Non-parametric Bayesian networks (NPBNs) are graphical tools for statistical inference widely used for reliability analysis and risk assessment and present several advantages, such as the embedded uncertainty quantification and limited computational time for the inference process. However, their implementation in hydrological studies is still scarce. Hence, to increase our understanding of their applicability and extend their use in hydrology, we explore the potential of NPBNs to reproduce catchment-scale hydrological dynamics. Long-term data from 240 river catchments with contrasting climates across the United States from the Catchment Attributes and Meteorology for Large-sample Studies (CAMELS) data set will be used as actual means to test the utility of NPBNs as descriptive models and to evaluate them as predictive models for maximum daily river discharge in any given month. We analyse the performance of three networks, one unsaturated (hereafter UN-1), one saturated (hereafter SN-1), both defined only by hydro-meteorological variables and their bivariate correlations, and one saturated network (hereafter SN-C), consisting of the SN-1 network and including physical catchments' attributes. The results indicate that the UN-1 network is suitable for catchments with a positive dependence between precipitation and discharge, while the SN-1 network can also reproduce discharge in catchments with negative dependence. The latter can reproduce statistical characteristics of discharge (tested via the Kolmogorov–Smirnov statistic) and have a Nash–Sutcliffe efficiency (NSE)

Strategies for water resources management and planning mostly rely on predictions from hydrological models

Data-driven models in general differ on the input–output technique implemented, which might not have a conventional physical interpretation

A wide range of scientific publications illustrates progress in formulations and implementations of both process-based and data-driven hydrological models, highlighting their respective potentials. However, among data-driven models, less attention has so far been given to explicitly representing the interdependence between inflow and outflow via high-dimensional probability functions. Bi- and multivariate probability functions, such as copulas, have been mostly implemented to derive critical flood design values when multiple flood characteristics are of interest

In the scientific literature, Bayesian networks (BNs) have been implemented in multiple fields to model the probabilistic relationship between variables.

NPBNs, similar to BNs, are probabilistic graphical models representing high-dimensional probability distribution functions of system properties with complex dependence structures

Starting from these premises, the main objective of this study is to further explore and test the suitability of NPBNs as a tool to reproduce catchment-scale hydrological dynamics and to explore challenges involved when inferring maximum daily river discharge in any given month. More specifically, long-term data from 240 river catchments across the United States from the Catchment Attributes and Meteorology for Large-sample Studies

For this study, we make use of the CAMELS data set

Catchments' attributes from the CAMELS database were used without further processing. The attribute aridity, Ar[–], refers to the ratio of long-term means of potential evapotranspiration calculated using Priestley–Taylor formulation and precipitation, where values higher/lower than 1 indicate water-/energy-limited regions. The attribute precipitation seasonality

Hydro-meteorological data and catchment attributes used in this study.

Catchments located in the eastern and central-eastern United States (56 %) are characterised by an average size of about 94 km

Catchment attributes extracted from the CAMELS database:

In the majority of the catchments selected (86 %), the correlation between maximum daily discharge (

Catchment attributes based on the correlation between maximum daily discharge,

Hydro-meteorological variables and catchment attributes described so far are used in the following as input to reproduce catchment-scale hydrological dynamics via NPBNs.

A BN is defined by two components

BNs differ on how nodes and arcs are quantified, and the inference process depends on this quantification. Discrete BNs specify the source nodes, i.e. nodes without parents, as discrete random variables and conditional probability tables for child nodes

Figure

Illustrative Bayesian network with three nodes.

The set of variables

To each arc of the network, the quantity

The correlation matrix

NPBNs based on the normal copula assumption are implemented in the open-source MATLAB toolbox BANSHEE

The aim of this study is to investigate the suitability of NPBNs to reproduce catchment-scale hydrological dynamics. The rationale adopted to identify suitable DAG consists of representing a catchment as a system in which discharge is generated by the interaction between the input of the system, for example, precipitation, the state of the system, for example, soil moisture, and the output of the system, for example, river discharge (Fig.

