the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 25 Aug 2026
From prediction improvement to uncertainty propagation structure: revisiting hybrid hydrological models
Abstract. Hybrid hydrological models integrating embedded neural networks (ENNs) have demonstrated strong potential for improving streamflow prediction. However, how uncertainties induced by ENNs propagate through hydrological processes and internal system states remains poorly understood. This limits a mechanistic understanding of the stability and reliability of hybrid hydrological systems. This study therefore moves beyond performance evaluation and focuses on the internal uncertainty propagation structure of fully coupled hybrid hydrological systems. We investigate the internal uncertainty propagation mechanisms of hybrid hydrological models with different levels of ENN embedding across 31 cold-region basins. An ensemble-based framework driven by stochastic optimization was used to quantify uncertainty across hydrological fluxes and internal state variables. Multiple complementary metrics were employed to characterize uncertainty magnitude, ensemble consistency, coverage and spread. Results show that ENNs improve streamflow simulation performance but alter the internal uncertainty structure of hydrological systems. Uncertainty amplification is primarily localized within ENN-replaced hydrological processes, while propagation to physically based unreplaced processes remains limited. Instead, internal state variables with memory effects act as key “uncertainty reservoirs”, where uncertainty accumulates over time. This reveals a structured rather than uniform pattern of uncertainty propagation in hybrid hydrological systems. Furthermore, increased ensemble coverage is accompanied by wider uncertainty bands, indicating a trade-off between predictive reliability and sharpness induced by stochastic optimization. Overall, this study provides new insights into how ENNs reshape uncertainty propagation pathways in hydrological models. The results highlight the importance of jointly considering predictive performance, uncertainty structure and internal system stability in hydrological modeling.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Hydrology and Earth System Sciences.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
(1166 KB) - Metadata XML
- BibTeX
- EndNote
Status: final response (author comments only)
-
RC1: 'Comment on egusphere-2026-4628', Anonymous Referee #1, 22 Sep 2026
-
AC1: 'Reply on RC1', Bu Li, 02 Oct 2026
This manuscript examines how replacing selected hydrological processes with embedded neural networks affects both streamflow prediction and variability in internal fluxes and states. The topic is interesting, and the 50 stochastic realizations across 31 basins provide useful information beyond conventional performance metrics. My main concern is that several interpretations are broader than what the current experiments directly support. The study mainly quantifies variability arising from random initialization and stochastic optimization, while the manuscript often discusses general “uncertainty propagation.” In addition, the proposed buffering and temporal accumulation mechanisms, and the attribution of increased variability specifically to ENN embedding, require stronger evidence. I therefore recommend major revision.
Response: Thanks for your recognition and valuable suggestions. We provide the response to the comment starting with “Response”.
- Scope and definition of uncertainty. The manuscript should distinguish optimization-induced/stochastic-training variability from broader hydrological or predictive uncertainty. Because forcing, observations, and model structure within each configuration are fixed, the ensemble does not represent forcing, observational, structural, or complete predictive uncertainty. Different uncertainty sources can affect physical and machine-learning models differently (Li et al., 2020; Li et al., 2025), and probabilistic uncertainty estimation generally requires more than stochastic-training spread alone (Klotz et al., 2022). Meteorological forcing products can also introduce substantial variability (Kratzert et al., 2021; Li et al., 2025). The title, abstract, and conclusions should therefore either use more specific terminology or better justify the broader framing of “uncertainty propagation.”
Response: We agree that the ensemble variability examined in this study need to be more clearly distinguished from broader hydrological and predictive uncertainty. References to internal uncertainty propagation should be qualified as concerning optimization-induced variability across hydrological processes and states. Although Sect. 2.2 defined the scope as optimization-induced uncertainty, this distinction was not consistently maintained in the title, abstract, introduction, and conclusions. In the revised manuscript, we will consistently use the term “optimization-induced uncertainty” to describe variability arising from random parameter initialization and stochastic optimization. We will also strengthen the limitations discussion to explicitly acknowledge that forcing, observational, and structural uncertainty are not comprehensively assessed by the present ensemble framework. Furthermore, we will clarify that the ensemble percentile bands and their coverage statistics serve as diagnostics of the stochastic optimization ensemble, rather than providing a complete probabilistic characterization of predictive uncertainty.
- Evidence for temporal uncertainty accumulation. The manuscript states that internal state variables act as “uncertainty reservoirs” in which uncertainty accumulates over time. However, a larger state-variable IQR demonstrates greater ensemble disagreement, but does not by itself establish temporal accumulation. A time-varying analysis of ensemble spread would provide more direct evidence of whether variability introduced by ENN-replaced processes persists or grows in storage states. Otherwise, the conclusion should be moderated to state that the results are consistent with memory-related uncertainty accumulation rather than directly demonstrating it.
