I found the following statement in here regarding the effect of twice lagged differences of CO2 ($\Delta C$) in the atmosphere on the once lagged values, i.e.
$$\Delta C_{\text{ @ }t=-1}= 0.83 \times \Delta C_{\text{ @ }t=-2}+0.00018$$
The data inform us that once CO2 changes got going into a particular direction, they tended to continue the same direction. CO2 changes had strong internal dynamics, almost random-walk like: whenever CO2 increased over 100 years, it strongly tended to increase over the next 100 years again, and by almost as much. Ergo, when a shock to CO2 occurs, it has long-term effects, far beyond a century. The half-life of shocks to changes in CO2 is about $−\log(2)/\log(0.83)≈3.7$ centuries.
almost intuitive based on log rules and the coefficient of the regression line indicating the effect of one unit of change in CO2. But how is this approximation calculated?
I bet this is a common approximation in econometrics for the duration of a system alteration or change.