We appreciate the reminder that many studies use circular/directional statistics to characterize the seasonality of extremes, as evidenced by the suggested references. In our manuscript, we acknowledge this development by stating:  

“Directional statistics have been widely used to characterize the seasonality of extreme flows (e.g., Burn et al., 1997; Young et al., 2000; Merz & Blöschl, 2003; Laaha and Blöschl, 2006; Blöschl et al., 2017; Berghuijs et al., 2016, 2019; Floriancic et al., 2021; Chagas et al., 2022).”  

We will incorporate some of the suggested references, although an exhaustive list of all studies that could be cited here would be far more extensive than what is provided by us and these suggestions.

Importantly, our manuscript develops and applies directional statistics to characterize (continous) seasonal flow regimes, rather than focusing on the seasonality of annual extremes or peaks-over-threshold events. Our approach thereby unifies the concepts of seasonal flow regimes, center of mass, and directional statistics—which is not covered by the referenced studies. These developments are probably not complicated to someone familiar with directional statistics for characterizing extreme events, but have not been shown in the (suggested) literature, and do have many potential applications (also beyond hydrology). 

Response to Reviewer #1 (in bold)

This study proposed a new metric based on directional statistics to quantify the streamflow seasonality, which is a significant contribution of hydrology since the streamflow seasonality is an important characteristic of hydrograph. The paper is overall well written and organized. However, I would like to post some questions and suggestions regarding the applicability, novelty and robustness of the proposed metrics. The major concerns are as followed:

We thank the reviewer for this constructive report and address these individual points below

1. Could the authors provide a discrete form of the calculation equation in addition to the integral form? This would make the adopting of the equation more practical, since the streamflow data is always in discrete form, such as daily.

We will add a discrete form of these equations in the revised manuscript.

1. I’ve seen a similar metric called concentration ratio (CR) and concentration period (CP) in some studies (e.g., Nan & Tian 2024; Jiang et al., 2022). If I understand correctly, CR and CP seems to be a simplified form of the R and tQ proposed in this study, which only uses the monthly data. I think the authors can conducted more literature reviews to illustrate the origin and the novelty of the proposed metric.

We indeed did not notice the interesting works by Nan & Tian (2024) and Jiang et al. (2022).

In these works, a simplified form of mass center and concentration are provided, and we will explicitly acknowledge that in our revised manuscript.

We will state how Jiang et al. (2022) already provides a (monthly) version of the proposed work, which is later adopted by Nan & Tian (2024).

We, however, believe that providing a more generalized form, including several applications, and robustness tests, can catalyze its uptake in hydrological (and other fields’) research.

1. The authors illustrate that tQ remains stable regardless of the shift start data, which is indeed an advantage. However, it seems that when interpreting the calculated tQ, we still cannot clealy understand the seasonality pattern if we don’t know the start date of water year. For example, if the tQ of catchment A and B are 0.4 and 0.5, we cannot say the mass center of A catchment is earlier than B catchment in the corresponding water year if we don’t know the start date.

For questions that require timings compared to a water year, it is inherently impossible to do this without defining the start date of that water year. For particular questions or problems, this may be very helpful to do.

To understand mass center shifts or differences between places (either in space or time), one can compare differences in the timing (irrespective of the water year start).

We will better emphasize both these aspects in the revised manuscript.

1. Regarding the robustness of seasonality strengths, a normalization based on the maximum variation range of each metric is needed. Otherwise, for the example in Figure 3, it’s difficult to say whether the change of Is3 from 14.53 to 3.21 is more significant than the change of R from 0.427 to 0.421.

In the revised manuscript, we will provide a normalized comparison of all metrics. Additionally, we will highlight that, regardless of the specific outcomes of this normalization, directional statistics metrics have the advantage of yielding values that remain unaffected by the data's time resolution, enhancing communication and interpretability of seasonality strength.

Nan, Y., & Tian, F. (2024). Glaciers determine the sensitivity of hydrological processes to perturbed climate in a large mountainous basin on the Tibetan Plateau. Hydrology and Earth System Sciences, 28(3), 669-689.

Jiang, Y., Xu, Z., & Xiong, L. (2022). Runoff variation and response to precipitation on multi-spatial and temporal scales in the southern Tibetan Plateau. Journal of Hydrology: Regional Studies, 42, 101157.

No further comments.

We thank the Reviewer for this helpful review.

We will provide detailed versions of all our changes once the discussion is closed and if the editor indicates we can revise our manuscript.

No further reply needed

No further reply needed

We thank the reviewer for this positive and constructive evaluation

This study proposes the use of directional statistics to characterize both the timing and strength of the seasonal streamflow cycle. It begins with a series of idealized comparisons to illustrate the methodology and then applies the statistics to real-world streamflow data. While prior studies (e.g., Nan & Tian, 2024; Jiang et al., 2022, indicated by another reviewer) have employed similar forms of directional statistics, this work advances the field by presenting a more generalized framework that is applicable to any temporal resolution. Additionally, it provides a comprehensive discussion on the advantages of directional statistics, including its ability to robustly capture seasonal timings and shifts (Section 3.1) and seasonal strength (Section 3.2). I believe this study makes a valuable contribution to the field of streamflow seasonality research.

Thank you.

I have only one suggestion for further improvement. Sections 4.1–4.3 effectively demonstrate the application of directional statistics. However, it would be highly informative to include a comparison with non-directional metrics, similar to what was done in Figures 1 and 2. Such a comparison would provide deeper insights into the practical differences between these metrics in streamflow analysis. In particular, I would be interested in seeing these comparisons extended to Figures 6 and 7, as they would help readers better understand the unique advantages of directional statistics in real-world scenarios.

We could provide a comparison with non-directional statistics for timing using the half-flow date and the center of mass (Eqs. 1 and 2). However, displaying these metrics in maps in the main paper seems problematic, as these new maps would lack physical meaning. This is because either the metrics use different starting dates of the water year regionally (and then the units of the metric differ, which makes them incomparable) or they begin at unrealistic times of the year (e.g., not matching the low flow season) which is not informative either. Consequently, we are hesitant to present these metrics in a Figure, as they may distract from the more meaningful examples shown (which is the purpose of Sections 4.1-4.3). Instead, we propose addressing this issue through a textual discussion in Sections 4.2 and 4.3.