This is really a metrology exercise in terms of trying to deconvolve the forcing signal from a combined signal, given limited or noisy measurement data. Like if you take a pendulum and infer the force of gravity from the measured period. To make an analogy, a Mathieu formulation can model an inverted pendulum that is is positioned on a moving platform. If the underlying platform motion is set to the right range of periods, then the inverted pendulum can undergo a stable yet complex oscillatory pattern as the pair of focings interact with the non-linear natural response. That's similar to what this ENSO sloshing model describes.

As far as extracting the periodic signal, the technical challenge is similar to this common test: Consider that a 60 Hz noise source is generated and you are trying to detect it with as short an interval as possible.

Using the Solver technique, input a model paremeter set consisting of a 60 Hz sinusoid with a varying phase and amplitude, along with harmonics as coonstraints. This represents the unknown 60 Hz "Hum" noise. With a sample interval just a bit stronger and longer then the rectified period, this can reveal the underlying full-wave modulated signal .

Using a hybrid $\frac{CC}{\Delta Err}$ objective to get the best fit
![](http://imageshack.com/a/img921/1637/3N37MX.png)

So for fitting ENSO, I am doing something akin to this. Only considering known tidal and wobble periods, along with the seasonal harmonics $\omega_o \pm n2\pi$ (which differ from the traditional harmonics, where $n\omega_0$ ), we can get a very good fit and one that validates on the intervals that are outside the training intervals.