the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 04 Sep 2026
Ensemble Kalman-guided model predictive path integral control for spatially localized suppression of extremes in chaotic geophysical flows
Haru Kuroki
Kazumune Hashimoto
Yuki Uehara
Yohei Sawada
Duc Le
Masashi Minamide
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- Final revised paper (published on 04 Sep 2026)
- Preprint (discussion started on 03 Feb 2026)
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2026-419', Victor Bertret, 23 Feb 2026
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CC1: 'Reply on RC1', Kazumune Hashimoto, 21 Mar 2026
Author Response to Referee #1
We thank the Referee for the thorough review and constructive comments. To address these comments, we have performed additional experiments and analyses, and in the revised manuscript we will make the following main changes:
1. Clarify the rationale for using MPPI under single-step actuation, and more explicitly position the present work as a fundamental method toward multi-step and multi-actuator extensions.
2. Improve the fairness of the baseline comparison by expanding the sensitivity study of BO-based baselines, including a BO-informed MPPI variant.
3. Add computational profiling that separates the EnKC-prior stage from the MPPI rollout stage, and discuss the effect of parallelization.
4. Improve the clarity of the notation by explaining the embedding functions f_loc, g_loc, f_mag, and g_mag when they are first introduced.Below, we respond point by point.
---
Referee Comment 1: Single-step actuation vs. sequential trajectory optimization
Referee comment:
The authors explicitly state that the control perturbation is applied only at the first rollout step, with no further input during the remaining horizon. Because the problem is reduced to optimizing a single initial impulse, the use of MPPI seems oversized. Discuss why MPPI was chosen over CMA-ES or Bayesian Optimization. If the intention is groundwork for multi-step actuation, state this more explicitly.Response:
We thank the Referee for this important point. We agree that the current implementation applies the perturbation only at the first rollout step, with zero input thereafter. Thus, at each control decision, the optimization reduces to choosing a single localized impulse. Indeed, this design was motivated by the current CSE setting: in chaotic flows, even a small localized perturbation can produce a meaningful downstream effect over the prediction horizon, while restricting the actuation to a single step reduces the effective control dimension and improves sampling efficiency.Nevertheless, we agree that a clearer justification for using the MPPI formalism is necessary. In the revised manuscript, we will emphasize that MPPI still offers several advantages in this setting:
1. Parallel sampling: MPPI evaluates a batch of rollouts whose costs and weights can be computed in parallel. This is advantageous when forward simulations are the dominant cost and can be parallelized on multi-core CPUs or GPUs.
2. Avoiding sequential acquisition loops: Bayesian optimization typically involves a sequential propose-evaluate loop, which can be less straightforward for real-time implementation.
3. Extensions to multi-step control: As the Referee points out, one motivation for adopting MPPI is that it can be generalized to multi-step control sequences and multiple actuators within the same algorithmic pipeline. The single-step setting in this manuscript is a proof of concept toward that goal.Following the Referee's suggestion, we have additionally performed a BO-based comparison for the present single-step optimization problem. In this comparison, we use a standalone BO baseline. In our BO implementation, the Expected Improvement (EI) acquisition function is used, and 10 BO iterations are carried out at each control decision. Although 10 iterations is a modest BO budget, we adopted this setting because the resulting per-decision computation time is already substantially larger than that of the proposed method.
The results show that EKG-MPPI outperforms BO in the key metrics:
- Number of exceedances (X > 12): 5547 (EKG-MPPI) vs. 6386 (BO)
- Mean input magnitude: 0.564264 (EKG-MPPI) vs. 0.611567 (BO)
- Mean computation time: 0.000253 s (EKG-MPPI) vs. 0.122630 s (BO)These results indicate that even in the single-step setting, MPPI provides a more efficient and effective solution than BO.
Furthermore, we clarify in the revised manuscript that one of the main motivations for adopting MPPI is its natural extensibility to multi-step control and multiple actuators within a unified framework. The current single-step formulation should be regarded as a proof of concept toward this broader objective.
---
Referee Comment 2: Fairness in baseline comparison (hyperparameter optimization)
Referee comment:
The comparison against vanilla MPPI lacks rigorous hyperparameter tuning. Vanilla MPPI uses fixed lambda = 0.7 while EKG-MPPI uses lambda = 0.1. Vanilla MPPI uses a static sampling distribution. EKG-MPPI adapts mean/variance via EnKC and thus has a massive advantage. Baseline MPPI should undergo hyperparameter tuning or sensitivity analysis.Response:
We thank the Referee for this important comment. We agree that the comparison with vanilla MPPI should be strengthened by examining hyperparameter sensitivity and by considering a stronger adaptive baseline. In addition to the original vanilla MPPI comparison, we also introduce a stronger MPPI baseline to examine whether the advantage of EKG-MPPI persists under a more adaptive baseline design.Specifically, we construct an additional BO-informed MPPI baseline, denoted BO-MPPI, in which Bayesian optimization is used to estimate prior information for the intervention location and magnitude, and these estimates are then supplied to vanilla MPPI.
For BO-MPPI, the BO objective balances the predicted threshold exceedance over the rollout horizon and the intervention magnitude, where the parameter alpha controls the trade-off between control effectiveness and actuation cost. Here, alpha is the BO-side trade-off parameter in the objective function, whereas lambda is the MPPI temperature parameter. These two parameters play different roles and are varied separately in our experiments.
To ensure a fair comparison with EKG-MPPI, BO-MPPI uses the same MPPI temperature lambda and the same sampling variances for intervention location and magnitude as EKG-MPPI. For each fixed value of lambda, we run BO-MPPI with alpha = 100, 10, 1, 0.1, 0.01. The BO budget is fixed at 10 iterations at each control decision, since even with this modest budget the BO-based baseline already requires a substantially longer computation time than the proposed method. We then evaluate the resulting Lorenz-96 simulations in terms of the number of threshold exceedances and the average intervention magnitude.
