A Fit to Global Temperature Data

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 [4].

The data is HadCrut3  temperature anomaly data [1] 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 [2]. 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.

Fourier Transform showing harmonic with periods 60, 11 and 9.3 year

Fourier Transform showing harmonic with periods 60, 11 and 9.3 year

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.

Fig 2: Residue temperature data showing fit to 3 harmonics

Fig 2: Residue temperature data showing fit to 3 harmonics

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.

Overall fit to 162 years of global temperature data

Fig 3: Overall fit to 162 years of global temperature data

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.

Possible Interpretations:

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 [4] 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.

Future Scenarios:

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[3]. 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.

Predicted temperature rises for two possible CO2 emission scenarios

Predicted temperature rises for two possible CO2 emission scenarios

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.


[1] Hadcrut Data http://www.cru.uea.ac.uk/cru/data/temperature/

[2] Logarithmic dependence of global temperature data on CO2 levels : https://clivebest.com/blog/?p=2241

[3] http://sres.ciesin.columbia.edu/

[4] Nicola Scafetta, Empirical Evidence for a celestial origin of the cimate oscillations and its implications. http://arxiv.org/pdf/1005.4639

About Clive Best

PhD High Energy Physics Worked at CERN, Rutherford Lab, JET, JRC, OSVision
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23 Responses to A Fit to Global Temperature Data

  1. Just read about this on wuwt

    Of interest I have done something similar to you
    Looked at ffts for frequencies – didnt work very well
    so then used a bandpass filter of fixed percentage bw which i swepth through 0 to 250 ish years peaks were noted and a cosine of the frequency generate with variable centre freq, phase and amplitude. These 3 vars were then adjusted to give a best fit over the whole data period.

    I did this for a polynomial increase and for pure cosine data.
    Both remain flat from 2000 to 2012. The all cycle then begins to fall whereas the polynomial stays flat until 2020 when rapid increase begins. The pure cycle form also begins to fail with the early 1800s data.
    NO PREDICTIVE ABILITY IS CLAIMED it simply shows a pause is very possible despite an ever increasing temperature driver.

    The evolution of plots can be seen here
    there are a couple of pages!

  2. Pingback: Models overestimate 60 year decadal trends | Watts Up With That?

  3. Greg Goodman says:

    This is rather old but since you linked it on WUWT today… Nice post BTW.

    I don’t understand your 11y peak marked on the the graph here. It looks more like 22y , it’s way too far away from 9 to be 11y.

    I did something similar with detrended AMO and used 9,22,60y

    Subtracting log CO2 is similar to detrending, so the two approaches are quite similar.

    I tend to avoid Hadley based datasets for this kind of think because their gridding process messes up the 9y periodicity.

    • Clive Best says:

      Originally I was thinking about the 11y solar cycle and the 18.6y lunar cycle. That was why I was looking for signals with these pathlengths – and there appeared to be a small signal. The Fourier analysis is in frequency space which is ~ 1/period. The top axis is the frequency not wavelength.

      Anyway I then just did a least squares fit to the H4 data including the 3 terms and that was the result I got.

      B is essentially zero so there is no real 9y effect. The main term is the 60y harmonic, but the 11y term is not negligible/ Maybe there really is a solar cycle effect.

  4. Greg Goodman says:

    BTW I think Scafetta’s 9.1 ( also reported by BEST/Curry paper ) is a mix of 9.3 and 8.85 , if you work out the mean frequency of those two it’s 9.05 or so.

    Due to noise and sampling error these two are usually not resolvable, though some ocean basins seem to show one or the other. IIRC Indian Ocean is 9.3y.

  5. charplum says:

    I do not post comments to sites very often. One of the reasons for that is that I have not learned how to post figures in my comments. Without figures sometimes the comments are just meaningless.
    I had a 35 year career in rotating equipment and used FFT extensively to solve my problems. With the modest temperature record that is available I used more of a brute force technique and just tried fitting about 10 sine waves to the measurements.
    The program I used was one I was very familiar with called TKsolver. I input guesses for frequency amplitude and phase and the program would iterate to a solution and come up with the answers.
    My interest in this came from recurring mention of a 60 year cycle.
    I was tired of sitting on the sidelines and decided I would try to make a contribution in an area I was familiar with. While inputting some of my guesses for frequencies I was aware of solar cycles around 1000 and 350 years so I input something near those periods.
    Anyway, the fit curve had a correlation with the data of 0.87. It was after doing this myself that I was starting to be convinced that the climate could, for the most part, be explained by natural cycles.
    The periods of the cycles that I found in the data were.
    1003, 64.8, 346.3, 205.8, 148.6, 103.5, 21.1, 9, 4 and 1.
    The surprising thing to me is when I took this curve fit back in time it revealed the MWP and the LIA in about the right places. As to the rest I don’t know.

