Climate Variability and Inferring Global Warming

The Earth’s historical global mean temperature record shows much variability, often changing by tenths of a degree from year-to-year. If this variability is somewhat cyclical, one could reason that the long term would approach a mean value of zero, indicating that as many positive temperature fluctuations occur as negative fluctuations.  That describes the reality of noisy fluctuations — noise is a detectable signal but it does not necessarily impact the outcome of a trend.

A bias-free noise should not contribute to the long term trend in the global temperature time series either.  The premise is that if we can remove this noise from the historical record, then we can make a more precise estimate of the underlying trend [1].

The candidates for noisy components are:

  1. The Southern Oscillation Index (SOI), which measures a large scale pressure differential
  2. Volcanic disturbances generating reflecting upper atmosphere aerosols
  3. Total Solar Insolation (TSI) differences from sun-spot activity
  4. Atlantic Meridional Oscillations (AMO) which are detrended fluctuations in the sea-surface temperature (SST)

These data sets are all easily available from sources such as NCAR and NOAA. The resolution of SOI is monthly data points stretching back to 1880.

The strongest contributing components of the set are (1) SOI and (2) volcanic disturbances.

The SOI is a red noise-like process which shows reversion to the mean properties.  Since the SOI is a measure of the sea-level barometric pressure difference between Tahiti and Darwin, this is not expected to vary from zero when taken as a long term average. That makes it a potential candidate for variance reduction, as it will add no bias to the underlying trend estimation.  The connection between SOI and global temperature is known to exist, showing a lag of around 7 months [2].

SOI impact

Figure 1:   The SOI and its trendless mean, scaled to reflect its approximate contribution to the global temperature anomaly.

Volcanic disturbances are sporadic, appearing more like a shot-noise behavior.  Only the largest of the disturbances add significant aerosols to the atmosphere, which will cause the atmosphere to cool temporarily. There are estimates for the amount of sulfates emitted and for the mean damped exponential residence time of the aerosols in the upper atmosphere [3][4].

The following figure shows how well the application of a scaled, lagged SOI and then a scaled, damped volcanic disturbance mapped to a growing trend matches that of the NASA’s GISS temperature record.

Figure 2:  Adding SOI and then volcanic disturbances to a upward trending curve to match the GISS temperature time series.

One can use a tool such as Eureqa to determine the proportional contributions to the fluctuations while at the same time fitting to the underlying trend.


Figure 3: The results from Eureqa show the best and least complex fit to the data. The top curve result is able to find a small contribution due to the TSI fluctuation. The bottom curve is volcanic disturbances, v, alone. The scaling factors were estimated prior to a Eureqa run, which explains the unity scaling factors.

The second order contributions come from (1) TSI sunspot variations and (2) AMO.

The TSI contribution is estimated to be around 0.05C for a 1 W/m^2 variation in solar variation, which essentially comes from the Plank response to an incoming thermal stimulus.

Delta T = 0.7 times 1/4 times frac{T}{4P} Delta P

The Eureqa solution shown in Figure 3 recommended this value as well, which is an imperceptible perturbation to the SOI and volcanic signals.

The last contribution is due to the AMO index. This has some issues relating to the fact that the detrending and temperature fluctuations are “baked in” to the index since the AMO itself is derived from SST time series.  However, Eureqa will find that the addition of the AMO signal to the mix will reduce the error of the model fit. The significant bumps and dips are further reduced as shown in the figure below.


Figure 4: Adding the AMO signal to the mix will reduce the variability further. The 15-year hiatus in temperature increase disappears at the end of the temperature record mostly due to compensating the decline of the SOI signal.

The real objective of this exercise is to demonstrate how the noise artifact known as the “pause”  or “hiatus” which has occurred in the last 15 years is simply that, an artifact of statistical fluctuations. By compensating the GISS temperature record with an unbiased estimator of the major noise terms, the true underlying trend is revealed and a flat noise spectrum is all that remains (see below).


Figure 5: Residual temperature after subtracting the components

The next obvious step is to estimate the log sensitivity of these de-fluctuated curves to the growing concentration of atmospheric CO2. The following chart shows log-sensitivity plots of TCR and ECS based on global mean surface temperature records. TCR uses global temperature from GISS while the estimation of ECS uses the land-only BEST temperature record (which eliminates the slow feedback caused by the ocean acting as a heat sink). Both these numbers match the mean estimates determined from paleo and modern instrumental evidence as well as first-principles physics models.


