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Generative Atmospheric Super-Resolution from Heterogeneous In Situ Observations through Composable Interfaces

Yang Xu, Dibyajyoti Chakraborty, Haiwen Guan, Sen Wang, Romit Maulik

arXiv:2609.29027v1cs.LGphysics.ao-ph

TL;DR

Sparse, heterogeneous in situ observations are difficult to combine consistently with models trained on regularly gridded atmospheric states. This paper introduces source-specific interfaces for composing those observations with one pretrained atmospheric diffusion prior, reducing ERA5-referenced RMSE by 9.24% across all 13 CONUS state variables versus R-only conditioning. The results also show lower CRPS and reduced errors at held-out aircraft and surface-station targets.

  • Problem

    In situ sources differ in geometry and sampling density, while generative atmospheric models learn gridded states, leaving their consistent combination in a common Bayesian inference framework unresolved.

    Source: §1 · p. 1

  • Method

    Composable source-specific interfaces connect radiosonde, aircraft, and surface observations to one pretrained 13-variable atmospheric prior through choices about retained measurements, state mapping, residual counting, and likelihood weighting.

    Source: §1 · p. 2, §10 · p. 26

  • Results

    9.24% mean reduction in ERA5-referenced RMSE across all 13 CONUS state variables for R+A+S versus R-only conditioning, with lower CRPS and reduced errors at held-out aircraft and surface-station targets.

    Source: §10 · p. 26

  • Takeaways & Limitations

    Explicit composable interfaces provide a practical route to reuse a pretrained atmospheric prior across heterogeneous in situ sources while keeping each source’s connection to the gridded state inspectable.

    Source: §10 · p. 27

    The evaluation uses one pretrained 13-variable ERA5 prior and sampler settings, and does not establish superiority over classical data-assimilation systems or other diffusion-based approaches.

    Source: §9.4 · p. 26

Abstract

from arXiv · show

Atmospheric observations are sparse, heterogeneous, and unevenly distributed, whereas many generative atmospheric models learn distributions over regularly gridded multivariate states. Once pretrained, diffusion models can supply atmospheric priors that can be combined with observation-derived likelihood factors in a Bayesian formulation. However, these observation sources differ substantially in geometry and sampling density, complicating the consistent use of their observations within a common inference framework. Here, we formulate this reconstruction problem as generative atmospheric super-resolution and introduce composable observation interfaces for conditioning a single pretrained 13-variable atmospheric diffusion model. The interfaces convert sparse radiosonde (R), clustered aircraft (A), and dense irregular surface-station (S) observations into source-specific likelihood factors that specify where observations constrain the gridded state, how residuals are counted under uneven sampling, and how strongly each source guides posterior sampling. We developed the aircraft and surface observation interfaces using 2019 observations and evaluated the selected interfaces throughout 2020 without further tuning. Compared with reconstructions conditioned only on radiosonde observations, the composed R+A+S interface reduces RMSE evaluated against ERA5 by $9.24\%$ across all 13 state variables over the CONUS domain. The aircraft and surface factors provide complementary improvements in upper-air and surface variables. The R+A+S combination also lowers the Continuous Ranked Probability Score (CRPS), while evaluations at held-out aircraft and surface-station observations show reduced prediction errors. Together, these results demonstrate a modular route for conditioning a pretrained atmospheric generative prior on heterogeneous in situ observations without retraining the underlying model.

1 Introduction

The paper frames reconstruction from heterogeneous in situ observations as generative atmospheric super-resolution, using source-specific likelihood interfaces to connect observations to a gridded state. It evaluates composing radiosonde, aircraft, and surface-station factors around one pretrained prior without retraining.

1 Introduction

  • The central problem is how to combine likelihood factors consistently when in situ sources differ in geometry and sampling density, while the pretrained prior models gridded-state relationships.The reconstruction is framed as generative atmospheric super-resolution in a Bayesian formulation.Source: §1 · pp. 1–2
  • The interfaces average aircraft and surface reports within each model-grid cell by variable, preventing densely reported cells from receiving greater weight solely because they contain more reports.Source: §1 · p. 2
  • The selected aircraft and surface interfaces are evaluated at 723 matched 2020 analysis times without further tuning, and R+A+S reduces ERA5-referenced RMSE versus R-only across all 13 CONUS state variables.The 2019 development cases were used to select the interfaces and calibrate their likelihood parameters.Source: §1 · p. 3
  • The study evaluates source-specific radiosonde, aircraft, and surface-station interfaces composed around one pretrained 13-variable diffusion model without retraining it when source combinations change.The interfaces specify retained measurements, state mapping, residual counting, and likelihood weighting.Source: §1 · pp. 2–3

2 Related Work

Related work spans learned observation conditioning, inference-time conditioning of pretrained diffusion priors, analysis and forecasting systems, and models that operate directly in observation space. This paper positions its contribution as explicit, composable likelihood interfaces for heterogeneous observations around one fixed gridded-state prior.

