This paper presents a new method for hydrograph separation. It is well-known that all hydrological methods aiming at separating streamflow into baseflow – its slow or delayed component – and quickflow – its non-delayed component – present large imperfections, and we do not claim to provide here a perfect solution. However, the method described here is at least (i) impartial in the determination of its two parameters (a quadratic reservoir capacity and a response time), (ii) coherent in time (as assessed by a split-sample test) and (iii) geologically coherent (an exhaustive validation on 1664 French catchments shows a good match with what we know of France's hydrogeology). With these characteristics, the method can be used to perform a general assessment of hydroclimatic memory of catchments. Last, an R package is provided to ensure reproducibility of the results presented.

Hydrograph separation and the identification of the baseflow contribution to streamflow is definitely not a new subject in hydrology. This age-old topic

To assess baseflow, direct measurement is generally impossible, since it is a conceptual quantity, not a physical one. Proxy approaches involving chemical tracer-based procedures are efficient but need chemical data and involve their load of assumptions. Most approaches rely on solving an

It is perhaps impossible to propose a physically based baseflow separation procedure (just because of the multiplicity of flow paths that makes the procedure fundamentally equifinal), and we will not argue about this point. But we believe that even the imperfect conceptual–mathematical–empirical methods in use could receive a non-arbitrary, impartial, repeatable parametrisation that could be used as a general-purpose study tool over large catchment sets.

This paper focuses on a hydrograph separation method that is based only on quantitative streamflow data and climate descriptors and does not require a priori physical parametrisation, presented in Sect. 2. The originality of this method lies in its parametrisation strategy, which we discuss in detail in Sect. 2.2. The application of this procedure to a set of 1664 catchments, its geological coherence and its stability in time are presented in Sect. 3.

Hydrograph separation is based on the following assumption: streamflow can be divided into two components,

A commonly found definition of hydrograph separation is splitting streamflow between surface runoff and groundwater contribution. This may be an acceptable interpretation of chemicophysical methods that are based on the difference – of composition or temperature, for instance – between different sources of water. However, process interpretation of hydrograph separation is a generally hazardous practice

Existing procedures for hydrograph separation and baseflow assessment can be sorted into three categories

Chemicophysical methods are based on the fact that total flow is a

The most common methods of this family are tracer-based baseflow separations. A passive tracer whose concentration is different in groundwater, subsurface, bank and surface runoff water – for instance, isotopes

There are three main drawbacks in chemicophysical methods. First, the inherent uncertainties: there are measurement errors; concentrations can have spatial and temporal heterogeneity, even within the same rainstorm; and the passiveness of tracers may not be a valid hypothesis even in short events.

Several classic baseflow separation methods are not based on hydrological considerations, but they rely on processing the hydrograph as a signal. Most of these methods are based on the hypothesis that the transfer time of surface runoff is much shorter than that of groundwater, and that this time is relatively constant between rainfall events. The first methods of this type were graphical. After identifying peaks along the hydrograph and estimating the surface runoff time constant – let us say

Within the same set of hypotheses, low-pass numerical filtering of the hydrograph has been used as a baseflow separation method. As in electronic signal filtering, the highest frequencies of the signal are dropped; and the low ones are kept. Fourier-transform-based filters implement this framework, but they are intended for data with stationary periodic processes, whereas hydrological signals operate on a very large range of timescales. Even

Numerical methods all need one or several parameters, with or without hydrological meaning, which have to be valued before processing: time interval width, coefficients of a linear filter, formula of the transformation function, etc. Even though authors generally give advice about the possible values of these parameters, determining them involves adding new hypotheses for each catchment examined

In addition to chemicophysical and numerical methods, another way of separating hydrographs has been developed, starting from the idea that the slow parts of catchments can be represented as conceptual reservoirs, whose outflow is baseflow. Such an approach comes from the analysis of long recession curves and the inference of depletion laws, which rule streamflow during long, dry periods of low flows. By knowing the outflow law of the conceptual reservoir, assuming that its input is zero, it is possible to get the full recession curve by the next significant rainfall event

