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Interesting discussion and I'm already fearing results in the upcoming volume post...

But (why is there always a but?) what I miss a little in this post is a more in-depth discussion on the supposed physical reasons for a selection of the Gompertz curve to do the prediction. Afterall, the choice for Gompertz is critical to the outcome so this seems to warrant a bit more discussion then just a comparison to a linear trend or suggesting that it is intuitively appealing.

Can you elaborate a bit on this aspect?



Many Thanks for presenting different graphical displays of the "model numbers" with commentary.

Looking forward to your following post on
"Gompertz model to trends in volume"

Jack Taylor


+1 for the volume post (but I would say that...).

If I read correctly, your linear fit indicates the NSIDC extent is declining at 80,000 km^2 per year (witin the limitations of a linear fit). Just for what its worth, I've analysed IJIS extent data to produce a daily anomaly. The linear trend there from mid-2002 to today is -70,712 km^2 per year. The noise is high (being daily data), but linear is a good fit to that data (ie other curves do not greatly reduce the R^2, as they do for the NSIDC data).

Similar, but (interestingly?) different.

Peter Ellis

Frank: are you fitting the IJIS data using only September data, or data for the whole year?

L. Hamilton

FrankD, I get a steeper trend, -148,000/year, regressing mean IJIS September extent on years 2002-present. Over that same period the linear NSIDC trend I see is -177,000 year, so both are much steeper than the linear 1979-2010 slope.

Peter Ellis

I suspect Frank's fitting the full data set, not just September. That effectively averages together the summer trend (steep, getting steeper) and the winter trend (not so steep, staying constant). Net effect is a medium-steepness trend with no obvious acceleration.

L. Hamilton

cynicus, good question, I'll elaborate in the follow-up volume post. Hopefully in the next couple of days.

For a volume model, I'm working with Zhang's mean September values, which differ in details from the graph-derived estimates.


Unlike a linear model, which falls to zero at a steady rate and then keeps on going to make impossible predictions of negative extent,

I still not see that there is anything strange with such behavior. There are many examples of discontinuous phenomena in physics where phase transitions ( like here the water/ice transition) are involved.

Imagine the following experiment:
Take a beaker with ice; using an electric immersion heater feed a constant flow of energy (eg 100 Watt) to the melting ice/water mixture; plot the amount of ice as function of time.
The result is a straight line as long as there is any ice in the beaker "predicting" negative amounts in the future.

I see there may be valid reasons why the curve will level out to some degree. But that is probably because the real world is a bit more complicated than a beaker with water and ice. Some ices may be harder to melt, because they are thicker, older, lay in more sheltered places. I am not convinced that we can see such effects yet.


Since the area is about 1million sq km less than extent and obviously when area hits zero extent must also doesn't it make more sense to work with area?


Peter Ellis

... you could say the same for volume, which I suspect is what the next post is about, at least in part!


Agreed, but we have longer data sequences for area than for volume.



I'd like to second Wipneus' point. Maybe a thought experiment with two beakers will help ...

Add 10g of ice at 0C to 100ml of water at 9C, and wait. The last of the ice will hang around for quite some time, because the eventual equilibrium is (if I've got my figures right) only barely above the melting point. Melting will tail off as the temperature difference between ice and water becomes smaller and smaller - the difference declines by three-quarters as melting proceeds

Compare adding 10g of ice at 0C to 100ml of water at 90C. The ice will melt out at a (very nearly) steady rate, with the beaker nowhere near equilibrium when the last ice melts - the temperature difference declines less than 10% and thus melting barely slows at all.

(Wipneus - have I understood you right?)

L. Hamilton

Regarding extent: Uni Bremen has uncorrected estimates back to 1972. The trend in their data broadly resembles that shown for NSIDC extent above.

Regarding area: There are NSIDC and CT time series back to 1979. I might include one or both graphs with the next post. First impression, area is dropping at a similar rate to extent.

Regarding volume: PIOMAS estimates go back to 1978, those look a bit different from the area and extent curves.

Gas Glo

>"Unlike a linear model, which falls to zero at a steady rate and then keeps on going to make impossible predictions of negative extent,"

>"I still [do] not see that there is anything strange with such behavior."