In graph type I, the following continuous hydro-meteorological variables will be considered here:

Network selection, i.e. moving from a graph to a DAG by selecting arcs connecting a given set of nodes to model dependence, is challenging due to the high number of possible configurations describing a given set of variables. In this study, we selected two DAGs a priori: a DAG in which the variables are parent nodes, with one child being the variable

In this study, we investigate two networks, one unsaturated (hereafter UN-1), as shown in Fig.

To further explore the applicability of NPBNs in hydrological studies, we investigate the potential of a single saturated network (hereafter SN-C, Fig.

The joint distribution function associated with each network (UN-1, SN-1, and SN-C) is derived following the protocol presented in

Graphs and qualitative networks (DAGs) used in this study. Panels

To assess the potential of NPBNs as probabilistic models for catchment dynamics, we first test the networks (UN-1, SN-1, and SN-C) as descriptive models. Subsequently, we evaluate the networks as predictive models. In this study, the term

In both the testing and evaluation process, we first test the assumption of the joint normal copula for modelling the bivariate dependence via the Cramér–von Mises test. Then, we use the

The Cramér–von Mises (CvM) statistic

The

The two-sample Kolmogorov–Smirnov (KS) is a non-parametric
hypothesis testing technique assessing whether two samples,

The null hypothesis

The Nash–Sutcliffe efficiency coefficient (NSE)

NPBN treats hydro-meteorological data and catchment attributes as random variables. This implies that during the inference process, the NPBN returns, at each time step, a conditional distribution function of the target variable, i.e. the distribution of maximum daily river discharge conditioned on the remaining hydro-meteorological data and attributes. From this conditional distribution of river discharge, 1000 possible discharge realisations are sampled, and the 50th percentile is taken as the estimated discharge value for that particular combination of hydro-meteorological data and attributes. Similarly, the confidence interval (CI) of the estimated discharge value is determined as the 5th and the 95th percentile of the 1000 realisations of the conditional distribution.

In this section, we first show the potential of NPBNs in estimating maximum daily river discharge when a catchment is modelled as single elements. Afterwards, we present the capability of NPBNs to model catchments in a cluster to eventually infer river discharge of an ungauged basin given its attributes.

We first analyse the performances of the UN-1 and SN-1 networks as descriptive models. In Fig.

The preliminary analysis on the descriptive capabilities of the UN-1 and SN-1 networks suggests that the SN-1 network is better suited for describing the dynamics of river discharge compared to UN-1. However, when we look more in depth into SN-1 network performances, we can observe that only 66 % of the catchments have a NSE higher than

Results of the testing process when the UN-1 and SN-1 networks are used as descriptive models of the 240 catchments considered as single elements. Panel

Performances of the SN-1 network in terms of NSE for the 159 catchments with NSE

To further investigate the ability of NPBNs to estimate maximum daily river discharge, we evaluate the performances of the SN-1 network as a predictive model. We limit the investigation to the SN-1 network since the above results suggest that it is a descriptive model for a larger number of catchments with contrasting characteristics compared to the UN-1 network. The

NPBNs provide a quantification of the uncertainty around the estimated river discharge values. We then quantify the uncertainty of the estimated maximum river discharge. On average and across all catchments, observed discharge in the test set falls within the simulated confidence interval (5th and 95th percentile) about 63 % of the time, ranging between a minimum of 45 % and a maximum of 78 % (Fig.

Comparison between maximum daily river discharge simulations from run 1 and observations of three different catchments. The grey shaded areas indicate years belonging to the test set. The shaded red areas represent the simulation confidence interval evaluated as the 5th and the 95th percentile.

To further evaluate the results of the SN-1 network in estimating maximum daily river discharge, the hydrograph of three stations, i.e. no. 6746095 (Colorado), no. 11481200 (California), and no. 14306340 (Oregon), with contrasting characteristics are shown in Fig.

