Guest Post by Willis Eschenbach
I came across a curious graph and claim today in a peer-reviewed scientific paper. Here’s the graph relating sunspots and the change in sea level:
And here is the claim about the graph:
Sea level change and solar activity
A stronger effect related to solar cycles is seen in Fig. 2, where the yearly averaged sunspot numbers are plotted together with the yearly change in coastal sea level (Holgate, 2007). The sea level rates are calculated from nine distributed tidal gauges with long records, which were compared with a larger set of data from 177 stations available in the last part of the century. In most of the century the sea level varied in phase with the solar activity, with the Sun leading the ocean, but in the beginning of the century they were in opposite phases, and during SC17 and 19 the sea level increased before the solar activity.
Let me see if I have this straight. At the start of the record, sunspots and sea level moved in opposite directions. Then for most of the time they were in phase. In both those cases, sunspots were leading sea level, suggesting the possibility that sunspots might affect sea level … except in opposite directions at different times. And in addition, in about 20% of the data, the sea level moved first, followed by the sunspots, suggesting the possibility that at times, the sea level might affect the number of sunspots …
Now, when I see a claim like that, after I get done laughing, I look around for some numerical measure of how similar the two series actually are. This is usually the “R2” (R squared) value, which varies from zero (no relationship) to 1 (they always move proportionately). Accompanying this R2 measure there is usually a “p-value”. The p-value measures how likely it is that we’re just seeing random variations. In other words, the p-value is the odds that the outcome has occurred by chance. A p-value of 0.05, for example, means that the odds are one in twenty that it’s a random occurrence.
So … what did the author of the paper put forwards as the R2 and p-value for this relationship?
Sad to relate, that part of the analysis seems to have slipped his mind. He doesn’t give us any guess as to how correlated the two series are, or whether we’re just looking at a random relationship.
So I thought, well, I’ll just get his data and measure the relationship myself. However, despite the journal’s policy requiring public archiving of the data necessary for replication, as is too common these days there was no public data, no code, and not even a Supplementary Online Information.
However, years of messing around with recalcitrant climate scientists has shown me that digitizing data is both fast and easy, so I simply digitized the graph of the data so I could analyze it. It’s quite accurate when done carefully.
And what did I find? Well, the R2 between sunspots and sea level is a mere 0.13, very little relationship. And even worse, the p-value of the relationship is 0.08 … sorry, no cigar. There is no statistically significant relationship between the two. In part this is because both datasets are so highly auto-correlated (~0.8 for both), and in part it’s because … well, it’s because as near as we can tell, sunspots [or whatever sunspots are a proxy for] don’t affect the sea level.
My conclusions from this, in no particular order, are:
• If this is the author’s “stronger effect related to solar cycles”, I’m not gonna worry about his weaker effect.
• This is not science in any sense of the word. There is no data. There is no code. There is no mathematical analysis of any kind, just bald assertions of a “stronger” relationship.
• Seems to me the idea that sunspots rule sea level would be pretty much scuttled by sunspot cycles 17 and 19 where the sea level moves first and sunspots follow … as well as by the phase reversal in the early data. At a minimum, you’d have to explain those large anomalies to make the case for a relationship. However, the author makes no effort to do so.
• The reviewers, as is far too often the case these days, were asleep at the switch. This study needs serious revision and buttressing to meet even the most minimal scientific standards.
• The editor bears responsibility as well, because the study is not replicable without the data as used, and the editor has not required the author to archive the data.
So … why am I bothering with a case of pseudo-science that is so easy to refute?
Because it is one of the papers in the Special Issue of the Copernicus journal, Pattern Recognition in Physics … and by no means the worst of the lot. There has been much disturbance in the farce lately regarding the journal being shut down, with many people saying that it was closed for political reasons. And perhaps that is the case.
However, if I ran Copernicus, I would have shut the journal down myself, but not for political reasons. I’d have closed it as soon as possible, for both scientific and business reasons.
I’d have shut it for scientific reasons because as we see in this example, peer-review was absent, the editorial actions were laughable, the authors reviewed each others papers, and the result was lots of handwaving and very little science.
And I’d have shut it for business reasons because Copernicus, as a publisher of scientific journals, cannot afford to become known as a place where reviewers don’t review and editors don’t edit. It would make them the laughing stock of the journal world, and being the butt of that kind of joke is something that no journal publisher can survive.
