2.3.1 v1.4.1: best-estimate historical emissions 1750–2022

fair-calibrate v1.4.1 uses up-to-date historical emissions as far as possible, and the emissions are as follows:

• CO₂ emissions for fossil fuel and industrial (FFI) and agriculture, forestry, and other land use (AFOLU) CO₂ are from the Global Carbon Project 2023 v1.0 (Friedlingstein et al., 2023).

• CH₄ and N₂O from non-biomass-burning sources, plus SF₆, NF₃, and aggregated hydrofluorocarbons (HFCs) and perfluorocarbons (PFCs), are from PRIMAP-Hist v2.5 (Gütschow and Pflüger, 2023; Gütschow et al., 2016), prioritising third-party (TP) data sources over country reported emissions.

• Short-lived climate forcers, comprising black carbon (BC), organic carbon (OC), sulfur dioxide (SO₂), nitrogen oxides (NO_x), ammonia (NH₃), carbon monoxide (CO), and volatile organic compounds (VOCs) from fossil, industrial, and agricultural sources, are from the Community Emissions Data System (CEDS) v2021.04.06 (O'Rourke et al., 2021; Hoesly et al., 2018).

• Biomass burning emissions of CH₄, N₂O, and short-lived climate forcers (SLCFs) are taken from the Global Fire Emissions Database (GFED) (van der Werf et al., 2017) v4.1, which includes the BB4CMIP dataset prepared for CMIP6 historical simulations (van Marle et al., 2017).

• Emissions of Montreal Protocol greenhouse gases (CFCs, HCFCs, halons, and chlorinated and brominated gases), along with SO₂F₂, are estimated using inverse greenhouse gas concentrations that have been prepared for the IGCC (Forster et al., 2023), as no inventories of these emission datasets are available to our knowledge.

All emission datasets are produced for 1750–2022, except CEDS, which has a 2019 end-date. To extend SLCFs from CEDS to 2022, we use the “2-year blip” scenario that estimates the decline and recovery from emissions due to COVID-19 from Forster et al. (2020) and is extended by Lamboll et al. (2021), based on proxy activity data. We take the ratios of SLCF emission species over 2020–2022 to 2019 in the 2-year blip scenario and apply them as a scaling factor to CEDS emissions in 2019. Such a version-controlled strategy allows for the calibration to be updated as newer emission data become available. Emission data prepared to the end of 2023 will be available over the course of 2024, and an anticipated update to CEDS should also bring non-biomass-burning SLCFs until at least the end of 2022 (Hoesly et al., 2023). This demonstrates that “operational” calibrations are often a moving target.

We use the “third-party” emissions from PRIMAP-Hist rather than country-reported values, based on the assumption that we expect solely country-reported values to be an underestimate of true emissions. We demonstrate that third-party emissions still appear to be an underestimate for many species, based on best-estimate greenhouse gas lifetimes and concentration estimates.

2.3.2 v1.4.0: RCMIP historical emissions prepared for AR6 (1750–2014)

For consistency and comparison with the FaIR projections used in the IPCC AR6, we produce a calibration using historical emissions from RCMIP (Nicholls et al., 2020, 2021) using v5.1.0 of the Reduced Complexity Model Intercomparison Project (RCMIP) emission dataset available from Nicholls and Lewis (2021). The RCMIP emissions contain global annual total emissions of CO₂ and SLCFs that were prepared for running CMIP6 models. Emissions of non-CO₂ greenhouse gases were back-calculated to reproduce the CMIP6 best-estimate historical concentrations (Meinshausen et al., 2017). These concentrations time series were also used to drive CMIP6 models.

For SSP scenarios, emissions from 2015 to 2100 were produced using IAMs, which were then extended to 2500 using simplified assumptions (Meinshausen et al., 2020). We use the same climate constraints on GMST, CO₂ concentration, and OHC as for v1.4.1 (Sect. 3.3) datasets, which run to 2022. For the bridging period 2015–2022 between the end of the CMIP6 historical and the observational climate data, we use emissions from SSP2-4.5, expected to be the closest Tier 1 SSP to current policies (Hausfather and Peters, 2020) and, as shown later, the closest Tier 1 scenario to post-2015 emissions.

One adjustment is made to the RCMIP emissions to correct NO_x. For accounting purposes, we express NO_x in units of Tg NO₂ yr⁻¹. The source datasets for RCMIP were earlier versions of CEDS, which reports emissions in Tg NO₂ yr⁻¹ for fossil fuel and agricultural emissions, and GFED, which reports emissions in Tg NO yr⁻¹ for biomass burning. The conversion for GFED emission data was not made in RCMIP v5.1.0.

