The Effect of Gleissberg’s “Secular Smoothing”

Guest Post by Willis Eschenbach

sft full sunspot and just maximaABSTRACT: Slow Fourier Transform (SFT) periodograms reveal the strength of the cycles in the full sunspot dataset (n=314), in the sunspot cycle maxima data alone (n=28), and the sunspot cycle maxima after they have been “secularly smoothed” using the method of Gleissberg (n = 24). In all three datasets, there is no sign of the purported 80-year “Gleissberg Cycle”. In addition, the effect on the periodograms of missing data is investigated.

Continuing my investigations of the non-existence of the purported “Gleissberg Cycle”, at the suggestion of a commenter I’ve now done periodograms of the full sunspot dataset, the maxima only, and the “secular smoothed” maxima using Gleissberg’s method. I’ve also re-written the code for my “slow Fourier transform” so that it deals properly with irregularly spaced data. To get started, let’s look at the data itself, including the maxima (red line) and the minima (blue line):

full sunspot maxima and secular smoothedFigure 1. SIDC sunspot data since 1700. Red line shows the maximum value of each cycle. Blue line shows the cycle maxima after smoothing with Gleissberg’s “secular smooth”, a 1-2-2-2-1 trapezoidal filter.

For one thing, this would serve as the first real-world test for how well my “slow Fourier transform” performs when using a dataset that is both greatly reduced, and also irregular in time. So without further introduction, here are the periodograms of the sunspot data itself, and of the irregularly-spaced cycle peaks.

sft full sunspot and just maximaFigure 2. Periodograms of the total sunspot dataset (gold) and of the cycle maxima (red).

To begin with, let me say that I am amazed at how much information is contained in just the cycle peaks alone. Remember, the red line represents a mere 9% of the data, 91% of the data has been removed.

Next, looking at the full three centuries of sunspot data (gold), there are three main peaks, at 11 years, 102 years, and 52 years. There is no sign of Gleissberg’s 80-year cycle.

So how does using just the cycle peaks affect the results? Well, everything but the size of the main 11-year cycle has seen an increase in the reported strength of the cycle. This is because there is less data to constrain the fitting of the various lengths of sine waves, so they almost invariably end up larger than the corresponding cycle strength of the full dataset.

Despite all of that, however, the correlation between the two (red and gold) is impressively high, at 0.88. And it suggests that I should be able to further improve the results … more on that later, once I actually try it …

In any case, for purposes of investigating long-term results, there is little difference between using the full dataset and just the cycle peaks. Both of them, for example, show that rather than there being any strong “Gleissberg Cycle”, in fact 80 years is near the bottom of a dip in the cycle strength … and both the cycle peaks and the full dataset put the peak in the long-term cycles at about 100 years …

Having seen the results for the full data and the cycle maxima, what happens when we do the same analysis of Gleissberg’s “secularly smoothed” cycle maxima data? Figure 3 shows that result …

sft full sunspot maxima and secular smoothedFigure 3. Periodograms of the total sunspot dataset (gold), of the cycle maxima (red, grayed out), and of the “secularly smoothed” cycle maxima).

Like I said, I had no idea what the periodogram of the “secularly smoothed” data would look like. One real surprise was that it totally wiped out the peak that exists at around 55 years in both the full and cycle maxima periodograms. It has also knocked out almost all of the power in the cycles from about 15-50 years. I wouldn’t have guessed either of those.

Curiously, the part that the “secular averaging” didn’t affect are the cycles of 70 years and longer. Well, it pushed the peak back to about 99 years instead of 102 years, but other than that all three tell the same tale.

And the tale they are all telling is that there is no such thing as an 80-year “Gleissberg Cycle”. Doesn’t exist in the sunspot data, even using Gleissberg’s crazy method.

Now, I’m sure people will jump up and down and say “but, but, but there are 80-year cycles in the Nile river data” or some other dataset … but so what? There is no 80-year cycle in the sunspot data, so if anything, your 80-year cycles in the Nile river data show that the sunspot cycles don’t affect the Nile river levels.

That’s what I started out to do regarding the lack of the Gleissberg Cycle, so I’ll leave the story there …

However, having seen how well my slow Fourier transform (SFT) performs when using the cycle maxima data, I’ve got to try randomly knocking out parts of the sunspot data to see how well the SFT performs … hang on while I go do that. … OK, here’s what happens when I randomly knock out 10% of the sunspot data.

periodogram sidc sunspot data monte 10 percentFigure 4. Periodogram of the sunspot data, along with 30 instances of periodograms of the sunspot data with 10% of the data removed.

As can be seen, the loss of 10% of the data makes little practical difference to the results. This is quite encouraging. Next, here’s the same situation but with 50% of the data removed instead of 10% …

periodogram sidc sunspot data monte 50 percentFigure 5. Periodogram of the sunspot data, along with 30 instances of periodograms of the sunspot data with 50% of the data removed.

Obviously, there’s much more variation with half of the data being missing, it’s getting sketchier, but the results still might be useable.

