An energy balance model (EBM) of the annual global mean surface temperature is described and calibrated to the sensitivity and temporal dynamics of the Goddard Institute for Space Studies modelE global climate model (GCM). The effective radiative forcings of 10 agents are estimated over the past 2009 years and used as inputs to the model. Temperatures are relatively stable from around A.D. 300 until a “Medieval Climate Anomaly” starting around A.D. 1050. This is ended by a massive volcanic eruption in A.D. 1258, which initiates a multicentury era of low and relatively variable global mean temperatures, including a “Little Ice Age” A.D. 1588–1720. This era only ends at the beginning of the 20th century. The model estimate of forced centennial variability is smaller than the observed variability in reconstructions over the past two millennia. Also, the default parameterization results in less warming than observed over A.D. 1910–1944. Prediction uncertainty in the pre-industrial era is dominated by solar forcing, with the climate feedback factor and volcanic aerosols also playing important roles. In contrast, prediction uncertainty post–A.D. 1750 is much higher and dominated by uncertainties in direct and indirect aerosol and land use forcings. Improving estimates of these will greatly increase our ability to attribute observed temperature variability to contemporary forcings.
 Understanding current warming trends on Earth require that we separate the impacts of anthropogenic forcings from those of natural factors such as solar variability and massive volcanic eruptions [Hegerl et al., 2007]. Energy balance models (EBMs) enable rapid and tractable assessments of these roles as well as analysis of uncertainty arising from imperfect knowledge of forcings and system sensitivity, with the latter potentially prescribed from more complex models such as global climate models (GCMs). Furthermore, EBMs can be developed into simple Earth system models through the inclusion of biogeochemical parameterizations, and be used thereby to improve understanding of coupled feedbacks.
 This paper describes a new simple EBM and the development of a comprehensive mean annual radiative forcing data set consisting of ten agents over the past 2009 years. The EBM is calibrated to the global mean behavior of a GCM and predicted annual variability in global mean surface temperature is compared to reconstructions and the instrumental record. An uncertainty analysis is performed to determine the relative importance of the uncertainty in different forcings and model parameters for model prediction.
2. Energy Balance Model
 An EBM is used here to calculate the evolution of Earth's global mean surface temperature in response to changing radiative forcing, and is based on the analysis of Hansen et al.  (hereinafter H84). H84 separate the equilibrium surface temperature response to a change in solar forcing into a component that restores radiative equilibrium with space and a component that results from internal system feedbacks such as changes in the vertical atmospheric temperature structure, water vapor, ground albedo, and clouds
where ΔTeq is the change in surface temperature required to restore radiative equilibrium following a change in solar irradiance, ΔTo is the change in equilibrium surface temperature in the absence of feedbacks, and f is the net feedback factor due to feedbacks initiated by changes in solar irradiance (dimensionless). ΔTo is the same as the change in the effective radiating temperature of Earth, Te. At equilibrium this effective temperature is that which balances absorbed total solar radiation with longwave energy loss to space, and therefore Te is given by (H84)
where So is solar irradiance, A is Earth's albedo, s is the mean flux of absorbed radiation per unit area, and σ is the Stefan-Boltzmann constant. H84 used this relationship between radiative temperature and absorbed solar radiation to derive the rate of change in effective temperature with change in forcing
 Therefore the relative change in the effective temperature equals 0.25× the relative change in solar forcing. This framework is extended here to include the response of Earth's surface temperature to changes in greenhouse gas mixing ratios as well as solar irradiance and thereby to other nonsolar forcings.
 If there were no feedbacks, following a sufficient length of time after a change in the mixing ratio of a greenhouse gas the global mean surface temperature would reach an equilibrium temperature analogous to To (To,g). If feedbacks are treated as for the response to solar irradiance, then the total change in surface temperature at equilibrium due to changed mixing ratio of a greenhouse gas (Teq,g) is given by
where fg is the net feedback factor due to feedbacks initiated by changes in greenhouse gas mixing ratios (dimensionless).
 Using a GCM, H84 computed that the global mean equilibrium surface temperature response to a +2% change in solar irradiance or a doubling of atmospheric CO2 is very similar (i.e., ∼4°C). In these experiments, a 2% increase in solar irradiance corresponded to a radiative forcing at the tropopause of +4.8 W m−2, and a doubling of CO2 corresponded to a global mean forcing of about 4 W m−2, with peak flux into the ocean surface of 4–5 W m−2 in both experiments. This similarity between surface temperature responses to both solar and greenhouse gas forcings is used here to extend the description of the solar response given above to also include nonsolar forcings. Applying the relationship between solar forcing and surface temperature given in equation (3) to nonsolar forcings does not mean to imply that the model assumes that the physics of the system responds the same to all forcings, but rather that the magnitude and functional form of the relationship is similar.
 The EBM accounts for the response of the equilibrium Earth radiative temperature to solar forcing using (3). The similarity between the response to solar and greenhouse gas forcings is used to justify extending this equation to calculate the equilibrium surface temperature response to simultaneous solar and nonsolar forcings in the absence of feedbacks (To,sF)
where ΔF(GHG) is nonsolar forcing. The change in the total equilibrium surface temperature in response to solar and nonsolar forcings, Teq,sF, is then calculated using the net feedback factor as in equation (4),
 All changes in the physical components of the Earth system that result in feedbacks, such as atmospheric water vapor, surface albedo, and cloud height, are subsumed into the net feedback factor.
 In the EBM, changes in Te in response to changes in solar irradiance, and changes in To,sg in response to total forcing both occur with relationships conditioned on Te/(4s), and total surface temperature changes result from the amplification of To,sg by the net feedback factor (equation (6)). This construction reproduces the equilibrium global mean surface temperature responses to solar and CO2 forcings simulated in H84.
 As indicated, H84 found that the radiative forcing from doubled CO2 is lower than from +2% solar irradiance. However this was more than compensated by a higher feedback factor associated with changes in CO2 (i.e., fg = 3.45 compared with f = 2.96). Therefore, when all feedbacks are considered, the equilibrium global mean surface temperature increased slightly more with doubled CO2 (i.e., +4.2°C) than it did with increased solar irradiance (i.e., +4.0°C). This difference in the magnitude of the surface temperature response to different forcings is incorporated into the EBM through the use of forcing efficacies, as described below. The different feedback factors suggested by the results of H84 are not incorporated into the formulation explicitly. However, fg − f = 0.49 is well within the range of parameter values used in the uncertainty analysis (see below).
