CSALT model

(added note: this is the initial post covering the CSALT model,  index to other posts)

It would be nice to have a simple climatology tool that  illustrates the trend of modern-day global warming, while also providing a view of the underlying mechanisms. What I came up with is the CSALT model, which is based on some of the analysis from a recent blog post titled Climate Variability and Inferring Global Warming.

What CSALT does right now is act as a reverse weather forecaster that can hindcast the global average temperature accurately without using any direct measures of temperature.

All CSALT requires is historical time-series of these characteristics:

  • CO2 concentration in the atmosphere
  • SOI (Southern Oscillation Index) as defined by the difference in values of atmospheric pressure in Tahiti and Darwin
  • Aerosol concentration in the atmosphere as generated by volcanic events (and potentially man-made events such as armed conflicts).
  • LOD (Length-of-Day) as defined by the correction in the Length Of Day measure in seconds. (edit: this is related to the Stadium Wave hypothesis of Wyatt&Curry)
  • TSI (Total Solar Irradiance) as a measure of the variation in solar insolation due to quasi-periodic sunspot activity.

The parameters form the mnemonic CSALT, with the CO2 acting as the main driving force to the rising temperature trend (the “control knob” ) and the other SALT terms adding fluctuations (i.e. natural uncertainty) to the trend.

Figure 1: CSALT output using the GISS global temperature time series. Top panel : Web interface . Bottom panel: Two different intervals of the same dataset using the interactive graphics

The CSALT model works very well over a time period stretching back to 1880. You can try it out here and play with the tuning parameters:

Foundation of the CSALT model

As always physics rules when it comes to defining natural behavior, and so we start by looking at energy balance. Consider a Gibbs energy formulation as a variational approach:

dG = V dP - S dT - {dE}_a - {dE}_b - {dE}_c

where S can describe heat capacity terms and the extra energy terms can represent external forces such as EM and induced terms such as the GHG effect reducing the radiative outflow.

So we have:

LOD representing differential internal free energy of the spinning planet as it gains or releases angular momentum → dG
SOI representing a pressure differential → V dP
Aerosols representing a reflective EM → dEa
TSI representing an external EM driving force → dEb
ln(CO2) representing a suppressive EM → dEc
Temperature and heat capacity (S) combine → S dT

We then have enough information to try to solve the first equation as a variational problem, where each of the terms needs to be scaled to get the best fit.

[ $ latex LOI = SOI – S dT – TSI – Aerosols – ln(CO2) $ ]

LOD = SOI - S dT - TSI - Aerosols - ln(CO2)

or rearranging to make it more convenient to a solver

dT = c_1 ln(CO2) + c_2 SOI - c_3 Aerosols + c_4 LOD - c_5 TSI

That turns into multiple linear regression problem as we use data from each month from 1880-2013 to give a best fit with respect to the coefficients ci.

That is essentially what goes into the R linear model solver, where we use the data from a temperature time-series such as GISS for dT and appropriate data sets for the other factors:

lm( dT ~ C + S + A + L + T )

and then out pop the coefficients, c1 … c5

The reconstructed function is created from these coefficients applied to the individual data sets.

So to do a good job of forecasting climate in the future, all we need is a good projection of CO2 levels to get the elevated temperature anomaly and the SALT parameters to get at the possible fluctuations.

In summary, the basic premise of the CSALT model is that are using a variational energy balance approach and then perturbing the system by looking at the energy differential with respect to various contributions to the system. Nature will then pick a path that has a stationary point of action that conserves energy, essentially treating the summed energy components as an invariant. One only has to account for all the ways that energy can transform so as not to lose track of any of the major pathways.

The residual, or what is left over, is noise. This noise is either systemic (caused by measurement errors, etc) or aleatory which consists of contributions from everything else we don’t yet fully comprehend, or can’t because of the shear number of other paths.

