I was reading Tamino's latest post, and one of the commenters wrote about losses in Arctic sea ice volume, and how some other people had said it could be as high as 75% over the last 30 years. So I thought I'd check that using the latest PIOMAS data.

But when I do calculate something like this, I always wonder/worry about what if the first few years in the data series were high or low relative to the overall trend, or what if the last year was an unusual fluctuation -- that would skew the results and give a number that wasn't really indicative of the long-term trend.

But if you assume the trend is linear, you can, with just a little algebra, get a useful result for the percentage loss (or gain) based on just the linear trend. (I'm sure this isn't original, but I haven't encountered it before.)

Take the PIOMAS data and calculate its slope

*m*via the usual linear regression.

The long-term percentage loss L, in the linear model, is

L = (y

_{2}-y

_{1})/y

_{1}

where y

_{1}and y

_{2}are the values from the linear fit: y

_{1}=mx

_{1}+b and y

_{2}=mx

_{2}+b, where m is the slope of the trend line as determined by linear regression, b is the intercept, and x

_{1}and x

_{2}the endpoints of the line.

With a little algebra you find, defining the data's average value as A [= (y

_{1}+y

_{2})/2]

where Δx (= x

_{2 }- x

_{1}) is just the length of the data record. [If you need help deriving this, just note that you have two equations with two variables, y

_{1}and y

_{2}. Solve as usual.] So you only have to calculate "bulk properties" of the data: m, A and Δx. Pretty neat.

The PIOMAS data (Δx = 33.4 yrs) gives A = 20.64 Kkm

^{3}, m = -0.30 Kkm

^{3}/yr, so

L = -39.0%

or -1.2% per year.

(A linear fit probably isn't the "best" fit -- my spreadsheet finds a slightly higher R

^{2}for a power law fit -- but this is just blog-work, not rocket science.)

Like I said, I'm sure this isn't original, but it's a little useful (at least to numbers geeks like me), and it was fun to work out.

(Mostly it was fun to work out.)

## 11 comments:

According to the IPCC FAR in 1990, the data from 1979 was high. In the early 1970s, it was lower. In other words, its probably a cycle. Picking 1979 after this report, in my opinion, is picking the high point.

http://www.ipcc.ch/ipccreports/far/wg_I/ipcc_far_wg_I_full_report.pdf

One minor point on your analysis.

When looking for trends in a wave function, you should start and stop in the same place in the cycle.

Staring in the upswing of the cycle, and finishing in the down swing, will always give you a negative trend, even in a sine wave with equal amplitude troughs and waves.

It would be best to go peak to peak or trough to trough.

Granted, it won't make much difference in this case.

The statistic of primary interest is minimum volume. If you redo your analysis using the annual minima (either actual minimum value, or September averages), you will get an estimate much closer to 75%.

Just looking at the first and last annual cycles, the maximum has only gone down by about 33% and the minimum about 75%. And if you read the comment you pointed to, it clearly refers to the summer minimum.

The linear trend of the summer minimum might show a little less decline, but not much.

The minimum has gone down about 75% from the

March 1979 maximum, according to the Greenpeace study.One should not compare minimum to maximum. It will always be negative.

The 75% figure refers to the decline in Spetember minimum, as I already explained and should be obvious from the actual PIOMAS figure in any case. But don't just take my word for it; read what PIOMAS scientist Axel Schweiger says (as reported at Carbonbrief.org).

To check, we contacted PIOMAS scientist Dr. Axel Schweiger. He explained that the 75 per cent figure was actually comparing minimum to minimum - it refers to a comparison between the September 1979 minimum ice volume and the September 2011 minimum ice volume.

So, as Dr. Schweiger told us:

"[W]hen referring to one particular measure of "Arctic sea ice loss" : PIOMAS ice volume, then that statement [the 75% figure from the PIOMAS website] is correct."

You're welcome.

Except if 1979 was a statistical fluke, or 2011 was, then the percentage decrease would be deceptive and not indicative of the long-term trend.

For example, the 1981 PIOMAS minimum was 25% below 1979's, but by 1986 the minimum has increased compared to that and was only 5% below 1979's.

I agree that the trend would be more accurate. However, in this case there is good evidence that summer minimum decline is accelerating and not linear (i.e. recent points are below the linear trend). A loess or gompertz fit would give a better indication of the trend (and the gompertz may also be a pretty good naive predictor). This is in line with the expected curve of summer sea ice decline, although it's happening much faster than AR4 models projected.

The linear trend gives a decline of about 66%. But as I say that appears to be overly conservative.

Yes, I agree that the decline is now more than linear, so my method in the post isn't that great in this case (as I noted).

Do you know if there is any theoretical expectation of what the functional form of the decline should now be? (Or a back of the envelope argument from basic physics?)

The reason the power law regression worked better is because of positive feedback mechanisms on arctic temperature change. The average rate of arctic melt is accelerating as less arctic ice covers the ocean and less sunlight is reflected back into space (aldebo).

The reason the power law regression worked better is because of positive feedback mechanisms on arctic temperature change. The average rate of arctic melt is accelerating as less arctic ice covers the ocean and less sunlight is reflected back into space (aldebo).

try plotting only the minimum value for each year.

http://stevengoddard.files.wordpress.com/2012/06/paintimage5214.jpg

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