The Tip of the Gleissberg

Guest Post by Willis Eschenbach

A look at Gleissberg’s famous solar cycle reveals that it is constructed from some dubious signal analysis methods. This purported 80-year “Gleissberg cycle” in the sunspot numbers has excited much interest since Gleissberg’s original work. However, the claimed length of the cycle has varied widely.

Back in the 1940s, a man named Wolfgang Gleissberg was studying sunspot cycles. To do so, in his own words, he introduced a new method, viz:

When I introduced the method of secular smoothing into the study of the variations of sunspot frequency (GLEISSBERG, 1944) I published a table containing the secularly smoothed epochs and ordinates of sunspot minima and maxima which I had deduced from the data published by BRUNNER in 1939. Since then, secular smoothing has proved to be one of the principal methods for investigating the properties of the 80-year cycle of solar activity (cf. RUBASHEV, 1964).

 Figure 1. SIDC sunspot data, along with the “best-fit” sine wave for each cycle length from 40 years (orange, in back) to 120 years (blue, in front). Heavy black and heavy red horizontal sine waves show respectively the strength of the 80-year “Gleissberg Cycle” and the 102-year maximum-amplitude cycle. 

This purported 80-year “Gleissberg cycle” in the sunspot numbers has excited much interest since Gleissberg’s original work. However, the claimed length of the cycle has varied widely. One source says:

In different studies the length of the period of the secular variation was determined to be equal to 95 years, 65 years, 55 years, 58 years, 83 years, 78.8 years, 87 years [Siscoe, 1980; Feynman and Fougere, 1984]. That situation is understandable, because the longest record of direct observations of solar activity was and still is the sunspot numbers which provides more or less reliable information since 1700 (see below). That gives one only 300 years of time span by now which encompasses ~3.4 periods of Gleissberg cycle which is quite low for its statistical analysis.

So what was Gleissberg’s “secular smoothing” method that he “introduced” in 1944? Well, it turns out to be a simple 1-2-2-2-1 trapezoidal filter … but one which he employed in a most idiosyncratic and incorrect manner.

Let’s start, though, by looking up at Figure 1. It shows the three centuries of sunspot data in black, along with actual best fit sine waves in color, year by year, for each cycle length from forty years (colored orange, in the back) to one hundred twenty years (colored blue, in the front). Of particular interest are the 80-year cycle proposed by Gleissberg (heavy wavy horizontal black line), and the largest long-term cycle, which is 102 years in length (heavy wavy horizontal red line). As you can see, the 80-year “Gleissberg cycle” is not distinguished in any way.

So … does this mean that in fact there is a 102-year cycle in the sunspot data? Well, no. We still only have data enough for three 102-year cycles. And in natural data, that’s not very reliable. The problem is that nature appears to be chaotic on all timescales, so I’m not trusting the 102-year cycle to stick around. But in any case … just how did Gleissberg get to his 80-year number? Therein lies a tale …

First, Gleissberg decided that what we’re looking at in Figure 1 is an amplitude modulated signal. So he figured he only had to deal with the envelope of the signal, which looks like this:

Figure 2. Envelope of the sunspot record shown in color. As an aside, it turns out to be a curiously tricky algorithm that is needed to identify true maxima, or true minima.

Having gotten that far, he threw away everything but the envelope, leaving only the following information:

Figure 3. Envelope only of the sunspot record, maximum envelope shown in red, minimum envelope shown in blue.

And that poor misbegotten stepchild of a once-proud record was what he analyzed to get his 80-year cycle … sorry, just kidding. That would be far too simple. You see, the problem is that when you look at that envelope data in Figure 3, there are no evident long-term cycles in there at all. It’s just not happening.

To get around the minor issue that the data has no obvious cycles, Gleissberg applies his whiz-bang “secular smoothing” algorithm to the maximum and minimum envelope data, which gives the following result. Remember, there are no obvious cycles in the actual envelope data itself …

Figure 4. Result of “secular smoothing” of the maxima and minima envelopes of the sunspot data. Dotted vertical line marks 1944, the year that Gleissberg introduced “secular smoothing” to the world.

And voilá! Problem solved.

