2.1. Climate model and observational data

We use monthly near-surface air temperature (tas) and precipitation (pr) from 25 CMIP6 global climate models, covering the historical period 1948–2014. The selected models span a wide range of modeling centers and include both Earth system models and coupled general circulation models. All model outputs are regridded to a common $ 1{}^{\circ}\times 1{}^{\circ} $ latitude–longitude grid to ensure spatial consistency across models.

As observational reference, we use the ERA5 reanalysis product, which provides globally complete and physically consistent estimates of atmospheric variables based on data assimilation of satellite and in situ observations. ERA5 is treated as the target for supervised learning and model evaluation. For both CMIP6 and ERA5, we compute global area-weighted means using cosine latitude weights.

2.2. Climatological feature construction

To characterize systematic differences among climate models, we compute each model’s long-term climatological mean for temperature and precipitation. Specifically, for each CMIP6 model $ m $ , we define

(2.1) $$ {\overline{x}}_m=\frac{1}{T}\sum \limits_{t=1}^T\frac{\sum_{i,j}{w}_{ij}\hskip0.1em {x}_{m,t,i,j}}{\sum_{i,j}{w}_{ij}}, $$

where $ {x}_{m,t,i,j} $ denotes the variable value at time $ t $ and grid cell $ \left(i,j\right) $ , and $ {w}_{ij}=\cos \left({\phi}_{ij}\right) $ are latitude-based area weights. These climatological summaries provide a low-dimensional representation of each model’s large-scale behavior and are used as features for clustering.

We intentionally adopt a low-dimensional representation based on long-term global climatological means. Our goal is to construct a coarse, interpretable partition that can be directly related to how ensemble learning redistributes weights. A scalar global mean is the simplest such feature and the most transparent for tracing learned weights. This choice reflects a trade-off between representational richness and interpretability: while higher-dimensional features (e.g., regional or decadal statistics) may capture more detailed structure, they also make it more difficult to attribute learned weights back to physically meaningful differences among models. We acknowledge this limitation explicitly; exploring richer feature spaces is an important direction for future work.

2.3. Train–validation–test splitting

All supervised learning experiments follow a time-ordered split to avoid temporal leakage. The full time series is divided into a training period (1948–1998), a validation period (1999–2005), and a test period (2006–2014). Machine learning models are trained on the training set, hyperparameters are selected using the validation set, and all reported performance metrics are computed on the held-out test period.

2.4. Ensemble learning methods

We compare four ensemble post-processing strategies. As a baseline, the arithmetic multi-model ensemble (AMME) prediction is computed as the simple average across all CMIP6 models:

(2.2) $$ {\hat{y}}_t^{\mathrm{AMME}}=\frac{1}{M}\sum \limits_{m=1}^M{x}_{m,t}, $$

where $ M $ denotes the number of models and $ {x}_{m,t} $ the prediction from model $ m $ at time $ t $ .

We then fit a Ridge regression model to learn a global linear weighting of models,

(2.3) $$ {\hat{y}}_t^{\mathrm{Ridge}}=\sum \limits_{m=1}^M{w}_m\hskip0.1em {x}_{m,t}+b, $$

where weights $ {w}_m $ are estimated by minimizing squared error subject to an $ {\mathrm{\ell}}_2 $ regularization penalty. Hyperparameter $ \alpha $ is selected from $ \left\{\mathrm{0.01,0.1,1.0,10.0,100.0}\right\} $ on the validation set. This formulation yields an interpretable mapping between individual climate models and their contributions to the ensemble prediction. The $ {\mathrm{\ell}}_2 $ penalty distributes weight broadly across models, preventing any single model from dominating; a sparser alternative such as LASSO ( $ {\mathrm{\ell}}_1 $ ) would perform automatic model selection and is a natural direction for future work.

To capture nonlinear interactions among models, we also apply an RF regressor using model outputs as input features. We use $ {n}_{\mathrm{estimators}}=600 $ trees with $ \min \_\mathrm{samples}\_\mathrm{leaf}=2 $ . Variable importance scores are computed as the mean decrease in impurity (MDI) averaged over all trees and normalized to sum to 1; they quantify the relative contribution of each CMIP6 model to the RF prediction.

