This study introduces two new end-to-end probabilistic deep learning (DL) models for predicting semi-continuous hydrological variables with a physical lower bound of zero. Both models couple long short-term memory networks (LSTMs) with probabilistic output layers to predict the parameters of a probability distribution fit to observed streamflow (i.e., the target), respecting its physical lower bound of zero. The first model is based on the rectified Gaussian (RG) distribution, which is akin to a Gaussian distribution that converts negative values to zero. The second model is a hurdle model, which includes two parts. The first part predicts the probability of the target being non-zero (the hurdle) while the second part predicts the probability distribution over non-zero values. Both RG and hurdle models are benchmarked against a state-of-the-art deterministic LSTM and evaluated on a daily streamflow prediction case study across 421 catchments in the contiguous United States. The hurdle model improved upon the mean normalized root mean squared error of RG and deterministic models by 9%. RG and hurdle models outperformed the deterministic model in mean low flow accuracy by 11% and 24%, respectively. When compared against third-party benchmarks, the hurdle model outperformed a hybrid process-based DL model by over 35% in low flow root mean squared error and performed competitively with a probabilistic benchmark in terms of reliability, sharpness, and efficiency metrics. Overall, the RG and hurdle models are promising probabilistic DL models for predicting semi-continuous hydrological variables. However, the hurdle model is recommended as a new probabilistic benchmark for daily streamflow prediction across CONUS given its state-of-the-art probabilistic performance, exceptional low flow accuracy, and inherent ability to respect the physical lower bound of streamflow.

Input variable selection (IVS) is an integral part of building data-driven models for hydrological applications. Carefully chosen input variables enable data-driven models to discern relevant patterns and relationships within data, improving their predictive accuracy. Moreover, the optimal choice of input variables can enhance the computational efficiency of data-driven models, reduce overfitting, and contribute to a more interpretable and parsimonious model. Meanwhile, including irrelevant and/or redundant input variables can introduce noise to the model and hinder its generalization ability. Three probabilistic IVS methods, namely Edgeworth approximation-based conditional mutual information (EA), double-layer extreme learning machine (DLELM), and gradient mapping (GM), were used for IVS and then coupled with a long short-term memory (LSTM)-based probabilistic deep learning model for daily streamflow prediction. While the EA method is an effective IVS method, DLELM and GM are examples of probabilistic neural network-based IVS methods that have not yet been explored for hydrological prediction. DLELM selects input variables through sparse Bayesian learning, pruning both input and output layer weights of a committee of neural networks. GM is based on saliency mapping, an explainable AI technique commonly used in computer vision that can be coupled with probabilistic neural networks. Both DLELM and GM involve randomization during parameter initialization and/or training thereby introducing stochasticity into the IVS procedure, which has been shown to improve the predictive performance of data-driven models. The IVS methods were coupled with a LSTM-based probabilistic deep learning model and applied to a streamflow prediction case study using 420 basins spread across the continental United States. The dataset includes 37 candidate input variables derived from the daily-averaged ERA-5 reanalysis data. Comparing the most frequently selected input variables by EA, DLELM, and GM across the 420 basins revealed that all three models select a similar number of input variables. For example, the top 15 input variables selected by all methods included nine variables that were similar. The input variables selected by EA, DLELM, and GM were then used in the LSTM-based probabilistic deep learning models for streamflow prediction across the 420 basins. The probabilistic deep learning models were developed and optimized using the top 10 variables selected by each IVS method. The results were compared to a benchmark scenario that used all 37 ERA-5 variables in the prediction model. Overall, the findings show that the GM method results in higher prediction accuracy (Kling-Gupta efficiency; KGE) compared to the other two IVS methods. A median KGE of 0.63 was obtained for GM, whereas for the EA, DLELM, and all input variables’ scenario, KGE scores of 0.61, 0.60, and 0.62 were obtained, respectively. DLELM and GM are two AI-based techniques that introduce elements of interpretability and stochasticity to the IVS process. The results of the current study are expected to contribute to the evolving landscape of data-driven hydrological modeling by introducing hitherto unexplored neural network-based IVS to pursue more parsimonious, efficient, and interpretable probabilistic deep learning models.

