Technical Note: Calculating state dependent equilibrium climate sensitivity from palaeodata

The evidence from both data and models indicate that specific equilibrium climate sensitivity S[X] – the global annual mean surface temperature change (∆Tg) as a response to a change in radiative forcing X (∆R[X]) – is state dependent. Such a state dependency implies that the best fit in the scatter plot of ∆Tg versus ∆R[X] is not a linear regression, but for instance a higher order polynomial. While for the conventional linear case the slope (gradient) of the regression is correctly interpreted as the specific equilibrium climate sensitivity S[X], the interpretation is not straightforward in the non-linear case. We here elaborate how such a state dependent scatter plot needs to be interpreted, and provide a theoretical understanding how to calculate S[X] in the non-linear case.