Orbital forcings in the CSALT model : explain the pause?

A set of orbital forcing cycles inspired by the persistent publications of Scafetta [1] was added to the CSALT model (also see Related).  This set was grouped into two parts. The first set comprises the identified luni-solar periods identified by Scafetta and others. These are pure sine waves with a phase giving the best residual fit. Interesting that they do indeed have a significant impact on the model fit, raising the correlation coefficient above 0.992 for a Pratt 12-9-7 triple running filter [2]. The other factor is a sun barycentric velocity that Scafetta has identified.  This also has an impact on improving the fit as seen in Figure 1.

Fig. 1:  CSALT interface including the orbital periods.  See Figure 5 for a description of the Pratt filter.

Overall, these extra cyclic terms do improve the goodness of fit as shown in Figure 2, while being understood as subtle factors which influence the free energy state of the earth’s climate.  Whether these gravitational effects are actually great enough to influence the climate in as strong a fashion as Milankovitch cycles remains to be seen, but they are presented here in the interest of open-mindedness. (Edit: see [3] for a book length analysis of the effects)

Fig 2 : Error bars on chart correspond to +/- 2 standard errors to the model/data fit over the entire range of values.

What is very interesting is how well the orbital forcings help to recreate the pause in recent years.  In a previous post, the C&W hybrid correction was invoked as a plausible mechanism to explain the recent diversion between the model and data temperature trend. Yet, as one can see in Figure 3 (below) that the recent extended pause coincides with a strong negative cycle excursion in the sum of the orbital cycles.

Fig 3: The set of orbital forcings shown in the top panel have undergone a recent negative excursion, which has extended the pause as shown in the lower panel. Given the relentless forcing of CO2, this cannot continue and one can see that the orbital forcing appears close to a minimum.

The attribution to CO2 is still just as strong as shown in Figure 4. The pause is barely observable once the fluctuation terms are removed leaving behind a signal with a very slight residual fluctuation.

Fig 4: The defluctuated signal fit to the ln(CO2) sensitivity remains at a 2.1C TCR for doubling of CO2.

Figure 5 is a view of the effects of the Vaughan Pratt 12-9-7 triple running mean filter. Note how it removes the sinc-related harmonic distortion induced by using a 12-month box-window moving average filter.

Fig 5: Use of Vaughan Pratt’s triple running mean removes the harmonics due to the shape of a 12 month sliding window

Whatever residual noise that is left is still highly-influenced by the strong temperature excursions. The alignment with the original peaks indicates that this is partly amplitude-dependent noise as might be observed with phase noise.

Fig 6 : Residuals from the CSALT model fit (bottom panel, expanded vertical scale) still show alignment with the original noisy temperature excursions (top panel), showing the difficulty in isolating further frequency components.

The CSALT model has much in common with historical climate data reanalysis, which one can gather from reading this comment.


CSALT model posts


[1] N. Scafetta, “Discussion on climate oscillations: CMIP5 general circulation models versus a semi-empirical harmonic model based on astronomical cycles,” Earth-Science Reviews, vol. 126, pp. 321–357, Nov. 2013.
[2] V. R. Pratt, “Multidecadal climate to within a millikelvin,” 2012. AGU poster.
[3] Sidorenkov, Nikolay S. The interaction between Earth’s rotation and geophysical processes. Wiley. com, 2009 (pdf).


40 thoughts on “Orbital forcings in the CSALT model : explain the pause?

  1. Pingback: Tidal component to CSALT | context/Earth

  2. Pingback: CSALT model and the Hale cycle | context/Earth

  3. Hi Paul,

    Is it possible that the fluctuations in LOD can be explained by the various orbital parameters? If so, it seems that LOD could be eliminated from the CSALT with orbital forcing model, or it might be interesting to try.

    Another thought, it would be interesting, if the user could force various coefficients to be zero (including for CO2) to see how this affects the overall fit.

    Some claim that CO2 has no effect on climate and that everything can be explained by natural fluctuations. A simple test would be to allow the user to force the ln(CO2) coefficient to equal zero and then come up with new coefficients for the other model components and see how well the model agrees with historical temperatures relative to a model that includes CO2 as a forcing.

    Just a thought.



    • Thanks Paul.

      I haven’t been able to access the entroplet server for a few days.

      It has always been a problem with IE but Firefox worked in the past. Do you know if Chrome orks or is the server down?



  4. Hi Paul,

    It worked for me for a session. It seems you can leave either SOI or AAM in the model without the other and it makes little difference. If you use the orbital forcings without LOD the fit is not very good, so it seems LOD is needed with the orbital parameters for the best fit. The server seems to have crashed again. I don’t know what I am doing to mess it up, sorry.



    • DC, It is not your fault. The CSALT model is the most intensive of the calculations running on the server and the Amazon folks don’t like it when their free instances use up too many cycles or too much memory stack, so they let the apps crash unceremoniously.

