Providing quality uncertainty estimates for streamflow predictions is critically important, particularly in applications such as operational drought simulation, water resource management, and flood mitigation where significant stakes are involved . Poorly quantified or overly confident predictions can lead to misinformed decisions, potentially resulting in economic losses, infrastructure damage, or even threats to public safety. To address this, various approaches have been proposed in the hydrological community for streamflow uncertainty quantification, including multi-model ensembles , Bayesian inference , and hydrological error modeling , referred to as post-processing. Post-processing involves a process-based model followed by a statistical approach to correct errors and quantify the associated uncertainties. With the post-processing procedure, uncertainties can be quantified by modeling the error patterns based on an archive of past data.

Hydrological post-processing techniques were adopted early through methods such as the hydrological uncertainty processor (HUP) and model conditional processor (MCP) , but recent machine learning (ML)-based approaches have emerged as powerful tools for hydrological post-processing. Although less interpretable, ML-based approaches can potentially produce reliable and more informative uncertainty estimates . Methods such as quantile regression (QR) , conformal prediction , and random forests have been used for streamflow post-processing with promising results. However, ML algorithms can produce different uncertainty estimates depending on how they are trained – particularly on which catchments are included in the training dataset. Since hydrological conditions vary significantly across catchments, the selection of catchments used for training can influence the uncertainty estimates of the ML model. In our study, we aim to explore whether including different catchments, referred to as regional or multi-site learning, may improve ML-based post-processing for error-correction and uncertainty prediction, and specifically for the quantile random forests (QRF) model. A QRF model for error-correction and uncertainty prediction trained in a regional setting can leverage information across several catchments. In contrast, local at-site approaches train models independently for each catchment and cannot benefit from error patterns shared in hydrologically similar catchments.

Random forest (RF) and its probabilistic variant, quantile random forest (QRF) are extensively used and are considered state-of-the-art in many hydrological applications. Recently, compared the QRF model and the countable mixtures of asymmetric Laplacians long short-term memory (CMAL-LSTM) model to probabilistically post-process streamflow simulations across 522 catchments. The QRF and CMAL-LSTM models were comparable in terms of uncertainty estimates, but the CMAL-LSTM deep learning (DL) model performed better in catchments with large flow accumulation areas. QRF has also been applied in hydrologically informed post-processing approaches. used in an RF framework and leveraged internal state variables to correct PCR-GLOBWB (PCRaster Global Water Balance, a global hydrological model) simulations at three stations in the Rhine Basin. They found that the use of hydrological model states as input features of RFs provides additional information that may not be included in the model simulations. However, challenges remain, particularly in modeling errors during high streamflow periods. expanded the same approach at a global scale, using PCR-GLOBWB model simulations and internal states, in conjunction with static catchment attributes, to train a single RF model on a global database of streamflow simulations and measurements. They found that improvements were independent of the availability of streamflow data, indicating the power of regional learning methods in poorly gauged and ungauged catchments.

Prediction in ungauged basins is not the only benefit of training a single ML model on data from multiple catchments. advocated the use of regional approaches to fit a deterministic Long Short-term Memory (LSTM) DL model for streamflow simulations. They found that larger LSTM models trained on all available basins outperform smaller models trained on a limited set of catchments. This is because some ML approaches can properly use the additional information contained in larger training sets to perform better than models specialized for individual catchments. Furthermore, found that hydrological model performance depends on catchment characteristics, indicating the presence of regional bias, where hydrological errors exhibit similar properties for neighbouring catchments. This can be harnessed to improve uncertainty estimation using a post-processing model with a regional parametrisation and trained on hydrological errors from multiple sites. We intend to consider these catchment characteristics as additional input features within the proposed post-processing model.

This study addresses uncertainty estimation of process-based hydrological model simulations, with a focus on streamflow reconstruction and future projection scenarios. We investigate the added value of multi-site post-processing applied to a quantile random forests (QRF) model. The main contributions of this work are: (i) to understand the impact of including different catchments in the training process of QRF (multi-site) on the quality if its uncertainty estimates and (ii) to investigate the importance of catchment characteristics for these multi-site QRFs. Although multi-site approaches can benefit the modelling of ungauged catchments, our work specifically addresses improvements in uncertainty estimation for catchments with available streamflow measurements.

