Digital Diatribes

A presentation of data on climate and other stuff

Curve-Fitting the ENSO Index Suggests We Have the Highest El Nino Event since 1998

Posted by The Diatribe Guy on January 28, 2010

ENSO 200912

ENSO Index and Fitted Curves @ 12/31/2009

I’ve continued to work, as I have the time to, on pulling the Oceanic Oscillation data, developing a fitting spreadsheet for each index, to get some idea of what the underlying cyclical nature of the oscillations may be.

I decided to post the above chart on ENSO, since I (a) completed it, (b) we’re currently in the midst of an El Nino, and (c) most amateur climatologists know about it and occasionally like to take a look at it.

The source for the data can be found on the right side of this page. The largest hurdle in the curve-fitting is that good(?) data only goes back to 1950. Even this may be a little questionable, and from what I’ve read, any proxied data prior to that is even less reliable. But, we’ll run with the data we have with the caveat that is always there about analysis only being as good as the accuracy of the data and all that.

I’ve been very interested in trying to understand the tendency of these oscillations to cycle (or not cycle, as the case may be). The ENSO index, in particular, can have the appearance of a random variation with no particular pattern.

At this point, I need to explain what I did here, and then I need to explain what I am saying and what I am not saying regarding the conclusions.


1) I have simultaneouosly fitted a long-period wave and a short-period wave to the ENSO data
2) The following elements have been fitted:

For the long period wave:

  1.  Best fit sine wave with a period of at least 30 years
  2. A scale factor to determine amplitude of the wave
  3. A phase increment amount to determine the length of the wave
  4. A starting point on the wave cycle to be fitted at the beginning of the data
  5. A vertical shift, to account for bias in the zero-anomaly base assumption
  6. A slope of linear trend

For the short wave:

  1. Best fit sine wave with a period less than 30 years
  2. A scale factor for amplitude
  3. A phase increment amount to determine cycle length
  4. A start point on the wave at the beginning of the data

There is no shift or trend determined on the short wave determination, since this will follow the path of the long wave.

Through a simultaneous and recursive process, all these elements are simultaneously solved to produce the minimum value of squared differences from the point on the short wave to actual ENSO index readings.   The ultimate solution is not necessarily incorporating the best-fitted long wave taken in isolation.   I initially ran the long-wave fit first, and then separately ran the best fit short wave along that curve.   Moving to running everything simultaneously helped the overall fit and actually reduced the length of the long-wave.    The difference is not huge, but since it is the best fit and the results appear reasonable, I went with that.


The results of this analysis show a 50.5 year ENSO cycle that underlies the shorter-term variations.   I have shown this before, and it is an interesting consideration in evaluating the relative magnitude of certain ENSO events, not so much as it relates to the zero value, but as it relates to the long-term wave.   The current long-term wave is on a decline, and may, in fact, be bottoming out in another four or five years.

The starting point is just past the halfway mark of the cycle, so we see a lower-index period at the beginning of the chart.  The amplitude of the wave is about 0.32 at its peak.   So, from top to bottom (with no linear trend) there is a difference of 0.64 in the magnitude contributed to ENSO events from one period to another, depending on where the long-term cycle is at.

There seems to be, in addition to the cycle, a linear trend in the data for which the long-term cycle moves along, at least since 1950.   It may be that there is a third cycle that is substantially longer that is being mistaken for a linear trend.   This may matter in the long run, but for a 60-year time period the linear approximation should suffice.   However, it may well be that we need a number of additional years of data to better fit this and judge whether or not there is a linear trend, or some other cycle at work.  For now, though, I go with the best fit, and for that the rate of change is 0.7 degrees Celsius per century.

This long wave is being fit simultaneously with a short wave placed on its path.   The period of this short wave is 4.93 years.  Its scale is 0.49.   So, at the top of this wave, plus the top of the long wave, we are adding 0.81 degrees to the ENSO index.   The first wave starts at about the 220 degree mark of a cycle.

Key Assumptions (What I’m doing versus what I’m suggesting)

There are a couple key assumptions here.  The primary assumption is the selection of a sine wave for fitting.  I am not saying this is the best assumption.  All I am saying is, given this assumption, there is a best fit.   As I look at the data, it actually doesn’t do too bad a job in aligning with peaks and valleys.  However, it is far from perfect.   There are other ways to manipulate this, if desired.  One can select a skewing assumption so that the wave peaks earlier or later in the cycle than at 90/270 degrees.   Or, once can assume that there is a factor that compresses or expands length over time (or both in some oscillating pattern over years).   Another thing to look at is to see if the length of sunspot cycles impacts the difference in timing of ENSO peaks/valleys, as I’ve seen suggested.

All these are potential refinements that could improve results.   However, all that said, I still think there are some interesting results here.   The most significant El Nino events do seem to correspond well with peaks in both waves.

The other assumption is that there are two cycles to consider.  A third assumption that can be questioned is the validity of a linear trend in the data.


One may think that the waves represent the anticipated direction of the ENSO index.  That is actually not what the waves imply.  The red wave pattern marks the “starting point” for the current index.  From there, deviations may go up or down.    This may be the most confusing aspect on how to read the chart.   It’s not so much about predicting El Nino or La Nina, it’s about showing how the ultimate magnitude is affected.


June 1955 ENSO index = -2.270; Wave value = -0.76.  Negative deviation = -1.51

September 1973 ENSO index = -1.71; Wave value = +0.20.  Negative deviation = -1.91.

One could argue that the 1973 event was substantially more profound than the 1955 event, even though the actual ENSO index reading was lower in 1955.


The most profound El Nino event, in terms of a deviation from the underlying wave, actually occurred in 1983 – not 1998.  The March 1983 reading = 3.11, and the wave value = 0.77, for a differential of 2.34.    The maximum deviation during the “Super El Nino” was actually August 1997 – a difference of 2.03.

In fact, we actually have a fairly significant event occurring right now.   The December 2009 deviation from the wave is +1.54, which is the largest difference since April of 1998!


4 Responses to “Curve-Fitting the ENSO Index Suggests We Have the Highest El Nino Event since 1998”

  1. Fluffy Clouds (Tim L) said

    I know how dumb this sounds but here it goes…

    use an “ABS” absolute value on the data then place a wave on that results the wave will be faster.

    thank you for your efforts.

  2. Mike Haseler said

    WALL OF DEATH – What’s happened to HADCRUT3.

    My webbrowser told me today that the HADCRUT3 data had finally been posted. I’ve just looked the changes – there’s very little white – which would be OK, if white meant: “the same”, and yellow didn’t mean change.

    Before I start entering 160 new temperatures, thought I would warn you!

  3. The Diatribe Guy said

    You enter them?

    I would recommend “cut and paste”!

  4. bancuri said


    […]Curve-Fitting the ENSO Index Suggests We Have the Highest El Nino Event since 1998 « Digital Diatribes[…]…

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