The objective of this post is to identify harmonic signals present in the 162 year series of global temperature data, and then to make a fit to the data in order to quantify their amplitudes and relative phases. Then we make an overall comparison to the full dataset, assuming a logarithmic dependence of temperature on CO2 levels and natural decadal scale oscillations, and finally use the fitted formula to predict future warming based on two different CO2 emission scenarios used by the IPCC. This work has been influenced by a previous study which I recommend reading .
The data is HadCrut3 temperature anomaly data  from 1850 to 2011. In a previous post it was argued that a simple energy balance argument for the greenhouse effect of increasing CO2 levels leads to a logarithmic temperature rise on CO2 . The overall trend is well fitted by the formula DT = -0.34 +2.5ln(C/C0). However a clear ~60 year oscillation is evident over and above this slow trend.
To study this oscillation further I subtracted the logarithmic term from the data to leave the residue data which can be seen in Figure 2. To identify harmonics in the data I did a fourier analysis of this residue which is shown below. The upper X-axis is the frequency times time period and the power spectrum in shown in black.The first three peaks correspond to frequencies of 60 years, 11years and 9.3years. The magenta peaks are simulations of sine waves with these frequencies over the 162 year period. The dashed peak is that for a 9 year period showing that the data better fit a 9.3years period.
Next I did a least squares fit to the residue data including all the 3 frequencies with 5 parameters. The phase for the 60 year oscillation was fixed as in the previous post. The fitted formula was :
DT = A*sin(0.105(x-1865)) + B*sin(0.57(x-C)) +D*sin(0.68(x-E))
The best fit is shown below as the blue curve below with A=0.14, B=-0.003, C=1943.5, D=-0.02, E=1879.5. The fit is not perfect before 1860 but many of the main features evident are reproduced rather well. The overall goodness of fit is examined next.
By comparing the UAH data from satellite measurements of the lower atmosphere temperatures with the HadCrut data overlapping the same time period we can estimate the individual errors on a single yearly measurement. This works out at about +- 0.05C for the period 1979-2011. We will assume that the error on each measurement is 0.05C over the full 162 year timespan for Hadcrut, although we might expect the error to be higher in early years. So taking the logarithmic temperature increase assumed to be due to AGW and the oscillations fitted above, we compare the full time series to the fits as shown in figure 3.
How good is this fit ? Calculating the reduced Chisquared between the curve and the datapoints gives 4.0 for errors of 0.05C and about 2.0 for errors of 0.07C which shows that the fit is a reasonably good, especially since other natural effects are also likely present.
The 11 year cycle lines up with the solar cycle of sunspot activity and corresponding variation of around 0.1% in energy output. The 9.3 year oscillation lines up with the twice per period crossing of the Earth-Sun ecliptic plane with the lunar orbit plane precession every 18.6 years. The major 60 year global cycle is still not understood but appears to be linked to the Pacific Decadel oscillation (PDO) and the Atlantic Multidecadal Oscillation (AMO). What actually causes this and whether there is also here an astronomical origin for this clear effect is still not clear. However reference  identifies a 60 year cycle for the relative position of the sun with respect to the overall centre of mass caused by the orbits of Jupiter and Saturn.
The fit to global temperature data over the last 160 years can be used to predict temperature responses to future increases in CO2 emissions. The IPCC use CEISIN scenarios for emissions based on different socio-economic models. The scenario A1B assumes an increasing reliance on fossil fuels with a growing world economiy leading to levels of around 700ppm by the end of the century. It is quite likely however that other sources of energy will be developed over the next 90 years due to environmental concerns and for economic reasons as fossil fuels become more expensive. Scenario B1 assumes a levelling off of emissions leading to CO2 levels of around 550 ppm by 2100. Figure 4. uses these two different CO2 scenarios to predict future temperature increases this century based on the fit described above.
Note that in both cases little or no temperature rise is expected between now and 2025, followed afterwards by a sharper rise until 2060 and another levelling off, all caused by the oscillations contained in the fit. For the lower scenario the overall temperature rise from today is just 0.6 degrees, whereas in the “business as usual” A1B scenario the overall rise is still a “modest” 1.2 degrees. These are smaller than IPCC predictions of 1.7 to 4.4 for scenario A1B and 1.1 to 3.0C for scenario B1.
 Hadcrut Data http://www.cru.uea.ac.uk/cru/data/temperature/
 Logarithmic dependence of global temperature data on CO2 levels : https://clivebest.com/blog/?p=2241