This is not the optimal situation, as not knowing all the combinations makes it an incomplete solution, but picking up the most significant patterns is very useful. It is indeed remarkable how well dT can be reconstructed from the other energy constituents.


The Tunable Parameters are mainly exponentially damped responses to the datasets. A lag of this kind gives the energy term time to propagate to the rest of the earth, or if its a discrete event the amount of time it takes to dissipate.

  • SOI lag  6 months to model the propagation across the earth
  • Aerosol lag  Between 2 to 3 years to dissipate an individual discrete event, otherwise use the Sato dataset from GISS
  • LOD lag  About 7-8 years for the internal energy to propagate and disperse  from the deep ocean or where ever it may be originating [4][5][6].
  • TSI lag another short lag to model response from latent and transient energy stores.
  • CO2   no lag on this because it is a continuous increase in globally dispersed atmospheric concentration.

The Fitted Parameters are what the solver finds as scaling factors for the lagged regressor variables.

  • SOI scaling This is scaled to fit most of the subdecadal fluctuations
  • Aerosol scaling This is scaled to fit the major volcanic events such as Pinatubo, which result in a temporary transient reduction in temperature.
  • LOD scaling A subtle factor which effects the ocean temperatures (2× to 4×) more than the land. This captures the PDO effect which impacts SST.
  • TSI scaling This turns out to agree with that predicted from the proportional radiative heating, about 0.05C heating due to a 1 W/m^2 change in solar irradiance at the top of the atmosphere
  • ln(CO2) scaling  This drops out to agree with the accepted value of TCR=2°C for global warming and 3°C for land.

Again, all the fitted parameters are due to the solution of the multiple regression. The scaling factors of the fitted parameters are shown at the top of the graph in blue cells. A correlation coefficent for the goodness of fit is also included.


If the surface of the earth releases aerosols in the air as a discrete event or impulse from a volcanic eruption, the effect will temporally lag from the action as the aerosols disperse.  This has to be in a direction that obeys causality.

As a rough approximation we can always say that the lag function is an exponential damped response on the stimulus.   The term lag is used to describe an “exponentially damped response” or “first-order impulse response function” .

The October 1943 spike is the single anthro effect I added apart from CO2. You can turn that on/off by the anthro aerosols checkbox. I added that because since the overall agreement is so good, one can really start to look at particular points in time for further evaluation.

All the lags can be modified by the user. You can turn all of them off by setting the lags to zero. The agreement is still good.

The application of the LOD (length of day) is crucial as a proxy for multidecadal oscillations. This ensures conservation of energy and conservation of momentum according to work by Dickey et al at NASA JPL [3][4][5]. The fluctuations in kinetic energy have to go somewhere and of course changes in temperature are one place for this dissipation.  So as of now, the LOD is in the category of a phenomenological behavior that can be described by a heuristic of a ~60 year cycle.  If the cycle does not reoccur in the future, that heuristic will get shot down.  But that is the nature of a heuristic, in that it is a stop-gap measure to describe something that is not yet completely understood.

Some insight to the relative importance of the LOD is seen by its impact depending on the temperature series used.  If we use a global series such as GISS-LOTI or HadCrut, then the LOD influence is clearly there.  However, if we use the Land-based CruTEM or BEST or GISS-land (dTs)  series, the LOD influence is a factor of 2× to 4× lower in strength.

Figure 2: The contribution of LOD is much greater if ocean surface temperatures are included in the globally averaged time series (above)  than if a land-based record (lower) is fitted.

In fact, the contribution of the LOD is still there because many of the land observations are near the ocean but it is suppressed. Yet, bottomline, the LOD does not seem to have as big an impact on land temperatures as on SST.

As far as volcanic disturbances, I initially took the 10 significant events from the BEST spreadsheet of Muller [6] and then later added Sato’s optical depth profile for volcanic activity [2].

Also note that several different temperature time series are included, and combined in various ways. A low-pass first-order exponential smoothing filter is provided to see how well the model works when the remaining red noise is removed.

