the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
PowerLaw between the Apparent Drainage Density and the Pruning Area
Abstract. Selfsimilar structures of river networks have been quantified as diverse scaling laws. Among them we investigated a power functional relationship between the pruning area A_{p} and the associated apparent drainage density ρ_{a} with an exponent η. We analytically derived the relationship between η and other scaling exponents known for fractal river networks. The derivation is supported by analysis of four real river networks. The relationship between η and noninteger fractal dimensions found for natural river networks is suggested. Synthesis of our findings through the lens of fractal dimensions provides an insight that the exponent η has fundamental roots in fractal dimension for the whole river network organization.
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RC1: 'Comment on hess2022237', Anonymous Referee #1, 16 Aug 2022
The authors investigated a power functional relationship between the pruning area A_{p} and the associated apparent drainage density ρ_{a} identifying the scaling exponent of such relationship. Moreover, the exponent η is then linked to other scaling exponents representative of the fractal characteristics of the river basin. Topic may be of interest, but I have some doubt about the reliability of some of the outcomes of the research.
Major comments:
 The entire manuscript is based on the derivation of an analytical relationship for the derivation of the scaling exponent η. The authors propose three different relationships which leads to very different results. In particular, the first two expressions have analytical basis, but not very good results. The last equation seems to be an empirical one derived from data provides better results. In my experience, the analytical relationships should hold in all environments, and this represents a great advantage for their use. On the other hand, the empirical relationship is very case specific and limit significantly the impact of the research.
 Personally, this kind of analysis require a limited computational effort. Therefore, it would be much interesting to explore a larger number of case studies.
 The range of variability of the exponent η is relatively limited (0.420.47). The mean value is 0.445, which probably performs better than any of the relationship proposed. I have tested the error associate to the use of a mean value which is of the order of 5%. Therefore, I would like to know what is the added value of the proposed relationships.
Citation: https://doi.org/10.5194/hess2022237RC1 
AC1: 'Reply on RC1', Kyungrock Paik, 06 Oct 2022
We thank Referee 1 (R1) for constructive comments. Our response to each comment is listed below. We will reflect any change addressed below into the revised manuscript which is requested in the subsequent step.
R1: The entire manuscript is based on the derivation of an analytical relationship for the derivation of the scaling exponent η. The authors propose three different relationships which leads to very different results. In particular, the first two expressions have analytical basis, but not very good results. The last equation seems to be an empirical one derived from data provides better results. In my experience, the analytical relationships should hold in all environments, and this represents a great advantage for their use. On the other hand, the empirical relationship is very case specific and limit significantly the impact of the research.
Reply: In general, as R1 commented, analytical relationships hold broader applicability to diverse environments in consistent ways. However, their applicability is also constraint by assumptions set for deriving them. Two analytical relationships of Eq. (21) and (24) are derived on the basis of the assumption employed to derive two fractal dimensions as the associated functions of Horton’s ratios. The specific assumption is too ideal, resulting in the inevitable discrepancy between the estimates and the observed. We will add the following sentences in Sec. 4 of the revised manuscript to describe this: “It is interesting that the simple Eq. (25) is well supported by analysis results, at least as good as the other two expressions. Theoretical derivations of Eqs. (20) and (23) rely on a fundamental assumption, i.e., Horton’s laws hold precisely at all scales of a unit length to measure (La Barbera and Rosso, 1989; Rosso et al., 1991). Indeed, the assumption is too ideal to be satisfied for real river networks, as corroborated in the nonperfect straight fits when estimating Horton’s ratios of our studied networks (Fig. S2 in SI). Moreover, the importance of fulfilling the assumption to employ Eq. (23) is demonstrated by Phillips (1993) studying very small catchments in the Southern Appalachians in the USA. This is the likely reason that Eqs. (21) and (24), derived on the basis of Eqs. (20) and (23), show greater deviations from the observed η values, than Eq. (25). As for Eq. (25), we suppose that incorporating the empirical approach into the theoretical D_{b} expression mitigates the likelihood of discrepancy between the estimated and the observed η values, compared to the other two expressions.”
