I found an annual reconstruction of CO2 atmospheric concentrations that goes from 1832 to 1978. It is made available by CDIAC and it comes from Etheridge et al. (1998). There's a more than adequate match between this data and the data collected at Mauna Loa, Hawaii for the range 1958 to 1978.
Naturally, I thought this CO2 data would be more accurate than that estimated from emissions, which I had used in my calculation of climate sensitivity to CO2 doubling. (BTW, that calculation was based on 150 years of data). So I reran the analysis, and the following is the new formula for the rate of temperature change (R) given a CO2 concentration in ppmv (C) and a temperature anomaly in degrees Celsius (T).
R = 0.0857 ( 10.398 log C - 26 - T )
The equilibrium temperature (T') is calculated as follows.
T' = 10.398 log C - 26
This means that climate sensitivity to CO2 doubling (based on this model which only considers this one forcing) is most likely 3.13 degrees Celsius.
I also rebuilt the hindcast graph, which follows.
I think this is a subjectively better hindcast than the original. Note that it even predicts a nearly flat temperature trend in the 1950s. This is simply what the more accurate CO2 data does. While sensitivity is lower (I had originally estimated it at 3.46C), the range of CO2 concentrations is wider. Estimations based on emissions produce a concentration of about 295 ppmv in 1850. Etheridge et al. (1998) determines the concentration is 283.5 ppvm at that point.
The model predicts that the rate of temperature change should be about 2.1C / century in 2007.
I also wanted to attempt a 1000-year hindcast. I had previously discussed the 1781-year temperature reconstruction that is the product of Mann & Jones (2003). It just so happens that there's also a 1000-year CO2 reconstruction from Etheridge et al. (1998). Well, this more ambitious hindcast didn't turn out to be as accurate. At first I thought this is just what happens when you fail to consider other important climate forcings. But then I went back and examined other 1000-year temperature reconstructions. I'm sure readers have seen that graph many times. It turns out that there's considerable uncertainty in these types of reconstructions.
Either way, I will post my first attempt at a 1000-year hindcast below. The red line is the reconstruction from Mann & Jones (2003). I also added a green line, which is a reconstruction based on glacier records that comes from Oerlemans (2005).
It could be better. I'm now curious as to what would happen if other major climate forcings were considered.