Figure 3: Comparison of global temperature time series, in this case comparing GIS (gistemp) and HADCRUT4.


Related Work

Climate Variability and Inferring Global Warming


[1] A. A. Lacis, G. A. Schmidt, D. Rind, and R. A. Ruedy, “Atmospheric CO2: principal control knob governing Earth’s temperature,” Science, vol. 330, no. 6002, pp. 356–359, 2010.
[2] “Data.GISS: Forcings in GISS Climate Model: Stratospheric Aerosol Optical Thickness.” [Online]. Available: http://data.giss.nasa.gov/modelforce/strataer/. [Accessed: 24-Oct-2013].
[3] J. Dickey and C. Keppenne, “Interannual Length-of-Day Variations and the ENSO Phenomenon: Insights via Singular Spectral Analysis,” JPL Technical Report, 1997.
[4] J. O. Dickey, S. L. Marcus, and O. de Viron, “Closure in the Earth’s angular momentum budget observed from subseasonal periods down to four days: No core effects needed,” Geophys. Res. Lett., vol. 37, no. 3, p. L03307, Feb. 2010.
[5] J. O. Dickey, S. L. Marcus, and O. de Viron, “Air Temperature and Anthropogenic Forcing: Insights from the Solid Earth,” Journal of Climate, vol. 24, no. 2, pp. 569–574, 2011.
[6] BEST, “FAQ|Berkeley Earth,” BERKELY Earth Surface Temperature, 2013. [Online]. Available: http://berkeleyearth.org/faq/. [Accessed: 12-Mar-2013].

47 thoughts on “CSALT model

  1. This is a typical feedback to the CSALT model:

    DocMartyn | October 26, 2013 at 6:52 pm |

    Capt, when you are not cocooned by an alien with acid for blood work out the degrees of freedom in a fit based on five components, each with a chosen ‘lag’, from 6 to 90 months.

    Doc, Most of the lags don’t matter much at all — 4 out of the 5 lags are 6 months or less which is a small amount needed to propagate a stimulus over the globe in one year, and which is essentially the resolution of the time series. The long lag is due to LOD which is a long-term ocean propagating factor. Dickey of JPL among others noticed that it had an 8 year delay ~ 96 months if it is used to match the gradual SST variations ( see ref [5] — “Here the CAM and LOD (with sign reversed) have both been lagged by 8 yr, and all series have been detrended over the interval shown.”)

    As far as scale terms, each of the 5 are determined by regression. One of them, that due to TSI, agrees remarkably well with the theory for a temperature increase. The ln(CO2) agrees with the consensus value of TCR=2C. The 2 parametric values for volcanic Aerosols and SOI are there to fit the complicated subdecadal fluctuations and do a remarkable job. The last parameter for LOD is there to match the multi-decadal scale variations and is the most unconstrained of the parameters.

    So the CSALT model really doesn’t has as many free parameters as you think. I can fit it by hand in fact, which is not something easily done with 2*5=10 degrees of freedom, unless you know what most of them are beforehand.


  2. How much worse does the fit get if you leave out the LOD effect? I’ve never heard of that being a significant effect, so my guess is that it’s quite small, and the fit would still be good without it.


    • John,
      The LOD effect will increase the correlation coefficient from 0.93 to 0.97 if included. As you know, when the CC gets closer to 1, it gets harder to incrementally make advances.

      I should probably include a checkbox to eliminate each of the factors so someone could see the effect for themselves.

      As it is, you can see the effect of each of the components by looking at “View the fluctuation components”.


      • The lags for the different factors have differing sensitivities. CO2 and TSI barely budge around the 6 month lags. The other 3 factors, SOI, Aerosols, LOD, benefit from the nominal lag provided.
        Click on the image below to enlarge:


  3. Hi Paul,

    LOI = SOI – S dT – TSI – Aerosols – ln(CO2)
    or rearranging to make it more convenient to a solver

    I may be mistaken, but should “LOI” be “LOD”?

