Inverting non-autonomous functions

This is an algorithm based on minimum entropy (i.e. negative entropy) considerations which is essentially an offshoot of this paper Entropic Complexity Measured in Context Switching.

The objective is to apply negative entropy to find an optimal solution to a deterministically ordered pattern. To start, let us contrast the behavior of autonomous vs non-autonomous differential equations. One way to think about the distinction is that the transfer function for non-autonomous only depends on the presenting input. Thus, it acts like an op-amp with infinite bandwidth. Or below saturation it gives perfectly linear amplification, so that as shown on the graph to the right, the x-axis input produces an amplified y-axis output as long as the input is within reasonable limits.

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Low #DOF ENSO Model

Given two models of a physical behavior, the “better” model has the highest correlation (or lowest error) to the data and the lowest number of degrees of freedom (#DOF) in terms of tunable parameters. This ratio CC/#DOF of correlation coefficient over DOF is routinely used in automated symbolic regression algorithms and for scoring of online programming contests. A balance between a good error metric and a low complexity score is often referred to as a Pareto frontier.

So for modeling ENSO, the challenge is to fit the quasi-periodic NINO34 time-series with a minimal number of tunable parameters. For a 140 year fitting interval (1880-1920), a naive Fourier series fit could easily take 50-100 sine waves of varying frequencies, amplitudes, and phase to match a low-pass filtered version of the data (any high-frequency components may take many more). However that is horribly complex model and obviously prone to over-fitting. Obviously we need to apply some physics to reduce the #DOF.

Since we know that ENSO is essentially a model of equatorial fluid dynamics in response to a tidal forcing, all that is needed is the gravitational potential along the equator. The paper by Na [1] has software for computing the orbital dynamics of the moon (i.e. lunar ephemerides) and a 1st-order approximation for tidal potential:

The software contains well over 100 sinusoidal terms (each consisting of amplitude, frequency, and phase) to internally model the lunar orbit precisely. Thus, that many DOF are removed, with a corresponding huge reduction in complexity score for any reasonable fit. So instead of a huge set of factors to manipulate (as with many detailed harmonic tidal analyses), what one is given is a range (r = R) and a declination ( ψ=delta) time-series. These are combined in a manner following the figure from Na shown above, essentially adjusting the amplitudes of R and delta while introducing an additional tangential or tractional projection of delta (sin instead of cos). The latter is important as described in NOAA’s tide producing forces page.

Although I roughly calibrated this earlier [2] via NASA’s HORIZONS ephemerides page (input parameters shown on the right), the Na software allows better flexibility in use. The two calculations essentially give identical outputs and independent verification that the numbers are as expected.

As this post is already getting too long, this is the result of doing a Laplace’s Tidal Equation fit (adding a few more DOF), demonstrating that the limited #DOF prevents over-fitting on a short training interval while cross-validating outside of this band.

or this

This low complexity and high accuracy solution would win ANY competition, including the competition for best seasonal prediction with a measly prize of 15,000 Swiss francs [3]. A good ENSO model is worth billions of $$ given the amount it will save in agricultural planning and its potential for mitigation of human suffering in predicting the timing of climate extremes.


[1] Na, S.-H. Chapter 19 – Prediction of Earth tide. in Basics of Computational Geophysics (eds. Samui, P., Dixon, B. & Tien Bui, D.) 351–372 (Elsevier, 2021). doi:10.1016/B978-0-12-820513-6.00022-9.

[2] Pukite, P.R. et al “Ephemeris calibration of Laplace’s tidal equation model for ENSO” AGU Fall Meeting, 2018. doi:10.1002/essoar.10500568.1

[3] 1 CHF ~ $1 so 15K = chump change.

Added: High resolution power spectra of ENSO forcing
see link

Nonlinear long-period tidal forcing with application to ENSO, QBO, and Chandler wobble

Model fitting process for ENSO

Back to EGU abstract and presentation

Addendum: After this presentation was submitted, a ground-breaking paper by a group at the University of Paris came on-line. Their paper, “On the Shoulders of Laplace” covers much the same ground as the EGU presentation linked above.

Their main thesis is that Pierre-Simon Laplace in 1799 correctly theorized that the wobble in the Earth’s rotation is due to the moon and sun, described in the treatise “Traité de Mécanique Céleste (Treatise of Celestial Mechanics)“.

Excerpts from the paper “On the shoulders of Laplace”

Moreover Lopes et al claim that this celestial gravitational forcing carries over to controlling cyclic climate indices, following Laplace’s mathematical formulation (now known as Laplace’s Tidal Equations) for describing oceanic tides.