The significance of probabilistic hydrological forecasting has grown in recent years, offering crucial insights for risk-based decision-making and effective flood management. This study explores generative deep learning models, specifically the conditional variational autoencoder (CVAE), for probabilistic streamflow forecasting. This innovative approach is applied for forecasting streamflow one to seven days (s) ahead in 75 Canadian basins included in the open-source Canadian model parameter experiment (CANOPEX) database. CVAE is compared against two benchmark quantile-based deep learning models: the quantile-based encoder-decoder (ED) and the quantile-based CVAE (QCVAE). Over 9000 deep learning models are developed based on different input variables, basin characteristics, and model structures and evaluated regarding point forecast accuracy and forecast reliability. Results highlight CVAE‘s superior reliability, showing a median reliability of 92.49% compared to 87.35% for ED and 84.59% for QCVAE (considering a desired 90% confidence level). However, quantile-based forecast models exhibit marginally better point forecasts, as evidenced by Kling-Gupta efficiency (KGE), with a median KGE of 0.90 for ED and QCVAE (compared to 0.88 for CVAE). Notably, the CVAE model provides reliable probabilistic forecasts in basins with low point forecast accuracy. The developed generative deep learning models can be used as a benchmark for probabilistic streamflow forecasting due to the use of the open-source CANOPEX dataset. Overall, the results of this study contribute to the expanding field of generative deep learning models in hydrological forecasting, offering a general framework that applies to forecasting other hydrological variables as well (precipitation and soil moisture).

Probabilistic hydrological forecasting has gained increasing importance in recent years, as it offers essential information for risk-based decision-making and flood management. Traditional hydrological models often produce deterministic forecasts, which do not account for the inherent uncertainties in hydrological systems. Although many previous studies have investigated using deep learning (DL) models for hydrological prediction, the development of probabilistic DL models (especially, generative models) has not yet been thoroughly examined for hydrological forecasting. The present study investigates the efficacy of a generative DL model, namely, conditional variational auto-encoder (CVAE). CVAE is applied for one-seven day(s) ahead probabilistic streamflow forecasting in 75 basins from the Canadian model parameter experiment (CANOPEX) database. The CVAE forecast model, which outputs forecasts in the form of a probability distribution, was benchmarked against two state-of-the-art quantile-based DL models: quantile-based encoder-decoder (ED) and quantile-based CVAE (QCVAE). The latter outputs forecasts for specific quantiles of a probability distribution (here, q = 0.05, 0.5, 0.95). More than 9000 models were developed based on different basins, input variable sets, and model structures. The models were evaluated in terms of point forecast accuracy and forecast reliability. The results indicate that the CVAE model generally outperforms the benchmark models in terms of reliability at a 90 % confidence level (median reliability of 92.49 % compared to 87.35 % and 84.59 % for ED and QCVAE, respectively). However, the quantile-based forecast models produce slightly more accurate point forecasts than the CVAE (median Kling-Gupta efficiency (KGE) of 0.88 compared to 0.90 for both ED and QCVAE). Notably, the CVAE model exhibits superior probabilistic forecasts in basins with poor point forecast accuracy, highlighting its usefulness over benchmark methods in difficult-to-forecast basins. Overall, CVAE is a promising probabilistic DL model for streamflow forecasting, and it can be readily applied for forecasting other hydrological variables (evapotranspiration, water level, etc.). The findings of this study provide a basis for future research on probabilistic forecasting of hydrological variables using generative DL models.

Accurate urban water demand (UWD) forecasts are key to the effective management of water distribution systems. This research explores the potential of encoder–decoder models, specifically sequence-to-sequence (S2S) deep learning models, for UWD forecasting. Two models were developed as follows: one based on long short-term memory (LSTM) networks and another using transformers. The models were trained on data from ten district metered areas (DMAs) in Northeast Italy. The results confirmed that the transformer models consistently outperformed the LSTM models across all DMAs, with an average (across all DMAs) improvement in mean absolute error of 15.3%.