Finally, we implement a continuous gating mixture model that learns a convex combination of Ridge and RF predictions. The gated prediction is

(2.4) $$ {\hat{y}}_{t,i,j}^{\mathrm{Gate}}={g}_{t,i,j}\hskip0.1em {\hat{y}}_{t,i,j}^{\mathrm{Ridge}}+\left(1-{g}_{t,i,j}\right)\hskip0.1em {\hat{y}}_{t,i,j}^{\mathrm{RF}}, $$

where subscripts $ \left(i,j\right) $ denote grid cell and $ t $ denotes time (global mean analyses use scalar $ {g}_t $ without spatial indices). For each validation time step, the optimal gate value is

(2.5) $$ {g}_t^{\ast }=\mathrm{clip}\;\left(\frac{\left({y}_t-{\hat{y}}_t^{\mathrm{RF}}\right)\left({\hat{y}}_t^{\mathrm{Ridge}}-{\hat{y}}_t^{\mathrm{RF}}\right)}{{\left({\hat{y}}_t^{\mathrm{Ridge}}-{\hat{y}}_t^{\mathrm{RF}}\right)}^2+\varepsilon },0,1\right), $$

where $ \varepsilon ={10}^{-12} $ ensures numerical stability. A second Ridge regression model is then fitted on the validation set to predict $ {g}_t^{\ast } $ from the standardized CMIP6 inputs $ {\mathbf{x}}_t $ . At test time, the gate $ {g}_t $ is evaluated by applying this fitted model to the (standardized) test inputs, so no test-period information is used in the gate.

The effective per-model gating weight is defined as

(2.6) $$ {w}_m^{\mathrm{Gate}}=\overline{g}\hskip0.1em {\tilde{w}}_m^{\mathrm{Ridge}}+\left(1-\overline{g}\right)\hskip0.1em {I}_m^{\mathrm{RF}}, $$

where $ \overline{g} $ is the time-averaged gate value, $ {\tilde{w}}_m^{\mathrm{Ridge}}=\mid {w}_m\mid /{\sum}_{m^{\prime }}\mid {w}_{m^{\prime }}\mid $ is the normalized Ridge weight, and $ {I}_m^{\mathrm{RF}} $ is the RF importance for model $ m $ . These effective weights are used in the spatial analyses of Section 5.

2.5. Evaluation metrics

Model performance is evaluated using multiple complementary metrics, including root mean square error (RMSE), mean absolute error (MAE), Pearson correlation, Nash–Sutcliffe efficiency (NSE), variance ratio between predictions and observations, and predictive skill relative to the arithmetic ensemble:

(2.7) $$ \mathrm{Skill}=1-\frac{{\mathrm{RMSE}}_{\mathrm{method}}}{{\mathrm{RMSE}}_{\mathrm{AMME}}}. $$

Positive skill indicates improvement over AMME.

For precipitation, we additionally report distribution-aware metrics. Wet-day frequency RMSE is the RMSE of the fraction of time steps exceeding a threshold of 0.1 mm day $ {}^{-1} $ , computed between each method’s monthly predictions and ERA5 over the test period. Upper-tail RMSE is the RMSE restricted to time steps where ERA5 exceeds its 90th percentile of the observed precipitation distribution, providing a measure of extreme-event representation.

Trust entropy is defined as

(2.8) $$ {H}_{i,j}=-\sum \limits_{m=1}^M{p}_{m,i,j}\hskip0.1em \log \left({p}_{m,i,j}+\varepsilon \right),\hskip0.5em {p}_{m,i,j}=\frac{\mid {w}_{m,i,j}\mid }{\sum_{m^{\prime }}\mid {w}_{m^{\prime },i,j}\mid +\varepsilon }, $$

where $ {w}_{m,i,j} $ is the effective weight of model $ m $ at grid cell $ \left(i,j\right) $ . The effective ensemble size is $ {N}_{\mathrm{eff},i,j}=\exp \left({H}_{i,j}\right) $ , ranging from 1 (full concentration on a single model) to $ M $ (uniform trust across all models).

4.1. Temperature

All learning-based methods outperform AMME for global mean temperature (Table 2). Ridge reduces RMSE by more than half relative to AMME, and RF also yields substantial improvement. The gating mixture achieves the best overall performance, with the lowest RMSE (0.196) and highest NSE (0.980). Correlations with ERA5 remain very high across methods, indicating that improvements reflect better tracking of temporal variability rather than simple variance rescaling.