My final conclusion is that my method deals quite well with missing data. My next project? Well, now that I’ve modified my code to not require regular dates for the time series, I want to take a look at the ice core records …

Onwards … always more to learn.

w.

Like I’ve Said Before: If you disagree with something I or someone else has said, please quote the exact words you disagree with. This avoids many misunderstandings.

Data: The adjusted SIDC data, along with the R slow Fourier transform functions to do the periodograms, are both available in a zipped folder here. In accordance with the advice of Leif Svalgaard, all sunspot values before 1947 have been increased by 20% to account for the change in sunspot counting methods. It makes little difference to this analysis. I believe the R code to be complete and turnkey. I’ve included an example with the functions.

 

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54 Responses to The Effect of Gleissberg’s “Secular Smoothing”

  1. David Byrne’s lyrics seem to me to have a skeptical bend to them, though I sincerely doubt he himself is – this is an excerpt from “Cross eyed and Painless.” Seems to fit your recent analysis amazingly well…
    ____________________________

    I’m ready to leave
    I push the fact in front of me
    Facts lost
    Facts are never what they seem to be

    Ah, nothing there
    No information left of any kind
    Lifting my head
    Looking for the danger signs

    And there was a line, there was a formula
    Sharp as a knife, facts cut a hole in us
    There was a line, there was a formula
    Sharp as a knife, facts cut a hole in us

    ____________________________

  2. Greg says:

    getting interesting.

    The red line is noisier since it has a lot less data, but the main peaks are confirmed.

    “Curiously, the part that the “secular averaging” didn’t affect are the cycles of 70 years and longer. ”

    Not curious. You passed a short period low-pass and it passed the low frequency components.

    “One real surprise was that it totally wiped out the peak that exists at around 55 years in both the full and cycle maxima periodograms.”

    You had data with roughly 10-11 year spacing and passed a slightly modified 5pt running mean. Not a real surprise either. By 60 years it’s starting to let some signal pass giving the false impression of a peak around 60.

    What both the gold and the red plots show is a strong signal at 10 and 11 (on your course scaling) it may be worth a more detailed look at that portion with higher resolution, More detailed work shows periods are round numbers of years.

    now if you add a your 10y period and your 11y period, it will produce an amplitude modulation of 10.5 modulated by 100 years (plus a residual of whichever short period is stronger).

    The fact that you see both in the period analysis means that the process is non-linear. In fact this can be seen by eye that simple SSN counting looks like a “rectified” signal. The count is the amplitude of an underlying process where the sign matters.

    In fact Ray Tomes has suggested that the noise level also indicates that it may be the square of whatever is the root cause.

  3. Greg Goodman says:

    ” By 60 years it’s starting to let some signal pass giving the false impression of a peak around 60.”

    Correction. On a closer look, it is the red line with less data points that is failing to resolve the 50 and 60 y periods in the full data, merging them in to an apparent 55 y peak. Having filtered out the 50y cmpt with the modified runny mean, the attenuated 60y comes through.

    It may be better to regard this as a false peak with the resolution left after all the chopping , it is more a matter of chance that it lines up a real peak in the full data.

    All in all, I would agree that there’s not much evidence of an 80 y cycle whatever way you dice it.

    However, it is quite a good demonstration that there is a long period amplitude modulation of about 102 years, resulting from the combination of circa 10 and 11 year periods.

    What I have never seen discussed in any mainstream work on SSN is what is needed to get to the root cause of this signal.

    I was initially critical of Ray’s suggestion that the data should be square-rooted because it was an arbitrary choice ( typical econometrics way of doing things, not science ). But I think he is basically correct. It we be good to have a proper scientific reason, based on a physical mechanism to take the square root.

    The ultimate need being for a way to ‘unfold’ the data and regain the signed time series. Once that is done the fourier analysis will become more meaningful and should tell us a lot more about the real underlying periodicities in the sun and that may finally start to give clues as to the cause.

  4. HenryP says:

    henry@willis

    as stated before, I don’t trust SSN as I found that there is a definitive trend upwards when measured against time. This is probably due to more stringent observation rules and better magnification technigues as time went by. Also, in addition, you even added data from years that Gleissberg never had, seeing as you took it to 2013. How do you justify that seeing as that I warned you before about the upward trend in SSN as time progresses?
    Surely, you must see that this could affect your results?

