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Reverse Engineering the Moon's Orbit from ENSO Behavior

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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Improved Solver Target Error Metric

In addition to the algorithm used for solving optimization problems, an important criteria is the form of the metric used to minimize the error or maximize the similarity between data and model.

The commonly used forms such as error variance (i.e. mean squared error) have issues related to how well they can navigate the search space. Other forms such as correlation coefficient (CC) often work better, but at the expense of losing track of scale.  This indicates that CC is better at matching the general characteristics of a specific shape than a pure error criteria. And if weighted, it can deal with noisy intervals.

In fact, the symbolic reasoner Eureqa features a proprietary metric referred to as a hybrid correlation coefficient.  From my experiences with the tool, hybrid version does qualitatively work better.

So in my quest to find an alternative metric, I came up with something related to the Cosine Similarity (CS) measure. As defined, CS is not that different from Pearson's correlation coefficient as it does not subtract the mean. But with a slight modification it's an excellent "starter" metric for initial exploration.

The new metric is essentially a +/- excursion matching criteria (EMC), which is important for a behavior as cyclically erratic about the origin as ENSO.

The algorithm for the EMC can be described as a ratio of two factors. The numerator is the sum of the multiplications of the model and data values. The denominator is the normalizing factor, which is the sum of the multiplication of the absolute values of each value.

EMC =  \frac{\sum x_i \cdot y_i}{\sum |x_i| \cdot |y_i|}

The resulting metric ranges from -1 to 1, with 1 being a perfect sign excursion matching, and -1 if all excursions had the sane magnitude but were reversed in sign.

This of course is not a perfect criteria as it will tend to force the minimal excursions to zero while maximizing the maximum excursions, instead of first normalizing them as the true CS does.

The evidence to how well it works is mainly based on observations in massive reductions in search time. For ENSO model optimization search, the EMC reduces the time it takes to get in the ballpark by 100×, so what could take an hour reduces to about a minute of computational time. It is important not to let it overfit, so wait until the metric starts to slow in its improvement before stopping the search and switching to the CC metric for the final stages optimization.

As it is so fast I have been using it for minimally filtered ENSO time series, where I can start from minimally seed sets of parameters. This gives more confidence that results are not correlated from one search optimization run to the next.

The EMC is therefore a great metric for randomizing searches. I can imagine using it in a scenario with different initialized seed values and then waiting a fixed time to return an interim solution, and then using the best of these in a more refined CC search.

Why it works so well is something I am still trying to explain. It is a more efficient computation than CC, but that is not enough to explain 100x.




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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The Earth's invisible "Saturn ring": The QBO

This is puzzling:

“This temperature and zonal wind structure resembles those of Earth’s quasi-biennial oscillation (QBO) and Jupiter’s quasiquadrennial oscillation (QQO), in which temperature anomalies and eastward/westward winds alternate in altitude”
Fouchet, T., et al. “An equatorial oscillation in Saturn’s middle atmosphere.” Nature 453.7192 (2008): 200.

And recently the final results of the Cassini spacecraft mission were in the news:

“The density wave is generated by the gravitational pull of Saturn’s moon Janus.”
Wild! Cassini Probe Spots Weird Waves in Saturn's Rings September 11, 2017 https://www.space.com/38114-weird-waves-saturn-rings-cassini-photo.html

But no one in the research literature has made the connection of the moon's orbit to the dynamics of the QBO.

From Fouchet et al, again

On Earth, the alternating wind regimes repeat at intervals that vary from 22 to 34 months, with an average period of about 28 months. On Jupiter, the equatorial stratospheric temperature exhibits a 4.4-year period and the equatorial zonal winds in the upper troposphere oscillate with a 4.5-year period. Long-term ground-based monitoring reveals a period of 14.7±0.9 terrestrial years on Saturn. The observational similarities between Saturn’s oscillation and the QBO and QQO are the strong equatorial confinement of temperature minima and maxima and associated shear layers, a stronger eastward than westward shear layer, and the bounding of the equatorial oscillation at latitudes 15–20° north and south. Temperatures near these latitudes are relatively high when equatorial temperatures are relatively low, and vice versa.
On Earth and Jupiter, the quasi-periodic oscillations are triggered by the interaction between upwardly propagating waves and the mean zonal flow.

