Symbolic regression-based regionalization of baseflow separation parameter using catchment-scale characteristics

Lin, Yongen; Wang, Dagang; Mei, Yiwen; Zhu, Jinxin; Wu, Huan; Wang, Shuo; Xu, Zhonghou; Shamseldin, Asaad Y.; Anagnostou, Emmanouil N.

doi:10.5194/egusphere-2025-6011

Preprints

https://doi.org/10.5194/egusphere-2025-6011

Preprints

12 Dec 2025

| 12 Dec 2025

Symbolic regression-based regionalization of baseflow separation parameter using catchment-scale characteristics

Yongen Lin, Dagang Wang, Yiwen Mei, Jinxin Zhu, Huan Wu, Shuo Wang, Zhonghou Xu, Asaad Y. Shamseldin, and Emmanouil N. Anagnostou

Abstract. Accurate separation of baseflow from streamflow is of utmost importance for understanding catchment hydrological processes and supporting effective water resource management. The Smooth Minima Method is a common baseflow separation technique with a segment length parameter (N) representing the catchment average flow event duration. N is usually predicted by a power function with catchment area or default to 5 days. Yet these estimations are insufficient given the multivariate nature of N with other catchment attributes. In this study, we employ symbolic regression (SR) to search for possible formulation of N with a range of catchment attributes based on 855 catchments across the Contiguous United States. We ultimately identify three mathematical expressions of increasing complexity, achieving R² values of 0.49, 0.50, and 0.54, compared to 0.23 and −0.84 for the power function and constant values. The three expressions reveal that increases exponentially with catchment area (A) and catchment-averaged soil saturated hydraulic conductivity (K_sat) with decreasing rates, while it increases linearly with snow day fraction (f_SWE). The effects of K_sat and f_SWE on N are particularly pronounced for larger values (K_sat > 25 mm/h and f_SWE> 0.4) and smaller area (A < 100 km²). The different calculations of N are also evaluated in baseflow separation, revealing higher medians of Kling-Gupta Efficiency of at least 0.84, outperforming the literature-suggested formulas for a maximum increment of 0.22. This study highlights the potential of SR for uncovering physically meaningful formulas in optimal baseflow separation.

Received: 03 Dec 2025 – Discussion started: 12 Dec 2025

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Journal article(s) based on this preprint

03 Jun 2026

Symbolic regression-based regionalization of baseflow separation parameter using catchment-scale characteristics

Yongen Lin, Dagang Wang, Yiwen Mei, Jinxin Zhu, Huan Wu, Shuo Wang, Zhonghou Xu, Asaad Y. Shamseldin, and Emmanouil N. Anagnostou

Hydrol. Earth Syst. Sci., 30, 3425–3438, https://doi.org/10.5194/hess-30-3425-2026,https://doi.org/10.5194/hess-30-3425-2026, 2026

Short summary

Yongen Lin, Dagang Wang, Yiwen Mei, Jinxin Zhu, Huan Wu, Shuo Wang, Zhonghou Xu, Asaad Y. Shamseldin, and Emmanouil N. Anagnostou

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-6011', Anonymous Referee #1, 13 Jan 2026

see pdf-file

Citation: https://doi.org/10.5194/egusphere-2025-6011-RC1
- AC1: 'Reply on RC1', Dagang Wang, 05 Feb 2026
  
  We sincerely thanks for the time and effort the review invested in carefully evaluating our work and providing many insightful suggestions. We are encouraged that you find the overall research direction relevant and valuable. We also greatly appreciate your detailed comments regarding the methodological choices (e.g., the use of genetic programming versus alternative approaches), benchmarking strategies, uncertainty description of the baseflow separation procedure, and clarification of the generated functional space and unit consistency. These remarks are highly helpful and will substantially improve the clarity and scientific rigor of the manuscript.
  In the revised version, we will:
  
  (1) provide a clearer justification for the choice of the genetic programming-based symbolic regression framework and discuss alternative approaches mentioned by the reviewer;
  
  (2) expand the benchmark comparison by including additional model structures and discussing performance boundaries in more detail;
  
  (3) substantially revise the methodological description of the baseflow separation procedure;
  
  (4) clarify the formulation space, functional operators, and constraints used in the symbolic regression process;
  
  (5) carefully revise equations, units, and technical details to ensure dimensional consistency and reproducibility;
  
