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Is there a graph which overlaps the Gompertz model, the quadratic model, and the linear model, along with their standard deviations?

I'm curious about the difference between the "inside" and "outside" projections for crossing the 2K threshold on the various models.

(Though it might easily be this year, in which case all academic argument is moot.)

Great post, as always.

L. Hamilton

A graph with all 3 models sounds over-busy, but it's easier to answer with numbers. Predicted September PIOMAS ice volume falls below 2,000 km^3 in the three models as follows. Limits given are predictions plus or minus twice the residual standard deviation.

linear, 2029 (2020 to 2039)

quadratic, 2017 (2013 to 2020)

Gompertz, 2017 (2013 to never, because 0 is the lowest prediction and +2sd is slightly greater than 2. I kind of like that.)


Awesome post, Larry. Great too that Zhang sent you the PIOMAS data.

Peter Ellis

Your final sentence, translated:

"Where various predictions most differ is in [the future], which [hasn't happened yet]."

... I find it hard to disagree. :-)

L. Hamilton

It does sound wishy washy, now that you mention it. I think I meant something *slightly* deeper, like the data don't yet hint whether Arctic ice will end with a bang (exponential) or a whimper (Gompertz). All the trends point down, though.


Larry I'd like to see a plot of ice thickness, i.e. volume/area, my back of the envelope calc suggests that it will soon be less than 1m which might indicate that it would be very prone to weather induced breakup.


Gas Glo


Have you seen my numbers for monthly Gompertz fits to Wipneus's PIOMAS volume numbers

and I also did monthly NSIDC area Gompertz fits to each month. Using the Smooth Gompertz fits numbers, PIOMAS volume divided by NSISC areas produced smoothed average ice thickness (in meters) as:

Plenty of numbers to be plotted as you wish there. 2014 for September to fall below 1 meter. October and November fall below 1 meter in 2012.

L. Hamilton

I took a quick look at Sep PIOMAS volume/Sep NSIDC area. That number drops below 2m for the first time in 2009, and below 1.5m in 2010.

Artful Dodger

Ice thickness starts the annual melt at a critical depth this year. It is typical for 1.5 m of sea ice to melt by the end of Summer, and even more for new sea ice.

There *IS* IceSat level 3 data now for Oct 2009. Even thought the campaign was not run the planned 6 weeks due to the last laser failing, the collected data is still valid and is timestamped...

Otto Lehikoinen

The lag in the iniation of photosynthesis and in the decay of living matter during dark times in steadily rising temperatures could possibly produce such a curve. The methane in the bottom could be regarded only as a by-product of this competion, and hence be regarder as a feedback. But anyway, enzyme kinetics is such that one must produce the whole reaction curve experimentally before applying the function to practical designs. And still one may get occasional mutations/invasive species in the fermentation vessel.

Artful Dodger

Larry, in your Figure 6 above, the data point plot for Sep 2010 PIOMAS volume appears to be at least 4,400 km^3. This differs substantially from the text on the PIOMAS page, which says Sep 2010 avg volume was 4,000 km^3.

Can you provide a table of the volume data you've used to plot this graph? Cheers!

L. Hamilton

With Dr. Zhang's permission, here are the mean September PIOMAS volume estimates (1,000s of km^3) I used to draw Figure 6.



Thanks for the numbers.

Interesting that both quadratic and Gompertz estimates give the 2013-2017 inside range. I guess that's what life on the tail of the beast looks like.

Thanks also for the area/volume numbers. That's where my mind went next. :D

Artful Dodger

Thank-you Larry, and to Dr. Zhang and the UWash team.

Frank, Wipneus, do these September values differ significantly from your estimates based on the PIOMAS graph?

L. Hamilton

There's a .99 correlation between Wipneus' graph-based numbers and the data Zhang provided (September only), but the values are noticeably different -- don't know why. Here they are for comparison:


Artful Dodger

Well, there seems to be about a 900 km^3 Standard Deviation between the two data sets.

