I have been cranking on the Universal ENSO Proxy records (top of this thread) some more. This is over 300 years worth of ENSO proxy data taken at yearly intervals.

In the modern day non-Proxy ENSO records, I [found three significant frequencies](http://contextearth.com/2014/05/27/the-soim-differential-equation/) in the context of a [Mathieu-like DiffEq](http://mathworld.wolfram.com/MathieuDifferentialEquation.html) formulation.

There is the nonlinear modulation of about 8.3 years, and a combined forcing frequency of a 6 year period and a 28 month = 2.33 year period. The latter 28 month period is common to the [Quasi-Biennial Oscillation](http://en.wikipedia.org/wiki/Quasi-biennial_oscillation) of stratospheric winds where the periodicity is quite striking. The 6 year period has no obvious connection but similar periods occur when looking at [periodic jerks](http://phys.org/news/2013-07-pair-year-oscillations-length-day.html) in the rotation of the earth, the [Chandler wobble](http://www.technologyreview.com/view/415093/earths-chandler-wobble-changed-dramatically-in-2005/), and the beat frequency between the anomalistic and draconic [lunar month ](http://en.wikipedia.org/wiki/Lunar_month).

This plot is where I broke up the UEP time series in two approximately equal intervals, with the break point at the year 1818. Since this is a yearly sample, I did not filter the data any further.


* The 1.537 number is the Mathieu equation modulation in rads/year on the LHS
* The 1.02/1.01 numbers are the first and second half of the ~6-year forcing period on the RHS
* The 2.67/2.68 numbers are the first and second half of the QBO forcing period on the RHS

I took the liberty of trying to modulate the QBO with another small cyclic term to model what appears to be a variation in the QBO forcing itself.

I believe that even though the correlation coefficient is "only" 0.42, this is a deceptively good fit and it is consistent with the model I fit to the ENSO SOI data. I am not sure how much further I can tweak the fit, as it seems to be close to converging.

The noise in the data seems to be a factor as the EUP is an ensemble of 10 different records, and there is considerable variance in the data from the different records. There is thus likely a "ceiling" to the correlation coefficient even if the fundamental underlying waveform is discovered. This correlation coefficient ceiling may in fact may be as low as 0.6 --a guess based on what I have seen in the past with such a busy waveform.

The remaining issue is that there may be other combinations of parameters that provide an even better fit.
Dara's evolutionary strategy may help out here.

Paul Pukite