Sunspot cycle length, instantaneous period

Posted: July 13, 2012 by tchannon in Analysis, Cycles, Dataset, methodology, Solar physics

Figure 1

The length of sunspot cycles is usually calculated by measuring peak to peak or trough to trough but this necessarily involves guessing the true min and max, an interpretation, as well as only providing a few spot figures.

An alternative exists in the field of signal processing where a contiguous figure for the period of a varying wave can be computed, called instantaneous frequency (in this case the reciprocal is more useful, the period).

I was going to explain in detail but Rog thought what I had written looked too complicated, fair comment. This is a cut down version, better than nothing.

To do this we need to produce the analytic signal from the sunspot data. This is simple in concept, a pair of signals where one is 90 degrees phase shifted relative to the other, the twist here, is at all frequencies. One way of visualising this is by delaying the signal by an amount directly proportional to each frequency component.

If that sounds hairy, it is yet in many ways it is trivially simple. A very specific kind of filter can do this and the general name for the procedure is the Hilbert Transform.

I can explain some more if asked.


The sunspot data is “difficult” so simply passing that wave through such a filter produces a mess where we can’t get what we want.

  1. bandpass filter the sunspot data keeping say 7 year to 19 year information, no hard and fast rules here. Lower traces of figure 1 show ssn and the bandpass filtered version.
  2. Apply the filter to the data
  3. Do various computations on the resultant pair of signals.

Ever seen Lissajous figures, XY plots of A against B?

In phase signals give a 45 degree slope straight line, antiphase -45 degree straight line and between the two it opens out. At 90 degree shift you get a circle.


Figure 2

Like this bandpass filtered ssn data.

A party trick is that sin and cos are complementary where for a pair at 90 degrees they sum to unity. If you take the RMS value sqrt(x^2+y^2) you get the envelope (amplitude) of the original signal, shown in figure 1, simple as that point by point.

It is also possible to compute the instantaneous phase. At this point I am going to point you at Liverpool John Moore university where I hope some insight can be gleaned, pdf and with simple plots. This is an extremely important topic in for example medical imaging yet there is no exact solution, only heuristics.

If you have the phase you can also compute the frequency. The meaning of that is vague, hence bandpass filtering has restricted the window onto what is of interest.

As an aside I’ve experimentally used a completely different method as part of a software development, it gives roughly the same result.


Figure 3

This is noisier but is less fooled by the 1790 event, moreover the end data shown is trustworthy. (keep in mind the early ssn data is very poor goodness)

Commentary on result

Figure 1, there is a spike in period around 1800, probably quite wrong, clip the plot, figure is >30 years. However, there is good if disputed data which suggests there was a sunspot cycle which is missing because the data back then is very poor. If true there is a short cycle thereabouts.

In broad terms this agrees with the piecewise estimations. It also tends to show amplitude and period are complementary, nothing new.

Simple XLS spreadsheet of raw data and results, here.  (appears empty, is a wide sheet, go right)

Post by Tim Channon, co-moderator

  1. Ray Tomes says:

    Yes, I do similar things with data to work out instantaneous phase, amplitude etc. In general the position in the phase diagram can be computed by using the smoothed data and the smoothed rate of change for the x and y co-ordinates.

  2. tchannon says:

    Wouldn’t surprise me if ad-hoc methods work on noisy data.

    A feel for this probably arrives from STDFT or Wavelet which can trace a peak amplitude but no actual single trace.

    Most major packages with signal processing toolboxes have this stuff. Not my scene, too complex and arms length.

    The sun? It does seem in unusual territory at the same time as unusual weather patterns. Mid summer here in southern England yet the heating will turn on first thing in the morning, no sun, lots of rain (8 hours today), and chilly. Day after day and the house core temperature drops too low to hold temperature overnight.

  3. Bart Leplae says:

    The stronger a solar sunspot cycle, the more rapid the increase/decrease in number of sunspots: so the shorter the resulting periodicity. The Dalton minimum clearly correlates with a longer periodicity.

    Other increases in periodicity can be explained follows:

    The 1792 event (the missing solar cycle) occurred right at the moment when the sun had a minimum velocity relative to the solar system (in other words: when the sun was about the accelerate)

    In 1851 and 1971 the same effect occured, although to a lesser extent.

    In other words: whenever the Sun velocity changes from acceleration to deceleration or vice/versa, (while the number of sunspots is decreasing) the solar cycle periodicity is extended.

    Between 1895 and 1900, the rate by which the velocity of the Sun decreased was minimal.
    Between 2003 and 2011, the rate by which the velocity of the Sun increased was minimal.
    So the lower acceleration/deleration of the Sun looks to lengthen the periodicity of the Sunspot cycle as well.

    I described a number of correlations in :
    Variations of the Sun Velocity correlate in various ways with the Solar Cycles

  4. Tenuc says:

    Thanks for the excellent charts Tim, which clearly show that long cycles tend to be weak, and vice versa.