I agree there is nothing strange with your imersion heated thermos flask. The negative ice volume can be taken as a proxy for temperature rise.

However, we have a situation where predicting the elimination of ice using two methods (volume and extent) gives very different answers.

You can achieve different or similar things in the thermos flask. By using a cube of ice where there is more edge melting than bottom melting causing rotation of ice and both volume and extent predict similar time to elimination of ice.

However, suppose a very thin even layer of ice is used so there is little edge melting and lots of bottom melting. Predicting using extent will predict a very much longer time before the ice melts and using volume is a much better prediction method.

This suggests we should ignore the extent prediction and use the volume prediction which gets it right for either of above.

In reality however, things are rather more complicated. Ice thickness varies and if you imagine a circle of ice with thickness increasing linearly towards centre then the extent prediction would not be too bad.

A Convex or concave lens of ice could make the extent prediction wrong in either direction.

The big problem is we do not have an imersion heater providing a steady flow of heat. More like a heat pump that pumps in heat in summer but pumps out heat in winter. Worse the heat pumped out in winter is likely to depend on the thickness of the ice and heat pumped in during summer depends on the ice extent. This means the volume prediction can be wrong as well as the extent prediction potentially being wrong.

Then what should we believe?

Quadratic fit is better than linear. The Gompertz fit may be better but it is not better by very much and I think there are more degrees of freedom so we could get better fit with Gompertz even if the underlying trend is really quadratic.

I like the Gompertz fits because
1) heat loss in winter will be greater with thinner ice allowing more ice formation. This could be outdone by increasing albedo effect but I don't want to raise unnecessary alarm.
2) There is less difference in the predictions using volume and area. I think this is a desirable feature even if the lesson of the 'thin ice with even thickness' case above suggests it isn't necessary.

L. Hamilton

Regarding Gompertz vs. quadratic:
- They are equally complex: both require 3 estimated parameters (a linear model just has 2).
- The Gompertz fit to extent in Fig 4 above is presently falling more steeply than a quadratic fit to the same data. Gompertz predicts 2011 extent of 4.4, vs. 4.6 for quadratic (and 5.2 for linear).
- Quadratics rise and then fall. A quadratic for September extent 1979-2010 will be rising slightly at the start, although the linear trend for those early years is gently down. If we fit a longer time series such as Uni Bremen (1972-2010) the unrealistic early behavior of a quadratic becomes more noticeable.
- Gompertz allows for a slow decline through the early period, more or less what we see. But the Gompertz does assume an asymptotic initial level, implying longterm stability that might not have existed.
- Relatively small areas of summer ice along the Archipelago and N Greenland might disappear more slowly than open-sea ice, in which case the asymptotic approach toward 0 would be more realistic.
- We don’t see evidence of slowing yet in the data. Nor should we expect to if these naive models happen to be accurate.

I don’t want to oversell the Gompertz, but it seems an intriguing model to try out.


The PIOMAS volume data has been updated.
Anomalies are based on 1979-2010 averages, while the average data are still 1979-2009.
I have recalculated those averages as best as I could myself.
Here are the updated data: http://snipt.org/wplni

Peter Ellis


Well, looks like Maslowski's updated his models and is sticking to the forecast of 2016 +/- 3 years.

Reporting thus far is painting it as a retraction of his supposed previous prediction of 2013. All that proves is that they either misunderstand (BBC) or wilfully misinterpret (Goddard) error bars.

At least he's been more careful this time to control the narrative and not let the reporters run fast and loose with only the lower bound.

Gas Glo

D'oh sorry Larry- it would help if I could count the number of parameters correctly. I got confused counting your ((2036-year)-18.912) as 2 but it can obviously be simplified to (2017.088-year).

However, I am not at all sure about whether you can reverse that to (year-2017) as shown on recent graphs.

Thank you Wipneus.

So, it looks like this year is now a -1310 km^3 fall below last year. This is a big increase in the fall compared to last month only being a -828 km^3 fall. The previous 4 months were between -646 and -942 km^3 falls but the Sept fall was -1828km^3.

