Given that my recent posts about climate change have generated some interest, I thought I would write something about the analysis of some of the climate change data and address some of the claims made by some climate change skeptics. One issue I thought I would address is the claim that most of the warming that has happened between 1880 and now, happened prior to 1940. This claim is made on a site called Global Warming Lies, amongst others. The other claim I thought I would address is that there has been no warming since 1998. I should add, that I am no expert at this, but am an active scientists, so (as usual) am always happy to be corrected by those who genuinely know better.

To do the analysis, I downloaded the GISS Surface Temperature Analysis data from NASA. This gives the monthly temperature anomaly since 1880. The temperature anomaly is simply the difference between the global surface temperature and some mean value. For this data set, the anomalies are computed relative to the base period 1951-1980. What I have done is essentially repeat what you can do using the Skeptical Science temperature trend calculator. I could have done all of this using their calculator, but at least this way I can claim to have done some of the analysis myself.

What I did was to write a Fortran code that read in the data and firstly produced a running average. It then used linear regression to produce the trend and the error in the trend. I should acknowledge here that my errors are uncorrelated errors, when in fact the data is correlated. My errors are a factor of 3 or so lower than they should be, but that doesn’t really change what I want to illustrate. Below is the first figure that I wanted to show. The small dots are the monthly temperature anomalies. The thick line is the 60 month running mean of this data. The solid straight line is the trend which I get to be 0.064^{o}C per decade. This is exactly the same as Skeptical Science’s trend for this data set and for this time period. The error in the trend I get is ± 0.0023^{o}C per decade which is about 3 times smaller than that given by Skeptical Science’s trend calculator, but this is because I’m assuming uncorrelated errors. This difference seems about constant so will assume that I can just multiply my errors by 3. I know I should do better than this, but I have to go back to work tomorrow so can’t really spend much more time on this. If I can find a proper way to do the errors, I’ll fix this later.

What I’ve also drawn on the above figure are two dashed line. These are meant to illustrate the error in the trend. I’ve overestimated it by about a factor of 2 (as it’s hard to see otherwise). Essentially the trend is simply the gradient of the solid line. The gradient is simply

where, for the solid line, *y _{2}* and

*y*are the y-values of the line at

_{1}*x*=2012 and

_{2}*x*=1880. If we consider the steeper dashed line, then the gradient will be

_{1}

where I’ve assumed that the error in *y* is approximately constant. One could do the same for the shallower dashed line, but all I’m trying to illustrate is that the error in the trend is roughly related to the error in *y*. I know that it’s a bit more complicated than this when doing linear regression, but this is just meant to be illustrative. The error in *y* depends on the scatter in the data and (as should be clear) is largely independent of the time interval considered.

If I consider the equation above, the second term will therefore depend on the time interval (*x _{2}* –

*x*) and will increase as the time interval decreases. The first term is essentially the trend, which doesn’t really depend on the time interval. If I consider the figure above, the trend is 0.064

_{1}^{o}C per decade. The standard deviation of the scatter in the data (i.e., the range that would contain about 2/3 of the data points) is probably about 0.1

^{o}C. In order for the trend to be statistically significant, we would want the second term in the above equation to be smaller than the first (the trend). If Δ

*y*= 0.1

^{o}C then we would need to consider a time interval of at least 20 years in order to determine a statistically significant trend. Having looked at this a little, I don’t see any reason why one would ever want to consider shorter than necessary time intervals. We have data from 1880. All we need to do is see how the trend changes as we collect more and more data.

Let’s consider the claim that most of the warming since 1880 happened prior to 1940. Below is a figure showing the data and trend from 1880 till 1940. The trend is 0.039 ± 0.019^{o}C per decade. If we consider the first figure in this post, it shows the anomaly from 1880 till 2012 which has a trend of 0.064 ± 0.007^{o}C per decade. If there was more warming prior to 1940 than after 1940, surely the warming trend per decade should get smaller as we consider data beyond 1940. Furthermore, this result suggests that (considering the mean trend) the surface warmed by 0.85^{o}C from 1880 till 2012, while from 1880 till 1940 it warmed by 0.234. Significantly more warming occurred after 1940 then before. If we now include the errors, there is a 95% chance that from 1880 to 1940 the warming was less than 0.35^{o}C and a 95% chance that the warming from 1880 till 2012 exceeded 0.75^{o}C. Even in this extreme case (assume highest likely warming from 1880 till 1940 but lowest likely from 1880 till 2012) more warming happened after 1940 than before.

What about the claim that there has been no warming since 1998. What some have done is consider the data from 1998 till 2012. As I discussed earlier, this has the problem that the trend is likely to be statistically insignificant given that the error in the trend is almost certainly going to be similar to the trend itself. How can anyone claim that there is no warming if the interval they’ve chosen is almost certainly going to result in a statistically insignificant trend? Very convenient! Let’s do the same as we’ve done before. Compare the trend from 1880 till 1998 with that from 1880 till 2012. If there has been no warming since 1998 the later trend should be smaller than the earlier. Below is the figure showing the data from 1880 till 1998 with the trend line which is 0.051 ± 0.007^{o}C per decade. The trend for the data from 1880 till 2012 is (as shown in the first figure) 0.064 ± 0.007^{o}C per decade. How can the trend have increased if there’s been no warming since 1998. If we include the errors, there is a 95% chance that the warming from 1880 till 1998 was smaller than 0.68^{o}C, while from 1880 till 2012 there is a 95% chance that it exceeded 0.75^{o}C. Basically this claim is not correct. The data very clearly suggests that there has indeed been warming from 1998 till 2012, we just can’t determine this using this time period alone. We have to look at how the trend has changed from 1998 till 2012, not just consider the trend from 1998 till 2012.

So, there we go. I’ve tried to actually do some of the data analysis myself, rather than rely on others. I’ve addressed the claims that there has been no warming since 1998 and that most of the warming since 1880 occurred prior to 1940. As far as I can tell, neither of these claims stand up to scrutiny. I’m more than happy to take sensible and reasonable comments or corrections. I would just ask that you address what I’ve done here, rather than just making *ad hominen* attacks simply because I’m writing about climate change.

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