Accurate predictions of seasonal low flows are critical for a number of water management tasks that require inferences about water quality and the ecological status of water bodies. This paper proposes an extreme gradient tree boosting model (XGBoost) for predicting monthly low flow in ungauged catchments. Particular emphasis is placed on the lowest values (in the magnitude of annual low flows and below) by implementing the expectile loss function to the XGBoost model. For this purpose, we test expectile loss functions based on decreasing expectiles (from

Prediction of low flow in ungauged basins is a basic requirement for many water management tasks

Recently, data-driven models have gained interest for prediction of daily discharge in ungauged basins because of their fast implementation and good prediction performance. These approaches consider a wide range of models, e.g., long short-term memories

In this study, we propose to use the extreme gradient boosting model

The objective of our study is to develop such a spatiotemporal low-flow model and to evaluate its performance when predicting at ungauged sites. The following research questions will be addressed: (i) To what extent can spatiotemporal monthly low flow be modeled by one single gradient boosting model? (ii) How does expectile regression perform compared to traditional loss functions (mean absolute error, median absolute error)? (iii) How accurate are low extremes modeled by different expectiles? (iv) Which spatial and spatiotemporal variables are used for different expectiles? Our analysis will be performed on a comprehensive Austrian streamflow dataset representing a range of seasonal low-flow regimes.

Our study area covers 260 gauging stations in Austria with different low-flow seasonality. The dataset was already used in a wide range of studies

Example of the calculation of the monthly

Additionally, we used specific discharge quantiles with

Spatiotemporal modeling requires predictor variables, of which two types can be distinguished. The first type consists of climate and catchment characteristics representing the long-term average hydrological conditions. This type corresponds to typical predictors in low-flow regionalization models such as regional regression approaches

Descriptions of static predictors used in the study, structured in topological, land-use, geological, and meteorological characteristics. Abbreviations are further used in plots. Precipitation, climatic water balance, potential evapotranspiration, aridity index, snowmelt, and temperature variables are used on an annual and a summer/winter half-year basis. These different accumulation periods are indicated in the subscript: no subscript for annual characteristics (e.g.,

For the static predictors, we used a set of climate and catchment characteristics of precedent rationalization studies in Austria

For spatiotemporal covariates, the initial choice of the variables is also important, as it can affect model performance and interpretability of results. In a preliminary assessment (not shown in this paper), we tested several combinations of spatiotemporal covariates, including monthly precipitation, climatic water balance (CWB), temperature, snowmelt, solid precipitation, or soil moisture characteristics. All these combinations were tested by a nested 10-fold cross validation (CV; see Sect.

Example of the transformations of the climate water balance (CWB) with no lag at station Hollenstein. Panel

Description of the different lags and transformations for the CWB. Center means centering per station and month and the standardized drought index (SDI) is transforming the CWB to an SDI per station and month.

The monthly CWB series are further coded as time lags (

Apart from the static predictions and the spatiotemporal covariates, some variables were added to capture the temporal periodicity. The numeric variable of the month (

Extreme gradient tree boosting

One crucial point in our study is the application of a suitable loss function. The loss function has to be a twice differentiable convex function. Since our main aim is to model the low flows in the range of annual minima corresponding to the lower tail of the monthly

Comparison of different loss functions. Expectile loss functions are shown for various

Model evaluation is performed by a nested 10-fold CV

The nested cross-validation (CV) procedure as adapted from the double CV-scheme of

Model evaluation is performed by several error metrics. First, we quantify the overall performance of the model by means of four error metrics, the median absolute error (MDAE), the mean absolute error (MAE), the root mean squared error (RMSE):

Second, we assess the predictive performance for individual stations by calculating the

For a more comprehensive evaluation of the performance, we perform a decomposition of the station-wise prediction errors on different time scales. The purpose is to assess the extent to which the model errors occur at the annual, seasonal, and monthly level, and which part of the error is due to a systematic error (i.e., bias). This will allow us to get insight into structural strength and weaknesses of the models. This decomposition is done by a three-way ANOVA, which was applied by e.g.,

The last part of our model evaluation specifically emphasizes the extreme low flows. This is assessed by three performance metrics. First, we filter the observations at each station by a specific quantile and calculate the overall

Finally, we want to capture the model ability of classifying extreme low-flow events. For this purpose, the specific low-flow quantiles

The data analysis was performed in R

Table

Overview of the error metrics for all loss functions. Shown are the MDAE, MAE, RMSE, and

A more detailed examination of our results is realized by analyzing the performance per station. Table

Performance per station summarized by the median

Empirical cumulative distribution function of station-wise

For a better understanding of the error at the individual stations, we decompose the model error at each station into monthly, seasonal, annual fractions and a component representing the average error (prediction bias). Figure

Error components of various expectile models. Panel

An additional perspective comes from analyzing the share of stations showing only moderate or weak performance, as indicated by an

One objective of this study was the application of different expectiles for improving predictions at low extremes. This section will consequently focus on their evaluation. In a first step, we will filter our observations by only considering the observations at each station below a specific quantile. The filtered observations are then used to calculate an overall

Global

EL

As a second assessment of the predictive performance for low extremes, we compute the relative expectile error (

The predictions for the

In a final assessment of the model performance, we analyze the skill of the models to classify extreme events. For this purpose, we focus on the hit score (

The precision

The implemented RFE algorithm leads to a substantial reduction of variables. On median, the different expectiles require between

Selection of spatiotemporal variables that are used for different expectiles. For each expectile it is shown whether the variable is used for the final prediction of the fold or not.

Figure

In this paper, we extended the extreme gradient boosting model XGBoost with the expectile loss function to develop a spatiotemporal model for low-flow predictions. We applied different

Overview of the error metrics EL

Performance evaluation in light of the existing literature is not straightforward, as we are not aware of any studies evaluating monthly low-flow models. Nevertheless, some studies did model the mean monthly streamflow, which we will use for comparison. Comparisons will be made on the NSE reported in these studies, which is an analogue measure to the

Panels

A more qualitative embedding to the scientific literature can be made by integrating our findings into the comparative evaluation of regionalization procedures of

In this paper, we analyzed the performance of a single spatiotemporal XGBoost model on the prediction of monthly low flow for a comprehensive dataset of

Weak-performing stations can also be found for the

Despite the low global performance of small expectiles (

We demonstrate that the expectile loss is a suitable alternative to common loss functions in spatiotemporal low-flow models. However, its application is not limited to statistical learning models such as XGBoost, but can also be considered for hydrological models when their focus is on predicting hydrological extremes. As with all models, there is a trade-off between overall predictive performance, accuracy on the tails of the distribution, and identification of extreme events that should be considered in model applications.

Data and code can be made available on personal request to johannes.laimighofer@boku.ac.at.

JL designed the research layout and GL contributed to its conceptualization. JL performed the formal analyses, and JL and GL prepared the draft paper. MM supported the analyses. GL supervised the overall study. All the authors contributed to the interpretation of the results and writing of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Johannes Laimighofer is a recipient of a DOC fellowship (grant number 25819) of the Austrian Academy of Sciences, which is gratefully thanked for financial support.

Data provision by the Central Institute for Meteorology and Geodynamics (ZAMG) and the Hydrographic Service of Austria (HZB) was highly appreciated. This research supports the work of the UNESCO-IHP VIII FRIEND-Water program (FWP).

This research has been supported by the Österreichischen Akademie der Wissenschaften (grant no. 25819).

This paper was edited by Elena Toth and reviewed by three anonymous referees.