To me, it’s a huge tragedy, for two reasons. One is that I and other skeptical researchers get tarred with the same brush. The media commentary never says “a bunch of fringe pseudo-scientists” brought the journal down. No, it’s “climate skeptics” who get the blame, with no distinctions made despite the fact that we’ve falsified some of the claims of the Special Issue authors here on WUWT.
The other reason it’s a tragedy is that they were offered an unparalleled opportunity, the control of special issue of a reputable journal. I would give much to have the chance that they had. And they simply threw that away with nepotistic reviewing, inept editorship, wildly overblown claims, and a wholesale lack of science.
It’s a tragedy because you can be sure that if I, or many other skeptical researchers, got the chance to shape such a special issue, we wouldn’t give the publisher any reason to be unhappy with the quality of the peer-review, the strength of the editorship, or the scientific quality of the papers. The Copernicus folks might not like the conclusions, but they would be well researched, cited, and supported, with all data and code made public.
Ah, well … sic transit gloria monday, it’s already tuesday, and the struggle continues …
w.
PS—Based on … well, I’m not exactly sure what he’s basing it on, but the author says in the abstract:
The recent global warming may be interpreted as a rising branch of a millennium cycle, identified in ice cores and sediments and also recorded in history. This cycle peaks in the second half of this century, and then a 500 yr cooling trend will start.
Glad that’s settled. I was concerned about the next half millennium … you see what I mean about the absence of science in the Special Edition.
PPS—The usual request. I can defend my own words. I can’t defend your interpretation of my words. If you disagree with something I or anyone has written, please quote the exact words that you object to, and then tell us your objections. It prevents a host of misunderstandings, and it makes it clear just what you think is wrong, and why.
vukcevic says:
January 26, 2014 at 8:12 am
I forgot to ask: have you ever updated the Fig. 23 in http://www.leif.org/research/suipr699.pdf
No, as there really is no need to elaborate further on this well-known result. But the effect shows up in modern data as well, of course, see e.g. Figure 17 of http://www.leif.org/research/2007JA012437.pdf and discussion in para [38]. See also section 5 of http://www.leif.org/research/Semiannual-Comment.pdf . In any case, it is just a small, second-order effect.
If so I would like to see a copy of the graph, while data would be greatly appreciated
The data are just the sunspot number and the aa-index which you can get from many places.
I have to suffer further ‘pseudoscience’ lapses
There is nothing wrong with pseudoscience [and make-believe] as long as you know it is pseudoscience. The problem comes in when you think it is [and advocate it as] science.
vukcevic says:
January 26, 2014 at 8:12 am
I forgot to ask: have you ever updated the Fig. 23…
The 22-year cycle was discovered by Ed Chernosky in 1966. Here is his original paper on that http://www.leif.org/EOS/JZ071i003p00965.pdf
If you go to his Figure 4 and look at the lower panel showing the sunspot number you may see that for the period covered odd sunspot cycles were more active in the first half of the cycle. If you ‘slide’ the B-curves down to match the A-curves during the first half of the cycles, you may see that the enhancement of activity in the first half of the odd cycles is simply due to those cycles being more active [per chance for the period covered]. The 22-year cycle in geomagnetic activity then becomes only something that happens in the last half of the cycle [after polar field reversal]. See also the discussion in para [20] of http://www.leif.org/research/Asymmetric%20Rosenberg-Coleman%20Effect.pdf
lsvalgaard says:
January 26, 2014 at 9:01 am
The 22-year cycle in geomagnetic activity then becomes only something that happens in the last half of the cycle [after polar field reversal].
I said that clumsily. The 22-year cycle enhancement is something that only occurs in one half of the 22-year cycle, namely from polar field reversal in even cycles to polar field reversal in the next odd cycle.
Thanks !
vukcevic says:
January 26, 2014 at 9:33 am
Thanks !
On second thought, it might be a good idea to re-visit the 22-year cycle some day as there are some misconceptions floating around…
The 22-year cycle in solar activity is called the Gnevyshev-Ohl rule [that odd cycles are intrinsically more active than even cycles]. Cycle 23 is a good example of the breakdown of the G-O rule and were cycle 18 and cycle 5 among others, so the ‘rule’ is not ‘robust’ [another word for ‘real’].
