LoRA-EnVar: Parameter-Efficient Hybrid Ensemble Variational Assimilation for Weather Forecasting

Thank you very much for your thoughtful and encouraging review. We are glad that you found the proposed approach and its evaluation to be of high quality and significance.

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Response to Q1&W1: Spatial/temporal mean for overall evaluation

In response to your helpful suggestion, we have included overall accuracy metrics in the table below, where each entry represents the spatial-temporal mean RMSE for a key variable (corresponding to Figure 3&4 in the main text and Figure 8&9 in the Appendix). Bold values indicate the best performance, and italic values indicate the second-best. As shown, LoRA-EnVar consistently ranks either first or second, closely matching or outperforming both Full Finetune (w/ reset) and Hybrid 3DEnVar, depending on the variable. This further confirms that the improvements observed in the space/time summaries also hold under global scalar evaluation.

Response to W1: Novelty of the VAE approach

Regarding your note on the familiarity of the VAE-based approach, we fully agree that the use of VAEs in data assimilation is established. Our contribution focuses on incorporating LoRA modules for efficient online adaptation, which we believe adds a novel and practically valuable enhancement to this VAE-Var framework.

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We appreciate your feedback once again and it helped us strengthen the overall presentation and completeness of the paper.

Thank you very much for your constructive and detailed comments. We greatly appreciate your positive assessment of the novelty and overall presentation of our work.

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Response to Q1&W4: comparison with other architectures such as DiffDA

By stating that "they do not incorporate climatological priors or flow-dependent ensembles," we mean that SDA does neither, while DiffDA incorporates a learned climatological prior but does not leverage flow-dependent ensembles.

More formally, data assimilation aims to estimate the posterior $p(x^i \mid x_b^i, y^i)$, given the prior state $x_b^i$ and observations $y^i$ at time step $i$. One common approximation is to replace the time-specific distribution with a time-averaged conditional prior $p(x \mid x_b)$, which encodes climatological structure. DiffDA follows this approach and uses a diffusion model to learn a fixed mapping from $x_b$ to $x$, approximating $p(x \mid x_b)$. However, this mapping is static and cannot reflect the time-varying nature of $p(x^i \mid x_b^i)$ under dynamically evolving conditions.

In contrast, VAE-Var, inspired by traditional variational data assimilation methods, assumes the forecast error $e^i = x^i - x_b^i$ is independent of $x_b^i$ and instead learns a time-averaged climatological error distribution $p(e)$. LoRA-EnVar builds upon this by incorporating flow-dependent adjustments to the error distribution. Specifically, it constructs online error samples using time-lagged forecasts and encodes them through low-rank updates, allowing the model to approximate time-varying distributions $p_i(e^i)$ and better capture regime-dependent uncertainties (e.g., stronger jet activity in winter).

Importantly, this modeling choice has architectural and practical implications. DiffDA requires paired data $(x_b^i, x^i)$ to train its diffusion model, which can be difficult or even infeasible to obtain in online or operational settings. In contrast, LoRA-EnVar only requires access to error samples $e^i$, which can be easily estimated from ensembles using established methods such as time-lagging. Therefore, LoRA-EnVar offers greater flexibility and is better suited for real-time deployment.

Response to Q2: How training error samples are constructed using the NMC method

To construct training error samples using the NMC method, we first select historical ERA5 reanalysis states at 12-hour intervals. For each selected state, we run the forecast model to generate two forecasts for the same target time: one with a 12-hour lead time and another with a 24-hour lead time. The difference between these two forecasts forms an error sample. This process is repeated over a 37-year span of ERA5 data, resulting in a large and diverse training set. Because the errors are aggregated over many years and meteorological regimes, the learned distribution reflects averaged, climatological characteristics rather than case-specific, flow-dependent variations. In contrast, the samples used during finetuning are constructed online from recent ensemble forecasts and are therefore sensitive to current atmospheric conditions. These are referred to as “flow-dependent” errors in our framework.

Response to Q3: Discussion on the computational cost

Thank you for the question.

