An obvious clue from tidal data

One of the interesting traits of climate science is the way it gives away obvious clues. This recent paper by Iz

Iz, H Bâki. “The Effect of Regional Sea Level Atmospheric Pressure on Sea Level Variations at Globally Distributed Tide Gauge Stations with Long Records.” Journal of Geodetic Science 8, no. 1 (n.d.): 55–71.
shows such a breathtakingly obvious characteristic that it’s a wonder why everyone isn’t all over it.  The author seems to be understating the feature, which is essentially showing that for certain tidal records, the atmospheric pressure (recorded in the tidal measurement location) is pseudo-quantized to a set of specific values.  In other words, for a New York City tidal gauge station, there are 12 values of atmospheric pressure between 1000 and 1035 mb that are heavily favored over all other values.
One can see it in the raw data here where clear horizontal lines are apparent in the data points:

Raw data for NYC station  (Iz, H Bâki. “The Effect of Regional Sea Level Atmospheric Pressure on Sea Level Variations at Globally Distributed Tide Gauge Stations with Long Records.” Journal of Geodetic Science 8, no. 1 (n.d.): 55–71.)

and for the transformed data shown in the histogram below, where I believe the waviness in the lines is compensated by fitting to long-period tidal signal factors (such as 18.6 year, 9.3 year periods, etc).

Histogram for transformed data for NYC station  Iz, H Bâki. “The Effect of Regional Sea Level Atmospheric Pressure on Sea Level Variations at Globally Distributed Tide Gauge Stations with Long Records.” Journal of Geodetic Science 8, no. 1 (n.d.): 55–71.

The author isn’t calling it a quantization, and doesn’t really call attention to it with a specific name other than clustering, yet it is obvious from the raw data and even more from the histograms of the transformed data.

The first temptation is to attribute the pattern to a measurement artifact. These are monthly readings and there are 12 separate discrete values identified so that connection seems causal. The author says

“It was shown that random component of regional atmospheric pressure tends to cluster at monthly intervals. The clusters are likely to be caused by the intraannual seasonal atmospheric temperature changes, which may also act as random beats in generating sub-harmonics observed in sea level changes as another mechanism.”
Nearer the equator, the pattern is not readily evident. The fundamental connection between tidal value and atmospheric pressure is due to the inverse barometric effect
“At any fixed location, the sea level record is a function of time, involving periodic components as well as continuous random fluctuations. The periodic motion is mostly due to the gravitational effects of the sun-earth-moon system as well as because of solar radiation upon the atmosphere and the ocean as discussed before. Sometimes the random fluctuations are of meteorological origin and reflect the effect of ’weather’ upon the sea surface but reflect also the inverse barometric effect of atmospheric pressure at sea level.”
So the bottom-line impact is that the underlying tidal signal is viably measured even though it is at a monthly resolution and not the diurnal or semi-diurnal resolution typically associated with tides.
Why this effect is not as evident closer to the equator is rationalized by smaller annual amplification
“Stations closer to the equator are also exposed to yearly periodic variations but with smaller amplitudes. Large adjusted R2 values show that the models explain most of the variations in atmospheric pressure  observed at the sea level at the corresponding stations. For those stations closer to the equator, the amplitudes of the annual and semiannual changes are considerably smaller and overwhelmed by random excursions. Stations in Europe experience similar regional variations because of their proximities to each other”
So, for the Sydney Harbor tidal data the pattern is not observed

Sydney histogram does not show a clear delineated quantization

Whereas, I previously showed the clear impact of the ENSO signal on the Sydney tidal data after a specific transform in this post. The erratic ENSO signal (with a huge inverse barometric effect as measured via the SOI readings of atmospheric pressure) competes with the annual signal so that the monthly quantization is obscured. Yet, if the ENSO behavior is also connected to the tidal forcing at these long-period levels, there may be a tidal unification yet to be drawn from these clues.

