San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

Analysis of the Spectrum of Temperature Anomalies
for a Radiating Surface and its Application to the
Spectral Analysis of Average Global Temperatures

The change in the temperature of a body is determined the net heat energy input to it; i.e.,

C·(dT/dt) = U(t)

where T is absolute temperature, t is time, C is the heat capacity of the body and U(t) is the net heat input.

Important insights into the statistical properties of temperature as function of time can be gained by considering U(t) to be random fluctuations about some mean value, perhaps zero. However the net heat input includes the effect of the radiation of energy away from the body that depends upon the temperature, specifically it depends upon the fourth power of temperature. Thus the more accurate equation for temperature change is

C·(dT/dt) = U(t) − σT4

Over time the temperature will move to a level T* where the outgoing radiative energy matches the net incoming heat energy. Thus

0 = U* − σT*4

where U* is the average value of U(t).

When this equation is subtracted from the previous equation the result is

C·(dΔT/dt) = u(t) − σ[(T*+ΔT)4−T*4]

where u(t) is equal to U(t)−U* and ΔT is T(t)−T* and is usually called the temperature anomaly.

For small values of ΔT the difference in the fourth powers can be closely approximated by 4T*³ΔT. Thus the equation for the dynamics of temperature is

C·(dΔT/dt) = u − σ(3T*³)ΔT
or, equivalently
(dΔT/dt) + αΔT = v(t)

where α4σT*³/C and v(t)=u(t)/C. If the above equation is multiplied by exp(αt) the result is

(dΔT/dt)exp(αt) + αexp(αt)ΔT = v(t)exp(αt)
which is the same as
d[ΔT·exp(αt)]/dt = v(t)exp(αt)

This equation can be integrated from 0 to t to give

ΔT(t)·exp(αt) − ΔT(0) = ∫0tv(s)exp(αs)ds
and, equivalently
ΔT(t) − ΔT(0)·exp(−αt) = exp(−αt)∫0tv(s)exp(αs)ds
ΔT(t) − ΔT(0)·exp(−αt) = ∫0tv(s)exp[−α(t-s)]ds

In contrast to the case in which the thermal radiation is ignored and the temperature anomaly is the simple integral of white noise, the case which includes thermal radiation has the temperature anomaly being equal to an exponentially weighted integral of white noise. This has the effect of shifting the spectrum by an amount equal to α.

This is seen by taking the Fourier transform of the equation

(dΔT/dt) + αΔT = v(t)
which is
iωFΔT(ω) + αFΔT(ω) = Fv
and thus
FΔT(ω) = Fv(ω)/(iω+α)

Suppose v(t) is white noise so Fv(ω)=c, a constant, up to some maximum ω. Then

FΔT(ω) = c/(iω+α)

The Fourier transform of a function is a complex function of the frequency. The spectrum is the absolute value of the Fourier transform; i.e., the positive square root of the value times its complex conjugate.

The Fourier transform is symmetric in the sense that the Fourier transform of −ω is equal to the complex conjugate of the Fourier transform of ω; i.e.,

Fy(−ω) = F*y(ω)
and this means the spectrum for −ω
is the same as for +ω
|Fy(−ω)| = |F*y(ω)| = |F*y(ω)|

Therefore the spectrum of ΔT is

|FΔT(ω)| = c/√(ω²+α²) = c/(ω²+α²)½

Moving Averages

The taking of a moving average involves a process of the form

ΔT(t) = ∫−∞ h(s)ΔT(t−s)ds

This is an instance of what is called the convolution of two functions; in this case the temperature anomaly ΔT(t) and the weighting function h(s) for the moving average. There is a theorem concerning Fourier transforms that says that the transform of a convolution of two functions is equal to the product of the transforms of the functions. In this case

FΔT(ω) = FΔT(ω)·Fh(ω)

There is another theorem concerning Fourier transform that says the transform of the derivative of a function is equal to iω times the transform of the function. If α=0 then the moving average of the cumulative sum of white noise has a Fourier transform of

Fy(ω) = (c/iω)Fh(ω)


Fdy/dt(ω) = (iω)(c/iω)Fh(ω)
= cFh(ω)

Thus the spectrum of the derivative of the moving average of the cumulative sum of white noise is entirely the spectrum of the weighting function for the moving average.

When α≠0 something similar but more complicated occurs.

Fdy/dt(ω) = (iω)(c/(α²+ω²)½)Fh(ω)
= c(iω/(α²+ω²)½)Fh(ω)
|Fdy/dt(ω)| = c(ω/(α²+ω²)½)|Fh(ω)|

In this case the spectrum goes to 0 as ω goes to 0 and goes to Fh(ω) as ω becomes large relative to α.

