Recursive digital filtering of hydrographs is a widely used method to identify streamflow components, which react to precipitation with varying
degrees of attenuation and delay. In this context, a distinction is often made between physically based and non-physically based algorithms. A
well-known example of a physically based algorithm is that of Furey and Gupta (2001). In this paper, it is contrasted with the widely used algorithm
of Eckhardt (2005). This algorithm is often considered merely a non-physically based low-pass filter. However, the comparison shows that both
algorithms largely agree. The algorithm of Eckhardt (2005) differs from the algorithm of Furey and Gupta (2001) only in the time delay assumed
between precipitation and the exfiltration of baseflow into surface waters and in the fact that two parameters are combined into one,

A catchment can be understood as a signal converter. The precipitation is the input signal that is converted into the output signal, streamflow. In the course of this signal conversion, the water takes different paths through the catchment and is subject to different hydrological processes. This results in streamflow components that are attenuated and delayed to varying degrees compared to the input signal, the precipitation. Usually, two components are distinguished: on the one hand, the so-called baseflow as a low-frequency signal component and, on the other hand, higher-frequency signal components that are generated more quickly and are less attenuated in response to precipitation events, the so-called direct runoff. From this idea, it is obvious that low-pass filtering of streamflow hydrographs can be used to identify these components.

This approach has been followed since Lyne and Hollick (1979) introduced the recursive digital low-pass filter to hydrology. The term “digital” refers to the fact that discrete, equidistant-in-time data of the streamflow are used, the processing of which can be easily automated by using a computer. The term “recursive” refers to the fact that the signals of the preceding time steps are included in the calculation of the output signal in the current time step.

Several such recursive digital low-pass filters were subsequently presented. In the following, the filter developed by Eckhardt (2005) is considered in particular. It is now one of the established methods of hydrograph separation – for example, as part of the US Geological Survey Hydrologic Toolbox (Barlow et al., 2022).

The “Eckhardt filter”, as it is oftentimes called, is usually counted among the non-physical or “purely empirical” (Healy, 2010, p. 87) methods of hydrograph separation. The apparent lack of a physical basis repeatedly raises doubts about the justification of the recursive digital filtering: “Most hydrograph separations (apart from tracer-based separations) lack a physical basis. […] Therefore, choosing one method or the other introduces an undesirable element of uncertainty and randomness into the analysis and comparison of runoff coefficients” (Blume et al., 2007). “The digital filter methods have no physical meaning” (Kang et al. 2022). However, without a physically meaningful interpretation, it becomes impossible to objectively determine the parameters of the filter algorithms: “parameters used in the RDF [recursive digital filtering] method are often determined arbitrarily, resulting in high uncertainty of the estimated baseflow rate.” (Zhang et al., 2013), “quantitative results of the filtering change with the value of the parameters. Although the shape of the hydrograph separation can be visually consistent with the conceptualisation of a hydrograph separation, it is basically impossible to draw any conclusion from it.” (Pelletier and Andréassian, 2020), “To accurately separate the baseflow from streamflow with the digital filter methods, appropriate filter parameters must be estimated by trial and error, which act as a difficulty or limitation on their use.” (Kang et al., 2022).

Is this criticism justified? Does the widespread recursive digital filtering, especially that with Eckhardt's algorithm, really lack a physical, hydrologically plausible explanation, and does the choice of parameter values remain arbitrary?

In order to shed light on the answers to these questions, Eckhardt's filter is compared below with the algorithm of Furey and Gupta (2001). The latter
has been developed explicitly from hydrological principles. Its developers therefore – rightly – describe it as physically based and emphasise the
difference to the previously mentioned low-pass filters: “Unlike other filters, our filter is not founded on the assumption that base flow and
overland flow are the low- and high-frequency components of streamflow, respectively.” The analysis shows that there is nevertheless a close
relationship between the Eckhardt (2005) and Furey and Gupta (2001) filters and thus provides a clue as to how the parameter

The equation of this low-pass filter is

Even though the filter of Eckhardt (2005) is contrasted here with the filter of Furey and Gupta (2001), which is explicitly described as physically
based, it is nevertheless also itself based on plausible assumptions:

The information about the baseflow

Baseflow is runoff from a linear reservoir, i.e. it is proportional to the amount of water stored in this reservoir. The filter
parameter

The algorithm of Lyne and Hollick (1979) has been criticised as hydrologically implausible, since it shows a constant streamflow

This equation can be simplified to

This is exactly the equation that describes the exponential decrease in runoff from a linear reservoir.

The second filter parameter

The calculation with Eckhardt's algorithm requires streamflow data and the values of two parameters, with the streamflow data allowing one of the two
parameters, the recession constant

Furey and Gupta formulated their filter algorithm as

The calculation of the baseflow according to Furey and Gupta (2001) requires streamflow and precipitation data and the values of four parameters:

In deriving their filter equation, Furey and Gupta (2001) assume that the baseflow in the current time step is a function of baseflow and groundwater
recharge one time step in the past (their Eq.

If instead it is assumed that baseflow occurs in the same time step as groundwater recharge and that groundwater recharge is not delayed compared to precipitation (in other words, if it is assumed that the delay between precipitation and baseflow is smaller than one time step), then

This equation can be transformed to

Equation (

The solution of this system of equations results in

In other words, a single assumption, namely that baseflow still begins at the same time step as precipitation, is sufficient to transform the
algorithm of Furey and Gupta (2001) into the algorithm of Eckhardt (2005), where Eq. (

Eckhardt's algorithm represents a whole class of recursive digital filters that only differ by the value of

Furey and Gupta (2001) introduced the parameter

The streamflow component calculated with Eq. (

In this publication, the algorithm of Eckhardt (2005) is compared to the model ideas of Furey and Gupta (2001) on the formation of baseflow. It
is not compared to the reality. If the baseflow calculated with Eq. (

If one assumes that (a) there is no inflow or outflow of groundwater below the surface boundaries of the catchment and that (b) there is no
evapotranspiration from groundwater or surface waters, then the sum of overland flow and groundwater recharge corresponds approximately to the
streamflow:

Streamflow is given. Consequently, “only” a method for estimating mean groundwater recharge is needed to approximate

The recursive digital filter of Eckhardt (2005) largely coincides with the physically based algorithm of Furey and Gupta (2001). As Eckhardt (2005)
has pointed out, his filter is identical to the filter of Boughton (1993) and passes for different values of the parameter

The preceding considerations also suggest a way in which the parameter

No data were re-evaluated for the present study.

The author has declared that there are no competing interests.

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This paper was edited by Thom Bogaard and reviewed by two anonymous referees.