    Also I am not sure what EM is, it brings electromagnetic to mind, but I am pretty sure that is not right. Very nice model. Is the air pollution of the 60s and 70s in North America and W. Europe, and the recent air pollution in Asia a factor in masking a temperature rise from carbon dioxide or is that an urban myth? Is there any aerosol data that could be used to include this effect (if it in fact exists)?



    • DC, Thanks, I fixed the typo on LOD.

      EM does mean electromagnetic energy which is essentially decribes TSI or solar radiation. I was trying to classify the energies according to the categories that they belong to.

      The reflective aerosols are definitely an influence. Hansen, Lacis and others at GISS think that the log(CO2) sensitivity is much higher and the anthropogenic aerosols are countering the GHG effect. I simply absorbed that factor into the log(CO2) factor for now as they seem to scale with industrial output, but in opposite directions. It does mask the temperature rise though.


  4. Part of the problem with typical economic development is that at first dirty sources of energy like coal and wood are used and then people start to demand clean air, or we start to run short of coal and/or wood and this masking of the underlying CO2 forcing is then revealed. It would be interesting to attempt to incorporate your shock model into the CSALT model. For example take estimates of world coal from Rutledge or Steve Mohr and use your shock model to estimate the coal peak and then make some big assumptions about a correlation between optical depth and coal consumed, include man-made aerosols explicitly a la Hansen and then predictions could be made about how much warming we are likely to see when coal peaks or consumption is reduced for health reasons. The 2011 paper by Hansen et al was quite interesting.



  5. On LOD see
    Mazzarella A., A. Giuliacci and N. Scafetta, 2013. Quantifying the Multivariate ENSO Index (MEI) coupling to CO2 concentration and to the length of day variations. Theoretical and Applied Climatology 111, 601-607.
    DOI: 10.1007/s00704-012-0696-9.

    Click to access CO2-MEI-LOD.pdf


    • Scafetta and company miss the obvious angle that the seasonal ripple and variation in atmospheric [CO2] are due to outgassing of CO2 that follows the ocean and biosphere temperature

      They try to prove that:
      ENSO => [CO2]

      Where in reality it is:
      Yearly solar insolation + ENSO + FF => dT => d[CO2]

      where FF is other forcing functions, which van include [CO2], making it a positive feedback.

      In the end they say this:
      “Equally, an increase of sea surface temperature causes a smaller solubility of CO2 in the ocean and so a higher concentration in the atmosphere.”

      So why didn’t they just correlate against SST in the first place?

      They do get the sign right on the LOD though.

      Scafetta has to work this ground carefully, because he can’t give away that the obvious warming culprit is CO2. That would not be good for his planetary gravitational theory that he goes on about elsewhere.


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    • That’s pretty funny, considering that I also included a check box to use the GISS aerosol model developed by Sato:

      As a means to understand that model, I first applied the simple BEST model provided by Muller

      All this does is use the top 10 volcanic eruptions in terms of estimated aerosol production and provides sporadic impulse responses. I placed the start of the eruption date accurate to the month on the time series.

      The Sato GISS aerosol model agrees with this simple model, which is not surprising considering that large volcanic eruptions are fat-tail distributions that contain most of the aerosol production in the rarest of events.

      Like I said it is funny that you think I fudge factored any of this, when the GISS aerosl data checkbox (GISS volc aerosol) is sitting right there in front of your eyes.

      Accusations of fudging at this level usually come about because someone is getting close to the answer, and someone else does not like it.


    • If a volcano were to erupt with tons of CO2 instead of injecting reflective aerosols high into the atmosphere, then the effect would be reversed.

      That is why Hansen says the data has uncertainty. It is unpredictable, and the effects of past events have to be inferred.