Also, there is an interesting skew in Wipeneus-APL (W-A): all the Years 1979-2006 have a Positive value for W-A, AND all the years 2007-2010 have a Negative value.

The Max diff in W-A is 1981: +2.61 Million km^3, min diff is 2009: -1.61 Million km^3.

I wonder if APL has systematically moved the Daily Averaged Arctic Sea Ice Volume curve from year-to-year? (Frank has introduced this issue previously in 2009 vs 2010 average).

Year	W-A
1979	+1.20
1980	+1.76
1981	+2.61
1982	+2.18
1983	+1.47
1984	+1.21
1985	+1.45
1986	+1.63
1987	+2.10
1988	+2.19
1989	+0.92
1990	+0.89
1991	+1.10
1992	+0.98
1993	+1.17
1994	+0.01
1995	+0.39
1996	+0.45
1997	+0.48
1998	+1.23
1999	+1.48
2000	+1.37
2001	+0.90
2002	+0.61
2003	+0.49
2004	+0.20
2005	+0.80
2006	+0.61
2007	-0.43
2008	-0.94
2009	-1.31
2010	-0.69

Larry, Artful Dodger:

I looked at the numbers and I cannot explain.

Recalculating Dr. Zhang's numbers, taking September mean volume as 13,463 km3 (my best estimate), plotting them on the well known PIOMAS anomaly plot gives this:
http://img718.imageshack.us/img718/4974/piomasalt1.png. It is clear that the exact September mean volume is not important for the conclusion that this is not the cause that those points do not fit.

Of course I can make mistakes (happened before), so let me check again.
Take September 1979, Dr Zhang gives an average ice volume of 16.8 [1000 km3]
From http://psc.apl.washington.edu/ArcticSeaiceVolume/images/PIOMAS_daily_mean.png I read a September average cover of 13.6 [1000 km3].
That gives an anomaly 16.8-13.6->3.2 [1000km3].
Check with the anomaly plot and the conclusion is that the 3.2 point does lie far from the graph. ( The mean 2009/2010 issue introduces an error of about 0.3 [1000 km3], much smaller than this discrepancy.)

If I did not make a mistake, then it looks that the Y-scales look different. How that can be explained I do not know yet.


Another thing that I cannot explain is that the 1979-2099 average of Dr. Zhang's number (drop the 1978 and 2010 values) is 12.48.

That is at least 1000 km3, from the daily mean graph.


Wipneus, I've fixed your first link to the PIOMAS anomaly plot.

I hope you guys find out what is causing the discrepancy. Maybe the graphs on the UWAPLPSC web page aren't accurate?

L. Hamilton

My guess is that the discrepancy reflects some difference in how the two series are calculated or defined.

Both correlate equally well with NSIDC extent and area. A Gompertz curve based on Wipneus' values looks generally similar to Figure 6, but starts 1,000 km^3 higher and falls faster. The Wipneus-based curve drops below 2,000 km^3 in 2015, bringing 2012 (or never) within the +/-2sd range.


I can see one source of difference - the method Wipneus and I used to calculate the baseline for the monthly mean. We both used the daily figure for the mid point in the month, which, for September, would read consistently below the actual monthly mean.

That does not explain the real difference though, only a small portion of it. I'm pretty confident we've interpolated the data accurately, which can only mean that the data Dr Zhang has supplied differs from the data displayed on their graphs (which I realise is a bit presumptuous of me).

Wipneus and I independently read values that differ by a maximum of 129 km^3, with an average difference of 46 km^3 (that's the average of the absolute difference - considering the sign, Wipneus tracks about 10 km^3 below mine for the 32 September datapoints). This leads me to conclude that we are reading the graph accurately - or at least that we are making the same mistake independently.