 Taking A = 0.31 and So = 1360.78 W m−2, Earth's radiative temperature, Te, is initially taken to be 253.66 K in the simulations described below. This value of So used here is lower than that assumed by H84, and results from improved observations made by the TIM instrument flying on the SORCE, and is taken as the solar minimum reported by Kopp et al. . Surface temperature is initially assumed to be at its equilibrium value, taken to be 33 K higher than Te (i.e., 286.66 K) (H84).
 The default value for the net feedback factor is calculated from the sensitivity of the model III version of the NASA Goddard Institute for Space Studies (GISS) modelE GCM as reported by Hansen et al.  (hereinafter H05). This model is described by Schmidt et al. . The GCM equilibrium global mean surface temperature sensitivity to doubled CO2 from 291 ppm is ∼2.7°C (H05). The effective (see below) global mean radiative forcing due to doubled CO2 in this model is 4.22 W m−2, which yields ΔTo = 1.14 K, and therefore fg (= f) = 2.7/1.13 = 2.37.
 Global mean surface temperature is assumed to continually decay towards Teq,sg, with temporal dynamics calibrated to the simulation results of the GISS modelE GCM. H05 investigated the sensitivity and dynamical responses of the GCM to doubled CO2 using alternative ocean representations. The mean of ensemble GCM simulations minus their controls obtained using a 300 yr doubled CO2 GCM experiment with the coupled dynamical ocean of Russell et al.  is used here (Figure 1).
 The GCM surface temperature dynamical response could not be captured with a single heat capacity. The simplest approximation that retains sufficient skill is to assume three components to the temperature response: fast, medium, and slow. Assuming these three components of the thermal response scale by orders of magnitude, their e-folding times are therefore related to one another by
where τf, τm, and τsl are the e-folding times of the fast, medium, and slow components, respectively. This construction is not intended to have any direct physical significance, although the three thermal components might notionally correspond to the land and ocean surface, ocean mixed layer, and the deep ocean. The intention is simply to fit the GCM results, and could easily be reparameterized to fit other GCMs.
 Each contributing component is assumed to decay towards Teq,sF according to
where Ti is the contribution to the surface temperature from component i such that
where α, β, and γ are the fractions of the total thermal response due to the respective components (i.e. α + β + γ = 1), and τi is the e-folding time of component i. An excellent fit to the GCM is obtained with τf = 2 yr, α = 0.4, and β = 0.16 (Figure 1), and these values are taken as default parameterizations in the simulations presented here.
3. Radiative Forcings
 The annual global mean radiative forcings due to the well-mixed anthropogenic greenhouse gases CO2, CH4, N2O, and CFC-type halogenated compounds, ozone (O3), CH4-derived stratospheric water vapor, solar irradiance, land use, changes in snow and ice albedo due to deposition of black carbon (BC), stratospheric aerosols of volcanic origin, tropospheric BC, and the direct and indirect effects of reflective aerosols are estimated here for A.D. 1–2009.
 Forcings vary in their effectiveness at producing global surface temperature change, due primarily to spatial variability. It is therefore more relevant to calculate an effective forcing for input to an EBM such as used here. Effective forcing, Fe (W m−2), is defined by H05 as
where Ea is the efficacy of the forcing agent, Fa is the adjusted forcing, ΔTs is the equilibrium global temperature change, and γ is the equilibrium climate sensitivity to CO2 forcing (°C per W m−2). The concept can also be usefully applied to nonequilibrium conditions, as long as the standard response is carefully quantified. Effective forcing is then the radiative forcing by CO2 that would cause the same global mean surface temperature change after the same time period, and Ea therefore depends on the time period considered for adjustment of surface temperature. Adjusted forcing, Fa, is the forcing after the stratospheric temperature has been allowed to adjust to the perturbation and is the standard forcing employed by the Intergovernmental Panel on Climate Change (IPCC) .
H05 used the GISS modelE GCM to compute the global mean surface temperature response to a standard CO2 forcing against which to measure the efficacies of other forcing agents at producing surface temperature change. This standard forcing response was computed as the global mean surface temperature change resulting from a 1.5× increase in CO2 mixing ratio from the A.D. 1880 value of 291 ppm, computed over model years 81–120. Surface temperature increased by 1.103°C and Fa was computed to be 2.39 W m−2 when using the WMO tropopause definition, and 2.37 W m−2 for a fixed pressure-height definition (H05), giving ΔTs/Fa = γ ∼ 0.463°C (W m−2)−1. Transient effective forcings were calculated relative to this value by Hansen et al.  (hereinafter H07) and M. Sato and J. Hansen (Global Mean Effective Forcing, accessed 10 June 2010, http://data.giss.nasa.gov/modelforce/RadF.txt) (hereinafter Sato and Hansen, online data set, 2010), and are used here to derive estimates of all ten forcings over A.D. 1–2009 as described in subsequent sections.
3.1. Atmospheric CO2
3.1.1. CO2 Radiative Forcing
 The GISS modelE GCM simulation results reported by H05 are used here to compute the effective annual global mean radiative forcing by CO2 over A.D. 1–2009 from estimates of changes in annual mean atmospheric CO2 mixing ratio. H05 computed the effective radiative forcing of a wide range of CO2 mixing ratios relative to an unperturbed state computed using a value of 291 ppm (i.e. its value in A.D. 1880) (Figure 2a). A range of simplified expressions for the dependency of global mean forcing on CO2 mixing ratio are given by IPCC [2001, Table 6.2], and the best fit to the simulations of H05 is obtained here with fitted coefficients to
where C is the varying CO2 mixing ratio (ppm) and Co is its unperturbed value. Fitted coefficients are α = 3.461 and β = 0.2514, and the fitted relationship is shown in Figure 2a together with the GCM simulation results. Fitting was achieved by minimizing the sum of the squares of the errors, and this fitting technique is used throughout this paper unless otherwise stated.