Learned conditioning

  • Learned conditioning methods train the relationship between observations and gridded atmospheric states, unlike this paper’s inference-time combination of explicit likelihood factors with a separately pretrained prior.Examples include SLAMS, MODS, and PhyDA.Source: §2 · pp. 3–4

Inference-time conditioning

  • Inference-time conditioning reuses pretrained diffusion priors with measurement constraints, but standard DPS may not faithfully approximate the posterior and can reduce sample diversity.Alternative posterior-sampling routes include sequential Monte Carlo and MCMC.Source: §2 · p. 4
  • Prior atmospheric work demonstrated zero-shot conditioning of the 13-variable prior on coarse ERA5 fields, radiosondes, and emulator outputs, but did not make heterogeneous in situ likelihood construction the focus.Source: §2 · p. 4

Observation-based analysis and forecasting; Models operating in observation space

  • Observation-based analysis and forecasting systems produce analyses or forecasts, while observation-space models forecast observations; this study instead reconstructs a gridded state at one analysis time.Source: §2 · p. 4

Positioning

  • The contribution is organizing retained observations, state mapping, residual counting, and likelihood weighting as composable interfaces, rather than proposing any one ingredient in isolation.Classical data assimilation provides precedents for observation operators, averaging dense observations, and error weighting.Source: §2 · pp. 4–5

3 Problem Setup

The setup combines a fixed pretrained atmospheric prior with source-specific observation likelihoods. During sampling, DPS evaluates their constraints on a denoised state and pulls the resulting guidance back to the noisy sampler state.

3 Problem Setup

  • The prior is a pretrained 13-variable ERA5 denoiser with a fixed land-sea-mask condition, used without retraining; the radiosonde interface follows prior work, while aircraft and surface interfaces are developed here.Source: §3 · p. 5
  • Each source contributes a likelihood factor with retained standardized observations, an observation operator, and a weight controlling its contribution to the combined likelihood.The atmospheric state contains 13 channels on a 128 × 256 grid.Source: §3 · p. 5, Table 1 · p. 7
  • Residual counting gives equal total weight to each active channel and equal weight to retained observations or occupied cells within that channel.The matrix W_m encodes this counting structure.Source: §3 · p. 6
  • At each reverse-diffusion noise level, the observation sources are treated as conditionally independent given the atmospheric state, and their likelihood factors are combined in clean-state coordinates.Source: §3 · p. 6
  • DPS evaluates the observation likelihood at the denoised estimate and differentiates it with respect to the current noisy state to supply the sampler’s likelihood correction.The source-specific observation construction determines the targets, operators, and residual-counting structure used in that likelihood.Source: §3 · p. 6

4 Observation Sources

The study combines global radiosonde profiles with CONUS aircraft and surface-station observations, whose sampling patterns and target variables differ. Aircraft and surface reports are filtered and mapped to selected state variables before cell-level aggregation where specified.

Observation sources

  • The three sources have distinct sampling geometries, and aircraft and surface reports are averaged by variable within each model-grid cell so each occupied cell contributes one residual per variable.The study uses global radiosonde observations and CONUS aircraft and surface observations.Source: §4 · p. 7

Radiosonde observations (R)

  • Radiosondes provide global vertical profiles and serve as the reference configuration for evaluating aircraft and surface additions.The interface maps surface temperature and winds to near-surface channels and reported 500/850 hPa variables to corresponding pressure-level channels.Source: §4 · p. 8

Aircraft observations (A)

  • Aircraft observations are individual reports along flight trajectories, with the final interface retaining ACARS direct, MDCRS/ARINC, and Canadian AMDAR streams after quality and physical-range checks.Eligible reports are assigned to 500 or 850 hPa channels when within ±25 hPa of the level.Source: §4 · pp. 8–9

Surface-station observations (S)

  • Surface-station observations constrain 2 m temperature and 10 m winds within CONUS using quality-controlled reports within ±60 minutes of each 00/12 UTC analysis time.Retained reports are averaged within model-grid cells.Source: §4 · p. 9

5 Composable Observation-Interface Factorization

Source-specific interfaces define how observations become likelihood factors, allowing radiosonde, aircraft, and surface-station constraints to be composed. Aircraft and surface reports are aggregated by grid cell, while DPS maps and combines their likelihood guidance during sampling.