Accordingly, we use temporally varying information (predicted streamflows and model states) in addition to catchment dependent characteristics to estimate uncertainties in hydrological model predictions. We chose to focus on QRF due to its balance of performance, interpretability, and popularity in the hydrological community. To the best of our knowledge, no prior study has explored the impact of multi-site learning with the QRF algorithm for uncertainty estimation, particularly when post-processing a hydrological model calibrated separately for each catchment. To that end, we fit different QRF variants on the internal states of a hydrological model, on meteorological variables, and on readily available catchment characteristics. The proposed regional QRF variants are evaluated across a large sample of 564 French catchments to identify when multi-site learning may be beneficial and to offer practical considerations for multi-site QRF applications.

The paper is organized as follows: We first introduce the dataset and describe the QRF algorithm, its variants, and the probabilistic evaluation framework. Then, we present and discuss the results before summarizing the key findings along with implications for future work.

We used a set of 564 catchments distributed throughout France (Fig. ). These catchments represent a wide range of hydrological regimes and simulation contexts. We selected these catchments from the CAMELS-FR hydroclimatic dataset . The criteria for selecting these catchments were as follows: (i) low anthropogenic influence, (ii) good data quality for all flow regimes, and (iii) an available time series longer than 21 years. Streamflow data were obtained from the national HydroPortail archive at a daily time step for the period 1977–2021. Meteorological forcings (precipitation and temperature) were provided by Météo-France's daily SAFRAN grid reanalysis . Potential evaporation (PET) is calculated using the formula proposed by and requires two inputs: extraterrestrial radiation (MJm-2d-1) and mean daily air temperature (°C). Extraterrestrial radiation is computed as a function of the localization of the basin and Julian day values and the temperature is the only dynamical meteorological input used to estimate PET. Since our interest is in developing a multi-site QRF post-processor, we used several static basin-averaged attributes describing climate, topography, and geology. All of these attributes are included in the CAMELS-FR dataset and are listed in Table .

Hydrological calibration was performed independently for each catchment over the 1977–2021 period. Subsequently, the QRF variants were implemented using a standard train–validation–test split over the 1990–2021 period:

P1: training period from 1990 to 2004, used to train the QRF post-processor.

We used discharge simulations obtained with the GR6J rainfall–runoff model , a daily 6-parameter conceptual lumped model. GR6J has been applied in several studies across a large number of catchments and hydroclimatic contexts e.g.. The GR6J model is based on several state variables that control its simulations, in particular, the production and routing store levels, as well as intercatchment exchange fluxes. We intend to use these state variables as predictors in the QRF algorithm. successfully used internal state variables as predictors in an RF framework to correct hydrological model errors. They found that internal state variables provided valuable information for the RF, enabling it to detect and correct for systematic hydrological model errors. To account for the influence of snow in some catchments, we incorporated Cemaneige , a snow accumulation and melt model, with constant parameters for all catchments.

The Cemaneige-GR6J model was calibrated using the airGR R package using the built-in calibration algorithm. To ensure good performance across a wide range of streamflow conditions, the target optimization criterion was a combination of KGE based criteria : an equal weighting of KGE criteria with power transformations of 0.5 and -0.5 applied to streamflow, as detailed in Appendix . This calibration approach was implemented to obtain the six parameters of GR6J: production store capacity (mm), groundwater exchange coefficient (mmd-1), routing store capacity (mm), time constant of unit hydrograph (days), groundwater exchange threshold (–), and exponential store coefficient (mm).

Random forest is a non-parametric ensemble tree-based model that offers good performance and provides certain interpretability through its feature importance estimates . RF and its probabilistic version QRF are used extensively in the hydrometeorological domain. An important advantage of QRF is that it provides full distributional estimates without the need to estimate each quantile separately, as is required in quantile regression . QRF has been applied to complex and heteroscedastic cases, including hydrometeorological ensemble forecasts , post-processing of streamflow simulation , and estimation of the limits of acceptability for hydrological models . Further details on the construction of RF and QRF can be found in , but, QRF can be viewed as an analog method that performs a weighted nearest-neighbor search for analogous events. Similarly to a classic RF, QRF grows a number of trees n, with each tree trained on a bootstrapped subsample of the original training data.