• The primary benefit of VAE-Var over traditional 3DVar is its ability to capture non-Gaussian error structures via a learned latent representation. While 3DVar applies a linear transformation to a Gaussian control variable, VAE-Var maps a latent variable to physical space using a neural network decoder. As such, the computational complexity of VAE-Var is not lower; in fact, it involves more floating-point operations. However, VAE-Var, implemented with optimized neural network operators, is highly compute-efficient on GPUs. In our experiments, one assimilation cycle using VAE-Var takes approximately 18 seconds on a single NVIDIA A100 GPU, compared to about 4 minutes per cycle using a CPU-based 3DVar implementation running on 16 cores.
• Although LoRA updates only a small subset of parameters, the backward pass still traverses the entire network to compute gradients. In practice, LoRA reduces the fine-tuning time modestly, from ~10s per cycle in full fine-tuning to ~9s with LoRA. Nevertheless, The key advantage of LoRA lies not in time savings but in memory efficiency and modularity. Because LoRA modifies only a few low-rank matrices, the adapted model can be stored and transferred compactly, which is especially valuable when tracking adaptations over time or switching between flow regimes. Moreover, LoRA enables efficient online updates without altering the backbone, simplifing deployment and mitigating catastrophic forgetting.

Response to W1: Comparison with 4DVar

To evaluate our framework under 4DVar settings, we constructed an extended observation scenario where, in addition to the standard 6-hourly observations (e.g., at 0h, 6h, 12h), we introduce intermediate observations at 1h, 7h, 13h, etc. We employ a 1-hour resolution version of the FengWu forecast model to resolve temporal dynamics within each 6-hour assimilation window.

We compared eight methods and the results over the first 30 days of January are summarized in the tables below. As expected, all 4D variants outperform their 3D counterparts, reflecting the benefit of temporal assimilation. Importantly, LoRA-4DEnVar consistently outperforms 4DVar and Hybrid 4DEnVar, demonstrating both the extensibility and effectiveness of our framework in fully 4D assimilation settings.

Response to W2: Mathematical definition of two types of error

We thank the reviewer for the valuable suggestion. To clarify the conceptual distinction, we provide a more precise mathematical formulation. We denote the forecast error at time step $i$ as $e^i = x^i - x_b^i$, where $x^i$ is the true state and $x_b^i$ is the background forecast. Then:

• The flow-dependent error distribution refers to the conditional error distribution at a particular time $i$, given the background flow conditions $\mathcal{F}_i$ (e.g., the synoptic-scale state of the atmosphere): $p_i(e) := p(e^i \mid \mathcal{F}_i)$. This formulation acknowledges that the forecast error structure depends on the current dynamical regime. For instance, during periods with strong jet streams or tropical cyclones, the forecast uncertainty tends to be larger and more anisotropic.
• The climatological error distribution is defined as the time-averaged distribution of forecast errors: $p_{\text{clim}}(e) := \mathbb{E}_i \, p_i(e)$. This captures the average statistical structure of forecast errors across a long historical period and is often used in traditional climatology-based assimilation schemes.

While both types of error indeed arise from model forecast errors, we respectfully note that the distinction is not solely based on time scales. The key difference, particularly in the context of ensemble-based assimilation, lies in whether the error statistics are conditioned on the current flow state. Flow-dependent errors reflect state-specific uncertainties, while climatological errors represent the marginal distribution averaged over many flow regimes.

Response to W3: Scope to strengthen mathematical formulation and conceptual links

Thank you for the thoughtful suggestion. While many recent works in the SciML community focus on general-purpose physical surrogates or PDE solvers, our work centers on improving probabilistic data assimilation in a large-scale, operational setting—namely, numerical weather prediction. Our adoption of LoRA is driven not only by considerations of efficiency, but also by the need for spatially localized, flow-dependent adaptation within a fixed VAE backbone. We agree that connecting our approach more clearly to the broader frameworks of online learning and meta-learning would enhance the conceptual framing of the paper. In particular, we appreciate the pointer to recent work on low-rank meta-learning architectures for physical systems, which indeed parallels our use of LoRA as a mechanism for cycle-wise adaptation. Although our primary motivation is practical—capturing flow-dependent uncertainty efficiently in high-dimensional systems—we recognize that the underlying problem structure closely aligns with meta-adaptation paradigms, where compact parameter updates enable generalization across evolving regimes. We will revise the manuscript to clarify these connections, formalize the VAE and LoRA components more precisely, and cite the suggested references accordingly.

Response to W5: Missing Hybrid EnVar citation

Thank you for pointing this out. The implementation we adopt for Hybrid 3DEnVar follows the formulation described in [1]. We will include this citation in the revised version.

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We appreciate your feedback once again and it helped us strengthen the overall presentation and completeness of the paper.