Separately, we repeat this comparison for lambda = 0.1, 1, 10, and provide the corresponding plots in the supplementary material. Across all tested values of lambda, the EKG-MPPI result lies below the empirical lower envelope (Pareto frontier-like trade-off curve) of the BO-MPPI results obtained by varying $\alpha$. These additional results show that the advantage of EKG-MPPI is robust to the choice of lambda and is not attributable to a single hyperparameter setting.
---Referee Comment 3: Computational scalability and profiling
Referee comment:
EKG-MPPI is the most computationally expensive. Provide a breakdown of time spent computing the EnKC prior vs. evaluating MPPI rollouts to clarify bottlenecks and scaling feasibility.Response:
We thank the Referee for this useful suggestion and agree that such a runtime breakdown is important for identifying the main computational bottlenecks and assessing scalability. We therefore measured the computation time separately for the EnKC-prior stage and the MPPI rollout/weight stage, and we have added this information to the revised manuscript.
In the Lorenz-96 experiment with K_MPPI = 40, the mean runtime per control decision was 0.000044 s for the EnKC-prior computation and 0.000222 s for the MPPI rollout/weight computation.
The total mean runtime was therefore 0.000266 s per decision step, with approximately 17% of the cost attributable to the EnKC-prior stage and 83% to the rollout and weight stage.
This indicates that the main computational bottleneck is the rollout evaluation rather than the EnKC prior itself. In other words, the additional overhead introduced by the EnKC-based prior is modest in the present setting. Moreover, the total runtime remains well below the sampling interval used in the experiment, supporting the computational feasibility of the proposed method in the current online-control setting.
---Referee Comment 4: Technical correction on embedding-function notation
Referee comment:
In Section 3.2, Step 3, the embedding functions f_loc, g_loc, f_mag, and g_mag are introduced. Add a brief sentence clarifying their exact mathematical definitions or references earlier.Response:
We thank the Referee for this helpful comment. In the revised manuscript, we will clarify that:
- f_loc and f_mag map the EnKC output to the mean of the sampling distribution (location and magnitude),
- g_loc and g_mag define the corresponding sampling variance or covariance used by MPPI.---
Summary
We again thank the Referee for the constructive feedback. We believe these revisions significantly strengthen the manuscript by clarifying the methodological rationale, improving the fairness of the comparisons, and providing a more transparent analysis of computational costs.
Sincerely,
Haru Kuroki, Kazumune Hashimoto, Yuki Uehara, Yohei Sawada, Duc Le, and Masashi Minamide-
AC1: 'Reply on RC2', Kazumune Hashimoto, 25 Mar 2026
Author Response to Referee #2
We thank the Referee for the careful review and for the positive assessment of the manuscript, which helped us improve the clarity of the paper.
In the revised manuscript, we address all of the points raised by the Referee. In particular, we have (i) added uncertainty information to the performance summaries,
(ii) clarified the role of vanilla MPPI and strengthened the baseline comparison, (iii) improved the figure labels, legends, and captions,
and (v) corrected the notational and bibliographic issues listed by the Referee.Below, we respond point by point.
--------------------------------------------------
Referee Comment 1:
In both the Lorenz-96 and SQG sections, the conclusions are reported without uncertainty estimates. For Lorenz-96, Table 1 reports extreme event suppressions; for SQG, the conclusions are drawn from raw peak wind and mean input comparisons. But in both cases no confidence intervals or variability estimates are provided for these performance diagnostics. Since for both metrics the improvement is moderate (less than 10%), it is then difficult to assess how significant the improvement of EKG-MPPI over EnKC is.Response:
We thank the Referee for pointing out the lack of uncertainty quantification in the original manuscript.
In the revised manuscript, we address this concern by performing repeated experiments with different random seeds and reporting the mean and standard deviation of the relevant performance metrics.For the Lorenz–96 experiments, we repeated the simulations 10 times with different random seeds (using the temperature parameter λ = 0.25 for EKG-MPPI). We evaluated both the number of threshold exceedances (extreme-event count) and the magnitude of the control input. The results are as follows:
- Extreme-event count:
EKG-MPPI: 7622 ± 146
EnKC: 7820 ± 142- Control input magnitude:
EKG-MPPI: 0.166 ± 0.002
EnKC: 0.191 ± 0.001These results indicate that EKG-MPPI consistently achieves a reduction in extreme-event occurrences relative to EnKC, while also requiring a smaller control input. The difference in the mean values is larger than the corresponding standard deviations, suggesting that the observed improvement is not solely attributable to stochastic variability.
For the SQG experiments, we conducted 5 independent runs with different random seeds for Scenario 5 (using λ = 0.1). We computed the mean and standard deviation for the maximum wind speed and the control input magnitude:
- Maximum wind speed:
EKG-MPPI: 3.341 ± 0.050
EnKC: 3.403 ± 0.005- Control input magnitude:
EKG-MPPI: 0.0135 ± 0.0038
EnKC: 0.0137 ± 0.0015In this case as well, EKG-MPPI yields a lower mean peak wind speed compared to EnKC, while maintaining a comparable level of control effort.
Although the variability in EKG-MPPI is somewhat larger for the control input, the reduction in peak wind speed remains consistent across runs.Based on these additional experiments, we will revise the manuscript to include uncertainty estimates (mean ± standard deviation) for all reported performance metrics.
--------------------------------------------------
Referee Comment 2:
In the Lorenz-96 experiments, EKG-MPPI uses lambda = 0.1 and an EnKC-informed proposal distribution, while vanilla MPPI uses lambda = 0.7 and uninformed Gaussian sampling for magnitude and location. As a result, the reported difference does not isolate the effect of the proposed guidance alone. I suggest clarifying that vanilla MPPI is included as a simple reference only, and that the principal comparison is with EnKC.Response:
We thank the Referee for this important comment. We agree with the Referee’s interpretation. In the revised manuscript, we now clarify explicitly that vanilla MPPI is included only as a simple reference with an uninformed prior, and not as the principal baseline for evaluating the proposed EnKC-guided mechanism.