  (6) address all minor and technical comments.
  We believe these revisions will significantly strengthen the manuscript. A detailed, point-by-point response to all comments will be provided together with the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-6011-AC1
RC2:
'Comment on egusphere-2025-6011', Anonymous Referee #2, 28 Jan 2026

The manuscript titled addresses the challenge of accurately estimating the segment length parameter (N) for the Smooth Minima Method (SMM) in baseflow separation. By employing Symbolic Regression (SR) on a dataset of 855 catchments across the Contiguous United States, the authors aim to overcome the limitations of fixed default values or simple power-law functions traditionally used in the literature. The study successfully identifies three interpretable mathematical expressions that link parameter N to catchment area, saturated hydraulic conductivity, and snow day fraction. The validation results demonstrate that the SR-derived formulas significantly improve predictive performance (improving R2 from 0.23 to 0.54) and achieve higher Kling-Gupta Efficiency in baseflow separation compared to existing methods. Additionally, the use of Specific Electrical Conductance (SEC) mass balance serves as a robust independent validation of the proposed regionalization approach. Overall, this work provides valuable insights into the physical controls of baseflow processes and demonstrates the utility of interpretable machine learning in hydrology.

However, I have several concerns regarding the physical consistency of the derived formulas, the justification for choosing SR over other methods, and the robustness of the performance evaluation thresholds. These issues, along with comments on terminology and methodology, are detailed below.

1. The abstract states that “N increases exponentially with catchment area...”. However, the derived formulas are power-law functions, not exponential functions.

2. The introduction of SR in the current manuscript is relatively brief (lines 61-71). While the authors highlight the interpretability of SR compared to Random Forest (RF), there is a lack of review regarding the application of SR in the broader field of hydrology. It would strengthen the rationale of the study if the authors could briefly mention successful applications of SR in other hydrological contexts (Chadalawada et al., 2020) before narrowing down to baseflow separation. This would demonstrate that SR is a proven tool in the domain.

3. The Introduction highlights the role of environmental tracer data in parameter optimization (Line 48) but lacks specificity. I suggest briefly listing common tracers, specifically including SEC, in this paragraph. Since the reference dataset used in this study is SEC-optimized (as detailed in Section 2), explicitly mentioning SEC early on would provide the reader with necessary context regarding the physical basis of the ground truth parameters.

4. The statement in Line 84 that optimal N shows “no clear spatial patterns” is ambiguous, particularly since the subsequent analysis links N to spatially structured attributes. I suggest clarifying this phrasing to avoid apparent contradiction.

5. Section 3.1 describes the parameters N and M well but briefly lacks a description of the algorithmic procedure. Adding one or two sentences explaining how the method iterates (e.g., dividing the hydrograph into blocks of N days, identifying minima, and comparing adjacent minima using M) would make the method section more self-contained.

6. In Section 3.2, the manuscript does not specify whether the input variables (catchment characteristics) were normalized or standardized prior to SR training. Based on the discussion in Section 4.1 (where specific thresholds like A < 100 km2 are mentioned), it implies that raw data with physical units were used.

If raw data were used, the term \left(A+K_{sat}\right) in Formulas F_2 and F_3 is dimensionally inhomogeneous, adding Area (L^2) to Hydraulic Conductivity (L/T). This makes the formula valid only for the specific combination of units chosen in this study (km2 and mm/h) and undermines the claim of physical meaningfulness.

Even if the variables were normalized (rendering them dimensionless), the authors should interpret the physical meaning of the additive structure \left(A+K_{sat}\right). This structure implies a direct substitutability between catchment size and soil permeability in generating baseflow delay. What is the hydrological justification for this specific interaction form, rather than, for example, a multiplicative interaction \left(A+K_{sat}\right) which is more common in hydraulic process laws?

7. Section 3.2 outlines the SR setup but omits key hyperparameters required for reproducibility, such as population size, number of generations, and mutation/crossover rates used in the PySR configuration. I suggest including these details, perhaps in the Appendix.

8. The legend in Figure 7b includes categories like F_1&F_2, implying a tie in performance. However, the manuscript lacks an explicit definition of the threshold used to categorize these ties (e.g.,\Delta R^2<0.02?). Crucially, the choice of this threshold is intrinsic to the discussion on the complexity-accuracy trade-off (Section 5.3). A strict threshold (e.g., \Delta R^2=0) favors complex models F3 even for negligible gains, while a practical threshold (e.g., \Delta R^2<0.01) might reveal that the simpler F1 is good enough for a much larger portion of the CONUS.Therefore, beyond simply clarifying the current threshold, I suggest the authors perform a brief sensitivity analysis on this threshold. For instance, how does the spatial pattern of the best formula change if the tolerance for a tie is set to 0.01 vs. 0.03? This would provide a more robust and transparent visualization of where the added complexity of F3 yields substantial benefits versus marginal statistical gains.