Individual trees are trained according to the algorithm by minimizing a loss function and making successive splits with a predefined number of features p. This tree-building process enables QRF to account for strongly correlated features, which is important given the strong correlation of some of the features used. For the purpose of this study, we use mean squared error (MSE) as a loss function to calculate the homogeneity of each group. The procedure continues recursively, with each resulting group split further until a minimum number of data points m in child splits is reached.

In the classic Random Forest (RF) algorithm, predictions from individual trees are averaged to produce a single deterministic output. In contrast, Quantile Regression Forest (QRF) leverages the leaf nodes of trees to compute proximity measures between a test input and training instances. For a prediction at time t and given input xt, each QRF tree is traversed using binary splits to reach a corresponding leaf node. A proximity weight ωi(xt) is then defined for each training instance i , which is then used to estimate the cumulative distribution function (CDF) of the prediction uncertainty: 2F^(ϵ|xt)=∑i=1nωi(xt)1{ϵi≤ϵ} where ωi(xt)>0 and ∑i=1nωi(xt)=1, ϵi denotes the hydrological error of training instance i, and F^ is the estimated CDF of uncertainty for xt.

To provide reliable and sharp uncertainty estimates, we considered three hyperparameters for optimization: (i) The minimum number of samples at child nodes m, which affects tree depth and strongly impacts reliability and sharpness. Setting high values for the minimum samples per leaf might yield high reliability, but can lead to poor performance, as the trees are too general and information is lost. Low values result in overfitting and unreliable uncertainty estimates. (ii) The number of features per split, p, which also shapes the QRF uncertainty estimates. Higher values can lead to under-dispersed uncertainties, while lower values may reduce sharpness. (iii) The number of trees n, which controls precision and stability. A larger number of trees improves the quality of uncertainty estimates, but improvements diminish as computational cost increases, especially in larger models. Further details on hyperparameter values and selection are provided in Sect. . Additionally, we also use a K-nearest neighbor (K-NN) approach as a benchmark for the QRF methods used in the study. Like QRF, K-NN aims to find analogous events but based on the Euclidean distance between features. Here, K-NN is fitted locally on the same variables as for QRF. Further details on the fitting process and the hyperparameters used are provided in Appendix .

Given Eq. (), the estimated CDF is bounded by the training sample. QRF is unable to predict a quantile higher than the maximum observed in the training sample, which implies that QRF trained on a single basin is constrained by the range of errors in its training data. A more hydrologically diverse training dataset would alleviate this problem and enable QRF to adapt to more extreme events, provided that QRF is able to use the additional information properly.

Figure  presents the framework employed for the QRF configurations analyzed in this study. The local approach (QRF-local) refers to training the QRF algorithm on data from a single basin. Given their construction, spatial features are constant for each individual catchment and only time-series features can be fed to QRF in the local setup. QRF-local yields 564 independently trained QRF models, each specific to its respective catchment. Next, spatial variability is added as we extend QRF to a multi-site setting. The objective here is twofold: (i) to examine whether spatial diversity can improve the uncertainty estimates of QRF. This can be challenging, particularly since the used GR6J hydrological model is calibrated on an on-site basis; and (ii) to determine the optimal number of catchments to include in the training set for effectively capturing hydrological diversity and improving QRF predictions. We test QRF with two spatial settings: (i) a regional approach (QRF-region), where QRF is trained on data from catchments that are geographically close and thus potentially have similar error dynamics. In total, 15 regional QRF models are developed for the hydrological regions of the study (see Fig. ), based on hydroclimatological groupings of French catchments; (ii) a global approach (QRF-national), in which a QRF is trained on data from all catchments in the dataset. Both static and dynamic descriptors are used in the training process of QRF-region and QRF-national. However, QRF-national uses the catchment's hydrological region as an additional input feature, which cannot be used for QRF-region. Intuitively, in cases where QRF is unable to transfer information from different basins or when there are no useful analogs in similar catchments, QRF-local would yield better performance, as no information from other catchments is used to build the model. To assess the usefulness of static features for the multi-site QRF setup, we included QRF-basic, a global QRF approach fitted on all catchments of the study, but only with dynamic features. This experiment is expected to highlight whether dynamic time series features are sufficient to improve multi-site QRF predictions or that static features are essential for multi-site post-processing. Table  presents the features used in the three configurations. The QRF models used in this study was fitted using the quantile-forest Python library .

