Guest Post by Willis Eschenbach
In my last post on the purported existence of the elusive ~60-year cycle in sea levels as claimed in the recent paper “Is there a 60-year oscillation in global mean sea level?”, I used a tool called “periodicity analysis” (discussed here) to investigate cycles in the sea level. However, some people said I wasn’t using the right tool for the job. And since I didn’t find the elusive 60-year cycle, I figured they might be right about periodicity analysis. In the process, however I found a more sensitive tool, which is to just fit a sine wave to the tidal data at each cycle length and measure the peak-to-peak amplitude of the best-fit sine wave. I call this procedure “sinusoidal periodicity”, for a simple reason—I’m a self-taught mathematician, so I don’t know the right name for the procedure. I’m sure this analysis method is known, but since I made it up I don’t know what it’s actually called.
I like to start with a look at the rawest view of the data. In this case, here’s the long-term Stockholm tide gauge record itself, before any further analysis. This is the longest complete monthly tidal gauge record I know of, at 200 years.
Figure 1. Stockholm monthly average sea level. This is a relative sea level, measured against an arbitrary zero point.
As you can see, Stockholm is (geologically speaking) rapidly leaping upwards after the removal of the huge burden of ice and glaciers about 12,000 years ago. As a result, the relative sea level (ocean relative to the land) has been falling steadily for the last 200 years, at a surprisingly stable rate of about 4 mm per year.
In any case, here’s what the sinusoidal periodicity analysis looks like for the Stockholm tide data, both with and without the annual cycle:
Figure 1a. “Sinusoidal Periodicity” of the Stockholm tide gauge data, showing the peak-to-peak amplitude (in millimetres) of the best-fit sine wave at each period length. Upper panel shows the data including the annual variations. In all cases, the underlying dataset is linearly detrended before sinusoidal periodicity analysis. Note the different scales of the two panels.
Now, I could get fond of this kind of sinusoidal analysis. To begin with, it shares one advantage of periodicity analysis, which is that the result is linear in period, rather than linear with frequency as is the case with Fourier transforms and spectral analysis. This means that from monthly data you get results in monthly increments of cycle length. Next, it outperforms periodicity analysis in respect of the removal of the short-period signals. As you can see above, unlike with periodicity analysis, removing the annual signal does not affect the results for the longer-term cycles. The longer cycles are totally unchanged by the removal of the annual cycle. Finally, I very much like the fact that the results are in the same units as the input data, which in this case is millimetres. I can intuitively get a sense of a 150-mm (6 inch) annual swing in the Stockholm sea level as shown above, or a 40 mm (1.5 inch) swing at both ~5.5 and ~31 years.
Let me start with a few comments on the Stockholm results above. The first one is that there is no significant power in the ~ 11-year period of the sunspot cycle, or the 22-year Hale solar cycle, as many people have claimed. There is a small peak at 21 years, but it is weak. After removal of the annual cycle, the next strongest cycles peak at ~5.5, 31.75, and 15 years.
Next, there are clearly cycle lengths which have very little power, such as 19.5, 26.5, and 35 years.
Finally, in this record I don’t see much sign of the proverbial ~60 cycle. In this record, at least, there isn’t much power in any of the longer cycles.
My tentative conclusion from the sinusoidal analysis of the Stockholm tide record is that we are looking at the resonant frequencies (and non-resonant frequencies) of the horizontal movement of the ocean within its surrounding basin.
So let me go through all of the datasets that are 120 years long or longer, using this tool, to see what we find.
So lets move on to the other 22 long-term tidal datasets that I linked to in my last post. I chose 120 years because I’m forced to use shorter datasets than I like. Normally, I wouldn’t consider results from a period less than three times the length of the cycle in question to be significant. However, there’s very few datasets that long, so the next step down is to require at least 120 years of data to look for a 60-year cycle. Less than that and you’re just fooling yourself. So without further ado, here are the strengths of the sinusoidal cycles for the first eight of the 22 datasets …
Figure 2. Sinusoidal amplitude, first eight of the 22 long-term (>120 year) datasets in the PSMSL database. Note that the units are different in different panels.
The first thing that strikes me about these results? The incredible variety. A few examples. Brest has lots of power in the longer-term cycles, with a clear peak at ~65 years. Wismar 2, on the other hand, has very little power in the long-term cycles, but a clear cycle at ~ 28 years. San Francisco has a 55-year peak, but the strongest peak there is at 13 years. In New York, on the other hand, the ~51 year peak is the strongest cycle after the annual cycle. Cuxhaven 2 has a low spot between 55 and 65 years, as does Warnemunde 2, which goes to zero at about 56 years … go figure.
