Tuesday, July 15, 2008

Here's How You Can Estimate CO2 Climate Sensitivity From Historic Data

Most likely value = 3.46C

[Note: Revised 08/02/2008]

When I first became interested in the science of Global Warming (which was not too long ago) I had some substantial misconceptions. For example, I thought the current temperature anomaly (about 0.6C globally) was due to the current levels of greenhouse gases in the atmosphere, primarily CO2 (about 380 ppmv). Reality is more complicated. The issue is not that there's some lag between greenhouse gas concentrations and temperature either – it's a bit more complicated that this.

I've been learning about a concept called CO2 climate sensitivity, which is defined as the equilibrium temperature increase expected if the atmospheric concentration of CO2 were to double. The word equilibrium needs to be emphasized. At current CO2 concentrations, I would estimate the equilibrium temperature anomaly should be 0.89C, but the actual temperature anomaly is only about 0.6C. There's a significant imbalance, and the imbalance is corrected by temperature change. Simplifying, the mechanism that causes temperature change is called CO2 forcing.

There is much debate and uncertainty about the most likely climate sensitivity value. For a good overview, see James' Empty Blog.

What I want to do in this post is go over a relatively simple analysis where we estimate climate sensitivity by using publicly available historic data. We will also come up with formulas that tell us the most likely equilibrium temperature for a given CO2 concentration, and the most likely temperature change rate for a given actual temperature and CO2 concentration. The plausibility of these results will be illustrated with a graph.

First, let's go over some of the underlying theory. Given the way climate sensitivity is defined, it's clear that the expected equilibrium temperature change is the same for any doubling of CO2 concentrations, be it from 100 to 200 ppmv, or 1000 to 2000 ppmv. This tells me there's a logarithmic relationship between temperature and CO2 concentrations (assuming all else is equal) as follows:

T' = a log C + b

T' is the equilibrium temperature and C is the atmospheric concentration of CO2; a and b are constants. Climate sensitivity is thus

S = (a log 2C + b) - (a log C + b) = a log 2

When the observed temperature (T) differs from the equilibrium temperature (T'), there's imbalance. We will define imbalance (I) as follows.

I = T' - T

Further, I put forth that temperature change rate is given by

R = d I

where d is a constant. We're guessing a bit here, but the above is consistent with Newton's Law of Cooling.

Finally, let me define a construct (J) that I will use in the analysis. It is simply the imbalance minus the constant b, as follows.

J = I - b = a log C - T

If we know S, then we know a. When we have S, a and C for any given year, we can calculate J for any given year. Since we should be able to determine the temperature change rate (R) for any given year, we can model J vs. R (a linear relationship). The relationship between J and R should be equivalent to the relationship between I and R, except for a shift given by the constant b.

Here's the plan. We need to test different hypotheses on the value of S. The way we determine a hypothesis is good is by checking if the resulting relationship between I and R is suitable. And we measure this by means of the "goodness of fit" of the linear association between J and R. (This methodology is called "selection of hypotheses by goodness of fit" and it seems adequate in this case, judging by Figure 3, which I will mention shortly).

Before I get into the nuances of the analysis (which are important) I wanted to show the reader how I chose the best value of S. Figure 1 models S vs. the goodness of fit of the linear association between J and R.

This tells us that the value of S that makes most sense is 3.46.

After we have determined the most likely value of S, we can calculate the constant b. The linear association between J and R is as follows.

R = 0.09152J - 2.63281

The slope should be the same in the association between I and R, except here the intercept must be zero.

R = 0.09152I

Therefore, b may be calculated as follows.

0.091521I = 0.091521(I - b) - 2.63281
b = 2.63281 / 0.091521 = -28.768

Figure 2 is the scatter graph that illustrates the association between imbalance (I) and temperature change rate (R) when we assume S=3.46. This confirms the slope of the linear fit and the "goodness of fit" we had previously found.

co2 climate sensitivity

A very important graph is one that shows the R and I time series side by side, under the same assumption (S=3.46). See Figure 3.

co2 climate sensitivity

Figure 3 validates much of the underlying theory. It's one of those graphs that, once again, show anthropogenic global warming to be an unequivocal reality.

Figure 3 can also be used to visually check different values of S. When S is less than 3.46, you will see the imbalance (I) time series rotate in a clockwise direction. When it is greater than 3.46, it will rotate in a counter-clockwise direction. This provides subjective confidence about the adequacy of the hypothesis selection methodology.

Note that the imbalance (I) time series in Figure 3 is shifted three years to the right. An initial inspection of the graph clearly showed there was a lag of 3 years between imbalance and temperature change rate. I would've expected the effect to be immediate, but that's why it's important to put your data in graphs. I couldn't begin to theorize why it takes time for imbalance to take effect, but this finding needs to be taken into account in the analysis; otherwise the results won't make sense.

