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For the quick understanding of trends, I have always liked mathematical formulas which possess the correct asymptotic limits (imposed by the underlying physical processes).

I have many past experiences building phenomenological models of mass/heat/momentum transfer processes where such mathematical fits/approximations are more than enough to capture the essence of the phenomena at hand, and many times allow for the design of real-life processes and equipment.

Models based on sound physics are irreplaceable, but in the meantime when there is no alternative fits such as your's are invaluable.

I would not call them, therefore, naive.

Now that I am thinking more about the involved processes, the choice of a sigmoid-like function with assymetric asymptotic approach rates is justified, since the processes involved at the beginning of the ice decline (70's) and the end (2020 - 2050) are different.

The choice of a sigmoid is absolutely correct. Furthermore, Gompertz and Weibull for example, are used in survival analysis, e.g. failure mechanisms, which is in my mind appropriate for the "failure" of ice to survive more than a few years. The average ice age is now 3 - 4 years and it will drop further, until stabilized. At the extreme limit, ice will survive for a single year. After that limit is reached, the ice area/extent will become much more erratic from year to year since it will become more linked to weather. This is starting to become evident with the large fluctuations from year to year.

Al Rodger

Could you not make it less "naive" by pre-casting the output of your method for previous years & so testing it for more than just last year? How does the method fare using prior data to predict, say, years 2000-2010? Even if error bars are larger with the reduced data available in the early part of this peroid, even if the pre-casts are initially badly out, such analysis would demonstrate some sort of tangible level of confidence.

Yvan Dutil

Naive is not the right adjective. Least square fitting are the most effective way to extract information from data (assuming a Gaussian distribution). Hence, your model is very sound. In theory, you could push thing further by examining the relationship between residual observed in September and residual observed sooner in the year. I think that at least one group is using a similar technique (Bremen?)

Craig Cyphers

Regarding the modeling of sea ice extent/area, I would like to provide an alternative view. That is that we are using sea ice extend/area as a proxy for Arctic heat content. If so, should not the modeling tools to be used, be more useful for modeling heat content than sea ice extent/area per se? Specifically with the concept that the extent/area trend must slow, over the years, towards the end. Just a question for consideration.

L. Hamilton

I call this approach "naive" because the model is not physics-based, and ingests no data except single numbers from each of the last n Septembers. This seems to make a useful null hypothesis that physical models, or better statistical ones, should be able to beat as the melt season advances and they (unlike my naive models) ingest further data.

From performance last year, it looks like these predictions would be tougher to beat using data through the last September only, without further information -- but that should be possible too. Anyway, it's a challenge.

We'll see soon enough whether the relatively close Gompertz predictions for 2011 (on everything but PIOMAS) were beginner's luck. Especially valuable would be noting which prediction strategies that out-performed the Gompertz last year can do so again.

L. Hamilton

DrTskoul, you've articulated better than I did the appeal of this approach.

Al, I hadn't tried that approach but after seeing your note made some trials with the CT daily min. The model remains pretty stable if I set aside years until we're down to 1979-2007 only. But earlier than that the 3-parameter Gompertz is not much better than a 2-parameter linear, and the estimation becomes unstable. We're down to 28 observations or fewer by then.

Another way of putting that is we can't tell the decline from linear until 2007, but from that point onwards it becomes clear (statistically) that something nonlinear is happening.

Aaron Lewis

I think your work with Gompertz curves is outstanding by ANY standard.

I would encourage you to fit your curves to ice sheets.


"othesis to which more sophisticated physical models might be compared. Can other models from October the year before outperform this approach? One might hope so, but it seems a good challenge.

Continuing in this vein, the same approach yields a September 2012 mean NSIDC area prediction of 3 million km2, with confidence interval from 2.2 to 3.7 (Figure 2).

Figure 2

The PIOMAS volume prediction is 4 thousand km3, with confidence interval from 2 to 5.9 (Figure 3).

Figure 3

But all these are monthly means. What about daily values? The dailies have higher entertainment value on this blog and elsewhere, although they are often disdained (by me among others) as being too random to reasonably predict. In a sporting mood I tried it out anyway, coming up with a minimum 1-day Cryosphere Today prediction of 2.7 million km2, with confidence interval from 2.1 to 3.3 (Figure 4).