Neither v1.4.1 nor v1.4.0 of fair-calibrate includes forcing from aviation contrails. Forcing from contrails and its temperature impact were assessed in the IPCC AR6 WG1 (Forster et al., 2021), with best-estimate contributions to present-day forcing of 0.06 W m⁻² and warming of 0.02 °C, and were included in the WG1 calibration of FaIR. However, contrail forcing was excluded from the WG3 IAM emission projections, rendering the WG1 and WG3 projection sets slightly inconsistent. To project contrail forcing into the future requires estimates of aviation activity. FaIR can accept a time series of contrails forcing directly or estimate it, using a linear combination of emission species. By default, FaIR uses NO_x emissions from the aviation sector to estimate contrail forcing (Smith et al., 2018). Neither aviation activity nor NO_x emissions from aviation are provided in IAM scenarios in general, so contrail forcing could not be assessed in WG3. Aviation NO_x emissions are provided in the RCMIP historical and SSP future emissions and could be included in fair-calibrate v1.4.0. However, in order to apply the calibrations consistently to as many scenarios as possible, we calibrate without them.

3.1.1 Climate response

The climate response module of FaIR v2.1.3 is an impulse response formulation of the three-layer stochastic energy balance model of Cummins et al. (2020). We calibrate this model using 150-year 4×CO₂ experiments from 49 CMIP6 models and using GMST (ΔT₁) and the top-of-atmosphere energy imbalance (ΔN) as anomalies relative to each model's pre-industrial control run, subtracting a linear trend from the appropriate branch point of each model's control to account for any residual drift. This calibration is performed using the maximum likelihood method of Cummins et al. (2020), and the EBM R package that accompanies Cummins et al. (2020) is used in the fair-calibrate process chain (Cummins, 2021).

The three-layer stochastic energy balance model is written as

In Eqs. (1)–(3), T₁, T₂, and T₃ are the temperature anomalies of the three ocean layers (starting from the surface); C₁, C₂, and C₃ are their heat capacities; κ_j represents the heat transfer coefficients between layers j−1 and j for j≥2; −κ₁ is the climate feedback parameter (often denoted λ); ε is the deep-ocean efficacy parameter (Held et al., 2010; Winton et al., 2010; Geoffroy et al., 2013); ξ is a stochastic disturbance term in the temperature response that does not affect the top-of-atmosphere energy imbalance; and F is the effective radiative forcing (ERF).

The effective radiative forcing is the sum of a deterministic and stochastic component $F=F_{\text{det}}+\mathit{\zeta }$. The stochastic forcing component ζ is modelled as a continuous-time red noise process

where η is white noise, and γ>0 controls the strength of temporal auto-correlation (Cummins et al., 2020). In FaIR, the stochastic behaviour can be switched off, and Eqs. (1)–(4) reduce to a deterministic energy balance model when $\mathit{\xi }=\mathit{\eta }=\mathrm{0}$ (Geoffroy et al., 2013; Leach et al., 2021).

The top-of-atmosphere energy imbalance N is given as

and the Earth's energy uptake, used as a model constraint, is the time integral of N.

For each of the 49 CMIP6 models, we obtain a set of 11 parameters $\mathit{\{}{C}_{\mathrm{1}},C_{\mathrm{2}},C_{\mathrm{3}},{\mathit{\kappa }}_{\mathrm{1}},{\mathit{\kappa }}_{\mathrm{2}},{\mathit{\kappa }}_{\mathrm{3}},\mathit{\epsilon },\mathit{\gamma },{\mathit{\sigma }}_{\mathit{\xi }},{\mathit{\sigma }}_{\mathit{\eta }},F_{\mathrm{4}\times {\mathrm{CO}}_{\mathrm{2}}}\mathit{\}}$ that describes the magnitude and rate of warming to a 4×CO₂ forcing and the behaviour of internal variability, where σ_ξ and σ_η are the standard deviations of ξ and η around the zero mean. $F_{\mathrm{4}\times {\mathrm{CO}}_{\mathrm{2}}}$ is the effective radiative forcing from a quadrupling of pre-industrial CO₂ concentrations. The comparison of one stochastic realisation of each model's energy balance model calibration (black) compared to the actual CMIP6 model (red) for the temperature response to an abrupt 4×CO₂ forcing is shown in Fig. 2. In almost all cases, the FaIR calibration is an excellent representation of the underlying CMIP6 model. The calibrated parameters are shown in Table S1.

Figure 2Comparison of temperature projections from abrupt 4×CO₂ simulations as calibrated in FaIR (black) to the original CMIP6 model results (red) for 49 CMIP6 models. For FaIR, we show one realisation with stochastic internal variability included; different random seeds would produce different internal variability profiles.

Download

The energy balance model parameters can be written as a matrix equation that describes the time evolution of each temperature layer (Cummins et al., 2020; Leach et al., 2021). The impulse response form of the temperature evolution in each layer can be calculated from the eigenvalues and eigenvectors of the energy balance matrix. From this, the ECS and “theoretical” TCR for each model calibration can be directly estimated from the impulse response coefficients as described in Leach et al. (2021, Sect. 2.4). The ECS calculated here is a true equilibrium value rather than as a regression over a 150-year simulation as usually performed from ESM output (the so-called effective sensitivity, EffCS). The theoretical TCR is not precisely what each model would predict after 70 years of a 1 % compound increase in atmospheric CO₂ concentrations but is usually close and has the advantage that model simulations do not need to be run to determine this value (Fig. S1b in the Supplement).