Another important aspect of the analysis is that time series noise needs to be reduced, otherwise you probably won't notice details like the 3 year lag. I calculated central moving averages of period 7 from the CRUTEM3v global data set. For example, the "smooth" temperature for 1953 is calculated as the average between 1950 and 1956. Additionally, the temperature change rate (R) is calculated based on the "smooth" temperatures, looking 4 years ahead and 4 years in the past. If you also consider the 3 year imbalance lag, this leaves us with a workable time range spanning 1859 to 2000.

How do I get CO2 concentration data spanning that time frame? I discussed how I estimate that here. Basically, I try to find the best possible constant half-life of extra CO2 by matching emission data with the Hawaii data. The best half-life is 70 years or so.

I should note that this technique produces pre-industrial CO2 concentrations that are higher than I believe is generally accepted. My estimate gives about 294 ppmv for the 1700s. From ice cores, I understand the concentration has been determined to be 284 ppmv circa 1830. However, I can report that I tried a different estimation method that produces a value closer to 284 ppmv in the early 1800s, and this data produces much poorer fits in the analysis. For this reason, I went with my original estimation based on a constant half-life.

Let's look at the results of the analysis.

S = 3.46

T' = 11.494 log C - 28.768

R = 0.0915I [ 95% CI 0.074I to 0.109I ]

Temperatures are given as anomalies in degrees Celsius, as defined in the CRUTEM3v data set. The rate of change (R) is given in degrees per year.

What's the confidence interval on S? We'll leave that as an unsolved exercise. It's not only that there's uncertainty on the various data sets used, but it's unclear how we would calculate the uncertainty on the best "goodness of fit." It's not a matter of calculating confidence intervals on R2 values, which is easy. We basically have to determine the likelihood that the best "goodness of fit" is other than the one we found. This seems non-trivial, but maybe a reader can suggest a method. From what I've seen in a visual inspection of Figure 3, I would say S is unlikely to fall outside the range 2.8 to 4.0. Of course, things might happen in the future which invalidate these results, as they are applicable to historic data.

Next up: We'll see how well these results hind-cast.


Pekko said...

Hi Joseph, very interesting post.

I fit T = a * log(C) + b to temperature (T) and CO2 (C) data (GISTEMP & Mauna Loa + Law Dome) and got sensitivity S = a * log(2) = 2.03 K.

I want to try your method with the same data. However, I got stuck and need some clarification.

I understand that you select some sensitivity S from which you get the constant a. Then you use this constant to calculate J. Finally you regress R against J and record the correlation coefficient for the chosen S.

How do you calculate R? This is where I got stuck.

Joseph said...

Hi Pekko. R is the rate of temperature change in any given year, so this is a known value (albeit noisy). You can see this value in figure 3. It's smoothed a bit as I recall. First, I got central moving averages for temperatures. On top of that, to estimate the rate of temperature change I looked forward and back a couple years. (To do the math it's probably a no-no to smooth, but to look at it in a graph, it helps to smooth it).

So you start out knowing R, C and T for all years in the series. Then you can look for the best value for S that gives the best fit of J vs. R. (J is calculated from the hypothesized value of S of course).

Do take a look at my updated analysis where I use better CO2 data for the last 150 years.

Bob Webster said...


You conclude: "Figure 3 validates much of the underlying theory. It's one of those graphs that, once again, show anthropogenic global warming to be an unequivocal reality."

On what basis do you conclude AGW has anything to do with the charts you've shown? You've outlined a model based on certain assumptions, then shown your model appears to "work" by comparing it with some historical data. But just what part of your model represents an AGW component vs. some natural component?

Seems to me that, in the end, the alarmist camp seems to simply equate any episode of warming to human origin without ever understanding the natural component of climate change. That is a good definition for "hubris" in my book.

Take a look at the chart on page 7 of the March 30, 2009 paper "Two Natural Components of the Recent Climate Change" (2 Mb pdf) by Dr. Syun-Ichi Akasofu at http://www.webcommentary.com/docs/2_natural_components_recent_climate_change.pdf and identify the AGW component of the climate trend.

Since it is not disputed that human emissions of greenhouse gases were insufficient to raise a question of climate change prior to 1950, the long linear trend of upward temperature cannot be attributed to AGW. The cyclic warm/cold/warm/cold multi-decade changes reflects the stability of the system in operation and cannot be attributed to AGW. So where is the AGW signal in the temperature record? (Note the absurd IPCC projections vs. reality)

Interesting work you've done, but you do seem to take a leap in identifying AGW as having anything to do with the material you present.

Bob Webster said...


The link to the article seems to have been truncated, it can be found under the heading "Excellent Climate Change Papers/Discussions" on the page:



Bob Webster

Joseph said...

It's like this. I assume that a theoretical model works. Then I check how the predictions of that model match actual data, and it matches quite well.

If you don't consider green house gases, the temperature trend is not easy to explain.

How the model hindcasts can be seen in this post.

I'm not familiar with a model (I'd have to see formulas) purely based on natural forcings which hindcasts like this.

Bob Webster said...


