From a careful analysis of Length-of-Day (LOD) measurements, we can glean lots of information about phenomenon such as tides, earthquakes, core/mantle motion, and ENSO [1]. In **Figure 1** below, one can see how this is manifested at different time scales.

I revisited the LOD data to see if I could detect the angular momentum changes that appear to force the QBO and ENSO oscillations, similar to what Wang et al do in [2]. In each of these natural behaviors, there is a clear indication that the 2.369 year seasonally-aliased long period draconic tide is a primary forcing mechanism.

There are two sets of data to look at. The original set from Gross at JPL called LUNAR97 [3] is not online in a machine-readable format, but it is included in the paper. So it was a simple matter to OCR a GIF of the paper's table. Another more recent set is available from IERS here.

The issue with the more recent set is that the site claims that they do a *"5-point quadratic convolute"*, at least up to 1955. I don't want any filtering, so chose the Gross LUNAR97 set. You can see the difference between the two data sets in **Figure 2** above.

To draw out the fine features even more, I performed a straightforward second-derivative on the LUNAR97 time-series (same approach as used on the QBO see result). This was then fed into a symbolic regression machine learning tool to determine the principle components. The results are shown in** Figure 3** below.

After about an hour of machine learning, the solution included a strong frequency at 22.481 rads/year. In** Figure 3**, that solution was one of the simplest along the Pareto front, shown by the red dot. This frequency is aliased to the slower frequency of 2.3694 rads/year, which is correspondingly seasonally aliased to an approximate lunar month of** 27.2121** days. Incredibly, but perhaps not surprisingly, this compares to the known draconic or nodal lunar month length of **27.21222** days. The other somewhat weaker term is very close to aliasing the anomalistic lunar month of 27.55 days, in the fortnight mode of 13.777 days.

That was the first run I attempted and subsequent runs are picking out some other potential long-period lunar periods (and also the 6.45 year Chandler wobble), but this is really enough to demonstrate a salient result -- that the same draconic period that the QBO model uses and that the ENSO model uses as the primary angular momentum forcing term also appears in the most direct and practical measurement of angular momentum known in Earth geophysics. And I do not think this is a instance of artificial sampling aliasing, since each of the LOD measurements was averaged over at least a month. It is likely a real seasonal aliasing behavior and serves to further substantiate the theorized mechanisms forcing the QBO and ENSO behavior. Read the short paper that I placed into arXiv to see how the other parts fall into place.

And a continuing lesson to be learned is to not filter data unless you can be sure that what you are filtering out is nuisance noise. In this case, it appears the filtering performed by IERS was too aggressive and removed a very important signal in the time series! Contrary to what the statistician Tamino has been preaching, these time-series do not have a lot of statistical content. I donated money to his lessons on time-series analysis, but largely disagree on the applicability of his approaches to earth data such as we see in QBO, ENSO, Chandler wobble, and now LOD. If you followed his instructions, you might not find any of the periodic cycles buried in the data, and you might dismiss it all as red noise. Yet, there is likely nothing statistical or complex about these behaviors, as they are largely driven by known deterministic forces. Stay tuned, as the next post will be titled "Needless Complexity".

## References

[1] B. Fong Chao, “Excitation of Earth’s polar motion by atmospheric angular momentum variations, 1980–1990,” *Geophysical research letters*, vol. 20, no. 3, pp. 253–256, 1993.

[2] Shen, Wen-Jun Wang Wen-Bin, and Han-Wei Zhang. "Verifications for Multiple Solutions of Triaxial Earth Rotation." PDF

*Note: the authors in [2] also find a 14-year signal in the LUNAR97 data (see figure below). This is likely a result of a wavelet filter that isolated periods between 8 and 16 years as they described the windowing procedure. This 14 year period is critical in the ENSO model. *

[3] Gross, Richard S. "A combined length-of-day series spanning 1832–1997: LUNAR97." *Physics of the Earth and Planetary Interiors* 123.1 (2001): 65-76. PDF

another simple solution with balanced draconic and anomalistic content

The aliased frequency is high because it's leveraging small differences in ffrequency to compensate for phase changes. Fold it back, expand the beat, and the numbers are close to the draconic and anomalistic.

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Performed an equivalent analysis via a spreadsheet using Excel's built-in Solver, and it found essentially the same factors

It's now more evident that the fit is primarily applying post-1930 data, as the weighting in the machine learning was focused on more recent data.

Tried a LOD first-derivative fit via machine learning. This should find frequencies slower than the 2 to 3 year periods than the 2nd-derivative fit found.

Sure enough, it uncovers the 14-yr modulation, as a 13.9 year period buried in a beat pattern, 2*pi/(0.276+0.176) = 13.9

Can see it a little better here:

Some other crazy frequency modulation, but it is definitely there, substantiating the Shen analysis.

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The second derivative of the wave-equation-transformed ENSO data, plotted against model. The lower curve is the correlation coefficient, calculated over 4 year window widths.

The CC only goes negative (anti-correlation) around 1921.

This indicates where some of the ENSO data was interpolated because of missing entries -- marked by -999 NOAA SOI

Both the ENSO model and LOD shows a glitch after 1983

Can see the 2.33 year period in Gross's paper, spectrum at 0.42 cycles/year

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