    The 88 year cycle is there, for a number of reasons
    1) because I found it myself in the data on maximum temperatures
    2) William Arnold found it and we both linked it to certain planetary configurations
    this makes it very easy for me to predict the turning point in the cycle.
    http://www.cyclesresearchinstitute.org/cycles-astronomy/arnold_theory_order.pdf
    (note the cycle times given for the flooding of the Nile. He thought 1990 was the end of warming, I say it was 1995, when looking at energy in. Apparently he was also about 7 years out on the planetary configuration, compared to what actually happened on the sun.)
    3)http://www.nonlin-processes-geophys.net/17/585/2010/npg-17-585-2010.html
    shows both the Gleisberg (86.5) and De Vries cycles (208)
    4)ersistence of the Gleissberg 88-year solar cycle over the last ˜12,000 years: Evidence from cosmogenic isotopes

    Peristykh, Alexei N.; Damon, Paul E.
    Journal of Geophysical Research (Space Physics), Volume 108, Issue A1, pp. SSH 1-1, CiteID 1003, DOI 10.1029/2002JA009390
    Among other longer-than-22-year periods in Fourier spectra of various solar-terrestrial records, the 88-year cycle is unique, because it can be directly linked to the cyclic activity of sunspot formation. Variations of amplitude as well as of period of the Schwabe 11-year cycle of sunspot activity have actually been known for a long time and a ca. 80-year cycle was detected in those variations. Manifestations of such secular periodic processes were reported in a broad variety of solar, solar-terrestrial, and terrestrial climatic phenomena. Confirmation of the existence of the Gleissberg cycle in long solar-terrestrial records as well as the question of its stability is of great significance for solar dynamo theories. For that perspective, we examined the longest detailed cosmogenic isotope record—INTCAL98 calibration record of atmospheric 14C abundance. The most detailed precisely dated part of the record extends back to ˜11,854 years B.P. During this whole period, the Gleissberg cycle in 14C concentration has a period of 87.8 years and an average amplitude of ˜1‰ (in Δ14C units). Spectral analysis indicates in frequency domain by sidebands of the combination tones at periods of ≈91.5 ± 0.1 and ≈84.6 ± 0.1 years that the amplitude of the Gleissberg cycle appears to be modulated by other long-term quasiperiodic process of timescale ˜2000 years. This is confirmed directly in time domain by bandpass filtering and time-frequency analysis of the record. Also, there is additional evidence in the frequency domain for the modulation of the Gleissberg cycle by other millennial scale processes.
    end quote
    5) I have shown you about 30 proxies that have a recurring 80-100 year cycle. The reason why time differs in these proxies is because of the intricacy whereby earth stores energy and/or releases energy and/or causes obscurity due to volcanic eruptions.

    Like I said before, for real scientists it is time to leave SSN alone and let it be…..
    Rather try to figure out this graph here

    http://ice-period.com/wp-content/uploads/2013/03/sun2013.png

    The declining magnetic fields allows more energetic particles to escape from the sun. Note the binomial you can draw from the top (hyperbolic) and the bottom (parabolic) showing that we must come to the bottom of the field strength somewhere in 2016. I hope my planets will arrive in time, as otherwise I donot know where we will end up with this graph.

    Have you thought about that? Come to think about it myself, if anything were to happen to the planets in our solar system so that they do not arrive in time (2016), we are all buggered here. We will all freeze to death……

    .

  5. Greg Goodman says:

    “as stated before, I don’t trust SSN as I found that there is a definitive trend upwards when measured against time. This is probably due to more stringent observation rules and better magnification technigues as time went by.”

    Just because a dataset has a finite “trend” we are supposed to “distrust” it? So we should only study data that are flat because everything else is nature is flat too , right?

    DUH.

    Also your comment about better magnification shows that you are totally ignorant of the subject. Considerable efforts are made to take this into account including using a small, antique telescope like used for early observations. This is used to ensure that counting methods from recent high quality observations are comparable to centuries old ones.

  6. HenryP says:

    Greg says
    Considerable efforts are made to take this into account including using a small, antique telescope like used for early observations

    henry says
    yet the upward trend is there
    which should not be, if we have recurring cycles.
    also there is no correlation as to why it should go up, so it was randomly going up.

    If various methods say 86.5 to 88 (including my own method) then why would you trust SSN ( showing 100) more? Surely, SSN is a very subjective discretionary observation, depending, for example, even on the strength of somebody’s eyes?

  7. Ulric Lyons says:

    “The wavelet power of the Gleissberg cycle is especially remarkable during 4750– 1400 BC, the power reaching peaks around 4215 BC, 3075 BC, 2755 BC, 2545 BC, 2075 BC and 1535 BC, with a statistical significance level higher than 95%. After 1400 BC, the wavelet power of the Gleissberg cycle becomes relatively weak.
    Ogurtsov et al. (2002) proposed that the Gleissberg cycle has a wide frequency band with a double structure, i.e., 50–80 year and 90–140 year periodicities.”
    http://ir.bao.ac.cn/bitstream/311011/930/2/3.pdf

  8. Just a minor quibble: As I am sure you know well, the “sunspot number” is not the number of sunspots. Yet, a few places, notably the graph axis labels, are written as if SSN and sunspots are interchangeable.

  9. Anthony Watts says:

    Thanks Willis, for using the new posting format, without being prompted.

  10. Greg Goodman says:

    Yes, the abstract is a good idea. It makes it quick to see whether an article is of interest from the front page. Nice move.