R1: Personally, this kind of analysis require a limited computational effort. Therefore, it would be much interesting to explore a larger number of case studies.
Reply: We have doubled the number of river networks analyzed, from four to eight. They exhibit diverse hydroclimatic, geomorphologic, and geologic conditions. We found consistent results about the estimation of the exponent η, supporting our arguments, from all river networks. Details will be given in the revised manuscript.
R1: The range of variability of the exponent η is relatively limited (0.420.47). The mean value is 0.445, which probably performs better than any of the relationship proposed. I have tested the error associate to the use of a mean value which is of the order of 5%. Therefore, I would like to know what is the added value of the proposed relationships.
Reply: One of the main goals of this study is to unravel the dimensional inconsistency in ρ_{a}A_{p }relationship, in other words, the deviation of η from 0.5. This is associated with the fractal dimension, by three relationships: two 'analytical' relationships based on earlier studies with assumptions, and a new relationship which is inspired empirically. We have no intention to claim which relationship is better than the other. We have successfully demonstrated the close linkage between η and fractal dimensions, on the basis of observed data. All three equations show 'generally' good relationships, which already addresses our purpose. The idea behind Eq. (25) is another contribution in that it provides a new scientific insight of the potential applicability of the 'quarterpower' scaling into the geomorphic system, which is worthy of being disseminated in the hydrology community.
Citation: https://doi.org/10.5194/hess2022237AC1

RC2: 'Comment on hess2022237', Anonymous Referee #2, 24 Aug 2022
General comment
The authors investigated on the relationship between the pruning area of a river basin (A_{p}) and the apparent drainage density (ρ_{a}) presenting two analytical derivations demonstrating that the exponent η of the pruning area is equal to the exponent ε of the powerlaw exceedance probability distribution of drainage area. The authors tested the derivation on four US river basins and, subsequently tested the relationship of the exponent η with the fractal dimension of the river basins considering two literature expressions and proposing a new expression.
The work has a scientific value, but I have two main concerns related to the analysis.
 The hypothesis behind the two derivations of the exponent η. Section 2.1.1. The Derivation 1 is developed in the hypothesis that A_{p} = A_{ð} (Page 5, line 123). However, eq (14) is derived in the hypothesis that A_{p} << A_{Ω}. The authors should discuss on this assumption. Does it mean that the expression could be scale relevant and not appliable if ω is close to â¦? The same assumption (A_{p} = A_{ð}) is done in Section 2.2.2 (Derivation 2), but then the equation (16) is generalized regardless the scale of the basin. Is equation (16) still valid if A_{p} << A_{ð}?
 It seems that the authors proposed a new expression (eq. 25) of η as function of the fractal dimension D_{b} just “observing the real river network” and finding a support of this choice considering that quarterpower scaling laws are widely found in biological systems. However, the proposed equation, even if it performs better in the four case studies as respect to the literature expressions, seems to always underestimate the η value. Why fixing an “a priori” coefficient? Even if the sample size is very low, why not deriving the coefficient, minimizing the error between observed and estimated values and proposing, as future development, a testing of the new expression with a wider sample size?
Minor comments
Page 5, line 134. “… the overland flow length W…” Do the authors mean flow width?
Page 11, line 259260. What does “satisfactory results” mean? Did they calculate any performance indicator? The sample seems quite small (4 basins). Can the analysis be considered robust?
Citation: https://doi.org/10.5194/hess2022237RC2 
AC2: 'Reply on RC2', Kyungrock Paik, 06 Oct 2022
We thank Referee 2 (R2) for helpful comments. Our replies to all comments are listed below. We will reflect any change addressed below into the revised manuscript which is requested in the subsequent step.
R2: The hypothesis behind the two derivations of the exponent η. Section 2.1.1. The Derivation 1 is developed in the hypothesis that A_{p} = A_{ω} (Page 5, line 123). However, eq (14) is derived in the hypothesis that A_{p} << A_{Ω}. The authors should discuss on this assumption. Does it mean that the expression could be scale relevant and not appliable if ω is close to Ω? The same assumption (A_{p} = A_{ω}) is done in Section 2.2.2 (Derivation 2), but then the equation (16) is generalized regardless the scale of the basin. Is equation (16) still valid if A_{p} << A_{ω}?
