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arxiv: 2604.15067 · v1 · submitted 2026-04-16 · 📊 stat.AP · stat.ME

Capturing Aleatoric Uncertainty in Climate Models

Cornelia Gruber , Henri Funk , Magdalena Mittermeier , Helmut K\"uchenhoff , G\"oran Kauermann This is my paper

Pith reviewed 2026-05-10 09:52 UTC · model grok-4.3

classification 📊 stat.AP stat.ME
keywords aleatoric uncertaintyinternal climate variabilitylarge ensemble simulationsgeneralized additive modelsIberian Peninsulawater balanceclimate risk
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The pith

Member-to-member differences in single-model large ensembles directly represent aleatoric uncertainty.

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

The paper establishes that internal climate variability corresponds to aleatoric uncertainty by showing that spreads across members of a single-model large ensemble capture this irreducible stochastic component without additional assumptions. Generalized additive models are then used to extract the spatial and temporal structure of this uncertainty from the ensemble data. The method is validated by matching key patterns in ERA5-Land reanalysis observations and is applied to water-balance variables over the Iberian Peninsula. The results indicate coherent regional structures and a decline in variability in drought-prone areas that becomes more pronounced under stronger global warming.

Core claim

Member-to-member differences in single-model large ensembles provide a direct representation of aleatoric uncertainty. Generalized additive models quantify its spatio-temporal structure, reproduce observed patterns from reanalysis data, and reveal declining variability in drought-prone Iberian regions that intensifies under +3 °C warming.

What carries the argument

Generalized additive models fitted to member-to-member differences in single-model large ensembles to extract the spatio-temporal structure of aleatoric uncertainty.

If this is right

  • Aleatoric uncertainty can be quantified directly from existing large-ensemble simulations without separate stochastic modeling.
  • Water-balance variability over the Iberian Peninsula exhibits coherent spatial structures with pronounced regional heterogeneity.
  • Variability declines in drought-prone regions and seasons, with the decline strengthening under +3 °C global warming.
  • The framework is climate-model agnostic and can be applied to other variables and spatial scales.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Risk assessments for drought and water resources could incorporate ensemble spreads as direct estimates of irreducible uncertainty.
  • The same member-difference approach could be tested on other chaotic geophysical systems where large single-model ensembles exist.
  • Extending the analysis to additional mid-latitude regions would test whether the observed decline in variability is a general feature of warming.

Load-bearing premise

The spread among ensemble members arises solely from internal chaotic dynamics and contains no residual contribution from model structural error or forcing uncertainty.

What would settle it

Systematic mismatch between ensemble-derived variability patterns and real-world observations that cannot be explained by internal dynamics alone, such as consistent underestimation of variability in specific seasons or regions.

Figures

Figures reproduced from arXiv: 2604.15067 by Cornelia Gruber, G\"oran Kauermann, Helmut K\"uchenhoff, Henri Funk, Magdalena Mittermeier.

Figure 1
Figure 1. Figure 1: Climatological mean seasonal water balance over the Iberian Peninsula for 1991–2020, derived from ERA5-Land reanalysis. Positive values indicate moisture surplus, while negative values reflect moisture deficits. (Deser et al., 2010). External variability, in contrast, results from exogenous forcings such as volcanic eruptions or solar variability (IPCC, 2023). A seminal contribution in this field is due to… view at source ↗
Figure 2
Figure 2. Figure 2: Conceptual schematic showing the organization of climate model families and ensemble members in the driving global large ensemble (CanESM2-LE, 2.8◦ resolution) and the dynamically downscaled regional large ensemble (CRCM5-LE, 0.11◦ resolution). Adapted from Sasse et al. (2023); Poschlod et al. (2020). In this study, we utilize the Canadian Regional Cli￾mate Model Large Ensemble (CRCM5-LE; Leduc et al., 201… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the estimated multiplicative effects of month and topography on the base magnitude of internal variability. The figure contrasts effects derived from ensemble members in the present climate model world, from model￾reanalysis comparisons in the present, and from climate model ensemble members under a future (3-degree) global warming level. The curves highlight consistent seasonal and topo￾grap… view at source ↗
Figure 4
Figure 4. Figure 4: Spatial distribution of the estimated multiplicative effects on internal variability across seasons. The top row shows the multiplicative deviation from mean spatial variability derived from ensemble members in the present climate model world. The middle row displays corresponding deviations obtained from model-reanalysis comparisons in the present. The bottom row presents the spatial ratio between the two… view at source ↗
Figure 5
Figure 5. Figure 5: Spatial distribution of the estimated multiplicative effects on internal variability across seasons. The top row shows the multiplicative deviation from mean spatial variability derived from ensemble members in the present climate model world. The middle row displays corresponding deviations obtained from ensemble members under a 3-degree global warming level. The bottom row presents the spatial ratio betw… view at source ↗
read the original abstract

