the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Nov 2025
Capturing the extremes: a quasi-comonotonicity-based algorithm for disaggregating daily to hourly rainfall
Abstract. Disaggregating daily precipitation data into hourly time scale is crucial for hydrological modelling, urban drainage design, and extreme rainfall risk assessment. This study presents Q-CODA, a novel Quasi-Comonotonicity-based Disaggregation Algorithm that leverages the quasi-comonotonic relationship between daily precipitation totals and their sub-daily maxima to generate hourly rainfall sequences consistent with observed extremes. The method first assumes the Fréchet–Hoeffding upper bound copula to simulate sub-daily maximum precipitation values conditioned on daily totals, which serve as target constraints. An initial hourly pattern is obtained via a K-Nearest Neighbours (KNN) approach and subsequently refined through an iterative adjustment algorithm to ensure coherence with both the daily precipitation total and previously calculated multiple sub-daily target constraints. We evaluated Q-CODA through a rigorous 5-fold cross-validation over 91 meteorological stations across Spain, spanning 1996–2024. Performance was benchmarked against state-of-the-art approaches including nearest-neighbour resampling methods, Poisson cluster-based rainfall generators, multiplicative random cascades models and deep learning techniques. Evaluation metrics were tailored to different aspects of the rainfall data: distributional distance (1-D Wasserstein), accuracy measures (mean absolute error, Nash–Sutcliffe efficiency), and extreme quantiles (90th to 99.9th percentiles) were computed on the hourly maximum precipitation over 1-hour windows, focusing on the representation of rainfall extremes. In contrast, autocorrelation and rainfall event duration statistics were calculated on the entire hourly rainfall time series to assess temporal coherence and event structure. Additionally, intensity-duration-frequency (IDF) curves were analysed. Results demonstrate that Q-CODA substantially improves the representation of rainfall extremes while maintaining temporal coherence and event structure. This approach offers a robust, data-driven framework for accurate sub-daily rainfall disaggregation, with significant implications for hydrometeorological applications and infrastructure design.
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- AC1: 'Comment on egusphere-2025-4469 - Additional materials for reproducibility', Carlos Correa, 11 Dec 2025 reply
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RC1: 'Comment on egusphere-2025-4469', Anonymous Referee #1, 19 Jan 2026
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This study presents the new algorithm Q-CODA for disaggregating precipitation data from daily to sub-daily values. The method leverages the relationship between daily precipitation totals and sub-daily maximum values, which is assumed and shown to be quasi-comonotonic, meaning that high daily precipitation totals are strongly associated with high sub-daily maximum values. The quasi-comonotonicity assumption is incorporated into the methodology via an upper bound copula. The algorithm is applied to 91 observational stations in Spain, and multiple other methodologies are included in the study as benchmarks.
A wide variety of evaluation metrics are applied, shedding light on which aspects of sub-daily precipitation the various methodologies handle well and which they struggle with (e.g., temporal consistency, the fraction of dry days, representation of extreme rainfall events).
The paper is generally well written and, as far as I can tell, methodologically sound. It appears that both Q-CODA and the benchmark methods are implemented according to best practices, which ensures a fair comparison.
I recommend that the paper be accepted with minor revisions, but have some suggestions that could strengthen the existing analysis.
Specific Comments
Page 4, lines 120-121
There are some references missing from the list at the back. I found two, Lee et al. (2022) and Chodhury et al. (2025), but there may be more throughout the document. I recommend that the authors check all citations and ensure they are included in the reference list.Page 4, lines 126-127
Could you please describe the quality control that was applied to the observational data? It doesn’t have to be a full and detailed account, but a brief mention of what type of homogeneity testing was applied or a reference to the dataset would be helpful.Page 5-6, Section 3
The methodology could benefit from clearer presentation in certain sections. For example, on line 150-151, you write about “the upper bound copula C+, representing perfect positive dependence (comonotonicity)...” which is then presented in Equation (2). After the equation, you write: “This copula implies that X and Y increase together almost surely, which aligns with the quasi-comonotonic behaviour observed between daily precipitation totals and sub-daily maxima.” The transition from perfect dependence to quasi-comonotonic behaviour is not entirely clear.Additionally, while the benchmarks are described clearly, the description of the Q-CODA methodology is more difficult to follow, perhaps due to the level of detail. The flowchart certainly helps, and after reading Section 3 a couple of times it does make sense, but it could be more accessible. Could you add a simplified example showing how the comonotonic upper bound is translated into target maxima and used during the iterative adjustment process? This is key to understanding the method.
