues commonly encountered in aquifer characterization. In this study, we introduce a novel DA approach rooted in deep learning (DL), referred to as ESDL, aimed at effectively characterizing coastal aquifers with varying levels of heterogeneity. We systematically investigate a range of factors that impact the performance of ESDL, including the number and types of observations, the degree of aquifer heterogeneity, the structure and training options of the DL models. Our findings reveal that ESDL excels in characterizing heterogeneous aquifers under non-linear and non-Gaussian conditions. Comparison between ESDL and ESK under different experimentation settings underscores the robustness of ESDL. Conversely, in certain scenarios, ESK displays noticeable biases in the characterization results, especially when measurement data from non-linear and discontinuous processes are used. To optimize the efficacy of ESDL, attention must be given to the design of the DL model and the selection of observational data, which are crucial to ensure the universal applicability of this DA method.

Seawater intrusion poses a substantial threat to water security in coastal regions, where numerical models play a pivotal role in supporting groundwater management and protection. However, the inherent heterogeneity of coastal aquifers introduces significant uncertainties into model predictions, potentially diminishing their effectiveness in management decisions. Data assimilation (DA) offers a solution by incorporating various types of observational data to characterize these heterogeneous coastal aquifers. Traditional DA techniques, like ensemble smoother using the Kalman formula (ESK) and Markov chain Monte Carlo, face challenges when confronted with the non-linearity, non-Gaussianity, and high-dimensionality issues commonly encountered in aquifer characterization. In this study, we introduce a novel DA approach rooted in deep learning (DL), referred to as ESDL, aimed at effectively characterizing coastal aquifers with varying levels of heterogeneity. We systematically investigate a range of factors that impact the performance of ESDL, including the number and types of observations, the degree of aquifer heterogeneity, the structure and training options of the DL models, etc. Our findings reveal that ESDL excels in characterizing heterogeneous aquifers, particularly when faced with non-Gaussian conditions. Comparison between ESDL and ESK under different experimentation settings underscores the robustness of ESDL. Conversely, in certain scenarios, ESK displays noticeable biases in the characterizing results, especially when measurement data from nonlinear and discontinuous processes are used. To optimize the efficacy of ESDL, meticulous attention must be given to the design of the DL model and the selection of training options, which are crucial to ensure the universal applicability of this DA method.

In hydrological research, data assimilation (DA) is widely used to fuse the information contained in process-based models and observational data to reduce simulation uncertainty. However, many popular DA methods are limited by low computational efficiency or their reliance on the Gaussian assumption. To address these limitations, we propose a novel DA method called DA(DL), which leverages the capabilities of deep learning (DL) to model non-linear relationships and recognize complex patterns. DA(DL) first generates a large volume of training data from the prior ensemble, and then trains a DL model to update the system knowledge (e.g., model parameters in this study) from multiple predictors. For highly non-linear models, an iterative form of DA(DL) can be implemented. Additionally, strategies of data augmentation and local updating are proposed to enhance DA(DL) for problems involving small ensemble size and the equifinality issue, respectively. In two hydrological DA cases involving Gaussian and non-Gaussian distributions, DA(DL) shows promising performance compared to two ensemble smoother (ES) methods, i.e., ES(K) and ES(DL), which respectively apply the Kalman- and DL-based updates. Potential improvements to DA(DL) can be made by designing better DL model architectures, imposing physical constraints to the training of the DL model, and further updating other important variables like model states, forcings and error terms.

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.