Reply on RC1

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.

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 non-perfect 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." 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 hydro-climatic, 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. η is relatively limited (0.42-0.47). 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.

R1: The range of variability of the exponent
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 'quarter-power' scaling into the geomorphic system, which is worthy of being disseminated in the hydrology community.