With this analysis, I wanted to demonstrate the underlying order of the most concise SOI Model. This model characterizes the salient fit parameters:

- Two slightly offset forcing sinusoids which match the average QBO forcing cycle
- A forcing sinusoid that maps to the frequency of the Chandler wobble beat
- A Mathieu modulation perturbing the 2nd-order DiffEq with a periodicity of about 8 years

This set of four parameters was used to model both modern day records corresponding to the atmospheric pressure data describing the Southern Oscillation Index, as well as to proxy records of historical coral data. The parameters seem to match closely over widely separated time intervals (see Figure 5 in the latter link).

**Figure 1** is the modern-day SOI record, suitably filtered to show the multi-year excursions.

It is amazing that this erratic a waveform can be modeled by a limited set of parameters that actually make some physical sense, but that is nature for you and the idea behind "sloppy modeling" -- models that use just a few parameters to accurately describe a behavior. The simplified model strongly suggests that there is a hidden harmony acting to drive ENSO.