To confirm that these rankings are not artifacts of a single training run, we repeated the experiment with $ n=10 $ different random seeds for the RF (Ridge is deterministic given a fixed training set). Table 4 reports mean RMSE and standard deviation across seeds. The standard deviation is near zero for all methods (Ridge: 0.000, RF: 0.002, Gating: 0.001), confirming that the ranking Gating > Ridge > RF > AMME is stable and not attributable to random variation in tree construction.

Table 4.

Stability of global temperature RMSE across $ n=10 $ random seeds

Table 4. Long description

The table consists of four columns and three rows.

Column headers from left to right are:

- Empty cell for row labels

- Ridge

- R F

- Gating

Row 1, Mean R M S E:

- Ridge: 0.209

- R F: 0.227

- Gating: 0.196

Row 2, Std R M S E:

- Ridge: 0.000

- R F: 0.002

- Gating: 0.001

A note below the table states that Ridge is deterministic with a standard deviation of 0, and the near-zero standard deviations confirm that the method ranking is not due to random variation.

Note. Ridge is deterministic (std = 0). The near-zero standard deviations confirm that the method ranking is not attributable to random variation.

4.2. Precipitation

For precipitation, we report distribution-aware skill using RMSE, wet-day frequency RMSE, and upper-tail RMSE (Table 3). All learning-based approaches improve over AMME across these metrics. Ridge provides strong gains relative to AMME, while RF improves less consistently. The gating mixture yields the best overall performance globally, improving not only mean error but also wet-day frequency and tail behavior. Compared to temperature, precipitation gains are smaller and more sensitive to method choice, consistent with higher intermittency and noise in the precipitation field.

Figure 2 provides a qualitative illustration of the test-period performance. Panel (a) shows all 25 CMIP6 model time series color coded by Ridge regression weight: the most-trusted models (red) track ERA5 closely, while less-trusted models (pale/blue) deviate more. Panel (b) uses the same time series but codes color by RF importance, revealing which models the nonlinear learner relies on most. Panel (c) compares predicted vs. observed global mean temperature for all four methods on the test period (2006–2014): all learning-based methods closely track ERA5, while AMME shows a systematic warm bias reflecting the multi-model mean offset.

Figure 2.

Global mean near-surface temperature time series and test-period comparison. (a) All 25 CMIP6 model time series color coded by normalized Ridge regression weight: red indicates high positive weight, blue indicates negative or near-zero weight. ERA5 is shown in black. Training/validation/test periods are delineated by vertical dashed lines. (b) Same time series color coded by RF variable importance (yellow = low, red = high). (c) Predicted vs. observed global mean temperature for AMME (dashed pink) and Gating (solid red) on the test period 2006–2014. Ridge and Gating closely track ERA5, while AMME exhibits a systematic warm bias.

Figure 2. Long description

A multi-panel figure with three vertically stacked line graphs. All panels share a common Y-axis labeled Global mean tas in degrees Celsius, ranging from 11 to 18, and an X-axis representing years from 1950 to 2014.

Panel a displays C M I P 6 model time series color-coded by Ridge regression weight. A vertical dashed line at 1999 marks the Val start and a dotted line at 2006 marks the Test start. High positive weights are shown in red, while negative or near-zero weights are in blue. A thick black line represents E R A 5 data, which oscillates seasonally between 12 and 16 degrees Celsius. A color bar on the right indicates Ridge weight from minus 0.05 to 0.05.

Panel b shows the same time series but color-coded by R F feature importance. Yellow indicates low importance and red indicates high importance. The R F importance color bar on the right ranges from 0.00 to 0.20. The seasonal oscillations remain consistent with panel a.

Panel c displays predicted versus observed global mean near-surface temperature. The background is shaded in light blue for Train (1950 to 1999), light green for Val (2000 to 2006), and light red for Test (2007 to 2014). The legend identifies E R A 5 as a solid black line, A M M E full as a solid light purple line, A M M E test as a dashed purple line, Ridge test as a blue line, R F test as an orange line, and Gating test as a solid red line. In the test period, the Gating and Ridge lines closely overlap the E R A 5 black line, while the A M M E lines show a distinct upward shift, indicating a warm bias.