In recent years, extreme learning machines (ELM) have been used to accurately predict a variety of hydrological variables (e.g., streamflow, precipitation, river water quality). Using the same model structure, ELM often obtains similar performance to multi-layer perceptron (MLP) networks without the need for an iterative learning process (backpropagation), resulting in faster training. However, despite the increasing popularity of ELM, the hydrology literature has not focused on training algorithms that can be used to generate probabilistic predictions for this method. This is an interesting research gap, as it is generally accepted that quantifying hydrological prediction uncertainty and producing probabilistic predictions (instead of point predictions or mean value predictions) is a prerequisite for reliable water resource management. Thus, for the first time, Bayesian ELM (BELM) and sparse BELM (SBELM) methods are adopted and applied for probabilistic streamflow simulation and multi-step ahead forecasting (1-3 days), using as a case study four watersheds from Mexico, Germany, Canada, and Belgium. Using deterministic and probabilistic metrics, BELM and SBELM are compared against Bayesian linear regression (BLR), MLP combined with Monte-Carlo dropout weights, and a deep learning method: long short-term memory network (LSTM) coupled with Monte-Carlo dropout weights. Adding time-lagged observations of streamflow and meteorological variables (up to 14 days), such as precipitation and potential evapotranspiration, improves the simulation accuracy up to a factor of 10. In general, both BELM and SBELM show more accurate point predictions than MLP, BLR, and LSTM. BELM and SBELM show similar performance in terms of accuracy and reliability. Although, BELM marginally outperforms SBELM by generating, on average, a narrower prediction interval width. The sparsity feature of SBELM reduces (on average) the network size by 14-83 %. BELM and SBELM produce more accurate and reliable predictions than LSTM and are up to 122 and 125 times more computationally efficient to train, respectively. The case study suggests that BELM and SBELM are promising probabilistic machine learning models for hydrological prediction that are attractive alternatives to physical (e.g., lumped conceptual) models and common deep learning models (e.g., LSTM).

Deep neural network (DNN) models have become increasingly popular in the hydrology community. However, most studies are related to (rainfall-) runoff simulation and comparatively fewer studies have focused on runoff forecasting. In this study, quantile-based (q = 0.05, 0.5, 0.95) encoder-decoder (ED) models that use long short -term memory network (LSTM) and dense network (DN) blocks were developed for three and five days ahead runoff forecasting. Through linear (LW) and non-linear (NLW) wavelet selection, hybrid models LSTM-DN, LSTM-DN-LW, LSTM-DN-NLW, ED, ED-LW, and ED-NLW were developed. For each lead time (LT = 3, 5) and value of q, different model configurations were created using different input lag lengths (IL = 15, 45, 180). The developed models were tested for runoff forecasting using three basins (with different characteristics) from the Catchment Attributes and MEteorology for Large-sample Studies (CAMELS) dataset. The models were compared using deterministic (e.g., the Kling-Gupta efficiency [KGE] metric) and probabilistic (e.g., reliability) statistical metrics. While the models showed high variability in performance across the three basins (KGE = 0.308-0.979 for the q = 0.5 models), very high accuracy (up to KGE = 0.979) was achieved for one of the basins with high snowmelt. The ED-NLW model was found to generally outperform the other models. Although the LSTM-DN model had the highest median KGE (0.434 across all configurations), the ED and ED-NLW models had higher reliability than LSTM-DN (90% and 91%, respectively, considering a 90% confidence level). Models coupled with NLW performed superior to those that used LW. All ED models had high reliability despite two of the basins achieving median KGE values of similar to 0.390, highlighting that quantile-based models can generate reliable forecast intervals even when the KGE of the median forecast (q = 0.5) is low. An additional experiment generated synthetic precipitation forecasts with varying degrees of accuracy. The models were trained using accurate precipitation forecasts and tested using both accurate and inaccurate precipitation forecasts. While up to a 120% improvement in KGE was found when accurate precipitation forecasts were used as input to the models, using inaccurate precipitation forecasts resulted in a substantial decrease in reliability. Overall, the results of this study can serve as a benchmark for future studies developing probabilistic DNN models for runoff forecasting.

Recently, the conceptual data-driven approach (CDDA) was proposed to correct residuals of ensemble hydrological models (HMs) using data-driven models (DDMs), followed by the stochastic CDDA (SCDDA) that used HM simulations as input to DDMs within a stochastic framework - both approaches improved ensemble HMs' simulations. Here, a new SCDDA is introduced where CDDA uncertainty is estimated (instead of DDM uncertainty in the original SCDDA). Using nine HM-DDM combinations for daily streamflow simulation in three Swiss catchments, the new SCDDA improved CDDA's mean continuous ranked probability score up to 15% and performed similarly without a snow-routine in a snowy catchment, suggesting that SCDDA may account for missing processes in HMs. The stochastic framework can convert unreliable ensemble models into more reliable (stochastic) models at the cost of simulation sharpness. The coverage probability plot is proposed as a diagnostic tool, predicting SCDDA's out-of-sample reliability using validation set data (CDDA simulations and observations).