The catchment in Colorado is located in a water-limited area (Ar

These results show the potential of the SN-1 network to model the river discharge generation process in catchments with contrasting climate exploiting information from the interaction between the different inputs of the system catchment, i.e.

We implement the SN-C network on a subsample of 133 catchments with a positive correlation between

We first test the performance of the SN-C network as a descriptive model. Similar to the results obtained previously, Frank and Gumbel copulas are the best theoretical copulas for about 50 % of the pairs, supporting the choice of the NPBN. The

Results from the KS test on the SN-C network. Panel

We note that removing one station from the overall pool of observations has a very small effect on the empirical correlation matrix of the empirical variables, the correlation matrix associated with the network, and the cumulative distribution of each node: the observations belonging to one catchment are around 0.8 % of the total observations from all the catchments. This shows that the SN-C network is quite robust. Hence, we further evaluate the robustness of the SN-C network performances as a predictive model by leave-one-out cross-validation. The KS test is performed for each catchment using the value of maximum daily river discharge observed and simulated via the SN-C network, calibrated without the information of the catchment analysed (evaluation process). This is done to assess the potential of such a network in exploiting the information from catchments with similar attributes. The descriptive and the predictive models perform similarly, suggesting that the SN-C network is quite robust. The KS test results show that in only 15 % of the subsample of catchments analysed here (Fig.

To further analyse the results, we look at one catchment in California (no. 11481200), where the

The performances of NPBNs indicate that the interdependence between hydro-meteorological information should be explicitly modelled to better capture the river discharge characteristics at the catchment level: the SN-1 network provides higher NSE values compared to the UN-1 network. Additionally, it suggests that at least the networks trained at the catchment scale, i.e. SN-1 and UN-1, show potential to describe the hydrological response, while more research will be needed to develop meaningful NPBNs trained across a range of multiple catchments. Indeed, in our study, river discharge could only be poorly captured by this network type, i.e. SN-C, which reflects the common issue of information transfer in hydrological modelling.

By further considering the results just obtained we identify six issues related to both NPBN properties (i.e. data quality and quantity, independence of weather events, feasibility of testing procedure, and Gaussian-copula assumption) and the river discharge generating process (i.e. catchment heterogeneity, and interacting spatial and temporal scales) that we believe have influenced the networks' performances. We discuss these issues in details, and, when possible, we propose a way to address them.

The main objective of this study was to further explore and test the suitability of NPBNs as a tool to reproduce catchment-scale hydrological dynamics and to explore challenges involved when inferring maximum daily discharge, since applications of NPBNs in hydrology are still limited. In this study, we investigated 240 catchments across the United States, obtained from the CAMELS data set, aiming at testing the ability of NPBNs to estimate maximum daily river discharge. We showed that, once a NPBN is defined, it is straightforward to infer any of its variables, i.e. discharge, when the remaining variables are known and extend the network itself with additional variables, i.e. going from the SN-1 network containing only hydro-meteorological variables to the SN-C network containing hydro-meteorological variables and catchments' attributes. The NPBNs individually trained to specific catchments showed potential to reproduce maximum daily river discharge in a wide range of environments with an average NSE of 0.59 (predictive models), while in the literature the performances of regression models for average monthly river discharge showed NSEs

NPBNs were modelled using the MATLAB toolbox BANSHEE (

The data used in this study are from the CAMELS project and can be found at

The supplement related to this article is available online at:

ER, MH, and OMN developed the study. ER carried out the numerical analyses and prepared the manuscript preliminary draft. MH and OMN contributed to the final version of the paper and the discussion of the results.

At least one of the (co-)authors is a member of the editorial board of

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

We would like to thank the editor and the anonymous reviewers for taking the time to review this study and for providing valuable comments. This project has received funding from the European Union’s Horizon 2020 Research and Innovation programme under the Marie Skłodowska-Curie Action.

This research has been supported by the H2020 Marie Skłodowska-Curie Actions (grant no. 707404).

This paper was edited by Fuqiang Tian and reviewed by Yingzhao Ma and two anonymous referees.