The initial value,
the exponential time constant, the equilibrium value, and the PDOI smoothing and variation are all governed by independent free variables.
Unsurprisingly, if he sets the equilibrium to current temperatures, the initial value to 1880 temperatures, and chooses
the exponential time constant and the PDOI parameters, he can get a good fit to temperatures since 1880, and finds that it projects constant future temperatures (because the exponential vanishes).
Not exact matches
One approach to forecasting the natural long - term climate trend is to estimate the
time constants of response necessary to explain the observed phase relationships between orbital variation and climatic change, and then to use those
time constants in the
exponential - response model.
Exponential growth in this context is related to a fixed
time for doubling or a
constant percentage growth.
Regardless of the value of the decay
constant, «a», be it millions of years or millionths of a second, the
time for an
exponential curve to go to zero is infinity.
That can't be characterized by an e-folding
time because it is not the result of a single
exponential decay function, but rather by a series of curves with different
time constants varying from a few decades to hundreds of thousands of years — the latter for restoration of oceanic carbonate stores from the weathering of terrestrial silicate and carbonate rocks.
Gloor et al (2010)-LRB--RRB- estimate a half life of ~ 30 years for CO2, and find an
exponential decay model with unvarying
time constant fits the data.
Some functions have naturally long tails and are not
exponential and hence have no meaningful
time constant.
Only once the forcing leaves the
exponential track (which it has roughly followed for over a century) will one have any hope of getting the
time constant from just the temperature series alone.
Another option is to regress T on lagged F. I've found that using an
exponential forcing decay with a
time constant of ~ 2 years works well (gives the best fit) for GISS - E2 - R.