It is possible to calibrate such a model by fitting it on

To filter the whole hydrograph with this conceptual idea of baseflow, a backward filter was developed by

Although it is impossible to perform an absolute evaluation of a hydrograph separation method, since baseflow cannot be measured and compared with a simulated value, several comparative studies have challenged results from various methods. Chemicophysical and non-calibrated graphical and empirical algorithms – i.e. without parameters or whose parameters are determined a priori and do not vary between catchments – have given very different results

To overcome this issue, several studies use parametric graphical and empirical methods and calibrate them with chemicophysical data

When performing a large-scale baseflow analysis over a territory, it is possible to extend calibrated parameters at gauged catchments to ungauged ones through regionalisation models.

Most of the methods above rely on one or several parameters whose value has a strong influence on hydrograph separation; as it is impossible to calibrate this – or these – parameter(s) on the basis of a measurement, some physical hypotheses must be made on the desirable properties of baseflow, in relation to the memory of the catchment. Non-calibrated algorithms are not usable as general-purpose analysis tools and existing calibration procedures rely on the availability of chemicophysical data. In this work, we propose a calibration procedure of a conceptual hydrograph separation method that relies only on hydroclimatic data: streamflow, rainfall and temperature.

Frugality and objectivity do not only result in a small number of parameters. The choice of modelling options can also be a source of arbitrariness and useless complication. A modelling alternative can, moreover, be solved only through a personal choice of the modeller; one way to create a trustworthy framework is to build it upon well-known and well-validated elements that have proven themselves relevant in lumped conceptual hydrology. Beyond the complexity of elements, there is the complexity of the modelling chain itself. The simpler the model is, the more readable and thus the less questionable it is; in this work, it was considered that a baseflow separation process should be much simpler than, for instance, a lumped conceptual model designed for flow simulation.

In this paper, we postulate that the memory and smoothing effect of a catchment – which underpins the concept of baseflow – can be represented by a conceptual reservoir, whose outflow will depict the delayed contribution to streamflow, which we will assimilate into baseflow. To be applied in practice, this postulate must be complemented by an answer to the two following questions: what should the input to the reservoir be? What should the relationship between the level of the reservoir and its outflow be?

The first question asks what should refill the conceptual reservoir (i.e. what recharges the catchment with water that will be baseflow afterwards). The part of rainfall that does not contribute to surface runoff or to evaporation is generally named

The general contribution of this

The filtering part of the method is a conceptual reservoir, whose inflow is a fraction of the total observed flow at the gauging station, and the outflow is regarded as baseflow. The content of this reservoir is managed by an outflow function and a continuous differential equation: since data are obviously available as discrete time series with a time step

The notations listed in Table

List of variables.

According to the definition of a reservoir,

linear reservoir:

quadratic reservoir:

exponential reservoir:

In order to discretise the filter, we define discrete versions of continuous variables cited above. If

Although it is done in most conceptual hydrological models, integrating differential equations using this method in a discrete hydrological model could be criticised

Thus, just before

The filtering process detailed above raises an issue: baseflow is by definition a fraction of total flow; yet, Eq. (

If

This update process takes water out of the reservoir when computed baseflow is too high; but do we not need to correct baseflow when it is too low? Indeed, quickflow is supposed to be zero during long, rainless, dry periods. For the sake of simplicity, a straightforward hypothesis is added: baseflow must be equal to total flow at least once in a hydrological year, when measured streamflow reaches its yearly minimum. The value of the latter can be affected by measurement errors, but it is hard to take it into account in a general-purpose analysis tool: the structure of the error is determined by the idiosyncrasies of each catchment. Moreover, the two ways of updating the reservoir helps the model to forget about previous errors; therefore, we prefer not to add another hypothesis about the low-flow measurement uncertainty.