 The adjusted forcing by CO2 (i.e. Fa(CO2)) reported by H05 is also shown on Figure 2a, and is close to Fe(CO2) up to ∼1000 ppm, above which Fe(CO2) increases discernibly faster than Fa(CO2). Fitted coefficients for Fa(CO2) are α = 4.704 and β = 0.1204. This divergence reflects the increasing efficacy of CO2 in producing surface temperature change as its mixing ratio increases, and is the result of the increasing strength of climate feedbacks, such as the water vapor feedback, with forcing (H05). It needs to be borne in mind that because the relative importance of different feedbacks is time scale–dependent, the details of the relationships derived from the GCM are to some extent dependent on the time scale used (H05 calculated a 100 year response). Also shown on Figure 2a is a commonly used estimate of radiative forcing by CO2 [Myhre et al., 1998], computed relative to 291 ppm. At 400 ppm, Fe(CO2) is 10.2% greater than the estimate of Myhre et al.  (see inset) and therefore, given the importance of CO2 as a climate forcing on decadal and greater time scales, this difference will likely have implications for the attribution of surface temperature to forcing agents, as well as predictions of future impacts of rising CO2.
3.1.2. CO2 Mixing Ratios
 Annual global mean tropospheric CO2 mixing ratios for A.D. 1–1979 are constructed from ice core measurements and for A.D. 1980–2009 from atmospheric measurements.
 Measurements made on the ice and firn collected at Law Dome in Antarctica [MacFarling Meure et al., 2006] are used to calculate global mean values over A.D. 1–1979 using a derived relationship with fossil CO2 emissions. The latitudinal gradient in atmospheric CO2 is changing with time as fossil CO2, of primarily northern hemispheric origin, is superimposed on the natural gradient. The difference between the global mean CO2 mixing ratio and the value measured in the Law Dome ice core is assumed to increase linearly with the rate of fossil CO2 emissions, following Taylor and Orr ,
where χD (ppm) is the global mean–South Pole surface CO2 mixing ratio difference (ppm), fCO2 is the rate of fossil CO2 emissions (Pg C yr−1), α is the sensitivity of the difference to the flux of fossil CO2 (ppm (Pg C yr−1)−1), and β is the difference under conditions of no fossil CO2 (ppm). β represents the effect of the natural component of the latitudinal CO2 gradient, and is assumed constant in time (but see Taylor and Orr  for arguments why some variation might be expected).
 The coefficients in equation (12) are derived here using the relationships between the globally averaged marine surface annual mean CO2 mixing ratio of the NOAA Earth System Research Laboratory [Tans, 2010], the annual mean flask air measurements of CO2 mixing ratio at the South Pole by the Scripps Institution of Oceanography [Keeling et al., 2008], and the global sum of fossil fuel burning, cement manufacturing, and gas flaring-related CO2 emissions taken from Boden et al. , with emissions for A.D. 2008 and A.D. 2009 taken from BP (BP Statistical Review of World Energy, http://www.bp.com/productlanding.do?categoryId=6929&contentId=7044622) and values prior to A.D. 1751 estimated from global human population as follows.
 Global mean annual human population for A.D. 1–1940 is derived here as a linear interpolation of the lower estimates of the compilation summary of U.S. Census Bureau, Population Division [2010a], and merged with annual values for A.D. 1950–2010 from U.S. Census Bureau, Population Division [2010b]. Both human population and fossil CO2 emissions increase exponentially from about A.D. 1750, but the greater rate of increase in emissions than population results in an exponential relationship between the two (Figure 3a). An exponential equation is fitted between the fossil CO2 emissions and human population over A.D. 1751–1829 (a strong discontinuity in emissions occurs in A.D. 1830), with the following providing an excellent fit:
 The relationship between χD and over A.D. 1980–2007, the years when the three data sets coincide, is shown in Figure 3b. A best fit linear regression of equation (12) to this relationship is obtained with α = −0.165 ppm (Pg C yr−1)−1 and β = 0.297 ppm (Figure 3b).
 The Law Dome ice core measurements of MacFarling Meure et al.  are here linearly interpolated to annual values and, together with the annual fossil CO2 emissions and equation (12), are used to estimate global mean CO2 mixing ratios for A.D. 1–1979. The resulting value for A.D. 1750 is 276.29 ppm. The ice core-derived and direct atmospheric measurement-derived estimates of Tans  are merged to produce a complete A.D. 1–2009 annual time series of global mean atmospheric CO2 mixing ratios (Figure 4a).
 These mixing ratios, together with equation (11) parameterized as described above, yield an effective forcing of 1.92 W m−2 over A.D. 1–2009 (1.65 W m−2 over A.D. 1750–2000) (Table 1 and Figure 5a). IPCC  presented a value of 1.46 W m−2 for the adjusted CO2 forcing over A.D. 1750–2000, whereas the value calculated here is 1.67 W m−2, 14.4% higher.
Table 1. Effective Radiative Forcings (W m−2) due to Different Agentsa
Net is sum (actual net used in simulations ×0.89 this value).
3.2. Atmospheric CH4
3.2.1. CH4 Radiative Forcing
 The simplified expression for the direct forcing due to CH4 given in Table 6.2 of IPCC  is fitted here to the values computed by H05 (Figure 2b). This expression includes dependencies on both CH4 and N2O mixing ratios
where M is the CH4 mixing ratio (ppb), Mo and No are the unperturbed mixing ratios of CH4 and N2O, and f(M, N) = β ln[1 + 2.01 × 10−5(MN)0.75 + 5.31 × 10−15M(MN)1.52], where N is the N2O mixing ratio (ppb). H05 computed Fe(CH4) for two large changes in mixing ratio from A.D. 1880 values, but one is larger than the valid range of equation (14) [Shine et al., 1990]. Therefore, to further constrain the relationship at lower mixing ratios, a value of Fe(CH4) = 0.58 W m−2 for A.D. 1750–2000 is included in the regression, estimated from Figure 28 of H05. Equation (14) is then fitted to these two values, giving α = −0.00365 and β = −4.50 (Figure 2b). It proved impossible to improve the fit by changing any of the other coefficients. The resulting forcings are compared with those reported by IPCC  in Figure 2b, together with the simulated and estimated values of Fa(CH4) (fitted regression coefficients for Fa(CH4) are α = 0.00596 and β = −3.18).
 The efficacy of CH4 at producing surface temperature change is substantially greater than 100%, and therefore Fe(CH4) increases faster with mixing ratio than Fa(CH4). IPCC  report a forcing of 0.48 W m−2 over A.D. 1750–2000, whereas H05 computed an adjusted direct forcing of 0.55 W m−2, ∼15% greater. This difference increases to ∼21% when comparing the effective forcing from H05 with the IPCC  value.