Both Jupiter and Saturn have 4 significant moons,making the collective lunar orbit difficult to describe. It's possible that Saturn's and Jupiter's "QBO" are more like the Earth's upper stratosphere oscillations, which align to the semiannual period (0.5 year period). This is suggestive as the values for Jupiter and Saturn's "QBO" period are closer to 1/2 the planet's full calendar year period, as show below

Planet "year" length "QBO" period "QBO upper" period
Earth 1 year 2.37 years 0.5 year
Saturn 29.46 years
14.7±0.9 years
Jupiter 11.86 years
4.5 years



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.

But it has really not been clearly established that the measure CW period is a resonant frequency.  I have a detailed rationale for a lunar forcing of CW in this post, and Robert Grumbine of NASA has a related view here.

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).

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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.

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After spending several years (edit: part-time) on formulating a model of ENSO (then and now), I decided to test out the formulation on another standing wave oscillation — specifically, the Atlantic Multidecadal Oscillation (AMO). All the optimization infrastructure was in place, with the tidal factors fully parameterized for automated model fitting.

This fit is for a training interval 1900-1980:

The ~60 year oscillation is a hallmark of AMO, and according to the results, this arises primarily from the anomalistic lunar forcing cycle modulated by a biennial seasonal modulation.  Because of the spiked biennial modulation, we do not get a single long-period cycle but one that is also modulated by the forcing monthly tidal periods. As with ENSO, second-order effects in the anomalistic cycle described by lunar evection and variation is critical.

Outside of the training interval, the cross-validated test interval matches the AMO data arguably well. Since AMO is based on SST anomalies, it's possible that strong ENSO episodes and volcanic perturbations (e.g. post 1991 Pinatubo eruption) can have an impact on the AMO measure.

This is a typical fit over the entire interval.

This is the day after I started working on the AMO model, so these results are preliminary but also promising.  AMO has a completely different character than ENSO and is more of an upper latitude phenomenon, which means that the tidal forces have a different impact than the equatorial ENSO cycle. Some more work may reveal whether the volcanic or ENSO forcing overrides the tidal forcing in certain intervals.

Identification of Lunar Parameters and Noise

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).

f = Year/LunarMonth - 13

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.

Yellow shading at intervals around 1930, 1936, 1948  indicate discrepancies between the NINO34 data in green and the ENSO model in red.


ENSO tidal forcing validated by LOD data

This is a straightforward validation of the forcing used on the lunar-driven ENSO model.

The paper by Chao et al [1] provides a comprehensive spectral analysis of the earth's length of day (LOD) variations using both a wavelet analysis and a power spectrum analysis. The wavelet analysis provides insight into the richness of the LOD cyclic variations (c.f. the Chao ref 6 in a recent post) :

Both the wavelet and the power spectrum (below) show the 6-year Fourier component that appears in the ENSO model as a mixed tidal forcing.

The original premise is that the change in LOD via the equivalent angular momentum change will impart a forcing on the Pacific ocean thermocline as per a reduced-gravity model:

Taken from presentation for ref [2]

Calculating a spectral analysis of the best fit ENSO model forcing, note that all of the model peaks (in RED) match those found by Chao et al in their ΔLOD analysis :

There are additional peaks not found by Chao but those are reduced in magnitude, as can be inferred from the log (i.e. dB) scale. If these actually exist in the Chao spectrum, they may be buried in the background noise.  Also, the missing Sa and Ssa peaks are the seasonal LOD variations that are taken into account separately by the model, as most ENSO data sets are typically filtered to remove seasonal data.

The tidal constituents shown above in the Chao power spectra are defined in the following Doodson table [3]. Chao likely is unable to discriminate the tropical values from the draconic and anomalistic values, being so close in value. On the other hand, the ENSO model needs to know these values precisely.  Each of the primary Mm, Mf, Mtm, and Mqm and satellite Msm, Msf, Mstm, Msqm factors align with the first 4 harmonics of the mixed nonlinear ENSO model with the 2nd order satellites arising from the anomalistic correction.

Tidal constituent coefficients taken from ref [3]

This is an excellent validation test because this particular LOD power spectrum has not been used previously in the ENSO model fitting process. If the peaks did not match up, then the original premise for LOD forcing would need to be reconsidered.



[1] B. F. Chao, W. Chung, Z. Shih, and Y. Hsieh, “Earth’s rotation variations: a wavelet analysis,” Terra Nova, vol. 26, no. 4, pp. 260–264, 2014.

[2] A. Capotondi, “El Niño–Southern Oscillation ocean dynamics: Simulation by coupled general circulation models,” Climate Dynamics: Why Does Climate Vary?, pp. 105–122, 2013.

[3] D. D. McCarthy (ed.): IERS Conventions (1996) (IERS Technical Note No. 21) :
Chapter 6