Thursday, May 31, 2012

Sea Ice anomalies

Since Jaxa resumed posting daily results of Arctic Sea Ice extent, I've been processing the data and posting plots here. I'm thinking of doing an interactive plotter for the various ice measures, similar to the climate plotter.You could choose which years to show.

So I started looking at means and anomalies, and thinking about trend calculations. The general method for anomalies seems to be to take a day-by-day average over some years and calculate discrepancies from that. The number of years is not huge, and there is a spike of variability near the maximum melt, so the daily mean is not free of the fluctuations from which an anomaly is sought. So I tried a multiple regression using trig functions and a secular trend.

An anomaly plot is useful for plotting in that it removes the seasonal oscillation, making the key information more visible. With the trig functions I can use a sidereal year, which gets rid of a minor but awkward leap year issue. I found four harmonics was sufficient as the magnitudes taper quite fast; using five made negligible difference.

I looked at the Jaxa data, and also the Arctic and Antarctic sea extent data from UIUC (Cryosphere Today).

There are two anomalies calculated - without and with trend adjustment. All curves have been very slightly smoothed, with a 5-point binomial filter.


Here's the familiar Jaxa plot with the fitted periodic annual curve.

2012 is marked in black.

Here is the anomaly plot with trend, shown over the years from 2002:

The downtrend is about 65000 sq/m/year. Last years minimum was about 4500000sq km; at that rate of attrition it will be nearly 70 years before it disappears. However, there's no reason to expect that trend to apply over the period, and indeed the minima seem more variable than the general trend.

Now here is the same data plotted over a one year interval. Again 2012 is black. It shows that the anomaly so far this year is mid-range for the decade.

However, plotting the anomaly after allowing for trend shows that so far, 2012 is rather higher than expected.

UIUC Cryosphere Today Arctic

UIUC published data over a much longer period. They do their own anomaly calculations, which are in the data set and published on their website. But these are from the trig function regression. Here is the anomaly plot over years:

The trend is somewhat less than Jaxa, since the period is three times longer. Now the same plot superimposed.

There are now a lot of curves, so I've shown the last ten years with greater thickness.

Finally the trend-removed anomaly plot. Again 2012 is currently fairly high relative to recent years.

UIUC Cryosphere Today Antarctic

Finally, for completeness, the Antarctic data. They are of course six months out of phase, and the trend is up rather than down, though the magnitude is less than Arctic. Here are the anomalies:

The same over years:

and with trend removed:


  1. I'v done some FFT's on the UIUC dailies and found peaks (or spikes) at the 1st thru 4th (as you have mentioned), nothing at the 5th, but did see a peak at the 6th (~every 2 moths), and no significant peaks at higher frequencies.

    These asymmetries are most likely due to "basin effects" meaning that the land boundaries impose seasonal temporal constraints onto the annual arctic sea ice extent/area time series (e. g. when Hudson Bay is fully iced over vs when it is full melted out is not symmetric with the annual cycle).

    This gets us into super and sub harmonics of the system, where the super harmonics (two annual cycles that are slightly different in phase/shape produce superharmoics at 2X, 3X, ...) are nultiples of the annual cycle.

    This is somewhat similar to what we see in harbor resonance modeling (HOT or higher order terms appear in the spectra from the fundimental pumping frequency, but in that case it is due to shallow water effects and at basin harmonics of each basin's primary pumping frewuency (I've directly observed this for the 2nd harmonic at the shoreline of a laboratory physical model where those long waves break, in doing harmonic analyses of these same data sets (at gage locations elsewhere in the harbor proper) we saw at least the 1st 6 super harmonics, but the difference frequency is ~0 for closely spaced (in frequency space) monoccromatic waves).

    We were never much interested in those diffreence frequencies (very low frequency), in harbor resonance modeling, since those occur so low in the resulting spectra as to be unimportant to induce ship motions (our harbor resonance studies were confined to long waves of 30-400 sec durations).

  2. EFS,
    Thanks for those observations. I think the frequencies are fascinating when you can relate them to geometry and oscillation modes.

    I tried the sixth harmonic. I find that fourth and higher are small, though they diminish rather slowly, But I didn't see a spike at six.

  3. Nick, did you use "raw" or "anomalized" data?

    I used my combined GSFC /JAXA record to generate this spectrum.

    I'm assuming EFS is referring to the non-uniform drop-off of harmonics, right? You'd expect harmonics to be present because solar forcing itself isn't strictly sinusoid in the Arctic. Right?

    1. Carrick,
      I used raw data - I'm using the first four harmonics to form the mean that I subtract to get the anomalies. Your plot looks very like what I found - the harmonics attenuate rapidly to the fourth, then fifth and sixth bounce around. Since they make little difference to the mean I didn't look into this too much.

      Yes, you'd expect some harmonics, but with rapid attenuation. There's no obvious natural source of high frequencies.

      I'm not using Fourier analysis, though - just least squares fitting. The reason is that the data isn't exactly sampled at fractions of a period (calendar vs sidereal) and I use the fragments of years as well. Also I want to couple the linear trend in a multiple regression.