3.1.2 Minor greenhouse gas emissions

This section describes the emission adjustment procedure in fair-calibrate v1.4.1 for emissions of minor greenhouse gases. In this context, “minor” means any species that is not CO₂ or CH₄. This includes N₂O, hydrofluorocarbons (HFCs), perfluorocarbons (PFCs), SF₆, and NF₃. This emission adjustment is not required in v1.4.0, where emissions from all species are provided by the RCMIP emission datasets (Sect. 2.3.2).

HFCs and PFCs are provided in PRIMAP-Hist as aggregate values reported in CO₂-equivalent (AR6 GWP₁₀₀) emissions. We disaggregate these emissions by scaling the annual historical emission totals in CO₂-equivalent emissions from RCMIP historical + SSP2-4.5 for 1750–2022 to the PRIMAP-Hist reported values and then by multiplying this scaling by the RCMIP individual species emission value in each year. Table S2 details the HFC and PFC gases included in the disaggregation.

The following step calculates atmospheric concentrations when run forward using a single time-constant decay model with the PRIMAP-Hist emission and time constants equal to atmospheric lifetimes assessed in IPCC AR6 (Smith et al., 2021b). The calculated concentration time series is compared to the best-estimate historical concentrations from Forster et al. (2023), which is an update of the AR6 concentrations in IPCC (2021) to 2022 using recent AGAGE and NOAA station data. In many cases, the calculated and observed concentrations differ substantially, and the calculated concentrations are usually lower than the observed. This implies that either the reported emissions in PRIMAP-Hist do not capture all true emissions or that the reported atmospheric lifetimes are too short (a third, less likely, possibility is that the reported concentrations are too high). A correction can be obtained by either lengthening the lifetimes or scaling up the emissions. We choose to adjust the emissions on the basis that countries under-reporting due to incomplete data is plausible, and scaling the emissions brings some species much closer to RCMIP estimates which are derived from inverting atmospheric concentrations. The scaling is performed in order to match the projected concentrations to the historical best estimates in 2019. In many cases the scaling is mild (for N₂O, emissions are scaled up by a factor of 1.08; Fig. 3a) but can be large (NF₃ is scaled by a factor of 7.5; Fig. S2). This implies that countries are severely under-reporting emissions of some greenhouse gases (GHGs) compared to the increasing stock of these gases observed in the atmosphere.

PRIMAP-Hist does not provide emissions of SO₂F₂ or of Montreal Protocol GHGs. We estimate their emissions by inverting the concentrations time series in Forster et al. (2023).

For future projections, we harmonise to 2022 (Gidden et al., 2018) the eight Tier 1 and Tier 2 SSP scenarios to our scaled calculated historical emissions. This produces SSPs that take into account the recent past. We can then compare the harmonised adjusted future concentration projections to those created for the SSP scenarios that used MAGICC6 (Meinshausen et al., 2020). Figure 3b shows recreated historical and future N₂O concentration projections to 2100 under eight SSP scenarios using the harmonised scaled emissions (thick lines) in FaIR and their comparison to the SSP concentrations time series (thin lines) from Meinshausen et al. (2017, 2020). Note that the historical concentrations differ between Fig. 3a and b as the dataset sources differ. For N₂O, the correspondence between FaIR and CMIP6 is very good for all eight SSPs for future projections.

Figure 3(a) Comparison of best-estimate historical N₂O emissions (black), the concentration projected from emissions in PRIMAP-Hist + GFED (dotted grey), and the concentrations after scaling up the emissions by a factor of 1.08 to get correct recent historical concentrations (solid grey). Note that a single lifetime cannot accurately reproduce best-estimate historical concentrations between 1850 and 1950. (b) Harmonised SSP projections using the scaled historical emissions (thick lines) compared to the SSP historical + future projections (thin lines) from Meinshausen et al. (2017, 2020).

Download

3.1.3 Methane lifetime

A new feature of FaIR introduced in v2.1.0 is a variable methane lifetime that depends on burdens of chemically reactive species and climate. This is an update from v2.0.0 that used a methane lifetime self-feedback (methane concentrations and temperature affect climate) and previous versions that did not modify the lifetime of methane at all.

A methane lifetime scaling factor ${\mathit{\alpha }}_{{\mathrm{CH}}_{\mathrm{4}}}$ is applied to the base lifetime ${\mathit{\tau }}_{{\mathrm{CH}}_{\mathrm{4}},\mathrm{base}}$ calculated as

In Eq. (6), S_i denotes a sensitivity to species i or GMST anomaly (ΔT₁), and ΔA_i represents abundances of species i (emissions rate for SLCFs and concentrations for GHGs) of chemically reactive species. If the anomalies in temperature and abundances are relative to the pre-industrial period, ${\mathit{\alpha }}_{{\mathrm{CH}}_{\mathrm{4}}}=\mathrm{1}$ in pre-industrial conditions and ${\mathit{\tau }}_{{\mathrm{CH}}_{\mathrm{4}},\mathrm{base}}$ is the pre-industrial lifetime.