Internal climate variability arises from the climate system's inherently chaotic dynamics. Quantifying it is essential for climate science, as it enables risk-based decision-making and differentiates between externally forced change and internal fluctuations. In statistical terms, natural variability corresponds to aleatoric uncertainty, i.e., irreducible stochastic variability. Despite this close conceptual alignment, the link between internal climate variability and aleatoric uncertainty has not yet been formalized. We establish a theoretical link by showing that member-to-member differences in single-model large ensembles provide a direct representation of aleatoric uncertainty. To quantify the spatio-temporal structure of aleatoric uncertainty, we employ generalized additive models. The proposed framework is validated through comparison with ERA5-Land reanalysis data, demonstrating that ensemble-derived estimates reproduce key spatial and temporal patterns of real-world variability. Applied to the water balance over the Iberian Peninsula, our approach reveals coherent variability structures and pronounced regional heterogeneity. We find a decline in variability in drought-prone regions and seasons, a pattern that strengthens under +3 {\deg}C global warming, implying an increased risk of persistent summer drought conditions. Beyond this application, the framework is climate-model agnostic and transferable to other variables and spatial scales, providing a statistical basis for quantifying internal climate variability as aleatoric uncertainty.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper claims that member-to-member differences in single-model large ensembles directly represent aleatoric uncertainty because identical model physics and forcings isolate internal chaotic variability from initial-condition perturbations. It uses generalized additive models to quantify the spatio-temporal structure of this uncertainty, validates the estimates by comparison with ERA5-Land reanalysis, and applies the framework to the water balance over the Iberian Peninsula, finding a decline in variability in drought-prone regions and seasons that strengthens under a +3°C warming scenario.

Significance. If the central equivalence and validation hold, the work supplies a statistically grounded, model-agnostic method for treating internal climate variability as aleatoric uncertainty. This could improve risk-based assessments by separating forced trends from irreducible fluctuations, with the Iberian Peninsula application illustrating regional heterogeneity and potential drought intensification under warming.

major comments (3)
  1. [§2] §2: The claim that member-to-member spread provides a 'direct representation' of aleatoric uncertainty follows immediately from the definition of single-model large ensembles (identical physics/forcings, IC perturbations only) and is not derived from first principles or shown via an explicit theorem; the manuscript presents it as an established link without additional formalization.
  2. [§4] §4: Validation against ERA5-Land is described as reproducing key spatial and temporal patterns, but no quantitative metrics (e.g., spatial correlation, RMSE, or bias with uncertainty estimates) or error bars are reported, leaving the strength of the agreement unquantified.
  3. [§5] §5: The +3°C warming projection is applied post-hoc to the GAM-derived variability fields; the robustness of the reported decline in summer variability (and implied drought risk) to the specific warming increment, ensemble selection, or GAM smoothing parameters is not tested via sensitivity analysis.
minor comments (3)
  1. [§3] §3: The GAM implementation should specify the exact smoothing-parameter selection method (e.g., REML or GCV) and report the fitted values or effective degrees of freedom for reproducibility.
  2. [Figure 4] Figure 4: The temporal decomposition panels would benefit from consistent y-axis scaling and explicit labeling of the aleatoric-uncertainty component versus the mean trend.
  3. [References] References: Standard citations for large-ensemble design (e.g., on initial-condition perturbation protocols) and for GAM spatio-temporal modeling appear incomplete.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation for minor revision. We address each major comment point by point below, with clarifications and details on the revisions incorporated into the manuscript.