When the moving average is a simple average over the interval t−½H to t+½H its Fourier transform is sinc(½Hω). The sinc(x) function has the following shape:

For the spectrum it is the absolute value of the sinc function which is relevant;

For the spectrum of the rate of change of the temperature anomaly the factor ω/(ω²+α²)½ must be included. For α=2.0 the spectrum is

For α=1.0 it is

The Magnitude of the Parameter α

The value of the Stefan-Boltzmann parameter is 5.67×10-8 (W/m²)/K4. Since T=300°K, 4σT³ = 4(27×106)(5.67×10-8=6.1236. The dimensions of this quantity is (W/m²)/K. The dimensions of heat capacity C is (W-year/m²)/K. Therefore the dimension of α is 1/Time.

An estimate of the heat capacity per unit area over the ocean is 14±6 (W year/m²). The heat capacity of the land is much smaller than that of the ocean, say 0. Since the oceans represent about 70 percent of the Earth's surface the overall average heat capacity would be 0.7(14)+0.3(0)= 9.8 (W year/m²), or rounding off, 10 (W year/m²). This means the value of α for the Earth's surface is about 0.6 per year.

The Spectral Analysis of the Average Global Temperature

The record of global average temperatures shown below displays some interesting characteristic. One of these is the hint of a cycle.

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 trend. From 1940 the average global temperature declined until about 1975. After 1975 the trend is upward until about 1998 and, although it is not shown in the graph, after that the trend has been downward. There is about a thirty year period for these periods of trends. This period from a minimum to a maximum or from a maximum to a minimum would be one half of a full cycle period. Thus the data indicate a full cycle period of about sixty years.

Spectral analysis is the method for identifying a cyclical period. The spectral analysis of the data from 1855 to 2003 is as shown.

The ratio of the peak heights for the peak at zero frequency and the next peak is about 1/5, more precisely 0.1936. The ratio for the sinc function is 2/(3π)=0.2122; not too bad of a correspondence.

The spectrum displays the sequence of peaks found previously from pure theory. The spectrum over a wider range of frequencies is shown below

The phenomenon of the peak at zero and the high values near zero is a manifestation of the fact that the average value is not zero. When the mean value is computed and subtracted from each datum the spectrum for the result is shown below.

As stated previously the period from a minimum to a maximum or from a maximum to a minimum is one half of a full cycle period. In the above spectrum the peak for a full cycle period at about 200 years corresponds to trend period of about 100 years. The data were not detrended so this peak may just represent the secular trend. The spectrum shows a peak for a full cycle period of 49 years, but this probably represents a cycle that has a period between 49 and about 55 years; roughly 52 years. This would correspond to a trend period of 26 years.

An extensive study of the spectra of time series relating to global temperature was carried out by Leonid B. Klyashtorin for the Russian Federation's Institute of Fisheries and Oceanography (FAO) in Moscow. The results were published in the FAO Fisheries Technical Paper No. 410, entitled "Climate change and long-term fluctuations of commercial catches: the possibility of forecasting." It was presented in Rome in 2001. Klyashtorin's study found a spectral peak for the global temperature anomalies for 1861 to 1998 for 55 years. This corresponds to a trend period of 27.5 years. For the spectral analysis of ice core temperatures over the 1421 year period from 552 to 1973 the dominant peak was for 54 years, corresponding to a trend period of 27 years. His analysis of the Atmospheric Circulation Index (ACI) which covered the Atlantic Eurasian region the dominant peak was for 50 years. He also analyzed indices for fluctuations in the anchovy mass and for tree ring temperature. For these the dominant peaks were 110 and 108, respectively, with secondary peaks at 51 and 56 years, respectively. Klyashtorin concluded that there is a climate cycle with average period of about 56 years. This corrsponds to a trend period of 28 years. This result came out of the analysis of long term series (c. 1500 years) as well as shorter term series of about 150 years.


The apparent linear trends in average global temperature lasting 25 to 30 years correspond to natural cycles in the Earth's climate system. Any forecasting of future average global temperatures must ignore these relatively short term trends. Roughly this would mean looking at the temperature at the midpoints of the trend periods. For the trend period in the 19th century the midpoint was about 1880 when the temperature anomaly was about −0.35 C°. The midpoint for the last full trend period was about 1990 when the anomaly was about +0.1 C°. This is an increase of about 0.45 C° in 110 years or roughly 0.4 C° per century. The rise in the temperature since 1990 was largely part of a natural cycle that is being reversed since 1998.

(To be continued.)

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