Unlike for minor GHGs, emissions are not scaled for CH₄ in fair-calibrate v1.4.1, and we instead calibrate the atmospheric chemical lifetime. Owing to dependence of the lifetime of several simultaneously changing emission species, as well as climate, there is not a unique invertible concentration to emission pathway for methane.

The UKESM1.0-LL, GFDL-ESM4, GISS-E2.1-G, and MRI-ESM2.0 Earth system models provide a complete set of results from the Aerosol Chemistry Model Intercomparison Project (AerChemMIP) single-forcing experiments that enable the estimation of the sensitivity in methane lifetime to climate (Thornhill et al., 2021a) and chemically reactive species (Thornhill et al., 2021b). We use results reported in Thornhill et al. (2021b) and Thornhill et al. (2021a) for methane lifetime in 1850 and its relative sensitivity to changes in CH₄, N₂O, and equivalent effective stratospheric chlorine (EESC) concentration; emissions of NO_x and VOCs; and global mean surface temperature between 1850 and 2014 in each of the four models. For each atmospheric species, the fractional change in lifetime in 2014 relative to 1850 is normalised by the burden change to provide lifetime changes in each model in terms of parts per billion (ppb) concentration change or Mt yr⁻¹ emissions. The four models that provide data are used as minimum and maximum ranges of a parameter search (in v1.4.1, we expand the search range by a factor of 2, since the PRIMAP-Hist methane emissions are again likely to be an underestimate and do not find suitable parameters within the model range) to minimise the difference between observed CH₄ concentrations from Forster et al. (2023) and those calculated from Eq. (6). The 1750 emissions are subtracted from the time series when performing the lifetime calibration, as it is assumed that pre-industrial concentrations of methane are in approximate equilibrium with pre-industrial emissions.

The historical best-estimate calibrations are shown in Table 1. It can be seen that the methane lifetime in fair-calibrate v1.4.1 is nearly 17 years in the pre-industrial period, which is much longer than typically determined from ESMs. The best-estimate lifetime in FaIR from historical emissions is shown in Fig. 4a (grey line) and is indeed longer than that calculated from the sensitivities in each CMIP6 model across most of the historical period, though close to the AR6 value in the present day. In Fig. 4b, the historical concentrations from Forster et al. (2023) (black) are compared to the best estimate from FaIR using the lifetime calculated in Fig. 4a and run forward with best-estimate historical emissions. In Fig. 4c, the SSP methane concentrations are projected with the harmonised emissions, starting in 2022, and compared to the SSP concentrations time series (Meinshausen et al., 2017, 2020). In general, the harmonised methane concentration projections from fair-calibrate v1.4.1 are lower than in CMIP6 for high-methane emission scenarios and higher for low-emission futures. This is due in part to the nearly 10 years of additional historical emissions in the best-estimate time series compared to the SSPs, which started to diverge from a common history in 2015. For these projections, we use the best-estimate GMST anomalies from the SSPs derived in Lee et al. (2021).

Figure 4Methane lifetime calibration (v1.4.1). (a) Methane lifetime in the historical + SSP3-7.0 scenario for four ESMs (colours) and the lifetime from the FaIR calibration (grey). (b) Methane concentration calculated from historical methane emissions from PRIMAP-Hist + biomass burning emissions using the lifetime in panel (a), using FaIR (grey), and using the observed atmospheric concentrations (black) for 1750–2022 from IGCC (Forster et al., 2023). (c) Methane concentrations calculated from methane emissions for the eight main SSP scenarios using the harmonised future emission projections (thick lines) compared to the SSP scenarios (thin lines) (Meinshausen et al., 2017, 2020).

Download

Table 1Baseline CH₄ lifetime and sensitives (S_i) in lifetime due to changes in greenhouse gas concentrations, short-lived climate forcer emissions, and temperature in calibrations v1.4.1 and v1.4.0. Note that ppt stands for parts per trillion.

Download Print Version | Download XLSX

The lifetimes, historical concentrations, and future concentrations for the RCMIP emissions (calibration v1.4.0) are shown in Fig. S3, where it is observed that lifetimes and concentration projections are much closer to AR6 and CMIP6. This demonstrates that, first, the calibration is plausible (CMIP6 emissions give CMIP6 concentrations) and, second, that the methane lifetime calibration is very sensitive to the historical emission time series used. In Fig. S4b, we compare the methane emissions from the v1.4.0 and v1.4.1 calibrations. As 1750 emissions are subtracted from the total to report changes away from a pre-industrial equilibrium, the change in emissions (1750–2022) in v1.4.1 from PRIMAP-Hist is smaller than in v1.4.0, leading to longer atmospheric lifetimes necessary to reproduce concentrations.

Unlike in versions of FaIR prior to 2.0.0, we do not assume any natural methane emissions. In v1.3 of FaIR, for example, natural emissions were back-calculated with the assumption of a constant methane lifetime and held constant for future projections (Smith et al., 2018). It is well-known that wetlands emit large quantities of methane, and it is very likely that this effect is climate-dependent (Zhang et al., 2017). As the climate continues to warm, biogenic methane will be released from permafrost soils and clathrates – sources that most ESMs do not include at present. Including these natural sources is a development priority for future versions of FaIR.

