the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 Sep 2025
NoahPy: A differentiable Noah land surface model for simulating permafrost thermo-hydrology
Abstract. Accurately representing permafrost in Earth System Models is a grand challenge that creates major uncertainty. A promising path forward is to create hybrid models that synergize process-based physics with deep learning, but this is fundamentally hindered by the non-differentiable nature of traditional land surface models (LSMs), which are incompatible with modern AI workflows. To overcome this limitation, we present NoahPy, a fully differentiable LSM developed by reconstructing the Noah LSM’s governing partial differential equations into a process-encapsulated Recurrent Neural Network (RNN). We first demonstrate that NoahPy perfectly replicates the numerical behaviour of the modified Noah LSM, achieving Nash-Sutcliffe Efficiency (NSE) coefficients above 0.99 for both soil temperature and liquid water. We then show that at a permafrost site, the calibrated NoahPy achieves robust simulation performance for for soil temperature (NSE > 0.9) and liquid water (NSE > 0.8). Critically, the differentiable workflow, when combined with the Adam optimizer, is significantly faster, more stable, and yields simulations with lower uncertainty compared to traditional SCE-UA calibration algorithm. NoahPy thus provides a foundational, "glass-box" framework that closes a key technical gap, enabling the development of the next generation of hybrid AI-physics models needed to more reliably predict the future of the cryosphere.
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Notice on discussion status
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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Preprint
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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.
- Preprint
(5445 KB) - Metadata XML
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-4253', Anonymous Referee #1, 16 Sep 2025
- AC1: 'Reply on RC1', Zhuotong Nan, 24 Oct 2025
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RC2: 'Comment on egusphere-2025-4253', Anonymous Referee #2, 10 Oct 2025
Tian et al. present NoahPy, a differentiable reformulation of the Noah land surface model (LSM) aimed at improving the representation of permafrost thermo-hydrology. The authors rewrite the traditional Fortran-based Noah LSM into a PyTorch-based, partially differentiable framework and demonstrate that it reproduces the original model’s numerical behavior while enabling gradient-based parameter optimization through backpropagation.
Strengths
- The paper provides a clear and rigorous implementation of a differentiable land surface model using PyTorch.
- The model reproduces the original Fortran Noah LSM with high fidelity (NSE > 0.99), indicating numerical equivalence.
- The differentiable structure allows efficient gradient-based calibration (using Adam), showing faster and more stable convergence than traditional SCE-UA optimization.
- The manuscript is well organized, the methodology transparent, and the validation experiments are convincing for the scope of the technical contribution.
Major comments
1. Scope of differentiability vs. the claim of “fully differentiable LSM”.
Although the abstract and conclusions describe NoahPy as a fully differentiable LSM, the actual implementation appears to make only the heat and moisture transport equations (the PDE solver) differentiable. Other key land-surface processes (e.g., those in Figure 1c) remain treated in their original, non-differentiable, piecewise form. As a result, the framework achieves gradient continuity for a subset of processes, but not necessarily full differentiability of the entire LSM.
The authors should clarify this scope explicitly in both the abstract and methods. Phrasing such as “a partially differentiable framework focusing on the heat–moisture solver” or “a differentiable core of Noah LSM” would be more accurate and prevent reader misinterpretation.
2. Gradient continuity within phase-dependent processes
The thermal conductivity λ, volumetric heat capacity Cₛ, and latent heat term Q exhibit abrupt transitions near the freezing point due to phase change. These discontinuities can interrupt or distort the backpropagated gradients, even if the overall framework is formally differentiable. The authors are encouraged to clarify how such non-smooth terms are handled in the current implementation.
Citation: https://doi.org/10.5194/egusphere-2025-4253-RC2 - AC2: 'Reply on RC2', Zhuotong Nan, 24 Oct 2025
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-4253', Anonymous Referee #1, 16 Sep 2025
The state-of-the-art land surface models (LSMs) have been reported to perform poorly in representing permafrost processes. To address this gap, the authors present NoahPy—a fully differentiable LSM developed by reconstructing the Noah LSM’s governing partial differential equations into a process-encapsulated recurrent neural network. NoahPy was compared with both the original and an improved version of the Noah LSM, and evaluated at a permafrost site. I find the model to be skillful and the results reasonable.
General Comments
1) Manuscript Structure
Introduction: It would be beneficial to restructure the introduction to better highlight the significance of permafrost, particularly as the authors aim to introduce the model to the permafrost research community. The section could begin by underscoring the importance of permafrost, followed by a critical review of how current LSMs represent permafrost processes, clearly outlining existing limitations. Addressing this gap, the authors should then introduce deep learning methods and explain how such approaches can provide an effective solution to improve permafrost modeling.
Discussion: The advantages and limitations are currently intermingled in this section. Please consider: (a) adding a brief outlook on future model development; and (b) using subsections to enhance the readability of the manuscript.