Reply: In the Derivation 1, we intended the condition of A_{p} << A_{Ω}_{}to imply the limited boundary condition for a pruning area variable. Reconciling with the orderbyorder A_{p} representation, it means that the powerlaw behavior in Eq. (13) with the exponent of Eq. (14) gets weaker as ω is close to Ω. Indeed, its behavior is caught as the finite size effect in the ρ_{a}A_{p} relationship (see Fig. 2 in the original manuscript). To clarify it, we will add following sentences in Sec. 2.2.1 : “Note that A_{p} << A_{Ω} is an extreme expression intended for the neat derivation of the powerlaw relationship. Notwithstanding, it is also supported from the empirical boundary condition of to characterize fluvial channel networks (McNamara et al., 2006; Montgomery and FoufoulaGeorgiou, 1993).”
In the Derivation 2, Eq. (16) is independent from the scale of the entire basin, because it is based on geometric representation of a river network. Please note that the treatment of A_{p} = A_{ω} is not an assumption but is merely to interprete A_{p} in the HortonStrahler ordered distrete series. It is not sure what R2 meant by the condition of A_{p} << A_{ω}_{}because it would also depend on ω.
R2: It seems that the authors proposed a new expression (eq. 25) of η as function of the fractal dimension D_{b} just “observing the real river network” and finding a support of this choice considering that quarterpower scaling laws are widely found in biological systems. However, the proposed equation, even if it performs better in the four case studies as respect to the literature expressions, seems to always underestimate the η value. Why fixing an “a priori” coefficient? Even if the sample size is very low, why not deriving the coefficient, minimizing the error between observed and estimated values and proposing, as future development, a testing of the new expression with a wider sample size?
Reply: Following R2's suggestion of "a testing of the new expression with a wider sample size," we have doubled the number of study networks from 4 to 8. On the basis of analysis of eight networks, our key results and conclusions remain the same. It is found that the seemingly underestimation from 4 networks was merely coincidence. Out of total 8 networks, 4 networks exhibit η greater than observed values, while the rest exhibit smaller η. These results will be reflected in the revised manuscript.
With the increased number of networks, following R2's suggestion, we find the bestfit coefficient. Interestingly, the bestfit coincides with the proposed coefficient of 1/4. To clarify the estimation processes, we will add the following sentences near Eq. (25) in the revised manuscript: “Note that the coefficient of 1/4 is based on our analysis of real studied river networks (0.25±0.02).”
R2: Page 5, line 134. “… the overland flow length W…” Do the authors mean flow width?
Reply: We will clarify the terminology in Sec. 2.2.2 : “W is the mean overland flow length”.
R2: Page 11, line 259260. What does “satisfactory results” mean? Did they calculate any performance indicator? The sample seems quite small (4 basins). Can the analysis be considered robust?
Reply: The original paragraph with the term of “satisfactory results” will be discarded in the revised manuscript. In the scope of this study, we don’t need to calculate a performance indicator. To present more robust arguments, we doubled study areas and conducted all relevant analyses. Further discussion on the updated results will be added in Sec. 4 of the revised manuscript as follows : “It is interesting that the simple Eq. (25) is well supported by analysis results, at least as good as the other two expressions. Theoretical derivations of Eqs. (20) and (23) rely on a fundamental assumption, i.e., Horton’s laws hold precisely at all scales of a unit length to measure (La Barbera and Rosso, 1989; Rosso et al., 1991). Indeed, the assumption is too ideal to be satisfied for real river networks, as corroborated in the nonperfect straight fits when estimating Horton’s ratios of our studied networks (Fig. S2 in SI). Moreover, the importance of fulfilling the assumption to employ Eq. (23) is demonstrated by Phillips (1993) studying very small catchments in the Southern Appalachians in the USA. This is the likely reason that Eqs. (21) and (24), derived on the basis of Eqs. (20) and (23), show greater deviations from the observed η values, than Eq. (25). As for Eq. (25), we suppose that incorporating the empirical approach into the theoretical D_{b} expression mitigates the likelihood of discrepancy between the estimated and the observed η values, compared to the other two expressions.”