It should be noted that the methane lifetimes derived are the best fits to observed concentrations across the 1750 to present-day period for each emission pathway and may not necessarily maintain an equilibrium concentration in 1750 with 1750 emissions. In v1.4.1, methane emissions in 1750 were around 38 Mt CH₄ and around 19 Mt CH₄ in v1.4.0, though v1.4.0 has a shorter lifetime for the same concentration. Methane emissions were not in equilibrium in 1750 and have steadily climbed over the last 2000 years (Meinshausen et al., 2017), with substantial variations due to agricultural and natural influences before then (Singarayer et al., 2011). Methane's relatively short lifetime and reactive nature make its calibration more difficult than longer-lived greenhouse gases such as CO₂ and N₂O, and the calibration strategy of the methane cycle depends on the goal of the user. In most cases using FaIR, this will be historical and future anthropogenic influences on climate for which the calibration that ensures historical emissions reproduce historical concentrations is most appropriate. Other use cases may require different calibration strategies.

3.1.4 Carbon cycle feedbacks

The carbon cycle is parameterised as a simple atmospheric decay model with four time constants, based on the impulse response functions of Joos et al. (2013). The time constants are scaled by a lifetime scaling factor that mimics the influence of carbon cycle feedbacks. This treatment is unchanged since the work of Leach et al. (2021, Sect. 2.1). A positive carbon cycle feedback reduces the efficacy of carbon sinks, thus effectively lengthening the atmospheric lifetime of CO₂.

The lifetime scaling factor is a function of the time-integrated airborne fraction of a CO₂ pulse over 100 years I₁₀₀ (Millar et al., 2017). I₁₀₀ is modified as

where r₀, r_U, r_T, and r_A are the pre-industrial time-integrated airborne fraction and its sensitivity to cumulative carbon uptake in land and ocean sinks ΔC_U, surface temperature anomaly ΔT, and airborne carbon ΔC_A respectively. Total cumulative emissions since pre-industrial is ΔC_A+ΔC_U.

The process for calibrating the carbon cycle feedbacks to 11 CMIP6 ESMs containing interactive carbon cycles is described in Leach et al. (2021, Sect. 3.2). The same coefficients derived in Leach et al. (2021) for the 11 ESMs are used in all calibrations to date.

3.1.5 Aerosol–cloud interactions

The effective radiative forcing due to aerosol–cloud interactions ERF_aci has been generalised:

where A_i is the emissions or concentration of a species, and the base subscript denotes its reference (usually pre-industrial) abundance. β is a scale factor, and s_i describes how sensitive a species is in contributing to ERF_aci. The generalisation allows for inclusion of more species that affect ERF_aci in addition to SO₂, BC, and OC that was modelled previously. The generalisation is useful as there is evidence of a large ERF_aci response to CH₄ in UKESM1-0-LL through methane's effect on competing for atmospheric oxidants, including OH, affecting the rate of new particle formation (O'Connor et al., 2022). As with earlier versions of FaIR, the form of Eq. (8) is inspired by Stevens (2015) but without any physical significance attached to the sensitivities s_i, allowing near-linear global mean responses in ERF_aci to changes in precursor abundances as postulated by some authors (Booth et al., 2018; Kretzschmar et al., 2017) and exhibited in some models (Smith et al., 2021a).

A total of 13 CMIP6 models provided results from transient aerosol experiments in AerChemMIP and the Radiative Forcing Model Intercomparison Project (RFMIP) (Table 2) that allow calculation of aerosol ERF. The breakdown of shortwave aerosol ERF into aerosol–radiation interactions (ERF_ari) and ERF_aci is performed using the approximate partial radiative perturbation (APRP) method (Taylor et al., 2007), following the logic of Zelinka et al. (2014, 2023). Longwave contributions to ERF_aci are estimated from the cloud radiative effect, with ERF_ari estimated as the difference between the longwave components of ERF and ERF_aci.

From the diagnosed ERF_aci in each model, a least squares curve fit of ERF_aci to historical emissions by fitting $s_{{\mathrm{SO}}_{\mathrm{2}}}$, s_BC, s_OC, and β is found (Table 2) using Eq. (8). The comparison of model-derived ERF_aci to the best fit from Eq. (8) is shown in Fig. 5.

Using Eq. (8), a wide range of ERF_aci trajectories are possible, and parameter estimates for β and individual species sensitivities span orders of magnitude. Where one or two of $s_{{\mathrm{SO}}_{\mathrm{2}}}$, s_BC, and s_OC are close to zero (CanESM5 and UKESM1-0-LL), this indicates that the species has little influence on ERF_aci in that model (e.g. UKESM1-0-LL's ERF_aci response is purely driven by sulfate in aerosol-only forcing experiments). Where all three of $s_{{\mathrm{SO}}_{\mathrm{2}}}$, s_BC, and s_OC are close to zero, and β has large magnitude (the two Geophysical Fluid Dynamics Laboratory (GFDL) models and NorESM2-LM), this indicates that ERF_aci behaves linearly in emissions from the Taylor expansion of log (1+x) for small x (Smith et al., 2021a). In the case of NorESM2-LM, the coefficient for BC is so small that it is effectively zero, with the ERF_aci response being linear with sulfate and OC.

