I'll start by summarizing predictors:
- Climatology 1979-2000: 7.03 million km^2
- Climatology 1979-2008: 6.67 million km^2
- Linear Trend 1979-2009: 5.37 million km^2
- Wu and Grumbine modeling: 5.13 million km^2
- Grumbine and Wu statistical ensemble: 4.78 million km^2
- Grumbine and Wu best fit statistical: 4.59 million km^2
The basis of the statistical prediction starts with my eyeball reaction to this figure (this particular copy is from Julienne Stroeve by way of the Weather Underground). It is comparing IPCC model estimates of ice against observations:
So what's going on?
To make a curve fit, as usual, you first consider what shape curve you would like (straight line, or something more involved), and then look for the best version of that curve. The curve of the models doesn't look like a straight line to me. Nor does the observed ice. One way of deciding that is that the straight line that my eye puts through the observations for the first half of the data record doesn't sit on the straight line my eye puts on the second half. We prefer to be quantitative, of course, and running the numbers for the first 16 years and the last 15 years gives slopes of -0.045 and -0.152, respectively. To be rigorous, we would then want to perform statistical significance tests. Still, those look quite different.
The curve that I chose is the logistic curve. Except it's upside down, so I had a little work to do of a sort that we often do in math and science.
The logistic curve is most common in areas like population growth. At least in mathematical idealizations of population growth. Early on, when the population is small, the growth is proportional to the population. This makes for exponential growth, for instance 2% per year. But, as time goes on, the population starts to get large enough that resource limits take over. Those limits slow down the growth. You then slowly approach a limiting population.
Considering the opposite of the thing of interest is often useful in math and science. In computing probabilities, it is often easier to determine directly how likely it is for something not to happen, than for it to happen. But, since the probabilities have to total to 1, if you know one side (happen versus not-happen), you can then easily get the other. So go with what is easiest to work with first. We'll come back to this point in thinking about when we might see an ice-free arctic (September average).
In my case here, the ice cover is not growing -- it's shrinking. So I can't do a logistic curve of ice cover growth. But the opposite of ice cover is open water. What I can do instead of considering ice cover is to consider the area of open water. This fits nicely the general idea of a logistic curve. In the early part of the growth (of open water), there is a feedback that more open water leads to more open water (due to increased heat absorption by the ocean, thinner ice, ...). The 'population' of open water is limited, though. You can only have as much open water as there was ice cover to begin with.
One thing which has bothered me about, say, linear extrapolation, or quadratic, etc., is that if you go far enough into the future, they say that ice cover will be negative. With the logistic curve growth of open water area, this will not happen. As you see above, the ice extent goes to zero and stays there. But it never becomes negative.
I then ran my program to search for the best logistic curve -- one that has the least squared errors. That gives my plot above. But, in talking with Xingren, we realized that quite a lot of logistic curves were very, very close to as good as the best one. We could ignore those others (the 'ensemble' of curves). But, given the uncertainties in the data, this didn't make much sense to us. What we did was to take all the logistic curves that were pretty good, and average their predictions of what 2010 would see. That's the ensemble prediction. For the Sea Ice Outlook, it is our preferred prediction (preferred over the best single logistic curve).
Notes to come:
Making predictions with an imperfect model
When might we see an ice free (September average) Arctic?
Reminder: I don't speak for my employer, whoever that is.