  11. HenryP says:

    Willis says
    http://wattsupwiththat.com/2014/05/17/the-tip-of-the-gleissberg/#comment-1640546
    henry says
    I hope I did cover my position here
    http://wattsupwiththat.com/2014/05/19/the-effect-of-gleissbergs-secular-smoothing/#comment-1640431

    if not, before you trash Gleissberg completely, show us the (uncorrected) SSN data that he was working on?

  12. HenryP says:

    Ulric says
    http://wattsupwiththat.com/2014/05/19/the-effect-of-gleissbergs-secular-smoothing/#comment-1640464
    henry says
    thanks for that!
    very interesting.
    As I was saying, I would never trust building a whole theory on one data set of results.
    I don’t think Gleiszberg did either. I am sure he also looked at other things besides SSN, like the level of lakes or rivers.
    I am going on a holiday soon down the river Rhine. Perhaps I will pick up something here or there on the subject….;

  13. Steven Mosher says:

    “I was initially critical of Ray’s suggestion that the data should be square-rooted because it was an arbitrary choice ( typical econometrics way of doing things, not science ). But I think he is basically correct. It we be good to have a proper scientific reason, based on a physical mechanism to take the square root.”

    sun spot count is not a proper physical unit. period. any equation you do with ‘spots’ will ever be dimensionally correct.

    you can transform it however you like, its not a physical unit.

    Now, if you want to switch to what matters.. like watts you’d be in a better position to understand physical systems.

    Spots and spot counts are an abstraction. the climate doesnt ‘see’ spots. humans see spots and then they devised a methodology for counting them. does the number matter? yes its a proxy. what about the size?

    if you drill down to the bottom and ask the simple question: what is a spot and how is it defined
    you will see that the real physics you are after doesnt quantify over spots. spots are a crude tool or metric used to understand something else..

  14. ren says:

    HenryP
    I am convinced that we are entering a deep solar minimum which none of us remembers. The sharp decline in activity after the last maximum is a clear signal from Cosmos. The activity will now fall rapidly.
    http://soho.nascom.nasa.gov/data/realtime/hmi_igr/512/latest.jpg

  15. HenryP says:

    Steven Mosher says

    http://wattsupwiththat.com/2014/05/19/the-effect-of-gleissbergs-secular-smoothing/#comment-1640655

    Henry@steven

    this must the first time ever that you post an argument that I absolute agree with….
    100%

    be blessed

  16. HenryP says:

    henry@ren
    don’t worry
    my planets will come in time,
    to switch the lever back to warming again (increasing solar magnetic polar strengths)
    unless something happens to any one of them….

  17. HenryP says:

    @ren
    I do agree, that now, more than I can remember since I live here,
    the sun is brighter/lighter than ever before
    Amazing is it not? that a brighter/lighter/warmer sun causes a cooler earth…
    it boggles the mind
    does it not?

  18. lsvalgaard says:

    Greg Goodman says:
    May 19, 2014 at 1:47 am
    However, it is quite a good demonstration that there is a long period amplitude modulation of about 102 years, resulting from the combination of circa 10 and 11 year periods.
    You have this backwards. There is a real 100-yr modulation of the amplitude of the cycle. This is a real physical effect: cycles vary in strength and we can understand why [or at least there are models that can 'explain' this]. The amplitude modulation then produces the side peaks at 10- and 12 years.

  19. lsvalgaard says:

    HenryP says:
    May 19, 2014 at 5:49 am
    Surely, SSN is a very subjective discretionary observation, depending, for example, even on the strength of somebody’s eyes?
    The SSN shows 98% correlation with the microwave flux from the Sun and therefore can be trusted to give a very close measure of real solar activity. Experienced observers agree closely.

  20. Willis Eschenbach says:

    Greg says:
    May 19, 2014 at 1:27 am

    getting interesting.

    The red line is noisier since it has a lot less data, but the main peaks are confirmed.

    “Curiously, the part that the “secular averaging” didn’t affect are the cycles of 70 years and longer. ”

    Not curious. You passed a short period low-pass and it passed the low frequency components.

    Thanks, Greg. Following Gleissberg, I used his 5-point filter. But in his method, the filter spans five sunspot cycle peaks, which is a varying period on the order of 44 years. Despite that, it has not eliminated the 11-year cycle much, and it has created a spurious 6-year cycle. So it is not acting like a normal low-pass filter, which would have left no cycles at all below 44 years or so.

    I think that the oddness of those results comes from the fact that the data points that are being smoothed are all cycle peaks, but hey, I was born yesterday, and unlike many commenters, I’ve never taken a single class in any of these matters. So I’m always willing to learn.

    “One real surprise was that it totally wiped out the peak that exists at around 55 years in both the full and cycle maxima periodograms.”

    You had data with roughly 10-11 year spacing and passed a slightly modified 5pt running mean. Not a real surprise either. By 60 years it’s starting to let some signal pass giving the false impression of a peak around 60.