ENSO model verification via Fourier analysis infill

Because the ENSO model generates precise temporal harmonics via a non-linear solution to Laplace’s Tidal Equations, it may in practice be trivially easy to verify. By only using higher-frequency harmonics (T<1.25y) during spectral training (with a small window of low-frequency signal to stabilize the solution, T>11y), the model essentially fills in the missing bulk of the signal frequency spectrum, 1.25y < T < 11y.  This is shown below in Figure 1.

Fig. 1: Bottom panel of amplitude ENSO SOI spectra shows the training windows.  A primarily low-amplitude spectral signal is used to fit the model (using least-squares on the error signal). Upper spectra shows the expanded view of the out-of-band fit. This rich spectra is all due to the non-linear harmonic solution of the ENSO Laplace’s Tidal Equation solution.

This agreement is statistically unlikely (nee impossible) to occur unless the out-of-band signal had knowledge of the fundamental harmonics (i.e. the highest amplitude terms in the meat of the spectra) that are contributing to the higher harmonics.

Figure 2 is the underlying temporal fit. Although not as good a fit as what we can achieve using more of the primary Fourier terms, it is still striking.

Fig. 2: Temporal model fit using only Fourier frequency terms shorter than 1.25 years and longer than 11 years. The correlation coefficient is 0.7 here

The consensus claim is that ENSO is a chaotic process with no long-term coherence. Yet, this shows excellent agreement with a forced lunisolar model showing very long-term coherence.   An issue to raise is: why has the obvious deterministic forcing model been abandoned as a plausible physical mechanism so long ago?

Last post on ENSO

The last of the ENSO charts.

This is how conventional tidal prediction is done:

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.

Thanks to Kevin and Keith for all their help, which will be remembered.

Correlation Coefficient of ENSO Power Spectra

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.

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).

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 [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.

NINO34 vs SOI

Experiment to compare training runs from 1880 to 1980 of the ENSO model against both the NINO34 time-series data and the SOI data. The solid red-curves are the extrapolated cross-validation interval..

NINO34

SOI

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.

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AGU 2017 posters

ED23D-0326: Knowledge-Based Environmental Context Modeling


click to enlarge


GC41B-1022: Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models


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.

The ENSO Forcing Potential – Cheaper, Faster, and Better

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.

F(t) propto frac{1}{(R_0 + a(t) + d(t))^2}'

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:

F(t) propto -frac{a'(t)+d'(t)}{(R_0 + a(t) + d(t))^3}

For a combination of monthly and fortnightly sinusoidal terms for a(t) and d(t) (suitably modified for nonlinear nodal and perigean corrections due to the synodic/tropical cycle)   the search routine rapidly converges to an optimal ENSO fit.  It does this more quickly than the harmonic analysis, which requires at least double the unknowns for the additional higher-order factors needed to capture the tidally forced response waveform. One of the keys is to collect the chain rule terms a'(t) and d'(t) in the numerator; without these, the necessary mixed terms which multiply the anomalistic and draconic signals do not emerge strongly.

As before, a strictly biennial modulation needs to be applied to this forcing to capture the measured ENSO dynamics — this is a period-doubling pattern observed in hydrodynamic systems with a strong fundamental (in this case annual) and is climatologically explained by a persistent year-to-year regenerative feedback in the SLP and SST anomalies.

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.

 

 

 

 

 

 

 

 

 

 

 

 

 

Approximating the ENSO Forcing Potential

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) :

F(r) = frac{1}{(R+r)^3}

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.

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Reverse Engineering the Moon’s Orbit from ENSO Behavior

[mathjax]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?

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Interface-Inflection Geophysics

This paper that a couple of people alerted me to is likely one of the most radical research findings that has been published in the climate science field for quite a while:

Topological origin of equatorial waves
Delplace, Pierre, J. B. Marston, and Antoine Venaille. Science (2017): eaan8819.

An earlier version on ARXIV was titled Topological Origin of Geophysical Waves, which is less targeted to the equator.

The scientific press releases are all interesting

  1. Science Magazine: Waves that drive global weather patterns finally explained, thanks to inspiration from bagel-shaped quantum matter
  2. Science Daily: What Earth’s climate system and topological insulators have in common
  3. Physics World: Do topological waves occur in the oceans?

What the science writers make of the research is clearly subjective and filtered through what they understand.

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