Figure 6 : De-fluctuated temperature records log-regressed against atmospheric CO2 concentrations.

Look at the bottom ECS plot and note the log-regression best fit formula. The temperature anomaly goes down another 25C as the CO2 approaches 1 PPM. This is close to how cold the earth would be if we didn’t have CO2 as a control knob  (to use Lacis’ term [5]).
Yet the ECS of 3C for doubling of CO2 is enough to keep Earth’s troposphere warm and fit enough to support biological life.

That is why it is important to understand and advance climate science.

The analysis in this presentation is not that complex — many people are following the same path that I outlined above [6] .


[1]  Y. Kosaka and S.-P. Xie, “Recent global-warming hiatus tied to equatorial Pacific surface cooling,” Nature, vol. 501, no. 7467, pp. 403–407, 2013.
[2]  J. D. McLean, C. R. de Freitas, and R. M. Carter, “Influence of the Southern Oscillation on tropospheric temperature,” Journal of Geophysical Research, vol. 114, no. D14, Jul. 2009.
[3] J. Hansen, M. Sato, P. Kharecha, and K. von Schuckmann, “Earth’s energy imbalance and implications,” Atmospheric Chemistry and Physics, vol. 11, no. 24, pp. 13421–13449, Dec. 2011.
[4] J. L. Lean and D. H. Rind, “How natural and anthropogenic influences alter global and regional surface temperatures: 1889 to 2006,” Geophysical Research Letters, vol. 35, no. 18, 2008.
[5] 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.
[6]  “The Pacific Ocean fills in another piece of the global warming puzzle, and puzzles Curry.” [Online]. Available: [Accessed: 24-Sep-2013].




26 thoughts on “Climate Variability and Inferring Global Warming

  1. One more important reference to add in

    G. Foster and S. Rahmstorf, “Global temperature evolution 1979–2010,” Environmental Research Letters, vol. 6, no. 4, p. 044022, Jan. 2011.

    This uses the SOI to remove the current pause, but is easily overlooked because a figure that shows how well the peaks and valleys line up is not included.


  2. From
    The length of day (LOD) correction is quite interesting. It’s an independent, factor that apparently has no effect on the underlying trend according to Dickey [1].

    So instead of correcting for SOI and AMO, we just correct for SOI and LOD (plus volcanic and the small TSI). All cycles and plateaus are removed, with just a few spikes remaining

    [1]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.


  3. Two comments: First, instead of plotting CO2 on the y-axis, you really need to plot CO2 forcing, which can be computed from the equation F=5.35 * ln(C/280), where F is forcing in Watts/m² and C is atmospheric CO2 in ppmv. That should remove the non-linearity, especially at the left end of the ECS graph.

    Second, the idea that length of day has an effect on climate is just plain nutty. Not only is there no physical basis for it, once you subtract out the effects of CO2 and other known climate forcings, there’s not even a correlation left.


    • KP, so the suggestion is to replace the fitting parameter of time with the parameter of natural log of CO2 concentration. I have no problem with that, since that is close to the log of a constant plus a power law, which is a power law to first order.

      The key new piece of info here is the correlation of dT to length of day (LOD), and how the LOD is representing changes in mass density in the earth — potentially the ocean’s density balance. For example, upwelling cold water will change the moment of inertia of the earth as the colder water has a different density than warmer water. Or glaciers and snow packs changing. This all sounds nutty until you try to understand what could cause changes in the earth’s rotation speed. The earth’s core as well could cause this, as magma and tectonic plates shift around.

      So what we do is apply the LOD as a correction to the global temperature anomaly and see how that removes more of the fluctuations observed. It is a small term, about +/-0.05C peak-to-peak yet does seem to address the subtle 60-year cycle observed, which means that the PDO and AMO are not needed.


  4. 1. You are trying to subtract the SOI which shows no trend to remove the noise. Since there is no trend this will just smooth the data not change the trend. By subtracting the SOI temperature anomaly from the GISS anomaly (which includes SST) you are essentially subtracting the temperature of some of the ocean from the temperature of most of the ocean.