Figure 4

I was going to follow Figure 4 with a remark about daily values being harder to predict, but then hesitated and wondered, are they really? The surprising answer is no, at least with respect to Cryosphere Today area. The Gompertz model for minimum daily area graphed above leaves a residual standard deviation of just .29 million km2. A very similar model for CT September mean has a slightly higher residual sd (.34). That wasn't what I expected, so I'll explore the idea more systematically in the future.

And for better or worse, I'll revisit these estimates in October and fill out a table:
Sep 2012 Sep 2012
Predicted Observed
NSIDC extent 4.3 ___
NSIDC area 3.0 ___
PIOMAS volume 4.0 ___
CT area 1-day 2.7 ___

Posted by L. Hamilton on June 07, 2012 at 06:03 | Permalink


For the quick understanding of trends, I have always liked mathematical formulas which possess the correct asymptotic limits (imposed by the underlying physical processes).



Nice work L. hamilton.


First off, L. Hamilton, I find your analyses most useful, and not at all naive.

In fact, I carefully consider your calculations when I make my own personal 'guesses' about upcoming Arctic Sea Ice Area and Extent on various online forums.

Nevertheless, in following the Arctic Sea Ice saga, as much directed to forum members than to you, it appears that so many multiple factors are interacting that no physical formula exists to accomodate how the arctic system is changing and responding as sea ice volume diminishes because so many operational factors remain undefined/uncharacterized.

For example, it appears that the fundamental processes driving Arctic Sea Ice melt, ie. the processes that add heat to the Arctic Sea Ice system, are not well articulated.

Okay, maybe it's just me being 'naive', and uneducated. If so, I would appreciate being enlightened. Is this system as complex as I perceive? How does it work?

Climate Changes


A good insight of things to come.


Read that too, AJP. I think it's correct when you keep in mind the config will produce severe, regional cold snaps. Not long cold winters on a hemispheric scale. And there will be extreme warm snaps too.

Climate Changes

And there will be extreme warm snaps too.


Hello. They are already occuring... O.O


Kaleschke and Bietsch from Klimacampus have put up their estimate graph again: ftp://ftp-projects.zmaw.de/seaice/prediction/2012/estimate.png

4.0 plus or minus 1.2 million square km.


Fine work, fascinating discussion, many thanks L. Hamilton et al.

And now for truly naive question.

What might be the physical process that would cause the summer ice decline to 'flatten out' as per a Gompertz model, rather than steepen, as an ice-free state was approached?

Patrice Monroe Pustavrh

Also, Ron Lindsay from PSC did his own prediction based on PIOMAS model for 4.06 +/- 0.42 million square kilometers:


Voyageur, I've asked the same question a year ago and Larry promised to expand on the physical choices for a Gompertz curve in the following post:



Weibull / Gompertz Statistics relate to mortality/failure processes. Since the ice is not uniform with respect to thickness and average seasonal local temperature, it does not melt at the same rate for different locations.

Let's consider the annual minimum (e.g. end of september). As the arctic climate warms up, the multiyear ice will give place to younger ice. The ice remaining at end of september every year will be the most difficult to melt at that particular point in time. Every year the remaining ice will be the one that has the most resistance to melt under a background of increased warming (CO2/sun etc. etc.). The increased difficulty or resistance to melt will slow the rate of the curve, reaching eventually a plateau. If the plateau corresponds to zero area or extent cannot be predicted and will depend on the exact climatic trajectory. However if the forcing is such that a zero area is inevitable, initially we are going to see fleeting moments of such events, growing in time duration as the climate keeps worming up.

Tor Bejnar

Part of the answer, in my understanding is:
Prevailing winds push some Arctic sea ice towards Greenland and Canada where it piles up. These piles, often creating ridges, are thicker – much thicker – than what will readily melt in a single season. The year will arrive (this year or 10 years from now?) when the summer will melt virtually all ice thinner than, say, 2.5 meters and the following winter first-year ice will grow to 2 meters or less; even so, a certain amount of ice will none-the-less remain because new ice piled up into 5 meters-or-more thick ridges. As long as ice forms and winds pile it up thickly, some ice is likely to remain through the melting season.

In the recent past (to a geologist, anyway), some of this piled up ice became (or was attached to) fast ice (sea ice fixed to the shore) and stayed around (and thickened) for decades or centuries. Recent warming in the Arctic has loosened virtually all of this very old and thick sea ice, and wind and currents move it out where it melts (mostly eastern Arctic or North Atlantic). This process of moving multiyear ice to ice boneyards, I suspect, will counter the above described delay of ice melting out.