      Either I can spend about $100/month on a compute-intensive Amazon instance or I can run the server on my PC, both of which are much less prone to crashing. On Monday, I will hook the PC back up.


  5. Thank you. The model is fascinating.

    It might be interesting to provide some of the statistics for the coefficients so one can evaluate which are most significant. If we eliminate ln(CO2) the R2 is still surprisingly high (70%), but the fit is not very good especially over the most recent decade. A (very) naive interpretation of this is that ln(CO2) explains about 25% of the temperature variation. If I tried to do something like CSALT using annual temperature data, I think I could make it work without the interpolation, quite a lot of the data is annual data. This would mean that the various orbital forcings would need to be averaged for the year. I think trying to match the data on a monthly basis is a little much to ask of this model. It is pretty amazing though.



    • DC,
      R provides a synopsis of the coefficients. The first 5 CSALT terms are

                    Estimate Std. Error  t value Pr(>|t|)    
      (Intercept) -1.779e+01  8.022e-02 -221.790  < 2e-16 ***
      c            3.086e+00  1.393e-02  221.486  < 2e-16 ***
      s           -3.216e-02  1.004e-03  -32.019  < 2e-16 ***
      a           -1.770e-01  5.853e-03  -30.233  < 2e-16 ***
      l           -5.619e-02  7.379e-04  -76.152  < 2e-16 ***
      t            5.655e-02  4.190e-03   13.496  < 2e-16 ***

      C for CO2 is very explanatory.

      I may try to expose this information to the user.


  6. Thanks Paul,

    You might also want to give a brief explanation of the meaning of std error and the t values because a lot of people know less than me about statistics.



    • And to be honest, I know very little. I took some econometrics in graduate school, but I have not used it much and that was long ago. I am trying to get back up to speed.



    • A good indicator is the ratio between the Std.Error and the Estimate. The smaller number this is, the more the regression is homing in on a good fit. The “t value” is close to the reciprocal of this value, so a larger number here is better.


      • Thanks Paul,

        I actually did know that and I read up on my statistics to make sure my recollection was correct. Thank you for answering questions that probably strike you as sophomoric. No doubt there are people out there that know even less than me.

        A quick question. I am using annual rather than monthly data. The fact that I have about 12 times fewer degrees of freedom will likely cause my R-squared (multiple) for my regressions to be lower than yours (if we are using the same regression method. Does that sound correct?



      • DC,
        I am being very pragmatic about how I am using the regression. I am thinking about it more as a solver for a variational problem than in a statistics sense.

        Today I responded to someone on the Climate Etc blog who said that I was not “explaining the pause”

        Matthew R Marler | January 7, 2014 at 2:55 am

        What do you mean be that? Your regressor variables do not represent any energy flows, even indirectly? Earth mean temperature rises and falls without fluctuations in energy? SOI rises and falls without fluctuations in energy?

        “and pause well explained”

        Actually, it is well “modeled”, not well “explained”. You do not have, for example, an explanation of why the regression coefficient for the SOI is what it is, what energy flow is represented by the regression on SOI, or anything else that counts as an “explanation.” You simply have an estimated regression between two observed variables, something that may someday be explained.

        I responded that he was confused between a physics-based model and what kind of mathematics applies to solve a physics-based model. I can give examples in mechanical statics or in chemical reaction kinetics or in a host of other physics disciplines, but solving an elementary electrical circuit is a very simple way of showing how to apply a regression model:

        So one can see that we can apply linear regression to solve for the unknowns. Physics students understand how this works from introductory courses where they can simply plot the variates and find a slope in the line. That is a form of linear regression.

        The part that is left over, the residual, is an indicator of how much that is unexplained. If this looks like white noise or measurement error, we have gotten as far as we can given the original premise of explaining the varriation by free energy factors.


      • Paul,

        I must be doing something wrong. Every time I use the CSALT model, I seem to crash the server. I really am not trying to do this, you can contact me by e-mail if you can see something specific I should try to avoid.

        I am trying to get up and running on R (which I have not used before). I have decided to use annual rather than monthly data (to avoid the need for interpolation in some of the datasets). As my first crack at it I did a simple OLS Regression on ln(CO2) vs NOAA Land-Ocean temperature from 1880 to 2013. I got the following summary:
        Min 1Q Median 3Q Max
        -0.32159 -0.09941 0.01008 0.10677 0.32407

        Estimate Std. Error t value Pr(>|t|)
        (Intercept) -0.4202 0.0247 -17.01 <2e-16 ***
        cdata$carbon 3.1129 0.1483 20.99 <2e-16 ***

        Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

        Residual standard error: 0.1456 on 132 degrees of freedom
        Multiple R-squared: 0.7695, Adjusted R-squared: 0.7678
        F-statistic: 440.7 on 1 and 132 DF, p-value: < 2.2e-16
        To produce the chart below I smoothed the NOAA temperature data by taking the 15 year centered moving avg and compared with the model. I plan on adding one component at a time without lags initially to keep the model simple and will use the t statistics that you posted as a guide, I will start with the components of greatest significance and work my way forward. I appreciate you putting up with my endless questions, I hope to pass on what I have learned to the math challenged masses (of which I am a member).