    A 5-point filter on data with 10-11 year spacing shouldn’t affect a 55-year cycle that much. That’s almost 25% longer than the filter width of 40-44 years, there should be little attenuation.

    What both the gold and the red plots show is a strong signal at 10 and 11 (on your course scaling) it may be worth a more detailed look at that portion with higher resolution, More detailed work shows periods are round numbers of years.

    I’ve done that, and not found that “periods are round numbers of years”. They fall where they fall. And there is no reason they should be in round years, as solar cycles know nothing of puny Earthlings and their curious time unit, “years” … hang on, let me do that for you. To get higher resolution with my slow Fourier transform, you just use data with higher resolution. Here’s the periodogram of the monthly (rather than annual) sunspot data.

    There are three prominent peaks, at ten years, eleven years, and eleven years ten months. There’s a much, much smaller cycle at twenty-one years one month. However, the fact that two of them are on even years is happenstance. Here’s the same analysis, starting 50 years later:

    Note that the peaks are no longer on the exact 10- and 11-year boundaries.

    now if you add a your 10y period and your 11y period, it will produce an amplitude modulation of 10.5 modulated by 100 years (plus a residual of whichever short period is stronger).

    The fact that you see both in the period analysis means that the process is non-linear.

    There’s where we part company. There is no “your 10y period and your 11y period”. There is a single cyclic phenomenon which peaks at intervals ranging from 9 to 15 years. So you can’t just pick a couple frequencies and note that their synoptic period is kinda sorta the same length as some other observable cycle. A variable cycle that peaks at intervals between 9 and 15 years doesn’t automatically create an underlying century-long cycle, that’s just numerical handwaving.

    In fact this can be seen by eye that simple SSN counting looks like a “rectified” signal. The count is the amplitude of an underlying process where the sign matters.

    Every rectified signal that I’ve ever seen started and ended every cycle at zero. That’s kinda the definition of “rectified” on my planet, that values below zero are converted to the same value above zero.

    But the sunspot data doesn’t do that. What it resembles is something like the length of the line of people waiting for a bank teller. It has a daily peak, and it has low spots, just like the peaks and lows of sunspots. And sometimes it goes to zero, again like sunspots. But that doesn’t make either of them into rectified signals.

    In fact Ray Tomes has suggested that the noise level also indicates that it may be the square of whatever is the root cause.

    Thanks for that idea. It has led me to the concept of a curious expansion of the type of direct spectrum analysis I’ve called the “slow Fourier transform” (SFT). In the SFT, I’m using sine waves to probe the individual cycle lengths.

    But your comment has led me to the idea that there is no reason to limit myself to sine waves. I could just as easily probe the data using log(sin), sin2, or other wave forms … always more adventure.

    Regards,

    w.

    PS – One of the more interesting ideas I’ve had about using other waveforms is first determining the actual wave-form of the predominant 11-year cycle, and then using that idiosyncratic wave-form to probe the data … man, not enough hours in a day.

  21. lsvalgaard says:

    Willis Eschenbach says:
    May 19, 2014 at 1:05 pm
    There’s where we part company. There is no “your 10y period and your 11y period”. There is a single cyclic phenomenon

    And Willis is correct here. The 102-yr real amplitude modulation creates the two side peaks at 10 and 12 years.

  22. Greg Goodman says:

    Damn! I’d just composed long reply and an accidental click zoomed you graphic. When I come back I’ve lost it.

    OK, I’ll resume. Yes, abs(cos(x)) would be very interesting, you’ll get periods about twice what you see above.

    Thanks for the high-res SFT, that is exactly what I was expecting. I wondered why there was not 11.8 in the original. 10y 1mo; 10y 11mo and 11y 10mo make up frequency side bands around the central period as Leif points out.

    Such triplets , as I have frequently pointed out are the result of modulation. Lief was right to pick me up on how I wrote that above. The two ways of expressing the data are equivalent. I was not intending to say which was causing which physically.

    Those numbers imply a modulation period of about 136 years, rather than 102, so maybe that something separate. It would be interesting to see what an abs(cos) SFT looks like. .

    When I said “rectified” don’t take it to be like a power socket. It is well known that Schwabe cycle has a polarity change. That does not mean that whole of the sun has to suddenly stop moving while it switches over. Each of the circa 11y bumps are just half of a longer cycle of which SSN is measure of the magnitude ( or according to Ray Tomes, the square of the magnitude ).

    You could try doing a sqrt(SSN) and fitting SFT .

    I wanted to try and abs(cos()) fit to Arctic ice, did you post your SFT code anywhere?

    Thanks.

  23. Greg Goodman says:

    Here’s what sqrt ( sun spot area ) looks like:
    http://climategrog.wordpress.com/?attachment_id=941

  24. Greg Goodman says:

    That sloping profile looks like it has a component twice as fast as the basic 11y at about 30% amplitude. probably what is picked up around 5.5 in your latest plots.

