Friday, May 29, 2009

Sea Ice Odds

I've been working out, finally, my thoughts on what the sea ice cover will be like for the Arctic, in September 2009. That for the average for the month, rather than minimum at any time.

But also a good time to return to some basics about sea ice. Two different questions we could ask about ice cover are: 1) what area of the earth (Arctic ocean, Hudson Bay, ...) has sea ice floating on it? 2) If I look at a grid of sea ice coverage, what is the area of the grid cells that has any noticeable ice in it? The former is sea ice area, and the latter is sea ice extent. See some of the web sites I link to on the right for samples.

In terms of measurement, extent is much easier to get than area. If there is sea ice in an area, it usually covers a large fraction of the surface around there. Extent doesn't worry about whether it was 70% covered or 95% covered. As long as, on the average, you can correctly distinguish between more than 15% and less than 15%, you're set. Sea ice area, on the other hand, is more challenging. Now you have to be confident that when the satellite says '10% water', all 10% are because of the ocean. Problem is, the water could also be ponds that are on the surface of the ice. (Melt ponds; we're not very creative in our vocabulary.) You can't have an ice floe entirely covered by melt ponds -- the water would run off. So extent is still safe, even if area could be off by 30% in a part of the pack with lots of melt ponds.

My prediction, then, regards the extent of ice, rather than the area. Predict numbers you can measure more reliably if you have a choice. For other reasons, it's for the average extent for the whole month. (As computed by the NSIDC, and, again, Arctic only).

The prediction is: 4.92 million km^2, with a standard deviation to it of 0.47 million km^2. 2008 showed 4.67 million km^2, and 2007 showed 4.30 million km^2 for the month's average. I'm not saying that the sea ice is 'recovering'. Actually, if my method (which, notice, I haven't told you what it is) is right, then I also expect the Arctic ice to be going to zero sometime after 2022, which is on the early end of estimates. How much after requires the next step of sophistication in the method. That'll come later. Recovery would mean 6.67 million km^2 (the mean of the last 30 years, arguably, it should mean more like 7.4 million). All that's involved is that 2007 and 2008 were extraordinary, which we already knew, and remain extraordinary even after going to a better :-) model for the progression.

I don't make cash bets, or recommend doing so. I will, however, bet honor points. (Quatloos, a group I was in named it.) One place I'll be making such a bet is Stoat, William Connolley's site. Then again, we'll have to see if he finds it attractive. Given my prediction, one obvious bet is even money that the September average will be less than (or more than) the number I gave, 4.92 million km^2. Someone who thinks the sea ice has recovered (which does not include William) should be leaping up with glee to take the 'more than' side of the bet. Actually, though, if they really believed that ice had recovered, they should go for even odds the the ice would be more than 6.67 million km^2.

This is where we get to the 'odds' mentioned. If someone believes that there's been no real change in the sea ice cover, just some bouncing around, or if they think it has recovered, this means they have a particular stastical model in mind. Namely, that sea ice in any given year should be about the mean. And there's a degree of spread (the standard deviation). This model says, given the last 30 years of observations, that we should see (every year) 6.67 million km^2, with a standard deviation of 0.87. The 0.87 is the measure of spread. One wouldn't be surprised to see ice cover be less than 5.8 million km^2, about 1 year in 6 should do so. On the other hand, only 1 year in 40 should be more than 2 standard deviations below normal, 4.9 million km^2. Someone who believes that the Arctic ice has recovered, then, should be offering me 20:1 odds that the Arctic ice extent will be more than 4.97 million km^2 (working this number out with full detail).

So I'll invite those who've been saying that Arctic sea ice has recovered the 20:1 bet -- they owe me 20 quatloos if the the Arctic ice extent averages less than 4.97 million km^2 for September, as computed by NSIDC. I give them 1 if it's more. I'll also invite you to refer people who say that the Arctic (or who just say 'sea ice') has recovered to this note. Or, if they prefer, I'll give 20 quatloos to their 1 that the figure will be less than 6.67 million km^2. By their thinking, it's even chance that it'll be more than 6.67. By mine, it's awfully unlikely.

