Water temperature in rivers is a crucial environmental factor with the ability to alter hydro-ecological as well as socio-economic conditions within a catchment. The development of modelling concepts for predicting river water temperature is and will be essential for effective integrated water management and the development of adaptation strategies to future global changes (e.g. climate change). This study tests the performance of six different machine-learning models: step-wise linear regression, random forest, eXtreme Gradient Boosting (XGBoost), feed-forward neural networks (FNNs), and two types of recurrent neural networks (RNNs). All models are applied using different data inputs for daily water temperature prediction in 10 Austrian catchments ranging from 200 to 96 000 km² and exhibiting a wide range of physiographic characteristics. The evaluated input data sets include combinations of daily means of air temperature, runoff, precipitation and global radiation. Bayesian optimization is applied to optimize the hyperparameters of all applied machine-learning models. To make the results comparable to previous studies, two widely used benchmark models are applied additionally: linear regression and air2stream. With a mean root mean squared error (RMSE) of 0.55 ^∘C, the tested models could significantly improve water temperature prediction compared to linear regression (1.55 ^∘C) and air2stream (0.98 ^∘C). In general, the results show a very similar performance of the tested machine-learning models, with a median RMSE difference of 0.08 ^∘C between the models. From the six tested machine-learning models both FNNs and XGBoost performed best in 4 of the 10 catchments. RNNs are the best-performing models in the largest catchment, indicating that RNNs mainly perform well when processes with long-term dependencies are important. Furthermore, a wide range of performance was observed for different hyperparameter sets for the tested models, showing the importance of hyperparameter optimization. Especially the FNN model results showed an extremely large RMSE standard deviation of 1.60 ^∘C due to the chosen hyperparameters. This study evaluates different sets of input variables, machine-learning models and training characteristics for daily stream water temperature prediction, acting as a basis for future development of regional multi-catchment water temperature prediction models. All preprocessing steps and models are implemented in the open-source R package wateRtemp to provide easy access to these modelling approaches and facilitate further research.

Estimating parameters for distributed hydrological models is a challenging and long studied task. Parameter transfer functions, which define model parameters as functions of geophysical properties of a catchment, might improve the calibration procedure, increase process realism, and can enable prediction in ungauged areas. We present the function space optimization (FSO), a symbolic regression method for estimating parameter transfer functions for distributed hydrological models. FSO is based on the idea of transferring the search for mathematical expressions into a continuous vector space that can be used for optimization. This is accomplished by using a text generating neural network with a variational autoencoder architecture that can learn to compress the information of mathematical functions. To evaluate the performance of FSO, we conducted a case study using a parsimonious hydrological model and synthetic discharge data. The case study consisted of two FSO applications: single‐criteria FSO, where only discharge was used for optimization, and multicriteria FSO, where additional spatiotemporal observations of model states were used for transfer function estimation. The results show that FSO is able to estimate transfer functions correctly or approximate them sufficiently. We observed a reduced fit of the parameter density functions resulting from the inferred transfer functions for less sensitive model parameters. For those it was sufficient to estimate functions resulting in parameter distributions with approximately the same mean parameter values as the real transfer functions. The results of the multicriteria FSO showed that using multiple spatiotemporal observations for optimization increased the quality of estimation considerably.

Parameter estimation is one of the most challenging tasks in large‐scale distributed modeling, because of the high dimensionality of the parameter space. Relating model parameters to catchment/landscape characteristics reduces the number of parameters, enhances physical realism, and allows the transfer of hydrological model parameters in time and space. This study presents the first large‐scale application of automatic parameter transfer function (TF) estimation for a complex hydrological model. The Function Space Optimization (FSO) method can automatically estimate TF structures and coefficients for distributed models. We apply FSO to the mesoscale Hydrologic Model (mHM, mhm-ufz.org), which is the only available distributed model that includes a priori defined TFs for all its parameters. FSO is used to estimate new TFs for the parameters “saturated hydraulic conductivity” and “field capacity,” which both influence a range of hydrological processes. The setup of mHM from a previous study serves as a benchmark. The estimated TFs resulted in predictions in 222 validation basins with a median NSE of 0.68, showing that even with 5 years of calibration data, high performance in ungauged basins can be achieved. The performance is similar to the benchmark results, showing that the automatic TFs can achieve comparable results to TFs that were developed over years using expert knowledge. In summary, the findings present a step toward automatic TF estimation of model parameters for distributed models.

Typical applications of process‐ or physically‐based models aim to gain a better process understanding or provide the basis for a decision‐making process. To adequately represent the physical system, models should include all essential processes. However, model errors can still occur. Other than large systematic observation errors, simplified, misrepresented, inadequately parametrised or missing processes are potential sources of errors. This study presents a set of methods and a proposed workflow for analysing errors of process‐based models as a basis for relating them to process representations. The evaluated approach consists of three steps: (1) training a machine‐learning (ML) error model using the input data of the process‐based model and other available variables, (2) estimation of local explanations (i.e., contributions of each variable to an individual prediction) for each predicted model error using SHapley Additive exPlanations (SHAP) in combination with principal component analysis, (3) clustering of SHAP values of all predicted errors to derive groups with similar error generation characteristics. By analysing these groups of different error‐variable association, hypotheses on error generation and corresponding processes can be formulated. That can ultimately lead to improvements in process understanding and prediction. The approach is applied to a process‐based stream water temperature model HFLUX in a case study for modelling an alpine stream in the Canadian Rocky Mountains. By using available meteorological and hydrological variables as inputs, the applied ML model is able to predict model residuals. Clustering of SHAP values results in three distinct error groups that are mainly related to shading and vegetation‐emitted long wave radiation. Model errors are rarely random and often contain valuable information. Assessing model error associations is ultimately a way of enhancing trust in implemented processes and of providing information on potential areas of improvement to the model.

To find the adequate spatial model discretization scheme, which balances the models capabilities and the demand for representing key features in reality, is a challenging task. It becomes even more challenging in high alpine catchments, where the variability of topography and meteorology over short distances strongly influences the distribution of snow cover, the dominant component in the alpine water cycle. For the high alpine Research Catchment Zugspitze (RCZ) a new method for objective delineation of hydrological response units (HRUs) using a time series of high resolution LIDAR derived snow depth maps and the physiographic properties of the RCZ is introduced. Via principle component analysis (PCA) of these maps, a dominant snow depth pattern, that turned out to be largely defined during the (winter) accumulation period was identified. This dominant pattern serves as a reference for HRU delineations on the basis of cluster analyses of the catchment’s physiographic properties. The method guarantees for an appropriate, objective, spatial discretization scheme, which allows for a reliable and meaningful reproduction of snow cover variability with the Cold Regions Hydrological Model—at the same time avoiding significant increase of computational demands. Different HRU schemes were evaluated with measured snow depth and the comparison of their model results identified significant differences in model output and best performance of the scheme which best represents measured snow depth distribution.