  6. Clive Best says:

    I think fitting harmonics to the temperature data is as good as a fourier analysis. However you have to allow for any underlying linear type trends. If you send me the plots I can add them. Otherwise just cut and paste a URL to the image into a comment on a single line

    What data did you fit to find 1000y and >100y cycles? The problem we have is that there is no consistent long trem temperature record. The best we have are benthic fora and the ice core record. The real mystery of course is the real cause of the 100,000y cycle in glaciations.

  7. charplum says:

    OK I am going to try with the figures. I put them on my one drive in a public directory.
    Just to let you know I have gone much further with this. I have closely watched the efforts of Dr. Evans and used his OFT analysis results and then used the results in my crude method of analysis. I am now up to 28 cycles and it does a remarkable job of depicting even El Ninos.

    Anyway here is my attempt to send the pictures. The first one seems to be a link to the first picture. The second seems to be a link to two picture. the third is a link to one picture and the last seems to be a link to the third picture plus the last picture. there are 4 in total.

    I hope I did this right.

  8. Keep trying, it will take you awhile to get to this point:

    The addition of 18.6 and 4.42 year cycles slightly improves the fit, but ENSO, volcanos are still the principal factors.

    • Hello telescope
      All this good weather around the planet must be very depressing for you… you have my simphaty! Even when is a hurricane, or earthquake somewhere; still on 97% of the planet is good weather – even though is same amount of CO2 everywhere… I’m saying: thank you, thank-you CO2, for all this beautiful weather – the birds are singing, people enjoy life; BUT for you must be alterable feelings…
      Keep daydreaming, eventually bad weather and misery some place will happen, its a big planet – when it happens -> your psychopathic genes will rejoice and you will have a lots of happiness – think positive: after all the good weather, bad weather will come for few days, somewhere – same as: after bad weather, good weather always follows, cheers!

    • Clive Best says:

      Only if you believe CMIP5 aerosol forcing scenarios, which strangely enough just happen to be the same as 2.5ln(C/C0) ,with a few aerosol wiggles carefully positioned to cover the oscillation.


  9. lgl says:

    Will deltaE=2.5 give a sensitivity of 2.5/5.3=0.47 K/(W/m2)?

    • Clive Best says:

      Unfortunately not it goes like 2.5*ln(2) or sensitivity = 1.7C

      I think TCR is more like 1.4C unless cloud feedbacks kick in.

      • lgl says:

        Sorry, I ment the 2.5 in your formula (deltaE was from that other post of yours so I confused myself) and 1.7C from a CO2 doubling gives a sensitivity of 0.46 K/(W/m2) so I guess we are both right 🙂
        Anyway, I worked from another direction and got almost the same result,
        but I have to agree with Greg on the ~22yrs.

  10. charplum says:

    My last link was incorrect in that it was the same picture twice. Since there was only 150+ years of data I decided to take the results back in time. Using only this much data to go back that far is certainly a stretch but this is what I got.


  11. charplum says:

    Since I retired some years ago and mostly what I did was read stuff on the WUWT website I finally decided to get off the couch. The 60 year cycle interested me because you could almost visualize it in the data. From other readings I was aware of what is shown in the figure. You might say I had the answer from the back of the book. So I thought I would try some of these solar cycles. I also remember reading about a 1000 year cycle on WUWT. No harm I thought I would try it.


    The figure comes from a paper by McCracken….

  12. charplum says:

    Not too long ago I did something quite similar to what you did with a few natural cycles and came up with the best fit to the data. I will try to locate that and add that information also. My recollection is that the ECS came out to be less than two.

    This is interesting but perhaps too confining. One thing I can recall fairly reliably is that the law dome CO2 levels are pretty flat going back in time from 1800 or 1750. It really does not matter which it is. If CO2 is the dominating factor then how can it justify the following graph.


    It would seem the only way to go after this one knowing the CO2 levels is to attach the validity of the proxy data. CO2 levels are unable to explain this.