9. In Section 5.1, the manuscript motivates the use of SR by contrasting it with the black box nature of RF. I suggest the authors provide a more precise definition of what constitutes interpretability in this context.

In hydrological literatures (Chen et al., 2022; Guillon et al., 2020), tree-based models are often considered interpretable via tools like SHAP values, which can quantify driver-response relationships much like the derivatives in SR. Given the significant drop in predictive skill (RF R2=0.80 vs. SR R2=0.54), it would be helpful to clarify why the explicit mathematical form of SR is preferred over the higher accuracy of RF. Is the interpretability here strictly defined as having a closed-form equation? A brief discussion about the definition of interpretability would provide a more balanced perspective.

References

Chadalawada, J., Herath, H.M.V.V., Babovic, V., 2020. Hydrologically Informed Machine Learning for Rainfall-Runoff Modeling: A Genetic Programming-Based Toolkit for Automatic Model Induction. Water Resour. Res. 56, e2019WR026933. https://doi.org/10.1029/2019WR026933

Chen, Y., Li, D., Zhao, Q., Cai, X., 2022. Developing a generic data-driven reservoir operation model. Adv. Water Resour. 167, 104274. https://doi.org/10.1016/j.advwatres.2022.104274

Guillon, H., Byrne, C.F., Lane, B.A., Sandoval Solis, S., Pasternack, G.B., 2020. Machine Learning Predicts Reach-Scale Channel Types From Coarse-Scale Geospatial Data in a Large River Basin. Water Resour. Res. 56, e2019WR026691. https://doi.org/10.1029/2019WR026691

Citation: https://doi.org/10.5194/egusphere-2025-6011-RC2
- AC2: 'Reply on RC2', Dagang Wang, 05 Feb 2026
  
  We sincerely thank you for the thorough, constructive, and encouraging review of our manuscript. We greatly appreciate your positive assessment of the study’s contribution to baseflow separation research and interpretable machine learning in hydrology. We are particularly grateful for your suggestions regarding the physical interpretation of the derived formulas, the clearer justification and contextualization of symbolic regression within hydrological applications, and the need to further clarify terminology, methodological details, and evaluation procedures. Your comments on dimensional consistency, algorithmic description, reproducibility of the SR configuration, and performance comparison thresholds are highly valuable and will help us substantially strengthen the manuscript.
  In the revised version, we will:
  
  (1) carefully revise the abstract and terminology to ensure consistency with the mathematical forms presented in the results;
  
  (2) expand the introduction to better position symbolic regression within the broader hydrological literature and clarify the motivation for its use relative to other machine learning approaches;
  
  (3) improve the description of environmental tracer data and explicitly introduce SEC earlier in the manuscript to provide clearer physical context;
  
  (4) revise methodological sections to provide a more self-contained explanation of the SMM algorithm and the SR workflow, including data preprocessing, hyperparameters, and reproducibility-related details;
  
  (5) carefully examine and clarify the dimensional consistency and physical interpretation of the derived equations and discuss the implications of different structural forms;
  
  (6) revise figure descriptions, performance thresholds, and evaluation procedures, and consider additional sensitivity analyses where appropriate;
  We believe that these revisions will significantly improve the clarity, robustness, and transparency of the study. A detailed, point-by-point response to all comments will be provided together with the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-6011-AC2
RC3:
'Comment on egusphere-2025-6011', Anonymous Referee #3, 28 Jan 2026

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-6011/egusphere-2025-6011-RC3-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2025-6011-RC3
- AC3: 'Reply on RC3', Dagang Wang, 05 Feb 2026
  
