the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Technical note: Hydrograph separation: How physically based is recursive digital filtering?
- University of Applied Sciences Weihenstephan-Triesdorf, Weidenbach, 91746, Germany
- University of Applied Sciences Weihenstephan-Triesdorf, Weidenbach, 91746, Germany
Abstract. Recursive digital filtering of hydrographs is a widely used method to identify the groundwater-borne portion of streamflow. In this context, a distinction is often made between physically based and non-physically based algorithms. The algorithm of Furey and Gupta (2001), for example, is counted among the former. In this paper, it is contrasted with the algorithm of Eckhardt (2005). This algorithm represents a whole class of recursive digital filters based on the assumption that the aquifer is a linear reservoir. It is shown that the algorithm of Eckhardt (2005) is not merely a low-pass filter, but that it is largely identical to the aforementioned physically based algorithm of Furey and Gupta (2001). 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 groundwater into surface waters, and in the fact that two parameters are combined into one, BFImax. This parameter can thus be interpreted physically and an approach for its calculation emerges.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
(247 KB)
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
Journal article(s) based on this preprint
Klaus Eckhardt
Interactive discussion
Status: closed
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CC1: 'Comment on hess-2022-186', Keith Beven, 23 Jun 2022
A very long time ago (in 1991) I published a review of hydrograph separation methods. The review had been requested for a meeting of the British Hydrological Society.  It included a section on Choosing a Hydrograph Seperation method.  That section consisted of a single word: "Don't."   The reason for that was mostly to avoid the types of problems represented by this paper - particularly the inference that some mathematical filter can be used to decide what is groundwater or not.  The author appears to think that baseflow and groundwater are equivalent.  This is particularly ironic when he references the use of tracer information to support the lack of a time delay in his own function on the basis that tracers show that there can be "a rapid release of so-called pre-event-water".  But just why should that pre-event water be baseflow (or groundwater)?  The tracer data generally undermine the whole idea of baseflow separation.  To suggest that something might be physically-based by comparing one mathematical function to another mathematical function based on a linear store is surely naive at best, and downright misleading at worst.  We might perhaps want to use a consistent mathematical filter to produce some "baseflow index" as a characteristic of catchment response but please do not relate it to any superficial process interpretation. Better still, please do not choose a baseflow separation method at all but try to understand the actual processes of catchment response.
Reference.
Beven, K.J. (1991), Hydrograph Separation?, Proc.BHS Third National Hydrology Symposium, Institute of Hydrology, Wallingford, 3.1-3.8.
Â
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AC1: 'Reply on CC1', Klaus Eckhardt, 28 Jun 2022
Keith Beven writes that the point of this article is to "to suggest that something might be physically-based by comparing one mathematical function to another mathematical function". Apparently, he does not concede the approach of Furey and Gupta (2001) to be physically based. I see it differently. In my view, Â their algorithm is physically based. One can argue about how accurate the physical basis is. However, a fundamental debate on whether hydrograph separation is useful or not goes far beyond the purpose of this technical note. There are, after all, a variety of methods of hydrograph separation and they are used. For more than four decades, recursive digital filtering has been one of them. The present contribution thus does exactly what HESS associates with a technical note: "Technical notes report [...] novel aspects of [...] theoretical methods and techniques which are relevant for scientific investigations within the journal scope." (https://www.hydrology-and-earth-system-sciences.net/about/manuscript_types.html).
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CC2: 'Reply on AC1', Keith Beven, 28 Jun 2022
OK, I will accept the argument that physically-based can be defined in terms of mathematics derived from explicit assumptions and you are certainly correct about the technical note. Â I am not in any way disputing that, only the utility of such analyses. Â The danger, as ever, is thinking that those assumptions represent the actual physics of the catchments we are interested in, as you seem to do when you equate baseflow and groundwater, and when you invoke tracer evidence for fast groundwater responses when surely that very information suggests that contributions of pre-event water are quite different from baseflow defined by those mathematics. Â That therefore suggests to me that the assumptions of the mathematics are wrong in terms of being a physically-based description of the actual processes. Â So create a baseflow index if you wish, but please do not call it groundwater (or even better, explicitly differentiate them to avoid others making a similar false equivalence). Â Â
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AC2: 'Reply on CC2', Klaus Eckhardt, 07 Jul 2022
I will gladly take into account the critical comments on the submitted contribution. I propose the following changes to the text, which will result in the terms "groundwater" and "pre-event water" being largely omitted. Only the term "groundwater recharge" remains in the context of its use by Furey and Gupta (2001).
