Temperature is something we deal with everyday. Is it cold outside? I’ll wear a jacket. Is it hot? Maybe I’ll wear a t-shirt. We all have an intrinsic idea of what temperature is. Temperature is really a measure of how fast something is moving. Consider a typical nitrogen molecule (N2) in the atmosphere. Near the equator, the molecule has more energy than if it were at the poles. This is a consequence of the Earth being nearly a sphere; more of the sun’s energy is intercepted near the equator than the poles.

The sun’s energy generally won’t go directly into the nitrogen molecule. Instead, it will be absorbed by the surface. The surface will warm, and the nitrogen molecules will bump into the surface. This transfers heat from the surface to the atmosphere. Because the nitrogen molecule has more energy, it will move faster. The translational kinetic energy of a particle is 1/2 multiplied by its mass and multiplied by the square of its velocity. Thus the temperature of the air is related to the amount of energy it holds.

What is Average?

The IPCC, and many others, show plots of the yearly average surface temperature anomaly. But what do they mean by “average”? Surely there is more than one way to compute an average. There are many ways that the IPCC could compute an average yearly temperature.

Let’s assume that there are N number of weather stations unevenly spaced throughout the world. The easiest way to take an average global temperature is to just take the arithmetic mean of the high temperature on one day. This has some obvious drawbacks. First, by averaging only on one day we’re going to have really noisy results because of the random weather. Second, only the high temperature was considered, which may influence results. And probably most importantly, because these stations are irregularly spaced, those areas which are not near a weather station will be under-represented in the yearly mean.

Now assume we have weather stations throughout the world at 1 degree spacing in both latitude and longitude over both land and ocean. Here we have the problem of over-sampling near the poles, but we can account for that if we assume the earth is roughly a sphere. Let’s futher assume that the temperature reported is the temporal average during the entire day at each location. Now we can compute a yearly average temperature at each location using the standard arithmetic mean. Then we can average each of the locations using an weighted average for how much area is represented by each location. The stations near the poles would get less weight than those at the equator. This would be a good value for the average temperature of the earth during a year.

Average or Average?

I obtained the surface air temperature from the NCEP/NCAR reanalysis for the years 1948 through 2007. This is a product that offers air temperatures on a 1 degree grid like the hypothetical above. Therefore, we can use the same method of temporally and spacially averaging get an average yearly temperature for the earth. We can then compare the temperature during each year for the last 60 years. But first, let’s look at what the average temperature of the earth looks like for one year.

This shows just the temporal averaging described above. This projection is used to not emphasize the polar regions. We can see that the equator is warmer than the poles, exactly like we would expect. In fact, this picture is quite boring. However, Dr. Pielke thinks that there is A Serious Problem With The Use Of A Global Average Surface Temperature Anomaly To Diagnose Global Warming.

Summary:  There are three [sic] main conclusions from Part I of our JGR paper. They are:

• To diagnose the magnitude of global warming using the global average surface temperature anomaly, T’  must be tightly coupled to that thermodynamic state of the climate system; however, this is not an accurate charaterization of the Earth’s climate;
• In constructing a global average of T’, its spatial distribution matters since T’ in regions with a baseline colder temperature have a significantly smaller effect on the return of heat energy to space (through infrared emission) than regions with a warmer baseline temperature
• The height that T’  is measured matters, since T’ at the actual surface is not the same as T’ even slightly higher; i.e., T’ is not, in general, height invariant near the surface.
• The use of dH/dt to diagnose global warming is a much more scientifically robust approach than using a global average surface temperature anomaly.

I will not deal with all four of the points made above, but only the second one. In the body of his post, Pielke argues that the temperature used in climate diagnostics must be the radiative temperature of the earth’s surface. He then argues that since the temperature is higher at the equator, a 1 degree rise in temperature at the equator will count more than a 1 degree rise at the poles. This is because the blackbody radiation depends upon temperature to the 4th power.

Okay, we’ll assume that is a good argument to make. What happens if we average T^4 instead of T in our calculations of a global average temperature?

As can be clearly seen, the differences between the two “average” temperatures is very small. By inspection, the red line does not appear to be rising at as fast a rate as the black line since the 1980s. However, it would be difficult to say that the average global temperatures are vastly different between the two averaging cases.

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