Excerpt from the paper “On the shoulders of Laplace”

This view also aligns with the way we model climate indices such as ENSO and QBO via a solution to Laplace’s Tidal Equations, as described in the linked EGU presentation above.

Review: Modeling of ocean equatorial currents in the phase of El Niño and La Niña!

The equatorial zone acts as a waveguide. As highlights they list the following bullet-points, taking advantage that the Coriolis effect at the equator vanishes or cancels.

This is a critical assertion, since — as shown in Mathematical Geoenergy –the Chandler wobble (a nutational oscillation) is forced by tides, then transitively so is the El Nino. So when the authors state the consequence is of both nutation and a gravity influence, it is actually the gravity influence of the moon and sun (and slightly Jupiter) that is the root cause.

The article has several equations that claim analytical solutions, but the generated PDF format has apparently not rendered the markup correctly. Many “+” signs are missing from equations. I have seen this issue before when I have tried to generate PDF pages from a markup doc, and assume that is what is happening. Assume the hard-copy version is OK so may have to go to the library to retrieve it, or perhaps ask the authors for a hard-copy.

main author:

Sergey А. Arsen’yev

Dept. of Earth and Planetary Physics of Schmidt’s Institute of the Earth’s Physics, Russian Academy of Sciences, 10 Bolshaya Gruzinskaya, Moscow, 123995, Russia

How to get students interested in climate science

William Kessler of NOAA wrote this:

I am a physical oceanographer who knows nothing about the Chandler wobble, is only slightly familiar with the QBO, but is a longtime expert on ENSO.
To be blunt, trying to shoehorn ENSO into a periodic tidal framework stretches reality to fit someone’s preconceived theory. Only the most motivated reasoning can believe this.

… (more stuff)

I am sorry to have wasted an hour on this.

Billy Kessler, NOAA/PMEL, Seattle

Interactive comment on Earth Syst. Dynam. Discuss.,,

Billy also wrote this on his web site (emphasis mine):

4. An idea for a science fair project.
Requested by a parent.

Here’s an idea. This experiment is similar to what actual scientists are doing right now.

The project is to construct some forecast models of El Niño’s development over the next few months. We don’t know what it will do. Will it get more intense?, weaken?, remain strong?, and if so for how long? These are the subject of much debate in the scientific community right now, and many efforts are under way to predict and understand it.

The models would be forecasts made using several assumptions, and the main result would be graphs showing how the forecasts compared with actual evolving conditions.

One model would be called “persistence”. That is, whatever conditions are occurring now, they will continue. Surprisingly, persistence is often a hard-to-beat forecast, and weather forecasters score themselves on how much better than persistence thay (sp) can do. A second model is continuation of the trend. That is, if the sea surface temperature (SST) is warming up it will continue to warm at the same rate. Obviously that can’t go on forever but in many ways a trend is a good indicator of future trends. A third model is random changes. Get a random number generator (or pick numbers out of a hat). Each day or week, use the random numbers to predict what the change of SST will be (scale the numbers to keep it reasonable). Those are three simple models that can be used to project forward from current conditions. Essentially that’s what weather forecast models do, just more sophisticatedly (see question 13). Maybe you can think of some other ways to make forecasts (if you get something that works, send it in!)

Choose a few buoys from our network in different regions of the tropical Pacific (for example, on the equator, off the equator, in the east, and the west). Get the data from our web page (click for detailed instructions to get this data). Make and graph predictions for each buoy chosen for a month or two ahead, then collect observations as they come in (the data files are updated daily). Graph the observations against the three predictions. My guess is that each model would be successful in some regions for some periods of time. Other extensions would be to compare forecasts beginning at different times. Perhaps a forecast begun with September comditions (sp) is good for 3 months, but one begun in December is only good for one month. Etc.

Another simple project is to determine how significant an effect El Niño has on your local region. Do this by gathering an assortment of local weather time series from your region (monthly rainfall, temperature, etc) (available at the web pages of the National Weather service). Then get an index of El Niño like the Southern Oscillation Index (see Question 17 for a description and graphic, and download the values at NOAA’s Climate Prediction Center. The specific data links are: values for 1951-today and 1882-1950. Note that the SOI monthly values are very jumpy and must be smoothed by a 5-month running mean). Compare the turns of the El Niño/La Niña cycle with changes in your local weather; this could either be through a listing of El Niño/La Niña years and good/bad local weather, or by correlation of the two time series (send me e-mail for how to do correlation). You will probably find out that some aspects of your local weather are related to the El Niño/La Niña cycle and some are not. Also that some strong El Niño or La Niña years make a difference but some do not. This reflects the fact that, far from the center of action in the tropical Pacific, El Niño is only one of many influences on weather.

If your (sp) are pretty good at math and computer programming (at least 8th-grade math), then I have a more advanced project that you can find here.