The correlation between annual March falls and September falls isn't very good (.25) but I do notice that the March fall of -1310 is the third largest fall in last 20 years. The two times the March fall was bigger were 1995 and 2007. In 1995 the March fall of -1681 grew to -2168 in Sept and in 2007 the March fall of -1589 grew to a fall of -3643 in September.

OTOH the 4th largest March fall of -1241 in 2004 reduced to a Sept fall of only -505.

I am probably trying to read too much into poorly correlated figures.


Regarding my linear trend: yes, Peter has it - full year data. I thought my reference to "daily anomalies" would indicate that but I should have said so explicitly.

"Net effect is a medium-steepness trend with no obvious acceleration."
Hmmm, I would have thought that the net effect should be a medium-steepness trend with a small but obvious acceleration. I think the lack of any acceleration is interesting, but probably explicable by, for example, the March behaviour over the last couple of years. With only 9 years of data, 2010 March-April by itself would be enough to boost the right-hand end of the curve to "flatness".


I have some qualitative remarks about Cynicus' question (without wishing to gazzump Larry's quantitative ones to come). Previously, I've remarks on two phenomena that would serve to prevent ice hitting zero in any given year:
1. Ice doesn't all melt at once: Shallow backwaters in the Canadian Archipelago with weak currents will screen some ice - warm water can't penetrate easily, wind and wave do little damage, there may even be some screening from direct sun. These local patches will require the average temperatures to get a fair way above zero before they finally disappear. We see this sort of thing every year on lakes, snowfields etc.
2. Ice will be added: Icebergs calved from glaciers are added to sea ice extent. While not truly "sea ice" they can hardly be excluded from the totals. If rates of calving off ice fields increases, it will add thousands (tens of thousands?) of square kilometres to extent.

Those two phenomena will help delay the achievement of zero in any given year. But over the long haul something else comes into play that validate the Gompertz curve as a reasonable approach: You can't have negative ice extent.

Lets consider Larry's linear projection - for the sake of the exercise, not because we think its right. Assuming the current linear trend holds good, in 60 years it will be 4.8 M km^2 lower. Suppose then that The 60's see the same pattern of variability as the noughties, just 4.8 M km^2 lower. So 2065 is 4.8 lower than 2005, 2066 is 4.8 lower than 2006, etc. Will the trend still be going down at 0.08 Mkm^2/yr? It should - all we've done is extend the existing trend.

Well, no, it won't. In this scenario, extent in 2007 will not be -600,000 km^2, but zero. I'll go out on a limb and put 2071 in the same basket. As more years that the trend says "should" be negatives are measured as zeroes, the trend necessarily flattens out.

So - aside from the factors above protecting any individual year - over the course of multiple years, weather / noise means the trend must flatten out gradually before hitting zero indicating permanently ice free Septembers. After the first ice free September, we will still see years where ice survives the year round, they will just become rarer and rarer.

Given the +/- 0.9 M km^2 weather-dependent noise about the average line, Larry's curve doesn't predict that we will lose less and less ice towards the end, but that years with ice free Septembers will gradually increase from 0% to 100%. That a September with ice will gradually become a "rare and exciting event" (to coin a phrase). And during that transition, while nominal negatives become actual zeroes, any trendline must flatten out as the Gompertz does.


Excellent Wipneus, thanks for the heads up.

I concur (well, I make it 19078, but that's within the expect range of difference given our differing methods).

By my reckoning, its a fall of 1238 km^3 from March last year (20316 - 19078).

@Gas Glo: "This is a big increase in the fall compared to last month"
In comparing Mar-Mar to Feb-Feb, one sees that March 2010 was a relatively good month and Feb 2010 relatively poor. I'd estimate about 50% of the increase you remark on reflects last years situation, and the rest reflects 2011 changes. So March, relatively was worse than February, but not hugely so.

Gas Glo

Don't know if anyone wants Gompertz parameters for each month and the estimates they produce:


But perhaps the excel speadsheet that produced them would be of more use but I am struggling to find anywhere I can upload that.