5 Composable Observation-Interface Factorization

  • Each interface specifies retained observations, state mapping, residual counting, and likelihood weighting; together these choices determine where and how strongly each source constrains the gridded state.Radiosonde targets remain station-level, while aircraft and surface targets are cell means; active channels and retained targets receive equal total and within-channel weights, respectively.Source: §5 · p. 9

5.1 Radiosonde Interface (R)

  • The radiosonde operator selects the observed variable and pressure-level channel and bilinearly interpolates it at each station, retaining one residual per available target.This preserves the locations of sparse radiosonde observations rather than averaging them onto the model grid.Source: §5.1 · p. 9

5.2 Equal-Cell Interfaces for Aircraft and Surface Stations (A/S)

  • Aircraft and surface reports are averaged within occupied model-grid cells, with one residual per occupied cell and observed variable, so report density alone does not increase likelihood influence.Aircraft targets cover temperature and horizontal winds at 500 and 850 hPa; surface targets cover t2m, u10, and v10.Source: §5.2 · p. 10, Figure 3 · p. 10

5.3 Inference-Time Composition of Observation Factors

  • Once each source’s targets, mapping, residual weights, and likelihood parameters are specified, it contributes an additive negative log-likelihood term.Radiosonde parameters are inherited from prior work; aircraft and surface parameters are selected for this study.Source: §5.3 · p. 11

5.4 From Observation Residuals to Runtime DPS Guidance

  • At each reverse step, DPS differentiates observation mismatch through the denoiser, sums source-specific likelihood gradients, clips the combined gradient once, and applies it to the reverse-solver update.The state mapping determines where a source acts directly, while the denoiser pullback couples that information across the gridded multivariate state.Source: §5.4 · pp. 11–12

1. Estimate the clean atmospheric state

Figure 4 depicts how source-specific observation interfaces produce a correction to the reverse-solver update.

1. Estimate the clean atmospheric state

  • Each observation source defines its retained targets, state mapping, and residual-counting rule.These choices produce likelihood gradients that are mapped through a shared denoiser, combined, clipped, and applied as a correction to the reverse-solver update.Source: Figure 4 · p. 12
  • The shared denoiser maps the resulting likelihood gradients before they are combined and clipped.The clipped gradients correct the reverse-solver update.Source: Figure 4 · p. 12
  • The corrected reverse-solver update completes the conditioning sequence shown in Figure 4.The figure links source-specific residual construction to the sampler update.Source: Figure 4 · p. 12

6 Evaluation Design and Metrics

The evaluation holds the pretrained prior and sampling setup fixed while developing aircraft and surface interfaces on 2019 cases and testing them independently in 2020. Matched comparisons, held-out observations, probabilistic metrics, and temporally aware uncertainty estimates assess the selected configurations.

6 Evaluation Design and Metrics

  • The 2020 evaluation uses 723 matched analysis times without further tuning, after aircraft and surface interfaces were selected and calibrated on 24 prespecified 2019 development cases.The prior was not trained or fine-tuned on either year's cases.Source: §6 · p. 13, §6.2 · p. 14

6.1 Fixed Prior, Matched Sampling, and Evaluation Domains

  • All configurations use the same 13-variable prior and sampler to generate 16-member ensembles with 50 denoising steps and matched member seeds.Gridded reconstruction errors are calculated against ERA5 on the CONUS, global, and global-excluding-CONUS domains; CONUS is a fixed 798-point latitude–longitude grid mask.Source: §6.1 · p. 13