Confused yet? Here’s another eight …
Figure 3. Sinusoidal periodicity, second eight of the 22 long-term (>120 year) datasets in the PSMSL database. Note that the units are different in different panels.
Again the unifying theme is the lack of a unifying theme. Vlissingen and Ijmuiden bottom out around 50 years. Helsinki has almost no power in the longer cycles, but the shorter cycles are up to 60 mm in amplitude.. Vlissingen is the reverse. The shorter cycles are down around 15-20 mm, and the longer cycles are up to 60 mm in amplitude. And so on … here’s the final group of six:
Figure 4. Sinusoidal periodicity, final six of the 22 long-term (>120 year) datasets in the PSMSL database. Note that the units are different in different panels.
Still loads of differences. As I noted in my previous post, the only one of the datasets that showed a clear peak at ~55-years was Poti, and I find the same here. Marseilles, on the other hand, has power in the longer term, but without a clear peak. And the other four all bottom out somewhere between 50 and 70 years, no joy there.
In short, although I do think this method of analysis gives a better view, I still cannot find the elusive 60-year cycle. Here’s an overview of all 22 of the datasets, you tell me what you see:
Figure 5. Sinusoidal periodicity, all twenty-two of the long-term tide gauge datasets.
Now, I got started on this quest because of the statement in Abstract of the underlying study, viz:
We find that there is a significant oscillation with a period around 60-years in the majority of the tide gauges examined during the 20th Century …
(As an aside, waffle-words like “a period around 60-years” drive me spare. The period that they actually tested for was 55-years … so why not state that in the abstract? Whenever one of these good cycle-folk says “a period around” I know they are investigating the upper end of the stress-strain curve of veracity … but I digress.)
So they claim a 55-year cycle in “the majority of the tide gauges” … sorry, I’m still not seeing it. The Poti record in violet in Figure 5 is about the only tide gauge to show a significant 55-year peak.
On average (black line), for these tide gauge records, the strongest cycle is 6 years 4 months. There is another peak at 18 years 1 month. All of them have low spots at 12-14 years and at 24 years … and other than that, they have very little in common. In particular, there seems to be no common cycles longer than about thirty years or so.
So once again, I have to throw this out as an opportunity for those of you who think the authors were right and who believe that there IS a 55-year cycle “in the majority of the tide gauges”. Here’s your chance to prove me wrong, that’s the game of science. Note again that I’m not saying there is no 55-year signal in the tide data. I’m saying I’ve looked for it in a couple of different ways now, and gotten the same negative result.
I threw out this same opportunity in my last post on the subject … to date, nobody has shown such a cycle exists in the tide data. Oh, there are the usual number of people who also can’t find the signal, but who insist on telling me how smart they are and how stupid I am for not finding it. Despite that, so far, nobody has demonstrated the 55-year signal exists in a majority of the tide gauges.
So please, folks. Yes, I’m a self-taught scientist. And yes, I’ve never taken a class in signal analysis. I’ve only taken two college science classes in my life, Introductory Physics 101 and Introductory Chemistry 101. I freely admit I have little formal education.
But if you can’t find the 55-year signal either, then please don’t bother telling me how smart you are or listing all the mistakes you think I’m making. If you’re so smart, find the signal first. Then you can explain to me where I went wrong.
What’s next for me? Calculating the 95% CIs for the sinusoidal periodicity, including autocorrelation. And finding a way to calculate it faster, as usual optimization is slow, double optimization (phase and amplitude) is slower, and each analysis requires about a thousand such optimizations. It takes about 20 seconds on my machine, doable, but I’d like some faster method.
Best regards to each of you,
w.
As Always: Please quote the exact words that you disagree with, it avoids endless misunderstandings.
Also: Claims without substantiation get little traction here. Please provide links, citations, locations, observations and the like, it’s science after all. I’m tired of people popping up all breathless to tell us about something they read somewhere about what happened some unknown amount of time ago in some unspecified location … links and facts are your friend.
Data: All PSMSL stations in one large Excel file, All Tide Data.xlsx
Just the 22 longest stations as shown in Figs. 2-4 as a CSV text file, Tide Data 22 Longest.csv .