  25. Greg Goodman says:

    replacing above sqrt(SSA) graph:
    http://climategrog.wordpress.com/?attachment_id=942

  26. Steven Mosher says:

    Henry, to understand my point see what leif says. Im not saying that spots are not important.
    Im suggesting that people not forget what the real physical units are at the bottom.
    It aint spots, its flux.

    “The SSN shows 98% correlation with the microwave flux from the Sun and therefore can be trusted to give a very close measure of real solar activity. ”

    flux gives you real units. units with a physical meaning — that is a meaning within the laws of physics..

    Why is that important?

    well when people talk about taking the sq root of sunspot numbers.. you really want to check does taking the square root of flux make any sense. maybe it does, dunno. If it doesnt then you are potentially fooling yourself by doing data manipulations.

  27. Greg Goodman says:

    It’s all a question of what causes the flux, or causes the change in flux.

    Perhaps Lief could clarify whether he means flux density or total flux. He also suggested there were some hypotheses about the cause of centennial scale modulation.

  28. Peter Sable says:

    from: http://arxiv.org/pdf/math/0305364v3.pdf (frequency analysis of quasi-periodic signals)

    “The Nyquist aliasing constraint means that to recover a given period, one needs
    to sample the data with at least two points per period. On the opposite, in order
    to determine precisely the long periods, one needs that the total interval length T
    is several time larger than these periods, in order to reach the asymptotic
    rates of theorems 1 and 2, or to be able to separate properly close frequencies.”

    Your cutoff for analysis is at 100 years, which is only 3 cycles on your 300 year data length. I don’t think you have enough data especially for a quasi-periodic signal of 80 years. (the varying cycle for 10-12 years shows it’s quasiperiodic). You are close though. Can you get another 100 years of data?

    Also I believe you are still failing to apply a window to the dataset, which means in the the actual math you’ve got a step function every 300 years from about 0 to 75 sunspots. That’s going to smear sinx/x of that step function over your frequency domain, and unfortunately the worst of that is going to appear right in the 80-150 year region which you were trying to analyze…

  29. lsvalgaard says:

    Greg Goodman says:
    May 19, 2014 at 3:59 pm
    It’s all a question of what causes the flux, or causes the change in flux.
    Perhaps Lief could clarify whether he means flux density or total flux.

    The microwave flux is causes by the solar magnetism. Since the microwaves is radiated in all directions they have to cross a surface of 4 pi around the Sun, so the flux density and the total flux are just different by a constant factor [the area of the surface].

  30. Greg Goodman says:

    Agreed, Peter, The long term part of the spectrum is unreliable. I think Willis would agree, he has made similar comments about >3x being required. What I’m not sure he appreciates is the effect of a long term rise. This will particularly effect periods that are a sub-multiple of the window length.

    A way to see this would be to fit a “linear trend” to the data and do the SFT of just that trend for the 300 years sample period.

    This site shows the FFT of a sawtooth. I think the 102y peak is largely a result of this, as is the circa 50y pk. Below that, the amplitudes make it small enough to ignore.

    http://acad.carleton.edu/courses/musc108-00-w13/pages/11-MTPresentation2/11SynthesizedWaveforms.html

    Windowing and detrending are classic solutions to reduce these artefacts but do themselves introduce distortions. So is diffing to remove autocorrelation. It’s not a simple situation with one correct solution. Like in most trades, this is where a bit of experience comes in.

    Gleissberg’s last peak was 1938 so his running average “secular” filter would have started in 1728 stopped in 1917. That shows two peaks and two minima in his filtered series. Really insufficient to draw conclusions about the long periods in such noisy data with so few data points.

    Willis is using all the data currently available .Willis’ 100y is 300/3 . What does the SFT look like if the data is cut off after the 1938 peak, as Gleissberg had it ?

  31. HenryP says:

    @GREG
    Don’t forget that after that, (leif will know when) they “corrected” the SSN data. So, if we hope to see what he had we must get the data and paper that he actually wrote.

  32. Greg Goodman says:

    Lief: “The microwave flux is causes by the solar magnetism. Since the microwaves is radiated in all directions they have to cross a surface of 4 pi around the Sun, so the flux density and the total flux are just different by a constant factor [the area of the surface].”

    That makes the rather unwarranted assumption that the microwave radiation is spherically symmetric in all directions and at all times, if you are going to extrapolate from point measurements to the whole sphere. IFAIK, we do not have the means to measure all directions from the sun.

    My question was about the measured quantities you said matched to within 98% . Now SSN is an Earth-based observation. We do not have a cloud of satellites around the sun measuring the flux, so presumably that is primarily Earth or Earth orbit measurements too.

    You made a specific comment that must relate to specific measurements in specific units. My guess is that these are flux density measurements. Could you clarify what the point of measurement was? Ground based, Earth orbit satellites, other?

    It is very useful to know SSN ties into hard unit measurements. But to be informative it needs to be a bit more specific than “flux”. Could you provide a bit more info about exactly what measurements were compared to SSN?

    Thanks.