Table 2Models used to calibrate forcing from aerosol–cloud radiation interactions and their parameter best fit values from Eq. (8).

Download Print Version | Download XLSX

Figure 5Calibrations of the ERF_aci relationship in FaIR (Eq. 8; coloured lines) to the derived ERF_aci from 13 CMIP6 models (grey lines). Extrapolation back to 1750 is shown in all cases, and extrapolation forward to 2100 is shown under SSP2-4.5 emissions where model simulations were not extended beyond 2014.

Download

3.1.6 Ozone

The best-estimate historical ozone ERF time series from Skeie et al. (2020) is used to calibrate the role of ozone precursors to ozone forcing. As in AR6, tropospheric and stratospheric ozone are not considered separately. Again following the AR6 methodology, we select six models from the 12 coupled historical CMIP6 models analysed in Skeie et al. (2020) that are relatively independent from each other, have full stratospheric and tropospheric chemistry enabled, and reproduce expected behaviour for the overall time history of ozone ERF. The six models used are BCC-ESM1, CESM2(WACCM6), GFDL-ESM4, GISS-E2-1-H, MRI-ESM2-0, and OsloCTM3. Skeie et al. (2020) provides historical ozone forcing for 1850–2010 in these models, and following Skeie et al. (2020), we add +0.03 W m⁻² to the time series to represent the change from 1750 to 1850. The Oslo-CTM3 model provided results under SSP2-4.5 to 2020, which was also used in calibration.

As ozone ERF includes a contribution from temperature change and is calibrated from coupled historical runs, historical warming is backed out using a temperature feedback of −0.037 W m⁻² K⁻¹ (Thornhill et al., 2021a) and historical GMST from Forster et al. (2023). For this “no-feedback” ERF time series, we find a least squares fit to the change in emissions of NO_x, VOC, and CO and concentrations of CH₄, N₂O, and EESC (Fig. 6). The lower and upper bounds of the search ranges for the parameter fits are the very likely range for each precursor in Thornhill et al. (2021b), which is also scaled up to account for the difference in best-estimate ozone forcing between models participating in AerChemMIP in Thornhill et al. (2021b) and the six-model subset in Skeie et al. (2020).

Figure 6Comparison of the ozone ERF time series from Skeie et al. (2020) (black) to the estimate from emissions and concentration precursors (grey). The estimated impact of temperature on ozone forcing has been backed out of the time series from Skeie et al. (2020) and is not included in the model fit.

Download

Similar to the methane lifetime calibration, we derive a coefficient for each precursor species relating emissions or concentrations of each to the ozone ERF. Uncertainty sampling for the prior distribution is described in Sect. 3.2.5.

3.2.1 Climate response

An 11-dimensional kernel density estimate is generated from the energy balance model parameters that were calibrated on 49 CMIP6 models (Fig. S5). $F_{\mathrm{4}\times {\text{CO}}_{\mathrm{2}}}$ is not used in the climate response of FaIR but is used in the theoretical calculation of ECS and TCR. All parameters of the energy balance model are strictly positive, so parameter sets containing negative values are discarded and redrawn until the 1.6 million threshold is reached. We also discard and redraw instances of κ₁<0.3 W m⁻² K⁻¹, C₁<1.8 W yr m⁻² K⁻¹, C₃<C₂, C₂<C₁, and γ<0.5. The κ₁ threshold puts an upper bound on the ECS prior of around 13 °C, and the other limits ensure model stability.

3.2.2 Aerosol–cloud interactions

Similar to the climate response, we draw correlated kernel density estimates for $\log (s_{{\mathrm{SO}}_{\mathrm{2}}}$), log (s_BC), and log (s_OC). We calculate an unscaled ERF_aci for the 2005–2014 mean relative to 1750 for each parameter set. The unscaled ERF_aci is then scaled to reproduce a draw from a trapezoid distribution with limits at −2.2 and +0.2 W m⁻² and plateau from −1.6 to −0.4 W m⁻² to represent the ERF_aci for 2005–2014 relative to 1750, which selects the β value to use for that parameter set. This process is similar to that of both Smith et al. (2021a) and AR6 (Forster et al., 2021). The prior distribution is chosen to give a wide but plausible range around the ERF_aci distribution for the present day assessed by the IPCC (Forster et al., 2021), which was −1.0 W m⁻² for a nominal 2014 date relative to 1750.