5.1. Global ridge weights and comparison with RF

Table 5 lists the ten models receiving the highest global Ridge weights for temperature, together with their RF importance scores. These models contribute heterogeneously to the ensemble prediction: the top-weighted models receive approximately $ 1.5\times $ the uniform AMME weight ( $ 1/25=0.040 $ ), while the lowest-weighted models receive correspondingly less. The degree of concentration is controlled by the $ {\mathrm{\ell}}_2 $ regularization strength: larger $ \alpha $ pushes weights toward uniformity, while smaller $ \alpha $ allows greater differentiation.

Table 5.

Top ten CMIP6 models by global Ridge weight for temperature, together with their cluster assignment (consistent with the fixed-seed clustering reported in Section 3)

Table 5. Long description

The table consists of four columns: Model, Ridge Weight, Cluster, and R F Importance. The data is as follows:

* C M C C-C M 2-S R 5: Ridge Weight 0.0613, Cluster 3, R F Importance 0.024.

* G I S S-E 2-1-H: Ridge Weight 0.0575, Cluster 3, R F Importance 0.236.

* C M C C-E S M 2: Ridge Weight 0.0573, Cluster 3, R F Importance 0.005.

* C A M S-C S M 1-0: Ridge Weight 0.0522, Cluster 1, R F Importance 0.000.

* A W I-C M-1-1-M R: Ridge Weight 0.0502, Cluster 1, R F Importance 0.158.

* M R I-E S M 2-0: Ridge Weight 0.0501, Cluster 1, R F Importance 0.123.

* G I S S-E 2-1-G: Ridge Weight 0.0495, Cluster 1, R F Importance 0.014.

* F I O-E S M-2-0: Ridge Weight 0.0464, Cluster 1, R F Importance 0.108.

* CanESM5: Ridge Weight 0.0461, Cluster 1, R F Importance 0.143.

* CanESM5-CanOE: Ridge Weight 0.0440, Cluster 1, R F Importance 0.007.

Note. The rightmost column shows the corresponding RF importance score for comparison.

Notably, the highest-weighted models are concentrated within Clusters 1 and 3 for Ridge, suggesting that Ridge regression preferentially amplifies specific model families rather than uniformly redistributing trust across the ensemble.

Comparing Ridge weights with RF importances (Table 5 and Figure 3) reveals partial but imperfect agreement (Pearson $ r=0.52 $ ). Six models receive substantially higher RF importance than Ridge weight: GISS-E2-1-H (RF: 0.236 vs. Ridge: 0.050), AWI-CM-1-1-MR (0.158 vs. 0.050), CanESM5 (0.143 vs. 0.046), MRI-ESM2–0 (0.123 vs. 0.050), FIO-ESM-2-0 (0.108 vs. 0.048), and FGOALS-g3 (0.106 vs. 0.044). These models appear to carry nonlinear information that the RF can exploit but that Ridge, as a globally linear model, cannot fully capture. Conversely, many models receive higher Ridge weight than RF importance, reflecting the regularized distribution of weights that Ridge enforces.

Figure 3.

(a) Normalized Ridge regression weights (blue) and RF variable importances (orange) for all 25 CMIP6 models, sorted by Ridge weight. The dashed gray line shows the uniform AMME weight (1/25). (b) Scatter of normalized Ridge weight vs. RF importance per model (Pearson $ r=0.52 $ ). Models above the diagonal are more trusted by RF than Ridge; the six models with RF importance $ >2\times $ Ridge weight are labeled. The partial agreement indicates that the two learners identify overlapping but non-identical sets of informative models.

Figure 3. Long description

Panel A is a bar chart titled Normalised Ridge weights versus R F importances. The y-axis represents Normalised weight or importance from 0.00 to 0.25. The x-axis lists 25 climate models. Blue bars represent Ridge weights, which remain relatively stable near the dashed gray uniform A M M E weight line at 0.040. Orange bars represent R F importance, showing high variability with significant peaks for models like F I O, A W I, M R I, C A M S, I I T M, and C A S.

Panel B is a scatter plot titled Ridge versus R F weight agreement with a Pearson r of 0.52. The x-axis is the absolute Ridge weight (normalised) and the y-axis is R F importance, both ranging from 0.00 to 0.25. A dashed diagonal line represents perfect agreement. Most models cluster in the bottom-left corner below the 0.05 mark on both axes. Six outlier models are labeled high above the diagonal, indicating much higher R F importance than Ridge weight: G I S S, A W I, Can E S M 5, M R I, F I O, and F G O A L S. Horizontal and vertical gray lines intersect at the 0.04 uniform weight mark.