[1] Wang, X., Snyder, C., & Hamill, T. M. (2007). On the theoretical equivalence of differently proposed ensemble–3DVAR hybrid analysis schemes. Monthly Weather Review, 135(1), 222-227.

We thank the authors for their clarifications.

The additional details provided in the rebuttal help clarify the method and would strengthen the paper if incorporated into a revised version. However, I still find that the lack of mathematical exposition of the core components, specifically the VAE-Var framework and the online distribution adaptation via LoRA, limits the clarity of the proposed approach.

Besides, could the authors provide more details on the adaptation of LoRA-EnVar to 4DVar?

Thank you very much for your constructive and detailed comments. We greatly appreciate your positive assessment of the overall methodology and experimental evaluation.

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Response to Q1: How the ensemble is obtained to finetune LoRA

Thank you for the question regarding the source of ensemble forecasts used to fine-tune the LoRA modules (“Other predictions” in Figure 1). The LoRA-EnVar framework is flexible and can support different strategies for ensemble forecast generation, such as perturbing initial conditions or using multiple forecast models. In our current implementation, we adopt a time-lagged ensemble approach, which is well adopted in use and fundamentally falls into the first category—based on different initial conditions. Specifically, all ensemble members are generated using the same forecast model (FengWu), but from earlier analysis times with different lead times (e.g., 1, 2, or 3 steps), such that they all arrive at the same target time. This strategy offers a computationally efficient way to represent flow-dependent uncertainty without requiring multiple models or explicit perturbations. We have provided the detailed implementation of this scheme in Appendix C.2, and the ensemble structure is illustrated in Figure 7.

Response to W1: Support for other autoencoder architectures

We fully agree that our current method relies on the assumption of a Gaussian latent space, which underpins the validity of applying 3DVar in latent space. While LoRA can be incorporated into more expressive autoencoders such as VQ-VAEs, particularly those based on Transformer architectures, we have not yet developed a data assimilation framework that is compatible with vector-quantized latent representations. Unlike VAEs with continuous latent spaces, the discrete nature of VQ-VAEs poses unique challenges when integrated with variational assimilation, including the absence of gradient flow in the discrete latent space and the difficulty of formulating background error priors. We will explicitly clarify this limitation and consider this an important direction for future work in the revised version.

Response to W2: Colors used in the plots

Thank you for pointing this out. We acknowledge that some color choices in the figures may make it difficult to distinguish between different curves. In the revised version, we will improve the figure design by optimizing the color palette and line styles to enhance clarity and accessibility.

Response to W3: Better explanation of the proposed methodology

We appreciate your suggestion to improve the presentation of the methodology. In the revised version, we will move the description of VAE-Var (currently in Appendix A) into the main text, so that the algorithmic pipeline can be more easily understood without requiring readers to consult the appendix.

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We appreciate your feedback once again and it helped us strengthen the overall presentation and completeness of the paper.

Thank you very much for your constructive and detailed comments. We greatly appreciate your positive assessment of the methodology of our work.

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Response to W1.A&W1.B&Q3: Comparison with modern DA tools

Comparison with traditional DA methods

We fully agree that stronger baselines, such as 4DVar and EnKF-based methods, are essential for a meaningful evaluation. In response, we extended our experiments to include:

• 4DVar and Hybrid 4DEnVar: To evaluate 4D variants, we introduce additional observations at 1h, 7h, 13h, etc., alongside standard 6-hourly observations. We use a 1-hour-resolution FengWu model to represent flow evolution within each assimilation window.
• LETKF: We implemented the Local Ensemble Transform Kalman Filter with 200 km horizontal localization and 0.3 vertical tapering. Ensemble size was kept consistent with other EnVar-based methods.

Comparison with learning-based DA methods

We carefully considered several recent ML-based DA baselines:

• DiffDA learns $p(x \mid x_b)$ via diffusion models, which requires paired $(x_b, x)$ data for training and adaptation. However, such labeled data is difficult to obtain online, making flow-dependent adaptation nontrivial. Our method instead models the error distribution $p(e = x - x_b)$, which is easier to sample in operational settings. We nonetheless include DiffDA (without flow-dependent ensembles) in our comparisons.
• SDA does not incorporate $x_b$ and cannot perform cyclical assimilation, making it incompatible with our setting.
• FuXi-En4DVar is a variant of Hybrid 4DEnVar with FuXi used for ensemble generation. We include a comparable Hybrid 4DEnVar baseline using FengWu.
• EnSF and Latent-EnSF: EnSF assumes complete observations and is not applicable to high-dimensional sparse settings. Latent-EnSF extends EnSF to sparse observations, but its results are only shown at low spatial resolution ($21 \times 72 \times 144$), and no official code is currently available for reproduction or scaling tests. We will include it in future work once the implementation becomes accessible.