A test study is performed on a set of catchments in continental France: thus, the low-flow hydrological year is taken from 1 April to 1 March, as streamflow minima do not generally occur during spring. The

This second update process adds water to the reservoir; it is thus possible to balance the updates of the reservoir and, thereby, to balance the total water budget of the filtering process. If

The yearly minimum update may look too strong and arbitrary, but tests have shown that it is necessary to avoid issues in the parametrisation strategy: removing it adds a third degree of freedom in calibrating the model – the value of

Our algorithm does not need a long warm-up period, as required by rainfall–runoff models. Since the algorithm presented above contains only one state and involves a regular update with observed data, this update erases the memory of the past states of the model. Therefore, a simple initialisation approach is possible. Several methods were tested, and the best compromise between simplicity and robustness was adopted. The reservoir level is initialised through Eq. (

Finally, the filtering algorithm can be summarised with the pseudo-code detailed in Algorithm 1, where

Synthesis of the filtering algorithm.

As stated above, the difficulty of parametrisation comes from the fact that baseflow cannot be measured. In the absence of measurement, we need to make a hypothesis concerning a desirable property of the computed baseflow, in order to be able to look for the parameter set that will best respect this property. Since baseflow accounts for the slow component of total flow, coming out of the storage parts of a river basin, we make the following hypothesis: baseflow should be correlated with past climatic events that have filled or emptied this storage; and since it is supposed to be slow, it should be correlated with events that happened quite a long time ago. The calibration criterion has to be a simple quantification of this idea, which corresponds to the concept of recharge stated earlier: a proxy of past recharge must be found.

The filtering algorithm detailed above depends on two parameters: capacity

After eliminating 1 degree of freedom, the correlation hypothesis stated above can be converted into a criterion: it computes Pearson's correlation between the computed baseflow time series – which depends on the reservoir's parameter

As far as the optimisation process is concerned, it is necessary to define search intervals for

Effective rainfall was computed with the Turc–Mezentsev formula

Therefore, we found an effective parametrisation strategy for choosing the three parameters

Several hydrographic regions of mainland France are influenced by large aquifers, which bear the memory of past climatic events and have a significant contribution to river flow. In the Paris basin, the chalk aquifer – composed of late Cretaceous formations – has a significant connection with many rivers in the Seine and the Somme basins, which notably led to major groundwater-induced floods after the extremely wet years of 1999 and 2000

Tests were performed on a set of 1664 catchments that were selected on the basis of diversity of area, climate, hydrological regime and completeness of the time series in the period 1967–2018. Streamflow data are from the national database Banque Hydro

Gaps in the streamflow time series were filled using simulated flows from the daily lumped rainfall–runoff model GR4J (modèle du Génie Rural à 4 paramètres Journalier; daily four-parameter model from the rural engineering service)

Tables

Geographic characteristics of the catchment dataset.

Hydrological characteristics of the catchment dataset. The aridity index is defined as the quotient of average rainfall by average PET. NB: precipitation yield greater than 100 % was encountered in a small karstic catchment where supplementary water is brought by a major resurgence.

Map of the catchments in the dataset. Dots represent the outlet of each catchment, with different colours according to the type of aquifer influence used in Fig.

For each catchment of the dataset, a two-variable grid search was performed to find the optimum of the criterion. Hydrograph separation was then computed with the obtained parameters.

As already mentioned in Sect. 2.2, for each catchment, there is a perfect bijection between values of the reservoir parameter

Two examples of surface plots are presented in Figs.

Surface plot of the optimisation criterion for the Vair River in Soulosse-sous-Saint-Élophe. The red cross indicates the numerical optimum.

Surface plot of the optimisation criterion for the Petit Thérain River in Saint-Omer-en-Chaussée. The red cross indicates the numerical optimum.

Hydrograph separation of the Vair River in Soulosse-sous-Saint-Élophe for 1995–2005.

Hydrograph separation of the Virène River in Vire-Normandie for 1995–2005.

Hydrograph separation of the Petit Thérain River in Saint-Omer-en-Chaussée for 1995–2005.

Figures

Figure

Figure

Among the dataset of 1664 catchments, optimisation of parameter

Table

Summary of the results from the set of 1664 catchments.

We found that no clear relationship can be highlighted regarding the correlations between these results and hydroclimatic values given in Table

Figure

Boxplots of BFI values and

Figure

Maps of resulting values of parameters on the catchment dataset.

Statistical assessment of the split-sample test through Pearson correlation and the Kolmogorov–Smirnov (KS) non-parametric test.

