For the ENSO model, there is an ambiguity in simultaneously identifying the lunar month duration (draconic, anomalistic, and tropical) *and* the duration of a year. The physical aliasing is such that the following *f *= frequency will give approximately equivalent fits for a range of year/month pairs (see this as well).

So that during the fitting process, if you allow the duration of the individual months and the year to co-vary, then the two should scale approximately by the number of lunar months in a year ~13.3 = 1/0.075. And sure enough, that's what is found, a set of year/month pairs that provide a maximized fit along a ridge line of possible solutions, but *only one* that is ultimately correct for the average year duration over the entire range:

By regressing on the combination of linear slopes, the value of the year that minimizes the error to each of the known lunar month values is 365.244 days. This lies within the interval defined by the value of the calendar year = 365.25 days — which includes a leap day every 4 years, and the more refined leap year calculation = 365.242 days — which includes the 100 and 400 year corrections (there are additional leap second corrections).

This analysis provides further confidence that the ENSO model is approaching the status of a metrology tool for gauging lunisolar cycles. The tropical month is estimated slow by about 1/2 a minute, while the draconic month is fast by a 1/2 a minute, and the average anomalistic month is spot on to within a second.

This is what the fit looks like for a 365.242 day long calendar year trained over the entire interval. It is the accumulation of the sharply matching peaks and valleys which allow the solver function to zone in so precisely to the known tidal factors.

About the only issue that hobbles our ability to achieve fits as good as ocean tidal analysis is the amount of noise near neutral ENSO conditions in the time-series data. The highlighted yellow regions in the comparison between NINO34 and SOI time-series data shown below indicate intervals whereby a sliding correlation coefficient drops closer to zero. (*The only odd comparison is the blue highlighted region around 1985, where SOI is extremely neutral while NINO34 appears La Nina-like. Is SOI pressure related to a second derivative of NINO34 temperature?*).

Those same yellow regions are also observed as discrepancies between the NINO34 data and the ENSO best model fit.

Yellow shading at intervals around 1930, 1936, 1948 indicate discrepancies between the NINO34 data in **green **and the ENSO model in **red**.

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*Related*

Large positive excursions in the NINO34 index appear to be triggering mechanisms that result in very poor correlations.

This is easier to see by plotting the sliding correlation alongside a rescaled NINO34. In this graph I used (NINO34^3)/2. A NINO34 value of 0.75 or higher seems to be a threshhold value (0.21 on my rescaled graph).

There are some interesting patterns

http://bit.ly/2xSEAgy

Wow, I think you are on to something. What I am applying is the Laplace's tidal equation solution that does sin(A*sin(wt)) -- this creates a kind of "amplitude folding" when the internal sinusoid attains too high an amplitude.

It appears that this effect is stronger on the

~~La Nina~~El Nino episodes. This may be because there is a +/- asymmetry in the sin(A*sin(wt)) modulation such that sin(A*sin(wt)+C), where C is the offset bias.Thanks Kev for the analysis

I've never heard of a 'sliding correlation coefficient' - much less ever used one. So the difficult part was figuring out how to reproduce your graph.

The noise reduction scheme was almost purely by accident. If the NINO34 signal wasn't so busy I might never have thought about it. Cubing it is obviously the quickest way to amplify signal-to-noise while retaining the sign. I've probably seen you do something similar on one of your spreadsheets.

The sliding CC does pick up stuff that is kind of hard to tell by eye.

Yes, this is a great example of where the sliding correlation coefficient tells you something that is not likely to be caught by eyeballing either the raw data or a graph of the raw data.

My guess is that the longer and noisier the series, the more useful the sliding correlation coefficient can be. It's always good to add another hammer to the analytical toolbox.

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Paul, what other variables other than the lengths of the lunar months have real world values that we should be targeting??

Kevin, The other important value is the time of the seasonal impulse, which appears to be sometime in November of each year.

There are also the differential equation parameters, including the delay weighting from the previous year, but that's about it. These are tuned so they don't come from a characterized value.