It is worth mentioning that in multi-site setups, the standardization procedure is an essential step that enables QRF to determine analogs across a set of diversified catchments, as the scales of streamflows and dynamic features (GR6J states and transformed variables) vary significantly. Standardization is important for a meaningful training process and for the identification of adequate analogous events. Initially, we standardized input data via the popular standard scaling method , which transforms dynamic features – for each catchment – so that the average and standard deviation are set to 0 and 1, respectively. However, the method resulted in inconsistencies for catchments with outliers, as the standard deviation is sensitive to extreme values. To solve this issue, we opted for robust standardization , which removes median values of dynamic features and the target errors ϵt defined in Sect. .

Since we use QRF for probabilistic predictions, hyperparameter selection was based on the mean of the alpha score and CRPSS values. This would enable a selection based on the quality of overall uncertainty estimates with an emphasis on reliability. For QRF-local, hyperparameters were tuned independently for each catchment and the set maximizing the aforementioned criteria during the validation period was selected. QRF-region and QRF-national hyperparameters were selected based on median criteria among the regions' catchments. The selection criteria for QRF-local are specific to each catchment, which is expected to enhance the results of the local model compared to approaches that use fixed hyperparameters across multiple catchments. Table  presents the hyperparameters selected for optimization.

The alpha score targets reliability. It calculates the closeness of predicted uncertainty distributions to the statistical distribution of observed streamflows. If the uncertainty distributions of streamflows are reliably quantified, the observations correspond to realisations from the uncertainty distributions of streamflows. In practice, the alpha score compares the empirical distribution of the probability integral transform (PIT) values to the uniform distribution on [0,1]. A perfect alpha score corresponds to uniform PIT distributions while deviations of PIT values from the uniform distribution indicate lower reliability. The values of the metric range from 0 (worst reliability) to 1 (perfect reliability).

For sharpness, we used the dispersion score calculated as a skill score following . The method consists in computing the continuous ranked probability score (CRPS; ) by comparing uncertainty distributions with their medians, as detailed in Appendix . This formulation targets the magnitude of the distributions instead of the agreement with the observations. To obtain a positively oriented score, the dispersion score is expressed as a skill score by dividing by the same quantity computed for the climatological distribution. The resulting metric scores 1 for a perfect point prediction; positive values indicate better sharpness compared to the climatological distribution, while negative values indicate worse performance.

We also compute CRPS by comparing uncertainty distributions to observed streamflow values, which allows to assess both reliability and sharpness. We express CRPS as a skill score (CRPSS) relative to the climatological distribution, and similarly to the dispersion score, CRPSS equals 1 for perfect point prediction; positive values indicate better performance than the climatological distribution, while negative values indicate worse performance. Throughout the study, the empirical distribution of observed streamflows is defined over the training period, as detailed in Appendix .

To provide a more comprehensive assessment of predictive uncertainty, evaluation metrics were calculated for prediction intervals at the 90 % and 95 % confidence levels. The coverage ratio (CR) is a measure of reliability that counts the proportion of observations that lie within the prediction intervals. Values closest to the desired coverage level (i.e., 90 % or 95 %) are best. Scoring lower values diminishes the utility of the model (under-coverage), while scoring higher values than the desired coverage is less problematic, but indicates that the model can provide sharper intervals (over-coverage). Moreover, it is worth highlighting that the two metrics used to assess reliability are closely related: the coverage ratio reflects reliability at a specific confidence level, while the alpha score aggregates coverage across all confidence levels. To assess the sharpness, we employ the average width metric (AW), which corresponds to the average width of the prediction interval during the evaluation period. We also evaluate the Winkler score (WS), which simultaneously includes both criteria, and enables an easy comparison between the variants of the study. Both AW and WS are presented as skill scores – AW skill score (AWSS) and Winkler skill score (WSS) – relative to the climatological distribution.

