# 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.

# AMO

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.

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# 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.

## References

[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

# Earthquakes, tides, and tsunami prediction

I've been wanting to try this for awhile — to see if the solver setup used for fitting to ENSO would work for conventional tidal analysis.  The following post demonstrates that if you give it the recommended tidal parameters and let the solver will grind away, it will eventually find the best fitting amplitudes and phases for each parameter.

The context for this analysis is an excellent survey paper on tsunami detection and how it relates to tidal prediction:

S. Consoli, D. R. Recupero, and V. Zavarella, “A survey on tidal analysis and forecasting methods for Tsunami detection,” J. Tsunami Soc. Int.
33 (1), 1–56.

The question the survey paper addresses is whether we can one use knowledge of tides to deconvolute and isolate a tsunami signal from the underlying tidal sea-level-height (SLH) signal. The practical application paper cites the survey paper above:

Percival, Donald B., et al. "Detiding DART® buoy data for real-time extraction of source coefficients for operational tsunami forecasting." Pure and Applied Geophysics 172.6 (2015): 1653-1678.

This is what the raw buoy SLH signal looks like, with the tsunami impulse shown at the end as a bolded line:

After removing the tidal signals with various approaches described by Consoli et al, the isolated tsunami impulse response (due to the 2011 Tohoku earthquake) appears as:

As noted in the caption, the simplest harmonic analysis was done with 6 constituent tidal parameters.

As a comparison, the ENSO solver was loaded with the same tidal waveform (after digitizing the plot) along with 4 major tidal parameters and 4 minor parameters to be optimized. The solver's goal was to maximize the correlation coefficient between the model and the tidal data.

The yellow region is training which reached almost a 0.99 correlation coefficient, with the validation region to the right reaching 0.92.

This is the complex Fourier spectrum (which is much less busy than the ENSO spectra):

The set of constituent coefficients we use is from the Wikipedia page where we need the periods only. Of the following 5 principal tidal constituents, only N2 is a minor factor in this case study.

In practice, multiple linear regression would provide faster results for tidal analysis as the constituents add linearly (see the CSALT model). In contrast, for ENSO there are several nonlinear steps required that precludes a quick regression solution.  Yet, this tidal analysis test shows how effective and precise a solution the solver supplies.

The entire analysis only took an evening of work, in comparison to the difficulty of dealing with ENSO data, which is much more noisy than the clean tidal data.  Moreover, the parameters for conventional tidal analysis stress the synodic/tropical/sidereal constituents — unfortunately, these are of little consequence for ENSO analysis, which requires the draconic and anomalistic parameters and the essential correction factors. The synodic tide is the red herring for the unwary when dealing with global phenomena such as ENSO, QBO, and LOD. The best way to think about it is that the synodic cycle impacts locally and most immediately, whereas the anomalistic and draconic cycles have global and more cumulative impacts.

# Limits to Goodness of Fit

Based on a comparison of local interval correlations between the NINO34 and SOI indices, there probably is a limit to how well a model can be fit to ENSO.  The lower chart displays a 4-year-windowed correlation coefficient (in RED) between the two indices (shown in upper chart):

Note that in the interval starting at 1930, the correlation is poor for about 7 years.

Next note that the ENSO model fit shows a poor correlation to the NINO34 data in nearly the same intervals (shown as dotted GREEN). This is an odd situation but potentially revealing. The fact that both the ENSO model and SOI don't match the NINO34 index over the same intervals, suggests that the model may match SOI better than it does NINO34.  Yet, because of the excessive noise in SOI, this is difficult to verify.

But more fundamentally, why would NINO34 not match SOI in these particular intervals? These regions do seem to be ENSO-neutral, not close to El Nino or La Nina episodes.  Some also seem to occupy regions of faster, noisy fluctuations in the index.

It could be that the ENSO lunar tidal model is revealing the true nature of the ENSO dynamics, and these noisier, neutral regions are reflecting some other behavior (such as amplitude folding) — but since they also appear to be obscured by noise, it makes it difficult to unearth.

The paper by Zajączkowska[1] also applies a local correlation to compare the lunar tidal cycles to plant growth dynamics. There's a treasure trove of recent research on this topic.

# Second-Order Effects in the ENSO Model

For ocean tidal predictions, once an agreement is reached on the essential lunisolar terms, then the second-order terms are refined. Early in the last century Doodson catalogued most of these terms:

"Since the mid-twentieth century further analysis has generated many more terms than Doodson's 388. About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many fewer can predict tides to useful accuracy."

That's possibly the stage we have reached in the ENSO model.  There are two primary terms for lunar forcing (the Draconic and Anomalistic) cycles, that when mixed with the annual and biannual cycles, will reproduce the essential ENSO behavior.  The second-order effects are the  modulation of these two lunar cycles with the Tropical/Synodic cycle.  This is most apparent in the modification of the Anomalistic cycle. Although not as important as in the calculation of the Total Solar Eclipse times, the perturbation is critical to validating the ENSO model and to eventually using it to make predictions.

The variation in the Anomalistic period is described at the NASA Goddard eclipse page. They provide two views of the variation, a time-domain view and a histogram view.

 Time domain view Histogram view

Since NASA Goddard doesn't provide an analytical form for this variation, we can see if the ENSO Model solver can effectively match it via a best-fit search to the ENSO data. This is truly an indirect method.

First we start with a parametric approximation to the variation, described by a pair of successive frequency modulated (and full-wave rectified) terms that incorporate the Tropical-modified term, wm. The Anomalistic term is wa.