The data Dr Zhang has supplied differs far more significantly - the average difference in absolute terms is 1100 km^3, and considering the sign, my data is 881.6 km^3 above APL. The deltas trend downward at 75.8 km^3 per year, falling from an average of 2000 km^3 above APL around 1980, to a current average discrepancy of around -300 km^3 (based on a linear trend through the individual deltas).

I can't say why. The fact that the difference changes over time in the way it does suggests to me a problem with baselining (and continuously re-baselining). However, the only scenario I can see here (the one Lodger refers to) is that APL are rebaselining without correcting previous anomalies. But that would produce the opposite effect: that Wipneus' and my data should, over time, decline less than the source data, rather than more. Continuous rebaselining with a downtrending dataset would reduce the amount of each new anomaly that is calculated (because the average falls each time) and, without recalculating old data would result in an incorrectly flat trend. If that were the case, the difference between us and APL should trend in the opposite direction to how it does)...

Artful Dodger

Wipneus, thanks for that plot. Just a small point: since Dr. Zhang's numbers are September means, rather than plotting points as you have done, you might consider plotting line segments of 30 day width. I recognize that this does not resolve the discrepancy...


FrankD: the way it does suggests to me a problem with baselining (and continuously re-baselining).

I can hardly believe that. It would not be meaningful as far as I can see. This is how it is described on the psc.apl website:
Anomalies for each day are calculated relative to the average over the 1979 -2009 period for that day to remove the annual cycle. The model mean seasonal cycle of sea ice volume ranges from 28,600 km3 in April to 13,400 km3 in September.

I just think the 2009 has been changed to 2010 and not been updated everywhere as it should have.

BTW, my collection of different curve fitting using the APL data is here: http://img851.imageshack.us/img851/5960/piomastrnd0.png

Compare with data extracted from their graphs: http://img220.imageshack.us/img220/4402/piomastrnd1.png

With the APL data:
- R2 scores are a bit lower
- R2 scores are a bit closer: it will probably take a couple of years before the curves differ significantly;
- (virtually) zero ice is predicted a few years later, a range 2013-2020 years.


Hi Wipneus,

Yes - I lost a little in editing and reediting the above. I had included a note that I didn't believe that that was *actually* the case, as it would be a pretty bad mistake.

It was just that baselining problems were the only sort of change that I could think of that would produce a delta that would change in this systematic way. In any case, as I noted, it would actually produce the opposite effect to what we see.

The source of the discrepancy remains a mystery for now.

Peter Ellis

It's not a baselining change, as that would shift all the points up and down to the same degree, or at least show a constant offset from the point the re-baselining was done.

It may be there's more subtlety in producing a "monthly average" than just averaging the daily figures together. If (say) the monthly average was generated on a pixel-by pixel basis and then summed to give the total volume, that might give quite different figures compared to generating the total volume for each day and then averaging.

Alternatively, perhaps they've re-run the hindcasts, but left the older figures up on the website for historical consistency?

L. Hamilton

I asked Dr. Zhang about the differences we were seeing, and learned the following.

"We run several ensembles to make sure the results are robust. The results you have are from one that has the best overall performance when compared with all available ice thickness measurements. This one will be denoted as our standard case. However, as more measurements become available, more will be used for model calibration."

Artful Dodger

Thanks for this Larry. This is entirely reasonable, and consistent with earlier statements of a 20% margin of error for volume measurements.

Are you able to provide monthly means for the other Months, in addition to September?

I have another curve fit mind... Indeed, next I need to develop a formula describing the shape of the Daily Averaged Arctic Sea Ice Volume curve.

With this, we can use all 12 months of PIOMAS estimates to further improve R^2 (384 data points instead of 32)

(gee it's nice working with grown-ups!)

L. Hamilton

Lodger, I was curious about other months as well, but Jinlun pointed out that absolute volume numbers make less sense in winter due to the model's limited domain (compared to that encompassed by extent or area data).