3.2.2. CH4 Mixing Ratios
 Annual global mean atmospheric CH4 mixing ratios for A.D. 1–2009 are estimated in a similar manner to CO2. Values for A.D. 1850–2008 are taken directly from J. Hansen and M. Sato (Global Mean CH4 Mixing Ratios, accessed 31 May 2010, http://data.giss.nasa.gov/modelforce/ghgases/Fig1B.ext.txt), which were derived from ice core and atmospheric measurements using methodology described by Hansen and Sato . A value for A.D. 2009 is estimated by adding the A.D. 2007–2008 change to the A.D. 2008 value.
 Pre–A.D. 1850 values are calculated from measurements made in the same ice core as used for CO2 [i.e., MacFarling Meure et al., 2006]. These measurements are first interpolated to annual values, and annual differences with the global values calculated for A.D. 1850–1950. These values are then regressed against fossil CO2 emissions, and the relationship fitted using equation (12), yielding α = 5.39 ppb (Pg C yr−1) and β = 15.4 ppb (Figure 3c). Pre–A.D. 1850 values are then obtained by adding the annual global–South Pole differences predicted from fossil CO2 emissions by equation (12) to the interpolated ice core measurements. These ice core–derived and direct atmospheric measurement–derived estimates are then merged to produce a complete A.D. 1–2009 time series of global mean atmospheric CH4 mixing ratios (Figure 4b).
 Comparing different ice core records, Chappellaz et al.  found that the Greenland to Antarctica CH4 gradient varied between 33 and 50 ppb over the Holocene due to changing roles of the tropics and boreal regions as sources. Most of the current gradient of about 140 ppb is in the Northern Hemisphere, and so if this were also true in pre-industrial conditions it would be consistent with the global mean–South Pole difference of ∼15 ppb calculated here.
 These estimated global mean CH4 mixing ratios, together with equation (14), yield Fe(CH4) = 0.63 W m−2 over A.D. 1–2009 (0.60 W m−2 over A.D. 1750–2000) (Table 1 and Figure 5b). CH4 adjusted forcing calculated here is 0.55 W m−2 over A.D. 1750–2000, 14.6% higher than the value reported by IPCC . CH4 also has indirect effects through stratospheric H2O and, especially, tropospheric ozone (O3); these are separately estimated below.
3.3. Atmospheric N2O
3.3.1. N2O Radiative Forcing
H05 computed the effective radiative forcing due to the N2O mixing ratio change 278–1898 ppb. They also computed Fa(N2O) = 0.15 W m−2 for 278–316 ppb, which gives Fe(N2O) = 0.16 W m−2, assuming the value of Ea (1.04) computed for the greater change. The resulting two values of Fe(N2O) are here fitted to equation (14), but with M and N swapped, yielding α = 0.0573 (β is kept to the same value as for CH4), and the result is shown in Figure 2c. Also shown are the curves for Fa(N2O) derived as for Fe(N2O) (α = 0.0744) and that reported by IPCC .
 The three estimates are very similar around contemporary mixing ratios (inset), but the H05-derived values increase noticeably more rapidly with mixing ratios above ∼0.5 ppm.
3.3.2. N2O Mixing Ratios
 Global mean atmospheric N2O mixing ratios for A.D. 1850–2008 are taken directly from J. Hansen and M. Sato (Global Mean N2O Mixing Ratios, accessed 1 June 2010, http://data.giss.nasa.gov/modelforce/ghgases/Fig1C.ext.txt), which were derived from ice core and atmospheric measurements using methodology described by Hansen and Sato . A value for A.D. 2009 is estimated by adding the A.D. 2007–2008 change to the A.D. 2008 value. Earlier values are taken directly from the (interpolated) ice core record of MacFarling Meure et al. , but with linear smoothing between the two records over A.D. 1829–1849. The resulting complete time series is shown in Figure 4c which, together with equation (14), yields an effective forcing of 0.20 W m−2 over A.D. 1–2009 (0.16 W m−2 over A.D. 1750–2000) (Table 1 and Figure 5c). The estimate of radiative forcing by N2O using the relation of IPCC  is almost identical to that computed here (Figure 5c).
3.4. Montreal Protocol Trace Gases and Other Trace Gases
3.4.1. Radiative Forcing by MPTGs+OTGs
 Radiative forcings of the Montreal Protocol trace gases (MPTGs) and other trace gases (OTGs) are calculated using the expression given in Table 6.2 of IPCC ,
where X is the changing mixing ratio (ppb), Xo is the unperturbed mixing ratio (ppb), and α is a constant (W m−2 ppb−1). IPCC [2001, Table 6.7] gave α(CFC-11) = 0.25 W m−2 ppb−1) and α (CFC-12) = 0.32 W m−2 ppb−1). These values would yield Fa(CFC-11 + CFC-12) = 0.946 W m−2 over A.D. 1880–2000 × 4, whereas H05 computed a value of 1.04 W m−2, and Fe(CFC-11 + CFC-12) = 1.37 W m−2 (i.e., Ea(CFCs) = 1.32). To approximate the effective forcing of all MPTGs and OTGs as simulated by H05, the constants (α) are therefore increased by ×1.45 from their IPCC  values, which are used for the adjusted forcings.