The principal comparison in the paper is with EnKC, since EKG-MPPI is designed as an EnKC-based nonlinear refinement.Moreover, to further strengthen the baseline comparison, we additionally introduced two more informative baselines in the revised manuscript: a standalone Bayesian optimization (BO) method and a BO-informed MPPI method. The standalone BO baseline directly optimizes the location and magnitude of the single-step perturbation under the same setting, and is therefore a relevant derivative-free comparator for the present low-dimensional problem.
The BO-informed MPPI baseline, in turn, uses BO to provide adaptive prior information for MPPI, thereby offering a stronger adaptive baseline than vanilla MPPI with a static uninformed proposal. These additional baselines make it possible to distinguish more clearly between the benefit of adaptive proposal guidance and the benefit of the optimizer itself. We have revised the corresponding discussion to make the role of each baseline explicit and to avoid overstating the significance of the vanilla MPPI comparison.--------------------------------------------------
Referee Comment 3:
Figure 4's caption states that the evaluation metric is the percentage difference ((EKG - EnKC)/EnKC) x 100, but the figure itself shows raw maximum wind speed and raw mean input magnitude by scenario. Either this text should be removed and replaced with useful information about the figure, or the results about the percentage difference should be added.Response:
We thank the Referee for pointing out this inconsistency. The original caption was misleading. In the revised manuscript, we correct the caption of Figure 4 so that it directly describes the quantities actually shown in the figure, i.e., the raw maximum wind speed and the raw mean input magnitude for each scenario. We remove the incorrect statement about the percentage difference.--------------------------------------------------
Referee Comment 4:
The figures could be made more informative. In figure 2, 3 and 5, the axes lack clear units or precise information on the quantities shown, and some legends remain too limited to guide the reader which needs to infer from the text outside of the figure. E.g. "X variables" is not defined, and most colour bars come without text. I would encourage the authors to improve the figure labels, legends, and captions so that each figure can be understood more directly on its own.Response:
We agree that these figures should be more self-contained. In the revised manuscript, we have improved not only the captions but also the axis labels, legends, and colour-bar annotations. The revised versions of these figures are provided in the Supplementary Material.
For Figure 2, we replaced the ambiguous y-axis label "X variables" with "State Value" and clarified in the caption that the plotted quantity is the Lorenz-96 state variable X_k, i.e. a dimensionless model variable. We also made the distinction between the full percentile range and the zoomed upper-tail range more explicit.For Figure 3, we added explicit labels for the quantities shown on the axes, namely the temperature parameter and the sample number, and descriptive colour-bar titles for all panels: "Number of Extreme Values," "Number of Interventions," and "Mean Intervention Magnitude." We also revised the caption so that the meanings of panels (a)-(c) can be understood without referring to the main text.
For Figure 5, we added spatial axis labels with units ("Distance [km]") and a labelled colour bar ("Wind Speed [m/s]"). In addition, the caption now explicitly states that the plotted field is the wind-speed magnitude and that the white rectangle denotes the target region.
We believe that these revisions make the figures substantially more informative and easier to interpret independently of the main text.
--------------------------------------------------
Referee Comment 5:
Also, here are some very small corrections that would make the manuscript easier to read and more complete:- In Algorithm 1, the Require line lists H but not H_c, although line 5 calls EnKC with H_c.
- In Eq. (1), the model equation uses q_{t-1}, while the text below defines q_t as the model error.
- In Eq. (10), the text just above introduces an ensemble {x_t^{alpha(i)}}_{i=1}^m, but Eq. (10) then uses x_{t+T_EnKC}^a as if it were a single, already defined state. It is unclear from a quick read whether this denotes one ensemble member, the ensemble mean, or another quantity.
- The switch from the generic EnKC horizon T_EnKC to T_c in the Lorenz-96 section could be made more explicit.
- The Gaussian sampling notation should be clarified. In Sect. 3, the second Gaussian parameter is introduced as a variance/covariance term, but in the numerical sections the paper sets g_mag(u_t^EnKC) = ||u_t^EnKC||_1 / 2 in Lorenz-96 and g_mag(u_t^EnKC) = ||u_t^EnKC||_1 in SQG, which seems like a natural choice for a standard deviation. It would help to clarify whether the authors mean m ~ N(mu, g_mag) or m ~ N(mu, g_mag^2).
- In the paragraph introducing Figure 5, the order “no-control, EKG-MPPI, and EnKC” should be made consistent with figure 5, where the order is “no-control, EnKC, and EKG-MPPI.”
- The introduction refers to Henderson as an early 4D-Var-based study, but the corresponding reference is missing from the bibliography.
- For the first CSE citation, the original CSE paper from 2022 is better suited.
- The manuscript currently cites Sawada (2024a) as a preprint; the authors may also update that entry to the published journal version.Response:
We thank the Referee for these careful and helpful comments. We revise the manuscript accordingly:
- In Algorithm 1, we add H_c to the Require line and explicitly define it as the operator that maps the state variables to the control criteria.
- In Eq. (1) and the text below it, we correct the notation so that the model-error term is written consistently as q_{t-1}.
- After introducing the analysis ensemble in Eqs. (8) and (9), we now state explicitly that \bar{x}_t^a denotes the analysis ensemble mean at time t. We also clarify the notation around Eq. (10) by specifying that the state at t + T_EnKC refers to the ensemble-mean forecast obtained by propagating the analysis ensemble to that lead time.
- In the Lorenz-96 section, we replace T_c with T_EnKC in order to unify the notation with the generic EnKC formulation.
- We clarify that the second parameter of the Gaussian distribution denotes a variance/covariance. Accordingly, we revise the definitions of g_mag so that they are given on the variance scale. Specifically, we now define g_mag(u_t^EnKC) = ||u_t^EnKC||_1^2 / 4 in the Lorenz-96 experiments and g_mag(u_t^EnKC) = ||u_t^EnKC||_1^2 in the SQG experiments.
- We correct the ordering in the paragraph introducing Figure 5 so that it matches the figure itself, namely: no control, EnKC, and EKG-MPPI.
- We add the missing Henderson reference to the bibliography: Henderson et al. (2005), "A 4D-Var study on the potential of weather control and exigent weather forecasting".