Figure 3 illustrates this comparison across all 25 models. Panel (a) shows normalized Ridge weights and RF importances as a grouped bar chart, with the uniform AMME baseline shown as a dashed line. Panel (b) shows a scatter plot of the two quantities; models above the identity line are RF-preferred, while models below are Ridge-preferred. The six strongly RF-preferred models cluster in the upper-left region. The partial agreement ( $ r=0.52 $ ) is interpreted in Section 5.2: the gating exploits both sets of signals by adaptively blending the two learners.

5.2. Spatial structure of gating and model dominance

To examine how the gating ensemble redistributes trust across space, we analyze the dominant-cluster and dominant-model at each grid cell, based on the mean per-model gating weight (i.e., the average absolute weight per model within each cluster, rather than the sum, to avoid bias toward whichever cluster has the most members).

For spatial reference, we define three large-scale regions: the Tropics ( $ 30{}^{\circ} $ S– $ 30{}^{\circ} $ N), Northern Hemisphere mid-latitudes ( $ 30{}^{\circ} $ N– $ 60{}^{\circ} $ N), and the Arctic ( $ 60{}^{\circ} $ N– $ 90{}^{\circ} $ N). The mean gating skill improvement over AMME is $ +0.418 $ in the Tropics, $ +0.148 $ in NH mid-latitudes, $ +0.319 $ in the Southern Hemisphere mid-latitudes, $ +0.298 $ in the Arctic, and $ +0.294 $ in the Antarctic.

The left panel of Figure 4 shows the dominant-model cluster per grid cell. Clear large-scale spatial organization emerges, with distinct cluster regimes in the tropics, mid-latitudes, and high-latitude oceans. Cluster 0 (MIROC models) dominates parts of the tropical and subtropical oceans; Cluster 3 (CMCC, GISS-H, ACCESS-ESM1–5 group) is preferred over continental interiors and parts of the Southern Ocean; Cluster 1 (the large 14-member group) dominates high-latitude regions. These spatial patterns are broadly consistent with known dynamical differences among climate model families: for example, the MIROC models are known for distinctive SST biases in the tropical Pacific, while the CMCC and GISS families differ in aerosol forcing and ocean heat uptake representations (Sanderson et al., Reference Sanderson, Knutti and Caldwell2015; Knutti et al., Reference Knutti, Sedláček, Sanderson, Lorenz, Fischer and Eyring2017). The right panel of Figure 4 shows the dominant individual CMIP6 model per grid cell. Rather than a single model dominating globally, the gating ensemble exhibits strong spatial diversity, with different models preferred in different regions.

Figure 4.

(Left) Dominant CMIP6 model cluster per grid cell, defined by the cluster whose models receive the highest mean per-model gating weight. (Right) Dominant individual CMIP6 model per grid cell, defined by the model with the maximum average gating weight. Results show strong spatial structure in ensemble trust, with different clusters and models preferred across distinct dynamical regimes. Cluster IDs: 0 = MIROC family (2 models), 1 = large mid-bias group (14 models), 2 = HadGEM–UKESM family (3 models), 3 = CMCC–GISS-H group (6 models).

Figure 4. Long description

A two-panel visualization of global climate model distribution.

Left Panel: Dominant cluster per grid cell (argmax |weight|).

* The map uses four distinct colors to represent Cluster I D values from 0 to 3.

* Red (Cluster 1) is the most widespread, covering most of North America, Eurasia, North Africa, and the Southern Ocean.

* Blue (Cluster 0) is concentrated in the tropical Pacific and parts of the Southern Hemisphere.

* Pink (Cluster 2) appears in the South Atlantic, Indian Ocean, and parts of Antarctica.

* Cyan (Cluster 3) is scattered across the North Pacific and high-latitude regions.

* A vertical color bar to the right labels the Cluster I D from 0 (bottom) to 3 (top).

Right Panel: Dominant C M I P 6 model per grid cell (argmax |Gating weight|).

* This map features a highly fragmented, multi-colored mosaic representing individual models.

* Large contiguous blocks are visible in the Southern Ocean (green and purple), the Sahara (light green), and the North Atlantic (red).

