ENSO redux

I've been getting push-back on the ENSO sloshing model that I have devised over the last year.  The push-back revolves mainly about my reluctance to use it for projection, as in immediately.  I know all the pitfalls of forecasting -- the main one being that if you initially make a wrong prediction, even with the usual caveats, you essentially don't get a second chance.   The other problem with forecasting is that it is not timely; in other words, one will have to wait around for years to prove the validity of a model.   Who has time for that ? 🙂

Yet, there are ways around forecasting into the future. One of which primarily involves using prior data as a training interval, and then using other data in the timeline (out-of-band data) as a check.

I will give an example of using training data of SOI from 1880 - 1913 (400 months of data points) to predict the SOI profile up to 1980 (800 months of data points). We know and other researchers [1] have confirmed that ENSO undergoes a transition around 1980, which obviously can't be forecast.   Other than that, this is a very aggressive training set, which relies on ancient historical data that some consider not the highest quality. The results are encouraging to say the least.

One other bit of push-back I have received concerns the use of TSI as a forcing parameter. In the past, I used it mainly as a means of fitting the 1980 transition, as the TSI profile seems to track the 1980 transition, with the TSI plateauing right around that time.  But in comparison to the other factors in the model, notably the QBO and the Chandler wobble forcing, it lacks an angular momentum basis. As the sloshing model is about considering factors that have a physical forcing basis, I removed TSI for consideration for the time being. So instead of the TSI, I applied the diurnal and semi-diurnal tidal beat frequencies of 18.6 years and 8.85 (and subharmonics) to supply alternate forcing fitting factors for the aggressive tact I am taking.  These are factors that supply angular momentum variations so they are in the same category as QBO and the Chandler wobble.

As an initial premise, the QBO and Chandler wobble are assumed to have a fixed fundamental frequency and some jitter around that period, fitted to the data available.  The tidal frequencies are stationary by definition.  In addition, the Quasi-triennial oscillation (QTO) frequencies identified by R. Kane are incorporated to provide periods between 3.5 to 3.8 years [2][3][4], as well a strict quadrennial oscillation (QO) of 4.0 years.

A critical factor in the fitting process is in being able to model the QBO and Chandler wobble as a stationary process.  The data for QBO is only available back to 1953, so the model to 1880 is a pure backward extrapolation.  The same holds for the Chandler wobble, where recent data is much more accurate than the older historical measures of wobble.

As the fitting interval is rather short, I also removed the Mathieu/Hill modulation factor from consideration.  I also let the fit find a characteristic period of 7.24 years instead of the 4.25 years I used in the paper (which is based on analysis by Allan Clarke [5]).  The correlation coefficient shown in Figure 1 below is 0.92.

Fig 1: Aggressive sloshing model fit to 1913. This is solving the sloshing wave equation via Mathematica. The yellow regions are regions of disagreement.

The aggressiveness of the fit is achieved by adjusting the forcing factor parameters (scale and phase) as tightly as possible to maximize the correlation coefficient. This can often backfire in curve-fitting applications, as the factor errors can quickly amplify and grow enormous beyond the fitting region.

Yet, continuing the integration beyond 1913 on to 1980, the fit shown in Figure 2 obviously degrades somewhat, yet the peak and valley positions are strongly evident.  The combined correlation coefficient is 0.73.  This is consistent with the factors providing a stationary process, with no accelerating error amplification observed.

Fig 2: Fit to 1980, the projected time series starts at the vertical tic marks indicated at 1913/

Only when the integration reaches 1981-1982 does a real discrepancy occur. See Figure 3 below. It appears that the strong El Nino of 1982-1983 becomes a La Nina according to the model.  This disturbance is consistent with the findings in Ref [1], who apply Takens embedding theorem to show that this short interval differs from the rest of the time series.

Fig 3: Complete fit of the SOI time series.  At around 2001-2002, the model becomes 180 degrees out-of-phase with the data. It appears to eventually recover.

It's becoming more and more evident that the sloshing model is heading in a direction of improved conciseness and greater simplification. The stationary angular momentum factors (QBO, Chandler wobble, tidal factors) are taking on greater significance, and some of the complicating factors such as TSI and the Mathieu factor may be fading in importance. The 1980 disturbance is likely the stumbling block that has prevented straightforward analysis of the ENSO time series in past work.


If you read through the series of ENSO ContextEarth posts from beginning to now, you will see something peculiar in the evolution of my entries, but a trend that is not totally unexpected.