Net result? Well,I'm pessimistic - and buy into the 2016 +/- 3 years to melt season minimum of less than 10^6 km^2, and an additional decade or two to get to 10^3. (No formulas, no graphs, just my gut - and therefore you should ignore this paragraph!)

Those of you who know much more about this: any tragic mistakes? (I know, pessimism is probably a mistake!)

L. Hamilton

"The June SEARCH Sea Ice Outlook reports are now available! The pan-Arctic Summary, Full Pan-Arctic Outlook, and Regional Outlook are
available at:


L. Hamilton

More, from the June SEARCH SIO:

"With 19 responses for the Pan-Arctic Outlook (plus 6 regional Outlook contributions), the June Sea Ice Outlook projects a September 2012 arctic sea extent median value of 4.4 million square kilometers, with quartiles of 4.3 and 4.7 million square kilometers (Figure 1). This compares to observed September values of 4.6 in 2011, 4.9 in 2010, and 5.4 in 2009. Both the 2012 quartile values and the range (4.1 to 4.9) are quite narrow. The 2012 June Outlook differs from all previous Outlooks in that there are no projections of extent greater than 5.0. It is always important to note for context that all 2012 estimates are well below the 1979–2007 September mean of 6.7 million square kilometers."

Bob Wallace

I'm with Tor. A fair amount of MYI gets transported out to melt.

In addition, the Sun is in a heating phase, CO2 and methane levels are rising, air transported in from out of the Arctic is warmer, oceans are warmer. All those things are cumulative.

Furthermore, if we have more quick-melt along the European side with its accompanying warmer water then less heat should be extracted from the North Atlantic Current allowing it to carry more heat deep into the Arctic where it will get added to the ice sloshing around in the Beaufort Gyre.

The hardest to get to, thickest ice will be the most resistant to melting, but melting forces are going to be freed to concentrate on melting it out. The outer defenses are falling.

L. Hamilton

For one example of Gompertz-like (or at least, asymmetrical sigmoid) curves emerging from physical models, see Figure 1 in Marika Holland et al. (2006):

Half a dozen others, Figure 1 in Wang & Overland (2009):
(Figure 2, showing estimated number of years before "ice-free" 1m km2 state is reached in different models, also worth checking out.)

More in Figure 1 of Boe, Hall and Qu (2009):

L. Hamilton

At the start of September, reality has been angling toward the low end of confidence intervals calculated for these naive predictions.

NSIDC extent Gompertz prediction: 4.3 (3.4-5.1)
The one-day NSIDC extent on 9/1 is 3.6.

NSIDC area Gompertz prediction: 3.0 (2.2-5.1)
One-day NSIDC area not available, but likely near CT area, 2.4 on 9/1

PIOMAS volume Gompertz prediction: 4.0 (2.0-5.9)
One-day PIOMAS volume on 8/25 is 3.6

CT one-day area Gompertz prediction: 2.7 (2.1-3.3)
One-day CT area on 9/1 is 2.4

Because none of the actual declines appear to be outside of the confidence bands (yet) I would not say the actual decline has been *significantly* steeper than predicted. After ingesting new data, however, next year's predictions will probably have steeper slopes -- and when extrapolated, cross the 1m lines sooner.

Otto Lehikoinen

Thanks L. Hamilton for the Gompertz values. I guess it's better to have a function around which the measured values dance, rather than guesstimating yearly variations in incoming warmth. At least it (should) make the computations easier.
O."does xx extra ppm of CO2 cause ice to lose 0,05cm of it's thickness during a year?" L.


PIOMAS Gompertz fit on ArctischePinguin:


With "predictions" as they would have worked out in previous years:



>"does xx extra ppm of CO2 cause ice to lose 0,05cm of it's thickness during a year?"

Despite Tiesche et al suggesting mere 2 years to revert to equilibrium level, I think it is unlikely to be like that.

If we somehow held GHG levels at current level for 50 years, ocean temperatures would still rise through thermal inertia. This will cause the summer ice to disappear despite the lack of CO2 extra ppm.

So as a minimum I think you would need a ghg forcing element and either a time or Atlantic water temperature element.

L. Hamilton

Worth noting: The June SEARCH Sea Ice Outlook predictions for NSIDC mean September extent ranged from 4.1 to 4.9.

My 4.3 marked the first quartile among these estimates, but with NSIDC at 3.6 (and still falling?) now they all look quaintly optimistic.