Artful Dodger

Well, even perfect knowledge of Sea Ice volume is only part of the puzzle... Salinity of sea ice greatly affects the latent heat of fusion, and hence the energy budget. Another wild-card is turbulence in the surface mixed layer, which as Anu points out can tap a vast amount of previously sequestered heat. For now, I'm satisfied with the +/- 20% uncertainty associated with APL volume estimates.

Artful Dodger

Hi Larry,

With the curve in Fig.6 above (and +/- 2 SD range), a September 2011 PIOMAS volume below 3,000 km^3 would be outside the 95% confidence interval. This would make the Gompertz curve less likely to be the correct underlying model...

... however, just out of curiosity Larry, are you able to find a best-fit Gompertz curve with the Y-value asymptote as a variable?

In other words, what limiting value of September sea ice volume (either positive or negative) produces a best fit Gompertz curve?

[EDIT]: folks, let's hold the "it's not physical" chorus for now... Just imagine a full melt-out occurring before, and freeze-up coming after, September. Cheers!

L. Hamilton

Lodger, it's easy enough to fit such a model (a "4-parameter Gompertz") but that gives an unrealistic result -- a large negative lower bound, rather than the small positive lower bound one might intuitively (even if correctly) expect.

The reason it doesn't work statistically is that the PIOMAS numbers so far show no signs of deceleration, so there's nothing telling the curve to start leveling out (unless the lower bound is constrained to be zero, as I did). At this point, curve-fitting can't distinguish between "crash to zero" and "asymptotically approach zero" behavior.

Artful Dodger

So, does a curve with a lower-bound of -2,200 km^3 have a better fit than one with a zero lower bound? Thanks, Larry!

L. Hamilton

Honestly, the 4-parameter Gompertz gives a lower limit of -240,000 km3 (that's not a typo) reached somewhere around 2070. In other words, the 4-parameter model falls much faster than my 3-parameter version in Fig 6. It drops below 2,000 km3 by 2013. I guess real volume might, but there's not much rationale for that model.

I did send to the SEARCH Sea Ice Outlook my pre-season prediction of 4.4 million km2 extent based on Fig 3 from the ice extent post

L. Hamilton

Correction, the lower limit would be -224,000 km3, there *was* a typo after all.

Artful Dodger

Yes, quite humourous in the extrema, but is the curve a better fit? :^)

L. Hamilton

Very slightly, but unimpressive as we've used up another degree of freedom. The simple correlations between observed and predicted values are:


FWIW, gom4 comes closest to predicting the Sep 2010 PIOMAS value. Linear and quadratic look too optimistic, compared with the past 4 years.

Which brings up an interesting point. With regard to temperature and related measures, there are good reasons to emphasize longer trends and not overinterpret short-term ups and downs. In the special case of PIOMAS volume, however, the downs have been getting closer to the floor. Maybe that's important.

Peter Ellis

Lodger, it's easy enough to fit such a model (a "4-parameter Gompertz") but that gives an unrealistic result -- a large negative lower bound, rather than the small positive lower bound one might intuitively (even if correctly) expect.

Why is a negative bound unrealistic? That may sound like a stupid question, but bear with me. Instead of thinking of ice volume, think of it as fitting the total heat content of the summer Arctic ice/ocean system. In that case, "zero ice" corresponds to water at freezing point - i.e. enough heat input to melt the ice with no extra heat left over. "Negative ice" corresponds to water above freezing point - i.e. enough heat input to melt the ice and then continue warming the top layers of the ocean.

My point is that "intuitive" arguments as to the shape of the curve are not really helpful - the overfitting argument is a much better reason to restrict the number of variables for now.

Kevin McKinney

Peter, I like this point.

Reminds me of Asimov on the history of negative numbers, 30 or more years ago--apparently there was at one point serious resistance to the idea. How could there ever be less than nothing? Baffling, from a physical point of view. But in the economic sphere, debt is effectively 'less than nothing.'

So, maybe "ice debt" isn't crazy.


"Ice debt". Nice!

