the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 Dec 2025
Symbolic regression-based regionalization of baseflow separation parameter using catchment-scale characteristics
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 (Ksat) with decreasing rates, while it increases linearly with snow day fraction (fSWE). The effects of Ksat and fSWE on N are particularly pronounced for larger values (Ksat > 25 mm/h and fSWE > 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.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2025-6011', Anonymous Referee #1, 13 Jan 2026
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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
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AC1: 'Reply on RC1', Dagang Wang, 05 Feb 2026
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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/10.1029/2019WR026691Citation: 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
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AC2: 'Reply on RC2', Dagang Wang, 05 Feb 2026
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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
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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
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AC3: 'Reply on RC3', Dagang Wang, 05 Feb 2026
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