San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The Nature of Trends in Global Average Temperature

The record of global average temperature displays one interesting characteristic, namely that the periods of trends appear to be linear.

Thus in the above display the global average temperature appears to be fluctuating about a rising linear trend until about 1880. From 1880 to 1910 there is a declining trend and then from 1910 to 1940 there is a rising linear trend. From 1940 the average global temperature declined linearly to about 1975. After 1975 the trend is upward but again linear.

The climatologist Patrick J. Michaels, in his book Meltdown and in other publications, makes the argument that the effect of increased concentration of greenhouse gases in the atmosphere is logarithmic and the concentration is going up exponentially so the temperature effect is the logarithm of an exponential function of time and thus the end result is that the temperature change is a linear function of time. What is presented below is an alternate explanation for linear trends in temperature. It is not in contradiction to Michael's explanation.

## The Statistical Characteristics of Changes in Average Global Temperature

The temperature change for any physical body is proportional to the net heat energy input to that body; i.e.,

#### C(dT/dt) = H or, equivalently (dT/dt) = H/C

where T is the temperature of the body, C is its heat capacity and H is the net input of heat energy. The statistical character of the net heat input can be examined by looking at the changes in aveage global temperature, as shown below.

A tabulation of the frequency distribution of these annual changes is

This distribution can be considered a finite sample from a normal distribution. Thus the annual temperature changes can considered a random variable. If the variation in the net heat input is a random variable then the temperature at any point in time is the cumulative sum of a random variable.

Below is a simulation of such a variable.

Each time the REFRESH button is clicked a new sample of random numbers is generated and their cumulative sum. In most cases the plot appears to show trends. However there are no long term trends in the model. These apparent short term trends generally appear linear. What is needed is an analysis of the patterns.

## A Simple, Yet Amazing Theorem Concerning Average Curvature of a Line

Let f(x) be a twice differentiable function. Then the average curvature between critical points of f(x) is equal to zero. (Critical points may be relative maximums, revative minimums or inflections points.)

Proof:
Curvature is the second derivative of the function, f"(x). For the moment let a and b be any two values of x. Then the average curvature between a and b is given by

#### (1/(b-a))∫abf"(x)dx = (1/(b-a))∫ab(df(x)/dx)dx which reduces to (1/(b-a))∫abdf'(x)= (1/(b-a))[f'(b)−f'(a)]

When a and b are for critical points of the function, f'(a)=0 and f'(b)=0. Therefore the average curvature between critical points is 0.

End of proof.

The theorem applies not just to adjacent critical points; it applies to any two critical points of the function be they maximums, minimums or inflection points.

For trends in global warming the points for a trend are a minimu and then a maximum or a maximum and then a minimum. The eye naturally averages the curvature and a thus it is no surprise that the trends then appear to be straight lines since straight lines have zero curvature.

## Extension of the Curvature Theorem to Moving Averages of Cumulative Sums of Random Disturbances

(To be continued.)