Although the main focus of this study is probabilistic post-processing, some decision-makers may require deterministic predictions. Therefore, we also evaluate mean predictions to compare the different post-processing variants of the study. We use the popular Nash–Sutcliffe efficiency (NSE) and Kling–Gupta efficiency (KGE) metrics to gauge the quality of deterministic predictions in multi-site learning setups.

We conducted a hyperparameter grid search for each QRF variant using an equal contribution of the alpha score and CRPSS (α+CRPSS2) as mentioned in Sect. . Figure  shows the hyperparameter tuning procedure for QRF-national's three hyperparameters: the minimum number of samples at child nodes, the maximum number of candidate variables to use for splitting at each tree node, and the size of the forest (number of trees). We aim to obtain a single set of hyperparameters, and the selection criterion is based on the median of the alpha score and CRPSS across all catchments of the study. Each subplot illustrates how the selection criterion varies with the hyperparameter under consideration, while the variability reflects the influence of the remaining hyperparameters. This allows to assess the sensitivity of the QRF-national to each hyperparameter during the optimisation process. Overall, the performance of QRF was most sensitive to the minimum number of samples at child nodes. QRF was trained with minimum number of samples at child nodes ranging from 5 to 600 data points.

It is notable that best results were recorded for lower values of the minimum samples at leaf nodes. The improvement slows for values lower than 25. As such, a minimum sample at leaf nodes of 10 is selected. Overall, QRF was found to be fairly insensitive to the number of candidate predictors used for splitting at each node. By default, the quantile-forest library uses the integer value of the square root (sqrt) of the total number of predictors for this parameter. With 31 total predictors for QRF-national, 6 would be the default. Figure  shows that using the default value of the square root was slightly better. For the number of trees parameter, a forest with more trees will generally be more skillful than one with fewer trees, as it can fit on the nuances of the training set, and there is a point when the rate of improvement with more trees is negligible, as noted in and . Most of the boxplot ranges overlap, and the results appear to be relatively insensitive to this QRF parameter over the range considered.. For the experimented values, Fig.  shows that a number of 400 trees allows for slightly better performances. We selected hyperparameters for the other QRF variants using a more specific basis: per catchment for QRF-local and per region for QRF-region. One can expect that catchment specific hyperparameter tuning might help model performance. The distribution of the selected hyperparameters for QRF-local and-region is provided in Appendix , as mentioned in Sect. 2.7.

We first present our results with distributional metrics in Fig. , which shows the cumulative distribution of reliability, sharpness, and CRPSS for the 564 catchments of the study. QRF-region and QRF-national slightly improve reliability compared to QRF-local. However, multi-site learning does not yield better alpha scores for well-calibrated stations with QRF-local. Figure  shows a direct comparison of the proposed variants and indicates that improvements were most noticeable for catchments where QRF-local provided low reliability, i.e., 25 % of the basins with the lowest alpha scores (the 25 % quantiles of the alpha score were 0.742 for QRF-local and 0.76 and 0.76 for QRF-region and QRF-national, respectively). In terms of sharpness, the different QRF variants performed similarly, while multi-site setups significantly improve CRPSS values. Among the QRF variants, QRF-national generally outperformed QRF-local, improving CRPSS by approximately 2 %, except in the case of four catchments, where QRF-local performed significantly better. Additionally, QRF-region improved CRPSS for 69 % of the catchments compared to QRF-local, while QRF-national showed improvements in 88 % of the catchments. Overall, the improvements are less apparent for reliability, but multi-site QRFs seem to improve performance for catchments with initially limited reliability in the local setup. As highlighted in Sect.  CRPSS improvements combine both reliability or sharpness. Given that the sharpness metric was nearly identical across the QRF variants in the study, we suspect that the CRPSS improvements are mainly due to improvements in reliability. This is noteworthy, as the objective of probabilistic predictions is to improve reliability prior to enhancing sharpness .