COS(wa*t+pa+c_1*ABS(SIN(wm*t+k_1*ABS(SIN(wm*t+k_2))+c_2)))

$\cos(\omega_a t+\phi_a+c_1 \cdot |\sin(\omega_m t+k_1 \cdot |\sin(\omega_m t+k_2)|+c_2)|)$

This can generate the cusped behavior observed, but the terms pa, c_1, c_2, k_1, and k_2 need to be adjusted to align to the NASA model. The solver will try to do this indirectly by fitting to the 1880-1950 ENSO interval.

Plotting in RED the Anomalistic time series and the histogram of frequencies embedded in the ENSO waveform, we get:

 Time domain view of model Histogram view of model

This captures the histogram view quite well, and the time-domain view roughly (in other cases it gives a better cusped fit).  The histogram view is arguably more important as it describes the frequency variation over a much wider interval than the 3-year interval shown.

What would be even more effective is to find the correct analytical representation of the Anomalistic frequency variation and then plug that directly into the ENSO model. That would provide another constraint to the solver, as it wouldn't need to spend time optimizing for a known effect.

Yet as a validation step, the fact that the solver detects the shape required to match the variation is remarkable in itself. The solver is obviously searching for the forcing needed to produce the ENSO waveform observed, and happens to use the precise parameters that also describe the second-order Anomalistic behavior.  That could happen by accident but in that case there have been too many happy accidents already, i.e. period match, LOD match, Eclipse match, QBO match, etc.

# Search for El Nino

The model for ENSO includes a nonlinear search feature that finds the best-fit tidal forcing parameters.  This is similar to what a conventional ocean tidal analysis program performs — finding the best-fitting lunar tidal parameters based on a measured historic interval of hundreds of cycles. Since tidal cycles are abundant — occurring at least once per day — it doesn't take much data collected over a course of time to do an analysis.  In contrast, the ENSO model cycles over the course of years, so we have to use as much data as we can, yet still allow test intervals.

What follows is the recipe (more involved than the short recipe) that will guarantee a deterministic best-fit from a clean slate each time. Very little initial condition information is needed to start with, so that the final result can be confidently recovered each time, independent of training interval.

# The Hawkmoth Effect

Contrasting to the well-known Butterfly Effect, there is another scientific modeling limitation known as the Hawkmoth Effect.  Instead of simulation results being sensitive to initial conditions, which is the Butterfly Effect, the Hawkmoth Effect is sensitive to model structure.  It's a more subtle argument for explaining why climate behavioral modeling is difficult to get right, and named after the hawkmoth because hawkmoths are "better camouflaged and less photogenic than butterflies".

Not everyone agrees that this is a real effect, or it just reveals shortcomings in correctly being able to model the behavior under study. So, if you have the wrong model or wrong parameters for the model, of course it may diverge from the data rather sharply.

In the context of the ENSO model, we already provided parameters for two orthogonal intervals of the data.  Since there is some noise in the ENSO data — perfectly illustrated by the fact that SOI and NINO34 only have a correlation coefficient of 0.79 — it is difficult to determine how much of the parameter differences are due to over-fitting of that noise.

In the figure below, the middle panel shows the difference between the SOI and NINO34 data, with yellow showing where the main discrepancies or uncertainties in the true ENSO value lie. Above and below are the model fits for the earlier (1880-1950 shaded in a yellow background) and later (1950-2016) training intervals. In certain cases, a poorer model fit may be able to be ascribed to uncertainty in the ENSO measurement, such as near ~1909., ~1932, and ~1948, where the dotted red lines align with trained and/or tested model regions. The question mark at 1985 is a curiosity, as the SOI remains neutral, while the model fits to more La Nina conditions of NINO34.

There is certainly nothing related to the Butterfly Effect in any of this, since the ENSO model is not forced by initial conditions, but by the guiding influence of the lunisolar cycles. So we are left to determine how much of the slight divergence we see is due to non-stationary variation of the model parameters over time, or whether it is due to missing some other vital structural model parameters. In other words, the Hawkmoth Effect is our only concern.

In the model shown below, we employ significant over-fitting of the model parameters. The ENSO model only has two forcing parameters — the Draconic (D) and Anomalistic (A) lunar periods, but like in conventional ocean tidal analysis, to make accurate predictions many more of the nonlinear harmonics need to be considered [see Footnote 1]. So we start with A and D, and then create all combinations up to order 5, resulting in the set [ A, D, AD, A2, D2, A2D, AD2, A3, D3, A2D2, A3D, AD3, A4, D4, A2D3, A3D2, A4D1, A1D4, A5, D5 ].

This looks like it has the potential for all the negative consequence of massive over-fitting, such as fast divergence in amplitude outside the training interval, yet the results don't show this at all.  Harmonics in general will not cause a divergence, because they remain in phase with the fundamental frequencies both inside and outside the training interval. Besides that, the higher order harmonics start having a diminished impact, so this set is apparently about right to create an excellent correlation outside the training interval.  The two other important constraints in the fit, are (1) the characteristic frequency modulation of the anomalistic period due to the synodic period (shown in the middle left inset) and (2) the calibrated lunar forcing based on LOD measurements (shown in the lower panel).

The resulting correlation of model to data is 0.75 inside the training interval (1880-1980) and 0.69 in the test interval (1980-2016).  So this gets close to the best agreement we can expect given that SOI and NINO34 only reaches 0.79.  Read this post for the structural model parameter variations for a reduced harmonic set to order 3 only.

Welcome to the stage of ENSO analysis where getting the rest of the details correct will provide only marginal benefits;  yet these are still important, since as with tidal analysis and eclipse models, the details are important for fine-tuning predictions.

Footnote:

1. For conventional tidal analysis, hundreds of resulting terms are the norm, so that commercial tidal prediction programs allow an unlimited number of components.

Switching between two models