Citation: https://doi.org/10.5194/hess2022237AC2
Status: closed

RC1: 'Comment on hess2022237', Anonymous Referee #1, 16 Aug 2022
The authors investigated a power functional relationship between the pruning area A_{p} and the associated apparent drainage density ρ_{a} identifying the scaling exponent of such relationship. Moreover, the exponent η is then linked to other scaling exponents representative of the fractal characteristics of the river basin. Topic may be of interest, but I have some doubt about the reliability of some of the outcomes of the research.
Major comments:
 The entire manuscript is based on the derivation of an analytical relationship for the derivation of the scaling exponent η. The authors propose three different relationships which leads to very different results. In particular, the first two expressions have analytical basis, but not very good results. The last equation seems to be an empirical one derived from data provides better results. In my experience, the analytical relationships should hold in all environments, and this represents a great advantage for their use. On the other hand, the empirical relationship is very case specific and limit significantly the impact of the research.
 Personally, this kind of analysis require a limited computational effort. Therefore, it would be much interesting to explore a larger number of case studies.
 The range of variability of the exponent η is relatively limited (0.420.47). The mean value is 0.445, which probably performs better than any of the relationship proposed. I have tested the error associate to the use of a mean value which is of the order of 5%. Therefore, I would like to know what is the added value of the proposed relationships.
Citation: https://doi.org/10.5194/hess2022237RC1 
AC1: 'Reply on RC1', Kyungrock Paik, 06 Oct 2022
We thank Referee 1 (R1) for constructive comments. Our response to each comment is listed below. We will reflect any change addressed below into the revised manuscript which is requested in the subsequent step.
R1: The entire manuscript is based on the derivation of an analytical relationship for the derivation of the scaling exponent η. The authors propose three different relationships which leads to very different results. In particular, the first two expressions have analytical basis, but not very good results. The last equation seems to be an empirical one derived from data provides better results. In my experience, the analytical relationships should hold in all environments, and this represents a great advantage for their use. On the other hand, the empirical relationship is very case specific and limit significantly the impact of the research.
Reply: In general, as R1 commented, analytical relationships hold broader applicability to diverse environments in consistent ways. However, their applicability is also constraint by assumptions set for deriving them. Two analytical relationships of Eq. (21) and (24) are derived on the basis of the assumption employed to derive two fractal dimensions as the associated functions of Horton’s ratios. The specific assumption is too ideal, resulting in the inevitable discrepancy between the estimates and the observed. We will add the following sentences in Sec. 4 of the revised manuscript to describe this: “It is interesting that the simple Eq. (25) is well supported by analysis results, at least as good as the other two expressions. Theoretical derivations of Eqs. (20) and (23) rely on a fundamental assumption, i.e., Horton’s laws hold precisely at all scales of a unit length to measure (La Barbera and Rosso, 1989; Rosso et al., 1991). Indeed, the assumption is too ideal to be satisfied for real river networks, as corroborated in the nonperfect straight fits when estimating Horton’s ratios of our studied networks (Fig. S2 in SI). Moreover, the importance of fulfilling the assumption to employ Eq. (23) is demonstrated by Phillips (1993) studying very small catchments in the Southern Appalachians in the USA. This is the likely reason that Eqs. (21) and (24), derived on the basis of Eqs. (20) and (23), show greater deviations from the observed η values, than Eq. (25). As for Eq. (25), we suppose that incorporating the empirical approach into the theoretical D_{b} expression mitigates the likelihood of discrepancy between the estimated and the observed η values, compared to the other two expressions.”
R1: Personally, this kind of analysis require a limited computational effort. Therefore, it would be much interesting to explore a larger number of case studies.
Reply: We have doubled the number of river networks analyzed, from four to eight. They exhibit diverse hydroclimatic, geomorphologic, and geologic conditions. We found consistent results about the estimation of the exponent η, supporting our arguments, from all river networks. Details will be given in the revised manuscript.
