Reliable characterization of subsurface structures is essential for earth sciences and related applications. Data assimilation‐based identification frameworks can reasonably estimate subsurface structures using available lithological (e.g., borehole core, well log) and dynamic (e.g., hydraulic head, solute concentration) observations. However, a reasonable selection of the observation type and frequency is essential for accurate structure identification. To achieve this, we extended a recently developed stage‐wise stochastic deep learning inversion framework by coupling it with non‐isothermal flow and transport simulations. With the extended framework, the worth of three common observations (hydraulic head, concentration, and temperature) are compared under different observation noise and frequency. The framework combines the emerging deep‐learning (DL)‐based framework with the traditional stochastic approaches. This combination makes it possible to simultaneously compare the ability of these two methods to assimilate observation data. Our results show that including at least one type of dynamic observation strongly improves subsurface structure identifiability and reduces the uncertainty. However, the DL‐based framework is able to identify subsurface structures more accurately than stochastic identification methods under the same scenarios. Assimilation of certain types of dynamic observations could reduce the prediction error for related dynamic responses, but not necessarily for other uncorrelated dynamic responses. Observation data worth is affected by the observation noise and frequency. High observation noise increases the uncertainty of the prediction and reduces the estimation accuracy. However, the higher observation frequency can significantly improve the temporal dynamic information of observations. This information can compensate for negative impacts of high observation noise.

Key Points: Addresses rebound concentrations in risk‐based framework. Fully analytical solutions are developed. Rebound concentration uncertainty

Quantifying the uncertainty of solute concentration in heterogeneous aquifers is an important step in both human health and ecological risk analysis. The need for a probabilistic representation of transport is justified by the incomplete characterization of the subsurface. We derive the one-point concentration cumulative distribution function (CDF) while taking into account the spatial statistical structure of the hydraulic conductivity, space dimensionality, the injection source size, the Péclet number, and the sampling volume at the monitoring location. The CDF is application oriented and derived at first order in the log-conductivity variance. We illustrate how several key parameters control the shape of the concentration CDF. The CDF shape is important since it reflects both uncertainty and the dilution state of the plume. The transition from a bimodal to a unimodal CDF is examined and results are further supported by analyzing the concentration coefficient of variation. Results indicate the significance of the statistical anisotropy ratio (i.e., the ratio between the hydraulic conductivity correlation scales) in determining the CDF shape. The importance of the sampling volume in the tails of the concentration CDF and a comparison between the proposed model with the β-CDF approach (i.e., beta distribution) are also shown. Finally, we illustrate how the framework could be used in applications by evaluating the human health risk CDF. Our results are formally valid for low to moderate heterogeneous aquifers and source sizes small as compared to the hydraulic conductivity correlation length. The proposed approach can serve as a benchmark tool for other methods.