  We sincerely thank you for your careful, constructive, and encouraging review of our manuscript. Your detailed comments and thoughtful suggestions are highly valuable and will help us significantly improve the clarity, physical consistency, and practical relevance of the work.
  We appreciate your important remarks regarding the mathematical structure and dimensional consistency of the derived formulas. We will carefully re-examine the formulation, clarify its interpretation and limitations, and provide additional discussion to ensure transparency regarding transferability and unit dependence.
  We also thank you for your suggestions concerning the methodological choices, including the use of genetic programming-based symbolic regression compared to more recent optimization strategies, as well as your request for a clearer discussion of the trade-off between interpretability and predictive performance relative to alternative machine learning approaches. In the revised manuscript, we will expand the methodological justification and provide a more balanced discussion of interpretability concepts and practical implications for water resources applications.
  In addition, we will:
  
  (1) ensure full consistency of units and notation throughout tables, figures, and text;
  
  (2) clarify the data filtering criteria, including the exclusion of low-KGE catchments and the implications for applicability across different hydrological regimes;
  
  (3) expand the physical interpretation of the relationships identified by the symbolic regression models;
  
  (4) revise the discussion of regional performance differences and clarify assumptions related to regulated or human-influenced basins;
  
  (5) improve figure readability and visualization clarity as suggested.
  We believe that these revisions will substantially strengthen the manuscript’s transparency and physical interpretation. A detailed, point-by-point response to all comments will be provided together with the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-6011-AC3

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-6011', Anonymous Referee #1, 13 Jan 2026

see pdf-file

Citation: https://doi.org/10.5194/egusphere-2025-6011-RC1
- AC1: 'Reply on RC1', Dagang Wang, 05 Feb 2026
  
  We sincerely thanks for the time and effort the review invested in carefully evaluating our work and providing many insightful suggestions. We are encouraged that you find the overall research direction relevant and valuable. We also greatly appreciate your detailed comments regarding the methodological choices (e.g., the use of genetic programming versus alternative approaches), benchmarking strategies, uncertainty description of the baseflow separation procedure, and clarification of the generated functional space and unit consistency. These remarks are highly helpful and will substantially improve the clarity and scientific rigor of the manuscript.
  In the revised version, we will:
  
  (1) provide a clearer justification for the choice of the genetic programming-based symbolic regression framework and discuss alternative approaches mentioned by the reviewer;
  
  (2) expand the benchmark comparison by including additional model structures and discussing performance boundaries in more detail;
  
  (3) substantially revise the methodological description of the baseflow separation procedure;
  
  (4) clarify the formulation space, functional operators, and constraints used in the symbolic regression process;
  
  (5) carefully revise equations, units, and technical details to ensure dimensional consistency and reproducibility;
  
  (6) address all minor and technical comments.
  We believe these revisions will significantly strengthen the manuscript. A detailed, point-by-point response to all comments will be provided together with the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-6011-AC1
RC2:
'Comment on egusphere-2025-6011', Anonymous Referee #2, 28 Jan 2026

The manuscript titled addresses the challenge of accurately estimating the segment length parameter (N) for the Smooth Minima Method (SMM) in baseflow separation. By employing Symbolic Regression (SR) on a dataset of 855 catchments across the Contiguous United States, the authors aim to overcome the limitations of fixed default values or simple power-law functions traditionally used in the literature. The study successfully identifies three interpretable mathematical expressions that link parameter N to catchment area, saturated hydraulic conductivity, and snow day fraction. The validation results demonstrate that the SR-derived formulas significantly improve predictive performance (improving R2 from 0.23 to 0.54) and achieve higher Kling-Gupta Efficiency in baseflow separation compared to existing methods. Additionally, the use of Specific Electrical Conductance (SEC) mass balance serves as a robust independent validation of the proposed regionalization approach. Overall, this work provides valuable insights into the physical controls of baseflow processes and demonstrates the utility of interpretable machine learning in hydrology.

However, I have several concerns regarding the physical consistency of the derived formulas, the justification for choosing SR over other methods, and the robustness of the performance evaluation thresholds. These issues, along with comments on terminology and methodology, are detailed below.

1. The abstract states that “N increases exponentially with catchment area...”. However, the derived formulas are power-law functions, not exponential functions.

2. The introduction of SR in the current manuscript is relatively brief (lines 61-71). While the authors highlight the interpretability of SR compared to Random Forest (RF), there is a lack of review regarding the application of SR in the broader field of hydrology. It would strengthen the rationale of the study if the authors could briefly mention successful applications of SR in other hydrological contexts (Chadalawada et al., 2020) before narrowing down to baseflow separation. This would demonstrate that SR is a proven tool in the domain.

