This is a quick rough and ready analysis of an QBO data since Paul Vaughan asked. Here are all the things you never wanted to know.
QBO ( Quasi-Biennial Oscillation) is a strange entity, wind direction alternates. The data is an index, artificial computation and in this case part of a reanalysis. This article might add some insight http://www.geo.fu-berlin.de/en/met/ag/strat/produkte/qbo
A problem I see with simple data such as at one pressure level and location is the many other changes over time. Perhaps the pressure level should vary a little. It’s a multi-dimension entity anyway.
I’m taking it as-is with no more comment.
Figure 1, straight plot of monthly data, which runs from 1948. No units are given so assume metres/second. Data used
The author has several bespoke software works useful for analysis, in C, unpublished so reproducing this work would be difficult.
On considering the data the period before 1970 is ignored. There is no information on technology changes or data reliability.
Paul Vaughan’s posit involves lunar modulation. If I take the largest 2.3 year term and find the amplitude modulation that gives 15.6 years which is not lunar. Not shown here.
The periodic terms circa 2.3 years are a triplet. Here is the evidence.
Figure 2. Evidence of a triplet.
Given the data length only vague Fourier spectra can be done but the author has an innovative method of analysis as shown in figure 2.
Blue trace is the 1970 onwards data, which shows a hump with signs of more than one item present. The software finds an exact best match in a least squares sense. (RMSE). Red trace is the residual after three terms have been subtracted from the data. Yellow trace is spectra of what was subtracted.
Windowing is necessary with a Fourier transform, the Kaiser-Bessel beta=3 windows used here is in the author’s opinion the best compromise which shows best detail of closely related periods, but rejection of artefacts is necessarily moderate.
The software outputs various files. Here is a plot of the above.
Figure 3, overlay of three term result and input data, hindcast has been allowed but is very unlikely to be correct.
In this case it seems reasonable to try and produce an analytic signal, a strange but important facility used in signal processing. The authors software has never been fully developed out for fringe use so this is pushing things given the awkward nature of climatic data.
The first move is bandpass filter the QBO data because high and low frequencies make a mess of things. This was done with diagnostics implying a good wanted signal from 1.8 years through 3.2 years.
What is an analytic signal? It is the signal and the 90 degree phase shifted version of the signal, with a twist, the 90 degrees applies to all frequencies. Mind bending things are close. This is usually produced via an Hilbert Transform[ation].
Now, if you happen to have a signal and it’s quadrature, another name for a 90 degree phase shifted version there are magical mathematics available.
First lets prove it worked. Best note the ends of the data go bad, I’ll plot them anyway.
Figure 5, plain plot of quadrature signals
Figure 6. Another way of looking at it. Should be circular.
Trick mathematics? Know that Pythagoras chap? Simple sine and cosine allow direct computation of the envelope, a continuous amplitude signal, nothing else needed, a direct method.
Figure 7, The envelope. As I mentioned earlier ignore the data ends.
Another family but more tricky is instantaneous frequency, phase and delay, all requiring phase unwrapping of the data, an indeterminate problem, very hard to do. This is why I bandpass filtered, reduces the problem. Did my code work?
Figure 8, Instantaneous frequency (actually period, bespoke software inverts frequency)
What happened 2001? Probably gone through a singularity.
Figure 9, phase, unwrapping was successful. (you’d see abrupt discontinuities)
Figure 10, the least interesting plot, a metric, usually a straight line
Finally answer a question
period phase amplitude
2.3396747739 0.7220279978 12.1855647167
2.1996734721 3.6371639858 6.624010329
2.5320954646 2.3449830156 5.6962092807
Phase reference January 1948
What to make of that lot?
Not really anything to add, no insight. If it means anything you tell me.
Post by Tim