    Regarding weather, visited the Wildfowl & Wetlands trust at Barnes earlier in the week and was surprised to find migratory birds such as curlew, godwits and oyster catchers are already arriving in good numbers, when these would normally not be with us until late summer / early autumn. Perhaps the Svensmark effect starting to kick in?

    It also makes me wonder if we’re in for another cold winter again, which is just what we need following the lousy summer! Reminds me of the weather we had back in the 70’s, when the news papers were full of dire warnings of the start of a new ice age. What goes around comes around.

  5. tchannon says:

    Oh no, not godwits! I tell you, ‘im up there has NO SENSE OF HUMOUR, chimes in DEATH.
    (Pratchet, for those unfamiliar, like many other writers with a sense of fun, has ploys, where a character called DEATH, always speaks in CAPITAL LETTERS, scythingly.)

    Like the greater Twain, Adams and others, sense is given in one phrase shortstyle, without of course the immortalising postscript of quotation, we get the add that. I thinks it’s the observation of absurd human behaviour.

    Pr. for example had the old policeman (Grimes) and young twirp constubule where on a barracade and fighting, young twirps asks (paraphrasing here, don’t recall exact words)

    Sarg, shouldn’t they be talking first?
    No lad, this is the tried and tested way.

    The unspoken implicit does apply around the world, smash each other first, a war or fight, only then talk.

  6. adolfogiurfa says:

    This is an intelligent approach as it reveals the law behind the phenomenon. While chaos only exists in the mind of the beholder.
    Would it be possible an extrapolation?
    M.Vukcevic´s curve shows also a regularity behind:

  7. Bart Leplae says:

    If we compare the years when Min, 0-Cross, Max occured for the :
    M.Vukcevic´s curve / Sun Acceleration / Observed Polar Field :

    Min: 1994 / 1996 / x
    0-Cross: 1968 / 1971 / 1968- 1971
    Max: 1974 / 1975 / 1976
    0-Cross: 1980 / 1982 / 1980
    Min: 1985 / 1987 / 1986-1988
    0-Cross: 1990 / 1991 / 1990-1991
    Max: 1996 / 1995 / 1993-1997
    0-Cross: 2001 / 2000 / 2000
    Min: 2006 / 2003 / 2003-2008
    0-Cross: 2012 / 2011 / 2012

  8. tchannon says:

    No extrapolation using that method, can’t go right to the end of the dataset either.

    A good heuristic if nothing else is known is what comes next is a rough repeat of history. Second guess the sun?

    The sun is behaving unusually.

    There is unusual weather and has been over much the same time.

    Seems there are echos early 1800s and 1900s. Today though man has a capability for much wider knowledge but I doubt any more in the way of wisdom.


    I’m writing this from an ancient dog of a computer on a just finished new install of openBSD, all starting from one floppy disk. Long story here.

  9. Bart Leplae says:

    Out of curiosity, I determined the relative positions of Jupiter/Saturn through the Solar Walk app:

    M.Vukcevic´s curve / Sun Acceleration / Observed Polar Field / JS relative positions

    Min: 1964 / 1966 / xxx / JS Quad 1965
    0-Cross: 1968 / 1971 / 1968- 1971 / JS Opp 1970
    Max: 1974 / 1975 / 1976 / JS Quad 1975
    0-Cross: 1980 / 1982 / 1980 / JS Conj 1981
    Min: 1985 / 1987 / 1986-1988 / JS Quad 1996
    0-Cross: 1990 / 1991 / 1990-1991 / JS Opp 1998
    Max: 1996 / 1995 / 1993-1997 / JS Quad 1995
    0-Cross: 2001 / 2000 / 2000 / JS Conj 2000
    Min: 2006 / 2003 / 2003-2008 / JS Quad 2006
    0-Cross: 2012 / 2011 / 2012 /JS Opp 2011

    Opp = at opposide sides of the Sun
    Conj = at the same side an aligned with the Sun
    Quad = at 90 degree angle relative to the Sun

    So following the pattern, the next expected events for the Sun Polar Magnetic Field would be:
    JS Quad > 2015 > Maximum
    JS Conj > 2020 > O-Cross
    JS Quad > 2025 > Minimum
    JS Opp > 2030 > O-Cross

    Since the Sun magnetic field correlates with its own acceleration/decelaration relative to the Solar System, it looks like the Sun magnetic field is induced through magnetic induction …

  10. Bart Leplae says:

    Min: 1964 / 1966 / xxx / JS Quad 1965
    0-Cross: 1968 / 1971 / 1968- 1971 / JS Opp 1970
    Max: 1974 / 1975 / 1976 / JS Quad 1975
    0-Cross: 1980 / 1982 / 1980 / JS Conj 1981
    Min: 1985 / 1987 / 1986-1988 / JS Quad 1986
    0-Cross: 1990 / 1991 / 1990-1991 / JS Opp 1990
    Max: 1996 / 1995 / 1993-1997 / JS Quad 1995
    0-Cross: 2001 / 2000 / 2000 / JS Conj 2000
    Min: 2006 / 2003 / 2003-2008 / JS Quad 2006
    0-Cross: 2012 / 2011 / 2012 /JS Opp 2011