2) Language
The language should be improved throughout for clarity and academic tone.Specific Comments:
- L18: Avoid using the word “perfectly,” as no model can be considered perfect. Please revise this throughout the manuscript (e.g., L224 and others). As noted on the GMD homepage:
“Essentially, all models are wrong, but some are useful.” (George E. P. Box, 1979) - L22: “SCE-UA” is not defined.
- L105: Eq.6: Clarify what “βF” and “βG” refer to.
- L204: While a reference is provided, please explain the principle of the Shuffled Complex Evolution (SCE-UA) algorithm. For example, what does the strength of the SCE-UA algorithm stem from?
- L255: This could be easily verified by additionally evaluating snow water equivalent or snow depth against observations. I assume snow depth data are available at the TGL site (e.g., Xiao et al., 2013).
- L296: There are two opening parentheses here. Similar typos occur elsewhere in the manuscript; please revise carefully.
- L407: What does the underline signify here?
References:Xiao, Y., Zhao, L., Dai, Y., Li, R., Pang, Q., and Yao, J.: Representing permafrost properties in CoLM for the Qinghai–Xizang (Tibetan) Plateau, Cold Regions Science and Technology, 87, 68–77, https://doi.org/10.1016/j.coldregions.2012.12.004, 2013.Citation: https://doi.org/10.5194/egusphere-2025-4253-RC1 - AC1: 'Reply on RC1', Zhuotong Nan, 24 Oct 2025
- L18: Avoid using the word “perfectly,” as no model can be considered perfect. Please revise this throughout the manuscript (e.g., L224 and others). As noted on the GMD homepage:
-
RC2: 'Comment on egusphere-2025-4253', Anonymous Referee #2, 10 Oct 2025
Tian et al. present NoahPy, a differentiable reformulation of the Noah land surface model (LSM) aimed at improving the representation of permafrost thermo-hydrology. The authors rewrite the traditional Fortran-based Noah LSM into a PyTorch-based, partially differentiable framework and demonstrate that it reproduces the original model’s numerical behavior while enabling gradient-based parameter optimization through backpropagation.
Strengths
- The paper provides a clear and rigorous implementation of a differentiable land surface model using PyTorch.
- The model reproduces the original Fortran Noah LSM with high fidelity (NSE > 0.99), indicating numerical equivalence.
- The differentiable structure allows efficient gradient-based calibration (using Adam), showing faster and more stable convergence than traditional SCE-UA optimization.
- The manuscript is well organized, the methodology transparent, and the validation experiments are convincing for the scope of the technical contribution.
Major comments
1. Scope of differentiability vs. the claim of “fully differentiable LSM”.
Although the abstract and conclusions describe NoahPy as a fully differentiable LSM, the actual implementation appears to make only the heat and moisture transport equations (the PDE solver) differentiable. Other key land-surface processes (e.g., those in Figure 1c) remain treated in their original, non-differentiable, piecewise form. As a result, the framework achieves gradient continuity for a subset of processes, but not necessarily full differentiability of the entire LSM.
The authors should clarify this scope explicitly in both the abstract and methods. Phrasing such as “a partially differentiable framework focusing on the heat–moisture solver” or “a differentiable core of Noah LSM” would be more accurate and prevent reader misinterpretation.
2. Gradient continuity within phase-dependent processes
The thermal conductivity λ, volumetric heat capacity Cₛ, and latent heat term Q exhibit abrupt transitions near the freezing point due to phase change. These discontinuities can interrupt or distort the backpropagated gradients, even if the overall framework is formally differentiable. The authors are encouraged to clarify how such non-smooth terms are handled in the current implementation.
Citation: https://doi.org/10.5194/egusphere-2025-4253-RC2 - AC2: 'Reply on RC2', Zhuotong Nan, 24 Oct 2025
Peer review completion
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Wenbiao Tian
Hu Yu
Yuhe Cao
Wenjun Yi
Jiwei Xu
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(5445 KB) - Metadata XML
The state-of-the-art land surface models (LSMs) have been reported to perform poorly in representing permafrost processes. To address this gap, the authors present NoahPy—a fully differentiable LSM developed by reconstructing the Noah LSM’s governing partial differential equations into a process-encapsulated recurrent neural network. NoahPy was compared with both the original and an improved version of the Noah LSM, and evaluated at a permafrost site. I find the model to be skillful and the results reasonable.
General Comments
1) Manuscript Structure
Introduction: It would be beneficial to restructure the introduction to better highlight the significance of permafrost, particularly as the authors aim to introduce the model to the permafrost research community. The section could begin by underscoring the importance of permafrost, followed by a critical review of how current LSMs represent permafrost processes, clearly outlining existing limitations. Addressing this gap, the authors should then introduce deep learning methods and explain how such approaches can provide an effective solution to improve permafrost modeling.
Discussion: The advantages and limitations are currently intermingled in this section. Please consider: (a) adding a brief outlook on future model development; and (b) using subsections to enhance the readability of the manuscript.
2) Language
The language should be improved throughout for clarity and academic tone.
Specific Comments:
“Essentially, all models are wrong, but some are useful.” (George E. P. Box, 1979)