* The tropical regions show high variability with small, pixelated color changes.

* A vertical color bar to the right, titled Model index, ranges from 0 to over 20, using a diverse categorical color palette to identify specific models.

To understand how gating relates to its two components, Figure 5 shows dominant-cluster maps and trust entropy maps for Ridge, RF, and Gating side by side. Ridge produces a spatially smooth, multi-cluster pattern (mean $ {N}_{\mathrm{eff}}=21.1 $ ), while RF is more concentrated (mean $ {N}_{\mathrm{eff}}=13.4 $ ), reflecting the implicit feature selection of tree-based methods. Gating achieves an intermediate effective ensemble size ( $ {N}_{\mathrm{eff}}=19.5 $ ), broadly following the Ridge spatial pattern but selectively concentrating trust in regions where RF’s nonlinear predictions are locally superior – consistent with the gating mechanism blending both learners. The spatial distribution of the per-pixel mean gate value $ \overline{g} $ (Figure 6) confirms this: 55% of pixels are Ridge-dominated ( $ \overline{g}>0.5 $ ) and 45% are RF-dominated ( $ \overline{g}<0.5 $ ), with RF dominance concentrated in tropical and Southern Ocean regions that also exhibit the strongest skill improvement.

Figure 5.

Dominant cluster (left column) and trust entropy (right column) for Ridge (top), RF (middle), and Gating (bottom). Ridge maintains high entropy (mean $ {N}_{\mathrm{eff}}=21.1 $ ) with a spatially smooth multi-cluster pattern. RF is more concentrated (mean $ {N}_{\mathrm{eff}}=13.4 $ ), with stronger Cluster-0 (MIROC) dominance in the tropics. Gating achieves an intermediate $ {N}_{\mathrm{eff}}=19.5 $ , selectively concentrating trust in regions where RF’s nonlinear predictions are locally superior.

Figure 5. Long description

A Robinson projection world map titled Spatial skill of Gating ensemble temperature. The map uses a diverging color scale to represent Skill vs A M M E.

* The color scale on the right ranges from negative 0.2 in dark blue to 0.6 in dark red.

* High skill values between 0.4 and 0.6 are concentrated in the tropical oceans, particularly across the central Pacific, the tropical Atlantic, and the Indian Ocean.

* High skill is also visible over large landmasses including central and southern Africa, South America, and parts of South Asia.

* Low or negative skill values between negative 0.2 and 0.1 are shown in light to dark blue, appearing predominantly in the high latitudes of the Northern Hemisphere, including the Arctic Ocean, northern North America, and northern Eurasia.

* Australia and the Southern Ocean show a mix of moderate skill in light red and low skill in light blue.

* The transition between high and low skill regions is granular, with a pixelated texture across most continental regions.

Figure 6.

Spatial distribution of the time-mean gate value $ \overline{g} $ (per pixel). Red ( $ \overline{g} $ near 1) indicates Ridge-dominated pixels; blue ( $ \overline{g} $ near 0) indicates RF-dominated pixels. Overall, 55% of pixels are Ridge-dominated and 45% are RF-dominated. RF dominance is concentrated in tropical oceans and the Southern Ocean, which also exhibit the strongest skill improvement relative to AMME (Figure 7).

Figure 6. Long description

The grid consists of three rows representing different methods: Ridge at the top, R F in the middle, and Gating at the bottom.

Left Column: Dominant cluster maps. These maps use a discrete color scale for Cluster I D from 0 to 3.

* Row 1 (Ridge): Shows a high concentration of red (Cluster 0) and brown (Cluster 2) across the globe.

* Row 2 (R F): Dominated heavily by red (Cluster 0) across almost all ocean and land masses, with small patches of purple (Cluster 1) and grey (Cluster 3).

* Row 3 (Gating): Similar to R F but with slightly more purple (Cluster 1) and grey (Cluster 3) distribution in the Southern Hemisphere and tropical regions.

Right Column: Trust entropy maps. These maps use a continuous diverging colormap from 0.0 to 3.0 nats.

* Row 1 (Ridge): Mean N sub eff equals 21.1. The map is mostly light yellow, indicating low entropy, with slightly darker shades in the Southern Ocean.

* Row 2 (R F): Mean N sub eff equals 13.4. This map shows significantly higher entropy (darker orange and red hues) across the tropical oceans and Southern Hemisphere compared to the other methods.