- We replace the first CSE citation with the original paper by Miyoshi and Sun (2022), as suggested.
- We update the Sawada (2024a) reference from the preprint entry to the published journal version in SOLA.--------------------------------------------------
Closing remarkWe thank the Referee again for the thoughtful comments. We believe that these revisions have improved the clarity and readability of the manuscript.
Sincerely,
Haru Kuroki, Kazumune Hashimoto, Yuki Uehara, Yohei Sawada, Duc Le, and Masashi Minamide
-
AC1: 'Reply on RC2', Kazumune Hashimoto, 25 Mar 2026
-
AC1: 'Reply on RC2', Kazumune Hashimoto, 25 Mar 2026
Author Response to Referee #2
We thank the Referee for the careful review and for the positive assessment of the manuscript, which helped us improve the clarity of the paper.
In the revised manuscript, we address all of the points raised by the Referee. In particular, we have (i) added uncertainty information to the performance summaries,
(ii) clarified the role of vanilla MPPI and strengthened the baseline comparison, (iii) improved the figure labels, legends, and captions,
and (v) corrected the notational and bibliographic issues listed by the Referee.Below, we respond point by point.
--------------------------------------------------
Referee Comment 1:
In both the Lorenz-96 and SQG sections, the conclusions are reported without uncertainty estimates. For Lorenz-96, Table 1 reports extreme event suppressions; for SQG, the conclusions are drawn from raw peak wind and mean input comparisons. But in both cases no confidence intervals or variability estimates are provided for these performance diagnostics. Since for both metrics the improvement is moderate (less than 10%), it is then difficult to assess how significant the improvement of EKG-MPPI over EnKC is.Response:
We thank the Referee for pointing out the lack of uncertainty quantification in the original manuscript.
In the revised manuscript, we address this concern by performing repeated experiments with different random seeds and reporting the mean and standard deviation of the relevant performance metrics.For the Lorenz–96 experiments, we repeated the simulations 10 times with different random seeds (using the temperature parameter λ = 0.25 for EKG-MPPI). We evaluated both the number of threshold exceedances (extreme-event count) and the magnitude of the control input. The results are as follows:
- Extreme-event count:
EKG-MPPI: 7622 ± 146
EnKC: 7820 ± 142- Control input magnitude:
EKG-MPPI: 0.166 ± 0.002
EnKC: 0.191 ± 0.001These results indicate that EKG-MPPI consistently achieves a reduction in extreme-event occurrences relative to EnKC, while also requiring a smaller control input. The difference in the mean values is larger than the corresponding standard deviations, suggesting that the observed improvement is not solely attributable to stochastic variability.
For the SQG experiments, we conducted 5 independent runs with different random seeds for Scenario 5 (using λ = 0.1). We computed the mean and standard deviation for the maximum wind speed and the control input magnitude:
- Maximum wind speed:
EKG-MPPI: 3.341 ± 0.050
EnKC: 3.403 ± 0.005- Control input magnitude:
EKG-MPPI: 0.0135 ± 0.0038
EnKC: 0.0137 ± 0.0015In this case as well, EKG-MPPI yields a lower mean peak wind speed compared to EnKC, while maintaining a comparable level of control effort.
Although the variability in EKG-MPPI is somewhat larger for the control input, the reduction in peak wind speed remains consistent across runs.Based on these additional experiments, we will revise the manuscript to include uncertainty estimates (mean ± standard deviation) for all reported performance metrics.
--------------------------------------------------
Referee Comment 2:
In the Lorenz-96 experiments, EKG-MPPI uses lambda = 0.1 and an EnKC-informed proposal distribution, while vanilla MPPI uses lambda = 0.7 and uninformed Gaussian sampling for magnitude and location. As a result, the reported difference does not isolate the effect of the proposed guidance alone. I suggest clarifying that vanilla MPPI is included as a simple reference only, and that the principal comparison is with EnKC.Response:
We thank the Referee for this important comment. We agree with the Referee’s interpretation. In the revised manuscript, we now clarify explicitly that vanilla MPPI is included only as a simple reference with an uninformed prior, and not as the principal baseline for evaluating the proposed EnKC-guided mechanism.
The principal comparison in the paper is with EnKC, since EKG-MPPI is designed as an EnKC-based nonlinear refinement.Moreover, to further strengthen the baseline comparison, we additionally introduced two more informative baselines in the revised manuscript: a standalone Bayesian optimization (BO) method and a BO-informed MPPI method. The standalone BO baseline directly optimizes the location and magnitude of the single-step perturbation under the same setting, and is therefore a relevant derivative-free comparator for the present low-dimensional problem.
The BO-informed MPPI baseline, in turn, uses BO to provide adaptive prior information for MPPI, thereby offering a stronger adaptive baseline than vanilla MPPI with a static uninformed proposal. These additional baselines make it possible to distinguish more clearly between the benefit of adaptive proposal guidance and the benefit of the optimizer itself. We have revised the corresponding discussion to make the role of each baseline explicit and to avoid overstating the significance of the vanilla MPPI comparison.--------------------------------------------------
Referee Comment 3:
Figure 4's caption states that the evaluation metric is the percentage difference ((EKG - EnKC)/EnKC) x 100, but the figure itself shows raw maximum wind speed and raw mean input magnitude by scenario. Either this text should be removed and replaced with useful information about the figure, or the results about the percentage difference should be added.Response:
We thank the Referee for pointing out this inconsistency. The original caption was misleading. In the revised manuscript, we correct the caption of Figure 4 so that it directly describes the quantities actually shown in the figure, i.e., the raw maximum wind speed and the raw mean input magnitude for each scenario. We remove the incorrect statement about the percentage difference.--------------------------------------------------
Referee Comment 4:
The figures could be made more informative. In figure 2, 3 and 5, the axes lack clear units or precise information on the quantities shown, and some legends remain too limited to guide the reader which needs to infer from the text outside of the figure. E.g. "X variables" is not defined, and most colour bars come without text. I would encourage the authors to improve the figure labels, legends, and captions so that each figure can be understood more directly on its own.Response:
We agree that these figures should be more self-contained. In the revised manuscript, we have improved not only the captions but also the axis labels, legends, and colour-bar annotations. The revised versions of these figures are provided in the Supplementary Material.