I started out by assuming the model had to be something more complicated than the obvious solution. After all, people had been working on the ENSO prediction problem for ages and no one had seemed to make any progress, right? Initially, I figured that the answer had to contain significant complexity, otherwise someone else would have come up with a solution. So I spent lots of spare time devising characterization strategies that could manage the complexity.

At first I tried out various automated optimization and fitting approaches, but then after a while, the modeled behavior became simpler and the amount of code and parameters involved began to decrease. That's the nature of first-order physics analysis, as the principal factors start to take hold.

Eventually I removed the automation and worked with the model manually. The more I worked it, the more concise and elegant the model has become.  What you see now is the skeleton of a canonical model that should be able to do projections.



[1] H. Astudillo, R. Abarca-del-Rio, and F. Borotto, “Long-term non-linear predictability of ENSO events over the 20th century,” arXiv preprint arXiv:1506.04066, 2015.

[2] R. Kane, Y. Sahai, and C. Casiccia, “Latitude dependence of the quasi‐biennial oscillation and quasi‐triennial oscillation characteristics of total ozone measured by TOMS,” Journal of Geophysical Research: Atmospheres (1984–2012), vol. 103, no. D7, pp. 8477–8490, 1998.

[3] R. Kane, “Quasi-biennial and quasi-triennial oscillations in the rainfall of Northeast Brazil,” Revista Brasileira de Geofísica, vol. 16, no. 1, pp. 37–52, 1998.

[4] R. Kane, “Differences in the quasi‐biennial oscillation and quasi‐triennial oscillation characteristics of the solar, interplanetary, and terrestrial parameters,” Journal of Geophysical Research: Space Physics (1978–2012), vol. 110, no. A1, 2005.

[5] A. J. Clarke, S. Van Gorder, and G. Colantuono, “Wind stress curl and ENSO discharge/recharge in the equatorial Pacific,” Journal of physical oceanography, vol. 37, no. 4, pp. 1077–1091, 2007.

8 thoughts on “ENSO redux

  1. If instead of the first 400 months as a training interval, I use the third 400 month interval, this is how it extrapolates back and forward in time. The disturbance at 1980 is very evident yet the profile starts to get back in phase within a few years.

    fit and extrapolation

  2. Getting push-back elsewhere, where my comments are being deleted.

    This is my comment:
    It's really not that complicated. The QBO and Chandler wobble waveforms are fitted separately as analytical functions so that the numerical integration algorithm is stable. The final model is therefore transitively constructed and those other parts are absorbed.

    The analogy is of emulating an audio filter. The forcing input waveform can be complex but the transfer function is not. The transfer function in this case is a second-order differential wave equation with a single parameter.

    " So given your excellent match through 1913 in the above, what happens after that?"

    Did you even read the entry completely? I used the interval from 1880 to 1913 as training. The data was evaluated then to 1980, with a very good match.

    The interesting feature is a disturbance that occurs right after 1980. Other researchers have found the same thing, see this Arxiv paper -- H. Astudillo, R. Abarca-del-Rio, and F. Borotto, “Long-term non-linear predictability of ENSO events over the 20th century,” arXiv preprint arXiv:1506.04066, 2015.

    Just received a reply from Dr. Abarca del Rio of the ENSO paper referred to above:
    Rodrigo Abarca del Rio
    Thanks for your interest !!
    Yes, it's under review. I know it is a breakthrough, but you know how the system is ...
    Keep waiting.

  3. > Only when the integration reaches 1981-1982 does a real discrepancy
    > occur. See Figure 3 ....

    So I'm metamodeling the slosh model as a simple pendulum, a swing occupied by a kid who is sitting without wiggling too much.

    Until 1981-1982, when the kid learns how to pump a swing by timing when to extend and when to pull in .... that is, the point at which the warming signal emerges.

    Too simple, I trust.

    • Hank, Could well be. I will post a graph of adding a significant forcing disturbance at 1980 and scaling the disturbance so the post-1980 years are the focus of the model fit. One way to do this is to adjust the boundary-conditions y(x) and y'(x) for x=1980 so as to find the best fit from 1980 to 2013.

      What is interesting about it is that if we back-propagate the disturbance we see that the fit is bad until it gets to 1880-1915, in which case the data and model get back in sync.

      The alternate way is to add a Dirac delta function disturbance at 1980, and assume that this is causal and so it won't propagate backwards.

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