3.4.2. Mixing Ratios of MPTGs+OTGs
 Annual global mean atmospheric mixing ratios of the MPTGs CFC-11 and CFC-12 for A.D. 1992–2008 are taken from J. Hansen and M. Sato (Mixing Ratios of 14 Gases Controlled by the Montreal Protocol: Mixing Ratios of 12 Gases Not Controlled by the Montreal Protocol, accessed 2 June 2010, http://data.giss.nasa.gov/modelforce/ghgases/TG_A.1992-2008.txt), and values for previous years are taken from J. Hansen and M. Sato (Global-Mean Greenhouse Gas Mixing Ratios Used in GISS 2004 GCM, accessed 2 June 2010, http://data.giss.nasa.gov/modelforce/ghgases/GHGs.1850-2000.txt) (CFC-11 and CFC-12 mixing ratios are zero prior to A.D. 1949 and A.D. 1938, respectively). Mixing ratios of the other 12 MPTGs (i.e., CFC-113, CFC-114, CFC-115, CCl4, CH3CCl3, HCFC-22, HCFC-141b, HCFC-142b, HCFC-123, H-1211, H-1301, and CH3Br) and the 12 OTGs (i.e., HFC-134a, SF6, CF4, C2F6, HFC-23, HFC-32, HFC-125, HFC-143a, HFC-152a, HFC-227ea, HFC-245ca, and HFC-43-10mee) for A.D. 1992–2008 are taken from Hansen and Sato (online data set, 2010). Values for previous years are taken from J. Hansen and M. Sato (Mixing Ratios of 14 Gases Controlled by the Montreal Protocol: Mixing Ratios of 12 Gases Not Controlled by the Montreal Protocol, accessed 2 June 2010, http://data.giss.nasa.gov/modelforce/ghgases/TG_A.1930-1990.txt) and linearly interpolated where necessary. All mixing ratios are extended to A.D. 2009 by linear extrapolation from the last known annual growth rate. The resulting mixing ratios of the MPTGs and OTGs for A.D. 1920–2009 are shown in Figure 4d.
 CF4 is the only gas in these two groups to have a radiatively significant presence in the global atmosphere prior to A.D. 1930, suggesting that it has a preindustrial, nonanthropogenic source. Its preindustrial mixing ratio was estimated by Worton et al.  to have been ∼34 ppt. It is assumed here to have been this value over A.D. 1–1900, with annual values for A.D. 1901–1929 estimated assuming a linear increase to the measured value of 40 ppt in A.D. 1930.
 Summing Fe of all MPTGs and OTGs over A.D. 1–2009 yields 0.50 W m−2 (0.48 W m−2 over A.D. 1750–2000) (Table 1 and Figure 5d). These values are significantly greater than those reported by IPCC  (Figure 5d) because of the high efficacy of these gases in causing surface temperature change (due in part to their lack of significant absorption band overlap with the primary GHGs, although this effect may be overestimated in H05).
3.5. All Well-Mixed Trace Gases
 Summing the effective radiative forcings of all well-mixed trace gases gives a net effective forcing of 3.24 W m−2 over A.D. 1–2009. H05 computed an equivalent value of 2.61 W m−2 over A.D. 1880–2000, and the value computed here for the same period through summing the effective forcings is almost identical (i.e., 2.63 W m−2), indicating consistency between the various regression fits used here and the original GCM simulations. These values are ∼18.5% greater than those reported by IPCC  due to the efficacies of the different agents being substantially greater than 100%.
 The series of annual well-mixed gas mixing ratios assembled here for A.D. 1–2009, and the calibrated equations for Fe, allow Earth's radiative forcing over the past two millennia to be related to the changing mixing ratios of the well-mixed anthropogenic GHGs.
 Ozone (O3) is a climate forcing in both the troposphere and stratosphere. Long-term increases have occurred in the troposphere due mainly to anthropogenic activities, particularly increases in emissions of CH4, NOx, CO, and VOCs (H07). In contrast, reductions have occurred in the stratosphere due to emissions of MPTGs. Overall global mean forcing is dominated by tropospheric changes, with perhaps half of the response due to the oxidation of CH4. H05 used estimated trends in tropospheric O3 from a chemical transport model [Shindell et al., 2003] within a GCM, together with the stratospheric O3 change from observations [Randel and Wu, 1999], to compute the effective direct radiative forcings due to changes in O3 for A.D. 1880–1990. They calculated a total O3 forcing of 0.23 W m−2 over A.D. 1880–2000. The global mean values computed by H07 (Sato and Hansen, online data set, 2010) are used here directly, with values for A.D. 1–1880 and A.D. 1990–2009 (the Shindell et al.  estimates stop at 1990) estimated from their correlation with fossil plus land use-related CO2 emissions.
 Land use–related CO2 emissions are based on the data set of Houghton , which covers A.D. 1850–2005. Values for A.D. 1–1849 are estimated from the relationship with global human population as for fossil CO2 emissions (see above). Land use–related CO2 emissions increase linearly with population over A.D. 1850–1931, after which emissions increase little while population increases exponentially. A best fit of = 0.466 P is obtained over A.D. 1850–1931, where is the land use–related flux (Pg C yr−1) (Figure 3d). This expression is used to estimate pre–A.D. 1850 emissions, and values for A.D. 2006–2009 are assumed equal to the A.D. 2005 rate of 1.47 Pg C yr−1.
 An extremely tight linear relationship exists between O3 forcing and total C emissions over A.D. 1880–1979 (Fe(O3) does not increase after 1979), with a best fit of Fe(O3) = −0.0330 + 0.0392 (fCO2 + lCO2) (Figure 3e). This relationship is used to estimate O3 forcings over A.D. 1–1879 and A.D. 1980–2009, yielding Fe(O3) = 0.39 W m−2 over A.D. 1–2009 (0.31 W m−2 over A.D. 1750–2000) (Table 1). It is assumed that Fa(O3) = Fe(O3)/0.82 (H05).
3.7. Stratospheric Water Vapor
 Oxidation of CH4 results in increased stratospheric H2O, producing a radiative forcing as an indirect consequence of CH4 emissions. The effective radiative forcing due to stratospheric H2O given by H07 (Sato and Hansen, online data set, 2010) for A.D. 1880–2003 is extended to the entire A.D. 1–2009 period using a linear regression on tropospheric CH4 mixing ratio. As the H2O production rate used in the GCM calculations was based on a linear scaling with CH4 abundance, the relationship is perfect (i.e., R2 = 1.00), with coefficients Fe(SW) = −0.0564 + 6.74 × 10−5M. Fe(SW) for A.D. 1–2009 is then 0.075 W m−2, and for the A.D. 1750–2000 period is 0.071 W m−2, equal to the ‘plausible’ estimate of H05 (Table 1). Efficacy of stratospheric water vapour derived from oxidation of CH4 (i.e., Fe(SW)/Fa(SW)) is assumed to be 0.96 (= Es(SW), see below) (H05).
