The model fit to ENSO takes place in the time domain. However, the correlation coefficient between model and data of the corresponding power spectra is higher than in the time series. Below in **Figure 1** the CC is 0.92, while the CC in the time series is 0.82.

The model allows only 3 fundamental lunar frequencies along with the annual cycle, plus the harmonics caused by the non-linear orbital path and the seasonally impulsed modulation.

What this implies is that almost all the peaks in the power spectra shown above are caused by interactions of these 4 fundamental frequencies. **Figure 2** shows a satellite view of peak splitting (also shown here).

One of the reasons that the power spectrum gives a higher correlation coefficient — *despite the fact that the spectrum wasn't used in the fit* — is that the lunar tides are precisely determined and thus all the harmonics should align well in the frequency domain. And that's what is observed with the multiple-peak alignment.

Furthermore, according to Ref [1], this result is definitely *not* a characteristic of noise-driven system, and it also possesses a very low dimension of chaotic content. The same frequency content is observed largely independent of the prediction time profile, i.e. training interval.

## References

1. Bhattacharya, Joydeep, and Partha P. Kanjilal. "Revisiting the role of correlation coefficient to distinguish chaos from noise." *The European Physical Journal B-Condensed Matter and Complex Systems* 13.2 (2000): 399-403.