We now consider the 90 % interval metrics. Figure  represents the box plots across the 564 catchments of the study for coverage ratio, average interval width, and Winkler skill scores. The multi-site learning setup was beneficial for QRF and improved the reliability of the predictive intervals. For instance, the median coverage ratios were set at 0.87, 0.89, and 0.89 for QRF-local, QRF-region, and QRF-national, respectively. Similar to the alpha score, the improvements in the multi-site QRF variants are most noticeable for stations with low reliability in the local approach. As shown in Fig. , prediction intervals from multi-site QRFs over-cover observed streamflows (coverage ratio greater than 0.9) for certain catchments. While not optimal, this is a preferable outcome compared to the local approach, where uncertainty intervals more frequently miss the observed streamflows and tends to underestimate uncertainty in some catchments. The improvements are also observed for Winkler skill score, where QRF-national provided the best results. The average interval width was similar for all the variants in the study, further indicating that improvements in multi-site learning in the case of QRF mainly relate to reliability. For the sake of completeness, we include interval metrics for the 95 % predictive uncertainty interval in Appendix , as the conclusions remain the same.

Figure  shows the cumulative distribution function for the deterministic metrics of Nash–Sutcliffe efficiency (NSE) and Kling–Gupta efficiency (KGE) scores for mean predictions. Multi-site learning improves NSE for most catchments, but for KGE, improvements are most apparent when the local approach yields low KGE values. For example, when investigating catchments at the lower 25 % percentile, QRF-region and QRF-national improved the median KGE by 6 %. However, for catchments where QRF-local provided decent KGE scores (top 25 % performers), multi-site setups yielded similar scores to a single-basin approach. This would highlight the equalizing effects of multi-site learning for QRF, as it is most impactful for catchments with limited performance in the single-basin post-processing.

Figure  also provides deterministic metrics for the uncorrected GR6J model predictions. The figure highlights that the proposed QRF methods can improve hydrological deterministic predictions, especially for NSE. For example, QRF-national produced better NSE performance compared to GR6J predictions (0.87 vs 0.86 in median NSE) and for 75 % of the study’s catchments. Overall, QRF variants had better NSE for the majority of the catchments. For KGE, the uncorrected GR6J estimates outperform all tested QRF approaches.

We argue that these results show: (i) the ability of QRF in its multi-site setup to identify and transfer useful information from neighbouring catchments; (ii) although the improvements relate to both deterministic and uncertainty predictions, they are most significant for coverage ratio, CRPSS, WSS, NSE, and KGE. Building on these findings, we investigated whether these benefits were more pronounced under specific hydrological conditions.

Here, we aim to understand how the proposed QRF approaches perform across different flow ranges. Table  summarizes the average values of the alpha, dispersion, CRPSS, and interval scores for three flow groups: high (> 67 % Qmed), medium (> 34 % Qmed and < 66 % Qmed), and low flows (< 33 % Qmed).

Performances were stratified based on the median values of the uncertainty distributions. Under low-flow conditions, the scores are similar, especially alpha and dispersion scores. But for higher simulated flows, multi-site QRFs provide better reliability (alpha score and coverage ratio) and better overall performance (CRPSS and WSS). Although QRF-local was able to provide narrower interval widths, especially for higher flows, it had lower reliability compared to multi-site QRFs. QRF-region and QRF-national adapt to higher-flow ranges by providing wider uncertainty estimates and enable better reliability and conditionality, as reflected in the improved CRPSS and Winkler scores.

To understand the impact of static catchment descriptors, Fig.  illustrates the distributional metrics for QRF-national and QRF-basic. QRF-basic is a multi-site QRF trained across all catchments of the study and using the same features as for QRF-local (no static features). Notable differences between the two variants are observed: in terms of reliability (median 0.827 vs. 0.806 across all catchments), sharpness (0.637 vs. 0.614), and CRPSS (0.706 vs. 0.691). Largest differences are observed for CRPSS, as QRF-national was better for 80 % of the stations of the study. Furthermore, Fig.  in Appendix  shows that QRF-basic had very similar CRPSS values as for QRF-local. These results suggest that the performance improvements in multi-site QRF models are not solely due to the inclusion of hydrological diversity in the training data. Static catchment descriptors play a significant role, and the selection of informative static features appears to be critical for effective multi-site QRF implementations.