Lines 7 - 8: "a widely used method to identify the groundwater-borne portion of streamflow" is replaced by "a widely used method to identify streamflow components, which react to precipitation with varying degrees of attenuation and delay".
Line 11: "the aquifer is a linear reservoir" is replaced by "baseflow is runoff from a linear reservoir".
Â
Line 14: "exfiltration of groundwater into surface waters" is replaced by "exfiltration of baseflow into surface waters".Lines 17 - 22 are replaced by
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 less attenuated in response to precipitation events, the so-called direct runoff. From this idea, it is obvious to low-pass filter streamflow hydrographs to identify these components.Lines 63 - 64: "(b) The aquifer is a linear reservoir, i.e. the discharge from the aquifer is proportional to the amount of water stored in it. Without further knowledge about the physical properties of the aquifer, this is the most obvious approach." is replaced by "(b) Baseflow is runoff from a linear reservoir, i.e. it is proportional to the amount of water stored in this reservoir.".
Line 115: "the assumption of the aquifer being a linear reservoir" is replaced by "the assumption that baseflow is runoff from a linear reservoir".
Lines 118 - 119: "the exfiltration of groundwater into surface waters" is replaced by "the exfiltration of baseflow into surface waters".
Lines 121 - 124 are replaced by
Furey and Gupta (2001) introduced the parameter d in Eq. (5) as the number of time steps between precipitation and groundwater recharge. A sensitivity analysis they conducted showed that the filter performance was "relatively insensitive to changes in d" so that d = 0 seemed to be an acceptable choice. Furthermore, when using Eq. (1), it is assumed that not only the groundwater recharge but also the generation of baseflow still occurs in the same time step as precipitation. When assessing this prerequisite, it should be noted that the streamflow component calculated with Eq. (1) is usually likely to consist not only of groundwater, but also of transient water sources, including interflow (Cartwright et al., 2014; Yang et al., 2021).Lines 136 - 137: "This is plausible. The parameter BFImax was introduced as the maximum value of the baseflow index that can be calculated. And the baseflow can at most correspond to the groundwater recharge." is deleted.
Lines 156 - 157 are deleted.
Lines 173 - 174 are deleted.Added are the references
Cartwright, I., Gilfedder, B., and Hofmann, H.: Contrasts between estimates of baseflow help discern multiple sources of water contributing to rivers, Hydrol. Earth Syst. Sci., 18, 15–30, https://doi.org/10.5194/hess-18-15-2014, 2014.Â
Yang, W., Xiao, C., Zhang, Z., and Liang, X.: Can the two-parameter recursive digital filter baseflow separation method really be calibrated by the conductivity mass balance method?, Hydrol. Earth Syst. Sci., 25, 1747–1760, https://doi.org/10.5194/hess-25-1747-2021, 2021.
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AC2: 'Reply on CC2', Klaus Eckhardt, 07 Jul 2022
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CC2: 'Reply on AC1', Keith Beven, 28 Jun 2022
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AC1: 'Reply on CC1', Klaus Eckhardt, 28 Jun 2022
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RC1: 'Comment on hess-2022-186', Anonymous Referee #1, 04 Aug 2022
 Comments:
- For equation 5 and 6, if no delay is assumed (d = 0), the subscript of y and b should be k-1 instead of k. Why is (yk – bk) in the manuscript? The baseflow in the current step should be related to the past step.
- For line 120, the assumption is that there is no major time lag between precipitation and the exfiltration of groundwater into surface waters. I don’t think this statement is always true, especially in the large storm or extreme soil infiltration.
-
AC3: 'Reply on RC1', Klaus Eckhardt, 06 Aug 2022
Reply to comment 1:
A precise distinction must be made as to which delay is being referred to. In the publication of Furey and Gupta (2001), d stands for the number of time steps between precipitation and groundwater recharge and not for the number of time steps between precipitation and the exfiltration of baseflow. Baseflow in the current time step is calculated from the groundwater recharge in the previous time step, among other things. Hence we have: time steps between precipitation and baseflow = d time steps between precipitation and groundwater recharge + 1 time step between groundwater recharge and baseflow. The statement that baseflow occurs in the same time step as precipitation thus means d + 1 = 0 or k - d - 1 = k - (d + 1) = k.