For the August SIO I switched to a regression method based on late-July data, yielding a notably lower 4.0 prediction. This moved toward the lower range of SEARCH predictions, then 3.9 to 4.9. But already, all those predictions look too optimistic as well.

It's fair to say the system is changing faster than many experts thought it would.


Thanks to Larry, crandles, Jim Petit, Wipneus and many others here who help us estimate and visualize a wealth of data sources.

Artful Dodger

Hi Folks,

Thanks to Larry Hamilton for leading our little tribe of curve-fitters!

What I'd be interested in knowing is, "Does 2012 data now allows us to favour one curve over the others?"

"Is it significant yet"? (ie: above the noise)

Let's compare some 'Arrh-Squares'...



Exponential and gompertz curves barely differed at all for the past, only really diverging in 2011. So R^2 values would naturally be close as well. I clearly favor exponential only because I see the point where all ice has melted not as a wall (from which you can only go backwards to having ice again) but as a borderline between icy seas and warming seas.

L. Hamilton

"What I'd be interested in knowing is, 'Does 2012 data now allows us to favour one curve over the others?'"

That's certainly something to check, but my hunch is the data can't decide yet -- quadratic and Gompertz curves remain close at this stage. And the quadratic actually predicts a higher value than Gompertz (4.43 vs. 4.26) for Sep 2012 mean NSIDC, so although both are too high, the Gompertz will likely to have a better R2.

Looking ahead ... the quadratic has that funny trick of going up in the early years, unrealistic and it might become more pronounced if the curve had to drop more steeply at the end. Quadratics like to fit the endpoints, which have highest leverage. That will give a better fit if extent does fall of a cliff, but the rest of the curve would look funny.

BTW, I put this example in my stats book, which should be published this week.

Artful Dodger

Cool Larry, (pun intended!)

I was more thinking about the exponential fit, which has previously led the pack according to r^2 values, per Wipneus.


L. Hamilton

Hmm, I don't think even a 3-parameter exponential does that with extent. Not for me, anyway; it's still less steep than the Gompertz.

PIOMAS, however, looks to be approaching the end game.

Artful Dodger

Fair enough, Larry. I'll defer to Wipneus to comment.


Artful Dodger

Hi Larry.

I wonder if you've posted updated 'Cycle Plots' lately? These are some of my favorite graphs to show people the decadal decline of Arctic sea ice.


L. Hamilton

Hmm, I do update the cycle plots now and then, but now that NSIDC August is out I should do that again. Perhaps a set of new plots (might as well do SH too) will be worth a short post -- stay tuned.


I'll defer to Wipneus to comment

Belated reply, this one was lost to me in the traffic.

An good exponential fit for volume does not mean a good exponential fit for area/extent.

In the first place mathematically. As Area=volume/thickness, the function of thickness comes into the equation.

Second, statistically the signal/noise ratio for area/extent is much worse than volume. Even for volume it maybe too early to discard the gompertz curve. Next year perhaps, if the estimate is too high again.

My own opinion is that volume is the leading parameter, extent and area follow from that. I have used and discussed this viewpoint in Nevens sea ice polls and in my sea ice outlook contribution:


We can look at 2002 to 2011 on quadratic, gompertz and exponential basis. Of the three, I think IIRC quadratic does noticably worse but it is close between exponential and gompertz.

If it comes out with exponential being better than gompertz 7 times out of 10, that could easily be the result of tossing a coin 10 times. Adding an eleventh year isn't going to help much and we might run out of ice before we can tell.

I am wondering if there is more sophisiticated analysis that can be more helpful. For example it appears that there is negative one year autocorrelation. If the trend is actually more like exponential than gompertz, would you expect the negative autocorrelation to be stronger with residuals from exponential trend than with residuals from gompertz trend? Does this provide a better test?

(I haven't tried this yet, and it need someone with far better maths skills than mine to say whether this could be a better test.)

Alan Clark

I don't think the quadratic curve has any merit because it says that there was no ice in the early 20th century, so it is a very poor fit, as Wipneus pointed out. I also think that we have to look at volume rather than area, because thickness cannot be ignored, and area can change very rapidly as the thickness suddenly goes to zero!

I was hoping that the volume this year would be incompatible with either exponential fit or Gompertz so we could eliminate one of them. That does not seem likely to happen, but it looks like being more compatible with exponential. Anyway, I expect we will have a much better indication next year which (if any) is reasonably correct, as the two curves diverge strongly.

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