Following up on the last post on the ENSO forcing, this note elaborates on the math. The tidal gravitational forcing function used follows an inverse power-law dependence, where a(t) is the anomalistic lunar distance and d(t) is the draconic or nodal perturbation to the distance.
Note the prime indicating that the forcing applied is the derivative of the conventional inverse squared Newtonian attraction. This generates an inverse cubic formulation corresponding to the consensus analysis describing a differential tidal force:
Here is the model fit for training from 1880-1980, with the extrapolated test region post-1980 showing a good correlation.
The geophysics is now canonically formulated, providing (1) a simpler and more concise expression, leading to (2) a more efficient computational solution, (3) less possibility of over-fitting, and (4) ultimately generating a much better correlation. Alternatively, stated in modeling terms, the resultant information metric is improved by reducing the complexity and improving the correlation -- the vaunted cheaper, faster, and better solution. Or, in other words: get the physics right, and all else follows.
From the last post, we tried to estimate the lunar tidal forcing potential from the fitted harmonics of the ENSO model. Two observations resulted from that exercise: (1) the possibility of over-fitting to the expanded Taylor series, and (2) the potential of fitting to the ENSO data directly from the inverse power law.
The Taylor's series of the forcing potential is a power-law polynomial corresponding to the lunar harmonic terms. The chief characteristic of the polynomial is the alternating sign for each successive power (see here), which has implications for convergence under certain regimes. What happens with the alternating sign is that each of the added harmonics will highly compensate the previous underlying harmonics, giving the impression that pulling one signal out will scramble the fit. This is conceptually no different than eliminating any one term from a sine or cosine Taylor's series, which are also all compensating with alternating sign.
The specific conditions that we need to be concerned with respect to series convergence is when r (perturbations to the lunar orbit) is a substantial fraction of R (distance from earth to moon) :
Because we need to keep those terms for high precision modeling, we also need to be wary of possible over-fitting of these terms — as the solver does not realize that the values for those terms have the constraint that they derive from the original Taylor's series. It's not really a problem for conventional tidal analysis, as the signals are so clean, but for the noisy ENSO time-series, this is an issue.
Of course the solution to this predicament is not to do the Taylor series harmonic fitting at all, but leave it in the form of the inverse power law. That makes a lot of sense — and the only reason for not doing this until now is probably due to the inertia of conventional wisdom, in that it wasn't necessary for tidal analysis where harmonics work adequately.
So this alternate and more fundamental formulation is what we show here.