  33. lsvalgaard says:

    Greg Goodman says:
    May 19, 2014 at 11:39 pm
    That makes the rather unwarranted assumption that the microwave radiation is spherically symmetric in all directions and at all times, if you are going to extrapolate from point measurements to the whole sphere.
    To the approximations involved the assumption is a good one. For once the F10.7 flux is measured on the ground as the total radiation received from visible part of the solar disk in narrow region around a wavelength of 10.7 cm. I don’t make ‘unwarranted assumptions’. You can learn about the flux here:
    http://www.leif.org/EOS/1994SoPh-Tapping.pdf

  34. HenryP says:

    @Steven Mosher @all
    I will share with you that I know that a lower solar field strength allows more of the shortest wave particles to escape (UV-C), which react TOA to form ozone, peroxides and nitrogenous oxides. In turn, the increase in these compounds TOA deflect more sunlight to space. This is basiccally how the wolf-gleiszberg cycle works. Paradoxically, a somewhat “brighter”-, “lighter” sun causes cooling on earth. It is a defence system that earth has, to stop UV-C reaching earth./
    So the important graph to watch is this one here:
    http://ice-period.com/wp-content/uploads/2013/03/sun2013.png

    Now, if anyone of you can tell me or guess what the next 44 or 46 years of that graph will look like, you are on your way to understand a big (important) part of the climate
    as witnessed in the tables 2 and 3 here:
    http://virtualacademia.com/pdf/cli267_293.pdf

    best wishes
    Henry

  35. Greg Goodman says:

    Lief, thanks for the reply. but…
    404 Error File Not Found

  36. lsvalgaard says:

    HenryP says:
    May 20, 2014 at 12:31 am
    I will share with you that I know that a lower solar field strength allows more of the shortest wave particles to escape (UV-C)
    No, HentyP that is absolutely positively wrong. You have several misconceptions about this and are totally confused. The graph you show is my graph of the polar fields of the Sun which have nothing to do with the ‘escape’ of UV waves [not particles]. What you ‘know’ is ‘not even wrong’.

    Greg Goodman says:
    May 20, 2014 at 3:03 am
    Lief, thanks for the reply. but…404 Error File Not Found
    Try now.

  37. HenryP says:

    leif says
    The graph you show is my graph of the polar fields of the Sun

    henry says
    I am using your graph? that’s funny. Then you are brilliant except for not interpreting it correctly yourself.

    The declining magnetic fields allows more energetic particles to escape from the sun. Note the binomial you can draw from the top (hyperbolic) and the bottom (parabolic) showing that we must come to the bottom of the field strengths somewhere in 2016 or 2017.

    Have you thought about that? Come to think about it myself, if anything were to happen to the planets in our solar system so that they do not arrive in time (2016), we are all buggered here. We will all freeze to death……

    Anyway, I am not really interested in having an argument with you. You can ignore whatever I say if you think “it is not even wrong”

  38. lsvalgaard says:

    HenryP says:
    May 20, 2014 at 10:36 am
    I am using your graph? that’s funny. Then you are brilliant except for not interpreting it correctly yourself.
    Perhaps you can accept that you are confused. If not, I have tried to tell you that you are, but it is for you to realize it for yourself. If not, just carry on as usual in your own private world without interference from the truth.

  39. HenryP says:

    henry said
    So the important graph to watch is this one here:
    http://ice-period.com/wp-content/uploads/2013/03/sun2013.png

    Now, if anyone of you can tell me or guess what the next 44 or 46 years of that graph will look like, you are on your way to understand a big (important) part of the climate
    as witnessed in the tables 2 and 3 here:
    http://virtualacademia.com/pdf/cli267_293.pdf

    henry says
    nobody even going to try and guess?

  40. ren says:

    HenryP, you’re right that the sun is brighter.

  41. HenryP says:

    @ren
    I am glad you agree.
    Have figured out yet what the next 44-46 years of that solar polar fields strength graph will look like?

  42. Dinostratus says:

    I didn’t get to comment to the first SFT post before it got to hundreds of responses but now I just want to say, I really like seeing the phrase SFT. I see people write FFT like it is somehow better than a Fourier Transform, like it provides more or better information, like it’s an FT turned up to eleven. They have no idea that the FFT is just an algorithm for giving the results of an FT.

    So I cheer the SFT. Let those noobs suck on their fastiness and pooh pooh the SFT not knowing that it’s just a means to the same end.

  43. ren says:

    HenryP judging from this chart, we have a very sharp decline in solar activity. At least the next cycle will be very weak in terms of of magnetic activity.
    http://oi59.tinypic.com/2qumddx.jpg
    The amount of magnetic storms shows a well scale changes in cycle 24.

  44. ren says:

    Taking into account that the magnetic storms are associated with CME (not with the amount of spots) you can see how weak the solar magnetic activity cycle 24.

  45. lsvalgaard says:

    ren says:
    May 21, 2014 at 7:43 am
    >i?Taking into account that the magnetic storms are associated with CME (not with the amount of spots) you can see how weak the solar magnetic activity cycle 24.
    The number of CMEs in cycle 24 is not lower than that in cycle 23.