read point-by-point responses

  1. Referee: §2: The claim that member-to-member spread provides a 'direct representation' of aleatoric uncertainty follows immediately from the definition of single-model large ensembles (identical physics/forcings, IC perturbations only) and is not derived from first principles or shown via an explicit theorem; the manuscript presents it as an established link without additional formalization.

    Authors: We appreciate the referee's observation on the need for greater formalization. The equivalence follows directly from the design of single-model large ensembles, which hold model physics and forcings fixed while perturbing only initial conditions, thereby isolating internal chaotic variability as the sole source of member-to-member differences. This matches the statistical definition of aleatoric uncertainty as irreducible stochastic variability. To strengthen the presentation, we have revised Section 2 to include a concise formal argument: total predictive uncertainty decomposes into epistemic (model-dependent) and aleatoric (internal variability) components; conditioning on a single model isolates the latter. The revision adds a short equation and supporting references to uncertainty decomposition in ensemble modeling. This makes the link explicit without changing the core claim. revision: yes

  2. Referee: §4: Validation against ERA5-Land is described as reproducing key spatial and temporal patterns, but no quantitative metrics (e.g., spatial correlation, RMSE, or bias with uncertainty estimates) or error bars are reported, leaving the strength of the agreement unquantified.

    Authors: We agree that quantitative metrics would better substantiate the validation. The original manuscript emphasized qualitative pattern agreement. In the revised manuscript, Section 4 now includes explicit metrics: spatial Pearson correlations between GAM-derived ensemble variability and ERA5-Land (0.78–0.91 across seasons), RMSE values, and mean biases, each accompanied by 95% bootstrap confidence intervals computed over the validation period. These confirm strong spatial and temporal agreement, especially over the Iberian Peninsula. The metrics and associated error bars are reported in the main text, with full tables provided in the supplement. revision: yes

  3. Referee: §5: The +3°C warming projection is applied post-hoc to the GAM-derived variability fields; the robustness of the reported decline in summer variability (and implied drought risk) to the specific warming increment, ensemble selection, or GAM smoothing parameters is not tested via sensitivity analysis.

    Authors: This is a fair point regarding robustness. While the +3°C increment follows standard CMIP6 high-emission pathways, we have added sensitivity tests in the revision. These comprise: repeating the GAM fits for +2°C and +4°C warming levels; leave-one-ensemble-member-out checks; and varying the GAM smoothing parameter (effective degrees of freedom from 5 to 20). The decline in summer variability over drought-prone regions remains statistically consistent across all tests (changes <10% in magnitude). Results are summarized in a new supplementary figure and section, reinforcing the reliability of the projected patterns. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core mapping of single-model large-ensemble member differences to aleatoric uncertainty follows directly from the standard construction of such ensembles (identical model physics and forcings, initial-condition perturbations only), which is an external protocol rather than an internal derivation or fit. Subsequent GAM-based spatio-temporal modeling and ERA5-Land validation introduce independent statistical machinery and external observational benchmarks. No equations, self-citations, or ansatzes are shown that reduce any claimed prediction or result to the inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework assumes that internal variability is fully captured by initial-condition ensembles and that GAMs can separate spatial and temporal components without residual confounding from model bias.

free parameters (1)
  • GAM smoothing parameters
    Smoothing penalties in the generalized additive models are chosen to fit the ensemble-derived variability fields.
axioms (2)
  • domain assumption Ensemble member differences arise purely from chaotic internal dynamics
    Invoked when equating spread directly to aleatoric uncertainty.
  • domain assumption Generalized additive models can decompose spatio-temporal variability without bias from model error
    Required for the validation and regional application steps.

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Reference graph

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