The errors of averaging Temperature

Physics is founded on differential equations. This also applies to climate physics in particular. Solving such equations requires spatial, temporal, or other integrations, which are a linear functionals. Therefore, in order reach a meaningful conclusions, sums or averages of contributions have to be considered.

In climate science, global temperature trends are of special interest. But there is a problem about temperature. Contrary to energy, temperature is not subject to a preservation law, and there is a lively debate whether the notion “global” temperature is meaningful at all. First of all temperature depends on the height above sea level due to lapse rate. Therefore by definition it is not meaningful to compare the absolute temperature of the mount Everest with that of the dead sea. When moving weather stations to a different elevation, the lapse rate corrected potential temperature is used to transform the temperature. The institutions calculating global temperatures have introduced anomalies to deal with these two problems, under the condition that lapse rate is assumed to be approximately constant over time: Anomalies are the average of the measured temperature difference of a location to a previous reference temperature at the same location. If there is a worldwide trend of temperature increase or decrease, it is reflected in this average.

Error of Stefan-Boltzmann equation by averaging Temperatur

The Stefan-Boltzmann equation is used to estimate a typical average equilibirum temperature of the earth by assuming an energy flow equlibrium betweein incoming shortwave radiation and outgoing longwave infrared radiation. There is vehement criticism about the legitimacy of averaging temperature in this way, due to the fact that the SB-equation relates the energy flow of the outgoing Infrared radiation to the 4th power of the local temperature. While infrared energy flow from different location adds up to the total energy flow and an average energy flow can be easily calculated, by definition temperature of different locations can’t be simply averaged, because the average of the 4th power is not same as the 4th power of the average.

Bevore we discard averaging temperature – as some people do – let’s ask the question: What happens when we do in fact use the average temperature? What is the actual error of the “wrong calculation”? If the error can be estimated and if it is smaller than e.g. the measurement uncertainties, the result may be acceptable, even when we know that from a theoretical point of view it is wrong.

There is a similar issue with Newtonian physics. We know today, that Newton’s mechanics is not mathematically correct, and has been replaced by special and general relativity. Nevertheless for most purposes the error of Newton’s equations is so small that it is irrelevant. In fact relativistic physics becomes Newtonian physics, when the relativistic terms of the Lorentz and Schwartzschild equations are expanded as Taylor series, restricted to the linear term.

So the key question is how large and possibly predictable the error is when linearizing the Stefan Boltzman equation.
Let’s assume an average temperature T. We calculate only 2 terms of the total averaging process. Let S be the energy flow of the Infrared radiation, not normalized with the Boltzmann constant:

Expanding the 4th powers leads to cancellations of the odd powers of T and adding up of the even powers:
‘
Using the average Temperatur for calculating the Infrared energy flow leads to . Subtracting this from the exact result and normalizing leads to the relative error:

Checking public sources like Berkeley Earth, the Standard Deviation of the daily and seasonal temperature measurements is less than 1 K. In order to be on the safe side, avoiding discussions about too small assumptions, we assume a 10 times higher value, . For the average temperature we assume . Therefore the relative error for the energy flow estimation from average temperature compared to averaging the energy flow from true temperatures, is:

This means, that even under extreme, unrealistically high assumptions about the spatial and temporal temperature variability, the relative error of the resulting energy flow is less than 1%. Considering all contributing error sources of climate data, this error is neglegible. Therefore it is legitimate to use average temperature, in particular in the form of anomalies.