Response: We agree that the current results do not directly demonstrate that internal state variables act as “uncertainty reservoirs” in which uncertainty accumulates over time. The ensemble comparisons show that greater optimization-induced variability in ENN-replaced processes is accompanied by greater variability in their corresponding state variables, whereas unreplaced processes show comparatively little change. Our intention was to suggest that increased optimization-induced uncertainty in ENN-replaced processes may propagate primarily to their corresponding state variables, with comparatively limited effects on unreplaced hydrological processes. We acknowledge that this is an interpretation of the observed patterns, rather than direct evidence of temporal accumulation.
We will therefore rename Sect. 3.3-3.4 and revise the corresponding statements in the abstract, discussion, and conclusions. Specifically, we will remove the “uncertainty reservoirs” metaphor and replace claims of temporal accumulation with descriptions of greater ensemble variability in the relevant internal states. The potential role of storage memory will be discussed as a possible explanation for these patterns, rather than as a mechanism demonstrated by the present analysis.
- Model flexibility, identifiability, and training strategy. It is unclear whether the larger ensemble spread is caused specifically by ENNs or more generally by the greater flexibility of the substituted model structure. Replacing a constrained physical formulation with an ENN changes both the representation type and the model’s effective degrees of freedom. A more flexible structure may admit more similarly performing solutions under limited calibration information, leading to weaker identifiability or equifinality and greater seed sensitivity. A more flexible physics-based representation could potentially show similar behavior. This issue may be amplified by site-by-site training. Each ENN is constrained by data from a single basin and mainly by the runoff objective, allowing multiple internal representations to achieve similar streamflow performance. Multi-basin training may provide stronger constraints through more diverse hydrological information and could reduce non-identifiability and seed sensitivity (Kratzert et al., 2024).
Response: We agree that the present experiments do not separate the effects of neural-network representation from those of increased model flexibility and changes in process constraints. Replacing a physical formulation with an ENN changes these aspects simultaneously. The larger ensemble spread should therefore be interpreted as a finding associated with the specific configurations investigated, rather than an intrinsic property of ENNs. More flexible physics-based formulations could potentially exhibit similar behavior, although this possibility was not evaluated in our experiments.
We also agree that basin-specific training and calibration against runoff alone may contribute to the observed variability. All three configurations were trained separately for each basin using the same observational information. This provides a consistent basis for comparison within the basin-specific setting, but does not necessarily constrain configurations with different flexibility equally. Different parameterizations and internal process representations may achieve similar runoff performance while producing different internal fluxes and storage trajectories.
We will explicitly discuss limited identifiability and equifinality as possible explanations for the observed ensemble spread, while clarifying that the present analysis does not directly establish these mechanisms. We will also discuss multi-basin training as a potential approach to introducing more diverse hydrological information, drawing on the benefits reported for LSTM models by Kratzert et al. (2024). Whether this approach reduces seed sensitivity and improves internal-process identifiability in the present hybrid configurations remains an open question requiring dedicated evaluation.
Our study highlights that improvements in runoff prediction can coexist with greater optimization-induced variability in internal processes and states. We will expand Sects. 4.1 and 4.2 and qualify the relevant conclusions to distinguish this observed pattern from its possible explanations, and to acknowledge the limitations imposed by model flexibility, runoff-only supervision, and basin-specific training.
Minor comments
- In Eqs. (5)–(6), appears to represent time steps although it was previously defined as the number of ensemble members.
Response: Thanks for your suggestion. We will replace with in Eqs. (5)–(6), where denotes the number of evaluation time steps. will remain the notation for the number of ensemble members, as defined in Eq. (4).
- Around lines 325–327, EXP appears to be incorrectly described as having an ENN runoff replacement.
Response: We will correct “EXP and ” to “ and ” in this sentence.
- The reference to Sect. 3.1.3 around line 331 appears incorrect.
Response: We will correct “Sect. 3.1.3” to “Sect. 3.3”.
- In Appendix Eq. (A1), the definitions of PsT appear unclear or malformed.
Response: We will correct the variable definitions: where , and denote snowfall, precipitation, and mean air temperature, respectively.
- Numerous additional grammatical and typographical issues remain; a careful language edit is recommended.
Response: We will carefully edit the manuscript to correct grammatical and typographical errors and improve clarity.