Stockholm data as an excel worksheet, eckman_2003_stockholm.xls
Code: The function I wrote to do the analysis is called “sinepower”, available here. If that link doesn’t work for you, try here. The function doesn’t call any external functions or packages … but it’s slow. There’s a worked example at the end of the file, after the function definition, that imports the 22-station CSV file. Suggestions welcome.
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Willis Eschenbach says:
“No, because it is linear in period where the FFT is linear in frequency.”
There’s no issue of linearity here. You are doing each optimisation at a fixed frequency (or period).
Here’s the math. You’re minimising
I = ∫ (f(t) – a*cos(ωt+b))² dt
over a and b, so dT/da=0
∫ (f(t) – a*cos(ωt+b)) cos(ωt+b) dt = 0
or a = 1/N ∫ f(t) cos(ωt+b) dt
where N is the length of your finite integration interval. Or expanding the cos:
a = 1/N( cos(b) ∫ f(t) cos(ωt) dt – sin(b)∫ f(t) sin(ωt) dt)
ie linear combination of cos and sin transform. When you optimise over b, you just have 1/N times the magnitude of the complex Fourier Transform. There’s a fuss about finite range, and that does lead to problems at low frequency. That’s inevitable – it follows from the finite range (200 years).
Greg says:
May 1, 2014 at 10:36 pm
Greg, let me suggest you re-read the head post, particularly the part where I said:
Is there some part of that which is unclear? I don’t care in the slightest about your brilliant ideas about the “fundmental point” that you think I’m missing. Come back when you can find the missing signal, and I’ll pay attention. Until then … not so much.
w.
PS—What on earth does this mean?
What “steady rise” are you talking about when all of the datasets are first detrended?
Ray Tomes says:
May 1, 2014 at 10:42 pm
Thanks, Ray. I note that your method finds about the same cycles as my method … including the fact that you do not find a 60-year cycle either.
w.
Greg says:
May 1, 2014 at 10:48 pm
Um … er … was my writing not clear? I said in the head post:
Please, please, please first read what you are criticizing, and only after you do that, then get all critical. Your repeated attempts to do it in the opposite order aren’t doing your reputation any favors.
w.
Willis
there is document at DEFRA with lots of links (some which I can’t get to work) to UK resources which you might find useful (if you’re not aware of them already)
http://chartingprogress.defra.gov.uk/feeder/Section_3.5_Sea_Level.pdf
One link leads here
http://www.ntslf.org/
Willis what was your point anyway. Cheerio.
I know you don´t love him… but 60 years?
https://www.youtube.com/watch?v=6R26PXRrgds#t=48
Ground is dropping up to 10 times faster than the sea level is rising in coastal megacities, a new study says: http://www.dailymail.co.uk/sciencetech/article-2616714/Forget-global-warming-groundwater-extraction-causing-megacities-SINK-beneath-sea-level.html
bushbunny says:
May 2, 2014 at 12:20 am
Without a quote to indicate which of my many points you might be referring to, I fear I can’t answer that.
Regards,
w.
SØREN BUNDGAARD says:
May 2, 2014 at 12:25 am
No clue what you’re talking about, Soren … and the odds of me watching a 48-minute video of Piers Corbyn in order to find out are either zero or zero, depending on the state of the tides.
w.
Sorry , I did not see that detrending comment in the caption of the graph. Perhaps it would have been better to put it in the text of your article describing the method. The description you gave just above the graph was this:
“In any case, here’s what the sinusoidal periodicity analysis looks like for the Stockholm tide data, both with and without the annual cycle:”
You forgot to say you’d removed the trend too, so I did not pay too much attention to the graph.
What is significant is that most of the graphs have still have rising energy at longer periods, which indicates that either there is still a trend of there is significant variability >=70 years. This can cause similar problems to a trend.
The usual remedy for this is a window function or “taper”. But if I explain what that means you’ll probably just say I’m trying to be smart, so I won’t bother.
The 22 station data file has actually 20 European stations plus 2 U.S. stations, with 0 elsewhere. (That wasn’t made blatant but required looking up the names; for example, if anyone wonders where the heck Swinoujscie is, it is just in Poland). Entire continents are not represented in any way in it. The reader should keep in mind how non-representative that is of what most people are most interested in: global average history.
As an analogy, would one judge the average temperature of planet Earth by having 10 stations in Antarctica averaged with 1 station in Alaska?