Following the results of the previous section, we found that multi-site learning can significantly degrade the performance for four cases across distinct hydrological regions; an example of such catchments is presented in Fig. . To investigate this, Table  presents the differences in performance (alpha score and CRPSS) between QRF-national and QRF-local based on the variability of errors ϵt for three groups: group 1 is characterized by important error variability (> 1.5), group 2 also has important error variability but to a lesser degree (0.77 < and  < 1.5), and group 3 (< 0.77) which can be seen as having normal variability. Values 1.5 and 0.77 are the 99 % and 90 % quantiles of the interquartile range used to standardize the errors ϵt. QRF-national performed poorly for catchments with significant variations in the target variable, with notable decreases in reliability and CRPSS compared to a single-basin approach. These results highlight that robust standardization of input variables and errors is key to delivering meaningful multi-site QRFs, since this enables the algorithm to find analogs from locally calibrated hydrological model inputs. Furthermore, the aforementioned scale discrepancies occurred specifically for catchments characterized by frequent zero values in simulated and observed streamflows. This can be problematic when using logarithmic relative hydrological errors. Figure  illustrates QRF-local and QRF-national predictions for the 5 % and 95 % quantiles alongside observed flows for the Y960000102 catchment. Clearly, QRF-national overestimates the upper quantile. The local approach thus yields better results, since the error dynamics of this catchment are unconventional compared to the other catchments of the study.

Although training QRF on local data yields good uncertainty estimates, as discussed in previous studies , using a multi-site setup can slightly improve QRF performance. Our results indicate that the best improvements are achieved with QRF-national, which includes all 564 catchments of the study. The improvements mainly concern (i) catchments where local information is not sufficient (i.e., limited QRF-local performance), (ii) CRPSS and WSS scores for nearly all catchments, and (iii) periods of higher flows.

The results presented in Sect.  indicate that multi-site learning improves the performance of QRF models, and that larger models yield better uncertainty estimates (QRF-national > QRF-region > QRF-local). We find this result interesting, since one might argue that regional models in the QRF-region approach provide an equilibrium in spatial variability by aggregating similar catchments and, possibly, similar error dynamics . Such an approach confines the QRF analog search to specific hydrological regions, without extending the search to the entire study area. QRF-national, however, is not constrained by the predefined hydro-climatological regions. Region indicators are included as input features for QRF-national, and the algorithm is trained to find analogous events using these indicators, but is not strictly limited by them. Spatial variability appears to be beneficial for QRF, provided that appropriate catchment descriptors are used, and incorporating explicit measures of catchment similarity could further improve multi-site learning with QRF .

Most improvements are noted for high and medium flows. QRF-national provided a better alpha score, i.e., reliability for 60 % of the catchments during high and medium flows compared to QRF-local, but performance was identical for low flows. As highlighted in and , high flows are generally more difficult to predict and some high-flow events cannot be predicted exclusively from local historical data. QRF makes use of information from neighboring catchments to provide uncertainty estimates for these events that can be more challenging to predict. Local information seems to be sufficient to characterize low-flow events.

The provided analyses highlighted a limitation of the multi-site QRFs which concerns catchments characterized by frequent zero flow values. Modelling hydrological model errors of ephemeral catchments is generally challenging , particularly with the use of a log-based error transformation. Figure  shows that multi-site learning can degrade the predictions for ephemeral catchments as uncertainties are overestimated. Although we have added the δ offset parameter, the use of an alternative transformation that is better suited to zero flow values (e.g., a Box–Cox transformation) could better stabilize hydrological errors used to train the QRF models for ephemeral catchments. Another solution could be the treatment of such catchments separately when training multi-site QRF. As showed in Fig. , QRF-local better managed the case of zero flow values. In the literature, other approaches use adapted catchment groupings based on clustering approaches grouping homogeneous catchments. The use the number of zero flow values or the clusters as inputs to QRF can allow QRF to better distinguish catchments characterized by this issue.