      • DC,
        The http://entroplet.com address has been switched to point to a different underlying IP address, so it may need to be refreshed in your browser’s cache.

        As far as your initial approach to OLS regression, it looks good. It is then a matter of adding more arrays of exploratory data to do the multiple linear regression.

        I will also add in that plug-in, thanks.


      • Hi Paul,

        I realized looking back at some of your posts that maybe you weren’t using all of the data in your regressions. When I use 1880 to 1960 for CO2 only the R2 falls to 37%. Is there any need to use more robust regression methods? I thought I would start out with OLS and then try some of the more robust methods. The problem is that my statistics skills are relatively weak and it will not always be clear when robust regression is called for. It seems to me that there are not a large number of outliers and that leverage points are not likely to be a problem.

        What did you use for a regression method and why? I have R so I have access to more powerful methods, once I determine how to use them properly.
        I now have a CL model with R2 of 81% using OLS on 1880 to 2013 with -0.1 C correction for 1940-1944.



  7. Thank you. The server is great. There seems to be something missing from the CSALT model. My understanding is that the current levels of carbon dioxide would result in continued warming due to an overall positive forcing on the climate system. The model would show no change in temperature if we held all inputs (CO2, aerosols, SOI, TSI, and LOD) constant. My guess is that the climate feedbacks are missing and no doubt these are not easily added to the model. Is it correct to say that this is a no feedback model?


    • DC,
      I know exactly what you mean. For now the CSALT works in the TCR (Transient Climate Response) mode when it comes to modeling the temperature change to a change in CO2. This is essentially a delta function response to a change in CO2 level.

      In practice the actual response is a fat-tailed diffusional lag, very similar to what we use for the Bakken response (to provide an analogy). This would then get us closer to an Equilibrium Climate Sensitivity (ECS) value, as the response would keep building if we suddenly shut CO2 down.

      No doubt we can do something better to model this effect, but as long as the CO2 input is growing, we can get by with the delta TCR as an approximation. The mathematical proof is that an acceleating power-function convolved with a weak 1/sqrt(t) response is close to that power-function. We kind of see that in the Bakken results as well.

      The complexity of adding this math is that someone can not easily recreate the dynamic response on their own. As it stands, CSALT generates the outputs as a linear combination of the inputs and corresponding scale factors determined by the fit. Plus an exponential lag if required. In contrast, most people do not know how to apply a diffusional lag to the CO2, or to any of the other values it they involve thermal diffusion into the ocean.

      BTW, you can play around with this idea by changing the CO2 lag greater than 0. This is an exponential lag, which isn’t quite right, as it does not model a diffusional response correctly, but you can see how it will move the TCR upward, as the effects of CO2 are deferred.

      In summary, adding in the ECS will get us closer to the slow-feedback model that we would like to see.


  8. Thanks. I had the not so bright idea of looking at what happens if CO2 levels stay at present levels and the other inputs were at their average 1880-2013 levels (I only have C+S+A+L+T in my model), I was disappointed to see no change in temperature and upon reflection realized that in a linear model if all the independent variables are constant, obviously the dependent variable (temperature in this case) remains the same. Instead I decided to see what happens if we ignore feedbacks and ln(CO2) rises at the 1970 to 2013 rate out to 2100. Temperature gets to about 2 C above preindustrial (-0.25C on NOAA scaling) or 1.75C above the 20th century average by about 2090. It would be interesting to add the diffusional element. I am not sure if a straight diffusion or OU Process would be more appropriate in this case. You are thinking more in terms of oceans, I was thinking clouds, but that would likely be more of a second order effect and is not as well understood. So I take it you are thinking about ocean heating and thermal diffusion coupled with reduced ocean uptake of CO2.
    Climate change is pretty complicated once you get to the carbon cycle and other feedbacks.



    • DC,
      Indeed I am thinking more in terms of the lagged thermal response of ocean heating rather than the faster feedback effects from clouds.

      Yes it does get more complicated with a more detailed carbon cycle model but with some effort I am sure we will make progress.

      And then there are the very long term effects due to long-term albedo changes (the very slow feedbacks) that James Hansen has described.


      • Is there ocean temperature datasets that could be used in such a model, it seems datasets are relatively short duration aside from sea surface temperature so we would need to make some assumptions about mixing with deeper ocean.

