[mathjax]An old game show called Name That Tune asked contestants to identify a song by listening to just a few notes.

This post is the QBO equivalent to that. How short an interval can we train on to reproduce the rest of the time series? The answer is not much is required.

Using only three fundamental frequencies on the QBO acceleration: (1) the strong aliased Draconic, (2) an annual signal, and (3) a semi-annual signal, the rest of the data can be fit by using an interval as short as 3 to 4 years.  The aliased Draconic signal is 2.36 years with the accompanying harmonics at 0.292, 0.413, 0.703, 1.73, 0.634 that sharpen the peaks on a seasonal basis.

Aliased Harmonic	Frequency	Period
Y/27.212-10	3.423	0.292
Y/27.212-11	2.423	0.413
Y/27.212-12	1.423	0.703
Y/27.212-13	0.423	2.363
Y/27.212-14	-0.577	-1.734
Y/27.212-15	-1.577	-0.634
Y/27.212-16	-2.577	-0.388

Where Y is the year in days. The shorter periods provide the fine structure and help to isolate the tidal frequency as they sharpen and define the acceleration peak amplitude on such a short interval (Figure 1).

The following training interval runs from 1970 to 1974 (Figure 2). Even though it trains on a different interval, the results on the validation interval looks very similar to the previous result of Figure 1! This indicates that there is an intrinsic stationary behavior common to the two intervals.

If one looks closely, the amount of detail that is replicated in the data by the model is also quite intricate in places. Yet there are intervals that show shifts in the peaks. It’s possible that some of the acceleration peaks are shifted due to transient volcanic effects on the stratosphere. Look at the discrepancy regions in yellow below and we can determine which may be associated with volcanic events.

So far the training intervals happen to be regions that are free of significant volcanic eruptions. The following fit also uses a largely volcano-free training interval from 1996-2006 and one can see the three significant volcanic events of the 20th century ( Agung – 1963, El Chichon – 1982, Pinatubo – 1991) immediately preceded regions that show discrepancy between the model fit and the data (the regions in yellow ♦).

In general, any deviations are likely due to whether the seasonal tidal impulse is enough to trigger the QBO reversal during that Draconic lunar cycle or a subsequent one.  So it is possible that volcanic particulates released to the stratosphere could impact the temperature of the layer, thus speeding up onset of a warm season, or delaying the transition to a cool season.

Importantly, we did not apply the Tropical and Anomalous lunar cycles to the fit, which could potentially improve the timing. But this would also require a longer training interval to avoid the problems of overfitting. The following is an Excel fit using the Solver plug-in  where the Draconic lunar period was allowed to vary.  The regions denoted by the blue horizontal bars were used as training intervals. Using the GRG Nonlinear solving method, the best fit for the Draconic period was 27.211 days, which compares favorably to the known value of 27.212 days  (Figure 5).  The difference of 0.001 day accumulates to less than a day over 50 years.

Another important point to note if these are indeed volcanic disturbances impacting the transient behavior, is that the QBO regains phase alignment with the lunar tides relatively quickly — as if it had no long-term effect.  This is a hallmark of an ergodic process forcibly driven by a guiding temporal boundary condition.

Also worth noting is that such a limited data series would have allowed Richard Lindzen to do this exercise back in the 1960’s and thus potentially project the QBO behavior over the ensuing years.  And that also implies he provided poor guidance to our understanding of QBO. Consider how many papers are trying to explain the recent QBO anomaly.   The fact that volcanic events, ENSO timing, and AGW could each impact the seasonal impact of the tidal forcing, having a solid foundational model for the stationary behavior is critical.

UPDATE – 10/11/2016

I fed the 1970-1974 and 1996-2006 QBO‘ 30-hPa training intervals into a symbolic regression machine learning tool to see what it would uncover, shown below.

The solution contains the ~2.66 rad/year fundamental with harmonics that are aliased multiples of $$2pi$$ rad/year spaced from that.  In particular, from an expanded view of the solution (by refactoring the $$cos(2pi x)^N$$ terms), the multiples range from $$-6pi$$, $$-4pi$$, $$-2pi$$, $$2pi$$, $$4pi$$, $$6pi$$. There is also a non-aliased harmonic of 2.66, i.e. precisely double that frequency, which is also predicted by the QBO cos(sin(wt)+constant) model. Any asymmetries in positive versus negative excursions will typically reveal that harmonic.

Aliased Harmonic	Frequency (1/y)	Period (y)
2.66 + $$6pi$$	3.423	0.292
2.66 + $$4pi$$	2.423	0.413
2.66 + $$2pi$$	1.423	0.703
2.66 = main harmonic	0.423	2.363
2.66 – $$2pi$$	-0.577	-1.734
2.66 – $$4pi$$	-1.577	-0.634
2.66 – $$6pi$$	-2.577	-0.388

The important point to note is that the table at the top of this post and the machine learning results are the same! Let that sink in. The top table was filled in based on the physical knowledge of aliasing the 27.212 day Draconic lunar tide against an annual signal would produce those specific harmonics. The bottom table was filled in by a symbolic reasoning engine finding the same terms by exhaustively testing millions of possibilities for the best and most canonical fit — only assuming that y = f(x), a completely arbitrary function.  In other words, the machine learning knows nothing about the physics yet finds the same set of terms.

As a second check, the closed form expression produced by the symbolic regression also fits right on top of the model optimized by the Excel solver.

As far as a finishing touch, it will not be difficult to add an impulse response to this model to fit the volcanic disturbances.  This will presumably apply the Sato stratospheric aerosol model to generate the appropriate delta functions.