For Figure 2, we replaced the ambiguous y-axis label "X variables" with "State Value" and clarified in the caption that the plotted quantity is the Lorenz-96 state variable X_k, i.e. a dimensionless model variable. We also made the distinction between the full percentile range and the zoomed upper-tail range more explicit.For Figure 3, we added explicit labels for the quantities shown on the axes, namely the temperature parameter and the sample number, and descriptive colour-bar titles for all panels: "Number of Extreme Values," "Number of Interventions," and "Mean Intervention Magnitude." We also revised the caption so that the meanings of panels (a)-(c) can be understood without referring to the main text.
For Figure 5, we added spatial axis labels with units ("Distance [km]") and a labelled colour bar ("Wind Speed [m/s]"). In addition, the caption now explicitly states that the plotted field is the wind-speed magnitude and that the white rectangle denotes the target region.
We believe that these revisions make the figures substantially more informative and easier to interpret independently of the main text.
--------------------------------------------------
Referee Comment 5:
Also, here are some very small corrections that would make the manuscript easier to read and more complete:- In Algorithm 1, the Require line lists H but not H_c, although line 5 calls EnKC with H_c.
- In Eq. (1), the model equation uses q_{t-1}, while the text below defines q_t as the model error.
- In Eq. (10), the text just above introduces an ensemble {x_t^{alpha(i)}}_{i=1}^m, but Eq. (10) then uses x_{t+T_EnKC}^a as if it were a single, already defined state. It is unclear from a quick read whether this denotes one ensemble member, the ensemble mean, or another quantity.
- The switch from the generic EnKC horizon T_EnKC to T_c in the Lorenz-96 section could be made more explicit.
- The Gaussian sampling notation should be clarified. In Sect. 3, the second Gaussian parameter is introduced as a variance/covariance term, but in the numerical sections the paper sets g_mag(u_t^EnKC) = ||u_t^EnKC||_1 / 2 in Lorenz-96 and g_mag(u_t^EnKC) = ||u_t^EnKC||_1 in SQG, which seems like a natural choice for a standard deviation. It would help to clarify whether the authors mean m ~ N(mu, g_mag) or m ~ N(mu, g_mag^2).
- In the paragraph introducing Figure 5, the order “no-control, EKG-MPPI, and EnKC” should be made consistent with figure 5, where the order is “no-control, EnKC, and EKG-MPPI.”
- The introduction refers to Henderson as an early 4D-Var-based study, but the corresponding reference is missing from the bibliography.
- For the first CSE citation, the original CSE paper from 2022 is better suited.
- The manuscript currently cites Sawada (2024a) as a preprint; the authors may also update that entry to the published journal version.Response:
We thank the Referee for these careful and helpful comments. We revise the manuscript accordingly:
- In Algorithm 1, we add H_c to the Require line and explicitly define it as the operator that maps the state variables to the control criteria.
- In Eq. (1) and the text below it, we correct the notation so that the model-error term is written consistently as q_{t-1}.
- After introducing the analysis ensemble in Eqs. (8) and (9), we now state explicitly that \bar{x}_t^a denotes the analysis ensemble mean at time t. We also clarify the notation around Eq. (10) by specifying that the state at t + T_EnKC refers to the ensemble-mean forecast obtained by propagating the analysis ensemble to that lead time.
- In the Lorenz-96 section, we replace T_c with T_EnKC in order to unify the notation with the generic EnKC formulation.
- We clarify that the second parameter of the Gaussian distribution denotes a variance/covariance. Accordingly, we revise the definitions of g_mag so that they are given on the variance scale. Specifically, we now define g_mag(u_t^EnKC) = ||u_t^EnKC||_1^2 / 4 in the Lorenz-96 experiments and g_mag(u_t^EnKC) = ||u_t^EnKC||_1^2 in the SQG experiments.
- We correct the ordering in the paragraph introducing Figure 5 so that it matches the figure itself, namely: no control, EnKC, and EKG-MPPI.
- We add the missing Henderson reference to the bibliography: Henderson et al. (2005), "A 4D-Var study on the potential of weather control and exigent weather forecasting".
- We replace the first CSE citation with the original paper by Miyoshi and Sun (2022), as suggested.
- We update the Sawada (2024a) reference from the preprint entry to the published journal version in SOLA.--------------------------------------------------
Closing remarkWe thank the Referee again for the thoughtful comments. We believe that these revisions have improved the clarity and readability of the manuscript.
Sincerely,
Haru Kuroki, Kazumune Hashimoto, Yuki Uehara, Yohei Sawada, Duc Le, and Masashi Minamide
-
CC1: 'Reply on RC1', Kazumune Hashimoto, 21 Mar 2026
-
RC2: 'Comment on egusphere-2026-419', Anonymous Referee #2, 11 Mar 2026
This manuscript introduces EKG-MPPI, a hybrid control approach for localized suppression of extremes in chaotic geophysical flows. The method combines EnKF for state estimation, EnKC for constructing candidate perturbations, and MPPI to improve on them through nonlinear refinement. The approach is tested on the Lorenz–96 and SQG models, where the authors report reductions in extreme-event, together with comparable or smaller control inputs relative to EnKC. Overall, the paper is well motivated, the methodological idea is interesting, and the combination of EnKC and MPPI is intuitive and potentially useful for this class of problems.
The paper shows clearly how the two methods are coupled in practice. The numerical experiments on Lorenz–96 and SQG support the claim that the method works and can improve on EnKC in the examples considered. However, the empirical advantage remains somewhat difficult to assess quantitatively, because the reported performance results in both numerical sections are small and presented without uncertainty estimates. This does not reduce the interest of the proposed method, but it does make the strength of the performance claim harder to evaluate in the present version.