I suggest to replace the lines 94 to 100 in my text as follows:
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. (10)). Further, they assume that the groundwater recharge is delayed by d time steps compared to precipitation (their Eq. (11)). In their model of the emergence of baseflow, the number of time steps between precipitation and baseflow is d + 1: d time steps between precipitation and groundwater recharge + 1 time step between groundwater recharge and baseflow. Hence, the index j – d – 1 in Eq. (4) or k – d – 1 in Eq. (5).
If instead it is assumed that baseflow occurs in the same time step as groundwater recharge and groundwater recharge is not delayed to precipitation, in other words, if it is assumed that the delay between precipitation and baseflow is smaller than one time step, then d + 1 = 0 and thus k - d - 1 = k - (d + 1) = k. Equation (5) is thenÂReply to the comment 2:
It is true that this point should be critically questioned. However, I cannot and do not want to go into it in depth in this technical note, which only deals with the comparison of two algorithms. Just two notes on this: (1) I have since corrected my text to the effect that I no longer equate baseflow with exfiltrating groundwater alone, see my reply 2 to CC1. (2) When I write that the generation of baseflow still occurs in the same time step as precipitation, this is a statement entirely within the framework of Furey and Gupta's (2001) modelling of how baseflow occurs. I am merely pointing out a difference to the algorithm of Furey and Gupta (2001). To what extent this algorithm itself accurately reflects reality is another question.
In my reply to CC2, I have already made a suggestion on how to replace lines 121 to 124. I modify this suggestion as follows:
Furey and Gupta (2001) introduced the parameter d in Eq. (5) as the number of time steps between precipitation and groundwater recharge. A sensitivity analysis they conducted showed that the filter performance was "relatively insensitive to changes in d" so that d = 0 seemed to be an acceptable choice. Furthermore, when using Eq. (1), it is assumed that not only the groundwater recharge but also the generation of baseflow still occurs in the same time step as precipitation. When assessing this prerequisite, two aspects should be considered:
(1) The streamflow component calculated with Eq. (1) is usually likely to consist not only of groundwater, but also of transient water sources, including interflow (Cartwright et al., 2014; Yang et al., 2021).Â
(2) In this publication, the algorithm of Eckhardt (2005) is compared to the model ideas of Furey and Gupta (2001) on the formation of baseflow, not to the reality. If the baseflow calculated with Eq. (1) occurs in Furey and Gupta's model world at the same time step as precipitation, this does not necessarily mean that it also corresponds to a runoff component in the real world that occurs without a relevant time lag to precipitation.Â
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RC2: 'Comment on hess-2022-186', Anonymous Referee #2, 11 Aug 2022
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess-2022-186/hess-2022-186-RC2-supplement.pdf
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AC4: 'Reply on RC2', Klaus Eckhardt, 15 Aug 2022
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess-2022-186/hess-2022-186-AC4-supplement.pdf
-
AC4: 'Reply on RC2', Klaus Eckhardt, 15 Aug 2022
Peer review completion
Interactive discussion
Status: closed
-
CC1: 'Comment on hess-2022-186', Keith Beven, 23 Jun 2022
A very long time ago (in 1991) I published a review of hydrograph separation methods. The review had been requested for a meeting of the British Hydrological Society.  It included a section on Choosing a Hydrograph Seperation method.  That section consisted of a single word: "Don't."   The reason for that was mostly to avoid the types of problems represented by this paper - particularly the inference that some mathematical filter can be used to decide what is groundwater or not.  The author appears to think that baseflow and groundwater are equivalent.  This is particularly ironic when he references the use of tracer information to support the lack of a time delay in his own function on the basis that tracers show that there can be "a rapid release of so-called pre-event-water".  But just why should that pre-event water be baseflow (or groundwater)?  The tracer data generally undermine the whole idea of baseflow separation.  To suggest that something might be physically-based by comparing one mathematical function to another mathematical function based on a linear store is surely naive at best, and downright misleading at worst.  We might perhaps want to use a consistent mathematical filter to produce some "baseflow index" as a characteristic of catchment response but please do not relate it to any superficial process interpretation. Better still, please do not choose a baseflow separation method at all but try to understand the actual processes of catchment response.
Reference.
Beven, K.J. (1991), Hydrograph Separation?, Proc.BHS Third National Hydrology Symposium, Institute of Hydrology, Wallingford, 3.1-3.8.