3.8. Solar Irradiance
 Variability in solar irradiance over the past two millennia is poorly constrained. H07 used the solar forcing of Lean  (based on sunspot areas and heliocentric locations), which includes both changes in the solar constant and solar spectral changes (which are particularly large at ultraviolet wavelengths) to compute the effective forcing due to solar variability (Fe(Solar)) for A.D. 1880–2003. These data include a trend in the long-term solar output, but the reality of this is highly uncertain [Lean, 2010].
 The more recent total solar irradiance (TSI) data set of Steinhilber et al.  (data set accessed 21 May 2010 at ftp://ftp.agu.org/apend/gl/2009gl040142/2009gl040142-ds01.txt) is ∼1/3rd as variable as that used by H07, with a maximum amplitude of 1.77 W m−2 (∼0.13%). This data set is used here to estimate annual TSI for A.D. 1–2007. The reconstruction spans much of the Holocene (i.e., A.D. −7362–2007) and is based on the abundance of cosmogenic radionuclide 10Be in polar ice. There is debate concerning the level of relevant information in this proxy [Lean, 2010], but it is currently the only fully consistent estimate over this time scale. The original data are 40 year running means resampled to 5 year intervals, and annual values for A.D. 1–2007 are derived here by linear interpolation. Anomalies are then applied to a recent value of the A.D. 1986 solar minimum estimated from Figure 3 of Kopp et al.  to be 1360.78 W m−2, rather than that reported by Steinhilber et al.  (i.e., 1365.57 W m−2). Annual mean TSI values for A.D. 2008 and A.D. 2009 are estimated from Figure 1 of Lean , yielding 1361.00 W m−2 and 1360.89 W m−2, respectively. The change in these TSI values over time is then used as direct input to the EBM. Fe(Solar)/Fa(Solar) is assumed to equal 0.92 (H05).
3.9. Land Use
H05 computed the effective radiative forcing due to land use changes for A.D. 1880–1990, as specified in the data set of Ramankutty and Foley , to be −0.09 W m−2. The major effect is due to increases in albedo following removal of natural vegetation for croplands in northern midlatitudes. The annual global mean land use radiative forcing for A.D. 1880–1980 reported by Sato and Hansen (online data set, 2010) and H07 is used here directly. It is also extended back to 1 A.D. and forward to A.D. 2009 using the relationship between the computed forcing and the estimated land use–related CO2 emissions, lCO2 (see above). This relationship is derived over A.D. 1880–1980 by hand-fitting the intercept to ensure continuity with values before A.D. 1880, giving Fe(LU) = 0.08 − 0.122 lCO2 (Figure 3f). Effective forcing due to land use changes over A.D. 1–2009 is then estimated to have been −0.17 W m−2 (−0.14 W m−2 over A.D. 1750–2000) (Table 1). Efficacy of land use radiative forcing is assumed to be 1.02 (H05).
3.10. Snow Albedo
 Reductions in snow and ice albedos due to deposition of black carbon (BC), while highly uncertain, are believed to be a significant climate forcing [Hansen and Nazarenko, 2004]. H07 computed the effective forcing due to these changes in snow albedo for A.D. 1880–2003 using albedo changes specified in the GCM in proportion to the time varying local BC deposition calculated off-line by Koch . Fe(SNW) was computed to be 0.15 W m−2 over A.D. 1880–1990, with the forcing held constant over A.D. 1990–2003 (Sato and Hansen, online data set, 2010). These estimates are extended here to the A.D. 1–2009 period using relationships with fossil CO2 emissions calculated as described above.
 Computed Fe(SNW) increases rapidly with emissions to A.D. 1890, with a best fit regression over A.D. 1880–1890 of Fe(SNW) = −0.0243 + 0.107 fCO2 (Figure 3g). This relation is used to extrapolate Fe(SNW) over A.D. 1–1879. Values for A.D. 1990–2009 are derived by extrapolating the computed relationship with fossil CO2 emissions in the latter part of the time series (Figure 3h). The relation Fe(SNW) = −0.0150 + 0.0258 fCO2 was obtained over A.D. 1983–1990 (Figure 3h). Effective forcing due to BC effects on snow and ice albedos is then estimate to have been 0.23 W m−2 over A.D. 1–2009 (0.18 W m−2 over A.D. 1750–2000) (Table 1). Efficacy of BC-affected albedo radiative forcing is assumed to be 2.7 (H07).
3.11. Stratospheric Aerosols
 Volcanic eruptions can cause large transitory perturbations to global tropospheric temperatures through injection of aerosols into the stratosphere. Their effective annual global mean radiative forcing for A.D. 1000–2001 is calculated here by summing the forcings of Hegerl et al.  over their four latitudinal regions. These estimates are derived from a composite sulphate flux derived from nine ice cores. Values for A.D. 1–999 are estimated from the ice core–derived data set of Ren et al. .
 A simple model of injection and decay is fitted to the effective forcing computed by H05 resulting from the Agung eruption in A.D. 1964
where sul (f/fT = relative to Tambora in 1815) is the sulphate record of Ren et al. , t is time (years), α = −5.31 W m−2 (f/fT)−1, and β = 0.44.
 The resulting time series of volcanic forcing displays fewer events prior to A.D. 1000 than after, when there are far more smaller events. This is presumably at least partly because of the lower sensitivity of the methodology employed by Ren et al. . Efficacy of radiative forcing due to stratospheric aerosols is assumed to be 0.91 (H05).
3.12. Black Carbon
 The direct effective forcing by BC tropospheric aerosols for A.D. 1–2009 is estimated as for snow albedo. Fe(BC) increases rapidly with emissions to A.D. 1900, with a best fit regression over A.D. 1880–1890 of Fe(BC) = −0.0624 + 0.274 (Figure 3g). This relation is used to extrapolate Fe(BC) over A.D. 1–1879. Values for A.D. 1990–2009 are derived by extrapolating the computed relationship with fossil CO2 emissions in the latter part of the time series. The relation Fe(BC) = −0.106 + 0.0864 is obtained over A.D. 1983–1990 (Figure 3h).
 Effective forcing due to direct tropospheric BC effects is then estimated to have been 0.69 W m−2 over A.D. 1–2009 (0.54 W m−2 over 1750–2000) (Table 1). BC efficacy is assumed to be 0.68 (H05).