Results

In the updated tables below, we report RMSEs for key variables (z500 and u500) across ten methods, including five traditional and five learning-based approaches (including both 3D and 4D variants). These comparisons demonstrate that our proposed LoRA-EnVar performs competitively against both advanced DA systems and state-of-the-art ML-based approaches.

Response to W1.C&Q1: Analysis of the calibration of the ensembles

Thank you for raising this important point. Our current work focuses on improving the analysis given a background ensemble, and we did not initially emphasize calibration analysis. To address this, we now include a new experiment to evaluate ensemble calibration. We ran both LoRA-EnVar and Hybrid 3DEnVar systems starting from July 1st and assessed the ensemble forecast initialized at 00:00 on July 15th, allowing sufficient spin-up for both methods. The results are summarized below:

These results show that LoRA-EnVar not only improves forecast accuracy (lower RMSE), but also yields better-calibrated ensembles, as indicated by the higher spread-to-RMSE ratios. This suggests that the flow-dependent latent prior learned by LoRA-EnVar more effectively captures uncertainty in the analysis compared to traditional hybrid methods. We will include additional calibration diagnostics in the revised version to further support this finding.

Response to W2: More extensive testing across time

Thank you for the suggestion. We fully agree that evaluating the system solely on January data is insufficient to demonstrate generalizability. In response, we have conducted an additional experiment using data from July, which represents a different seasonal regime.

Response to W3: Hardware efficiency gains

Thank you for the comment. While LoRA does not significantly reduce runtime (from $\sim$10s to $\sim$9s per cycle) or memory (~40.5GB to ~39.1GB) compared to full-finetune methods, its main advantage lies in dramatically lower storage cost. A single LoRA checkpoint is only ~648KB, enabling long-term retention of fine-tuned states (e.g., ~2.5GB for one year at 6-hour cycles), compared to ~2.3TB for full fine-tuning, which is over 1000× smaller. This makes LoRA-EnVar much more practical for retrospective analysis or downstream applications.

Response to W4: Key formulations not presented in main body

Thank you for the helpful suggestion. We agree that the variational procedure is important and should be briefly clarified in the main text. In variational data assimilation, the goal is to estimate the state that minimizes a cost function balancing prior and observational constraints, equivalent to a regularized negative log-posterior. In our case, this is formulated in a latent space with learned priors. We will add concise explanations in the revised version to improve clarity and self-containment.

Response to Q2: Concerns about realism of forecast distribution

Thank you for raising this important point. Our LoRA-EnVar framework does not rely on reducing high-frequency features or extremes to lower RMSE. To examine whether our analysis field preserves realistic distributional characteristics, rather than artificially smoothing the output, we compute two standard statistical metrics: standard deviation and skewness, which reflect the overall spread and asymmetry of the field values, respectively.

Due to limits on image uploads, we report numerical values at a representative time (2022.01.01 00:00) in the table below. It can be observed that:

• The standard deviation of the analysis field is similar to than that of the background field, indicating that the assimilation process does not suppress variance.
• The skewness values of both z500 and u500 are well aligned with those of the ground truth, suggesting that extreme values are retained, and the distributional shape is preserved.

The relative differences between the analysis and the ground truth are below 1%, confirming that our method does not distort the statistical structure of the field. In the revised version, we will include additional visualizations and statistical analyses in the appendix to further support this point.

Response to S3: Interesting but underexplored ablation insights

Thank you for highlighting this interesting finding. We also observed that LoRA yields the largest improvements for Z500, a relatively slow-evolving and spatially coherent variable. One possible reason is that Z500 reflects the large-scale circulation and tends to integrate information from multiple physical processes, making it more amenable to the structured, low-rank adaptation provided by LoRA. Conversely, the decline in performance with more finetunable parameters suggests potential overfitting or interference with pretrained representations. We will explore this further and add discussions in the revised version to strengthen the interpretation of these results.

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We appreciate your feedback once again and it helped us strengthen the overall presentation and completeness of the paper.