In order to evaluate the stability of the proposed algorithm and of its calibration procedure, a split-sample test was performed. The time extent of the time series of the dataset – 1 August 1958 to 31 July 2018 – was divided into two equal periods

Figure

Results of the split-sample test for BFI

The BFI obtained was compared with the one resulting from a simple graphical method, based on local minima on a sliding interval of 5 d. It is detailed in

Comparison between BFI obtained with the conceptual algorithm and a reference graphical method.

The hydrograph separation algorithm and its calibration procedure presented in this work have yielded credible results for a large set of test catchments. The algorithm has suitable characteristics for such a model: (i) frugality – only two parameters to calibrate; (ii) objectivity – the procedure is not supposed to require any intervention or interpretation from the user; (iii) generalisation ability – the procedure succeeded on the whole dataset, with a unique value found for the reservoir parameter; and (iv) repeatability – as assessed by the split-sample test performed on the test dataset. The analysis of the results in the light of the geological characteristics of catchments shows that the model is able to acknowledge the importance of groundwater flow in total flow generation without prior knowledge about the studied catchment, which is a significant step forward with respect to other non-tracer-based baseflow separation methods.

A conceptual and empirical baseflow separation method is neither intended to replace precise estimation of the elements of water balance at the scale of a given catchment, especially when in-field physical data are available nor can it precisely highlight particular hydrological processes occurring at the catchment or under-catchment scale, such as karstic non-linear transfers or human regulation of rivers. Yet, in the absence of in-field data, i.e. outside of specially instrumented catchments, conceptual baseflow separation is a useful tool for obtaining meaningful insights into the role of groundwater and aquifers inside a catchment. Even if the present method has limitations – which we summarise below – it could be applied as an objective, general-purpose, automated analysis tool for information on a large set of flow series. Beyond the inherent characteristics of the presented method, the results show that the hydrogeological processes of a catchment are deeply engraved in streamflow.

We used an optimisation procedure set which searches for values of

The split-sample test is a way to check temporal consistency of the baseflow separation method; the one performed on the test dataset can be considered as successful, at least concerning BFI results. Nevertheless, the spatial coherence of the method was not checked; it could be tested, for instance, with the propagation of baseflow sub-basins of a larger catchment, in order to test whether the sum of tributary baseflows makes up total baseflow. Our optimisation procedure does not take into account regional aspects while calibrating the model, although it could be a way of introducing objective prior knowledge into the method. On a sample of catchments where chemicophysical data are available, a comparison with tracer-based separation procedures would be a reliable way to validate the presented method.

Our algorithm is only intended for catchments that are not affected by human activity – pumping, regulation by dams or the use of canals – which excludes most large rivers in Europe. Finally, the generalisation capacity of the method has only been demonstrated for the hydroclimatic conditions of mainland France: even if the spatial extent of the present work has a substantial diversity of climates, a further study with a wider range of climates and geologies is needed to validate the method.

An R package named baseflow

Data for this work were provided by Météo France (climatic data) and the hydrometric services of the French government (flow data). Flow data is publicly available on the Banque Hydro website

The supplement related to this article is available online at:

VA and AP conceptualised the method. AP performed the tests on the catchment dataset and developed the computing code. The paper was written by both authors.

The authors declare that they have no conflict of interest.

Data for this work were provided by Météo France (climatic data) and the hydrometric services of the French government (flow data). We would like to thank Pierre Nicolle (Université Gustave Eiffel) and Benoît Génot (INRAE Antony) for the database creation and maintenance and José Tunqui Neira (Sorbonne Université) for his help in the literature review. We would also like to extend our warmest thanks to Charles Perrin (INRAE Antony), Lionel Berthet (DGPR) and Paul Astagneau (INRAE Antony) for their remarks and suggestions during the writing process. Last but not least, computing codes could not have been developed without the precious expertise and availability of Olivier Delaigue (INRAE Antony) and Denis Merigoux (INRIA Paris).

This paper was edited by Markus Hrachowitz and reviewed by Renata Romanowicz and Ian Cartwright.