3. The Introduction highlights the role of environmental tracer data in parameter optimization (Line 48) but lacks specificity. I suggest briefly listing common tracers, specifically including SEC, in this paragraph. Since the reference dataset used in this study is SEC-optimized (as detailed in Section 2), explicitly mentioning SEC early on would provide the reader with necessary context regarding the physical basis of the ground truth parameters.

4. The statement in Line 84 that optimal N shows “no clear spatial patterns” is ambiguous, particularly since the subsequent analysis links N to spatially structured attributes. I suggest clarifying this phrasing to avoid apparent contradiction.

5. Section 3.1 describes the parameters N and M well but briefly lacks a description of the algorithmic procedure. Adding one or two sentences explaining how the method iterates (e.g., dividing the hydrograph into blocks of N days, identifying minima, and comparing adjacent minima using M) would make the method section more self-contained.

6. In Section 3.2, the manuscript does not specify whether the input variables (catchment characteristics) were normalized or standardized prior to SR training. Based on the discussion in Section 4.1 (where specific thresholds like A < 100 km2 are mentioned), it implies that raw data with physical units were used.

If raw data were used, the term \left(A+K_{sat}\right) in Formulas F_2 and F_3 is dimensionally inhomogeneous, adding Area (L^2) to Hydraulic Conductivity (L/T). This makes the formula valid only for the specific combination of units chosen in this study (km2 and mm/h) and undermines the claim of physical meaningfulness.

Even if the variables were normalized (rendering them dimensionless), the authors should interpret the physical meaning of the additive structure \left(A+K_{sat}\right). This structure implies a direct substitutability between catchment size and soil permeability in generating baseflow delay. What is the hydrological justification for this specific interaction form, rather than, for example, a multiplicative interaction \left(A+K_{sat}\right) which is more common in hydraulic process laws?

7. Section 3.2 outlines the SR setup but omits key hyperparameters required for reproducibility, such as population size, number of generations, and mutation/crossover rates used in the PySR configuration. I suggest including these details, perhaps in the Appendix.

8. The legend in Figure 7b includes categories like F_1&F_2, implying a tie in performance. However, the manuscript lacks an explicit definition of the threshold used to categorize these ties (e.g.,\Delta R^2<0.02?). Crucially, the choice of this threshold is intrinsic to the discussion on the complexity-accuracy trade-off (Section 5.3). A strict threshold (e.g., \Delta R^2=0) favors complex models F3 even for negligible gains, while a practical threshold (e.g., \Delta R^2<0.01) might reveal that the simpler F1 is good enough for a much larger portion of the CONUS.Therefore, beyond simply clarifying the current threshold, I suggest the authors perform a brief sensitivity analysis on this threshold. For instance, how does the spatial pattern of the best formula change if the tolerance for a tie is set to 0.01 vs. 0.03? This would provide a more robust and transparent visualization of where the added complexity of F3 yields substantial benefits versus marginal statistical gains.

9. In Section 5.1, the manuscript motivates the use of SR by contrasting it with the black box nature of RF. I suggest the authors provide a more precise definition of what constitutes interpretability in this context.

In hydrological literatures (Chen et al., 2022; Guillon et al., 2020), tree-based models are often considered interpretable via tools like SHAP values, which can quantify driver-response relationships much like the derivatives in SR. Given the significant drop in predictive skill (RF R2=0.80 vs. SR R2=0.54), it would be helpful to clarify why the explicit mathematical form of SR is preferred over the higher accuracy of RF. Is the interpretability here strictly defined as having a closed-form equation? A brief discussion about the definition of interpretability would provide a more balanced perspective.

References

Chadalawada, J., Herath, H.M.V.V., Babovic, V., 2020. Hydrologically Informed Machine Learning for Rainfall-Runoff Modeling: A Genetic Programming-Based Toolkit for Automatic Model Induction. Water Resour. Res. 56, e2019WR026933. https://doi.org/10.1029/2019WR026933

Chen, Y., Li, D., Zhao, Q., Cai, X., 2022. Developing a generic data-driven reservoir operation model. Adv. Water Resour. 167, 104274. https://doi.org/10.1016/j.advwatres.2022.104274

Guillon, H., Byrne, C.F., Lane, B.A., Sandoval Solis, S., Pasternack, G.B., 2020. Machine Learning Predicts Reach-Scale Channel Types From Coarse-Scale Geospatial Data in a Large River Basin. Water Resour. Res. 56, e2019WR026691. https://doi.org/10.1029/2019WR026691