Thank you authors for the thorough response and updated comparisons. It's great to see that the method generalized to 4D approaches as well. Overall I'd say the rebuttal has moved in some very positive directions, but my concerns about comparisons and testing scope have not been addressed in a way that would make me comfortable increasing my rating. I have a few more questions based on the content of the responses.

Comparison with learning-based DA methods

Thanks for these detailed explanations, but it's not clear why SDA would not work in a cyclic assimilation setting - their approach is designed to work on sequences of observations. Presumably if the inclusion of a prior on $x$ is important, SDA would largely under-perform validating the hypothesis proposed in this submission.

Comparison with traditional DA methods

Is there a reasonable explanation for why the relative performance of the classical DA methods is not consistent with existing literature? 3DVar appears to be quite competitive here, while historical comparisons of 3DVar vs 4D approaches like the EnKF and 4DVar have generally strongly favored 4D both on small-scale global models [1] and regional models [2]. With respect to variational methods generally, the introduction of ensemble-driven covariances has been observed to consistently produce positive or non-harmful results [3][4] while the results here are mixed.

Some more information on the configurations of these methods would be helpful. Are these results given a particular budget for each (unusually small ensembles, low numbers of observations per window in 4DVar)? If so, that's OK, but it is important to explain the limitations of the analysis. What would also be interesting here is if these restrictions are lifted, does the relative performance change? Since generating the finetuning data requires extra forward evaluations of the model, if this budget were instead spent on larger ensembles, does the proposed method still have an advantage.

More extensive testing across time

Thanks for adding the additional month. Is there a reason why this can't be tested across the full test set? The seasonal analysis is extremely valuable, but it's hard to evaluate the consistency of the method from two data points.

Concerns about realism of forecast distribution

Since images are unavailable in responses I'm not going to evaluate this too heavily. More informative approaches here would be a spectral analysis (magnitude/cut off curve of high frequencies) or even just images of the fields.

Calibration of the ensembles

Thank you for this additional information. The results indicate that the DA approach (and the baselines) actually are not very well calibrated (post-calibration results aim to be around 1), but that is important information for the reader to know and it is not an issue that the method has limitations. The posted results are sufficient to address this concern.

[1] Kalnay, Eugenia, et al. "4-D-Var or ensemble Kalman filter?." Tellus A: Dynamic Meteorology and Oceanography 59.5 (2007): 758-773.
[2] Meng, Zhiyong, and Fuqing Zhang. "Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part III: Comparison with 3DVAR in a real-data case study." Monthly Weather Review 136.2 (2008): 522-540.
[3] Buehner, M., J. Morneau, and C. Charette. "Four-dimensional ensemble-variational data assimilation for global deterministic weather prediction." Nonlinear Processes in Geophysics 20.5 (2013): 669-682.
[4] Buehner, Mark, et al. "Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part II: One-month experiments with real observations." Monthly Weather Review 138.5 (2010): 1567-1586.

Thank you for your valuable comments. We address your concerns below, point by point.

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Comparison with learning-based DA methods

Thank you for the thoughtful comment. We agree that SDA is an interesting line of work and, in principle, is designed to handle sequences of observations. However, there are practical challenges that currently limit its application in large-scale, cyclic assimilation settings such as numerical weather prediction (NWP).

First, the original SDA paper was tested only on toy models (Lorenz96 [1] and a two-layer quasi-geostrophic model [2]), and its extension to high-dimensional geophysical systems presents significant engineering and training challenges. In particular, training score-based diffusion models on high-resolution atmospheric fields is much more complex than on low-dimensional toy systems. To the best of our knowledge, the most relevant recent work is Appa [3], which can be seen as an extension of SDA. However, this paper only appeared on arXiv in April 2025 and does not provide open-source code, making it difficult to reproduce under our experimental setup within the review timeframe. Furthermore, their observational setup differs significantly from ours: they simulate 11,000 observation locations based on satellite and ground stations—about 11 times more than ours—and inject artificial noise into the observations. Despite the denser observations, their reported performance is comparable to ours: RMSE on z500 stabilizes between 60–70 $\mathrm{m}^2/\mathrm{s}^2$, and on t850 around 1.2 $K$. By comparison, our LoRA-EnVar achieves z500 RMSE of 64.11 $\mathrm{m}^2/\mathrm{s}^2$ and t850 RMSE of 1.075 $K$, using fewer observations.

In addition, the Appa paper only evaluates performance over a 7-day assimilation period, whereas our system is designed and tested for month-long operational settings. It remains unclear how well score-based methods like SDA or Appa would scale under such extended assimilation windows and data volumes. Thus, while we recognize the conceptual appeal of SDA, practical barriers currently limit its deployment in realistic cyclic data assimilation workflows.