3.13. Reflective Aerosols
 The radiative forcing by reflective aerosols is highly uncertain. Values for their direct effective forcing are calculated here for A.D. 1–2009 based on the simulation results of H07, with a methodology as used for the direct effects of BC. A best fit regression over A.D. 1880–1890 of Fe(RA) = 0.0903 − 0.397 is obtained (Figure 3g). This relation is used to extrapolate Fe(RA) over A.D. 1–1879. Values for A.D. 1990–2009 are derived by extrapolating the computed relationship with fossil CO2 emissions in the latter part of the time series. The relation Fe(RA) = −0.226 − 0.131 is obtained over A.D. 1983–1990 (Figure 3h).
 Effective forcing due to reflective aerosols over A.D. 1–2009 is then estimated to have been −1.43 W m−2 (−1.20 W m−2 over A.D. 1750–2000) (Table 1). Efficacy of reflective aerosols is assumed to be 1.08 (H05).
3.14. Aerosol Indirect Effect
 The effective radiative forcing due the aerosol indirect effect (AIE) computed by H05 is extended over A.D. 1–2009 as for reflective aerosols by exploiting the tight linear relationships with anthropogenic CO2 emissions. A best fit regression over A.D. 1880–1890 of Fe(AIE) = 0.103 − 0.470 fCO2 is obtained (Figure 3g). This relation is used to extrapolate Fe(AIE) over A.D. 1–1879. Values for A.D. 1990–2009 are derived by extrapolating the computed relationship with fossil CO2 emissions in the latter part of the time series. The relation Fe(AIE) = −0.373 − 0.0580 is obtained over A.D. 1983–1990 (Figure 3h).
 Effective forcing due to the AIE over A.D. 1–2009 is then estimated to have been −0.97 W m−2 (−0.87 W m−2 over A.D. 1750–2000) (Table 1). H05 did not calculate the value of Ea for the AIE. However, they did report Es(AIE) (i.e., the efficacy using fixed SST, but allowing the stratosphere, troposphere, and land surface temperatures to adjust) for the cloud cover effect equal to 0.97 ± 0.08, and so Ea(AIE) is taken to be 1.0.
3.15. Net Forcing
 The magnitude of positive and negative forcings by the ten agents considered here are shown in Figure 6. Notable features are the highly variable contributions of CO2, solar, and volcanic activity over A.D. 1–1850 (Figure 6a), the coincidence of reduced CO2 forcing, reduced solar forcing, and increased volcanic forcing during the 17th century (Figure 6a), and the accelerating increase in anthropogenic forcings from the late 18th century, totally obscuring all natural forcings from the early 19th century apart from volcanic aerosols (Figure 6b).
 Summed effective radiative forcing over A.D. 1–2009 is 2.00 W m−2 (Table 1) (1.83 W m−2 over A.D. 1750–2000) (Table 1). H05 estimated a summed value of 1.82 W m−2 for A.D. 1750–2000 (calculated from their Figure 28). However, the similarity of this net figure with that in Table 1 hides some significant differences between individual forcings. In particular, Fe(Solar) is lower here than in H05 by 0.27 W m−2, whereas forcings due to CO2, MPTGs+OTGs, O3, reflective aerosols, and the AIE are all significantly higher.
H05 computed the effective forcing of all agents acting at once over A.D. 1880–2000 to be 1.69 W m−2. The difference with their arithmetic sum of individual forcings, 1.85 W m−2, is mainly due to spatial overlaps between different forcings. A correction to the combined (summed) effective forcings estimated here for input to the EBM is therefore necessary, and is taken as ×0.91, yielding Fe(Net). The resulting time series of total net effective forcing over A.D. 1–2009 is shown in Figure 7. Also shown in Figure 7 is the net adjusted forcing using the same correction factor for interactions between forcings. Fe is lower than Fa primarily because of the low efficacy of tropospheric BC.
4. Uncertainty Analysis
 Prediction uncertainty in annual temperature over A.D. 1–2009 was assessed with respect to all model inputs, including parameter values. It was assumed that each input can be varied independently and that each value within the stated range is equally likely (i.e., uniform distribution). Uncertainty with respect to alternative volcanic and solar forcing data sets was also assessed (see below).
 Where possible, the ranges of each model input were chosen on the basis of estimates from literature values, otherwise best guesses were made. Ranges are given in Table 2.
Table 2. Prediction Uncertainty Decomposition Into Model Parameters and Forcingsa
η2 (%) A.D. 1–1749
η2 (%) A.D. 1750–2009
Here η2, the correlation ratio, is the magnitude of the variance of the conditional expectation relative to the total prediction variance [Kendall and Stuart, 1979], and is given here as the mean over the stated period. Approximately 7% of total prediction variance is accounted for by nonlinear interactions between individual parameters and forcings.
 A range of possible values for the net feedback factor was calculated from the range of GCM sensitivities to doubled atmospheric CO2 given in Table 8.2 of Randall et al. , assuming no-feedback warming of +1.25 K.
 The default correction to the total net forcing derived from the GISS GCM simulation results is ×0.91. Without further knowledge, the possible range was simply specified as 10% around this value.
 The e-folding time of the fast thermal component was varied over 1–3 years. This then specifies the time constants of all components. The associated relative contributions from the fast and medium components were varied over the ranges 0.3–0.5 and 0.11–0.21, respectively. This combination of ranges specifies a wide range of thermal properties.
 Forcings were varied using a constant scalar applied to the anomalies across all years in each simulation. The scalar ranges for CO2, CH4, CFCs, ozone, land use, reflective aerosols, and the aerosol indirect effect were derived from the range of uncertainties discussed in previous sections and by H05. The scalar for solar forcing was specified as 0.5–3 to include the 3× greater forcing of the Lean  data set than the default used here. Uncertainties in forcings due to snow albedo, black carbon, and stratospheric aerosols were harder to ascertain, and so their scalars were simply set to 0.5–1.5 to cover a relatively wide range of possible values.
 Importance of model inputs for each annual prediction was computed as the ratio of the variance of the conditional expectation (VCE) to total prediction variance, the correlation ratio of Kendall and Stuart  (η2; computed as detailed by McKay ). Sampling of each input across its estimated range was made using a random number generator. Accurate estimates of η2 could be obtained with a sample size of 50,000 for the conditional annual expected values and 1,000 replicates of each input, giving 50,000,000 model runs for each annual estimate of η2. The results are given in Table 2 as means of η2 over the pre-industrial and industrial periods. In addition, the mean and 95% limits of the annual prediction probability distributions obtained when all inputs are varied are shown in Figures 8 and 9.