  7. When I looked at your website (at your invitation) one of the first things I saw was the model “equation” with the variable a(man) with a coefficient of 4+. My assumption which turned out to be correct is that it refers to manmade aerosols. I stopped reading right there and made my comments above because I already knew what Hansen said it in his January 15 paper – “…..aerosol forcing is extremely uncertain”. A model based on “guesses” (Hansen’s word) is subject to manipulation and isn’t useful.
    However, when you indicated that a(man) is not part of your current model I’ve gone back and read more. I like the idea that the model is based on measureable data. But I do have a question on LOD. What is your data source and where can one find the data?


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  10. A concrete question. When I go to your page here:


    …There’s a default set of lags for all of the right-hand-side variables except for nao, and if I press on the graph button I see a vector of estimated coefficients, including one for nao. Would you please tell me what lag (if any) of nao was used in the model with those estimated coefficients? Thanks.


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    • DC, I haven’t seen that particular one but am having a discussion with the author of the paper, Ian Wilson, at the Climate Etc blog.
      It seems that he does not like at all what I am doing with the CSALT model.

      I think he does not like it because the effects are not big enough for his tastes. Apparently he wants the orbital forcing elements to completely swamp out the CO2 and GHG effects, so as to disprove the AGW theory.

      So he has apparently become angry that someone like me has come along and quantified how small the orbital effects are.


  12. Interesting.

    I read your work first and thought maybe you had read his work and then integrated it into your CSalt model.

    Does your aerolsol data come from Goddard/Sato?



    • Hi Paul,

      I was trying to reconcile the data you use with data I have found on the web. For LOD I tried to reconcile data from JPL called LUNAR97 in a paper by Richard Gross in 1999 (link below):

      Click to access 99-1782.pdf

      see page 21.

      I assume your data in in seconds of difference from 86400 seconds in a day, the Gross table 1 gives this difference in milliseconds and it does not agree well with your LOD data. Where did you get the LOD data you use? Thanks.

      Dennis Coyne


      • DC,
        Thanks for that data. The offset doesn’t matter, all that matters is the relative scale, as the regression will remove offsets and absolute scales.

        In that case the two sets look like this when compared:

        The JPL data does have some extra features in the detail, which is useful if it will discriminate with respect to the global temperature record. But then again, I am using a 5-year exponential lag on the LOD effects to propagate as a temperature response (see the Stadium Wave paper by Wyatt & Curry) so that the detail won’t be that important.


    • DC asks
      “Does your aerolsol data come from Goddard/Sato?”

      The radio button labelled GISS aerosol model will switch in Sato’s data.

      I prefer using my estimates because it only uses the major volcanic events, and I can get a better understanding of the effects.


  13. Hi Paul,

    I have used the view fluctuations button to look at your underlying data.

    The LOD data may be too high by a factor of 10 based on the data from Scaffetta and from Gross. The difference in length of day should range from -3 to +3 milliseconds over the 1860 to 1997 period. Your LOD data ranges betweem -100 and +100 milliseconds where I have assumed your data is presented in seconds. Though there are no units in the fluctuation table so I could be wrong.



    • DC, Thanks for that. The units are arbitrary as the regression algorithm will rescale. So, as long as the relationship between the Scafetta or JPL/Gross data and the IERS data is linear, then it should be OK.


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  19. Hey, I just reread this. Good work! I actually came on to see what you used for data for solar irradiance? I just read a paper about UV B in the troposphere. It was from the late ninties but it gave a real good explaination for a non-scientist about UV flux and chemicals in the troposhere and the green house effect. Given that all the info on UV is not well understood, I wondered if you knew where the data on that was and fit it in to CSalt. Does Sorce have any? I do read your posts on Climate Etc and occasionally like to spar with you but don’t take anything I say too seriously.



    • I think it is too complicated to start to infer UV effects. CSALT will scale the TSI numbers to fit the curve and if there is some sort of amplification factor going on, we should see it.

      As it is the TSI fluctuations are around 0.05C, which is what is predicted from a Stefan-Boltzmann differential analysis.


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