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)

Using just a correlation cooefficient to get the best fit
![](http://imageshack.com/a/img924/9840/RuhzsA.png )

The full-wave recified signal is considered a good test case for checking the precision of the solver. A rectified signal is rich in harmonics due to the sharp reversal around the zero-crossing, but if these aren't handled properly they can end up producing artifacts due to overfitting. In this case the pure correlation coefficient target metric shows more artifacts than the hybrid metric.

If the training interval is twice as long, it captures the shape more precisely. This fit includes the fundamental 60Hz plus 7 harmonics.

![](http://imageshack.com/a/img921/3421/W5I6BK.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.