    2. Of course you can derive the coefficient for Fourier coefficients from least-squares fitting, but I get the point about simultaneously fitting to the linear trend. (It may not be that different though than fitting to the anomalized series, it'd be interesting to compare results.)

      I resampled the data to 365-samples per year before taking the Fourier transform.

      For the multiyear variability (ENSO/Arctic oscilaltion-driven portion) fitting to a quasipeirodic function is better than what I did here. Yes I have the results but I'd have to dig them out.)

      The trend estimate is still pretty interesting. Here's what I get for the slope. I think the wiggles are probably real---the periods of minimum ice loss correspond to El Niño years and maximum ice loss to La Niña years. (And yes there is definitely a coupling between ENSO and Arctic Oscillation.)

      THe "jagginess" of the figure comes from the step-size between windows, I could produce a smoother version if you were interested (I was sampling at the Nyquist frequency, which is more useful for curve fitting if you don't want to deal with artificially induced autocorrelation in your data.)

      Incidentally here's what a spectral analysis of the DMI temperature series looks like:

      There are just three harmonics there, so something else is generating the rest of them in the ice extent record. (The fact they don't monotonically decrease as I said is already evidence that more than one mechanism is at play here.)

  4. By the way, I didn't anomalize the data before taking the spectrum (otherwise most of the frequency components associated with the annual cycle gets subtracted off). Meant to say that.

  5. What's this nonsense "Yes, I think the Guardian is wrong. I think they are misinterpreting this calculation, which says that the minimum ice volume in Sep 2011 was 75% down on the maximum for 1979."? They're just pointing out that the ice volume has declined 75% since 1979. What do you think the Grauniad is wrong about?

    1. You're right that I've misinterpreted the PSC calc myself. I thought PSC were comparing Sep '11 with Mar '79 ("the maximum in 1979"). But of course if I'd done the arithmetic, that would be a much bigger ratio.

      Still, the Guardian should have said that they were comparing min (Sep) with min. And it would have helped to say that they were using volume as the measure. That's correct, of course, but many people think of extent.

      I'll have to petition WUWT for you to be unbanned, if I ever get out of the woods myself :).

    2. I was going to try to correct at WUWT but I see Phil Clarke has done that.

  6. Phil's also not got it quite right IMO. The uncertainties are large. The interannual variation in the minimum is large. Saying "75%" in a news report without qualification is misleading and should have been avoided by better vetting of the article.

    Also he says: "Now I don’t totally agree with the Greenpeace/Guardian choice of presenting just 75% as this is taken from the month showing the greatest loss (the mean is around 66%), though one could argue that for albedo changes this is the important number."

    Of course, ice volume has little to do with albedo other than very indirectly.

    I don't know how you can argue as Anthony appears to be doing that the Guardian was saying "extent" when they clearly labeled the ice loss as by volume for that figure.

  7. On another thread, Bart's left the reservation again and is on the warpath.

    Ferdinand is trying to help with some nice modeling. I admire his patience with all of the foolishness that goes on there.

    1. Oh boy, and it's still barely light here. A lot to catch up on here.

  8. Oh dear. Well, at least one person there has a clue. AW clearly doesn't like the obvious being pointed out though; and it really doesn't say much for the quality of the folk there that it took that far down the comment thread for someone to point out the bleedin' obvious, and that even once it was done they were too thick to understand. I sense a post coming on.

    1. No, quite a lot of people picked it up. here is JohnB, just after me.

    2. Looking more closely I agree with you: the commentors manage to get the right answer, despite AW's best attempts to lie to them ( Would it not be honourable for you to go back and correct your now-known-to-be-incorrect slur on the Graun?

    3. Well, a correction would be suitably humble. But I did link to the right calculation, so people could see the basis. I said that I thought the G had misinterpreted it - not a dreadful slur. In fact I had. But the numbers are there now, and several commenters have given the correct interpretation, vindicating the G. So I wouldn't be adding to the now past discussion.

      I would make this excuse for my error - the PSC said
      "Monthly averaged ice volume for September 2011 was 4,200 km3. This value is 66% lower than the mean over this period, 75% lower than the maximum in 1979"
      which does sound rather like they mean March 1979. But I should have checked the arithmetic.

      Oddly, here I am accused of spinning in the Guardian's defence!

    4. PG is a looney. No-one cares what she thinks. OTOH at least some people have in the past cared what you thought.

  9. Nick don't you agree the Guardian could (and should) have explained how they got that number and what it represents? It's hardly a reasonable argument to say "the average reader will spend three days vetting this 75% number and recognize all of the nuances that have to be applied before it's not a blatant falsehood."

    I'm not sure what I'm more bemused about here... AWs over the top reaction to an article that appeared in the MUSIC section of the Guardian (McKibben's opinion piece that gets ignored had a lot more fresh meat associated with it), or the somewhat over-the-top defense of what I view as a noobish journalistic mistake. As I said above:

    The uncertainties [in volume] are large. The interannual variation in the minimum is large. Saying "75%" in a news report without qualification is misleading and should have been avoided by better vetting of the article.

    Even though we can squint our eyes and hold our tongues *just so* and it's only technically true then, that hardly excuses journalistic ham-handedness. Yes even, or especially, in the music section (where the serious science gets discussed).