    2. You then plot SOI next to GISS where the SOI anomaly jumps up to to a .8 trend instead of no trend with no explanation why it was plotted this way. It appears you simply bent the curve with no justification.

    3. You then subtract the SOI to remove noise. But this is surely done incorrectly since the trend jumps .2 degrees over the period. If you subtract a trendless line (which you acknowledge) from another line you simply get a different curve with the same trend. Just look at the period after 2000 that has no volcanoes. The SOI increases and then decreases with no trend. The GISS temperature also shows no trend but your adjustments create a trend out of thin air.

    4. Volcanoes are probably the only adjustment that is justified here though you haven’t defined what size eruptions deserve adjustment. In reality this will give you some underlying trend value but if you want to then figure out climate sensitivity you would then have to put them back in unless you think the earth will never have have another volcanic eruption.

    5. It’s not clear what your doing with TSI since the graphs have no timescale but since TSI is only shown to vary enough to change temperatures by about .15 (by the formula you show) even if you had started at the lowest point and ended on the highest this would make little difference.

    6. You don’t show the AMO trend. AMO has been positive since 2000 so this should create an even lower trend than the flat temperature over the same period but somehow in your graph it doesn’t change the trend over the same period. However, since AMO like SOI is trendless it should once again make no difference in the overall trend. And what about the PDO?

    SOI, AMO, and even PDO do not change the energy budget of the globe, they simply move energy already here. That is why they are trendless over long periods where there is known warming. TSI does change the energy budget but it is recognized as too small of an effect and it is trendless. The sun may actually effect the energy budget (if Svensmark is correct) but not through TSI. Volcanoes change the energy budget. Once done correctly what you will discover is the temperature of the earth has risen bursts and fits slightly since the LIA. The problem is CO2 didn’t start being emitted at unprecedented rates until the 40’s so you then have to figure out what caused the warming prior to that time. And TSI doesn’t work. What you’ll inevitably end up with is that some of the warming (but nobody can say for sure how much and what mechanism) was natural and some was probably man made (but without knowing how much was natural you can’t figure out how much was man-made).


  5. Replying to shenan

    To #1

    Huh? The surface temperature holds no heat compared to the rest of the ocean. It is very sensitive to surface area heating and cooling. This generates a perturbation that propagates across the earth. This is well understood by everyone that has experienced El Ninos and La Ninas, and climatologists that study this.

    To #2
    Huh? What “jumps up” to 0.8? The underlying trend that is bent upward is a proxy for the piece that is missing, which most likely is caused by the GHG forcing, of which happens to bend upward over the 20th century.

    As an alternative, I work the problem out by using CO2 instead of a power-law acceleration term and find that this is also not that hard to analyze.
    1. Grab the GISS data stretching back to 1880, call that dT
    2. Get the CO2 estimated data from KNMI explorer, same years
    3. Get the SOI data from NCAR
    4. Get the LOD from the site that Curry referenced in the last paper
    5. Get estimated relative volcanic forcings from the BEST spreadsheet
    6. Get the TSI data from WFT

    Do a multiple regression of factors #2 though #6 against #1 after applying first-order response times on the fast varying components.
    dT = c1 * ln(CO2) + c2 * SOI + c3 * LOD + c4 * volcanic + c5 * TSI

    After you get the values of the scalar coefficients, reconstruct the best fit to dT.

    The underlying trend is due to CO2, the many sub-decadal variations are due to the SOI, the sporadic deeper valleys due to volcanic aerosols, a slight +/- 0.05 multi-decadal modulaton obseved via the LOD proxy, and a slight periodic variation due to SOI.

    Bottom line, the CO2 TCR is a little over 2C per doubling of CO2.

    The fit is even simpler in that case, because the CO2 does the acceleration for us.

    To #3
    Wrong. The 0.2 is just an offset that is needed to provide a base for the power-law fitting function which starts at zero. I am not the only one that has done this rather simple compensation. Tamino has done this correction, as have a few “ctizen scientists” on the SkS blog here:

    And of course Kosaka & Xie have done this correction in a highly praised paper.
    Y. Kosaka and S.-P. Xie, “Recent global-warming hiatus tied to equatorial Pacific surface cooling,” Nature, vol. 501, no. 7467, pp. 403–407, 2013.