Â
-
AC1: 'Reply on CC1', Klaus Eckhardt, 28 Jun 2022
Keith Beven writes that the point of this article is to "to suggest that something might be physically-based by comparing one mathematical function to another mathematical function". Apparently, he does not concede the approach of Furey and Gupta (2001) to be physically based. I see it differently. In my view, Â their algorithm is physically based. One can argue about how accurate the physical basis is. However, a fundamental debate on whether hydrograph separation is useful or not goes far beyond the purpose of this technical note. There are, after all, a variety of methods of hydrograph separation and they are used. For more than four decades, recursive digital filtering has been one of them. The present contribution thus does exactly what HESS associates with a technical note: "Technical notes report [...] novel aspects of [...] theoretical methods and techniques which are relevant for scientific investigations within the journal scope." (https://www.hydrology-and-earth-system-sciences.net/about/manuscript_types.html).
-
CC2: 'Reply on AC1', Keith Beven, 28 Jun 2022
OK, I will accept the argument that physically-based can be defined in terms of mathematics derived from explicit assumptions and you are certainly correct about the technical note. Â I am not in any way disputing that, only the utility of such analyses. Â The danger, as ever, is thinking that those assumptions represent the actual physics of the catchments we are interested in, as you seem to do when you equate baseflow and groundwater, and when you invoke tracer evidence for fast groundwater responses when surely that very information suggests that contributions of pre-event water are quite different from baseflow defined by those mathematics. Â That therefore suggests to me that the assumptions of the mathematics are wrong in terms of being a physically-based description of the actual processes. Â So create a baseflow index if you wish, but please do not call it groundwater (or even better, explicitly differentiate them to avoid others making a similar false equivalence). Â Â
-
AC2: 'Reply on CC2', Klaus Eckhardt, 07 Jul 2022
I will gladly take into account the critical comments on the submitted contribution. I propose the following changes to the text, which will result in the terms "groundwater" and "pre-event water" being largely omitted. Only the term "groundwater recharge" remains in the context of its use by Furey and Gupta (2001).
Lines 7 - 8: "a widely used method to identify the groundwater-borne portion of streamflow" is replaced by "a widely used method to identify streamflow components, which react to precipitation with varying degrees of attenuation and delay".
Line 11: "the aquifer is a linear reservoir" is replaced by "baseflow is runoff from a linear reservoir".
Â
Line 14: "exfiltration of groundwater into surface waters" is replaced by "exfiltration of baseflow into surface waters".Lines 17 - 22 are replaced by
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 less attenuated in response to precipitation events, the so-called direct runoff. From this idea, it is obvious to low-pass filter streamflow hydrographs to identify these components.Lines 63 - 64: "(b) The aquifer is a linear reservoir, i.e. the discharge from the aquifer is proportional to the amount of water stored in it. Without further knowledge about the physical properties of the aquifer, this is the most obvious approach." is replaced by "(b) Baseflow is runoff from a linear reservoir, i.e. it is proportional to the amount of water stored in this reservoir.".
Line 115: "the assumption of the aquifer being a linear reservoir" is replaced by "the assumption that baseflow is runoff from a linear reservoir".
Lines 118 - 119: "the exfiltration of groundwater into surface waters" is replaced by "the exfiltration of baseflow into surface waters".
Lines 121 - 124 are replaced by
Furey and Gupta (2001) introduced the parameter d in Eq. (5) as the number of time steps between precipitation and groundwater recharge. A sensitivity analysis they conducted showed that the filter performance was "relatively insensitive to changes in d" so that d = 0 seemed to be an acceptable choice. Furthermore, when using Eq. (1), it is assumed that not only the groundwater recharge but also the generation of baseflow still occurs in the same time step as precipitation. When assessing this prerequisite, it should be noted that the streamflow component calculated with Eq. (1) is usually likely to consist not only of groundwater, but also of transient water sources, including interflow (Cartwright et al., 2014; Yang et al., 2021).Lines 136 - 137: "This is plausible. The parameter BFImax was introduced as the maximum value of the baseflow index that can be calculated. And the baseflow can at most correspond to the groundwater recharge." is deleted.
Lines 156 - 157 are deleted.
Lines 173 - 174 are deleted.Added are the references
Cartwright, I., Gilfedder, B., and Hofmann, H.: Contrasts between estimates of baseflow help discern multiple sources of water contributing to rivers, Hydrol. Earth Syst. Sci., 18, 15–30, https://doi.org/10.5194/hess-18-15-2014, 2014.Â
Yang, W., Xiao, C., Zhang, Z., and Liang, X.: Can the two-parameter recursive digital filter baseflow separation method really be calibrated by the conductivity mass balance method?, Hydrol. Earth Syst. Sci., 25, 1747–1760, https://doi.org/10.5194/hess-25-1747-2021, 2021.