FAQ from

shorter: “your thay”

Face with rolling eyes

Nonlinear Generation of Power Spectrum : ENSO

Something I learned early on in my research career is that complicated frequency spectra can be generated from simple repeating structures. Consider the spatial frequency spectra produced as a diffraction pattern produced from a crystal lattice. Below is a reflected electron diffraction pattern of a reconstructed hexagonally reconstructed surface of a silicon (Si) single crystal with a lead (Pb) adlayer ( (a) and (b) are different alignments of the beam direction with respect to the lattice). Suffice to say, there is enough information in the patterns to be able to reverse engineer the structure of the surface as (c).

from link

Now consider the ENSO pattern. At first glance, neither the time-series signal nor the Fourier series power spectra appear to be produced by anything periodically regular. Even so, let’s assume that the underlying pattern is tidally regular, being comprised of the expected fortnightly 13.66 day tropical/synodic cycle and the monthly 27.55 day anomalistic cycle synchronized by an annual impulse. Then the forcing power spectrum of f(t) looks like the RED trace on the left-side of the figure below, F(ω). Clearly that is not enough of a frequency spectra (a few delta spikes) necessary to make up the empirically calculated Fourier series for the ENSO data comprising ~40 intricately placed peaks between 0 and 1 cycles/year in BLUE.

click to expand

Yet, if we modulate that with an Laplace’s Tidal Equation solution functional g(f(t)) that has a G(ω) as in the yellow inset above — a cyclic modulation of amplitudes where g(x) is described by two distinct sine-waves — then the complete ENSO spectra is fleshed out in BLACK in the figure above. The effective g(x) is shown in the figure below, where a slower modulation is superimposed over a faster modulation.

So essentially what this is suggesting is that a few tidal factors modulated by two sinusoids produces enough spectral detail to easily account for the ~40 peaks in the ENSO power spectra. It can do this because a modulating sinusoid is an efficient harmonics and cross-harmonics generator, as the Taylor’s series of a sinusoid contains an effectively infinite number of power terms.

To see this process in action, consider the following three figures, which features a slider that allows one to get an intuitive feel for how the LTE modulation adds richness via harmonics in the power spectra.

  1. Start with a mild LTE modulation and start to increase it as in the figure below. A few harmonics begin to emerge as satellites surrounding the forcing harmonics in RED.
drag slider right for less modulation and to the left for more modulation

2. Next, increase the LTE modulation so that it models the slower sinusoid — more harmonics emerge

3. Then add the faster sinusoid, to fully populate the empirically observed ENSO spectral peaks (and matching the time series).

It appears as if by magic, but this is the power of non-linear harmonic generation. Note that the peak labeled AB amongst others is derived from the original A and B as complicated satellite-cross terms, which can be accounted for by expanding all of the terms in the Taylor’s series of the sinusoids. This can be done with some difficulty, or left as is when doing the fit via solver software.

To complete the circle, it’s likely that being exposed to mind-blowing Fourier series early on makes Fourier analysis of climate data less intimidating, as one can apply all the tricks-of-the-trade, which, alas, are considered routine in other disciplines.

Individual charts


Modern Meteorology

This is what amounts to forecast meteorology — a seasoned weatherman will look at the current charts and try to interpret them based on patterns that have been associated with previous observations. They appear to have equal powers of memory recall, pattern matching, and the art of the bluff.

I don’t do that kind of stuff and don’t think I ever will.

If this comes out of a human mind, then that same information can be fed into a knowledgebase and either a backward or forward-chained inference engine could make similar assertions.

And that explains why I don’t do it — a machine should be able to do it better.

What makes an explanation good enough? by Santa Fe Institute

Plausibility and parsimony are the 2P’s of building a convincing argument for a scientific model. First one has to convince someone that the model is built on a plausible premise, and secondly that the premise is simpler and more parsimonious with the data that any other model offered. As physics models are not provable in the sense of math models, the model that wins is always based on a comparison to all other models. Annotated from  DOI: 10.1016/j.tics.2020.09.013


Chandler Wobble Forcing

Amazing finding from Alex’s group at NASA JPL

What’s also predictable is that the JPL team probably have a better handle of what causes the wobble on Mars than we have on what causes the Chandler wobble (CW) here on Earth. Such is the case when comparing a fresh model against a stale model based on an early consensus that becomes hard to shake with the passage of time.

Of course, we have our own parsimoniously plausible model of the Earth’s Chandler wobble (described in Chapter 13), that only gets further substantiated over time.

The latest refinement to the geoenergy model is the isolation of Chandler wobble spectral peaks related to the asymmetry of the northern node lunar torque relative to the southern node lunar torque.

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