San José State University
Thayer Watkins
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The Relationship Between the Second Derivative of
Global Temperature and Global Temperature and Time

There is a cycle in average global temperature along with a long term trend that goes back the 160 years for which there is data, as is shown below for the data from the National Oceanic and Atmospheric Administration (NOAA) which goes back to 1880.

The data from the Goddard Institute for Space Science also goes back to 1880.

The data from the Hadley Climate Research Unit (CRU) of the University of East Anglia goes back to 1850.

The cycle is an upswing of roughly 30 years and a downswing of roughly the same length. The pattern can be represented as a bent line, as in the diagram above, but a better representation would be

T(t) = A*sin(νt+φ) + Bt + u(t)

where T is the deviation of the average global temperature from its long term average, the temperature anomaly, and t is time. The parameter ν is the frequency of the cycle and φ is its phase. The variable u(t) is a random fluctuation.

The cycle period P is such that νP=2π and thus P=2π/ν.

One important property of the sinusoidal function y(t)=sin(ωt) is that d²y/dt² = −ω²y. Thus if there is a statistical relationship between the second derivative of temperature and temperature that will provide an estimate of the cycle frequency and the cycle period.

The first derivative of T(t) with respect to time is

dT/dt = Aνcos(νt+φ) + B + u'(t)

The second derivative is

d²T/dt² = −Aν²sin(νt+φ) + u"(t)

The quantity A*sin(νt+φ) is equal to [T(t)−Bt−u(t)]. Therefore

d²T/dt² = −ν²[T(t)−Bt−u(t)] + u"(t)
or, equivalently
d²T/dt² = −ν²T(t) + ν²Bt + U(t)

where U(t) = u"(t)−u(t). The expected value of U(t) is zero. Therefore there is no constant term in this equation.

The second derivative of T can be approximated as


where h is the time interval. For annual data h is equal to 1. Let this discrete approximation of the second derivative of T with respect to time be denoted as St. Then according to the above

St = −ν²Tt + ν²Bt + Ut

As a preliminary consider St plotted versus Tt, as shown below.

There is a great deal of noise (random fluctuation) in the relationship, but there is a perceptive negative slope.

The regression of St on Tt with a constant term gives

St = −0.15114Tt + 0.00079244

The coefficient of determination is an insignificant 0.06 but the t-ratio for regression coefficient is −2.8, which is significantly different from 0 at the 95 percent level confidence. The t-ratio for the constant term is not significantly different for zero at the 95 percent level of confidence.

If the constant term is dropped the regression results are:

St = −0.15106Tt

Again the coefficient of determination is an insignificant 0.06 and the t-ratio for the regression coefficient is a significant −2.8. So the results look promising.

If the coefficient of Tt is taken to be equal to −ν² then ν is equal to 0.38868 radians per year and the time period of the cycle is 16.17 years. This greatly at variance with the observed cycle period of about 60 years.

Regressions Using the Full Version of the Analysis

When St is regressed on Tt and t the results are:

St = −0.54365Tt + 0.0031243t −6.07238

The coefficient of determination is better but still only 0.22367. The t-ratio for the coefficient of Tt is −6 and that of the time variable is 5.1, both significantly different from zero at the 95 percent level of confidence.

The value of B, the long term trend in temperature is obtained by dividing the coefficient of t by the negative of the coefficient of Tt; i.e.,

B = 0.0031243/0.54365 = 0.005747 °C per year = 0.5747 °C per century

This is quite comparable to the estimate of 0.5°C per century found elsewhere using a different method.

(To be continued.)

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