Extreme Event Attribution: Fraction of Attributable Risk

Warming Forcing

Global mean warming ΔT 1.2°C

Extreme threshold (σ above 1850–1900 mean) 2.0σ

Variability change +10%

Ensemble size (per world) 600

Attribution Statistics

P(event) — today's world

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P(event) — pre-industrial

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Fraction of Attributable Risk

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Risk Ratio (RR)

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Factual exceedances / N

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Counterfactual exceedances / N

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Factual world (warmed by ΔT) — below threshold Factual world — exceeds threshold (attributable) Counterfactual (pre-industrial) — below threshold Counterfactual — exceeds threshold (natural rate)

How it works

Extreme-event attribution science — the method behind reports like World Weather Attribution and the evidence cited in cases such as Urgenda v. Netherlands and Milieudefensie v. Shell — compares two parallel worlds: the factual world with observed/warmed climate, and a counterfactual world with greenhouse-gas forcing removed. Event intensity in each world is modelled as a Gaussian ensemble of model runs:

Factual:        X₁ ~ N(μ₀ + ΔT·k,  σ₀²·(1+v))
Counterfactual:  X₀ ~ N(μ₀,         σ₀²)
P(event) = P(X > threshold) = 1 − Φ((threshold − μ)/σ)

The two probabilities are combined into the two standard courtroom/attribution metrics:

Risk Ratio   RR  = P₁ / P₀
Fraction of Attributable
Risk         FAR = 1 − P₀ / P₁ = 1 − 1/RR

ΔT slider — shifts the factual ensemble's mean via a fixed sensitivity k = 0.6σ per °C, mimicking how warming raises the odds of an extreme.
Threshold slider — the intensity that defines "extreme" (e.g. a specific heatwave or flood magnitude), in standard deviations above the 1850–1900 baseline mean.
Variability change — some extremes also become more (or less) variable under warming, independently of the mean shift.
Ensemble size / Resample — draws a fresh Monte-Carlo sample of model runs for each world; the two rows of bars are the actual histograms, while the FAR/RR readouts use the exact analytic Gaussian tail probability.

An FAR near 1 means the event was almost entirely attributable to human warming; an FAR near 0 means it was about as likely in a world without greenhouse-gas emissions. Real litigation uses peer-reviewed versions of exactly this comparison as quantitative evidence of causation.

Factual (warmed) Counterfactual (pre-industrial)

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