6.2 Interface Development and Independent-Year Evaluation

  • Development comparisons selected ±25-hPa pressure windows and equal-cell residual counting for aircraft, equal-cell counting for surface stations, and the ACARS direct, MDCRS/ARINC, and Canadian AMDAR streams.Additional non-TAMDAR streams supplied no or sparse eligible observations and did not materially change 2019 errors, while including TAMDAR increased them.Source: §6.2 · pp. 13–14, Table 3 · p. 14
  • The nine R+A+S candidate settings were compared against the best mean all-variable RMSE reduction relative to R-only, with paired-month bootstrap intervals guiding selection among indistinguishable candidates.The procedure selected the candidate closest to the tested parameter-range center, using lower combined likelihood weight as a tie-breaker.Source: §6.2 · pp. 13–14

6.3 Selected Interfaces and Conditioning Comparisons

  • Matched 2020 comparisons evaluate R-only, R+A, R+S, and R+A+S with the same prior, sampler settings, and random seeds.R+A and R+S isolate the respective additions to R-only conditioning; R+A+S evaluates their combined contribution.Source: §6.3 · p. 14

6.4 Held-Out Observation Evaluation

  • Held-out evaluations use 24 seasonally distributed 2020 cases, conditioning on 80% of aircraft or surface targets and evaluating predictions on the excluded 20%.The source-specific comparisons are R+A+S versus R+A for surface targets, and R+A+S versus R+S for aircraft targets.Source: §6.4 · p. 15

6.5 Metrics

  • RMSE changes are paired against R-only at matching times, variables, and domains; CRPS assesses probabilistic skill, while spread–skill ratios, rank histograms, and percentile coverage diagnose dispersion and calibration.Lower CRPS is preferred, and negative RMSE changes indicate lower error than R-only conditioning.Source: §6.5 · pp. 15–16

6.6 Temporal Uncertainty under Serial Correlation

  • Annual uncertainty intervals use a 14-day moving-block bootstrap with 10,000 replicates to account for temporal dependence among successive analysis times.The blocks preserve missing-data gaps, and all variable errors at each sampled time remain together.Source: §6.6 · p. 16

7 Results

Across the 2020 evaluation, adding aircraft and surface observations to radiosondes reduced CONUS errors, with the improvements supported by temporal, held-out-observation, and probabilistic evaluations.

7.1 Annual RMSE Changes from Composed Observation Sources

  • R+A+S reduced CONUS ERA5-referenced RMSE relative to R-only by 9.24% across all variables, 14.17% for surface targets, and 9.30% for upper-air temperature and wind.R+A and R+S separately reduced upper-air and surface-group RMSE by 6.74% and 13.31%, respectively.Source: §7.1 · p. 16, Figure 5 · p. 17, Table 4 · p. 18

7.2 Temporal Robustness under the 14-Day Moving-Block Bootstrap

  • −9.24% was the mean CONUS RMSE change for R+A+S across all 13 variables, with a 95% moving-block-bootstrap interval of [−9.57, −8.92]%.All 13 variable-specific CONUS intervals were below zero, while the all-variable mean outside CONUS remained near zero.Source: §7.2 · p. 19, Table 5 · p. 19

7.3 Prediction at Held-Out Observations

  • Adding retained observations reduced RMSE at excluded surface-station observations by 13.50% and at excluded aircraft observations by 11.71%.The held-out observations came from the same observing systems as the retained data, so this evaluates within-system interpolation rather than transfer to a new system.Source: §7.3 · p. 20, Table 6 · p. 20, Figure 9 · p. 20

7.4 Probabilistic Evaluation

  • R+A+S reduced CONUS CRPS relative to R-only by 9.98% across all variables, 15.01% for surface targets, and 10.45% for upper-air temperature and wind.All three 95% bootstrap intervals remained below zero.Source: Table 7 · p. 21, §7.4 · p. 21

8 Mechanism Diagnostics: From Source Likelihood Gradients to Runtime Guidance

The diagnostics trace source-specific gradients from clean-state variable support through denoiser pullback to runtime guidance. In one selected trajectory, the denoiser couples initially distinct source directions, and guidance relationships change during sampling.

