TANSTAAFL: there ain’t no such thing as a free lunch … but there’s always crumbs for the taking.

Machine learning won’t necessarily make a complete discovery by uncovering some ground-breaking pattern in isolation, but more likely a fragment or clue or signature that could lead somewhere. I always remind myself that there are infinitely many more non-linear formulations than linear ones potentially lurking in nature, yet humans are poorly-equipped to solve most non-linear relationships. ML has started to look at the tip of the non-linear iceberg and humans have to be alert when it uncovers a crumb. Recall that pattern recognition and signal processing are well-established disciplines in their own right, yet consider the situation of searching for patterns in signals hiding in the data but unknown in structure. That’s often all we are looking for — some foot-hold to start from.

The data in the set of QBO time-series offer many signatures to analyze in detail. Many are well known to people who do signal processing for a living — for example, looking at side-band peaks in the data, consideration of aliasing, and detecting long-range correlations. In Mathematical Geoenergy, Chapter 11, the general model of a tidally-forced QBO aliased response was outlined. Satellite sub-bands in the Fourier spectra should be at the locations specified in the Chap. 11 table:

Many of these are clearly evident in the spectra (there are valid reasons that not all will appear simultaneously¹). To see these, consider the satellite sub-band peaks grouped in (Brillouin) zones according to the annual harmonics 0/yr (completely aliased wavenumber=0), 1/yr (3rd vertical line below), 2/yr (6th vertical line), etc.

For QBO 30 hPa, can see a clear factor at 0.42/year (1st line) and 2.42/year (7th line) frequency and weaker at 1.42 (4th line).

For QBO 40 the clear factors are at 0.42/year (1st) and 1.42 (4th)

For QBO 50 the clear factors are at 0.42 (1st), 1.42 (4th), and 2.42/yr (7th).

The model fir QBO 50 spectra will pick up these factors as well. It also picks up the (-) sub-bands at 0.58/yr (2nd line) and 1.58/yr (5th line)

Can machine learning pick these sub-bands out? Certainly. Several years ago, when the symbolic regression tool Eureqa was available for use, it picked out radian frequencies of 15.18, 11.93, 3.198+5.857, 5.857-3.198 in the 2^nd derivative of the QBO 30 time-series (the 2^nd derivative will enhance higher frequencies).

Converting these 4 identified radian frequencies into cyclic frequencies gives 2.416, 1.899, 1.441, and 0.4232. Ignore the weakest at 1.899, and each of them ostensibly matches the *.423 pattern of aliased draconic frequencies expected from the QBO model of Table 11.1 in Mathematical GeoEnergy. This would certainly classify as a ML crumb revealed in the data. Yet, this pattern is meaningless apart from providing a fitting heuristic, unless a model is associated with it. That’s where the human reasoning comes in.

Yet, there is likely more to the story than the single draconic frequency and its aliased frequencies in the QBO time-series. To 2^nd-order, the draconic cycle is modulated by distance (lunar perigee/apogee) and by lunisolar alignment (lunar nodal cycle) as described in https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html#draconic . This is modeled by either a frequency or phase modulation. If the original sinusoidal waveform is: and the instantaneous frequency (derivative of the phase) is:

The NASA lunar site has an estimate of this instantaneous frequency on-line, which can be fit to using the cyclic factors 177 day (nodal), 206 day (perigee), and 6 year (mixed nodal/perigee).

This generates the above fit in red, which was trained with slight variations on the QBO 30, 40, & 60 time series for the time interval 1956 to 1973. The rationale for this approach is that both the fit to the training interval and the excluded test/validation interval would improve, which certainly checks out. The cross-validation for QBO 40 is shown below. where everything outside the yellow region was excluded from training:

The starred () times indicate intervals where both the alignment and shape of the QBO model match to the underlying data, which indicates that a long-range correlation and time stationarity of the process is maintained,

Certain QBO time intervals have been declared anomalous based on differing or unique characteristics observed including 2015/2016 and 2019/2020.² These are shown as question (?) marks, with noticeable phase and shape discrepancies in comparison to that expected according to the otherwise cross-validated predictive fit.

Here is another example with a slightly different draconic modulation:

Next, try a QBO 50 hPa time-series, which shows the characteristic time shift of a QBO measure taken at a different altitude. The anomalies at 2015 and 2019 still standout.

This is a longer training interval which also features a slight reduction in impulse time event every year. Based on the model structure, the time of the impulse event is the most sensitive selector of the phase alignment, and it’s plausible that the impulse has been occurring earlier each year, while the role of the Hunga Tonga volcanic in 2022 is a wild card.

Once again, the modulation of the draconic follows that expected with a 6-hour variation corresponding to approximately 1 rad on the 27.2122-day cycle of the draconic (see the sinusoid below, visualized essentially as a time-warped variation of a draconic sine-wave). This is slightly less than expected, yet the agreements should all be considered in the ballpark considering the conciseness with respect to degrees of freedom in the formulation. As always, it is tempting to overfit the model to data.

Which brings up the possibility of applying the Laplace’s tidal equation (LTE) modulation as described in Chapters 11 & 12. Although not as expected on a wavenumber=0 behavior such as QBO, more of that wave-breaking behavior can be expected at lower-altitudes/higher-density regimes. In particular, note the stronger modulation modeled in the negative excursions as shown below.

Finally, here is the QBO 30 hPa modeling analysis

(Click on any images to expand)

References

1. For example, the spectral shape of annual plateau and the sinc function (sin(x)/x) it creates may supply a zeroing node that cancels at that specific frequency. Or based on the pulse amplitude strength, such as balance between the 2 paired semi-annual pulses, such as with the Chandler wobble, see https://geoenergymath.com/2021/01/07/chandler-wobble-forcing/ ↩︎
2. Wang, Yue, et al. “A revisit and comparison of the quasi-biennial oscillation (QBO) disruption events in 2015/16 and 2019/20.” Atmospheric Research 294 (2023): 106970. ↩︎