    What you are probably not seeing is a reversion-to-the-mean of the SOI warming excursion. Once the reversion stops the compensating effects will halt.

    The addition of AMO is definitely a weak link and that is why a better alternative is to replace it with the LOD as a correction proxy term .

    The rest of the comment starting with “Once done correctly what you will discover is the temperature of the earth has risen bursts and fits slightly since the LIA. The problem is CO2 didn’t start being emitted at unprecedented rates until the 40’s so you then have to figure out what caused the warming prior to that time. And TSI doesn’t work. What you’ll inevitably end up with is that some of the warming (but nobody can say for sure how much and what mechanism) was natural and some was probably man made (but without knowing how much was natural you can’t figure out how much was man-made).”
    CO2 has been rising since 1850.


  6. “These data sets are all easily available from sources such as NCAR and NOAA.” Can you tell me the dataset you used for the SOI. I would like to try and reproduce you graphs. This, they say, is part of science!



      • The reference [2] is available on the web. That says that temperature changes lag SOI changes by 7 months. I assume that you used a 7 month lag? Whatg I do not understand is how you have saceled the SOI: “scaled to reflect its approximate contribution to the global temperature anomaly.” Can you describe what you have done to scale the index?


      • It is a 6-month lag as the the default to the CSALT model, close enough to 7 months.

        The scaling is done by the multiple linear regression solution. That figures out the strength of the SOI to best match across the entire time span while allowing the other components to vary in strength as well. You don’t want to do this by hand.


      • OK, in the end I followed the recipe you gave in reply to shenan with one difference.

        The TSI data I got from “WFT” (WoordForTrees) only went back to 1978. Instead I used the data at which is from an open access paper by C. de Jager and H. Nieuwenhuijzen ( This uses the magnetic state of sun rather than TSI. I got a chart looking like yours (at But I am doubtful that I have done things right. I used R and the lm function. But the resulting model had an R^2 of 0.6135 which does not seem too good though I am very much a learner in statistics! What is the R^2 on your model?

        Interestingly the de Jager paper says that solar magnetic influence can largely explain temperature changes up to 1970 after which time CO2 (presumably) becomes the main factor. In my model the solar co-efficient is nearly an order of magnitude larger than the others saving, of course, CO2. My chart (smoothed) does not have the period around 1910 when the temperature was colder than the model, but still shows the other anomaly around 1940 when temperature was hotter than the model. I see that there is an image upload button which I shall try ….


      • R^2 is not the same as R or correlation coefficient. When R gets squared obviously it drops in value.

        If you include the subdecadal variations of SOI, it will raise the value of R significantly. I find values of R^2 above 0.9.


      • The data is all there if you go to “fluctuation components” and choose Table.

        I think you are doing something egregiously wrong if you are using any solar data from de Jager and Nieuwenhuijzen. J&N talk about there being a “Grand Maximum” in the 20th Century, yet I have to defer to experts such as Leif Svalgaard who say that “there is no Grand Maximum in the 20th century, no matter what they call it.”

        Are you interested in doing this with the best scientific evidence or are you trying to cherry-pick some outcome that you want to see?

        Use the TSI data from here


      • I just wanted to reconstruct your model. But getting your data is like pulling teeth and now you are starting to show signs of paranoia so I’ll thank you for the TSI source and leave it at that.


      • The de Jager and Nieuwenhuijzen data does not include the 11-year cyclic variation. The cyclic information is critical because with that one can match to periodicities in the temperature record, much like one can pick up the temperature difference between night and day and between seasonal insolation characteristics.

        This is not paranoia, just common sense. I am sorry if I am accusing you of cherry-picking as opposed to excusable ignorance. I just found it bizarre that you would go to that source (de Jager) to pick up data that is so easy to google for.


  7. Pingback: CSALT model | context/Earth

  8. Pingback: Detailed Analysis of CSALT Model | context/Earth

  9. Pingback: CSALT and SST corrections | context/Earth

  10. Pingback: CSALT with CW Hybrid | context/Earth

  11. Pingback: Variational Principles in Thermodynamics | context/Earth

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s