Obtaining consistent forecasts at different timescales is important for reliable decision‐making. This study introduces and evaluates the benefits of utilizing temporal hierarchical reconciliation methods for water resources forecasting, with an application to precipitation. Original (precipitation) Forecasts (ORFs) were produced using “automatic” Exponential Time‐Series Smoothing (ETS), Artificial Neural Network (ANN), and Seasonal Auto‐Regressive Integrated Moving Average (SARIMA) models at six timescales, namely, monthly, 2‐monthly, quarterly, 4‐monthly, bi‐annual, and annual, for 84 basins extracted from the Canadian model parameter experiment. Temporal hierarchical reconciliation methods, including structural scaling‐based Weighted Least Squares (WLS), series variance scaling‐based WLS, and Ordinary Least Squares, along with the simple Bottom‐Up (BU) method, were applied to reconcile the forecasts. In general, ETS (direct forecasting) demonstrated better performance compared to ANN and SARIMA (recursive forecasting). The results confirmed that improvements in accuracy due to reconciliation is dependent on the basin, timescale, and the ORFs' accuracy. For different forecast models, the reconciliation methods showed different levels of performance. For ETS, BU was able to improve forecast accuracy to a greater extent than the temporal hierarchical reconciliation methods, while for ANN and SARIMA, forecast accuracy was improved through all temporal hierarchical reconciliation methods but not BU. The reconciled forecasts' accuracy was affected more by the ORFs' accuracy than by the reconciliation method. Different timescales showed dissimilar sensitivity to reconciliation. The presented results are anticipated to serve as a valuable benchmark for evaluating future developments in the promising area of temporal hierarchical reconciliation for water resources forecasting.

In a companion paper, Sikorska-Senoner and Quilty (2021) introduced the ensemble-based conceptual-data-driven approach (CDDA) for improving hydrological simulations. This approach consists of an ensemble of hydrological model (HM) simulations (generated via different parameter sets) whose residuals are 'corrected' by a data-driven model (one per HM parameter set), resulting in an improved ensemble simulation. Through a case study involving three Swiss catchments, it was demonstrated that CDDA generates significantly improved ensemble streamflow simulations when compared to the ensemble HM. In this follow-up study, a stochastic version of CDDA (SCDDA) is developed that, in addition to parameter uncertainty, accounts for input data, input variable selection, and model output uncertainty. Using several deterministic and probabilistic performance metrics, it is shown that SCDDA results in significantly more accurate and reliable ensemble-based streamflow simulations than the CDDA, ensemble and stochastic HMs, and a quantile regression-based approach, improving the mean interval score by 26–79%.

Water infiltration into soil is an important process in hydrologic cycle; however, its measurement is difficult, time-consuming and costly. Empirical and physical models have been developed to predict cumulative infiltration(CI), but are often inaccurate. In this study, several novel standalone machine learning algorithms(M5Prime(M5P), decision stump(DS), and sequential minimal optimization(SMO)) and hybrid algorithms based on additive regression(AR)(i.e., AR-M5P, AR-DS, and AR-SMO) and weighted instance handler wrapper(WIHW)(i.e., WIHW-M5P, WIHW-DS, and WIHW-SMO) were developed for CI prediction. The Soil Conservation Service(SCS) model developed by the United States Department of Agriculture(USDA), one of the most popular empirical models to predict CI, was considered as a benchmark. Overall, 154 measurements of CI(explanatory/input variables) were taken from 16 sites in a semi-arid region of Iran(Illam and Lorestan provinces). Six input variable combinations were considered based on Pearson correlations between candidate model inputs(time of measuring and soil bulk density, moisture content, and sand, clay, and silt percentages) and CI. The dataset was divided into two subgroups at random: 70% of the data were used for model building(training dataset) and the remaining 30% were used for model validation(testing dataset).The various models were evaluated using different graphical approaches(bar charts, scatter plots, violin plots, and Taylor diagrams) and quantitative measures(root mean square error(RMSE), mean absolute error(MAE), Nash-Sutcliffe efficiency(NSE), and percent bias(PBIAS)). Time of measuring had the highest correlation with CI in the study area. The best input combinations were different for different algorithms. The results showed that all hybrid algorithms enhanced the CI prediction accuracy compared to the standalone models. The AR-M5P model provided the most accurate CI predictions(RMSE = 0.75 cm,MAE = 0.59 cm, NSE = 0.98), while the SCS model had the lowest performance(RMSE = 4.77 cm, MAE = 2.64 cm, NSE = 0.23). The differences in RMSE between the best model(AR-M5P) and the second-best(WIHW-M5P) and worst(SCS) were 40% and 84%, respectively.