3.2.3 Aerosol–radiation interactions

The ERF_ari contributions are not sampled directly from CMIP6 models, though much of the basis of this assessment is rooted in AerChemMIP (Thornhill et al., 2021b). AR6 assessed that several species (CH₄, N₂O, halogenated compounds, sulfate, BC, OC, nitrate, and VOCs) contribute directly or indirectly to ERF_ari, though only sulfate, BC, OC, and NH₃ are significant. We use the contributions to ERF_ari assessed in AR6 with the relative uncertainty from each precursor (Szopa et al., 2021) as prior distributions (Table 3) and scale both the best-estimate and uncertainty range of the ERF_ari from each precursor to reproduce the IPCC AR6 distribution of $-\mathrm{0.3}\pm \mathrm{0.3}$ W m⁻² (Forster et al., 2021). All ranges quoted are for 5th to 95th percentile, unless otherwise stated.

Table 3Distributions of the contributions to the direct aerosol ERF sampled in fair-calibrate v1.4.1. Uncertainty ranges are shown as 90 % ranges and sampled from a Gaussian distribution.

Download Print Version | Download XLSX

3.2.4 Carbon cycle and initial CO₂ concentration

A four-dimensional kernel density estimate is drawn from the r₀, r_U, r_T, and r_A parameters from the 11 models calibrated in Leach et al. (2021). As part of the carbon cycle sampling, we draw CO₂ concentration values in 1750 using the IPCC AR6 best estimate and uncertainty of the 278.3±2.9 ppm (5 %–95 %) range (Gulev et al., 2021), using a Gaussian distribution.

3.2.5 Ozone

The coefficients relating emissions or concentrations of chemically relevant precursors to ozone ERF take their mean value from the bounded least square fit derived in Sect. 3.1.6, and their uncertainty values are sampled by applying the scaled 5 %–95 % uncertainty range from Thornhill et al. (2021b) to this best-estimate value. This means that some precursor ranges are outside the range of that described by Thornhill et al. (2021b), though only seven models (fewer for some precursors) provided the necessary experiments in Thornhill et al. (2021b), and thus AerChemMIP represents a small ensemble of opportunity.

Table 4Distributions of the contributions to the ozone ERF sampled in fair-calibrate v1.4.1. Uncertainty ranges are shown as 90 % ranges and sampled from a Gaussian distribution.

Download Print Version | Download XLSX

3.2.6 ERF scalings

Forcing uncertainties in ERF_ari, ERF_aci, and ozone are sampled from the contribution to total forcing from their precursor species, as described in previous sections. For other major categories of forcings, we use the IPCC AR6 ranges (Forster et al., 2021) as relative uncertainty factors to scale the ERF (Table 5).

For CO₂, we use the sampled $F_{\mathrm{4}\times {\text{CO}}_{\mathrm{2}}}$ value from the climate response calibration and perform a quantile mapping to derive a scaling factor for CO₂ forcing that is Gaussian. While this does not preserve the shape of the $F_{\mathrm{4}\times {\text{CO}}_{\mathrm{2}}}$ distribution kernel, it does map low 4×CO₂ forcings to low CO₂ scalings, and vice versa.

Table 5Forcing scaling factors used to translate the raw best estimate from FaIR to IPCC-assessed uncertainty ranges (Forster et al., 2021). Scaling uncertainty ranges are 5 %–95 %. Except for the solar trend, median distribution values are 1.

^* Contrail forcing is not used in v1.4.0 and v1.4.1 but is included in other versions.

Download Print Version | Download XLSX

3.3.1 Step 1: root mean squared difference with respect to historical

The root mean squared (rms) difference in each ensemble member's GMST anomaly projection compared to the historical values for 1850–2022 is used as a simple pass/fail criterion for ruling out parameter sets that are inconsistent with historical observed warming. Ensemble members that have an rms difference that is greater than 0.17 °C are rejected. The mean of four GMST datasets (HadCRUT5, Berkeley Earth, NOAAGlobalTemp, and Kadow) from Forster et al. (2023) is used as the historical GMST dataset for comparison. The choice of 0.17 °C is somewhat arbitrary, which balances sufficient variability in the historical record to allow for observational uncertainty with the need for projections that are true to observations. By design, this threshold roughly reproduces the uncertainty range in present-day GMST relative to the pre-industrial range assessed by the IPCC (Gulev et al., 2021), whereas a more stringent threshold may over-constrain both the historical observational uncertainty and scope for future climate projection uncertainty (Fig. 7). Internal variability is switched on for this historical comparison to allow for the possibility that the historical record can be well-simulated by chance in mean state climate configurations that would be warmer or cooler than expected (e.g. a strong pattern effect; Andrews et al., 2018). This step reduces the ensemble size from 1.6 million to 224 342, ruling out around 86 % of the original ensemble.

Figure 7 compares the 10 ensemble members with the lowest RMSE relative to observations (blue; RMSE ≈0.10 °C) with the 10 largest RMSE members that still meet the RMSE constraint (red; RMSE ≈0.17 °C). Figure 7 shows that runs with low internal variability tend to result in the closest correspondence with historical observed temperature, and therefore, the final ensemble could be biased towards ensemble members with smaller variability. A formal analysis of the internal variability characteristics in relation to observations is not performed in this version of fair-calibrate, though it could be added to the constraining criteria in the future.