Comparison with traditional DA methods

Clarification of the experimental settings

At first, we would like to clarify that our comparison experiments are designed to be fair and consistent across all methods.

1. For Hybrid 3DEnVar, Hybrid 4DEnVar, LoRA-3DEnVar, and LoRA-4DEnVar, the ensemble size is uniformly set to 8, and the ensembles are constructed using the same time-lagged approach. For the LETKF method, the ensemble size is also set to 8, and the ensembles are generated using the standard LETKF procedure.
2. In all 3D-based methods (3DVar, Hybrid 3DEnVar, LETKF, VAE-3DVar, LoRA-3DEnVar), observations are sampled every 6 hours (i.e., at 0h, 6h, 12h, and 18h), with 1000 points sampled on a 721x1440 grid at each time step. In all 4D-based methods (4DVar, Hybrid 4DEnVar, VAE-4DVar, LoRA-4DEnVar), observations are distributed across finer temporal intervals within the assimilation window (i.e., at 0h, 1h, 6h, 7h, 12h, 13h, 18h, and 19h).
3. All AI-based methods (VAE-3DVar, LoRA-3DEnVar, VAE-4DVar, LoRA-4DEnVar) share the same core pipeline as their classical counterparts (3DVar, Hybrid 3DEnVar, 4DVar, Hybrid 4DEnVar), except for the replacement of the background error covariance (typically modeled by a static or hybrid B matrix) with a learned latent representation via a VAE, optionally enhanced with LoRA-based online adaptation.

Therefore, we believe the comparisons in our study are fair, self-consistent, and controlled in terms of ensemble size, observation setup, and algorithmic design.

Regarding the experimental results

Our results are broadly consistent with prior literature:

1. 4DVar vs. 3DVar: All 4DVar-based methods (4DVar, Hybrid 4DEnVar, VAE-4DVar, LoRA-4DEnVar) outperform their 3DVar counterparts.

2. VAE vs. traditional covariance: VAE-based methods (VAE-3DVar, VAE-4DVar) consistently improve over their static-covariance baselines (3DVar, 4DVar).

By comparing LoRA-based methods (i.e., LoRA-3DEnVar, LoRA-4DEnVar) with their traditional hybrid ensemble method counterpart (i.e., Hybrid 3DEnVar, Hybrid 4DEnVar), we find that LoRA-based methods are better, which further supports the effectiveness of our proposed method.

There are, however, two aspects that may appear counterintuitive: (1) The strong performance of 3DVar, and (2) Mixed results from ensemble-driven covariance.

(1) Why 3DVar is so competitive?

The relative performance of DA methods depends heavily on the forecast model. In [4], EnKF only clearly outperforms 3DVar under perfect-model assumptions (e.g., SPEEDY); when using reanalysis-based observations (e.g., NCEP), this advantage diminishes significantly. Our experiments use FengWu, a deep learning-based forecast model with inherent model error, making our setting more comparable to the reanalysis-based case in [4]. Hence, the weaker advantage of ensemble methods over 3DVar is reasonable.

Additionally, our implementation of 3DVar differs slightly from traditional configurations due to the characteristics of the AI forecast model. Specifically, while most traditional NWP systems (e.g., WRF) treat geopotential as a diagnostic variable and infer it from primitive variables [5], the AI-based FengWu model predicts geopotential directly. This distinction necessitates adaptive modifications to the 3DVar formulation [6]. In our implementation based on WRF_DA with the cv5 option, we additionally incorporated the regression coefficients between geopotential and stream function to better handle this change. Because geopotential integrates multi-variable information and is directly corrected in our formulation, the effective degrees of freedom in the assimilation are reduced, potentially improving the 3DVar performance. In contrast, traditional models must rely on nonlinear observation operators to assimilate geopotential, making the task more difficult. As a result, the 3DVar baseline in our AI-model-based setup is inherently stronger, which may help explain its unexpectedly competitive performance.

(2) Why mixed results with ensemble covariances?

We would like to clarify that, overall, our experimental findings are consistent with the conclusions of prior literature: incorporating ensemble-driven background covariances generally improves or maintains performance across most variables. In our experiments, variables such as horizontal wind (u, v), temperature (t) exhibit positive or non-harmful improvements when transitioning from static covariance estimation to hybrid estimation. We demostrate results for other variables in the table below.