8 Mechanism Diagnostics: From Source Likelihood Gradients to Runtime Guidance

  • Before denoiser pullback, A assigns 100% of its squared clean-state gradient norm to upper-air temperature and wind variables, while S assigns 100% to surface-targeted variables.Their all-variable A–S cosine is zero because they act directly on disjoint variables; nonzero cosines involving R have magnitudes no greater than 0.23 in this case.Source: §8 · p. 22, Figure 10 · p. 22
  • After denoiser-Jacobian pullback, a pairwise runtime-guidance cosine reaches 0.93 at high noise, although the three pairwise cosines move back toward zero near the final step.At that final step, S accounts for 90.80% of the summed raw-guidance norms.Source: §8 · pp. 22–23, Figure 11 · p. 23
  • The diagnostics show source-guidance norms at the first, middle, and final reverse steps on a shared logarithmic scale, with much smaller raw-guidance magnitudes at the first step than near the trajectory’s end.The norms are computed before shared summation and clipping.Source: §8 · p. 23, Figure 12 · p. 24
  • These diagnostics describe one selected trajectory and do not decompose the final applied correction among sources or establish annual mechanism statistics.All three are computed before the shared summation and clipping operation.Source: §8 · p. 23

9 Discussion

R+A+S combines the distinct upper-air and surface benefits of aircraft and surface observations, while explicit interfaces control which reports contribute and how they are weighted. The evaluation supports improved CONUS reconstruction and probabilistic skill, with regional and modeling limits.

9.1 Source-Specific Improvements and Shared-State Coupling

  • R+A+S reduces CONUS RMSE for all 13 variables and improves on the stronger single-source combination by 3.89, 0.86, and 2.56 percentage points across all-variable, surface-targeted, and upper-air groups.The 9.24% all-variable reduction reflects information from both A and S; a single-case diagnostic shows a coupling pathway but not necessarily the same gradient alignment at every time.Source: §9.1 · p. 25

9.2 Why Explicit Observation Interfaces Matter

  • Including TAMDAR increased reconstruction errors, while counting one averaged residual per occupied cell and variable outperformed counting every A or S report under tested likelihood settings.Additional reports may refine cell-level targets, but report count alone should not give dense routes or station clusters greater likelihood influence.Source: §9.2 · p. 25

9.3 What the Evaluation Establishes

  • The 9.98% all-variable CRPS reduction and bootstrap intervals below zero for all grouped and per-variable RMSE changes support persistent annual improvements, while held-out gains concern interpolation within the same systems.Adding S to R+A reduced RMSE by 13.50% at excluded S cells; adding A to R+S reduced RMSE by 11.71% at excluded A cell–variable targets.Source: §9.3 · pp. 25–26
  • The 9.24% all-variable RMSE reduction is supported over CONUS, but the global-domain change outside CONUS is near zero and its interval includes zero.The interfaces directly constrain CONUS; this evidence does not establish a global improvement.Source: §9.3 · p. 26

9.4 Limitations and Future Directions

  • Conclusions are conditioned on one pretrained 13-variable ERA5 prior and sampler setting; aircraft guidance uses only 500 and 850 hPa levels with fixed pressure windows.The likelihood does not explicitly model correlated errors across sources or nearby observations beyond within-cell averaging, and mechanism diagnostics cover one analysis time and trajectory.Source: §9.4 · p. 26

10 Conclusion

The paper reconstructs a gridded atmospheric state from heterogeneous in situ observations by connecting each source to a pretrained diffusion prior through an explicit interface. Evaluated without further tuning in 2020, the composed interfaces reduce error against ERA5 and at held-out observation targets.

10 Conclusion

  • Source-specific interfaces define retained measurements, state mapping, residual aggregation, and likelihood weighting, allowing different observation geometries and densities to be combined without retraining the prior.Held-out evaluations also show reduced errors at aircraft and surface-station targets excluded from conditioning.Source: §10 · pp. 26–27
  • R+A+S lowers ERA5-referenced RMSE across all 13 CONUS state variables by a mean 9.24% and also lowers CRPS relative to R-only conditioning.The selected interfaces were evaluated at 723 matched 2020 analysis times after development and calibration on 24 prespecified 2019 cases, without further tuning.Source: §10 · p. 26

Code Availability

The supplementary material documents code, sampling and data processing, interface selection, and metric definitions. Development comparisons use prespecified 2019 cases, while the selected configuration and parameters are reported separately.