Alongside or instead of RMSE, a correlation metric could be used to evaluate goodness of fit between the observations and the model. However, RMSE encapsulates goodness of fit into a single number and is sensitive to model runs that overall warm too quickly or too slowly. Correlation coefficients would not differentiate simulations that had the right shape of historical warming but warmed too quickly or slowly.

Figure 7Comparison of the 10 ensemble members with the smallest RMSE error (blue) compared to the historical best-estimate GMST from the Indicators of Global Climate Change 2022 (Forster et al., 2023) (black) with 10 of the largest RMSEs (red) that passed this first historical constraining step.

Download

3.3.2 Step 2: reweighting based on observed and assessed climate metrics

The second constraining step takes the ensemble members that passed the RMSE threshold and simultaneously fits the projections to eight target distributions (Fig. 8). For each target distribution, either a Gaussian (if symmetric) or skew normal (if asymmetric) continuous probability distribution is constructed from the 5th, 50th, and 95th percentiles of the variable's uncertainty range. As a three-parameter distribution, a skew normal can uniquely fit three specified quantiles. For symmetric distributions, the number of degrees of freedom is reduced to two (by imposition of symmetry), and the Gaussian is a natural choice, as well as being a general form of the skew normal. The percentiles of the target distributions are shown in the first eight rows of Table 6. Emergent parameters (ECS, TCR, and aerosol forcing ranges) are taken from the IPCC AR6 WG1 Chap. 7 (Forster et al., 2021), and updated climate observations (GMST, OHC, and CO₂ concentrations) are taken from the Indicators of Global Climate Change 2022 (Forster et al., 2023).

Figure 8Comparison of distributions of key climate metrics (Table 6) in each step of the constraining process. The prior distributions from the 1.6 million member prior ensemble are in blue. The first constraining step using the RMSE comparison to historical temperature is in yellow. The second constraining step that reweights each distribution to its target is in red. The target distribution is in black. The goal is for the red distribution to be as close as possible to the black across all metrics.

Download

The ensemble size in the final reweighted constrained distribution is a user choice. Typically, ensemble sizes of a few hundred to a few thousand are used for projections using reduced-complexity models (Nicholls et al., 2021), which allows for full exploration of the uncertainty space while keeping the number of simulations small enough to allow for efficient computation. For the final posterior distribution in calibrations v1.4.0 and v1.4.1, we select 841 ensemble members from an effective ensemble size of 4356. Moreover, 841 is 1 more than a highly composite number and allows many quantiles of the full distribution to correspond to a single-ensemble member at each point in time.

The posterior ensemble size being one more than a highly composite number is simply an author preference; it is more important to ensure that the posterior is (1) large enough to provide a dense coverage of posterior constraint distributions and (2) small enough that it can provide an unbiased sample size after likelihood weighting. Condition (1) generally imposes a lower bound of around 500 ensemble members, and condition (2) suggests that the effective sample size should be around 5 or more times larger than the target posterior size. If both conditions cannot be simultaneously met, a larger or differently sampled prior or a relaxation of one or more constraints is required.

The evolution of GMST projections from the prior ensemble to the historical RMSE constraint, and finally the reweighted constrained ensemble, is shown in Fig. 9. The prior ensemble allows for a wide range of projections, the majority of which are clearly incompatible with historical GMST (Fig. 9a). The RMSE threshold step, alongside producing historically reasonable projections, substantially reduces the range in projected future warming (Fig. 9b). However, low and particularly very high future warmings pass the historical RMSE constraint. The reweighting step provides a narrower band on historical warming, as well as reducing the spread in future warming further (Fig. 9c). The 5 %–95 % ranges of future warming are similar between the RMSE constraint and the reweighted posterior, but the latter distribution constrains out much of the warm and cool tails of the distribution that passes the RMSE constraint.

Figure 9Progression of projections using the historical + harmonised SSP2-4.5 emissions for (a) all prior ensemble members, (b) the RMSE <0.17 °C first constraining step, and (c) the final reweighted and constrained posterior. In each plot, progressively darker shaded regions correspond to the minimum–maximum, 5 %–95 %, and 16 %–84 % ranges. The black line is the ensemble median, and the blue line is a historical best-estimate GMST from the Indicators of Global Climate Change 2022 (Forster et al., 2023).

Download

Figure 10 shows the distributions of the 45 parameters used to construct the prior samples (blue histograms) and the reweighted posterior (red histograms). Table S3 lists the parameters and the part of the model that is being affected, as well as its location within the paper. For some distributions, the constraining steps create posteriors that are differently shaped to the priors. Sometimes this is by design. For example, κ₁, the climate feedback parameter, is inversely related to ECS, and the IPCC constraint downweights the likelihood of “hot” combinations (noting that the prior distribution is constructed from CMIP6 models, many of which have higher climate sensitivity than the 95th percentile of 5 °C assessed in IPCC AR6). Occasionally, distributions are multi-modal, such as the parameters that define the ERF_aci shape, due to the model calibrations themselves spanning several orders of magnitude.

Figure 10Prior (blue) and reweighted posterior (red) distributions of the 45 parameters sampled. For a description of what the parameters correspond to, refer to Table S3.

Download