Below are suggestions of minor corrections that may improve the paper:-In both the Lorenz-96 and SQG sections, the conclusions are reported without uncertainty estimates. For Lorenz-96, Table 1 reports extreme event suppressions; for SQG, the conclusions are drawn from raw peak wind and mean input comparisons. But in both cases no confidence intervals or variability estimates are provided for these performance diagnostics. Since for both metrics the improvement is moderate (less than 10%), it is then difficult to assess how significant the improvement of EKG-MPPI over EnKC is.
-In the Lorenz-96 experiments, EKG-MPPI uses lambda = 0.1 and an EnKC-informed proposal distribution, while vanilla MPPI uses lambda = 0.7 and uninformed Gaussian sampling for magnitude and location. As a result, the reported difference does not isolate the effect of the proposed guidance alone. I suggest clarifying that vanilla MPPI is included as a simple reference only, and that the principal comparison is with EnKC.
-Figure 4's caption states that the evaluation metric is the percentage difference ((EKG - EnKC)/EnKC) x 100, but the figure itself shows raw maximum wind speed and raw mean input magnitude by scenario. Either this text should be removed and replaced with useful information about the figure, or the results about the percentage difference should be added.
-The figures could be made more informative. In figure 2, 3 and 5, the axes lack clear units or precise information on the quantities shown, and some legends remain too limited to guide the reader which needs to infer from the text outside of the figure. E.g. "X variables" is not defined, and most colour bars come without text. I would encourage the authors to improve the figure labels, legends, and captions so that each figure can be understood more directly on its own.
Also, here are some very small corrections that would make the manuscript easier to read and more complete:- In Algorithm 1, the Require line lists H but not H_c, although line 5 calls EnKC with H_c.
- In Eq. (1), the model equation uses q_{t-1}, while the text below defines q_t as the model error.
- In Eq. (10), the text just above introduces an ensemble {x_{t}^{alpha(i)}}{i=1}^m, but Eq. (10) then uses x{t+T_EnKC}^a as if it were a single, already defined state. It is unclear from a quick read whether this denotes one ensemble member, the ensemble mean, or another quantity.
- The switch from the generic EnKC horizon T_EnKC to T_c in the Lorenz-96 section could be made more explicit.
- The Gaussian sampling notation should be clarified. In Sect. 3, the second Gaussian parameter is introduced as a variance/covariance term, but in the numerical sections the paper sets g_mag(u_t^EnKC) = ||u_t^EnKC||_1 / 2 in Lorenz-96 and g_mag(u_t^EnKC) = ||u_t^EnKC||_1 in SQG, which seems like a natural choice for a standard deviation. It would help to clarify whether the authors mean m ~ N(mu, g_mag) or m ~ N(mu, g_mag^2).
- In the paragraph introducing Figure 5, the order “no-control, EKG-MPPI, and EnKC” should be made consistent with figure 5, where the order is “no-control, EnKC, and EKG-MPPI.”
- The introduction refers to Henderson as an early 4D-Var-based study, but the corresponding reference is missing from the bibliography. The citation should be added explicitly. From my quick research as: Henderson, J. M., Hoffman, R. N., Leidner, S. M., Nehrkorn, T., and Grassotti, C. (2005), “A 4D-Var study on the potential of weather control and exigent weather forecasting,” Quarterly Journal of the Royal Meteorological Society, 131(612), 3037-3051, doi:10.1256/qj.05.72.
- For the first CSE citation, line 34 , I think the original CSE paper from 2022 is better suited: Miyoshi, T. and Sun, Q. (2022), “Control simulation experiment with Lorenz’s butterfly attractor,” Nonlinear Processes in Geophysics, 29, 133-145.
- The manuscript currently cites Sawada (2024a) as a preprint, the authors may also update that entry to the published journal version: Sawada, Y. (2024), “Ensemble Kalman Filter Meets Model Predictive Control in Chaotic Systems,” SOLA, 20, 400-407, doi:10.2151/sola.2024-053
Overall, I believe this manuscript is suitable for publication after minor corrections. I thank the authors for an interesting and enjoyable contribution.Citation: https://doi.org/10.5194/egusphere-2026-419-RC2 -
AC1: 'Reply on RC2', Kazumune Hashimoto, 25 Mar 2026
Author Response to Referee #2
We thank the Referee for the careful review and for the positive assessment of the manuscript, which helped us improve the clarity of the paper.
In the revised manuscript, we address all of the points raised by the Referee. In particular, we have (i) added uncertainty information to the performance summaries,
(ii) clarified the role of vanilla MPPI and strengthened the baseline comparison, (iii) improved the figure labels, legends, and captions,
and (v) corrected the notational and bibliographic issues listed by the Referee.Below, we respond point by point.
--------------------------------------------------
Referee Comment 1:
In both the Lorenz-96 and SQG sections, the conclusions are reported without uncertainty estimates. For Lorenz-96, Table 1 reports extreme event suppressions; for SQG, the conclusions are drawn from raw peak wind and mean input comparisons. But in both cases no confidence intervals or variability estimates are provided for these performance diagnostics. Since for both metrics the improvement is moderate (less than 10%), it is then difficult to assess how significant the improvement of EKG-MPPI over EnKC is.Response:
We thank the Referee for pointing out the lack of uncertainty quantification in the original manuscript.
In the revised manuscript, we address this concern by performing repeated experiments with different random seeds and reporting the mean and standard deviation of the relevant performance metrics.For the Lorenz–96 experiments, we repeated the simulations 10 times with different random seeds (using the temperature parameter λ = 0.25 for EKG-MPPI). We evaluated both the number of threshold exceedances (extreme-event count) and the magnitude of the control input. The results are as follows:
- Extreme-event count:
EKG-MPPI: 7622 ± 146
EnKC: 7820 ± 142- Control input magnitude:
EKG-MPPI: 0.166 ± 0.002
EnKC: 0.191 ± 0.001These results indicate that EKG-MPPI consistently achieves a reduction in extreme-event occurrences relative to EnKC, while also requiring a smaller control input. The difference in the mean values is larger than the corresponding standard deviations, suggesting that the observed improvement is not solely attributable to stochastic variability.
