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Ensemble score filter with image inpainting for data assimilation in tracking surface quasi-geostrophic dynamics with partial observations

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read With inpainting, a training-free diffusion filter tracks turbulent flow from only 5% observed grid points.

desk verdict Useful proof-of-concept for training-free diffusion DA with partial observations, but the numerical validation needs error bars, a no-assimilation baseline, and a softened conclusion. read the letter →

arxiv 2501.12419 v1 pith:5GX3TU7O submitted 2025-01-20 physics.ao-ph cs.LGphysics.data-anphysics.flu-dynstat.ML

Siming Liang , Hoang Tran , Feng Bao , Hristo G. Chipilski , Peter Jan van Leeuwen , Guannan Zhang This is my paper · ORCID
classification physics.ao-phcs.LGphysics.data-anphysics.flu-dynstat.ML MSC 68Q2568R1068U05
keywords dataassimilationensemblescorefilterdiffusionmodelimageinpaintingpartialobservationssurfacequasi-geostrophicLETKFtraining-freegenerative
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a training-free diffusion-based ensemble score filter (EnSF), extended with image inpainting, can accurately track the chaotic surface quasi-geostrophic (SQG) model using only 5% of grid points observed through a nonlinear arctangent operator—conditions under which the tuned Local Ensemble Transform Kalman Filter (LETKF) fails completely. The key move is to split the Bayesian update: observed variables are updated through the diffusion score, while unobserved variables are filled in by inpainting from the updated observed field rather than by covariance-based cross-variable updates. If the claim holds, it would give geoscience data assimilation a method that handles high-dimensional, nonlinear, partial-observation problems without covariance localization, re-training, or parameter fine-tuning. The paper demonstrates this on 16 scenarios at 64×64 and 256×256 resolutions, with total root-mean-square error around 4.6 in the hardest case and near 2.8 at the higher resolution.

What carries the argument

The ensemble score filter (EnSF) is a training-free diffusion method in which the prior filtering distribution is transported to a standard Gaussian by a forward SDE and sampled by solving a reverse SDE whose score function is evaluated by Monte Carlo from the prior ensemble; observations enter by adding the likelihood gradient to the score with a time-damping weight h(t). The likelihood gradient has a nonzero block only for observed variables, so the unobserved block is untouched. The paper's addition is an inpainting step after each score update: the Navier-Stokes inpainting propagates isophotes from observed into unobserved pixels by a vorticity transport equation; biharmonic inpainting solves a fourth-order PDE for a smooth extension; and dictionary-learning inpainting represents the full state sparsely in a discrete-cosine basis thresholded by the prior ensemble and optimizes the coefficients against the updated observed field with a total-variation penalty. This inpainting step plays the role that the cross-covariance matrix plays in Kalman filters: it spreads observational information from observed to unobserved variables.

What would settle it

Run the same EnSF+inpainting on a system where unobserved variables are dynamically decoupled from observed variables (e.g., two independent SQG layers with observations only from one layer); if the unobserved-layer RMSE is not identical to a pure forecast, the inpainting step is fabricating correlation. Alternatively, in a 64×64, 5%-observation, nonlinear case, compare the EnSF+inpainting ensemble against a particle filter with a modest number of particles; if the ensemble spread/RMSE ratio deviates strongly from 1 or the filtered trajectories diverge, the inpainted ensemble is not a posterior sample.

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Extended reading notes

Core claim

The central discovery is that the zero likelihood gradient on unobserved variables in EnSF can be compensated, not by learning a covariance, but by treating the observed field as a partial image and reconstructing the full state with classical or dictionary-based inpainting at every filtering step. In the most demanding scenario tested—64×64 grid, 12-hour assimilation interval, 5% fixed-point observations through a nonlinear arctangent operator—the original EnSF and LETKF both diverge, while EnSF with any of the three inpainting variants keeps the total RMSE around 4.6 and visually tracks the main SQG dynamics. At 256×256 resolution the same setup yields total RMSE near 2.8. The paper further claims that inpainting is not always needed: with 3-hour assimilation and 25% observation coverage, EnSF without inpainting is already accurate, and the inpainting step can add small errors. The authors frame the result as a proof of concept for using image inpainting as a substitute for covariance-based cross-variable updates in data assimilation.

