10.5 Future Challenges

Step 1: Sources of Uncertainty in Climate Projections

Total uncertainty in a climate projection $Y(t)$ (e.g., SST in 2100) comes from three independent sources: scenario (forcing) uncertainty, model (structural) uncertainty, and internal (natural) variability:

$$\text{Var}[Y(t)] = \sigma_{\text{scenario}}^2(t) + \sigma_{\text{model}}^2(t) + \sigma_{\text{internal}}^2(t)$$

Step 2: ANOVA Decomposition (Hawkins & Sutton)

Following Hawkins and Sutton (2009), decompose the multi-model ensemble into these components. For $M$ models each run under $S$ scenarios with $R$ realisations, the total variance is partitioned by ANOVA:

$$Y_{msr}(t) = \mu(t) + \alpha_s(t) + \beta_m(t) + \epsilon_{msr}(t)$$

Step 3: Estimate Each Variance Component

The grand mean $\mu(t)$ is the forced response. Scenario uncertainty $\sigma_{\text{scenario}}^2$ is the variance of scenario means, model uncertainty $\sigma_{\text{model}}^2$ is the variance of model means within a scenario, and internal variability $\sigma_{\text{internal}}^2$ is the residual:

$$\sigma_{\text{scenario}}^2 = \frac{1}{S-1}\sum_s (\bar{Y}_{s\cdot\cdot} - \bar{Y}_{\cdot\cdot\cdot})^2, \quad \sigma_{\text{model}}^2 = \frac{1}{S(M-1)}\sum_{s,m}(\bar{Y}_{sm\cdot} - \bar{Y}_{s\cdot\cdot})^2$$

Step 4: Time Dependence of Uncertainty Fractions

Near-term (2020--2040): internal variability dominates (~60%). Mid-century (2040--2060): model uncertainty is largest (~50%). End-of-century (2080--2100): scenario uncertainty dominates (~70%) as forcing pathways diverge. This informs both science priorities and policy relevance:

$$f_{\text{source}}(t) = \frac{\sigma_{\text{source}}^2(t)}{\sigma_{\text{total}}^2(t)} \times 100\%$$

Step 1: Simple Multi-Model Mean (Democracy)

The simplest ensemble approach gives equal weight to each model. The multi-model mean (MMM) and its uncertainty are:

$$\bar{Y} = \frac{1}{M}\sum_{m=1}^{M} Y_m, \quad \sigma_{\bar{Y}} = \frac{1}{M}\sqrt{\sum_m (Y_m - \bar{Y})^2}$$

Step 2: Performance-Based Weighting

Models can be weighted by their skill in reproducing historical observations. Using a distance metric $D_m$ between model $m$ and observations, weights are assigned inversely proportional to squared distance:

$$w_m = \frac{\exp(-D_m^2 / (2\sigma_D^2))}{\sum_{k=1}^{M} \exp(-D_k^2 / (2\sigma_D^2))}, \quad \bar{Y}_w = \sum_m w_m Y_m$$

Step 3: Independence Weighting (ClimWIP)

Many CMIP models share code and development history, violating independence assumptions. The Climate model Weighting by Independence and Performance (ClimWIP) method down-weights similar models using an inter-model distance $S_{mk}$:

$$w_m \propto \frac{\exp(-D_m^2 / \sigma_D^2)}{\sum_{k \ne m} \exp(-S_{mk}^2 / \sigma_S^2) + 1}$$

Step 4: Bayesian Model Averaging (BMA)

BMA treats each model as a hypothesis and computes posterior weights from the likelihood of the observations given each model, multiplied by a prior weight:

$$p(Y|\text{obs}) = \sum_{m=1}^{M} p(Y|M_m) \cdot p(M_m|\text{obs}), \quad p(M_m|\text{obs}) \propto p(\text{obs}|M_m) \cdot p(M_m)$$

Step 5: Ensemble Spread vs. Skill

A well-calibrated ensemble has the property that its spread matches its actual prediction error, tested by the rank histogram. The reliability ratio $R = \sigma_{\text{ensemble}}^2 / \text{MSE}$ should be near 1. Under-dispersive ensembles ($R < 1$) underestimate uncertainty, which is common in CMIP projections and motivates the weighting approaches above.