Stochastic Analysis of the Marginal and Dependence Structure of Streamflows: From Fine-Scale Records to Multi-Centennial Paleoclimatic Reconstructions

Round 1

Reviewer 1 Report

This paper determines the fractal behavior and long-range dependence behavior of streamflow by adjusting for statistical bias. In general, the paper is well written, the method is clearly stated with sufficient evidence to support the conclusion. I see it a well prepared manuscript that can be accepted as it is. 

Author Response

We thank the anonymous Reviewer 1 for the kind words and appreciation of the manuscript. Despite this good assessment, the manuscript was revised and rewritten in parts where the authors found that a better context or explanation could have been given.

Reviewer 2 Report

In the introduction, you need to connect the state of the art to your paper goals. Please follow the literature review with a clear and concise state of the art analysis. This should clearly show the knowledge gaps identified and link them to your paper goals. Please reason both the novelty and the relevance of your paper goals.

An updated and complete literature review should be conducted. The originality of the paper needs to be further clarified. The present form does not have sufficient results to justify the novelty of a high-quality journal paper.

Also, recent papers should be included in state of the art - the present lack of recent references creates a wrong impression that the authors are not aware of the most recent development

Please eliminate all multiple references. Please check the manuscript thoroughly and eliminate ALL the lumps in the manuscript. This should be done by characterising each reference individually. This can be done by mentioning 1 or 2 phrases per reference to show how it is different from the others and why it deserves mentioning.

In the conclusions, in addition to summarising the actions taken and results, please strengthen the explanation of their significance. It is recommended to use quantitative reasoning comparing with appropriate benchmarks, especially those stemming from previous work.

1,000,000 rather than 1000000 or 1 000 000 or 1.000.000

and not 13.000 km²

Author Response

Obs. 1: In the introduction, you need to connect the state of the art to your paper goals. Please follow the literature review with a clear and concise state of the art analysis. This should clearly show the knowledge gaps identified and link them to your paper goals. Please reason both the novelty and the relevance of your paper goals.

Ans. 1: Thank you for pointing out this matter. We have restructured the Introduction, connecting the state-of-the-art to the research aims, and highlighted the contribution and novelty of the study. The updated paragraph reads now:

“Based on the mentioned above, the aim of this work is to estimate the stochastic structure of the streamflow process over an extensive range of temporal scales, from fine scales (of the order of minutes) to very large timescales (of the order of hundreds of years). To this aim, we use several global databases of finer-scale streamflow timeseries and larger-scale paleoclimatic reconstructions, for which the scientific interest has grown (e.g., see discussions and methods in recent works [25; 26; 27]) but without having performed any similar stochastic analysis. The stochastic dependence structure, and in particular, the importance of the LRD in hydrology and streamflow, has been highlighted in several works analyzing large-scale observational datasets (e.g.; [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]), yet the temporal coverage of these datasets is usually limited compared to the one of paleoclimatic reconstructions. To our knowledge, this is the first time that such an extensive range of temporal scales (spanning eight orders of magnitude) has been assembled and studied for the streamflow process from both observational records and paleoclimatic reconstructions. This range of scales and data enables unique insights into the process’s stochastic properties. The analysis also applies the K-moments [8] for the estimation of the first four moments, which present some advantages compared to the commonly used classical (raw or central) or L-moments [10] estimators. Also, the second-order dependence structure is expressed herein through the climacogram (i.e., the variance of the averaged process at the scale domain [11]), which is adjusted for estimation bias as opposed to the commonly used estimators (see definitions in Beran [12], and comparison among other estimators in [6, 7]). Finally, the resulting model form and parameters (such as the first four K-moments, and the fractal and Hurst parameters) are compared to the ones reported in other studies, where a higher number of streamflow stations were assessed. Worthy of mentioning is that published studies used shorter lengths, a smaller range of scales, and lower resolution.”

Obs. 2: An updated and complete literature review should be conducted. The originality of the paper needs to be further clarified. The present form does not have sufficient results to justify the novelty of a high-quality journal paper. Also, recent papers should be included in state of the art - the present lack of recent references creates a wrong impression that the authors are not aware of the most recent development.

Ans. 2: We thank the Reviewer for the comment and suggestion. Regarding the originality and novelty of the manuscript, please see Ans. 1. Regarding the updated literature review, we would like to remark that 18 of our 43 references were published in 2018-2022 (i.e., ~42% of the total number of references). Therefore, the manuscript provides a good description of the most recent research activities in this file, but if there are more specific suggestions we would be glad to consider additional references.  

 Obs. 3: Please eliminate all multiple references. Please check the manuscript thoroughly and eliminate ALL the lumps in the manuscript. This should be done by characterizing each reference individually. This can be done by mentioning 1 or 2 phrases per reference to show how it is different from the others and why it deserves mentioning.

Ans. 3: We appreciate this comment. However, we respectfully disagree with Obs. 3, since this is not a review paper and in non-review papers, multiple citations that support a single statement are common in any scientific field. Furthermore, our references support the scientific gap (motivating this study) and the methodology adopted.

Obs. 4: In the conclusions, in addition to summarising the actions taken and results, please strengthen the explanation of their significance. It is recommended to use quantitative reasoning comparing with appropriate benchmarks, especially those stemming from previous work.

Ans. 4: We thank the Reviewer for the suggestions. We note that we have made extended quantitative comparisons in the Discussion section, where we also elaborate on the significance of the findings. We now have extended the relevant parts in the updated Conclusions section, which reads:

“Streamflow time series (from observational records and paleoclimatic reconstructions, forming a unique range of temporal scales of the order of minutes to thousands of years) are analyzed to determine the stochastic properties of the process. Results show that the streamflow process is smooth at small scales, i.e. characterized on average by a fractal parameter of M = 0.8, and strongly persistent at large scales, i.e. characterized by long-range dependence with a parameter H = 0.8. Furthermore, the first four statistical moments (estimated from classic and K-moments) determine a useful relationship for operational and scientific practices. In particular, it is found that a power-law function can fit the cloud data and, more importantly, that a common stochastic behaviour is followed. The latter leads to concluding that the Pareto-Burr-Feller probability distribution function, found to be a good candidate distribution for various hydrological-cycle processes in the literature, can also describe these streamflow datasets.

These results complement previous studies on the stochastic structure of streamflow by significantly extending the empirical evidence from minute to centennial scales. In particular, they strengthen the hypothesis that the multi-scale streamflow process can generally be characterized by a Pareto-Burr-Feller marginal distribution and an HK-type dependence model, with the specific parameters determined by the local hydrometeorological processes and geophysical conditions. The identified common stochastic structure can be useful to support estimation at sites with limited data and serve as a basis for streamflow regionalization analyses. Furthermore, this study paves the way for the construction of a stochastic model for streamflow based on a common structure validated on a global spatial scale and a temporal scale ranging from minutes to centuries.”

Obs. 5: 1,000,000 rather than 1000000 or 1 000 000 or 1.000.000 and not 13.000 km2

Ans. 5: We Thank the Reviewer for the remark. We have adopted the suggested notations in the text.