So let’s start getting into some of the problems here if one wants to evaluate global climate history:
In a way, it is appropriate that this article begins with a gauge reading a sea level fall over the past 2 centuries (due to local land rise), for such implicitly highlights how that is very, very different from the global average.
To isolate a signal on the scale of mm/year, data for each station would need to be properly calibrated relative to local land rise or fall (postglacial rebound, subsidence, tectonic effects).
As analogy, not doing so would be a little like trying to measure the history of global climate (e.g. the 1/500th absolute temperature change in Kelvin which constituted global warming over the past century) by looking solely at the readings from a thermometer on some airline jet, without even compensating for variation in how much time the plane spent in different locations and at different altitudes.
Looking at a single station is not a way to judge global sea level history. If someone thought it was, the result would be entertaining if done analogously with temperature: Someone could depict global cooling over the past century or just about anything by choosing the station, as localities vary greatly. (As a further analogy, there may be some temperature proxy in some location which, if looked at in isolation, would have noise and local factors overwhelm the detection even of entire ice ages in global climate history).
What about simple averaging, such as with the 22 station data file? How much would averaging those 22 stations give a global average? Averaging all (those) stations then would only be misleading. About a third of the total are in the tiny country of the Netherlands, for example, thus grossly overweighted in such an average. Another several are in Germany. Zero of those 22 are outside the U.S. or Europe.
For this comment, I considered doing an illustration nevertheless with stations in that file, without overweighting Europe quite so much, but it’d take more time than desired, especially since the file is riddled with gaps of years of missing data in different series, which would have to be handled (better by interpolation than by treating them as 0s).
———
So what does give the actual picture better? As an example, which is not behind a paywall but as an available full-text PDF, let’s observe what is done within a paper at http://www.joelschwartz.com/pdfs/Holgate.pdf :
First of all, look at figure 1 in that link: When creating a 9 tide gauge average, the author has only 3 of 9 stations (not 20 of 22) be from Europe. He deliberately includes others spread out and far away, such as Hawaii and New Zealand. Europe is still relatively overrepresented, due to the history of modern society and to where technology like tide gauges got first installed long ago, but such isn’t as bad.
Secondly, notice that gauge data is “corrected for glacial isostatic adjustment and inverse barometer effects.”
Thirdly, notice that averaging is used to get the decadal rates, as opposed to plotting unaveraged data. As an analogy with temperature instead of sea level, if someone plotted surface temperature history in daily values without averaging, they could create a grand mess concealing the existence of about any real patterns in global climate history. (The 0.6K of global warming over the past century, a 1 in 500 part change in absolute temperature, requires averages to even be visible on a plot).
Fourthly, note the author compares the 9 station average to a 177 station average over a time when data is available from both (figure 2). As this is real world data, sometimes a peak or trough in one dataset happens before or after that in another dataset, but they are similar enough as could be seen by an unbiased observer, which is a good sign.
———
As for a 60 year cycle, that isn’t something I emphasize in climate history, let alone in sea level in particular, so mainly it isn’t something I’m here to support. As I’ve often noted, surface temperature in the 20th century followed more a double peak appearance (in original data) than a hockey stick, but such isn’t about implying peaks were always 60 years apart indefinitely further back.
However, amusingly, although few if any commenters here seem to have even noticed, earlier this very week there was a WUWT article showing a large plot of sea level rise rate history with a high near 1940, then another high towards the end of the 20th century (around 60 years later). The plot in the following is of 18-year trends, over about 2 decades at a time, so it hides the shorter oscillations in the Holgate plot, depicting a longer average:
http://wattsupwiththat.com/2014/04/28/sea-level-rise-slows-while-satellite-temperature-pause-dominates-measurement-record/
———
Note to readers: If this post is argued with, look at what is snipped and not quoted, as that can be most revealing of all.
“they are investigating the upper end of the stress-strain curve of veracity”
I’m sorry Willis, I’m pinching this…
Willis, can you tell us how “monthly” tide gauge readings are calculated? Calendar months, with their variable length, 30.44 days on average, complicated by leap years, are not the best time units for tidal data. True periods should be related to the synodic month (29.18 – 29.93 days, 29.53 days on average) and tropical year (12.368 synodic months or 365.2422 days). Also, at some locations, like Stockholm, max tidal range is small (~40 mm) compared to haphazard effects of wind driven swells (~500 mm).