-
AC2: 'Reply on CC2', Klaus Eckhardt, 07 Jul 2022
-
CC2: 'Reply on AC1', Keith Beven, 28 Jun 2022
-
AC1: 'Reply on CC1', Klaus Eckhardt, 28 Jun 2022
-
RC1: 'Comment on hess-2022-186', Anonymous Referee #1, 04 Aug 2022
 Comments:
- For equation 5 and 6, if no delay is assumed (d = 0), the subscript of y and b should be k-1 instead of k. Why is (yk – bk) in the manuscript? The baseflow in the current step should be related to the past step.
- For line 120, the assumption is that there is no major time lag between precipitation and the exfiltration of groundwater into surface waters. I don’t think this statement is always true, especially in the large storm or extreme soil infiltration.
-
AC3: 'Reply on RC1', Klaus Eckhardt, 06 Aug 2022
Reply to comment 1:
A precise distinction must be made as to which delay is being referred to. In the publication of Furey and Gupta (2001), d stands for the number of time steps between precipitation and groundwater recharge and not for the number of time steps between precipitation and the exfiltration of baseflow. Baseflow in the current time step is calculated from the groundwater recharge in the previous time step, among other things. Hence we have: time steps between precipitation and baseflow = d time steps between precipitation and groundwater recharge + 1 time step between groundwater recharge and baseflow. The statement that baseflow occurs in the same time step as precipitation thus means d + 1 = 0 or k - d - 1 = k - (d + 1) = k.
I suggest to replace the lines 94 to 100 in my text as follows:
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. (10)). Further, they assume that the groundwater recharge is delayed by d time steps compared to precipitation (their Eq. (11)). In their model of the emergence of baseflow, the number of time steps between precipitation and baseflow is d + 1: d time steps between precipitation and groundwater recharge + 1 time step between groundwater recharge and baseflow. Hence, the index j – d – 1 in Eq. (4) or k – d – 1 in Eq. (5).
If instead it is assumed that baseflow occurs in the same time step as groundwater recharge and groundwater recharge is not delayed to precipitation, in other words, if it is assumed that the delay between precipitation and baseflow is smaller than one time step, then d + 1 = 0 and thus k - d - 1 = k - (d + 1) = k. Equation (5) is thenÂReply to the comment 2:
It is true that this point should be critically questioned. However, I cannot and do not want to go into it in depth in this technical note, which only deals with the comparison of two algorithms. Just two notes on this: (1) I have since corrected my text to the effect that I no longer equate baseflow with exfiltrating groundwater alone, see my reply 2 to CC1. (2) When I write that the generation of baseflow still occurs in the same time step as precipitation, this is a statement entirely within the framework of Furey and Gupta's (2001) modelling of how baseflow occurs. I am merely pointing out a difference to the algorithm of Furey and Gupta (2001). To what extent this algorithm itself accurately reflects reality is another question.
In my reply to CC2, I have already made a suggestion on how to replace lines 121 to 124. I modify this suggestion as follows:
Furey and Gupta (2001) introduced the parameter d in Eq. (5) as the number of time steps between precipitation and groundwater recharge. A sensitivity analysis they conducted showed that the filter performance was "relatively insensitive to changes in d" so that d = 0 seemed to be an acceptable choice. Furthermore, when using Eq. (1), it is assumed that not only the groundwater recharge but also the generation of baseflow still occurs in the same time step as precipitation. When assessing this prerequisite, two aspects should be considered:
(1) The streamflow component calculated with Eq. (1) is usually likely to consist not only of groundwater, but also of transient water sources, including interflow (Cartwright et al., 2014; Yang et al., 2021).Â
(2) In this publication, the algorithm of Eckhardt (2005) is compared to the model ideas of Furey and Gupta (2001) on the formation of baseflow, not to the reality. If the baseflow calculated with Eq. (1) occurs in Furey and Gupta's model world at the same time step as precipitation, this does not necessarily mean that it also corresponds to a runoff component in the real world that occurs without a relevant time lag to precipitation.Â
-
RC2: 'Comment on hess-2022-186', Anonymous Referee #2, 11 Aug 2022
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess-2022-186/hess-2022-186-RC2-supplement.pdf
-
AC4: 'Reply on RC2', Klaus Eckhardt, 15 Aug 2022
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess-2022-186/hess-2022-186-AC4-supplement.pdf
-
AC4: 'Reply on RC2', Klaus Eckhardt, 15 Aug 2022
Peer review completion
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Klaus Eckhardt
Klaus Eckhardt
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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