* Row 3 (Gating): Mean N sub eff equals 19.5. The entropy levels are lower than R F, appearing mostly light yellow with moderate orange shading concentrated in the tropical Pacific and Indian Oceans.

Figure 7 shows the spatial distribution of skill improvement of the gating ensemble relative to AMME for near-surface temperature. Large contiguous regions exhibit substantial positive skill, particularly over tropical oceans, the Southern Ocean, and parts of the Indian Ocean basin. These regions correspond to areas of strong model disagreement and known systematic biases in coupled climate models, suggesting that gating is most effective where ensemble diversity is high and model errors are structured rather than random. Conversely, regions with weak or near-zero skill gains are primarily located over continental interiors and high-latitude land areas, where internal variability dominates and cross-model differences are less informative. This spatial pattern indicates that gating does not uniformly improve predictions everywhere but selectively enhances skill in regions where ensemble learning is epistemically meaningful.

Figure 7.

Spatial distribution of skill improvement of the gating ensemble relative to AMME for near-surface temperature (test period 2006–2014). The diverging colormap is centered at zero (white), with red indicating improvement and blue indicating degradation relative to AMME. Positive values indicate regions where gating reduces RMSE compared to the arithmetic ensemble. Strong improvements occur over tropical oceans and parts of the Southern Hemisphere; small degradations occur over some continental interiors and high-latitude land areas.

Figure 7. Long description

A world map in a Robinson-style projection displays mean gate weight g-bar per pixel. A color scale on the right ranges from 0.0 in dark red to 1.0 in dark blue.

* The scale indicates that 0 equals pure R F and 1 equals pure Ridge.

* Blue pixels, representing R F dominance, are heavily concentrated in the Southern Ocean surrounding Antarctica and across the tropical Pacific and Indian Oceans.

* Red pixels, representing Ridge dominance, are more prevalent over landmasses and in the high Northern latitudes, particularly the North Atlantic and Arctic regions.

* The transition zones between these regions appear in light white or pale orange, indicating a mix of both models.

* A title at the top explicitly states Blue equals R F-dominated and Red equals Ridge-dominated.

Compared to traditional performance-based weighting schemes – such as scalar skill-based weighting (Knutti et al., Reference Knutti, Sedláček, Sanderson, Lorenz, Fischer and Eyring2017) or the ClimWIP approach (Sanderson et al., Reference Sanderson, Knutti and Caldwell2015), which assign globally uniform scalar weights – the gating mechanism operates at the pixel level and is time-varying, allowing trust to be concentrated only where and when a sub-model is locally superior. The $ {\mathrm{\ell}}_2 $ regularization in Ridge explains why that component maintains high entropy (many models contribute); the tree-based selection of RF explains its lower entropy; and gating mediates between these extremes.

5.3. Trust dispersion and effective ensemble size

To quantify how concentrated or distributed model trust is under the gating ensemble, we compute the entropy of the gating weights at each grid cell (Section 2.5). Higher entropy indicates more evenly distributed trust across models, while lower entropy indicates strong concentration on a small number of models.

Figure 8 shows the spatial distribution of trust entropy for the gating ensemble. Most regions exhibit relatively high entropy, corresponding to a mean effective ensemble size of $ {N}_{\mathrm{eff}}=19.5 $ models. For comparison, Ridge yields $ {N}_{\mathrm{eff}}=21.1 $ and RF yields $ {N}_{\mathrm{eff}}=13.4 $ (see also Figure 5). This indicates that, despite spatially varying dominance patterns, gating generally maintains a broad ensemble rather than collapsing to a few preferred models.

Figure 8.

Spatial distribution of trust dispersion under the Gating ensemble, measured as (a) trust entropy $ H $ and (b) effective ensemble size $ {N}_{\mathrm{eff}}=\exp (H) $ [mean $ =19.5 $ models]. Higher values indicate more evenly distributed trust (larger effective ensemble size), while lower values indicate strong concentration on a small subset of models. For comparison, Ridge yields mean $ {N}_{\mathrm{eff}}=21.1 $ and RF yields mean $ {N}_{\mathrm{eff}}=13.4 $ ; see Figure 5 for side-by-side entropy maps of all three methods.

Figure 8. Long description

Two side-by-side global maps using a Robinson projection.