A recent nonlinear and multiscale framework, the Wavelet Data-Driven Forecasting Framework (WDDFF), was proposed for water resources forecasting. The main objective of this study is to explore the WDDFF for short-term urban water demand (UWD) forecasting over multiple lead times (1, 2, 3, 6, 12, 18, and 24 h ahead) by focusing on two separate issues that have yet to be considered within the framework: 1) a comparison of artificial neural network (ANN), least squares support vector machines (LSSVM), regularized extreme learning machines (RELM), and random forest (RF) and 2) two dataset partitioning approaches for reducing overfitting in deterministic and probabilistic machine learning (ML) models, a permutation-based approach for the deterministic models and a bootstrap-based approach for the probabilistic models. The secondary objective is to evaluate the usefulness of an input variable selection approach based on RF (RFIVS) for identifying the most important inputs to use in the ML models. The results of a real-world UWD forecasting case study in Qom, Iran demonstrate several noteworthy findings: 1) the probabilistic RF and its 'best' wavelet-based version provided the most accurate and reliable forecasts, with average test set Nash Sutcliffe Efficiency Index (NASH) coefficients (i.e., across all lead times) of similar to 0.80 and 0.81, respectively; 2) the permutation- and bootstrap-based dataset partitioning approaches demonstrated potential to reduce overfitting; 3) wavelet decomposition improved probabilistic and deterministic ML model performance, increasing test set NASH coefficients by 1-7% on average (across all lead times); 4) wavelet-based models provided approximately the same level of reliability as the non-wavelet-based models but the best performing wavelet-based models improved forecast sharpness by an average of 14-24% (across all lead times); and 5) RFIVS substantially reduced the number of input variables used in the ML models (e.g., the number of inputs used in the wavelet-based models was often reduced by 50%) while still leading to improved performance over the case where all input variables were used.

Despite significant efforts to improve the calibration of hydrological models, when applied to real-world case studies, model errors (residuals) remain. These residuals impair flow estimates and can lead to unreliable design, management, and operation of water resources systems. Since these residuals are auto-correlated, they should be treated with appropriate methods that do not require limiting assumptions (e.g., that the residuals follow a Gaussian distribution). This study introduces a novel data-driven framework to account for residuals of hydrological models. Our framework relies on a conceptual-data-driven approach (CDDA) that integrates two models, i.e., a hydrological model (HM) with a data-driven (i.e., machine learning) model (DDM), to simulate an ensemble of residuals from the HM. In the first part of the CDDA, a HM is used to generate an ensemble of streamflow simulations for different parameter sets. Afterwards, residuals associated with each simulation are computed and a DDM developed to predict the residuals. Finally, the original streamflow simulations are coupled with the DDM predictions to produce the CDDA output, an improved ensemble of streamflow simulations. The proposed CDDA is a useful approach since it respects hydrological processes via the HM and it profits from the DDM’s ability to simulate the complex (nonlinear) relationship between residuals and input variables. To explore the utility of CDDA, we focus principally on identifying the best DDM and input variables to mimic HM residuals. For this purpose, we have explored eight different DDM variants and multiple input variables (observed precipitation, air temperature, and streamflow) at different lag times prior to the simulation day. Based on a case study involving three Swiss catchments, the proposed CDDA framework is shown to be very promising at improving ensemble streamflow simulations, reducing the mean continuous ranked probability score by 16-29 % when compared to the standalone HM. It was found that eXtreme Gradient Boosting (XGB) and Random Forests (RF), each using 29 input variables, were the strongest predictors of the HM residuals. However, similar performance could be achieved by selecting only the six most important (of the original 29) input variables and re-training the XGB and RF models. Additional experimentation shows that by converting CDDA to a stochastic framework (i.e., to account for important uncertainty sources), significant gains in model performance can be achieved.

