### 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 CO

_{2} (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

*CO*_{2} climate sensitivity, which is defined as the

*equilibrium* temperature increase expected if the atmospheric concentration of CO

_{2} were to double. The word

*equilibrium* needs to be emphasized. At current CO

_{2} 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 CO

_{2} 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 CO

_{2} concentration, and the most likely temperature change rate for a given actual temperature and CO

_{2} 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 CO

_{2} concentrations, be it from 100 to 200 ppmv, or 1000 to 2000 ppmv. This tells me there's a logarithmic relationship between temperature and CO

_{2} 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 CO

_{2};

`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.

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.

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 CO

_{2} 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 CO

_{2} 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 CO

_{2} 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 R

^{2} 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.