The 22-year cycle in geomagnetic activity is a geometric effect that relies on the solar polar fields regulating the length of solar sectors in just the right way [the Rosenberg-Coleman effect] such as they maximize the Russell-McPherron effect for enhancing the reconnection-generated geomagnetic activity. These effects are small and insignificant as far as being causative for any effects on the Geosystem, but important in the sense that they show us a very important aspect of solar cycles, namely that the solar polar fields have reversed normally since at least the 1840s.
Greg Goodman says:
January 22, 2014 at 4:33 pm
Good morning, Greg. I don’t know how to read the significance level from your plot … which peaks are significant?
I ask because I just looked at the cross-correlation of the annual change in the Jevrejeva data, versus the SIDC sunspot numbers from Leif. I don’t find a significant correlation at any lag. The maximum correlation is with the sea-level change occurring ten years after the sunspots, R^2 = 0.005 …
Best regards, and many thanks for your participation, always valuable,
w.
A number of people have recommended the use of the cross-correlation function to determine if a relationship exists. While this is valuable, and I use it often enough that long ago I wrote an Excel function that does cross-correlation, often the claim of significance is overblown. Here’s an example of why.
The figure below shows the cross-correlation between two datasets. At a lag of -12, we find a strong negative correlation, -0.25. Upon examination, this relationship is statistically significant at the 95% level (p-value less than 0.05), so I say “Whoa … big news, we’ve found a significant correlation”.
But how can that be, this is a pair of random datasets?
The answer lies in the fact that to find that “significant” correlation, the cross-correlation does an automated data dredge through no less than 33 separate comparisons. And as a result, the significance level we now need to achieve goes through the roof. If our significance level for one comparison is 95% (p-value of 0.05), the usual in climate change, then if we are examining 33 separate realizations, to be considered significant we need to find a relationship with a p-value of .0015 … which means, of course, that the relationship that I found in the graph above is NOT significant, it’s just random chance at work.
Now, it’s not quite that bad. I consider it this way. I’m starting my search at lag 0. I only consider lags on the logical side (effect lags cause). And I stop when I find a significant result. So for example, if I find a significant relationship after I’ve looked at lags of 0, 1, 2, and 3 time units, then I’ve examined four series. Accordingly, to achieve the equivalent of a 95% significance level (climate science standard) my adjusted signficance level is X^4 = 0.95, which solves to x = 98.7% or a p-value of 0.013. The series at lag 3 needs to have a p-value of 0.013 or better to be considered statistically significant.
The net result of all this is that as I look at longer and longer lags, the required p-value gets harder and harder to achieve.
Now, that doesn’t mean I don’t use cross correlations. I do. But I also know from experience that if I don’t find a strong relationship within a fairly short lag period, significance gets hard to achieve pretty fast.
And of course, sometimes the relationship is quite strong, and so none of this is an issue. In the world of sunspots and sea level, however, this is not common …
Finally, regarding the Holgate data shown in Fig. 1 above, the strongest relationship is at lag 0, and it isn’t statistically significant … nor are the lagged relationships.
Best regards,
w.
lsvalgaard says:
January 26, 2014 at 10:30 am
……….
I certainly would be interested in your updated observations. I did quick superficial ‘survey’ of the annual Ap index ( cycles 10-23) and found that (for cumulative value) the odd cycles have on average of 6% more than the even ones (two exceptions are SC10 and SC23)
Willis, the plot you pulled up there was a late night data processing mix-up which I corrected in another post a few minutes later.
A better labelled version of the latter graph is here:
http://climategrog.wordpress.com/?attachment_id=759
Both of those were power spectra. The correlation plot with the signif estimations is here ( I’ve already posted this too but here again for clarity ).
Now we’re using different sunspot data but that SSN is not substantially different from sunspot area in general form. It should not make a major difference, so I’m a little confused that your x-correl looks to be about pi.2 out of phase with mine. It looks like one of us has made slip up on d/dt, or something.
The other thing I don’t understand in what you show is “R^2” goes negative. Usually squares are positive, I thought R^2 went from 0..1 , but I could be wrong. If not , why do you discount the negative values which look more significant than the +ve ones. Neg. correl is just as important as +ve one.
Also typo “R^2 = 0.005” has an extra zero.
Finally 0.10 is presumably your signif estimation. As lag increases the dataset gets shorter and the required correl coeff for significance should rise. Can you run your plot from -60…+60 to see whether you get the same long term modulation I found?
We should be getting substantially the same thing, or at least have a good explanation why not.
richardscourtney,
The moving of phase is an illusion that is the result of echo interference (to over-simplify).
