The theoretical relationship between climate forcing and climate sensitivity is obtained by considering Earth's global energy budget [e.g., Andreae et al., 2005],
which states that the forcing of the climate system, ΔQ, is balanced by energy escaping to space, λΔT, and energy stored in the oceans, H. Note that the total forcing of the system, ΔQ, is defined as the change in total radiative forcing between the end of the 20th century and the time period of the late 1800s. λ is the climate feedback parameter and is determined by the various feedback processes in the system. The climate sensitivity, ΔT2X, is defined as the change in equilibrium global surface-air temperature due to a doubling of carbon dioxide, where at equilibrium, H = 0, and (1) implies,
Using (2), λ = ΔQ2X/ΔT2X, which means (1) can be written as,
The forcing due to a doubling of carbon dioxide is 3.7 Wm−2 [Andreae et al., 2005], while the observed change in surface-air temperature is taken to be 0.6°C. The change in ocean energy storage is 0.7 Wm−2 [Wigley, 2005]. Substituting these values into (3) yields,
This expression indicates an inverse relationship between total forcing and climate sensitivity. With regards to the change in ocean heat storage, Hansen et al.  estimate an uncertainty in H of ±0.15 Wm−2. An analysis of coupled model simulations forced with a 1% per year increase in carbon dioxide indicates a spread in ocean energy storage, at the point of doubling, of 0.4 Wm−2. Based on these estimate, the present analysis will assume an uncertainty of ±0.2 Wm−2 in H.
 The model results for total anthropogenic forcing, ΔQ, and equilibrium climate sensitivity, ΔT2X, are shown in Figure 1, while the solid line is based on (4), and the dashed lines above and below this central line arise from the ±0.2 Wm−2 uncertainty in H. It is assumed that the natural forcing is much smaller than the anthropogenic forcing. These results clearly illustrate a strong inverse correlation between total anthropogenic forcing used for the 20th century and the model's climate sensitivity. Indicating that models with low climate sensitivity require a relatively higher total anthropogenic forcing than models with higher climate sensitivity.
 It may be argued that it would be more accurate to use a measure of the transient climate sensitivity in this analysis, since the simulation of the 20th century is a transient phenomenon. However, the transient climate response as defined as the change in global mean surface air temperature at the time of doubling in models assuming 1% per year increase in CO2 show a similar range as the equilibrium sensitivity (1.4 to 3.8°C, see Table 9.1 by Houghton et al. ). So whether equilibrium or transient sensitivity is used the results in Figure 1 will be unchanged.
 Note that the range in total anthropogenic forcing is slightly over a factor of 2, which is the same order as the uncertainty in climate sensitivity. These results explain to a large degree why models with such diverse climate sensitivities can all simulate the global anomaly in surface temperature. The magnitude of applied anthropogenic total forcing compensates for the model sensitivity.
 Although there is a clear inverse correlation between the forcing and the climate sensitivity there is some spread in the data points. Note that all the model results, except for one, fall close to or within the theoretical curves based on the estimated uncertainty in H. This strongly suggests that the scatter among the models is mostly due to the range in modeled change in ocean heat storage.
 What is the major reason for the large uncertainty in total anthropogenic forcing? Figure 2 shows the correlation between total anthropogenic forcing and forcing due to tropospheric aerosols. There is a strong positive correlation between these two quantities with a near 3-fold range in the magnitude of aerosol forcing applied over the 20th century. This large uncertainty in aerosol forcing has recently been noted [Anderson et al., 2003; Schwartz, 2004] as a significant challenge to the climate modeling community. Thus, the large uncertainty in aerosols over the past leads to a wide range in total anthropogenic forcing. Some of the models used in these simulations employed only the direct effect, while others used both direct and indirect effects of aerosols, which makes a more detailed comparison of simulated aerosol forcing difficult.