Load-bearing premise

The load-bearing premise is that the unobserved part of the state can be recovered from the observed part by image inpainting; since the likelihood gradient on unobserved variables is zero, the paper replaces their Bayesian update with a deterministic reconstruction and gives no proof that this produces samples from the posterior conditional on the observations.

Editorial extensions

If this is right

  • If correct, EnSF with inpainting offers a training-free alternative to localization-tuned ensemble Kalman filters for nonlinear, sparsely observed systems.
  • The method needs no re-tuning when resolution changes from 64×64 to 256×256, while LETKF's transferred hyperparameters degrade sharply.
  • Dictionary-learning inpainting is the better choice under 5% observations; PDE-based inpainting is competitive when 25% is observed.
  • The approach separates observed-state accuracy from unobserved-state accuracy, which could let operational systems report observed-variable reliability even when full-state RMSE is moderate.
  • Combining EnSF's nonlinear observation handling with a cheap deterministic reconstruction step suggests a path to assimilating radar or satellite operators similar to arctangent compression.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The inpainting step implicitly defines a conditional model for the unobserved given the observed field; if that conditional is misspecified, the ensemble will be overconfident, so the spread/RMSE ratio near 1 reported in the paper should be checked case by case.
  • The dictionary method's DCT basis is generic; replacing it with physics-informed bases (e.g., vorticity or wave modes) could improve the 5%-observation regime while preserving training-free operation.
  • The failure of inpainting in dense-observation cases suggests a hybrid controller could switch off inpainting when the forecast step is short and observation density is high, gaining EnSF-only accuracy without losing robustness.
  • Testable extension: apply the same split update to a primitive-equation or coupled model where the observed and unobserved variables live on different physical fields (e.g., sea-surface temperature observed, subsurface velocity unobserved), where the inpainting analogy to images is weaker.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes an extension of the ensemble score filter (EnSF) to data assimilation with partial observations by integrating image inpainting. At each filtering step, the observed state variables are updated via the EnSF likelihood score, and the unobserved variables are reconstructed using one of three inpainting techniques: biharmonic, Navier-Stokes, or dictionary-learning-based inpainting. The method is tested on the surface quasi-geostrophic (SQG) model in 16 scenarios that vary spatial resolution, observation operator (linear vs. arctangent), assimilation frequency, and observation sparsity (5% and 25%). The main claim is that EnSF with inpainting can track the SQG dynamics even under 5% observations and nonlinear observation operators, where the LETKF fails, with average total RMSE around 4.6 in the hardest 64x64 case (C7) and around 2.8 at 256x256 (C15). The paper also includes a comparison against the original EnSF without inpainting and a fine-tuned LETKF, and reports ensemble spread/RMSE ratios as a measure of uncertainty estimation.

Significance. If the main claim is fully supported, the paper makes a valuable methodological contribution: it demonstrates a training-free, localization-free diffusion-based data assimilation approach that can handle nonlinear observation operators and highly sparse observations in a chaotic geophysical model. The breadth of tested scenarios (16 cases across two resolutions, two observation operators, two frequencies, and two observation densities) is a strength, and the public availability of the code supports reproducibility. However, the current evidence has two important gaps: the absolute RMSE values are not calibrated against any unconstrained reference (free run, persistence, or climatology), and all experiments appear to be single realizations with no error bars. These gaps make the central 'can track' claim underdetermined and weaken the quantitative comparisons among inpainting variants. The theoretical justification for replacing the Bayesian update of unobserved states with inpainting is also not provided. With these issues addressed, the paper could be a solid proof-of-concept for a new class of DA methods.

