CLIMATE HISTORY AND GEOLOGY


6.  SPECTRAL ANALYSIS OF CLIMATE DATA:
A CAUTION

Commonly some form of time-series or spectral analysis is used for extracting periodic signatures from a set of data pertaining to climate variations.  However, spectral analysis is, like statistical analysis, a black art rather than a fully respectable branch of mathematics or science.  What comes out depends very much on how the analysis is applied.

In trying to extract weather and climate periodicities from data extending back many centuries, researchers have made claims for such a wide variety of periods that the results should be viewed with great skepticism.  Few of the analyses agree with each other.  Some find the expected solar and lunar cycles; others do not—even using the same kinds of data.

In extracting periodic signatures from climate data, three important factors must be observed:

Temporal inhomogeneity has always plagued long runs of climate data.  The data run is seldom long enough to extract the really interesting long periods.  Temporal events such as the Little Ice Age and other slow variations may disturb the shorter periods, or even establish a new state with totally different cyclic behavior.  The circulation and weather systems may have changed qualitatively with the end of the Little Ice Age; so the short-term variations—such as the ENSO—may be substantially different from what they were several hundred years ago.

Temporal inhomogeneity may distort historical climate data because the instruments and methods change over time.

Spatial inhomogeneity can be a serious problem whenever two sets of data from widely separated stations are mixed.  Sometimes—as in the analysis of tree ring data—investigators may combine data from a broad area in order to get a satisfactorily "clean" signal.  However, the weather patterns and climate cycles may be substantially different in, say, coastal areas, mountain areas, and continental interiors.  Results which are claimed to apply to a broad area, containing many kinds of topography and environment, are always suspect.

Methods for analysing temporal data have not been standardized, and investigators are often drawn to the newest, most "trendy" methods.  Keep in mind that spectral analysis methods were developed for extracting weak signals from very long runs of data.  They work very well when they are called on to extact the blips of a pulsar or a clandestine radio transmitter from days to months of radio transmissions; they don't work so well when the blip only occurs two or three times in the entire set of data.  Often a set of climate data does not span a long enough time to confidently extract any but the single strongest signal.  The results may suggest other signals, but they may be totally spurious.

As an example, consider several sets of tree ring data, which have been widely published and referenced [from the work of Fritz and Shao].  The figure below shows inferred precipitation (in light blue) and temperature (gray) for the Northern Great Plains.  At the top are bars, denoting the historical droughts since 1800, and inferred droughts for earlier times.  Three of the historical droughts were especially long-lived and severe.  The drought from 1840 to 1864 resulted in displacement and hardship for native peoples.  Several explorers came to the Northern Plains, including Isaac Stevens and John Palliser, and saw evidence of extreme aridity.  The years 1860 and 1861 were among the driest ever recorded there.  The dustbowl years of the 1920's and 1930's lasted almost as long, and included some extremely dry years.  The most recent drought, which began about 1996, is almost as severe; it could last as long as the others.  Over the past several centuries great historical droughts in the interior of North America have occurred at intervals of 75 years.

The data here are obviously very "ragged." The old Blackman and Tuckey (BT) method, based on simple Fourier deconstruction of the signal, is instructive, because it shows the need for very long series of data.  The BT method involves a fourier transform of the autocorrelation function, shown below for the data sets, plus a data set for the Southwest Deserts.

The plotted curves show little order at short lag times.  Only at lag times greater than 200 years do the autocorrelations begin to show regular variations that might yield a genuine spectral feature.  But the value of the analysis is called into question because the total data run is only slightly more than 200 years.  The data here have been folded to overcome the limitations of the data sets.  So the regular oscillations beyond 200 years may simply be an artifact of the method of analysis.  Such data folding may be implicit in spectral analysis methods, and is a potential source of aliasing at long periods.  Researchers should always examine the autocorrelation function to see whether they indeed have a long enough data sample that real periodicities begin to show up.

The cross correlations of the three data sets show that there is a high degree of synchronism at short periods and very long periods.  At lag times between 120 and 250 years the correlations are poor.

Finally, a fourier analysis of the Northern Plains autocorrelation yields a power spectrum, with peaks whose sharpness is dependent on the lag windows used.  The spectra are unnormalized, so the relative heights of peaks are not important.

For comparison, see the following spectrum, derived by one of the more "modern" methods, the Auto Regressive (AR) method (which is similar to Maximum Entropy methods).  The mathematical method here is quite different, so the peaks show a dramatically different shape.  In the AR method, choosing a large number of poles is equivalent to choosing a long lag window.  The secondary peaks appear sharper, but they are not necessarily more real. Presumably one could "refine" the analysis to produce ever sharper peaks, but there is a great risk of fooling ourselves.

Many papers have been published on drought and other climate cycles showing a series of sharp peaks at the frequencies determined by the mathematical analysis.  But the peaks shown above have a finite width.  The mathematics of Fourier analysis says that the power spectral frequency peaks cannot be narrower than

&Delta freq = 1 / (2 &pi T)

where T is the length of the data sample.  (The basis of this "uncertainty principle" is exactly the same as the basis of the famous Heisenberg Uncertainty Principle of quantum mechanics.)  The Blackman Tuckey, and other Fourier analysis methods yield frequency peaks with the right uncertainty; the AR and Maximum Entropy methods can produce artificially narrow peaks.  This may be acceptible in certain kinds of analysis of radio signals, but can give a misleading sense of the uncertainty of the results in applications to climate data.

The lesson:
Beware of published analysis which show spectral peaks represented by narrow lines!




Is There a 37 Year Weather Cycle
Related to Lunar Tides?


The data sets shown above are clearly not long enough to reveal a 75 year period; but the first pronounced peak is at half that period: 37 years (with an uncertainty of about 7 years).  Lesser peaks appear at about 18 years.  Though the autogression analysis suggests very sharp secondary peaks at the shorter periods—approximately 18 and 10 years; they are probably not meaningful. 

A useful test of the results of spectral analysis is to overlay the data with a periodic "comb," represented above by alternating green and brown bands.  (This is equivalent to the old "periodogram" method.)  Here some of the peaks and valleys (for the Northern Plains) agree moderately well with the comb.  With such a short run of data, the results of analysis can be trusted only if they can be demonstrated by such a simple "eyeball" procedure.

A physical cause for 18 and 37 year periods may be related to the lunar tidal cycle of 18.6 years.  There is no evidence here of an 11-year sunspot cycle.  See a discussion of droughts on the Great Plains.

The possibility of a 37 year resonance with the lunar tidal cycle is intriguing; confirmation would require a homogeneous data set running over many hundreds of years.  The tree ring data are probably not sufficiently homogeneous.  There is an excellent theoretical reason why such the lunar tidal resonance should be strongest at the double period of 37 years: in 37 years the tidal cycle repeats at the same time of the year (much like the 18 year Saros cycle of eclipses).  A plausible physical mechanism has not yet been found, though 35 to 40 year weather and precipitation cycles have been frequently noted, and several meteorologists have suggested that the lunar tidal cycle modifies ocean currents in the northern ocean basins.

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