Also open to other offers. Add them to the comments. I might be slow to respond, but offers will be considered through the end of June. If you think the trend is a simple straight line, your prediction is for 5.46 million km^2, with standard deviation 0.53 (again based on 30 years). You should offer your 5 quatloos to my 3 that the figure will be greater than my prediction of 4.92. I'll give 5 to your 3 that it will be less than 5.46. Notice that there's a symmetry to the odds.

Oh well. I hope I haven't just confused most everyone, and bored to tears the rest.

Currently having technical difficulties, but I'm adding a poll at the bottom about how it'll turn out. The numbers are centers of the ranges. 5 means 4.75 through 5.25, for instance.


Alastair said...

There is a Japanese site here from which you can download the daily data for the last few years. The data is updated twice a day but only the final value is saved.

What odds would you give me on the ice dropping below 40,000,000 km^2 this year?

Penguindreams said...

Considering that the global cover has not been as high as 40 million km^2 in recorded history, I can't see a bet. If you want to take the position that it'll be greater than 40 million km^2 for a monthly average for some (any) month in calendar 2009, I'll put 1000 quatloos against your 1 that you're wrong.

I thought, though, that you'd be offering a bet on being below 4 million. I don't think it'll be below 4. If you want to take the under' side of that, I'll put my 50 quatloos to your 1 that it'll be over 4. (If you run the numbers, you'll see that I'm hedging either bet in my favor.)

Alastair said...

Yes, an extra 0 slipped in there, possible since the current value of 11,500,000 sq km is in that order.

I'll take you bet, of 50:1 that it will reach less than 4 M sq km at some time this year. Or are these odds that you giving me for it being below 4 M sq km for a complete calendar month? I wouldn't rule that out for September, but I would like better odds than 50:1.

There is a good chance that it will not get down to 4 M until early September, but stay there through early October.

Belette said...

*I*'ll offer you real money for your under 4 million. Though I agree with PD that monthly averages are a better thing to look at, and quite probably extent too.

Bob: is this for ?

"Given my prediction, one obvious bet is even money that the September average will be less than (or more than) the number I gave, 4.92 million km^2" - isn't this rather an odd bet? You wouldn't care which side you took, since either side has (in your view) a 50% chance of winning. Isn't the obvious bet for you that the sept ice will be 4.92 +/- 1 SD?

Penguindreams said...

It's a fair bet -- meaning that I'm willing to take either side. 1:1 for over versus under 4.92 million km^2. I'd also take either side of a quatloo bet based on the standard deviation, so, what 19:1 that the extent will average within 1.645 * .47 million km^2 of the 4.92 (or 1:19 that it won't). (I forget whether 1.645 is the range for 95%, but we can look up that number.) Fair bet, so if you want the inside of the range, I'll take the outside. I do aim, though, at things which can be evaluated with the results this year. If, say, I were wrong about the standard deviation, it'd take quite a few years to establish that (especially since I don't think the mean is stationary).

On the other hand, haven't you been taking that even linear might be too agressive a curve? In that case, I'd want better odds from you :-)

Yes, that's what I'm referring to on ARCUS.

Belette said...

"It's a fair bet -- meaning that I'm willing to take either side" - yes that is what I meant. It is a fair bet so not very interesting to you. The game is to find a bet that you think is unfair, but someone else thinks is fair.

My own pet theory is that 2007/8 were outliers and that we expect recovery to the long term trend. But I haven't worked out the statistics of what that makes me expect. More ice than you expect, I suspect. Oh go on - you've got all the data and stats packages to hand - suppose I expect 2009 to be on the 1979-2006 trend line, what is my prediction and SD?

Alastair said...

I have just re-read your post and have to admit, as you feared, I am thoroughly confused :-(

But forget that.