Kratzert, F., Gauch, M., Klotz, D., & Nearing, G. (2024). HESS Opinions: Never train a Long Short-Term Memory (LSTM) network on a single basin. Hydrology and Earth System Sciences, 28, 4187–4201.
Citation: https://doi.org/10.5194/egusphere-2026-4628-AC1 -
AC3: 'PDF version of our response (AC1) with corrected numbering', Bu Li, 02 Oct 2026
Due to a formatting issue, all comment numbers in our previous online response were displayed as “1”. For clarity, we provide the same response in the attached PDF with the correct numbering and formatting. Please refer to the attached PDF for our point-by-point responses.
-
AC1: 'Reply on RC1', Bu Li, 02 Oct 2026
-
RC2: 'Comment on egusphere-2026-4628', Anonymous Referee #2, 23 Sep 2026
Dear Editor and authors,
The following comments detail my review of the manuscript "From prediction improvement to uncertainty propagation structure: revisiting hybrid hydrological models," submitted to *Hydrology and Earth System Sciences*.
## Overall Recommendation
The paper addresses a current and very active area of research regarding hybrid hydrological models. It is well written, and the results are supported by the evidence shown. However, I suggest **major revisions** before publication.
## Major Comments
* The authors need to better justify their selection of catchments. The initial selection of just HUCs 1, 4, and 17 seems arbitrary. Why not just filter for catchments with the described mean annual air temperatures regardless of the region? Furthermore, the authors need to describe the "substantial computational cost" to justify reducing their potential training set by 75%.
* Although the model hyperparameters and training configuration are likely detailed in previous work, I suggest including a summarized version in this paper for completeness, perhaps in the appendix.
* Of the models evaluated, I would insist on including a data-driven baseline, such as an LSTM. Such a baseline provides a point of comparison that simply fits the data directly, alongside all evaluated metrics against the observed data. This suggestion is also motivated by the large increase in the IQR of some of the internal processes for the NN_QSM models (S, M, S0). This indicates that the internal trajectories of these states and fluxes are not truly constrained, meaning the NN_QSM model is essentially one small step away from being a fully data-driven solution.
* Section 3.2: Particularly for the neural network ENN_Q, I believe it is a missed opportunity to only analyze state variables in this study. Höge et al. (2022) in Sect. 4 show how much replacing a process with a neural network changes the expected behavior of that process. Showing a similar response surface for ENN_Q in NN_Q and NN_QSM would complement this section nicely.
* Section 3.4: The idea of uncertainty "accumulating" is misleading, as the analysis does not aggregate over time and only overall statements can be made. Uncertainty "accumulating" implies that the model remains able to make accurate predictions over time despite greater variability in its internal states. Furthermore, the fact that this occurs (NN_QSM performs better than NN_Q, which performs better than EXP) points to an issue of equifinality that is not addressed at all in the paper.
* This uncertainty framework captures only optimization-induced uncertainty (random initialization/stochastic training), not parameter, forcing, or structural uncertainty. This is acknowledged as a limitation (L382–384), but given that the abstract and introduction frame the study broadly around "internal uncertainty propagation", it should be clarified in those sections too.## Minor Comments
* L103: the process 'partitioning S' is not labeled in Figure 1a.
* L104: the process 'evapotranspiration E' is labeled in Figure 1a as ET.
* L127: the use of 'day length' seems a bit unusual. Is it used only by the ENNs?
* L203: this normalization seems important, yet there is little detail provided. Do the authors mean normalized by the 'total' precipitation in a basin? Or, if it is a daily process, how were zero-precipitation days handled?
* Figure 3: outliers are barely visible in these plots. Additionally, the overall quality of the figure is poor.
* L226: typo: "compraison"
* L285: typo: "processess"
* L331: is this meant to refer to Sect. 3.3?
* L416: the variables and what they denote are incorrect here.## References
* Höge, M., Scheidegger, A., Baity-Jesi, M., Albert, C., and Fenicia, F.: Improving hydrologic models for predictions and process understanding using neural ODEs, Hydrol. Earth Syst. Sci., 26, 5085–5102, 2022.
Citation: https://doi.org/10.5194/egusphere-2026-4628-RC2 - AC2: 'Reply on RC2', Bu Li, 02 Oct 2026
Viewed
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 231 | 106 | 62 | 399 | 62 | 53 |
- HTML: 231
- PDF: 106
- XML: 62
- Total: 399
- BibTeX: 62
- EndNote: 53
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
This manuscript examines how replacing selected hydrological processes with embedded neural networks affects both streamflow prediction and variability in internal fluxes and states. The topic is interesting, and the 50 stochastic realizations across 31 basins provide useful information beyond conventional performance metrics.