The big difficulty, of course, is that smoothing data often creates entirely specious cycles out of thin air. Look at what happens with the maximum envelope at 1860. In the original maximum data (light red), this is a low point, with peaks on either side … but after the filter is applied (dark red), it has magically turned into a high point. Smoothing data very commonly results in totally factitious cycles which simply do not exist in the underlying data.

There are a couple of other problems. First, after such a procedure, we’re left with only 24 maximum and 24 minimum datapoints. In addition, they are strongly autocorrelated. As a result, whatever conclusions might be drawn from Gleissberg’s reduced dataset will be statistically meaningless.

Next, applying a trapezoidal filter to irregularly spaced data as though they were spaced regularly in time is a big no-no. A filter of that type is designed to be used only on regularly spaced data. It took me a while to wrap my head around just what his procedure does. It over-weights long sunspot cycles, and under-weights short cycles. As a result, you’re getting frequency information leaking in and mixing with your amplitude information … ugly.

Finally, if you read his description, you’ll find that not only has he applied secular smoothing to the amplitudes of the maxima and minima envelopes. Most curiously, he has also applied his wondrous secular smoothing to the times of the maxima and minima (not shown). Is this is an attempt to compensate for the problem of using a trapezoidal 1-2-2-2-1 filter on irregularly spaced data? Unknown. In any case, the differences are small, a year or so one way or the other makes little overall difference. However, it likely improves the (bogus) statistics of the results, because it puts the data at much more regular intervals.

CONCLUSIONS:

First, the method of Gleissberg is unworkable for a variety of reasons. It results in far too few datapoints which are highly autocorrelated. It manufactures cycles out of thin air. It mixes frequency information with amplitude information. It adjusts the time of the observations. No conclusions of any kind can be drawn from his work.

Next, is the 80-year cycle described by Gleissberg anywhere evident in the actual sunspot data? Not anywhere I can find. There is a very wide band of power in the century-long range in the sunspot data, as shown in Figure 1. However, I don’t trust it all that much, because it changes over time. For example, you’d think that things would kind of settle down over two centuries. So here’s the first two centuries of the sunspot data …

Figure 5. As in Figure 1, but only the earlier two centuries of the sunspot data. 

Note that in the early data shown in Figure 5, there is very little difference in amplitude between the 80-year Gleissberg cycle, and the 95-year maximum amplitude cycle. You can see how Gleissberg could have been misled by the early data.

Now, let’s look at the latter two centuries of the record. Remember that this pair of two-century datasets have the middle century of the data in common …

Figure 6. As in Figure 5, but for the latter two centuries of the data.

In this two-century segment, suddenly the maximum is up to 113 years, and it is 2.5 times the size of the 80-year Gleissberg cycle.

In none of these views, however, has the 80-year Gleissberg cycle been dominant, or even noteworthy.

Please note that I am NOT saying that there are no century-long cycles, either in the sunspot data or elsewhere. I am making a careful statement, which is that to date there appears to be power in the sunspot data in the 95-120 year range. We can also say that to date, the power in the 80-year cycle is much smaller than anything in the 95-120 year range, so an 80-year “Gleissberg cycle” is highly unlikely. But we simply don’t have the data to know if that power in the century-long range is going to last, or if it is ephemeral.

Note also that I am saying nothing about either 80-year Gleissberg cycles, or any other cycles, in any climate data. This is just the tip of the Gleissberg. So please, let me ask you to keep to the question at hand—the existence (or not) of a significant 80-year “Gleissberg cycle” in the sunspot data as Gleissberg claimed.

Finally, if you are talking about e.g. a 85 year cycle, that’s not a “pseudo-80 year cycle”. It’s an 85 year cycle. Please strive for specificity.

My best wishes to all,

w.

Claimer (the opposite of “disclaimer”?): If you disagree with anything I’ve written, which did actually happen once a couple years ago, please quote the exact words that you disagree with. Often heated disagreements stem from nothing more than simple misunderstandings.

Data: The adjusted SIDC data is available as SIDC Adjusted Sunspots 1700 2012.csv . In accordance with the advice of Leif Svalgaard, all values before 1947 have been increased by 20% to account for the change in sunspot counting methods. It makes little difference to this analysis