Page 11, line 290
The parenthesis in the cited paper should be after the name when it is in running text, i.e., “The ELU activation function was chosen by Bhattacharyya et al. (2024)...” rather than “(Bhattacharyya et al. 2024)”.This issue recurs later on lines 302, 303, and 344, but I may have missed other instances. While this is a minor formatting issue, I recommend that the authors carefully check all citations to ensure they follow the correct format.
General Comments
Evaluation Metrics
The evaluation metrics are described in the main text, but references or equations are not provided. While this makes the main text more readable, it would be useful to include an appendix with a fuller description of the metrics, including equations and references (e.g., for the Nash-Sutcliffe Efficiency and Wasserstein distance). This would improve the reproducibility of the study. I understand the code will also be made available, but it would still be good if the paper included all information needed to interpret and recreate the results.
Assumption of Quasi-Comonotonicity
The assumption of quasi-comonotonicity underpins the methodology, and the authors argue that the heterogeneity of precipitation regimes in Spain makes the results broadly applicable. While this argument is pretty convincing, I can think of precipitation regimes where the quasi-comonotonicity assumption might not hold as well. For example:
- In regions like northwestern Europe or the Pacific Northwest of the US and Canada, storm tracks bring sustained moderate precipitation over long periods of time, which may weaken the relationship between daily totals and sub-daily maxima.
- In monsoon regions, short-lived, strong convective systems could contribute to very high sub-daily maxima without corresponding high daily totals. This could weaken or break the dependence between daily totals and sub-daily maxima.
Could the authors expand on the discussion of quasi-comonotonicity and the applicability of Q-CODA in other regions? This might fit best in the Discussion section.
How sensitive is Q-CODA to deviations from the quasi-comonotonicity assumption? For example, could the authors test the algorithm on artificial datasets where the correlation between daily totals and sub-daily maxima is systematically reduced? This would help determine how far the assumption can deviate before the method’s performance degrades.
Performance Variability Across Stations
The Q-CODA algorithm performs well across most evaluation metrics, but some outliers appear (in Figure 6b and 6f for example), suggesting that the method struggles a bit at certain stations. This is to be expected, but it would be interesting to investigate these cases further. For example:- Do these outliers correspond to stations with lower correlation values in Figure 2, potentially indicating weaker quasi-comonotonic dependence?
- Are there locations where all methods perform poorly, potentially pointing to data quality issues or precipitation processes that are difficult to capture with disaggregation methods?
Computational Feasibility
The paper does not discuss the computational costs of Q-CODA relative to the benchmarks. Given the iterative nature of the algorithm, this could be an important consideration, particularly for larger datasets or real-time applications. Please add a discussion of the computational efficiency of Q-CODA as compared to the benchmark methods. Even better, you could perform a comparative test of the computational costs. I am thinking of runtime, where parallelization might be an option for some of these methods, but also computational costs and energy usage (which can be checked with tools like CodeCarbon in python).Citation: https://doi.org/10.5194/egusphere-2025-4469-RC1 -
AC2: 'Reply on RC1', Carlos Correa, 12 Feb 2026
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The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4469/egusphere-2025-4469-AC2-supplement.pdf
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RC2: 'Comment on egusphere-2025-4469', Anonymous Referee #2, 16 Feb 2026
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The manuscript presents Q-CODA, a novel copula-based framework for temporal disaggregation of daily precipitation into hourly time scales. The primary methodological contribution lies in exploiting the quasi-comonotonic dependence between daily total and sub-daily maxima though the Frechet-Hoeffding upper bound copula, combined with KNN based seeding and an iterative adjustment scheme. The approach is supported by a comprehensive multi-metric validation framework that takes into account accuracy, distributional fidelity, structural realism and hydrological relevance.