For the SQG experiments, we conducted 5 independent runs with different random seeds for Scenario 5 (using λ = 0.1). We computed the mean and standard deviation for the maximum wind speed and the control input magnitude:
- Maximum wind speed:
EKG-MPPI: 3.341 ± 0.050
EnKC: 3.403 ± 0.005- Control input magnitude:
EKG-MPPI: 0.0135 ± 0.0038
EnKC: 0.0137 ± 0.0015In this case as well, EKG-MPPI yields a lower mean peak wind speed compared to EnKC, while maintaining a comparable level of control effort.
Although the variability in EKG-MPPI is somewhat larger for the control input, the reduction in peak wind speed remains consistent across runs.Based on these additional experiments, we will revise the manuscript to include uncertainty estimates (mean ± standard deviation) for all reported performance metrics.
--------------------------------------------------
Referee Comment 2:
In the Lorenz-96 experiments, EKG-MPPI uses lambda = 0.1 and an EnKC-informed proposal distribution, while vanilla MPPI uses lambda = 0.7 and uninformed Gaussian sampling for magnitude and location. As a result, the reported difference does not isolate the effect of the proposed guidance alone. I suggest clarifying that vanilla MPPI is included as a simple reference only, and that the principal comparison is with EnKC.Response:
We thank the Referee for this important comment. We agree with the Referee’s interpretation. In the revised manuscript, we now clarify explicitly that vanilla MPPI is included only as a simple reference with an uninformed prior, and not as the principal baseline for evaluating the proposed EnKC-guided mechanism.
The principal comparison in the paper is with EnKC, since EKG-MPPI is designed as an EnKC-based nonlinear refinement.Moreover, to further strengthen the baseline comparison, we additionally introduced two more informative baselines in the revised manuscript: a standalone Bayesian optimization (BO) method and a BO-informed MPPI method. The standalone BO baseline directly optimizes the location and magnitude of the single-step perturbation under the same setting, and is therefore a relevant derivative-free comparator for the present low-dimensional problem.
The BO-informed MPPI baseline, in turn, uses BO to provide adaptive prior information for MPPI, thereby offering a stronger adaptive baseline than vanilla MPPI with a static uninformed proposal. These additional baselines make it possible to distinguish more clearly between the benefit of adaptive proposal guidance and the benefit of the optimizer itself. We have revised the corresponding discussion to make the role of each baseline explicit and to avoid overstating the significance of the vanilla MPPI comparison.--------------------------------------------------
Referee Comment 3:
Figure 4's caption states that the evaluation metric is the percentage difference ((EKG - EnKC)/EnKC) x 100, but the figure itself shows raw maximum wind speed and raw mean input magnitude by scenario. Either this text should be removed and replaced with useful information about the figure, or the results about the percentage difference should be added.Response:
We thank the Referee for pointing out this inconsistency. The original caption was misleading. In the revised manuscript, we correct the caption of Figure 4 so that it directly describes the quantities actually shown in the figure, i.e., the raw maximum wind speed and the raw mean input magnitude for each scenario. We remove the incorrect statement about the percentage difference.--------------------------------------------------
Referee Comment 4:
The figures could be made more informative. In figure 2, 3 and 5, the axes lack clear units or precise information on the quantities shown, and some legends remain too limited to guide the reader which needs to infer from the text outside of the figure. E.g. "X variables" is not defined, and most colour bars come without text. I would encourage the authors to improve the figure labels, legends, and captions so that each figure can be understood more directly on its own.Response:
We agree that these figures should be more self-contained. In the revised manuscript, we have improved not only the captions but also the axis labels, legends, and colour-bar annotations. The revised versions of these figures are provided in the Supplementary Material.
For Figure 2, we replaced the ambiguous y-axis label "X variables" with "State Value" and clarified in the caption that the plotted quantity is the Lorenz-96 state variable X_k, i.e. a dimensionless model variable. We also made the distinction between the full percentile range and the zoomed upper-tail range more explicit.For Figure 3, we added explicit labels for the quantities shown on the axes, namely the temperature parameter and the sample number, and descriptive colour-bar titles for all panels: "Number of Extreme Values," "Number of Interventions," and "Mean Intervention Magnitude." We also revised the caption so that the meanings of panels (a)-(c) can be understood without referring to the main text.
For Figure 5, we added spatial axis labels with units ("Distance [km]") and a labelled colour bar ("Wind Speed [m/s]"). In addition, the caption now explicitly states that the plotted field is the wind-speed magnitude and that the white rectangle denotes the target region.
We believe that these revisions make the figures substantially more informative and easier to interpret independently of the main text.
--------------------------------------------------
Referee Comment 5:
Also, here are some very small corrections that would make the manuscript easier to read and more complete:- In Algorithm 1, the Require line lists H but not H_c, although line 5 calls EnKC with H_c.
- In Eq. (1), the model equation uses q_{t-1}, while the text below defines q_t as the model error.
- In Eq. (10), the text just above introduces an ensemble {x_t^{alpha(i)}}_{i=1}^m, but Eq. (10) then uses x_{t+T_EnKC}^a as if it were a single, already defined state. It is unclear from a quick read whether this denotes one ensemble member, the ensemble mean, or another quantity.
- The switch from the generic EnKC horizon T_EnKC to T_c in the Lorenz-96 section could be made more explicit.
- The Gaussian sampling notation should be clarified. In Sect. 3, the second Gaussian parameter is introduced as a variance/covariance term, but in the numerical sections the paper sets g_mag(u_t^EnKC) = ||u_t^EnKC||_1 / 2 in Lorenz-96 and g_mag(u_t^EnKC) = ||u_t^EnKC||_1 in SQG, which seems like a natural choice for a standard deviation. It would help to clarify whether the authors mean m ~ N(mu, g_mag) or m ~ N(mu, g_mag^2).
- In the paragraph introducing Figure 5, the order “no-control, EKG-MPPI, and EnKC” should be made consistent with figure 5, where the order is “no-control, EnKC, and EKG-MPPI.”
- The introduction refers to Henderson as an early 4D-Var-based study, but the corresponding reference is missing from the bibliography.