Greg says:
May 2, 2014 at 12:49 am
Riiiight. You didn’t read all of what I wrote, and somehow your inattention is all my fault because I forgot to write it correctly …
Do you realize how foolish you look when you do that? You screwed up, and as a result you made wildly incorrect claims. Now me, if I did that I’d apologize and move on. Hey, it happens. When you’re wrong, say so, it’s the painful but honest way out, and I’ve been forced to take it more than I’ve wished. It happens.
But instead of apologizing for your incorrect assertions and getting on with life, you want to convince everyone that you not reading all the parts of a scientific work before lecturing the author is somehow my fault? What, like I control where your eyes wander?
Sorry, Greg, but that pig won’t fly …
w.
@kasuha
I looked at the wavelet site you listed, and it’s not done by someone who can write about mathematics. They are clearly regurgitating some other text. See formula 2.1, they use a complex exponential and then plot the result in a 2D space. Not possible! e^ix = cos(x) + i*sin(x). They clearly got this formula from some other place. The use of complex exponentials is required for electrodynamics and quantum mechanics where there really are two components to the field being represented. However, it’s unjustifiable for representing temperature, as they propose to do. Their introduction to their terms is non-existent. This sort of poor understanding of a field is everywhere in mathematics. So, I don’t recommend Willis read this site. Any others?
Henry Clark says:
May 2, 2014 at 12:56 am
Look, Henry, if you want to “evaluate global climate history”, that’s a fine thing … but this is not the place for it.
Here, I’m asking for assistance in evaluating a claim from a scientific study which said that “there is a significant oscillation with a period around 60-years in the majority of the tide gauges examined during the 20th Century”. To investigate that claim, I simply chose all the tide stations I could find with a record at least twice as long as the 60-year cycle I’m looking for. Records 120 years long are not really long enough to reliably identify a 60-year cycle, but it’s what we’ve got.
I am NOT doing this to “evaluate global climate history”. I’m evaluating individual tide gauge records to see if there is a ~60 year cycle in a majority of the individual records as the authors claim. I am saying absolutely nothing about global climate history, or global average temperature of the planet earth, or anything global of any kind. I’m looking at the characteristics of individual tidal records.
Let me ask that you do the same, and you leave “evaluating global climate history” for some other thread. One problem at a time, please.
w.
Willis, Piers Corbyn´s explanation of a 60-year cycle starts here:
[ http://youtu.be/6R26PXRrgds?t=23m42s ]
Willis,
As an aside about the Stockholm results -and I haven’t time this morning to read all of the comments – given that the land is rising could the periodicity of peaks be influenced by planetary gravitational forces ?
Just a thought.
I sometimes wonder why I do what I do …
Regards,
because unlike some of us old farts you really do enjoy playing with maths and you do it really well even if you are self taught although I see nothing degrading (not right word but but can’t think of the one I want) in being so.
One thing that is good about this pseudo-fourier approach is that it can work on data with breaks in it.
One notable difference is that you can not take the series of fitted amplitudes to rebuild the source file. It is not a transform in the same way as FFT is.
Henry Clark says:
May 2, 2014 at 12:56 am
Thanks for that, Henry. Unfortunately, Holgate et al. have used decadal running means, which have horrendous properties, often inverting peaks. I’d take a look at the underlying data to see how badly this affected the results … but heck, Holgate et al. didn’t bother with archiving their data.
Ah, well …
w.
michaelozanne says:
May 2, 2014 at 1:01 am
Glad someone appreciates it as a description of bending the truth until it breaks …
w.
“Now me, if I did that I’d apologize and move on. Hey, it happens. When you’re wrong, say so, it’s the painful but honest way out, and I’ve been forced to take it more than I’ve wished. It happens.”
What part of the first sentence are you having trouble reading?
Sorry , I did not see that detrending comment in the caption of the graph.
The fact that you forgot to say you’d subtracted the trend when you described your method but would be a good opertunity to take your own painful way out and move on. But that’s not really the way you work is it? Despite what you like to think.
I’m not going to scrupulously read every line and caption of something that is wrong according to the description of how it was dervied. At that point I start scanning.
As a result I missed what you put in the caption and appologised. Enought.
Get back to what the data may tell us.
Willis, Piers Corbyn´s explanation of a 60-year cycle starts at 23:40…
(I do not know why the video is not started at this point – sorry)
[it doesn’t matter, Corbyn has nothing useful to say -mod]