William Crump

Lines drawn using arctic-wide data for area and extent are flawed as many regions have reached a near zero level at the September minimum. These regions have nothing left to contribute to further declines in the level of ice at the September minimum as they can not have negative ice extent or area. While you can speculate that the remaining regions will see an acceleration in their rate of decline, there does not appear to be sufficient data showing this.

The Arctic Basin (per Cryosphere today)appears to be maintaining itself in spite of the loss of ice in surrounding regions.

The area and extent lines should only include regions which had at least 50% of the amount of ice remaining at the September 2010 minimum as they had compared to ice levels in 1979. Please plot an asymmetrical S-shape Gompertz curve using this data base and see when the Arctic is "ice free".

Peter Ellis

William: Go jump off a cliff. When you get to half way down, please plot a graph showing when you will hit the ground. Make sure to take account of the fact that you haven't yet reached the 1 metre mark, therefore your current rate of progress through that final metre is zero, and you are thus perfectly safe. Perfectly.

L. Hamilton

William, this post is about ice volume, not area or extent. Do you believe that central Arctic volume has not yet declined? On what evidence? All the data I've seen suggest otherwise. Moreover, the PIOMAS model covers a more restricted domain, well within winter area or extent outlines.

Regarding area or extent -- when the ice in my drink is half gone, its core might not yet have shrunk, but that's the wrong trend to measure for predicting its fate.

L. Hamilton

One point I'd like to bring back -- William asks when the Arctic will be "ice free," but that dropped a word "virtually" which I carefully used in the text. The model is asymptotic, never reaching zero. I specified 1 million km2 as the threshold to call "virtually ice free." My convention, though other Arctic researchers have made similar statements.

A line indicating this threshold appears in the area and extent graphs (Figures 4 & 5), and in the discussion.

The volume notes in this post likewise graph (Figure 6) and discuss a small round number, 2,000 km3, as a point to call "virtually ice free." So please recall the "virtually," and its definition, if you're going to quote this ten years hence when it's wrong.

I doubt that every last ice cube will melt in our lifetimes. But the way things have gone lately, a lot of the area/extent/volume could erode within the next few decades.

Artful Dodger

Thank-you for your thoughtful responses Larry. I believe the Gompertz model describes a transition from one steady state to another, caused by a change in Climate forcings. There is a time lag from that change until Sea Ice equilibrium is reestablished.

Our situation differs from these assumptions however in that the Arctic faces ever increasing, and possibly accelerating, Climate forcings.

So the asymptotic behavior at the top of the Gompertz curve may fit well with historical observations, but as long as Climate forcings continue to increase, a new equilibrium state can never be reached. Then the asymptotic behavior at the bottom of the Gompertz Curve may only apply if forcings stabilize.

Therefore I suggest an asymptotic approach to zero September Sea ice is problematic under an increasing forcings scenario.

L. Hamilton

Well, could be, I lean toward the not-totally-ice-free scenario partly because that's what recent modeling studies like Zhang & Rothrock find, and more intuitively because it will still get cold enough to freeze seawater in the Arctic each winter. Given the large interannual variability, I can picture at least some of that, some of the time, sitting out the summer near Ellesmere.

But that's just a guess, the Arctic seems to be heading toward a new state.

William Crump

Peter, you just don't get it that the graphs are averaging several items and not measuring the decline of a single item.

If you jump off a cliff without a parachute and I jump off the same cliff with a parachute we will not hit bottom at the same time and I will not fall at the average rate for the two of us measured at the time you go splat at the bottom of the cliff.

William Crump

L Hamilton:

I appreciate your comment concerning volume, but I can not get volume data for the Arctic Basin.

The only data I have for the thickness of first year ice is from ICESAT. While it showed significant declines in the thickness of multiple-year ice, the percentage declines for first year ice were much lower. This leads me to question whether graphs that mix the volume of first and second year ice with multiple year ice will show a more rapid rate of decline than will occur in the future.