major comments (4)
  1. [Section 4.2.2, Table 3] The central claim that EnSF with inpainting 'can track' the SQG dynamics is not calibrated by an appropriate reference scale. The RMSE values of about 4.6 (C7) and 2.8 (C15) are reported without comparison to any unconstrained baseline, such as a free run, persistence, or climatological forecast, for the same truth trajectory, observation mask, and error metric. Since the biharmonic, Navier-Stokes, and DCT-threshold inpainting methods are all spatial smoothing operations, a static or slowly varying field could achieve a moderate RMSE without tracking the time evolution of unobserved scales. Please add such a baseline or use a skill score (e.g., anomaly correlation) to calibrate the absolute RMSE and substantiate the word 'tracking'.
  2. [Section 4.2.2, Figures 6 and Table 3] All reported results appear to come from a single realization (one draw of observation noise and one model-error trajectory). Without repeated seeds, multiple observation-noise replicates, or error bars, the differences among EnSF+DL, EnSF+Bi, and EnSF+NS (e.g., 4.46 versus 4.61 in C7) cannot be distinguished from run-to-run variability. Please report statistics over at least several independent runs, or otherwise demonstrate that the relative ordering of methods is robust to stochastic variability.
  3. [Section 3.1 and Sections 3.2-3.3] The methodological core of the paper, which replaces the Bayesian update of unobserved states with image inpainting, lacks a theoretical or empirical justification that the resulting ensemble approximates the posterior p(X_{n+1}|Y_{1:n+1}). Eq. (3.7) shows that the likelihood gradient for unobserved variables is zero, and then the unobserved states are reconstructed by inpainting from the observed states without any proof that this corresponds to a Bayesian update. The paper should either prove that this reconstruction is asymptotically correct under specific priors (e.g., smoothness or sparsity priors) or demonstrate in a controlled synthetic experiment that the inpainting-based update yields calibrated posterior samples (e.g., via rank histograms or coverage probabilities).
  4. [Section 4.2.1] The LETKF at 256x256 resolution is evaluated with hyperparameters fine-tuned only at 64x64 and then transferred without retuning. This is acknowledged in the text, but it gives EnSF with inpainting a substantial advantage in the 256x256 experiments because the LETKF is not adapted to the finer resolution. The paper should either tune LETKF at 256x256 for at least a subset of cases (e.g., a coarse grid search or a few trial runs), or explicitly reframe the 256x256 comparisons as demonstrating sensitivity to parameter transfer rather than as a head-to-head method comparison. The current statement in Section 4.2.2 that 'LETKF performs poorly, as expected' conflates these two interpretations.
minor comments (5)
  1. [Section 3.3, Step 3] The values of the DCT threshold nu and the total variation weight gamma are not reported in the paper. Please provide these values, either in the main text or in a table, for reproducibility.
  2. [Section 3.4] The title of Section 3.4 contains a typo: 'inpaintitng' should be 'inpainting'.
  3. [Introduction and various figures] There are several typographical errors, including 'dyanmics' in the Introduction, 'pursue' in the Figure 2 caption, 'obserable' in the Figure 3 caption, and 'Spacial' in Section 4.1. These should be corrected.
  4. [Table 3] The text states that 'The best-performing method in each case is highlighted,' but the table as rendered does not show any highlighting. Please ensure the highlighting is visible or remove the statement.
  5. [Section 4.4] The paper's own admission that inpainting introduces additional error in the ideal cases (C2, C6, C10, C14) is a useful and honest limitation statement. Consider making this limitation more prominent in the abstract or conclusions, since it tempers the generality of the main claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EnSF-with-inpainting results are measured against an external synthetic truth and an independent LETKF baseline, and no fitted parameter is renamed as a prediction.

full rationale

The paper's central claim is empirical: EnSF with inpainting tracks the SQG model under partial, nonlinear observations with lower total RMSE than LETKF or EnSF without inpainting. Nothing in the construction forces this result. The EnSF score update in Eq. (3.6) uses the likelihood gradient from Eq. (3.7), which is zero on unobserved variables, and the inpainting step then reconstructs unobserved states from observed states using biharmonic, Navier-Stokes, or DCT-dictionary methods. These reconstructions are not fit to the truth trajectory; they use only current observed values and prior samples from the model. The reported RMSE numbers are compared against the synthetic nature run, so the evaluation is external to the method's own outputs. The self-citations to the authors' prior EnSF papers [2,3,4] supply the score-based diffusion framework and the choices of alpha_t and beta_t, but the target result (tracking SQG under partial observations) is not assumed by those references, and the key novelty of this paper, the inpainting extension, is implemented and tested here rather than imported. Table 3 also shows that inpainting can hurt relative to plain EnSF in the dense-observation cases C2, C6, C10, and C14, which further demonstrates that the inpainting results are not forced by construction. The absence of an unconstrained reference RMSE and of replicate runs is a correctness/statistical concern, not a circularity concern.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities, forces, or conserved quantities. It borrows score-based diffusion, PDE inpainting, and dictionary learning as existing tools. The main uncharged assumption is that image inpainting can stand in for the Bayesian dependence between observed and unobserved state variables, which is a heuristic rather than a derived property.

