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.