Citation: https://doi.org/10.5194/egusphere-2025-6011-RC2
- AC2: 'Reply on RC2', Dagang Wang, 05 Feb 2026
  
  We sincerely thank you for the thorough, constructive, and encouraging review of our manuscript. We greatly appreciate your positive assessment of the study’s contribution to baseflow separation research and interpretable machine learning in hydrology. We are particularly grateful for your suggestions regarding the physical interpretation of the derived formulas, the clearer justification and contextualization of symbolic regression within hydrological applications, and the need to further clarify terminology, methodological details, and evaluation procedures. Your comments on dimensional consistency, algorithmic description, reproducibility of the SR configuration, and performance comparison thresholds are highly valuable and will help us substantially strengthen the manuscript.
  In the revised version, we will:
  
  (1) carefully revise the abstract and terminology to ensure consistency with the mathematical forms presented in the results;
  
  (2) expand the introduction to better position symbolic regression within the broader hydrological literature and clarify the motivation for its use relative to other machine learning approaches;
  
  (3) improve the description of environmental tracer data and explicitly introduce SEC earlier in the manuscript to provide clearer physical context;
  
  (4) revise methodological sections to provide a more self-contained explanation of the SMM algorithm and the SR workflow, including data preprocessing, hyperparameters, and reproducibility-related details;
  
  (5) carefully examine and clarify the dimensional consistency and physical interpretation of the derived equations and discuss the implications of different structural forms;
  
  (6) revise figure descriptions, performance thresholds, and evaluation procedures, and consider additional sensitivity analyses where appropriate;
  We believe that these revisions will significantly improve the clarity, robustness, and transparency of the study. A detailed, point-by-point response to all comments will be provided together with the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-6011-AC2
RC3:
'Comment on egusphere-2025-6011', Anonymous Referee #3, 28 Jan 2026

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-6011/egusphere-2025-6011-RC3-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2025-6011-RC3
- AC3: 'Reply on RC3', Dagang Wang, 05 Feb 2026
  
  We sincerely thank you for your careful, constructive, and encouraging review of our manuscript. Your detailed comments and thoughtful suggestions are highly valuable and will help us significantly improve the clarity, physical consistency, and practical relevance of the work.
  We appreciate your important remarks regarding the mathematical structure and dimensional consistency of the derived formulas. We will carefully re-examine the formulation, clarify its interpretation and limitations, and provide additional discussion to ensure transparency regarding transferability and unit dependence.
  We also thank you for your suggestions concerning the methodological choices, including the use of genetic programming-based symbolic regression compared to more recent optimization strategies, as well as your request for a clearer discussion of the trade-off between interpretability and predictive performance relative to alternative machine learning approaches. In the revised manuscript, we will expand the methodological justification and provide a more balanced discussion of interpretability concepts and practical implications for water resources applications.
  In addition, we will:
  
  (1) ensure full consistency of units and notation throughout tables, figures, and text;
  
  (2) clarify the data filtering criteria, including the exclusion of low-KGE catchments and the implications for applicability across different hydrological regimes;
  
  (3) expand the physical interpretation of the relationships identified by the symbolic regression models;
  
  (4) revise the discussion of regional performance differences and clarify assumptions related to regulated or human-influenced basins;
  
  (5) improve figure readability and visualization clarity as suggested.
  We believe that these revisions will substantially strengthen the manuscript’s transparency and physical interpretation. A detailed, point-by-point response to all comments will be provided together with the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-6011-AC3

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

ED: Reconsider after major revisions (further review by editor and referees) (10 Feb 2026) by Rohini Kumar

AR by Dagang Wang on behalf of the Authors (04 Apr 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (07 Apr 2026) by Rohini Kumar

RR by Anonymous Referee #3 (04 May 2026)

RR by Anonymous Referee #2 (09 May 2026)