free parameters (5)
  • DCT coefficient threshold nu = not reported
    Defines the sparse mask in dictionary inpainting Step 2; controls how many cosine coefficients are kept per prior sample.
  • Total variation weight gamma = not reported
    Regularization in the dictionary inpainting objective Step 3; balances data fidelity against image smoothness.
  • Time damping function h(t) = h(t)=1-t (example)
    Controls how quickly likelihood information enters the reverse SDE; chosen monotonically decreasing, not derived from Bayes.
  • LETKF localization and RTPS inflation parameters = grid-searched at 64x64, transferred to 256x256
    Baseline filter tuning strongly affects the comparison and is expensive; the transfer is itself an assumption.
  • Ensemble size K = 20
    Number of ensemble members used for Monte Carlo score estimation and spread calculation; a hand-chosen experiment setting.
assumptions (5)
  • domain assumption Observation noise eps is Gaussian with covariance R
    Used to write the likelihood in Eq. (2.7) and the score update in Eqs. (3.6)-(3.7).
  • standard math The prior distribution can be represented by a diffusion bridge with alpha_t=1-t and beta_t^2=t
    Taken from the authors' prior EnSF papers [2,3]; not rederived or independently verified here.
  • ad hoc to paper Inpainting reproduces the conditional distribution of unobserved states given observed states
    Core of the proposed method; enters at Section 3.1 when the zero likelihood gradient for unobserved variables is replaced by image reconstruction.
  • domain assumption A dictionary learned from prior samples remains valid for the current state
    Dictionary inpainting Steps 1-2 use the current forecast ensemble as the source of basis functions; temporal transfer is assumed in Section 3.3.
  • domain assumption The SQG model is a representative testbed for geophysical turbulence
    Motivates conclusions for operational weather and ocean data assimilation; stated in Section 2.1.

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Cite this review

Pith. "Pith review of Ensemble score filter with image inpainting for data assimilation in tracking surface quasi-geostrophic dynamics with partial observations." pith.science (2026). https://pith.science/paper/5GX3TU7O

@misc{pith2026250112419,
  author       = {Pith},
  title        = {Pith review of: Ensemble score filter with image inpainting for data assimilation in tracking surface quasi-geostrophic dynamics with partial observations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5GX3TU7O}},
  note         = {Machine review of arXiv:2501.12419}
}
read the original abstract

Data assimilation plays a pivotal role in understanding and predicting turbulent systems within geoscience and weather forecasting, where data assimilation is used to address three fundamental challenges, i.e., high-dimensionality, nonlinearity, and partial observations. Recent advances in machine learning (ML)-based data assimilation methods have demonstrated encouraging results. In this work, we develop an ensemble score filter (EnSF) that integrates image inpainting to solve the data assimilation problems with partial observations. The EnSF method exploits an exclusively designed training-free diffusion models to solve high-dimensional nonlinear data assimilation problems. Its performance has been successfully demonstrated in the context of having full observations, i.e., all the state variables are directly or indirectly observed. However, because the EnSF does not use a covariance matrix to capture the dependence between the observed and unobserved state variables, it is nontrivial to extend the original EnSF method to the partial observation scenario. In this work, we incorporate various image inpainting techniques into the EnSF to predict the unobserved states during data assimilation. At each filtering step, we first use the diffusion model to estimate the observed states by integrating the likelihood information into the score function. Then, we use image inpainting methods to predict the unobserved state variables. We demonstrate the performance of the EnSF with inpainting by tracking the Surface Quasi-Geostrophic (SQG) model dynamics under a variety of scenarios. The successful proof of concept paves the way to more in-depth investigations on exploiting modern image inpainting techniques to advance data assimilation methodology for practical geoscience and weather forecasting problems.