My main concern is that several interpretations are broader than what the current experiments directly support. The study mainly quantifies variability arising from random initialization and stochastic optimization, while the manuscript often discusses general “uncertainty propagation.” In addition, the proposed buffering and temporal accumulation mechanisms, and the attribution of increased variability specifically to ENN embedding, require stronger evidence. I therefore recommend major revision.
1. Scope and definition of uncertainty
The manuscript should distinguish optimization-induced/stochastic-training variability from broader hydrological or predictive uncertainty. Because forcing, observations, and model structure within each configuration are fixed, the ensemble does not represent forcing, observational, structural, or complete predictive uncertainty.
Different uncertainty sources can affect physical and machine-learning models differently (Li et al., 2020; Li et al., 2025), and probabilistic uncertainty estimation generally requires more than stochastic-training spread alone (Klotz et al., 2022). Meteorological forcing products can also introduce substantial variability (Kratzert et al., 2021; Li et al., 2025).
The title, abstract, and conclusions should therefore either use more specific terminology or better justify the broader framing of “uncertainty propagation.”
2. Evidence for temporal uncertainty accumulation
The manuscript states that internal state variables act as “uncertainty reservoirs” in which uncertainty accumulates over time. However, a larger state-variable IQR demonstrates greater ensemble disagreement, but does not by itself establish temporal accumulation.
A time-varying analysis of ensemble spread would provide more direct evidence of whether variability introduced by ENN-replaced processes persists or grows in storage states. Otherwise, the conclusion should be moderated to state that the results are consistent with memory-related uncertainty accumulation rather than directly demonstrating it.
3. Model flexibility, identifiability, and training strategy
It is unclear whether the larger ensemble spread is caused specifically by ENNs or more generally by the greater flexibility of the substituted model structure. Replacing a constrained physical formulation with an ENN changes both the representation type and the model’s effective degrees of freedom.
A more flexible structure may admit more similarly performing solutions under limited calibration information, leading to weaker identifiability or equifinality and greater seed sensitivity. A more flexible physics-based representation could potentially show similar behavior.
This issue may be amplified by site-by-site training. Each ENN is constrained by data from a single basin and mainly by the runoff objective, allowing multiple internal representations to achieve similar streamflow performance. Multi-basin training may provide stronger constraints through more diverse hydrological information and could reduce non-identifiability and seed sensitivity (Kratzert et al., 2024).
Minor comments
1. In Eqs. (5)–(6), NN appears to represent time steps although it was previously defined as the number of ensemble members.
2. Around lines 325–327, EXP appears to be incorrectly described as having an ENN runoff replacement.
3. The reference to Sect. 3.1.3 around line 331 appears incorrect.
4. In Appendix Eq. (A1), the definitions of PsT appear unclear or malformed.
5. Numerous additional grammatical and typographical issues remain; a careful language edit is recommended.
References
Klotz, D., Kratzert, F., Gauch, M., Keefe Sampson, A., Brandstetter, J., Klambauer, G., Hochreiter, S., & Nearing, G. (2022). Uncertainty estimation with deep learning for rainfall–runoff modeling. Hydrology and Earth System Sciences, 26, 1673–1693. https://doi.org/10.5194/hess-26-1673-2022.
Kratzert, F., Gauch, M., Klotz, D., & Nearing, G. (2024). HESS Opinions: Never train a Long Short-Term Memory (LSTM) network on a single basin. Hydrology and Earth System Sciences, 28, 4187–4201. https://doi.org/10.5194/hess-28-4187-2024
Kratzert, F., Klotz, D., Hochreiter, S., & Nearing, G. S. (2021). A note on leveraging synergy in multiple meteorological data sets with deep learning for rainfall–runoff modeling. Hydrology and Earth System Sciences, 25, 2685–2703. https://doi.org/10.5194/hess-25-2685-2021.
Li, P., Zha, Y., Shi, L., Tso, C.-H. M., Zhang, Y., & Zeng, W. (2020). Comparison of the use of a physical-based model with data assimilation and machine learning methods for simulating soil water dynamics. Journal of Hydrology, 584, 124692. https://doi.org/10.1016/j.jhydrol.2020.124692.
Li, P., Song, Y., Pan, M., Lawson, K., & Shen, C. (2025). Ensembling differentiable process-based and data-driven models with diverse meteorological forcing datasets to advance streamflow simulation. Hydrology and Earth System Sciences, 29, 6829–6861. https://doi.org/10.5194/hess-29-6829-2025.