Imagine a signal
y(t) = Cos[w t]
Now imagine there is an signal an ‘echo’ signal
y(t) = 0.5 Cos[w (t-100)];
An echo could be at any delay and have any relative phase. This will interfere with our signal to move the phase of the output that we can see (even possibly moving it forward).
Now this is an exaggerated example of course and I’m just using it to illustrate how past signals can interfere. The reason that linearity is important is that echoes in linear systems will have a constant phase shift. In chaos we can not assume that. Amplitude, phase, and shape of the echo may be different every time.
Unfortunately correlation is all we have! When all you have is a hammer, every problem looks like a nail. We should still use it for chaotic systems (we don’t have a choice), but we should also be aware that things are harder to ‘see’ here. Correlation metrics may be lower than they would otherwise normally be.
vukcevic says:
January 26, 2014 at 11:58 am
I certainly would be interested in your updated observations. I did quick superficial ‘survey’ of the annual Ap index ( cycles 10-23) and found that (for cumulative value) the odd cycles have on average of 6% more than the even ones (two exceptions are SC10 and SC23)
The cycles should be counted from polar field reversal to polar field reversal which is roughly from max to max, not as normal cycles from min to min.
Willis,
Thanks for following up demonstrating a cross-correlation for the Jevrejeva data versus sunspots. You didn’t even have to bother with the random counter-example, but for completeness, sure. In signal processing if their are no spikes significantly above the background level – there is nothing there.
Now sorry if I’m confused, but is this the same data as in the graph above in your original post? I see different Scientists’ names and very different R^2 values being thrown around. I’ll assume it is, but just double checking.
OK, I’ve figured out your R^2=0.005 is from corr-coeff of -0.07 , fine.
BTW my comment was posted in between your two above and thus related to the first one January 26, 2014 at 10:52 am
Still don’t get the differences between our two graphs.
Ian Schumacher:
Thankyou for your explanation for me at January 26, 2014 at 12:15 pm.
Allow me to see that I have understood your mathematical illustration.
1.
The effect is periodic because its cause is periodic.
2.
Thus, the cause and the effect each consists of a series of peaks.
3.
A peak of the cause may have an ‘echo’ which is an additional small peak.
4.
The ‘echo’ induces an additional but small peak in the effect.
5.
The additional but small peak in the effect occurs shortly before the next periodic peak of the effect.
6.
Signals are additive, so the additional but small peak in the effect adds to the start of the next periodic peak of the effect.
7.
Thus, the next peak in the effect is distorted by the ‘echo’.
8.
The distorted peak in the effect occurs before the maximum of the next periodic peak in the cause.
9.
Thus the distorted peak seems to occur before its cause.
have I understood you correctly, please?
The reason I ask is that I have played around with your illustration and I fail to obtain an effect similar to cycle18 in the above graph. In that case a peak in SLR occurs before the solar peak but it is accompanied by a very small SLR peak coincident with the solar peak.
I appreciate that your maths were only an illustration and reality would differ. But I do not see how to amend the equations such that introduction of an ‘echo’ can both add to the effect before its next peak and also subtract from that next peak.
I am assuming you have been doing this stuff so would be grateful if you could explain this for me.
Richard
richardscourtney says:
I hope I understand your question properly. Here is a concrete I came up with. In Mathematica you can use
Plot[{Cos[t], Cos[t] + .5 Cos[t + .9]*Exp[-(t – 5)*(t – 5)]}, {t, 0, 14}]
There is an echo here that comes in ands fades away again.
The resulting plot looks like this:
http://imgur.com/Y23geir
The blue line is the unaltered signal. The purple line is the signal plus echo. [Weird that Mathematica chooses colors so close together].
Now I get an r value of 0.55 even with simple detrending for secular trend removal, consistent with Shaviv 2008, considering only data for the years 1919.5 – 2002.5.
Ian Schumacher:
I apologise if I am seeming obtuse. It is not deliberate. I am genuinely trying to get to the bottom of this in my understanding.
Thankyou for your fine illustration at January 26, 2014 at 1:37 pm .
Yes, your link does show what I found (several ways) as my points 1 to 9 state in my post at January 26, 2014 at 1:19 pm. As I there said
So, yes, you did answer my first question, and I apologise if I was not clear in saying that.
Thankyou for what you have provided. Clearly, I put you to additional and pointless work by failing to be clear and thanking you for what you have provided. Sorry.
But my playing around with the waveform additions posed another problem for me. And I asked a new question; viz.