The manuscript is well presented, methodologically sophisticated and addresses a significant hydrological problem. The key strength of the paper lies in the explicit use of quasi-comonotonicity as a structural constraint, which poses as an advancement over the prevalent temporal disaggregation procedures. The comparative evaluation against multiple temporal disaggregation models under a 5-fold cross-validation scheme enhances the robustness and credibility of the results. The benchmark disaggregation methodologies have been implemented carefully and appear to be applied in a technically sound manner. The inclusion of precipitation dataset from 91 meteorological stations lying across diverse climatic regimes of Spain is commendable. The evaluation framework appropriately considers both upper-tail behaviour and key statistical and temporal properties of the hourly series, thereby ensuring that model performance is assessed rigorously.
Overall, the manuscript appears to have the potential to make an impactful contribution to the field of stochastic rainfall disaggregation.
My primary concerns largely overlap with those raised by Referee #1, specifically with respect to 1) the robustness and general applicability of the quasi-comonotonicity assumption 2) the need for improved clarity in the explanation of the proposed methodological framework 3) inclusion of a more detailed description of the evaluation metrics employed in the present study 4) the issue of missing references
Based on the author’s response, in my assessment, these four concerns appear to have been adequately addressed. Therefore, I recommend acceptance of the manuscript, provided that the revisions described are fully reflected in the final version of the manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-4469-RC2 -
AC3: 'Reply on RC2', Carlos Correa, 16 Feb 2026
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We sincerely thank Referee #2 for the careful reading of our manuscript and for the positive and encouraging assessment of our work. We are grateful for the recognition of the methodological contribution of Q-CODA, the robustness of the validation framework, and the potential relevance of the study for stochastic rainfall disaggregation.
We especially appreciate the referee’s remark that the primary concerns identified largely overlap with those raised by Referee #1, namely: (1) the robustness and general applicability of the quasi-comonotonicity assumption, (2) the need for improved clarity in the explanation of the methodological framework, (3) the inclusion of a more detailed description of the evaluation metrics, and (4) the issue of missing references. The fact that both referees independently highlighted the same key aspects is particularly valuable for us. It helps to clearly identify the elements of the manuscript that most required strengthening and reinforces our confidence that focusing our revision efforts on these points was the appropriate and necessary course of action.
In response, and as detailed in our point-by-point reply to Referee #1 (https://doi.org/10.5194/egusphere-2025-4469-AC2), we have carefully revised the manuscript to address each of these aspects. Specifically, we have expanded and clarified the methodological description to improve accessibility, including the addition of a step-by-step illustrative disaggregation example using Q-CODA in a new Appendix A. We have also added a comprehensive description of evaluation metrics in a new Appendix B, completed and checked all references, and strengthened the discussion on the quasi-comonotonicity assumption and its broader applicability. In particular, we extended the analysis beyond Spain by incorporating results from additional climatic regions and a sensitivity test.
We confirm that all the revisions described in our previous response to Referee #1 will be fully incorporated into the updated manuscript. We thank the referee again for the positive evaluation and for the constructive feedback, which will help us refine and strengthen the study.
Citation: https://doi.org/10.5194/egusphere-2025-4469-AC3
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AC3: 'Reply on RC2', Carlos Correa, 16 Feb 2026
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Following the release of the preprint, I realised that the Code Availability section unintentionally omitted several important resources required for full transparency and reproducibility. In particular, the observational station datasets used in the validation experiments, the implementations of the benchmark disaggregation methods, and the scripts employed to reproduce the comparative evaluation and all figures in the manuscript were not included.
To address this oversight, I am providing here a Zenodo repository containing all missing materials:
https://doi.org/10.5281/zenodo.17896411
A detailed README.txt is included in the package, describing how to reproduce all results and figures.
These resources will be fully integrated into the Code and Data Availability section in the revised version of the manuscript. I apologise for the omission and hope this package facilitates full reproducibility of the study.