Data-driven models have been explored in numerous studies for solar radiation ( $${R}_{s}$$ ) prediction. However, the use of different input variable selection (IVS) methods for improving $${R}_{s}$$ prediction accuracy has mostly been neglected. This study explores various IVS methods, including Gamma test (GT), Procrustes analysis (PA) and Edgeworth approximation-based conditional mutual information (EA) and evaluates their ability to improve $${R}_{s}$$ prediction accuracy by coupling them with popular non-linear data-driven models, multilayer perceptron (MLP), support vector machine, extreme learning machine and multi-gene genetic programming (MGGP). The partial correlation input selection method was coupled with multiple linear regression to serve as a linear benchmark. Meteorological data from eight stations in northern Iran was used for building the $${R}_{s}$$ prediction models. The type and number of variables selected at each station was dissimilar and dependent on the IVS method. The models utilizing EA selected fewer variables compared to the GT method and had higher accuracy, while models using PA selected fewer variables than all methods but were not able to adequately predict $${R}_{s}$$ . It was also found that predictive performance substantially varied when pairing the IVS methods with different model types. For example, MLP, the model with the best average performance, when coupled with EA instead of PA resulted in a ~ 27% improvement (decrease) in the normalized root mean square error (nRMSE). The results also indicated that MGGP produced the least accurate predictions, where the nRMSE increased by up to 40% compared to MLP when the EA method was used for IVS. Finally, IVS hyper-parameter adjustment (which is routinely overlooked in the literature) profoundly affected the results and is recommended as a very important step to consider when developing data-driven models for solar radiation prediction.

Hydrological time series modeling is an important task in water resources planning and management. However, time series may include noise, which can result in an inaccurate model. Therefore, removing noise from time series is valuable to obtain accurate predictions. The aims of this study are i) to develop and compare Long-Short Term Memory (LSTM) and Gated Recurring Units (GRU) Deep Learning (DL) models to predict hydrological time series and ii) to integrate a preprocessing method, Gaussian Filter (GF), to smooth out time series and couple it with DL to improve prediction accuracy. Moreover, the DL models are benchmarked against statistical time series models (e.g., Seasonal Autoregressive Integrated Moving Average (SARIMA)) to assess their added value for hydrological time series modeling. To establish predictive models, several monthly hydrological time series including water level (e.g., from the Great Lakes in North America, including Lakes Michigan, Ontario, and Erie (1918-2019)) and streamflow (e.g., gauging stations at Umfreville, along the English River, Ontario, Canada (1921-2019), Rapides Fryers, along the Richelieu River, Quebec, Canada (1937-2020) and near Lethbridge, along the Oldman River, Alberta, Canada (1957-2019)) were explored. For developing non-GF- and GF-DL models, time series were partitioned into training (70% of the data) and testing (the remaining 30% of the data) subsets and the time series’ past measurements up to 12 months (t-1, t-2, ..., t-12) were served to the DL models (LSTM and GRU) to predict the time series at time t. The structure of the DL models was tuned using Bayesian optimization. The SARIMA models (i.e., non-GF- and GF-SARIMA) were also implemented and tuned using pmdarima's auto-arima function. After calibrating the models, the testing step was implemented and the performance of the models was evaluated using statistical indicators including correlation coefficient, root mean square error, mean absolute error, the Nash-Sutcliffe efficiency coefficient, and Willmot’s index. The results of the developed DL models showed that the GRU outperforms the LSTM models. Moreover, both LSTM and GRU have superior performance when compared to the SARIMA models. It is observed that GF preprocessing significantly improves the accuracy of the developed DL and SARIMA models. It is concluded that coupling GF preprocessing with DL, due to capturing both linear and nonlinear features of the time series, represents a promising tool for obtaining accurate hydrological time series predictions.

This paper carefully studies long short-term memory networks (LSTM) for rainfall-runoff prediction, using a large-sample of catchments in France. The key focus is on exploring local and regional models as well as the impact of the ‘lookback’ period, an important hyper-parameter of LSTM, with respect to predictive performance and physical understanding of the model results. The authors include well-thought out experiments to identify the impact of the lookback period and cases where local and regional LSTM models are best suited. The authors also benchmark LSTM with GR4J, due to its useful ability to capture ground water exchanges with aquifers and/or between catchments. The authors spend a considerable amount of effort on tying the performance of LSTM, locally and regionally, to a physical understanding of the results. Some examples include the comparison between local and regional LSTM models with GR4J in terms of a water balance exercise in Section 5.3 as well as the ability of LSTM to predict runoff in controlled catchments at a higher degree of accuracy than GR4J (in Section 5.4). This paper also presents findings (e.g., LSTM does not necessarily outperform simple conceptual rainfall-runoff models) that are counter to other recent studies on LSTM (Gauch et al., 2021; Kratzert et al., 2019; Lees et al., 2021); in all such cases, the authors take the time to carefully describe potential reasons for these differences. Overall, this paper is very strong and I could not find much to criticize. The methodology seems correct. The figures are very nice and easy to interpret and I did not find any of the content, tables, or figures to be superfluous. I suspect this paper will be very useful to other researchers interested in exploiting the general-ity of machine learning