The only notable exception is the geopotential height (e.g., z500), where hybrid EnVar underperforms 3DVar. We believe this discrepancy arises from structural differences in how z is treated in our AI-based forecasting model compared to traditional NWP systems. In conventional models, geopotential is typically a diagnostic variable, computed from mass and thermodynamic relationships under physical constraints (e.g., hydrostatic balance). In such settings, 3DVar can effectively correct z indirectly through adjustments to other state variables or via nonlinear observation operators. However, in our AI forecasting system, geopotential is modeled as a direct prognostic variable without such physical coupling. This allows 3DVar to explicitly correct z via direct gradient descent on the observed variable, which can sometimes outperform ensemble-based methods when the ensemble suffers from high-dimensional sampling noise or fails to fully capture the complex flow-dependent structure of z errors.

Regarding Computational Budget

We clarify that LoRA-3DEnVar does not incur additional model evaluations compared to Hybrid 3DVar. Both methods use the same ensemble of 8 members, generated via time-lagged forecasts. In Hybrid 3DVar, ensemble information is incorporated through modifications to the B matrix, whereas LoRA-3DEnVar uses finetuning to achieve a similar effect.

This finetuning step operates on the latent space and is computationally lightweight, requiring less than 10 seconds per assimilation window for an ensemble size of 8, without any extra forward model runs.

More extensive testing across time

Thank you for highlighting this important point. We apologize for the limited temporal coverage in the original submission. In response, we have extended our experiments to cover the entire year of 2022 and demonstrate the results in the table below. The results are generally consistent with what we have obtained for the period of one month.

Results for u10 ($m/s$)

Results for v850 ($m/s$)

Results for t850 ($K$)

Results for mslp ($Pa$)

Concerns about realism of forecast distribution

Thank you for your understanding. We will include visualizations of both the analysis fields and the corresponding spectral analysis in the revised version to better illustrate the high-frequency behavior.

Calibration of the ensembles

Thank you for the helpful clarification. We appreciate the acknowledgement that the posted results sufficiently address the concern. While improving calibration was not the primary focus of this work, we agree that reporting these metrics provides valuable context. We will clarify this limitation in the revised version to ensure transparency for the reader.

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References

[1] Rozet, F., & Louppe, G. (2023). Score-based data assimilation. Advances in Neural Information Processing Systems, 36, 40521-40541.

[2] Rozet, F., & Louppe, G. (2023). Score-based data assimilation for a two-layer quasi-geostrophic model. arXiv preprint arXiv:2310.01853.

[3] Andry, G., Rozet, F., Lewin, S., Rochman, O., Mangeleer, V., Pirlet, M., ... & Louppe, G. (2025). Appa: Bending weather dynamics with latent diffusion models for global data assimilation. arXiv preprint arXiv:2504.18720.

[4] Kalnay, E., Li, H., Miyoshi, T., Yang, S. C., & Ballabrera-Poy, J. (2007). 4-D-Var or ensemble Kalman filter?. Tellus A: Dynamic Meteorology and Oceanography, 59(5), 758-773.

[5] Meng, Z., & Zhang, F. (2008). Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part III: Comparison with 3DVAR in a real-data case study. Monthly Weather Review, 136(2), 522-540.

[6] Adrian, M., Sanz-Alonso, D., & Willett, R. (2025). Data assimilation with machine learning surrogate models: A case study with FourCastNet. Artificial Intelligence for the Earth Systems, 4(3), e240050.

Thank you very much for your constructive and detailed comments. We greatly appreciate your positive assessment of the methodology, novelty, and overall presentation of our work.

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Response to Q1: Selective LoRA integration on queries

Thank you for the insightful question regarding the selective integration of LoRA modules into the query projections only.

To address this, we conducted additional ablation experiments where LoRA was applied to different combinations of the query (Q), key (K), and value (V) projections. Specifically, we tested: (1) finetuning Q only (the default in our main results), (2) finetuning both Q and K, (3) finetuning both Q and V, and (4) finetuning Q, K, and V simultaneously.

The results over the first 30 days of January are summarized in the tables below. For each 3-day period, we computed the average RMSE across twelve 6-hour assimilation cycles. These experiments were conducted for two representative variables: z500 and u500. As shown, finetuning Q alone already achieves strong performance, and while incorporating additional LoRA modules into K and/or V leads to minor improvements in some periods, the relative performance fluctuates across different 3-day windows. Overall, the average gains are marginal and often fall within the variability observed across repeated runs.