Code Availability

  • The repository contains research code, compact numerical summaries, selected interface settings, 2020 evaluation manifests, and processing, sampling, and evaluation code.Source: Code Availability · p. 28

A Sampling Parameters and Selected Analysis Times

  • The sampling appendix gives the matched-member base seed as 17 and EDM schedule values σmin = 0.005, σmax = 80, and ρ = 7, and identifies dates used for single-case examples.The single-case dates are distinct from the annual and held-out evaluations supporting quantitative conclusions.Source: A Sampling Parameters and Selected… · p. 32

B.1 Aircraft Data Coverage, Timing, and Quality Control

  • The processed 2020 aircraft archive covers 732 nominal analysis times and contains 15,850,004 raw reports; the annual evaluation uses 723 times with all required inputs.After reporting-system selection and pressure matching, 889,024 reports remained; all passed the MADIS time check.Source: §B.1 · p. 32

B.2 Monthly Differences between Observations and ERA5

  • Monthly RMSE between retained surface and aircraft observations and ERA5 varied within comparable overall ranges in 2020, with no evident month-specific failure in the processed inputs.These diagnostics assess input stability, not posterior performance, and did not retune the interfaces.Source: §B.2 · p. 32, Figure 13 · p. 33, Figure 14 · p. 33

C Interface Selection and Calibration Details

  • The interface-selection comparisons use the same 24 prespecified 2019 development cases, with ERA5-referenced RMSE changes generally reported against R-only conditioning.The final selection uses all-variable RMSE as its primary score and surface-targeted and upper-air groups as secondary outcomes.Source: C Interface Selection and Calibration… · p. 34, Table 8 · p. 34

C.1 Interface-Design Comparisons

  • The selected A design used ACARS direct, MDCRS/ARINC, and Canadian AMDAR reports, ±25-hPa matching, and equal-cell mean residuals; the selected S residual representation was also equal-cell averaging.Under common pre-calibration parameters, A reduced mean development RMSE by 6.697% across all variables and 9.902% for upper-air temperature and wind; S reduced it by 5.828% across all variables and 13.887% for surface-targeted variables.Source: Figure 15 · p. 35, §C.1 · p. 35, Figure 16 · p. 36

C.2 Likelihood-Parameter Calibration and R+A+S Selection

  • The selected R+A+S configuration produced an 11.961% mean all-variable RMSE reduction in 2019, 0.037 percentage points below the largest observed reduction.Six candidates were statistically indistinguishable under paired-month bootstrap intervals; the selection rule chose the candidate nearest the tested parameter-range center, then used lower combined weight as a tie-breaker.Source: §C.2 · p. 36

D.1 Metric Definitions and Aggregation

  • Annual grouped RMSE and CRPS aggregation avoids averaging incompatible physical units; physical-unit annual RMSE is reported separately, with humidity RMSE in 10^-3 kg kg^-1.Source: §D.1 · p. 37, Table 9 · p. 38

D.2 Additional Temporal-Uncertainty Results

The annual uncertainty analyses find lower RMSE across all variables, but probabilistic diagnostics still show underdispersed ensembles. Spectral changes are variable-specific rather than uniform across scales, and coverage is interpreted comparatively.

D.2 Additional Temporal-Uncertainty Results

  • All 13 per-variable R+A+S RMSE-change intervals lie below zero over CONUS under the 14-day moving-block bootstrap.Figure 17 shows per-variable means and 95% intervals; negative changes indicate lower RMSE than R-only conditioning.Source: §D.2 · p. 38, Table 10 · p. 38, Figure 17 · p. 39

D.3 Spatial Spectral Shape

  • R+A+S leaves broad spectral shape close to R-only for t2m and q850, while moving several intermediate and long v500 scales toward ERA5, without uniformly suppressing relative power.The normalized-power diagnostic measures relative spectral shape, not absolute variance or phase accuracy; the limited CONUS grid constrains interpretation near grid scale.Source: §D.3 · p. 40, Figure 18 · p. 40

E.1 Finite-Ensemble Coverage Interpretation

  • The 5th–95th percentile coverage’s 0.90 value is a large-ensemble reference, not the exact expected coverage for this finite ensemble, so coverage is compared alongside other diagnostics.The interval is interpreted comparatively across conditioning configurations and together with spread–skill ratios and rank frequencies, rather than as a standalone calibration test.Source: §E.1 · p. 40

E.2 Annual Probabilistic Diagnostics

  • R+A+S lowers CRPS and ensemble-mean RMSE relative to R-only, but both configurations remain underdispersed, with spread–skill ratios below one and rank frequencies elevated near outer ranks.The fair-CRPS sensitivity leaves the reported configuration ordering unchanged.Source: §E.2 · p. 41, Figure 19 · p. 41, Figure 20 · p. 41
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