  46. ren says:
    May 21, 2014 at 7:43 am
    Taking into account that the magnetic storms are associated with CME (not with the amount of spots) you can see how weak the solar magnetic activity cycle 24.
    The number of CMEs in cycle 24 is not lower than that in cycle 23.

  47. Bernie Hutchins says:

    Dinostratus said in part May 21, 2014 at 2:34 am:
    “I didn’t get to comment to the first SFT post before it got to hundreds of responses but now I just want to say, I really like seeing the phrase SFT. I see people write FFT like it is somehow better than a Fourier Transform, like it provides more or better information, like it’s an FT turned up to eleven. They have no idea that the FFT is just an algorithm for giving the results of an FT…..”

    NOPE! The FFT is a fast algorithm for EXACTLY computing the DFT (Discrete Fourier Transform). Because we can do very few integrals, the FT (better called CTFT or Continuous-Time Fourier Transform) is largely incomputable. Except numerically. (Many times we don’t even have a functional expression for a signal.) This numerical computation leads to the DFT. But because of sampling in both domains, the DFT pairs become periodic (aliased). With proper precautions (such as appropriate zero padding) we can interpret an FFT (DFT) as approximating a CTFT. In general, if you start with actual data (not a functional expression) you can’t compute a CTFT; only interpret an FFT as a CTFT approximation. The FFT is “superior” to the CTFT in the sense that you can always do it. The CTFT of actual data is generally not available.

  48. ren says:

    Leif Svalgaard ,does the amount of magnetic storms is the same? The graph shows the number of magnetic storms. Is the graph is false?
    On the 50 the strongest storms in cycle 23 and 24, just 5 of in the cycle of 24.
    http://www.spaceweatherlive.com/en/solar-activity/top-50-solar-flares
    I did not write about the number of CME, but the amount of magnetic storms.

  49. ren says:

    Leif Svalgaard, do I have to explain? Not every CME cause magnetic storm on Earth.

  50. ren says:

    Number of spots is 126 but they are not magnetically active. Right? There will be no magnetic storm on Earth.

    Region Number of
    sunspots Class
    Magn. Class
    Spot
    2061 1 α HSX
    2065 2 β BXO
    2066 5 β DAO
    2069 1 α AXX
    2070 4 β BXO
    2071 4 β CRO
    2072 2 α AXX
    2073 1 α HSX

  51. TED VAUGHN says:

    The AP index proves your point REN.

  52. ren says:

    TED VAUGHN , may 23 arrived the “old” CME to Earth. Magnetosphere has reacted strongly. It was even Kp 5. Regards.

  53. Robert Cuffel says:

    Thank you for using the corrected sunspot numbers from Svalgaard’s working group. But you only included the rather ratty data form 1700 to 1750, not that back to nearly 1600 to provide 400 years of data with about 50 years of nearly all zeros to confuse your Fourier analysis!!! (I have no credentials in this field)

  54. HenryP says:

    Willis says
    climate is chaotic

    henry@RichardB, and @willis

    I have analysed the rainfall figures for Potchefstroom, South Africa.
    Setting the (total) yearly rainfall in mm out against time, I get that from 1927 – 2014 the linear trend line is nearly flat. I would expect this to happen, as the period 1927 until 2016 is in fact one complete Wolf-Gleissberg cycle.
    http://blogs.24.com/henryp/2012/10/02/best-sine-wave-fit-for-the-drop-in-global-maximum-temperatures/

    To check the pattern of rainfall, I divided 1927-2014 into 4 segments, as apparent from the 4 Hale cycles that make up one Gleissberg cycle and worked out the yearly average for each of these periodes.

    1927-1950 611.7
    1951-1971 587
    1972-1995 596.1
    1996-2013 641.2
    .
    Setting these periods out against rainfall, you will find a binomial with the following co-ordinates:
    17.45×2 – 77.49x + 671.85 (mm)
    correlation r2 = 0.9999

    Again, the rainfall pattern is exactly as predicted by me, namely in a warming period you will get less rainfall at the <[30] latitudes. (Potchefstroom lies at ca. 25)

    One is always tempted to see what happens when we forecast and back cast on such an equation, and, doing just that, one would be inclined to think that average yearly rainfall in 1904-1926 and from 2016-2039 will be around 680-700 mm. However, from my own data and from the data on field strengths we get from the sun, we realize that this cannot be done like this. In fact, we know that 1927 and 2016 are dead end stops. That means that if we want to back-cast the rainfall for the period 1904-1927 – and forecast what 2016- 2039 – looks like ,
    it would form a mirror, going down, instead of up, -
    i.e. namely,
    1904-1927 ca. 587
    2016-2038 ca. 596

    So, you see, there is nothing chaotic and unpredictable about the weather in Potchefstroom. It works just like a clock.

    I think I must become a weather forecaster?

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