Importantly, extending LoRA to K or V increases the number of trainable parameters by $2\times$ or $3\times$, which goes against our design goal of efficient online adaptation with minimal computational overhead. Thus, we chose to finetune only Q in our main experiments.

These findings are also consistent with the original LoRA paper [1], which reports that finetuning Q alone often suffices, and that additional gains from finetuning K and V are typically small. This reinforces our decision to prioritize efficiency by limiting updates to the query projections only.

Response to Q2: Relationship between the LoRA rank and an effective ensemble size

Thank you for the insightful question regarding the relationship between LoRA rank and ensemble size or background covariance rank.

While both the ensemble size in traditional data assimilation and LoRA rank in our framework reflect constraints on the representational capacity of uncertainty, they operate in fundamentally different ways. In traditional ensemble-based methods, ensemble size determines the number of samples used to estimate the flow-dependent structure of the background error. A small ensemble size can lead to an under-represented covariance with artificially low rank, failing to capture key dynamical modes. In general, assuming sufficient computational resources are available, larger ensembles lead to better performance, as they provide more accurate and higher-rank estimates of the background error.

In contrast, LoRA rank does not directly correspond to ensemble size, nor does it determine the rank of any explicit background covariance matrix. Instead, it controls the maximum number of directions in parameter space along which dynamical corrections can be introduced into a climatologically trained model. If the LoRA rank is too low, even a large ensemble cannot be fully leveraged, as the model lacks sufficient degrees of freedom to incorporate flow-dependent corrections. Conversely, an overly high rank may introduce redundant parameters with little benefit. In our experiments, a LoRA rank of 2 achieved a optimal balance between expressiveness and efficiency. Moreover, in LoRA-EnVar, flow-dependent information is incorporated via online learning, enabling the model to extract meaningful dynamical information even with a small ensemble, as long as the LoRA rank is sufficiently expressive.

Exploring a more explicit theoretical connection between LoRA rank, ensemble size, and the effective background error structure remains an interesting direction for future work.

Response to W1: Portability to backbones beyond Swin-V2

We thank the reviewer for raising the question of portability. Our current implementation of LoRA-EnVar is based on the Swin-V2, but the approach is in principle easily adaptable to other Transformer-based architectures. This flexibility is because LoRA modules operate over standard linear projections, such as query/key/value matrices, which are widely shared across Transformer variants.

For models like GraphCast, which are based on graph neural networks (GNNs) and do not contain Transformer-style attention mechanisms, a direct application of LoRA-EnVar is not straightforward. Such architectures lack the projection layers that LoRA typically targets. Adapting our framework to GNNs would require non-trivial modifications, such as developing alternative fine-tuning mechanisms tailored for message-passing networks. We consider this a promising direction for future work.

Response to W2: Inner-loop gradient updates requiring full transformer backpropagation

We appreciate the reviewer’s insightful observation. Indeed, our method requires running the Swin-V2 decoder in each inner-loop gradient evaluation during assimilation, reflecting the inherent computational cost of variational approaches that rely on gradient optimization. However, by updating only the lightweight LoRA modules, we significantly reduce the number of trainable parameters. While the full decoder remains part of the backward graph to allow gradient flow into the LoRA parameters, all backbone weights remain frozen, eliminating gradient storage for the backbone itself. This approach reduces memory usage and simplifies optimization, even though backpropagation still traverses the full network. In future work, we plan to explore further optimizations, such as updating only a subset of decoder layers or employing lighter-weight backbone architectures.

Response to S6 (last sentence): Comparison with other baselines, such as VAE-Var

We appreciate the reviewer’s comment. Indeed, even in traditional data assimilation methods, the gain from incorporating flow-dependent (dynamical) error structures is often incremental, particularly on some variables, when compared to strong climatological baselines (e.g., Hybrid 3DEnVar vs. 3DVar, as shown in Figure 3). Our goal is to introduce such capabilities in a lightweight and extensible way, and we believe LoRA-EnVar achieves this with minimal overhead.

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We appreciate your feedback once again and it helped us strengthen the overall presentation and completeness of the paper.

References:

[1] Hu, E. J., Shen, Y., Wallis, P., Allen-Zhu, Z., Li, Y., Wang, S., ... & Chen, W. (2022). Lora: Low-rank adaptation of large language models. ICLR, 1(2), 3.

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