Suggestions for revision or reasons for rejection

I commend the authors for their diligent and thorough revisions. My previous concerns regarding dimensional consistency, model interpretability, and evaluation thresholds have been adequately addressed. The manuscript has been substantially improved and is now in very good shape. I only have a few minor corrections and discussion points for the authors to consider before publication.
1. Physical implication of the additive vs. multiplicative structure (Section 5)
While the separation of A and \operatorname{K}_{sat} into independent terms successfully resolves the strict unit mismatch (as the empirical coefficients now consistently absorb the respective dimensions), it implies that the delays caused by catchment size and soil permeability are purely additive (i.e., independent parallel or sequential components). From a classical hydraulics perspective, travel time is fundamentally a multiplicative function of a length scale and inverse celerity. Specifically, considering that characteristic flow distance roughly scales with the square root of the catchment area (\operatorname{A}^{0.5}, e.g., Hack's law) (Hack, 1957) and subsurface flow velocity is proportional to hydraulic conductivity (\operatorname{K}_{sat} per Darcy's law) (Freeze and Cherry, 1979), the travel time could follow a coupled, multiplicative relationship (Time\ \propto Distance/Velocity\ \propto\operatorname{A}^{0.5}\cdot\operatorname{K}_{\operatorname{sat}}^{-1}).
I suggest expanding the discussion (e.g., in one section of Discussion) to briefly touch upon this difference. It would be insightful to hear the authors' thoughts on why the SR algorithm might have favored an additive form over a multiplicative one—for instance, was it due to the complexity penalties applied to nested operators during the SR search, or does the additive structure genuinely offer a superior empirical fit? Furthermore, does this SR-derived additive structure have a plausible physical justification (e.g., representing distinct delay mechanisms such as horizontal channel routing vs. vertical infiltration), or does it act primarily as an empirical approximation of a classically coupled process? Adding this perspective would deepen the current discussion and highlight the fascinating boundaries between data-driven equation discovery and theoretical hydrology.
2. Typographical and Grammatical Corrections
Please correct the following minor language issues throughout the manuscript:
Abstract: "default to 5 days" should be "defaults to 5 days".
Introduction: Change "gaged catchment lacking..." to plural "gaged catchments lacking...". Additionally (Line 89), the term "ungagged catchments" appears to be a typographical error and should be corrected to "ungauged" or "ungaged" catchments.
Section 3.1: "A smaller N result in..." should be "A smaller N results in...".
Section 3.2: "The function space is consisted of..." is grammatically incorrect; please change to "The function space consists of...".
Section 4.3: "This may be because the SR formulas were calibrated... at the CONUS scale, their region-specific performance..." is a run-on sentence. Please replace the comma with a semicolon (e.g., "...scale; thus, their...").
Section 5.2: "...making more rainfall excess to rout through..." contains a typo. "rout" should be corrected to "route".

References
Freeze, R.A., Cherry, J.A., 1979. Groundwater. Prentice-Hall, Englewood Cliffs, N.J.
Hack, J.T., 1957. Studies of longitudinal stream profiles in Virginia and Maryland (Report No. 294B), Professional Paper. https://doi.org/10.3133/pp294B

Hide

ED: Publish subject to minor revisions (review by editor) (09 May 2026) by Rohini Kumar

AR by Dagang Wang on behalf of the Authors (17 May 2026) Author's response Author's tracked changes Manuscript

ED: Publish as is (18 May 2026) by Rohini Kumar

AR by Dagang Wang on behalf of the Authors (21 May 2026) Manuscript

Journal article(s) based on this preprint

03 Jun 2026

Symbolic regression-based regionalization of baseflow separation parameter using catchment-scale characteristics

Yongen Lin, Dagang Wang, Yiwen Mei, Jinxin Zhu, Huan Wu, Shuo Wang, Zhonghou Xu, Asaad Y. Shamseldin, and Emmanouil N. Anagnostou

Hydrol. Earth Syst. Sci., 30, 3425–3438, https://doi.org/10.5194/hess-30-3425-2026,https://doi.org/10.5194/hess-30-3425-2026, 2026

Short summary

Yongen Lin, Dagang Wang, Yiwen Mei, Jinxin Zhu, Huan Wu, Shuo Wang, Zhonghou Xu, Asaad Y. Shamseldin, and Emmanouil N. Anagnostou

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Month	HTML	PDF	XML	Total
Dec 2025	614	312	52	978
Jan 2026	605	205	50	860
Feb 2026	450	116	55	621
Mar 2026	294	117	47	458
Apr 2026	72	39	0	111
May 2026	38	33	1	72
Jun 2026	5	2	1	8

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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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Understanding how baseflow contributes to river flow is essential for managing water resources. We studied a widely used method for separating baseflow and found that a key parameter was often estimated too simply. Using symbolic regression and data from 855 catchments, we uncovered new formulas that greatly improve accuracy and reveal how soil, snow, and catchment size jointly influence baseflow estimation.


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