Time-lapse electrical resistivity tomography (ERT) is a popular geophysical method to estimate three-dimensional (3D) permeability fields from electrical potential difference measurements. Traditional inversion and data assimilation methods are used to ingest this ERT data into hydrogeophysical models to estimate permeability. Due to ill-posedness and the curse of dimensionality, existing inversion strategies provide poor estimates and low resolution of the 3D permeability field. Recent advances in deep learning provide us with powerful algorithms to overcome this challenge. This paper presents a deep learning (DL) framework to estimate the 3D subsurface permeability from time-lapse ERT data. To test the feasibility of the proposed framework, we train DL-enabled inverse models on simulation data. Subsurface process models based on hydrogeophysics are used to generate this synthetic data for deep learning analyses. Results show that proposed weak supervised learning can capture salient spatial features in the 3D permeability field. Quantitatively, the average mean squared error (in terms of the natural log) on the strongly labeled training, validation, and test datasets is less than 0.5. The R2-score (global metric) is greater than 0.75, and the percent error in each cell (local metric) is less than 10%. Finally, an added benefit in terms of computational cost is that the proposed DL-based inverse model is at least O(104) times faster than running a forward model. Note that traditional inversion may require multiple forward model simulations (e.g., in the order of 10 to 1000), which are very expensive. This computational savings (O(105) - O(107)) makes the proposed DL-based inverse model attractive for subsurface imaging and real-time ERT monitoring applications due to fast and yet reasonably accurate estimations of the permeability field.

Watershed models such as the Soil and Water Assessment Tool (SWAT) consist of high-dimensional physical and empirical parameters. These parameters need to be accurately calibrated for models to produce reliable predictions for streamflow, evapotranspiration, snow water equivalent, and nutrient loading. Existing parameter estimation methods are time-consuming, inefficient, and computationally intensive, with reduced accuracy when estimating high-dimensional parameters. In this paper, we present a fast, accurate, and reliable methodology to calibrate the SWAT model (i.e., 21 parameters) using deep learning (DL). We develop DL-enabled inverse models based on convolutional neural networks to ingest streamflow data and estimate the SWAT model parameters. Hyperparameter tuning is performed to identify the optimal neural network architecture and the nine next best candidates. We use ensemble SWAT simulations to train, validate, and test the above DL models. We estimated the actual parameters of the SWAT model using observational data. We test and validate the proposed DL methodology on the American River Watershed, located in the Pacific Northwest-based Yakima River basin. Our results show that the DL models-based calibration is better than traditional parameter estimation methods, such as generalized likelihood uncertainty estimation (GLUE). The behavioral parameter sets estimated by DL have narrower ranges than GLUE and produce values within the sampling range even under high relative observational errors. This narrow range of parameters shows the reliability of the proposed workflow to estimate sensitive parameters accurately even under noise. Due to its fast and reasonably accurate estimations of process parameters, the proposed DL workflow is attractive for calibrating integrated hydrologic models for large spatial-scale applications.