R1: The range of variability of the exponent η is relatively limited (0.420.47). The mean value is 0.445, which probably performs better than any of the relationship proposed. I have tested the error associate to the use of a mean value which is of the order of 5%. Therefore, I would like to know what is the added value of the proposed relationships.
Reply: One of the main goals of this study is to unravel the dimensional inconsistency in ρ_{a}A_{p }relationship, in other words, the deviation of η from 0.5. This is associated with the fractal dimension, by three relationships: two 'analytical' relationships based on earlier studies with assumptions, and a new relationship which is inspired empirically. We have no intention to claim which relationship is better than the other. We have successfully demonstrated the close linkage between η and fractal dimensions, on the basis of observed data. All three equations show 'generally' good relationships, which already addresses our purpose. The idea behind Eq. (25) is another contribution in that it provides a new scientific insight of the potential applicability of the 'quarterpower' scaling into the geomorphic system, which is worthy of being disseminated in the hydrology community.
Citation: https://doi.org/10.5194/hess2022237AC1

RC2: 'Comment on hess2022237', Anonymous Referee #2, 24 Aug 2022
General comment
The authors investigated on the relationship between the pruning area of a river basin (A_{p}) and the apparent drainage density (ρ_{a}) presenting two analytical derivations demonstrating that the exponent η of the pruning area is equal to the exponent ε of the powerlaw exceedance probability distribution of drainage area. The authors tested the derivation on four US river basins and, subsequently tested the relationship of the exponent η with the fractal dimension of the river basins considering two literature expressions and proposing a new expression.
The work has a scientific value, but I have two main concerns related to the analysis.
 The hypothesis behind the two derivations of the exponent η. Section 2.1.1. The Derivation 1 is developed in the hypothesis that A_{p} = A_{ð} (Page 5, line 123). However, eq (14) is derived in the hypothesis that A_{p} << A_{Ω}. The authors should discuss on this assumption. Does it mean that the expression could be scale relevant and not appliable if ω is close to â¦? The same assumption (A_{p} = A_{ð}) is done in Section 2.2.2 (Derivation 2), but then the equation (16) is generalized regardless the scale of the basin. Is equation (16) still valid if A_{p} << A_{ð}?
 It seems that the authors proposed a new expression (eq. 25) of η as function of the fractal dimension D_{b} just “observing the real river network” and finding a support of this choice considering that quarterpower scaling laws are widely found in biological systems. However, the proposed equation, even if it performs better in the four case studies as respect to the literature expressions, seems to always underestimate the η value. Why fixing an “a priori” coefficient? Even if the sample size is very low, why not deriving the coefficient, minimizing the error between observed and estimated values and proposing, as future development, a testing of the new expression with a wider sample size?
Minor comments
Page 5, line 134. “… the overland flow length W…” Do the authors mean flow width?
Page 11, line 259260. What does “satisfactory results” mean? Did they calculate any performance indicator? The sample seems quite small (4 basins). Can the analysis be considered robust?
Citation: https://doi.org/10.5194/hess2022237RC2 
AC2: 'Reply on RC2', Kyungrock Paik, 06 Oct 2022
We thank Referee 2 (R2) for helpful comments. Our replies to all comments are listed below. We will reflect any change addressed below into the revised manuscript which is requested in the subsequent step.
R2: The hypothesis behind the two derivations of the exponent η. Section 2.1.1. The Derivation 1 is developed in the hypothesis that A_{p} = A_{ω} (Page 5, line 123). However, eq (14) is derived in the hypothesis that A_{p} << A_{Ω}. The authors should discuss on this assumption. Does it mean that the expression could be scale relevant and not appliable if ω is close to Ω? The same assumption (A_{p} = A_{ω}) is done in Section 2.2.2 (Derivation 2), but then the equation (16) is generalized regardless the scale of the basin. Is equation (16) still valid if A_{p} << A_{ω}?