Assuming theoretical models "work" is risky business, especially when models are built upon conjecture and assumption, as the IPCC tweaked GCMs have been. (See Monckton's piece for the ASU, Climate Sensitivity Revisited at http://www.webcommentary.com/climate/monckton.php)

While your work is interesting and reveals a keen mind, it appears to assume that the observed CO2 rise is the cause of the temperature rise when it is far more reasonable to expect that the CO2 rise is a consequence of temperature rise. The response of CO2 to temperature is well-documented and is to be expected because of the important role that oceans play in CO2 emission/absorption. There is no documented evidence that demonstrates any greenhouse gas, even the potent water vapor, has ever been a driver of climate change (going back more than half a billion years). Indeed, there is absolutely no correlation between atmospheric CO2 and global average temperature. So why make the effort to construct a model that indicates changes in CO2 can be a discernible driver of climate change?

As a mathematician, I am well aware that there are wonderful techniques for fitting a variety of curve types shaped by their parameters. The fact that a nice curve fit to data can be obtained does nothing to validate a cause and effect relationship, particularly when the observed record clearly indicates the only cause and effect between CO2 and temperature is that temperature drives CO2 levels.

How else does one explain these three charts based on observed temperatures? http://www.webcommentary.com/docs/3charts.pdf

Two are from Dr. Akasofu's recent paper, Two Natural Components of the Recent Climate Change, which pdf you can download from http://www.webcommentary.com/docs/2natural.pdf and Ray Evans' paper, Nine Facts About Climate Change, which pdf you can download from http://www.webcommentary.com/docs/9facts.pdf

These papers provide a clear demonstration of non-greenhouse gas natural climate variability. Dr. Patterson's paper, The Geologic Record and Climate Change, at http://www.tcsdaily.com/article.phpx?id=010405M demonstrates the observations and conclusions arrived at independently by climatologist Dr. Marcel Leroux in his excellent text, Global Warming/Myth or Reality? wherein he wrote:

"The marked warming of 1918-1940, of the same order of magnitude as that of recent decades, corresponded to only a slight rise in CO2 levels, of only 7 ppmv (from 301 to 308 ppmv). Between 1940 and 1970, CO2 levels rose by 18 ppmv (from 308 to 326 ppmv) but the temperature did not rise: on the contrary, it was supposed that a new 'Little Ice Age' might be on the way. Only the (presumed) rise in temperature towards the end of the century (RCO), from 1980 onwards, coincides with a rise in CO2 levels (of more than 22 ppmv). The greenhouse effect scenario therefore cannot be invoked to explain recent thermal evolution, since:

* It does not tally with the warming from 1918 to 1940.
* Neither does it account, a fortiori, for the cooling from 1940 to 1970.
* Therefore, there is only one period, at the end of several thousand years analysed, during which some 'relationship' between CO2 and temperature could be envisaged: the RCO, about thirty years long!"


"The possible causes, then, of climate change are:

* well-established orbital parameters on the palaeoclimatic scale, with climatic consequences slowed by the inertial effect of glacial accumulations

* solar activity, thought by some to be responsible for half of the 0.6°C rise in temperature, and by others to be responsible for all of it, which situation certainly calls for further analysis

* volcanism and its associated aerosols (and especially sulphates), whose (short-term) effects are indubitable, and

* far at the rear [my emphasis], the greenhouse effect, and in particular that caused by water vapour, the extent of its influence being unkown." [and indetectible]

Material quoted from pages 120 and 218.

Your interest in this topic is commendable, but there is a lot more to determining cause and effect than mere curve fitting, which, in the final analysis, is what you've done (and an excellent job of it).

I urge you to read Dr. Akasofu's paper (and the others, if your interest is piqued sufficiently) to better understand the natural causation of recent temperature variations.

Indeed, examining the three charts I've extracted from the Akasofu and Evans papers, where is the CO2 component? It isn't even detectible. There have been many past warming episodes that were just as rapid as that of the late 20th century and which resulted in warmer temperatures achieved by recent warming. Greenhouse gases had nothing to do with those warming episodes. Recent warming is characteristically the same and cannot be distinguished from many prior warming episodes. So why pick on CO2 now? Why has the IPCC gotten away with failing to identify natural components of climate change before making assumptions about human causation that, with sufficient model tweaking to match the assumed result, manages to do exactly what it was designed to do ... "prove" a human causation, regardless of what is really going on?

If I've missed something or you find a weakness in any of the material I've cited, please let me know. My only interest is in discovering what is true.


Bob Webster

Joseph said...

Note that this is not a model that directly associates CO2 with temperature. The model associates CO2 concentration with the rate of temperature change (which also depends on the actual temperature). The rate of temperature change will, given enough time, result in actual temperature variations.

If CO2 were to be fluctuating up and down, like in a sine wave, the model predicts that temperature will fluctuate in a sine wave, with a lag of approximately 10 years. This is entirely consistent with what I've previously seen in the detrended CO2 and temperature data.

What I'm getting at is that what I see in the model, and the hindcast, can't be explained by causation in the opposite direction. It doesn't fit as well, and in any case, temperature can cause changes in CO2, but only over very long periods of time.

Denying that CO2 concentration increases are essentially human-caused is a stretch, and I have a post about that here.

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