Figures

Figures reproduced from arXiv: 2501.12419 by the authors.

Figure 1
Figure 1. The proposed workflow. unobserved states based on updated observed states, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Illustration of PDE-based inpainting methods based on the Navier-Stokes [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Illustration of dictionary-learning-based inpainting for one snapshot of the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (36 more)
Figure 4
Figure 4. Figure 4: Illustration of nonlinear observation using arctangent operator. Subfigure (b) [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: The LETKF fine tuning chart at the 64 × 64 resolution, where the case labels are defined in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Comparison of the total RMSE for the 16 scenarios listed in Table [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Comparison of the uncertainty estimation for the 16 scenarios listed in Table [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Results of (C7): (a) The total RMSE including all state variables, (b) the RMSE of the observed variables (c) RMSE of only unobserved variables. (a) Truth (b) LETKF (c) EnSF Only (d) EnSF+Bi (e) EnSF+DL (f) EnSF+NS [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Results of (C7): snapshot at filtering step 100. LETKF (b) diverges to undefined values, EnSF without inpainting shows a total failure, EnSF inpainting can track the major dynamics. they lose some small-scale details. However, compared to (C7), the finer 256×256 grid r…
Figure 10
Figure 10. Figure 10: Results of (C15): The total RMSE (a) includes all state points, the observed RMSE (b) only includes observed state points, and the unobserved RMSE (c) only includes unobserved state points. (a) Truth (b) LETKF (c) EnSF Only (d) EnSF + Bi (e) EnSF + DL (f) EnSF + NS […
Figure 11
Figure 11. Figure 11: Results of (C15): Snapshot at filtering step 100. LETKF (b) captures the overall shape of the major dynamics, missing most extreme events (dark red and blue regions) and mismatching small-scale details. EnSF without inpainting shows a total failure, EnSF inpainting ca…
Figure 12
Figure 12. Figure 12: Results of (C6): The total RMSE (a) includes all state points, the observed RMSE (b) only includes observed state points, and the unobserved RMSE (c) only includes unobserved state points. 4.3.4. The case study on (C14). We increase the resolution from the coarse 64 ×…
Figure 13
Figure 13. Figure 13: Results of (C6): Snapshot at filtering step 100. LETKF (b) shows a total failure. EnSF captures both major dynamics and small-scale detail. EnSF inpainting can track the major dynamics but lose some small-scale details. 3, is achieved by EnSF combined with Biharmonic …
Figure 14
Figure 14. Figure 14: Results of (C14): The total RMSE (a) includes all state points, the observed RMSE (b) only includes observed state points, and the unobserved RMSE (c) only includes unobserved state points. 4.4. Additional discussion. In this work, we conducted 16 experiments to explo…
Figure 15
Figure 15. Figure 15: Results of (C14): snapshot at filtering step 100. LETKF (b) shows a total failure. EnSF and EnSF inpainting both capture major dynamics and small-scale detail. observation sparsity [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]
Figure 16
Figure 16. Figure 16: RMSE of (C1) (a) Truth (b) LETKF (c) EnSF Only (d) EnSF+Bi (e) EnSF+DL (f) EnSF+NS [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]