The inherent assumption in my question is that the effect seen in ‘cycle 18’ is caused by ‘echoes’. This, of course, may not be true because SLR variation may be induced by several factors. However, if that assumption is not valid then any assessments which assume effect of ‘echoes’ are all also not valid for the same reason.
I genuinely appreciate your efforts so far and, therefore, I know my continued questions are an imposition. But I take the liberty of pressing the matter because, as I said,
Thanking you in anticipation
Richard
Greg Goodman says:
January 26, 2014 at 11:59 am
My plot is not showing R^2, it’s showing correlation, so it goes negative. I just noted the R^2 for the maximum correlation, which indeed is 0.005.
To avoid endless discussion, let me post the sunspot data, the jevrejeva data, and the R code I used. Hang on … OK, here’s the complete code to make the graph:
thespots=ts(read.csv("SIDC Sunspots.csv")[,2],start=1700.5,frequency=1) thesea=ts(read.csv("jevrejeva sea level.csv")[,2],start=1700.5,frequency=1) ccf(diff(thesea),thespots, main="Cross-Correlation, Sunspots and ∆ Jevrejeva Sea Level", ylab="Correlation",col="salmon",lwd=3)It pulls the data from the two csv files, “SIDC Sunspots.csv“, and “jevrejeva sea level.csv“, which need to be in the current workspace. The sunspot data is modified per Leif Svalgaard (and the upcoming SIDC revision) by increasing all pre-1947 values by 20% to account for the change in counting methods. Jevrejeva data is from KNMI.
Let me know what you find … code as used and data as used, the simplest way to answer all such questions. English is generally inadequate for resolving this kind of question.
w.
One more thing, Greg. I suspect the best way to see if the sunspot vs. Jevrejeva results are significant would be to do a monte carlo analysis against red noise with the same characteristics as the Jevrejeva data. The underlying problem is the strongly repetitive nature of the sunspot data. It’s not like comparing two datasets that are both non-cyclical.
Anyhow, I may get to that today, I’ll post up the results.
w.
@ur momisugly richardscourtney
Sunspot 11 and 60+ years climate cycles among many others, are (in my view) forms of forced oscillations where the natural frequency (Fo) may be different from the forcing (driving) frequency Ff.
Under certain circumstances depending on the relationship of two frequencies (Ff = or > or < Fo) and when the driver isn’t very stable or contains number of higher harmonics i.e a pulse, the responding oscillator may occasionally (for a cycle or two) move in advance of the forcing one.
Greg, the question of a monte carlo analysis of the cross-correlation wouldn’t let me rest … it turns out that the cross-correlation between sunspots and the Jevrejeva data is nothing but chance. Here’s a typical run from a monte carlo analysis, many of them look like this:
For comparison, here’s the real cross-correlation:
I’m sure you can see the problem … as I said, the difficulty is that one of the two datasets is strongly cyclical, so it will generate cyclical cross-correlations against random data.
w.
[UPDATED TO ADD] The code, the code. I first measured the AR and MA components of the Jevrejeva data, and generated random ARIMA datasets with those values. Then I differentiated those proxy datasets to give me the delta values.
Then I simply measured the CCF of the sunspot data versus the random data. This should be turnkey, uses the same data as above:
thespots=ts(read.csv("SIDC Sunspots.csv")[,2],start=1700.5,frequency=1) thesea=ts(read.csv("jevrejeva sea level.csv")[,2],start=1700.5,frequency=1) detrend=function(x){ xlm=lm(x~c(0:(length(x)-1))) xlm$fitted.values=c(xlm$fitted.values,rep(NA,length(x)-length(xlm$fitted.values))) (x-xlm$fitted.values) } # get the arima variables seaarima=arima(detrend(as.vector(thesea)),c(1,0,1)) testsize=10 # make proxy data proxydata=arima.sim(list(ar= seaarima$coef[1],ma= seaarima$coef[2]),length(thespots)*testsize) # convert it to time series testbox=ts(matrix(proxydata,nrow=314,ncol=314*testsize),start=1700.5,frequency=1) # get the annual differences testdiff=diff(testbox) # cross-correlation ccf(testdiff[,3],thespots, main="Typical Cross-Correlation, Sunspots and Random Data", ylab="Correlation",col="salmon",lwd=3)</pre.vukcevic:
Thankyou for your post at January 26, 2014 at 2:38 pm.