Not having managed to get you to raise your odds I'll bet one quatloo that the average for the month of September will be less than 4 million sq. km. If it is less than 4 you pay me 50 quatloos, which I will donate to the charity of my choice, viz. The Robert Grumbine Education Trust, to be spent on the purchase of a copy of "The Black Swan" by Nassim Taleb, friend and colleague of Benoit Mandelbrot :-)

I did not mention earlier that there is yet another graph of ice extent that is not mentioned directly in your Science links here. I deliberately did not mention it fearing that after you had seen it you might have offered lower odds. Do you want to change your odds now?

We already have a bet where you offered 2:1 that this year would not be a record minimum. I've put £100 on it.

Of course I would love to put another £100 (which I can afford) at 50:1 that it will be below 4 M sq km at some time this year, but can you afford to pay out? I'll put 10P at those odds. My £5 winnings will pay for a celebratory beer when I collect the £200 for my other bet :-)

Gareth said...

My bet with William is for a new record (we were betting on CT numbers IIRC), so I guess I'm on the warm side of your estimate.

When will the ARCUS forecasts show up? Nothing at the web site yet... Presumably that's when we learn the nitty-gritty of your method ;-)

jyyh said...

FYI, on there's a poll of the same, like the ice would take a note. You might have some luck someone taking the bet there.

EliRabett said...

William an I have a principle disagreement. I think that 2007 was not an outlier, but a step change, mostly because sea ice has hysteresis since one year ice is much easier to melt than multiyear ice.

2007 was an abnormally warm year in the Arctic coupled with a perfect circulation storm. Absent another such, or an abnormally cold SERIES of years to regrow old ice and given the pressure of a gradually warming climate, I anticipate that the average minimum ice cover will resume its slow decline, but from a lower base. FWiW

Penguindreams said...

You're fairly close to the professional consensus in this. A major supporting element is the near-obliteration of thick ice in the Arctic in the late 1990s-present. Had it not been preconditioned that way, the 'perfect storm' of weather conditions in 2007 could not have clobbered so much area. Ignatius Rigor of the International Arctic Buoy Program gave a nice presentation on this at the recent Arctic conference hosted by the National Ice Center and the Arctic Research Commission. Jim Overland also favored the sentiment (Jim, with Hajo Eicken, coordinates the ARCUS prediction pool, among other things).

I'm taking a different view almost strictly because most people are taking the view that 2007 represented a step change in the system. If most thinking is in one direction, I'll look for a different direction. Mine shortens to "Ice is taking a sigmoidal approach to ice-free conditions." This still leaves 2007 (and 2008) as fairly drastic outliers, meaning it could be a substantially incorrect view. The data will show within a few years.

Ok, you're taking an even more different from the consensus view than I am. I'll work up what your model predicts and put it in part of a free-standing post for our bet. Whatever your prediction method comes up with, I'll make an even quatloos bet than September 2009 (as computed at NSIDC) extent is below the midpoint between my prediction and yours. We both go away chucking at what a lopsided bet the other accepted :-) But make it 50 quatloos.

Alastair: Ok, my 50 to your 1. I'll take that the September monthly average extent as computed by the NSIDC is over 4 million km^2. And by all means, feel free to contribute to my education. If it's book form, though, be forewarned that I have about 300 on my 'to read' bookcases.

I'll stand by the prediction. Observing that the extent loss picked up a few days later doesn't change the basis of my prediction. I figure I should stick with it for at least 1 season before abandoning it. I do link to the NSIDC, and see their figure or William Chapman's, on pretty regular basis.

Gareth: Jim Overland said earlier this week that he's expecting the ARCUS forecasts to be up in a couple of weeks. It takes time to bring the disparate submissions in to some vaguely consistent form.

In general: If Alastair is right, I take my winnings from the bet with William, and pay Alastair. Now I'm covered! :-) But it's still only quatloos.