        A couple of thoughts on CSALT, could changes ice volume be added to give a feel for the albedo feedback and maybe changes in the airborne fraction of CO2 to indicate carbon cycle feedbacks or do we need a completely different type of model.

        How do you do your modeling of the orbital forcings and the hale cycle? Do you just take the residual and try to fit a set of sine functions with frequencies matching the different cycles and the amplitude and phase determined by a regression of the residual vs several sine functions?
        Also do you use ols regression or something more robust? I ask because when I realized that I can do a multiple regression in excel I stopped using R, because it was quicker for me. My csalt model is coming up with slightly different results than yours. (I have used the -1 in the lags to eliminate everything except CSALT. The lag for GISS aerosols does not have any effect on entroplet, can you tell me what the lag is for aerosols when using GISS aerosol data? It my be a matter of using different data sets, I can adjust the temp time series to NOAA land ocean (which is what I used), but I may be using different CO2 series (I used Law Dome 20 yr spline fit up to 1959, Mauna Loa to 1979, and then NOAA global set 1980 to present), and my SOI and aerosol coefficients do not match well with yours. Also I used a linear interpolation for CO2 (up to 1959) and for LOD.

        My SOI data is at http://www.cgd.ucar.edu/cas/catalog/climind/SOI.signal.annstd.ascii
        found at http://www.cgd.ucar.edu/cas/catalog/climind/soiAnnual.html#download
        I uses the SOI file “SOI.signal.annstd.ascii”
        but before 1935 it might be better to just use the Darwin
        data rather than the SOI data, according to the website.


      • DC, The ocean data is deep so to speak. Have to figure out what depth range to use to get good signal-to-noise on the data.

        The regression on orbital cycles is close to what you are imagining. I place one Cosine factor and one Sine factor as orthogonal components for a particular frequency. This allows us to get the amplitude and phase of the best fit.

        I don’t have the lag on the GISS aerosols because those I perceive as being “baked in” to the analysis already. Perhaps I could reconsider this though.

        I find it very valuable that you are working at the problem from an independent angle. By all means redo the CO2 estimates to your liking.

        We are getting incredible correlation coefficients on some of the fits, above 0.996 with minimal filtering. I don’t know how much closer we can get but this is getting close enough for a kill.

        Any mistakes that I have made, if corrected, will in all likelihood only improve the fit. I am a firm believer that it it is not coincidence that the model works as well as it does. Giving the temperature the full thermodynamic treatment of all the forcing functions is something that no one has seriously attempted before. This could turn into a slam dunk for explanation and perhaps forecasting.

        BTW, one idea I have is to use all the data and coefficients to do a “hindcast” to years before 1880. This could go back to 1850 using the HadCrut data set.


  9. Paul,

    I was thinking of the hindcast idea as well, maybe compare with an average of several paloeclimate recontructions, The data on LOD goes back pretty far (1600’s I believe) on the SOI and TSI, you could use the orbital and hale cycles without the TSI and SOI data as a fit, maybe use the Law Dome 20 year spline for CO2 (all the way back to 1 AD), possibly a 60 year cycle could be substituted for LOD (in order to go back further in time). It would indeed be interesting.



    • DC,
      I am hindcasting back to 1850 for the moment. The SOI data only goes back to 1866 and the fit is fairly decent, but HadCrut shows a huge temperature spike in 1878 which is only partially compensated by the SOI spike occurring at the same time.

      I think there is a reason that GISS only goes back to 1880, as the temperature records get increasingly noisy before that time.


  10. Paul,

    I am doing this both to confirm your results and because I feel I get a firmer understanding of the data and analysis by actually working with it. I really appreciate your patience. One reason I asked about any lag on the GISS aerosols is because the aerosol coefficient and the SOI coefficients that pop out of my regressions are different from what I get from entroplet. Also I thought we might be using different datasets. Do you use the same SOI dataset as me (and so you use the Darwin only data before 1935 (supposedly the Tahiti data is not very good before 1935).



      • Hi Paul,

        I was just trying to reconcile the differences between your analysis and my own. I used the same dataset for SOI, the NCAR website suggests that before 1935, using the Darwin only data may be a better estimate of the SOI, I thought thet may have accounted for the difference, but the coefficients were still not aligned when I tried Darwin data for 1879.5 to 1934.98 and SOI data from 1935 to the present (6 month lag). I am using the Sato aerosol data (no lag) and the LOD data that you gave me a link to (with linear interpolation). I used the NOAA monthly Land-Ocean Data (with no rescaling). Maybe we are using different CO2 data, what did you use prior to 1959 for CO2? Also I used 278 ppm (1500 to 1750 average) for preindustrial CO2 so I am using ln(CO2/278) in the regression for “c”.



  11. Pingback: Reverse Forecasting via the CSALT model | context/Earth

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