 Prediction uncertainty with respect to volcanic and solar forcing data sets was assessed using default parameters and forcings. The data sets used are given in Table 3, and all possible combinations were tested. Results are included in Figures 8 and 9 as the total range of annual anomalies.
Table 3. Solar and Volcanic Data Sets Used to Examine Sensitivity of EBM Simulationsa
Data Set Code
See Schmidt et al.  for full documentation of these data sets. GSA supplemented with 14 Tg of H2SO4 in A.D. 1982 from El Chichón and converted to radiative forcing using Fe(SA) = (−20 W m−2) AOD [Wigley et al., 2005], and AOD = S/150 Tg [Stothers, 1984], where S is total sulphate injection. CEA is global mean AOD at 10 day time resolution, converted to Fe(SA) as for GSA and averaged over each year. Both “no-background” and “with background” TSI data sets used, with all TSI data sets calibrated to default data set using the calibration required for WSL in A.D. 2003. All data sets accessed 27 January 2010 through http://pmip3.lsce.ipsl.fr/.
 The EBM was run with an annual time step using default parameter values and driven by Fe(Net) and TSI as derived here for A.D. 1–2009. The computed evolution of Ts is shown in Figures 8 and 9.
 Prior to the massive volcanic eruption in A.D. 186 (attributed to the Taupo volcano [Ren et al., 2010]), simulated temperatures using default values are relatively high, whereas over A.D. 350–1050 they are lower but fairly stable. There is then evidence of a slightly warmer ‘Medieval Climate Anomaly’ (MCA) around A.D. 1050–1258, a ‘Little Ice Age’ (LIA) A.D. 1588–1720, and strong warming from around A.D. 1920, all in the context of large short-lived negative excursions due to volcanic eruptions. After A.D. 1258 temperatures generally remain relatively low until the recent rapid increase to present levels from about A.D. 1920. The volcanic eruption in A.D. 1258 was perhaps the largest of the past millennium, and is believed to have initiated radical social upheaval [Stothers, 2000].
 Also shown in Figure 8 are two reconstructions of past temperatures, together with their uncertainties. The reconstruction of Ljungqvist  is for the extratropical northern hemisphere only, whereas that of Mann et al.  is for both hemispheres, albeit with very few sites outside of North America or Europe. The reconstructions suggest greater long-term variability than is predicted by the EBM, in particular a cool period A.D. 300–800, a strong warming peaking around A.D. 950, and a stronger cooling during the LIA. It is likely that at least some of the discrepancy between the proxy-based record and the EBM is due to the bias towards northern midlatitudes in the former. Other plausible explanations are that the long-term forcings derived here are too weak, the model is not sensitive enough, and/or internal modes of variability on decadal to centennial time scales are important in the real system, whereas they are not treated by the EBM. The possible role of changes in Atlantic Meridional Overturning Circulation in explaining both the MCA and LIA has received recent attention [e.g., Palastanga et al., 2011; Trouet et al., 2009].
 Instrument-based measurements over A.D. 1880–2009 allow a more rigorous test of the EBM and forcings than is possible for earlier periods. Observed global mean surface temperatures from the GISTEMP data set [Hansen et al., 2010] are compared with those predicted by the EBM in Figure 9. While many features of the observed record are captured, especially from ∼A.D. 1945, the increase in temperatures over A.D. 1910–1944 is not fully simulated, although it is within the range of model uncertainties. It is possible that internal variability, such as due to ocean heat transport, could explain at least part of this discrepancy as this is not represented in the EBM.
5.2. Analysis of Input Importance
 The results of the analysis of input importance for A.D. 1–1759 and A.D. 1750–2009 are given in Table 2. During the pre-industrial period, the contribution to uncertainty in model prediction is fairly evenly balanced between model parameters and forcings, whereas in the industrial period uncertainty is almost entirely due to forcings. Uncertainty in solar forcing is the most important contributor to prediction uncertainty in the pre-industrial era, followed by the feedback factor and then forcing by volcanic aerosols. Land use forcing is next most important before A.D. 1749, but becomes much more important over A.D. 1750–2009. Only uncertainty in the AIE forcing is more important than the land use forcing uncertainty in the industrial period. The range of land use forcing anomalies is taken from Myhre and Myhre , who estimate a large range of −0.6 to +0.5 W m−2.
 Prior to A.D. 1750, prediction uncertainty is relatively low, but increases greatly in the industrial era (Figures 8 and 9). Improvement in model performance will depend largely on better constraints on the net feedback factor and solar forcing for the pre-industrial era, and on better constraints on land use and direct and indirect aerosol forcings post–A.D. 1750.
6. Conclusions and Outlook
 The model simulations presented here suggest that reconstructions of surface temperature variability over the pre-industrial era could be used to constrain model parameters, whereas this is not possible for the instrumental period when variability is dominated by poorly quantified forcings (i.e., aerosol direct and indirect effects, and land use). The lower predicted centennial variability than that observed in the reconstructions over the past two millennia suggests that solar forcing may be underestimated. However, it is also possible that model sensitivity is lower than in the actual Earth system, particularly as a consequence of the lack of treatment of internal modes of variability that operate on decadal to centennial time scales such as due to ocean heat transport.
 The EBM presented here has been constructed to allow simple reparameterization using the results of alternative GCM and observation-based sensitivities and temporal dynamics. This model is being further developed to include a prognostic carbon cycle as well as being downscaled to geographical regions by exploiting the strong coherence in climate patterns when expressed relative to effective forcings (H05). It is also planned to address the issue of modes of internal variability through the incorporation of additional processes in future versions of the EBM, in collaboration with the developers of the CLIMBER family of Earth system models of intermediate complexity (A. Ganopolski, personal communication, 2011).
 I thank Jim Hansen, Gilles Delaygue, Makiko Sato, Andy Lacis, Jérôme Ogee, Sönke Zaehle, and two anonymous reviewers for advice and suggestions concerning this work. I am also extremely grateful to all those who have made the various data sets available on which the majority of this work is based. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007–2013) under grant agreement 238366 (GREENCYCLESII: Anticipating climate change and biospheric feedbacks within the Earth system to 2200). Data sets derived here are available at http://www.greencycles.org/datasets.