Note how well it does in extrapolating a projection from a training interval.
This is an ENSO model fit to SOI data using an analytical solution to Navier-Stokes. The same algorithm is used to solve for the optimal forcing as in the tidal analysis solution above, but applying the annual solar cycle and monthly/fortnightly lunar cycles instead of the diurnal and semi-diurnal cycle.
The time scale transitions from a daily modulation to a much longer modulation due to the long-period tidal factors being invoked.
Next is an expanded view, with the correlation coefficient of 0.73:
This is a fit trained on the 1880-1950 interval (CC=0.76) and cross-validated on the post-1950 data
This is a fit trained on the post-1950 interval (CC=0.77) and cross-validated on the 1880-1950 data
Like conventional tidal prediction, very little over-fitting is observed. Most of what is considered noise in the SOI data is actually the tidal forcing signal. Not much more to say, except for others to refine.
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.
Fig.1 : Power spectra of ENSO data against model
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.
Fig 2: Frequency sideband plot identifying components created by modulation of a biennial cycle with the lunar cycles (originally described 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 , 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.
Many interesting inferences one can potentially draw from these comparisons. The SOI signal appears more noisy, but that could actually be signal. For example, the NINO34 extrapolation pulls out a split peak near 2013-2014, which does show up in the SOI data. And a discrepancy in the NINO34 data near 1934-1935 which predicts a minor peak, is essentially noise in the SOI data. The 1984-1986 flat valley region is much lower in NINO34 than in SOI, where it hovers around 0. The model splits the difference in that interval, doing a bit of both. And the 1991-1992 valley predicted in the model is not clear in the NINO34 data, but does show up in the SOI data.
Of course these are subjectively picked samples, yet there may be some better combination of SOI and NINO34 that one can conceive of to get a better handle on the true ENSO signal.
In the last month, two of the great citizen scientists that I will be forever personally grateful for have passed away. If anyone has followed climate science discussions on blogs and social media, you probably have seen their contributions.
Keith Pickering was an expert on computer science, astrophysics, energy, and history from my neck of the woods in Minnesota. He helped me so much in working out orbital calculations when I was first looking at lunar correlations. He provided source code that he developed and it was a great help to get up to speed. He was always there to tweet any progress made. Thanks Keith
Kevin O'Neill was a metrologist and an analysis whiz from Wisconsin. In the weeks before he passed, he told me that he had extra free time to help out with ENSO analysis. He wanted to use his remaining time to help out with the solver computations. I could not believe the effort he put in to his spreadsheet, and it really motivated me to spending more time in validating the model. He was up all the time working on it because he was unable to lay down. Kevin was also there to promote the research on other blogs, right to the end. Thanks Kevin.
There really aren't too many people willing to spend time working analysis on a scientific forum, and these two exemplified what it takes to really contribute to the advancement of ideas. Like us, they were not climate science insiders and so will only get credit if we remember them.
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.
With an ideal tidal analysis, one should be able to apply the gravitational forcing of the lunar orbit1 and use that as input to solve Laplace's tidal equations. This would generate tidal heights directly. But due to aleatory uncertainty with respect to other factors, it becomes much more practical to perform a harmonic analysis on the constituent tidal frequencies. This essentially allows an empirical fit to measured tidal heights over a training interval, which is then used to extrapolate the behavior over other intervals. This works very well for conventional tidal analysis.
For ENSO, we need to make the same decision: Do we attempt to work the detailed lunar forcing into the formulation or do we resort to an empirical bottoms-up harmonic analysis? What we have being do so far is a variation of a harmonic analysis that we verified here. This is an expansion of the lunar long-period tidal periods into their harmonic factors. So that works well. But could a geophysical model work too?
Now that we have strong evidence that AMO and PDO follows the biennial modulated lunar forcing found for ENSO, we can try modeling the Chandler wobble in detail. Most geophysicists argue that the Chandler wobble frequency is a resonant mode with a high-Q factor, and that random perturbations drive the wobble into its characteristic oscillation. This then interferes against the yearly wobble, generating the CW beat pattern.
The key to applying a lunar forcing is to multiply it by a extremely regular seasonal pulse, which introduces enough of a non-linearity to create a physically-aliased modulation of the lunar monthly signal (similar as what is done for ENSO, QBO, AMO, and PDO).
After spending several years on formulating a model of ENSO (then and now) and then spending a day or two on the AMO model, it's obvious to try the other well-known standing wave oscillation — specifically, the Pacific Decadal Oscillation (PDO). Again, all the optimization infrastructure was in place, with the tidal factors fully parameterized for automated model fitting.
This fit is for the entire PDO interval:
What's interesting about the PDO fit is that I used the AMO forcing directly as a seeding input. I didn't expect this to work very well since the AMO waveform is not similar to the PDO shape except for a vague sense with respect to a decadal fluctuation (whereas ENSO has no decadal variation to speak of).
Yet, by applying the AMO seed, the convergence to a more-than-adequate fit was rapid. And when we look at the primary lunar tidal parameters, they all match up closely. In fact, only a few of the secondary parameters don't align and these are related to the synodic/tropical/nodal related 18.6 year modulation and the Ms* series indexed tidal factors, in particular the Msf factor (the long-period lunisolar synodic fortnightly). This is rationalized by the fact that the Pacific and Atlantic will experience maximum nodal declination at different times in the 18.6 year cycle.