    This chart come from: http://wattsupwiththat.com/2012/10/17/new-paper-confirms-the-climate-was-warmer-1000-years-ago/

  13. charplum says:

    I went back and did some additional work. I combined CO2 with cycles and came up with an ECS of 1.04. The factor that reduces it to this value is the very long cycle that had a period of 279 years. Whether that is real or not can be questioned but that can be questioned as much as the role CO2 plays.

    The figures for this portion are here.


    I also did some additional work of trying to fit only 4 cycles to the data the results are given here.


    In the last one I included the proxy figure again. I remain surprised that out of only 160 years’ worth of data that I can derive a figure that approximately places the MWP and the LIA. The more I do in this area the more I am convinced that natural cycles seems to be doing a splendid job of explaining the behavior of the climate.

    BTW, while we have been discussing this we are indeed fortunate that a further reduction in ECS seems to be substantiated.


    Pay particular attention to the comment by rgbatduke. It is a good one.

    • Clive Best says:

      Your fit looks remarkably similar to climate4you . The trillion dollar question is what happens after 2020!

      I love rgbatduke’s comment! He is a retired prof of physics . The smart climate scientists will now be looking around for an exit strategy which leaves their reputation in tact.

  14. charplum says:

    Getting those scientists pinned down will be like trying to grab an eel wearing boxing gloves.

    I don’t think you will have to wait beyond 2020.

    As you can see I have been trying to fit sinusoids to the data. I think that I have done pretty well. I would say in May or June of last year I tried to post some of this same stuff in a comment on WUWT. However, as I related previously I did not know how to add pictures.

    I found Joanne Nova’s email address at the bottom and sent her an email with information similar to what I revealed here.

    She put me in contact with her husband, Dr. David Evans. We communicated infrequently but he did reference the McCracken paper which furnished information about solar cycles over the last 9000+ years. I tried using them in my curve fitting procedure and many of them worked.

    Dr. Evans is working on a notch delay theory. If you go to the website I have furnished below you will see an in-depth explanation and all the details. I am all in.


    He is further revising his work to address some identified issues. But in the figure below you can see you won’t have to wait until 2020.


    It gets better than that. Dr. Evans came up with an Optimized Fourier Transform (OFT). I used it to analyze the same data up to 28 cycles. The figures give the results. It is remarkable how well this method clarifies ENSO. There is no physical basis for this. It is what the signal analysis reveals extending it only a few years.


    The last figure roughly reveals the same behavior as Dr. Evans is showing in his notch delay. That fact that the signal analysis reveals it means it is baked in the cake.

    I fully appreciate the respect that Joanne and David gave to me. I am not a climate scientist. I solved problems on rotating equipment by treating the measured data as gold.

  15. charplum says:

    I simply wanted to provide an update on the additional things I have done. I modified the program that makes it easier for me to change the number of cycles I am using in my analysis.
    Here is an example on what one of my function sheets now looks like.

    pk=length(‘b0) b0 are the initial guesses for all the values
    for j=1 to kk
    for i=1 to n
    ‘aaa[j][i]=b[e]*sin(2*pi()*b[e+1]*x[i]+b[e+2]) This creates a matrix of all the waves used
    next i
    next j
    For i=1 to n
    co21=A*sin(2*pi()*B*’x[1]+C)+D*’x[1]^2+E*’x[1]+FF The initial value of CO2
    next i
    for i = 1 to n
    T = 0
    for j = 1 to kk+1
    T = T + ‘aaa[j][i]
    next j
    next i
    This makes things much easier for me. As you will see I have analyzed the latest RSS data using 89 sinusoids plus I have added the contribution of CO2.

    Since the data at Mauna Loa goes back to 1958 I can use that data in my analysis on the contribution of CO2.

    I approximated the CO2 data using an equation that looks like the one above.


    I think the results came out quite well.


    The correlation coefficient for this fit of the CO2 data was 0.9993.

    Previously in analyzing the RSS data I had only used natural cycles and was not looking for a contribution from CO2.

    That has changed. I now want to investigate if I can have both contributions in my analysis of the raw data. I am only going to show the fit with both contributions just to see what the ECS comes out to be to make it all work.

    I think it all worked as I hope to show you.



    In the figures I do show the contribution of CO2 as the green line.
    The green line does not have much of a slope.

    Here is how the SSE and the correlation turned out. Note how low the ECS value is.


    I have information on all the waves that make up the red line. Here are just the first three.


    Here is what the future might look like from CO2 alone.



    Judge for yourself but I think this is a better estimate of the future than what we are getting from the climate models.

    BTW, I have done the same for Hadcrut4 and the ECS value came out less than 0.4.

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