Recently, rainfall-runoff simulations in small headwater basins have been improved by methodological advances such as deep neural networks (NNs) and hybrid physics-NN models — particularly, a genre called differentiable modeling that intermingles NNs with physics to learn relationships between variables. However, hydrologic routing, necessary for simulating floods in stem rivers downstream of large heterogeneous basins, had not yet benefited from these advances and it was unclear if the routing process can be improved via coupled NNs. We present a novel differentiable routing model that mimics the classical Muskingum-Cunge routing model over a river network but embeds an NN to infer parameterizations for Manning’s roughness (n) and channel geometries from raw reach-scale attributes like catchment areas and sinuosity. The NN was trained solely on downstream hydrographs. Synthetic experiments show that while the channel geometry parameter was unidentifiable, n can be identified with moderate precision. With real-world data, the trained differentiable routing model produced more accurate long-term routing results for both the training gage and untrained inner gages for larger subbasins (>2,000 km2) than either a machine learning model assuming homogeneity, or simply using the sum of runoff from subbasins. The n parameterization trained on short periods gave high performance in other periods, despite significant errors in runoff inputs. The learned n pattern was consistent with literature expectations, demonstrating the framework’s potential for knowledge discovery, but the absolute values can vary depending on training periods. The trained n parameterization can be coupled with traditional models to improve national-scale flood simulations.

Recently, runoff simulations in small, headwater basins have been improved by methodological advances such as deep learning (DL). Hydrologic routing modules are typically needed to simulate flows in stem rivers downstream of large, heterogeneous basins, but obtaining suitable parameterization for them has previously been difficult. It is unclear if downstream daily discharge contains enough information to constrain spatially-distributed parameterization. Building on recent advances in differentiable modeling principles, here we propose a differentiable, learnable physics-based routing model. It mimics the classical Muskingum-Cunge routing model but embeds a neural network (NN) to provide parameterizations for Manning’s roughness coefficient (n) and channel geometries. The embedded NN, which uses (imperfect) DL-simulated runoffs as the forcing data and reach-scale attributes as inputs, was trained solely on downstream hydrographs. Our synthetic experiments show that while channel geometries cannot be identified, we can learn a parameterization scheme for n that captures the overall spatial pattern. Training on short real-world data showed that we could obtain highly accurate routing results for both the training and inner, untrained gages. For larger basins, our results are better than a DL model assuming homogeneity or the sum of runoff from subbasins. The parameterization learned from a short training period gave high performance in other periods, despite significant bias in runoff. This is the first time an interpretable, physics-based model is learned on the river network to infer spatially-distributed parameters. The trained n parameterization can be coupled to traditional runoff models and ported to traditional programming environments.

Stream water temperature (T) is a variable of critical importance and decision-making relevance to aquatic ecosystems, energy production, and human’s interaction with the river system. Here, we propose a basin-centric stream water temperature model based on the long short-term memory (LSTM) model trained over hundreds of basins over continental United States, providing a first continental-scale benchmark on this problem. This model was fed by atmospheric forcing data, static catchment attributes and optionally observed or simulated discharge data. The model achieved a high performance, delivering a high median root-mean-squared-error (RMSE) for the groups with extensive, intermediate and scarce temperature measurements, respectively. The median Nash Sutcliffe model efficiency coefficients were above 0.97 for all groups and above 0.91 after air temperature was subtracted, showing the model to capture most of the temporal dynamics. Reservoirs have a substantial impact on the pattern of water temperature and negative influence the model performance. The median RMSE was 0.69 and 0.99 for sites without major dams and with major dams, respectively, in groups with data availability larger than 90%. Additional experiments showed that observed or simulated streamflow data is useful as an input for basins without major dams but may increase prediction bias otherwise. Our results suggest a strong mapping exists between basin-averaged forcings variables and attributes and water temperature, but local measurements can strongly improve the model. This work provides the first benchmark and significant insights for future effort. However, challenges remain for basins with large dams which can be targeted in the future when more information of withdrawal timing and water ponding time were accessible.

Stream water temperature is considered a “master variable” in environmental processes and human activities. Existing process-based models have difficulties with defining true equation parameters, and sometimes simplifications like assuming constant values influence the accuracy of results. Machine learning models are a highly successful tool for simulating stream temperature, but it is challenging to learn about processes and dynamics from their success. Here we integrate process-based modeling (SNTEMP model) and machine learning by building on a recently developed framework for parameter learning. With this framework, we used a deep neural network to map raw information (like catchment attributes and meteorological forcings) to parameters, and then inspected and fed the results into SNTEMP equations which we implemented in a deep learning platform. We trained the deep neural network across many basins in the conterminous United States in order to maximize the capturing of physical relationships and avoid overfitting. The presented framework has the ability of providing dynamic parameters based on the response of basins to meteorological conditions. The goal of this framework is to minimize the differences between stream temperature observations and SNTEMP outputs in the new platform. Parameter learning allows us to learn model parameters on large scales, providing benefits in efficiency, performance, and generalizability through applying global constraints. This method has also been shown to provide more physically-sensible parameters due to applying a global constraint. This model improves our understanding of how to parameterize the physical processes related to water temperature.