Figure 17
Figure 17. Figure 17: Snapshot at filtering step 100 of (C1) (a) Total RMSE (b) observed RMSE (c) unobserved RMSE [PITH_FULL_IMAGE:figures/full_fig_p028_17.png]
Figure 18
Figure 18. Figure 18: RMSE of (C2) [PITH_FULL_IMAGE:figures/full_fig_p028_18.png]
Figure 19
Figure 19. Figure 19: Snapshot at filtering step 100 of (C2) (a) Total RMSE (b) observed RMSE (c) unobserved RMSE [PITH_FULL_IMAGE:figures/full_fig_p029_19.png]
Figure 20
Figure 20. Figure 20: RMSE of (C3) (a) Truth (b) LETKF (c) EnSF Only (d) EnSF+Bi (e) EnSF+DL (f) EnSF+NS [PITH_FULL_IMAGE:figures/full_fig_p029_20.png]
Figure 21
Figure 21. Figure 21: Snapshot at filtering step 100 of (C3) [PITH_FULL_IMAGE:figures/full_fig_p029_21.png]
Figure 22
Figure 22. Figure 22: RMSE of (C4) (a) Truth (b) LETKF (c) EnSF Only (d) EnSF+Bi (e) EnSF+DL (f) EnSF+NS [PITH_FULL_IMAGE:figures/full_fig_p030_22.png]
Figure 23
Figure 23. Figure 23: Snapshot at filtering step 100 of (C4) (a) Total RMSE (b) observed RMSE (c) unobserved RMSE [PITH_FULL_IMAGE:figures/full_fig_p030_23.png]
Figure 24
Figure 24. Figure 24: RMSE of (C5) [PITH_FULL_IMAGE:figures/full_fig_p030_24.png]
Figure 25
Figure 25. Figure 25: Snapshot at filtering step 100 of (C5) (a) Total RMSE (b) observed RMSE (c) unobserved RMSE [PITH_FULL_IMAGE:figures/full_fig_p031_25.png]
Figure 26
Figure 26. Figure 26: RMSE of (C8) (a) Truth (b) LETKF (c) EnSF Only (d) EnSF+Bi (e) EnSF+DL (f) EnSF+NS [PITH_FULL_IMAGE:figures/full_fig_p031_26.png]
Figure 27
Figure 27. Figure 27: Snapshot at filtering step 100 of (C8) [PITH_FULL_IMAGE:figures/full_fig_p031_27.png]
Figure 28
Figure 28. Figure 28: RMSE of (C9) (a) Truth (b) LETKF (c) EnSF Only (d) EnSF+Bi (e) EnSF+DL (f) EnSF+NS [PITH_FULL_IMAGE:figures/full_fig_p032_28.png]
Figure 29
Figure 29. Figure 29: Snapshot at filtering step 100 of (C9) (a) Total RMSE (b) observed RMSE (c) unobserved RMSE [PITH_FULL_IMAGE:figures/full_fig_p032_29.png]
Figure 30
Figure 30. Figure 30: RMSE of (C10) [PITH_FULL_IMAGE:figures/full_fig_p032_30.png]
Figure 31
Figure 31. Figure 31: Snapshot at filtering step 100 of (C10) (a) Total RMSE (b) observed RMSE (c) unobserved RMSE [PITH_FULL_IMAGE:figures/full_fig_p033_31.png]
Figure 32
Figure 32. Figure 32: RMSE of (C11) (a) Truth (b) LETKF (c) EnSF Only (d) EnSF+Bi (e) EnSF+DL (f) EnSF+NS [PITH_FULL_IMAGE:figures/full_fig_p033_32.png]
Figure 33
Figure 33. Figure 33: Snapshot at filtering step 100 of (C11) [PITH_FULL_IMAGE:figures/full_fig_p033_33.png]
Figure 34
Figure 34. Figure 34: RMSE of (C12) (a) Truth (b) LETKF (c) EnSF Only (d) EnSF+Bi (e) EnSF+DL (f) EnSF+NS [PITH_FULL_IMAGE:figures/full_fig_p034_34.png]
Figure 35
Figure 35. Figure 35: Snapshot at filtering step 100 of (C12) (a) Total RMSE (b) observed RMSE (c) unobserved RMSE [PITH_FULL_IMAGE:figures/full_fig_p034_35.png]
Figure 36
Figure 36. Figure 36: RMSE of (C13) [PITH_FULL_IMAGE:figures/full_fig_p034_36.png]
Figure 37
Figure 37. Figure 37: Snapshot at filtering step 100 of (C13) (a) Total RMSE (b) observed RMSE (c) unobserved RMSE [PITH_FULL_IMAGE:figures/full_fig_p035_37.png]
Figure 38
Figure 38. Figure 38: RMSE of (C16) (a) Truth (b) LETKF (c) EnSF Only (d) EnSF+Bi (e) EnSF+DL (f) EnSF+NS [PITH_FULL_IMAGE:figures/full_fig_p035_38.png]
Figure 39
Figure 39. Figure 39: Snapshot at filtering step 100 of (C16) [PITH_FULL_IMAGE:figures/full_fig_p035_39.png]

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Forward citations

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