Gas Glo

Certainly going a bit wild here but:

Correlation coefficient between March deviation from gompertz prediction to Sept deviation from Gompertz prediction is 0.5.

Using muliple linear regression to estimate the September deviation from Gompertz prediction using the deviations in Jan Feb and March from the Gompertz predictions and the year yields a prediction that Sept 2011 will be 656 below the Gompertz prediction.

So instead of Gompertz predicting a volume of 4270 km^3 for Sept 2011, the below Gompertz predictions so far this year suggests a prediction 656 lower at 3614 km^3. This isn't much below Sept 2010 figure of 3994 km^3 but 2010 was particularly far below the Gompertz predictions.

L. Hamilton

"However, I am not at all sure about whether you can reverse that to (year-2017) as shown on recent graphs."

Gas Glo, sorry for the confusion between my earliest versions of these graphs (posted earlier by Neven) and Figure 4 above. For the earlier version I made time run backward, giving the Gompertz parameters their conventional "growth curve" signs. In the cleaner Fig 4 version, time runs forward because I flipped the sign of the second parameter.

I've run the volume model with Jinlun Zhang's data, and sent results to him for any comments. Hopefully I'll have a post-able version this weekend.

Peter Ellis, thanks for that link about Maslowski's EGU paper. Very interesting, for reasons you'll soon see!

Gas Glo

Thanks Larry, I should have spotted the parameter sign switch.

If anyone is interested in seeing projected thicknesses using Gompertz fits of Wipneus' PIOMAS volume data and NSIDC area data, I arrived at:


Whether thicknesses falling to 18cm by end of decade are viable or not is up to you because I just don't know.

R. Gates

Wow, once more I amazed at the depth of your sea ice posts. Very informative, thanks!

One note that may be of interest: Currently the most important data we need is of course sea ice thickness, and the only real tool left to get this arctic wide is CryoSat 2. When that data starts to come be validated, we'll have a wealth of information about sea ice thickness, and that, combined with sea ice area will give us, for the first time, a good metric for sea ice volume. Sea ice volume will be very important data to have in looking a true year-to-year changes in the ice. ESA is trying to validate the CryoSat 2 data, and has set up an "on the ice" effort to do just that. A blog for that effort can be found here:



What does 'negative ice' mean?

Positive ice extent/area/volume reduces as sufficient heat is absorbed by ice to melt it.

If you re-plot ice extent in terms of heat values, then negative values of ice extent plot as ocean warming. We are seeing negative temperatures in ice turning into positive temperatures in water. Zero degrees C may be a human-chosen arbitrary datum, but it sure fits with what we are seeing: a swing through a zero ice datum.

Plots of degrees C or Joules would be a great way to show the Arctic switchover.

Anyone care to run with this?

Now hear this!
I would like to see (hear) a convertion of the 'tale of the tape' into an AIFF sound file. Why? The human ear picks out features in data that the eye misses. Besides the use as a tool, a 'tale of the tape' file posted on Youtube as a scope trace + wav file might be a cool way to show the world that the ice is melting ever faster.

A statistical analysis of the tape would also be useful. Anyone want to write an article showing trends with the working?

L. Hamilton

Something I like about the Gompertz for modeling ice extent, area or volume is that it matches a physical constraint: these quantities cannot fall below zero.

If heat content or temperature were our left-hand-side variable, however, we would need no physical floor and other models could be more realistic.

Artful Dodger

Hi Larry. Gompertz does work properly with sea ice, but with Winter Sea Ice approaching the asymptote... I'm working on a 3-D graph using Gas-Glo's month-by-month data, but feel free to dabble!

Gas Glo

Houston, err I mean Huntsville, do we have a problem?

With AMSU that is. Last three days channel 5 has risen more steeply than any other 3 day period. Previous top 5 largest rises were .242, .243, .251, .261, .264 so several clustered quit close to those values. Last 3 day rise 0.311 is a significant gap.

Also looking at end of

compared to appropriate way through

seems to have rather too consistent looking values.

If there is a problem, (I am not completely convinced there is), lets hope it is something fixable.

Perhaps treat AMSU products with caution until we know.

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