Panel a is titled Trust entropy gating weights. It displays a heat map where values are measured in nats. The color scale on the right ranges from 0.0 black to 3.0 light yellow. The map is predominantly light yellow and pale orange, indicating high entropy across most of the globe, with slightly lower values appearing as darker orange patches in the Southern Ocean and parts of the North Atlantic.

Panel b is titled Effective ensemble size N sub eff = exp entropy mean = 19.5. The color scale on the right ranges from 0 to 25, with dark purple representing low values and bright yellow representing high values. The map shows a more heterogeneous distribution than panel a. High values near 20 to 25 are concentrated in the tropical regions and mid-latitudes of both the Northern and Southern Hemispheres. Lower values, indicated by purple and dark pink patches, are visible in the North Atlantic, the Southern Ocean near Antarctica, and parts of the South Pacific, indicating a higher concentration of trust on fewer models in these specific geographical zones.

Regions with lower entropy appear primarily in parts of the tropical oceans and the Southern Ocean, where a small subset of models consistently outperforms others. These regions also correspond to areas of particularly strong skill improvement under gating (Figure 7), suggesting that localized trust concentration is associated with genuine predictive advantage.

Taken together, these results demonstrate that gating produces a structured redistribution of trust: locally selective where strong signals exist, but globally conservative in preserving ensemble diversity. This behavior supports the interpretation of gating as a principled trust reallocation mechanism rather than an aggressive model selection procedure.

5.4. Temporal and seasonal behavior of the gate

To assess the temporal consistency of the gating weights, Figure 9 shows the gate value $ {g}_t $ over the test period 2006–2014 together with a 12-month rolling mean and a per-season breakdown. The gate is Ridge-dominated for 92% of monthly time steps (global mean $ \overline{g}=0.685 $ ) but exhibits meaningful seasonal modulation. MAM (spring) has the highest mean gate value ( $ {\overline{g}}_{\mathrm{MAM}}=0.873 $ , almost entirely Ridge-dominated), while DJF (winter) has the lowest ( $ {\overline{g}}_{\mathrm{DJF}}=0.592 $ ), with JJA and SON at intermediate values (0.638 and 0.639, respectively). This pattern suggests that the global mean relationship between CMIP6 models and ERA5 is more linear in spring – when ensemble spread may be dominated by systematic rather than nonlinear biases – and that the RF nonlinearity contributes more in winter. We note that this temporal gate operates on the global mean signal; spatial variation of $ \overline{g} $ is shown in Figure 6. Seasonal effects on the spatial gating weights are a natural direction for future work.

Figure 9.

(a) Temporal evolution of the global gating weight $ {g}_t $ over the test period 2006–2014. Blue shading indicates Ridge-dominated months ( $ {g}_t>0.5 $ ); yellow shading indicates RF-dominated months ( $ {g}_t<0.5 $ ). The red curve shows a 12-month rolling mean; the dotted horizontal line shows the global mean $ \overline{g}=0.685 $ . (b) Seasonal distribution of $ {g}_t $ : MAM is strongly Ridge-dominated (mean 0.873), while DJF shows the greatest RF contribution (mean 0.592). Boxplots show the interquartile range and outliers across individual months within each season.

Figure 9. Long description

Panel a is a line graph titled Temporal evolution of the global gating weight g sub t. The y-axis is Gate value g sub t ranging from 0.0 to 1.0. The x-axis shows years from 2007 to 2015. A blue line fluctuates between 0.4 and 1.0. Areas where g sub t is greater than 0.5 are shaded blue, indicating Ridge-dominated months. Small areas where g sub t is less than 0.5 are shaded yellow, indicating R F-dominated months. A red curve represents a 12-month rolling mean, staying relatively stable between 0.6 and 0.8. A dotted horizontal line marks the global mean g-bar equals 0.685, and a dashed line marks the 0.5 equal mix threshold.

Panel b is a boxplot titled Seasonal distribution of gate value. The y-axis is g sub t and the x-axis lists four seasons.

- D J F is blue with a median near 0.6 and two outliers.

- M A M is green with the highest median near 0.9.

- J J A is orange with a median near 0.6 and a wide interquartile range.

- S O N is red with a median near 0.65 and one outlier.

The global mean and 0.5 threshold lines from panel a are extended across these boxplots.