Fig. 2. Linear relationship between $\frac{1}{\alpha }$ and $\frac{1}{\mathrm{TCRE}}$ across timescales.

a Amplitude spectrum from Fourier analysis for annual atmospheric CO₂ (ΔC_A) and cumulative anthropogenic CO₂ emissions (ΔC_E) for 1880–2017. b Same as (a) but for annual global-mean temperature from four observational datasets (HadCRUT4, GISTEMP, Berkeley Earth, and NOAA GlobalTemp) for 1880–2017. c Estimates of $\frac{1}{\alpha }$ (=ΔC_A/ΔT_A) across timescales against $\frac{1}{\mathrm{TCRE}}$ (=ΔC_E/ΔT_E) for four observational global-mean temperature datasets with atmospheric CO₂ records (or anthropogenic emissions) for 1880–2017. The solid line with the shaded area between the dashed lines is the linear regression of all datasets with a slope (p) of 2.52 ± 0.15 ppm ppm⁻¹, which indicates that the carbon-concentration feedback parameter β is 3.22 ± 0.32 GtC ppm⁻¹.