Yes, I thought of that some time ago but I was concerned that stating the possibility would imply I was ‘playing games’ with Ian Schumacher, and I am not.
Basically, it is possible that the SLR oscillates naturally for some reason at near to the frequency of the solar cycle, and the possible small solar influence causes the SLR variation to adjust to get into synchronicity with it.
If so,
(a) the solar cycle is causal of the synchronicity of the solar cycle and SLR
but
(b) the solar cycle is NOT causal of the SLR fluctuation.
It is a nice idea because it overcomes the problem of coherence: the synochronicity would not be precise and would drift around the frequency of the solar cycle (coupled pendulums can do this).
Please note that I am NOT suggesting this is true. I am recognising a possible relationship between the two parameters which would overcome the coherence problem. And it overcomes the ‘cycle 18’ problem because the cause of the SLR fluctuation is something other than the solar cycle.
And that is why I have not mentioned it in my discussion with Ian Schumacher: it rejects a causal effect of the solar cycle on the SLR fluctuation. Induced synchronicity between the variations of the two parameters is very different from one parameter causing the other. It rejects that the fluctuation of the SLR is caused by the solar cycle and assumes the SLR fluctuation is caused by something else.
I hope this makes clear that I am not rejecting your suggestion, but I would like to ‘get to the bottom’ of the ‘echo’ suggestion.
Richard
richardscourtney,
Is it the small coincident peak combined with the delayed peak that’s bothering you? The small peak may just be noise or something else. In fact all of it could be noise. Who knows 🙂
My example shows how an echo could make a peak occur ‘before’ the signal peak. The purple line occurs ‘before’ the cause (the blue line). This is an illusion of shifted causality due to the periodic nature of the signals, it’s not a real shift in causality. I’m not sure if I’m able to explain it better.
In addition to that, three things:
– There is background noise. We are not looking at the ‘pure’ result. It is obscured by random ups and downs. A ‘peak’ may look to be shifted merely because random noise added and took away from the signal at the right time. A small false peak may just be noise and nothing more.
– There are other external signals at work here. PDO, or whatever it’s call. Volcanic eruptions, hurricanes, tsunamis, etc I don’t know which if any of these is important as I’m not a climate expert, but presumably they all could be. Overlapped at the right time it could cause peaks to occur and obscure our signal anywhere they occur, or their echo occurs.
– There may actually be no real correlation and I have conceded that. It’s an interesting overlap, but it may just be coincidental.
Not sure if that helps or not, but I don’t think I can add much more of value.
Ian Schumacher says:
January 26, 2014 at 12:15 pm
Interesting … and noted. However, see below.
The problem I’m having is in understanding how we can have an echo of say sunspot data in the climate system. The thing about an echo is that it is a reflection of something. With say an electrical signal or a sound signal, that makes perfect sense. The sound goes out … the sound comes back. All it takes is a signal, and something for it to be reflected from …
But what does a “sunspot echo” look like or consist of? And what is it bouncing off of? Most people, myself included, see the only possible means of the sunspot cycle affecting the planet as being electromagnetic. You know, via cosmic rays, or even directly via the heliomagnetic field.
But if that is the case, what would the “echo” of that magnetic field look like? And what would reflect it? In particular, one characteristic of an echo is that whatever it is echoing off of needs to be physically removed from the location where the echo is measured. An electric signal might bounce off of an impedance mismatch further down the wire. A sound echoes off of the distant cliff or the far side of the auditorium.
But what would a change in magnetic field echo off of?
This is why I have been pressing people for some other real-world example of what people think they see in figure 1, some process that is out of phase, switches to in-phase, stays that way for decades, switches back again, and where cause sometimes leads and sometimes follows effect …
Yes, as good old Joe Fourier showed, you are right—name a signal, we can gin up a combination of cosines that matches it exactly. Or we can go old-school, model it with delays and transformations and echoes of a couple of signals, phase-shift it forwards and back, anything we want.
But in the real world, only a very small subset of those infinite possibilities actually occurs. As a result, although it is possible to come up with waveforms that lead to what we see in Figure 1, in the real world that kind of thing is very rare. I look, day after day, at natural waveforms of all kinds. I see them inside my eyelids in bed at night. That kind of frequency flip-flop is very unusual.
Can’t say fairer than that, I agree entirely.
w.
Ian Schumacher:
You end your post to me at January 26, 2014 at 3:13 pm saying
You have helped a lot and provided much of value.
Many thanks.
Richard