Developing accurate groundwater vulnerability maps is important for the sustainable management of groundwater resources. In this research, resampling methods [e.g., Bootstrap Aggregating (BA) and Disjoint Aggregating (DA)] are combined with machine learning (ML) models, namely eXtreme Gradient Boosting (XGBoost), Light Gradient Boosting Machine (LGBM), Adaptive Boosting (AdaBoost), Categorical Boosting (CatBoost), and Random Forest (RF), to improve the GALDIT groundwater vulnerability mapping framework that considers Groundwater occurrence (G) (i.e., aquifer type), Aquifer hydraulic conductivity (A), depth to groundwater Level (L), Distance from the seashore (D), Impact of existing seawater intrusion status (I), and aquifer Thickness (T). The proposed approach overcomes the subjectivity of the weights and ratings given to the six variables in the GALDIT framework (via the ML methods) and helps address the small dataset issue (via resampling methods) common to groundwater vulnerability predictive mapping. Considering the Shabestar Plain aquifer, situated in the northeast of Lake Urmia (Iran), the predicted vulnerability indices from GALDIT were adjusted using total dissolved solid (TDS, an indicator of drinking water quality) concentrations, and were modeled by the ML models. Pearson’s correlation coefficient (r) and distance correlation (DC) between the predicted vulnerability indices and TDS were used to validate the models. Using a validation set, the GALDIT framework (r = 0.447 and DC = 0.511) was compared against the best performing standalone (XGBoost-GALDIT, r = 0.613, DC = 0.647) and coupled resampling (BA-XGBoost-GALDIT, r = 0.659, DC = 0.699 and DA-RF-GALDIT, r = 0.616, DC = 0.662) ML models, revealing that the proposed framework significantly increases r and DC metrics. In general, the BA resampling method led to better performing ML models than DA. However, in all cases, it was found that integrating resampling methods and ML models are promising tools to improve the accuracy of GALDIT vulnerability models.

Random Tree (RT) and Iterative Classifier Optimizer (ICO) based on Alternating Model Tree (AMT) regressor machine learning (ML) algorithms coupled with Bagging (BA) or Additive Regression (AR) hybrid algorithms were applied to forecasting multistep ahead (up to three months) Lake Superior and Lake Michigan water level (WL). Partial autocorrelation (PACF) of each lake’s WL time series estimated the most important lag times — up to five months in both lakes — as potential inputs. The WL time series data was partitioned into training (from 1918 to 1988) and testing (from 1989 to 2018) for model building and evaluation, respectively. Developed algorithms were validated through statistically and visually based metric using testing data. Although both hybrid ensemble algorithms improved individual ML algorithms’ performance, the BA algorithm outperformed the AR algorithm. As a novel model in forecasting problems, the ICO algorithm was shown to have great potential in generating robust multistep lake WL forecasts.

This study introduces the maximal overlap discrete wavelet packet transform (MODWPT) for forecasting hydrological variables that exhibit change over multiple timescales (e.g., rainfall, streamflow). The advantage of the MODWPT over other recent wavelet decomposition methods (à trous algorithm (AT) and the maximal overlap discrete wavelet transform (MODWT)) is that it can extract finer scale information that may be important for improving forecast performance. Multiple wavelet decomposition methods (MODWPT, AT, MODWT) are integrated within the Wavelet Data-Driven Forecasting Framework (WDDFF), applied for forecasting monthly rainfall at six meteorological stations in the Awash River Basin (Ethiopia), and compared using eight statistical performance metrics. Results demonstrate that the MODWPT can be used to generate more accurate forecasts than the AT and MODWT for the majority of stations and performance metrics. Certain settings within the WDDFF (decomposition level, wavelet filter, input variable method, and data-driven model) lead to improved performance more often than others.