I do not think the dynamic processes are as simple as saying x volume decline occurred last year so x volume of decline will occur this year.

I would like to see someone show me the thickness decline rate for first and second year year ice, which may not be the same as the volume change. Volume for this type of ice may be increasing as its extent increases since it is replacing multiple-year ice.

Based on NSIDC charts, first and second year ice are becoming the predominant type of ice.


Per NSIDC, "This year the older, thicker ice has increased somewhat over last year, although it remains younger than the 1979 to 2000 average ice age. Data through the third week of March shows an increase in sea ice one to two years old, and older than two years old, compared to recent years. However, the amount of older ice remains much lower than in the mid-1980s, and there is still almost none of the oldest ice, older than four years old, that used to dominate much of the Arctic Ocean."

How can I tell if the volume decline is due to the loss of this older type of ice rather than a decline in the thickness of first and second year ice?

William Crump

Re: "virtually ice free"

One issue with the definition is that 2,000km3 of .75 meter ice generates an area of 3,000,000 km2 ice. Even worse, 2,000km3 of .5 meter ice generates an area of 4,000,000 km2. I am struggling to call these conditions a "virtually ice free" Arctic when I can see the ice in the satellite image. I agree the ice will be thin, but it will still be there.

As volume declines, the ice thickness declines, with the result that area and extent are maintained. At some point, volume declines will slow and extent loss will accelerate, but this may not happen until the Arctic ice minimum hits 750km3 or less.

William Crump

Oops, s/b 2,666,666 km2 not 3,000,000 km2


Hi William,

So glad to see that you're back. I was beginning to feel very lonely, as the only one of my kind here.

I hope you won't be pursuing your extremely alarmist thesis too much, though.

If you are so desperate to save the ice of the entirely artificially demarcated Arctic Basin, this necessarily means that the extra heat within the Arctic Ocean is distributed elsewhere within that Ocean.

Most likely, it will then be distributed along the fringes of the Artic, along the shallow continental shelf, which is extremely shallow, with a seabed composed of sedimentary silt, frozen into methane clathrates.

Hopefully, the nightmare scenario which you have repetitively described, whereby the central Arctic surface sea ice is preserved, and the heat is transported elsewhere, to the fringes of the Arctic Ocean, is just a worse-case scenario, which need not trouble us too much.

Honestly, it is far, far, far preferable that the extra heat within this system should act upon the surface sea ice, and not upon the methane clathrates of the seabed.

In sum, you may be right. Though I suspect not, as FrankD has quite patiently explained several dozen times now.

But I have to hope that you are wrong. For the sake of saving Santa Claus's workshop at the North Pole, your model seriously risks an accidental firing of the "methane clathrate gun".

...which might be a bit worse news than poor old Santa having to find a new workshop.

Kevin McKinney

Will, for various reasons, including the fact that advection of ice out of the Arctic is a leading mechanism for ice loss, ice is lost most rapidly at the edges of the pack. When the edge of the pack is mostly confined within the Arctic basin, do you really think that loss rates will decline markedly?

That seems to be what you are thinking will happen, and I must say that I see no reason why it should.


From a physical perspective, if the amount of energy in the Arctic increases linearly, ice volume should decline in a parabolic way (ie, as a 2nd order polynomial function).

I would suggest that the temperature data that we have for the Arctic (for example, via UAH here: http://vortex.nsstc.uah.edu/data/msu/t2lt/uahncdc.lt) shows that energy in the Artic is indeed increasing in linearly.

So the only thing that would prevent a parabolic decline would be if there was something that prevented this increasing energy from mixing well across the whole Arctic.

While I can certainly see sheltered bays and narrow channels - such at those in the Canadian archipelago - being barriers to energy mixing, there is nothing of the kind in the central Arctic basin.

Unless such a physical mechanism is presented, I am betting on parabolic decline in Arctic ice volume.

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