Reply: In the Derivation 1, we intended the condition of A_{p} << A_{Ω}_{}to imply the limited boundary condition for a pruning area variable. Reconciling with the orderbyorder A_{p} representation, it means that the powerlaw behavior in Eq. (13) with the exponent of Eq. (14) gets weaker as ω is close to Ω. Indeed, its behavior is caught as the finite size effect in the ρ_{a}A_{p} relationship (see Fig. 2 in the original manuscript). To clarify it, we will add following sentences in Sec. 2.2.1 : “Note that A_{p} << A_{Ω} is an extreme expression intended for the neat derivation of the powerlaw relationship. Notwithstanding, it is also supported from the empirical boundary condition of to characterize fluvial channel networks (McNamara et al., 2006; Montgomery and FoufoulaGeorgiou, 1993).”
In the Derivation 2, Eq. (16) is independent from the scale of the entire basin, because it is based on geometric representation of a river network. Please note that the treatment of A_{p} = A_{ω} is not an assumption but is merely to interprete A_{p} in the HortonStrahler ordered distrete series. It is not sure what R2 meant by the condition of A_{p} << A_{ω}_{}because it would also depend on ω.
R2: It seems that the authors proposed a new expression (eq. 25) of η as function of the fractal dimension D_{b} just “observing the real river network” and finding a support of this choice considering that quarterpower scaling laws are widely found in biological systems. However, the proposed equation, even if it performs better in the four case studies as respect to the literature expressions, seems to always underestimate the η value. Why fixing an “a priori” coefficient? Even if the sample size is very low, why not deriving the coefficient, minimizing the error between observed and estimated values and proposing, as future development, a testing of the new expression with a wider sample size?
Reply: Following R2's suggestion of "a testing of the new expression with a wider sample size," we have doubled the number of study networks from 4 to 8. On the basis of analysis of eight networks, our key results and conclusions remain the same. It is found that the seemingly underestimation from 4 networks was merely coincidence. Out of total 8 networks, 4 networks exhibit η greater than observed values, while the rest exhibit smaller η. These results will be reflected in the revised manuscript.
With the increased number of networks, following R2's suggestion, we find the bestfit coefficient. Interestingly, the bestfit coincides with the proposed coefficient of 1/4. To clarify the estimation processes, we will add the following sentences near Eq. (25) in the revised manuscript: “Note that the coefficient of 1/4 is based on our analysis of real studied river networks (0.25±0.02).”
R2: Page 5, line 134. “… the overland flow length W…” Do the authors mean flow width?
Reply: We will clarify the terminology in Sec. 2.2.2 : “W is the mean overland flow length”.
R2: Page 11, line 259260. What does “satisfactory results” mean? Did they calculate any performance indicator? The sample seems quite small (4 basins). Can the analysis be considered robust?
Reply: The original paragraph with the term of “satisfactory results” will be discarded in the revised manuscript. In the scope of this study, we don’t need to calculate a performance indicator. To present more robust arguments, we doubled study areas and conducted all relevant analyses. Further discussion on the updated results will be added in Sec. 4 of the revised manuscript as follows : “It is interesting that the simple Eq. (25) is well supported by analysis results, at least as good as the other two expressions. Theoretical derivations of Eqs. (20) and (23) rely on a fundamental assumption, i.e., Horton’s laws hold precisely at all scales of a unit length to measure (La Barbera and Rosso, 1989; Rosso et al., 1991). Indeed, the assumption is too ideal to be satisfied for real river networks, as corroborated in the nonperfect straight fits when estimating Horton’s ratios of our studied networks (Fig. S2 in SI). Moreover, the importance of fulfilling the assumption to employ Eq. (23) is demonstrated by Phillips (1993) studying very small catchments in the Southern Appalachians in the USA. This is the likely reason that Eqs. (21) and (24), derived on the basis of Eqs. (20) and (23), show greater deviations from the observed η values, than Eq. (25). As for Eq. (25), we suppose that incorporating the empirical approach into the theoretical D_{b} expression mitigates the likelihood of discrepancy between the estimated and the observed η values, compared to the other two expressions.”
Citation: https://doi.org/10.5194/hess2022237AC2
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