Also, we expected that the proposed variants would also improve KGE as was observed in the previous studies of . For example, used a closely related deterministic RF framework for hydrological error correction and found that the hybrid RF approach boosted streamflow predictions from a median of -0.03 to 0.51. Here, the post-processing was not beneficial for KGE performances, and we suggest that this can be attributed to how QRF hyperparameters were selected. The used selection criterion in this study aims to maximize the probabilistic performances of reliability and sharpness of the uncertainty estimates. While for the aforementioned studies, the RF post-processor was optimized specifically for deterministic error correction.

We showed in Fig.  that the improvements in QRF-national depend on the use of static descriptors. A nation-wide QRF variant with no catchment attributes (identical input descriptors to the local variant) performed worse than a classic single-catchment QRF. This indicates that increasing hydrological diversity and lumping more catchment data are not the primary drivers of performance improvements. The information shared within a multi-site setup is best used by the QRF algorithm in conjunction with quality catchment descriptors. This would enable better uncertainty characterization and improved analog searches in similar catchments. The catchment descriptors used are readily available in the CAMELS-FR and other CAMELS datasets, making the use of such descriptors straightforward. Furthermore, a globally parametrized QRF post-processor is able to extend its uncertainty estimates into ungauged catchments. found that RF is able to learn global mappings and improve deterministic estimates in poorly gauged and ungauged basins. Similar improvements could be obtained in an uncertainty estimation at ungauged catchments context, if appropriate catchment descriptors that do not rely on observed streamflows are selected.

We used a large sample dataset (CAMELS-FR) to train the QRF model, and many practical hydrological applications can be interested in applying QRF post-processing to catchments not included in the training set. Our study demonstrates the ability of the QRF model to make use of hydrological information from neighbouring catchments to improve uncertainty estimates. Building on this finding, we suggest that applying a multi-site QRF, supported by appropriate catchment descriptors, to catchments outside the training set is likely to yield improved uncertainty estimates compared to a QRF model trained with single catchment data. However, the quality of these estimates remains unexplored, as the generalizability of multi-site QRF variants would depend on the similarity between hydrological model errors between the training catchments and the target regions to which uncertainty estimates are extrapolated. A spatio-temporal cross-validation experiment that splits the training and testing by catchments and time periods can be used to understand the flexibility of multi-site QRF post-processing. Also, the proposed framework can be adapted for a prediction at ungauged basins. to explore this extension, the main practical difficulty lies in obtaining consistent hydrological model states for ungauged catchments, and to adapt the significant additional uncertainty usually associated with such settings . If this can be properly handled, the proposed QRF multi-site variants could provide meaningful uncertainty estimates in the context of uncertainty estimation for ungauged catchments. A comparison to LSTM based post-processing techniques would be also interesting, as LSTMs perform well in ungauged basin setting .

Table  presents the number of parameters for each QRF variant, calculated as the product of the number of trees and the parameters of each tree (e.g., split thresholds, used input features). QRF-region and QRF-local exhibit a similar number of parameters, despite QRF-region providing better uncertainty estimates. In contrast, QRF-national shows a 47 % increase in model complexity. This suggests that the performance gains of QRF-national come at the expense of increased computational cost which can be a drawback, especially since QRF stores not only the tree parameters but also samples used for training.

RF-based algorithms are CPU-intensive and suffer from memory voracity, especially for larger datasets . In the case of our study, we had no difficulty fitting QRF-local and QRF-region with an Intel(R) Core(TM) i7-4770 CPU (3.40 GHz) and 16 GBs of memory. However, because of memory issues, we trained QRF-national on Jean-Zay HPC, where a single node with two CPUs (at 2.5 GHz) and 128 GBs of memory was sufficient. With this configuration, it takes on average 25 min to fit QRF-national with a single parameter. While training time is longer for multi-site settings, inference/prediction times are very similar to those of QRF-local.