- For the first CSE citation, the original CSE paper from 2022 is better suited.
- The manuscript currently cites Sawada (2024a) as a preprint; the authors may also update that entry to the published journal version.Response:
We thank the Referee for these careful and helpful comments. We revise the manuscript accordingly:
- In Algorithm 1, we add H_c to the Require line and explicitly define it as the operator that maps the state variables to the control criteria.
- In Eq. (1) and the text below it, we correct the notation so that the model-error term is written consistently as q_{t-1}.
- After introducing the analysis ensemble in Eqs. (8) and (9), we now state explicitly that \bar{x}_t^a denotes the analysis ensemble mean at time t. We also clarify the notation around Eq. (10) by specifying that the state at t + T_EnKC refers to the ensemble-mean forecast obtained by propagating the analysis ensemble to that lead time.
- In the Lorenz-96 section, we replace T_c with T_EnKC in order to unify the notation with the generic EnKC formulation.
- We clarify that the second parameter of the Gaussian distribution denotes a variance/covariance. Accordingly, we revise the definitions of g_mag so that they are given on the variance scale. Specifically, we now define g_mag(u_t^EnKC) = ||u_t^EnKC||_1^2 / 4 in the Lorenz-96 experiments and g_mag(u_t^EnKC) = ||u_t^EnKC||_1^2 in the SQG experiments.
- We correct the ordering in the paragraph introducing Figure 5 so that it matches the figure itself, namely: no control, EnKC, and EKG-MPPI.
- We add the missing Henderson reference to the bibliography: Henderson et al. (2005), "A 4D-Var study on the potential of weather control and exigent weather forecasting".
- We replace the first CSE citation with the original paper by Miyoshi and Sun (2022), as suggested.
- We update the Sawada (2024a) reference from the preprint entry to the published journal version in SOLA.--------------------------------------------------
Closing remarkWe thank the Referee again for the thoughtful comments. We believe that these revisions have improved the clarity and readability of the manuscript.
Sincerely,
Haru Kuroki, Kazumune Hashimoto, Yuki Uehara, Yohei Sawada, Duc Le, and Masashi Minamide
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AC1: 'Reply on RC2', Kazumune Hashimoto, 25 Mar 2026
Peer review completion
This manuscript presents a novel and elegant hybrid control strategy, termed EKG-MPPI, aimed at spatially localized interventions in chaotic geophysical flows. By combining the statistical data assimilation capabilities of Ensemble Kalman Control (EnKC) with the nonlinear, derivative-free optimization of Model Predictive Path Integral (MPPI) control, the authors provide a compelling framework for "Control Simulation Experiments". From a systems and control engineering perspective, using the EnKC to generate a sparse, physics-informed prior to guide the sampling distribution of the MPPI is a brilliant approach to solving the sample inefficiency inherent to high-dimensional state spaces. The method is entirely matrix-free and avoids the computation of adjoint models or gradients, which is highly desirable for chaotic, non-linear atmospheric models. However, while the theoretical hybridization is sound, the current implementation simplifies the temporal complexity of the control problem to a single-step actuation. This reduces the framework to a static spatial optimization problem rather than true multi-step trajectory optimization. Furthermore, the comparison with baseline methods lacks rigorous hyperparameter tuning, which somewhat weakens the performance claims. The manuscript is a strong proof-of-concept, but addressing the specific points below would significantly improve its rigor and impact.
Specific Comments :
- Single-Step Actuation vs. Sequential Trajectory Optimization: The authors explicitly state that the control perturbation is applied only at the first rollout step, with no further input provided during the remaining prediction horizon ($T_{MPPI}$). Because the problem is reduced to optimizing a single initial impulse, the use of MPP, a method designed for sequential, multi-step trajectory optimization, seems algorithmically oversized. The authors should discuss why MPPI was chosen over standard derivative-free black-box optimization algorithms (e.g., CMA-ES or Bayesian Optimization) which are highly efficient for static, single-step parameter search. If the intention is to lay the groundwork for future multi-step actuation, this should be stated more explicitly as the primary justification for using the MPPI formalism.
- Fairness in Baseline Comparison (Hyperparameter Optimization): In Section 4.1.1, the proposed EKG-MPPI is compared against a "vanilla MPPI" baseline. However, the setup for the vanilla MPPI utilizes arbitrarily fixed hyperparameters without any apparent tuning. Specifically, the authors set the temperature parameter to $\lambda = 0.7$ for the vanilla MPPI, whereas EKG-MPPI operates with $\lambda = 0.1$. Furthermore, the vanilla MPPI relies on a completely static sampling distribution (variance of 0.5 for magnitude and 20.0 for location). Because EKG-MPPI inherently adapts both the mean and variance of its sampling distribution using the EnKC output, it naturally possesses a massive advantage. To ensure a rigorous comparative analysis, the baseline MPPI should ideally undergo a degree of hyperparameter tuning or at least a sensitivity analysis similar to the one performed for EKG-MPPI in Figure 3 to firmly demonstrate that its poorer performance is due to the lack of EnKC guidance rather than sub-optimal static parameters.
- Computational Scalability and Profiling: The results in Table 2 show that EKG-MPPI is the most computationally expensive method tested. While the authors correctly point out that MPPI rollouts are "embarrassingly parallel" and benefit directly from GPU acceleration, a brief computational profiling of the EKG-MPPI execution time would be highly valuable. Specifically, providing a breakdown of the time spent computing the EnKC prior versus the time spent evaluating the MPPI rollouts would clarify where the true bottleneck lies. Given the authors' stated intention in the conclusion to scale this approach to multi-actuator and multi-step scenarios, this profiling would help readers assess the operational feasibility of scaling the framework.
Technical Corrections :
- Equation notations: In Section 3.2, Step 3, the embedding functions $f_{loc}$, $g_{loc}$, $f_{mag}$, and $g_{mag}$ are introduced. It would be helpful to the reader to add a brief sentence clarifying their exact mathematical definitions or references earlier in the text, before they are fully explicitly defined for the experiments in Equations (39) and (50).