A previous post described the use of proxy records of ENSO to fit the Southern Oscillation Index Model (SOIM). This model fit used one specific set of data that featured a disconnected record of coral measurements from the past 1000 years, see Cobb [1].
As the focus of this post, another set of data (the Unified ENSO Proxy set) is available as an ensemble record of various proxy measurements since 1650 — giving an unbroken span of over 300 years to apply a SOIM fit [2]. This ensemble features 10 different sets, which includes the Cobb coral as a subset.
To fit over this long a time span is quite a challenge as it assumes that the time series is stationary over this interval. The data has a resolution of only one year, in comparison to the monthly data previously used, so it may not have the temporal detail as the other sets, yet still worthy of investigation. (an interactive version is available here).
For the initial attempt, the same Mathieu-like differential equation structure and parameters were applied as for the previous proxy and for the modern-day SOI data. The figure below is repeated from the previous proxy post.
Fig. 1: Previous SOIM fits to proxy records and modern day SOI data
Parameters approximating those in Figure 1 are used in the unified ENSO Proxy fit as shown below:
Fig 2: SOIM fit (red) to unified ENSO Proxy (blue) using parameters close to other proxy fit.
This correlation coefficient is only 0.256, which is not as good as the previous model fits over shorter time spans.
To see if this fit could be improved, the strategy chosen was to start from a reduced set of forcing data. So the forcing parameters corresponding to w1 and w2 were removed, leaving only w3. A small drag factor was also included as y'[x].
Fig. 3: Single forcing sinusoid fit
This improved the correlation coefficient from 0.256 to 0.336, which is not bad considering the length of data.
As a final fit, an additional forcing sinusoid was added corresponding to 1.548 rads/year (see Figure 4). This improved the correlation coefficient marginally to above 0.36.
Why do the values of w1 and w2 of around 2.5 to 3 rad/year not appear on this extended proxy record? As these are higher frequencies, it is entirely possible that coherence can not be maintained over the 300+ year range of proxy values that the SOIM model is fit against. The sampling resolution of 1 year also works against selecting higher frequencies.
Fig 4: An additional forcing sinusoid marginally improved the fit.
Recall that high correlation coefficients are hard to achieve on these oscillating time series. Even the correlation coefficient of Tahiti to Darwin is only -0.55 as shown in the previous post on SOIM. Any amount of noise will quickly reduce the coefficient from 1.0. However, the authors of the Unified ENSO Proxy study find high correlation coefficients ranging from 0.5 to 0.84 to the SOI and various Pacific equatorial SST data sets during the modern era. This indicates that the unified proxy reconstruction likely does what it is intended to — represent the actual dynamics of past history without having the use of thermometers or hygrometers.
The variance in the 10 proxy reconstructions is shown below in Figure 5.
Fig 5: Standard deviation in variability between proxy data sets is shown in (b) with the gray background in (a) showing the scatter.
An interesting feature of the Mathieu equation fit is how it captures the variance in the ENSO signal across decades. The early years show less variance in ENSO peak-to-peak modulation than the later years, and the Mathieu equation formulation can capture this growing modulation. See [3] for more.
Overall the results remain encouraging and it is good to know that so many proxy reconstructions are available for ENSO along with historical dates for El Nino events, and that this data can be further mined for fitting extended time-span models.
References
[1] Cobb, Kim M, Christopher D Charles, Hai Cheng, and R Lawrence Edwards. “El Nino/Southern Oscillation and Tropical Pacific Climate during the Last Millennium.” Nature 424, no. 6946 (2003): 271–76.
[2] McGregor, S., A. Timmermann, and O. Timm. “A Unified Proxy for ENSO and PDO Variability since 1650.” Clim. Past 6, no. 1 (January 5, 2010): 1–17. doi:10.5194/cp-6-1-2010. PDF
[3] Carré, Matthieu, Julian P Sachs, Sara Purca, Andrew J Schauer, Pascale Braconnot, Rommel Angeles Falcón, Michèle Julien, and Danièle Lavallée. “Holocene History of ENSO Variance and Asymmetry in the Eastern Tropical Pacific.” Science (New York, NY), 2014.
GeoEnergy Math 
Running Eureqa on the UEP data set reveals a strong factor along the Pareto front of 7.84 rads/year (from a modulated 7.82 with a side-band of 0.014). Since the data is only sampled once per year, this is likely an aliased period for for an intra-annual component.
The anomalistic tidal month of 27.55455 days results in 83.285 rads/year. But subtracting 12 * 2π from this gives 7.887 rads/year, which is very close to 7.84 rads/year.
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In the modern day non-Proxy SOI records, I found three significant frequencies (http://contextearth.com/2014/05/27/the-soim-differential-equation/) in the context of a Mathieu-like DiffEq (http://mathworld.wolfram.com/MathieuDifferentialEquation.html) formulation.
There is the nonlinear modulation of about 8.3 years, and a combined forcing frequency of a 6 year period and a 28 month = 2.33 year period. The latter 28 month period is common to the Quasi-Biennial Oscillation (http://en.wikipedia.org/wiki/Quasi-biennial_oscillation) of stratospheric winds where the periodicity is quite striking. The 6 year period has no obvious connection but similar periods occur when looking at periodic jerks (http://phys.org/news/2013-07-pair-year-oscillations-length-day.html) in the rotation of the earth, the Chandler wobble (http://www.technologyreview.com/view/415093/earths-chandler-wobble-changed-dramatically-in-2005/), and the beat frequency between the anomalistic and draconic lunar month (http://en.wikipedia.org/wiki/Lunar_month).
This plot is where I broke up the UEP time series in two approximately equal intervals, with the break point at the year 1818. Since this is a yearly sample, I did not filter the data any further.
* The 1.537 number is the Mathieu equation modulation in rads/year on the LHS
* The 1.02/1.01 numbers are the first and second half of the ~6-year forcing period on the RHS
* The 2.67/2.68 numbers are the first and second half of the QBO forcing period on the RHS
I took the liberty of trying to modulate the QBO with another small cyclic term to model what appears to be a variation in the QBO forcing itself.
I believe that even though the correlation coefficient is “only” 0.42, this is a deceptively good fit and it is consistent with the model I fit to the ENSO SOI data. I am not sure how much further I can tweak the fit, as it seems to be close to converging.
The noise in the data seems to be a factor as the EUP is an ensemble of 10 different records, and there is considerable variance in the data from the different records. There is thus likely a “ceiling” to the correlation coefficient even if the fundamental underlying waveform is discovered. This correlation coefficient ceiling may in fact may be as low as 0.6 –a guess based on what I have seen in the past with such a busy waveform.
The remaining issue is that there may be other combinations of parameters that provide an even better fit.
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