Guest Post by Willis Eschenbach
Yesterday, I discussed the Shepherd et al. paper, “Recent loss of floating ice and the consequent sea level contribution” (which I will call S2010). I also posted up a spreadsheet of their Table 1, showing the arithmetic errors in their Table.
Today, I’d like to discuss the problems with their method of calculating the loss in the Arctic Ice Pack. To start with, how big is the loss? Here is a graphic showing the change in area of the Arctic ice pack from one year’s loss of ice, with the ice pack area represented by a circle:
Figure 1. One-year change in the area of the Arctic Ice Pack, using the average annual loss which occurred 1996–2007. Note the obligatory polar bears, included to increase the pathos.
OK, so how do they calculate the Arctic ice loss in S2010?
Here is their description from the paper:
We estimated the trend in volume of Arctic sea ice by considering the effects of changes in both area and thickness. According to ERS and Envisat satellite altimeter observations, the 1993-2001 (average wintertime) thickness of Arctic sea ice was estimated to be 273 cm (Laxon et al., 2003), the thickness decreased by 6.7 ± 1.9 cm yr-1 between 1992 and 2001 (Laxon et al., 2003), and the thickness decreased by 4.8 ± 0.5 cm yr-1 between 2003 and 2008 (Giles et al., 2009).
We combined these datasets to produce a new estimate of the 1994-2008 thickness change. Published satellite microwave imager observations (Comiso et al., 2008) show that the 1996-2007 Arctic sea ice area trend was -111 ± 8 x 10^3 km2 yr-1 and, based upon our own analysis of these data, we estimate that the 1990-1999 average wintertime area of Arctic sea was 11.9 x 10^6 km2.
The combined reductions in Arctic sea ice area and thickness amount to a decrease in volume of 851 ± 110 km3 yr-1 during the period 1994 to 2007, with changes in thickness and area accounting for 65 % and 35 % of the overall loss, respectively.
What is the problem with that method?
The problem is that they have assumed that the ice is the same thickness over the entire area. As a result, the reduction in area is causing a large loss of ice, 35% of the loss by their estimate.
But the ice is not all the same thickness. The perimeter of the ice, where the loss occurs, is quite thin. As a result, they have overestimated the loss. Here is a typical example of the thickness of winter ice, from yesterday’s excellent article by Steve Goddard and Anthony Watts:
Figure 2. Ice thickness for May 2010. Note that the thickness of the ice generally tapers, from ~ 3.5 metres in the center to zero at the edges.
So their method will greatly overestimate the loss at the perimeter of the ice. Instead of being 273 cm thick as they have assumed, it will be very thin.
There is another way to estimate the change in ice volume from the melt. This is to use a different conceptual model of the ice, which is a cone which is thickest in the middle, and tapers to zero at the edges. This is shown in Figure 3
Figure 3. An alternative model for estimating Arctic ice pack volume loss.
Upon looking at this drawing, I realized that there is a way to see if my model fits with the facts. This is to use my model to estimate how much of a thickness change would be necessary to create the 111,000 square kilometre loss. It turns out that to get that amount of loss of area, it would require a ~4 cm ice loss over the entire surface … which is a good match to their estimate of ~ 5 cm of loss.
So, what difference does this make in the S2010 estimate of a global loss of 746 cubic kilometres per year? Lets run the numbers. First, I’ll use their method. I have used estimates of their numbers, as their description is not complete enough to give exact numbers.
Thickness loss: (11,900,000 km^2 – 111,000 )* 5 cm / (100,000 cm/km) = 589 cubic km (66 % of total).
Area loss: 111,000 km^2 * 273 cm / (100,000 cm/km) = 303 cubic km (34% of total)
Total: 892 cu km, which compares well with their answer of 851 cubic km. Also, the individual percentages I have calculated (66% and 34%) compare well with their numbers (65% and 35%). The difference is likely due to the decreasing area over the period of the analysis, which I have not accounted for.
So if we use a more realistic conceptual model of the ice (a conical shaped ice pack that is thick in the middle, and thin at the edges), what do we get?
The formula for the volume of a cone is
V (volume) = 1/3 * A (area of base) * h (height)
or
V = 1/3 * A * h
The difference in volume of two cones, therefore, is
V = 1/3 * (A1*h1 – A2*h2)
This means that the volume lost is
V = 1/3 * (11900000 km^2 * 273 cm – 11789000 km2 * 268 cm) / (100000 cm/km)
= 297 cubic km
This is much smaller than their estimate, which was 851 cubic km. And as a result, their estimate of global ice loss, 746 km^3, is reduced by 851 – 297 = 554 km^3, to give a final estimate of global ice loss of 192 cubic kilometres.
FINAL THOUGHTS
1. Is my estimate more accurate than theirs? I think so, because the simplistic assumption that the ice pack is equally thick everywhere is untenable.
2. How accurate is my estimate? I would put the 95% confidence interval (95%CI) at no less than ± 25%, or about ± 75 km^3. If I applied that same metric (±25%) to their estimate of 851 km^3, it would give a 95%CI of ±210 km^3. They give a 95%CI in their paper of ±215 km^3. So we are in agreement that this is not a WAG*, it is a SWAG*.
3. This exercise reveals some pitfalls of this kind of generally useful “back-of-the-envelope” calculation. First, since the final number is based on assumptions rather than data, it is crucial to be very clear about exactly what assumptions were made for the calculations. For example, from reading the paper it is not immediately evident that they are assuming constant thickness for the ice pack. Second, the change in the assumptions can make a huge change in the results. In this case, using my assumptions reduces the final value to a quarter of their global estimate, a value which is well outside their 95%CI estimate.
To close, I want to return to a separate point I made in my last post. This is that the S2010 paper has very large estimates of both gains and losses in the thickness of the Antarctic ice shelves. Now, I’m not an ice jockey, I’m a tropical boy, but it seems unlikely to me that the Venable ice shelf would lose a quarter of a kilometre in thickness in 15 years, as they claim.
Now it’s possible my math is wrong, but I don’t think so. So you colder clime folks out there … does it make sense that an ice shelf would lose 240 metres thickness in fifteen years, and another would gain 130 metres thickness in the same period? Because that is what they are claiming.
As mentioned above, I have posted their table, and my spreadsheet showing my calculations, here. here I’d appreciate anyone taking a look to make sure I have done the math correctly.
PS:
* WAG – Wild Assed Guess, 95%CI = ±100%
* SWAG – Scientific Wild Assed Guess, 95%CI = ±25%
[UPDATE] My thanks to several readers who have pointed out that I should not use 273 cm as the peak thickness of the ice. So following this NASA graphic of submarine measured winter and summer ice, I have recalculated the peak as being about 4 metres.
Using this value, I get arctic ice loss of 344 km^3, and a global ice loss of 239 km^3.
Did you mean to say in Figure 1 “…after one year’s ice loss” and NOT “…after once year’s ice loss”? I like the figure but I can’t stand it with the grammar…
willis
I have been a long time reader of this site and i simply believe you need to turn it down a notch
Willis, could you or Steve do a and overlay or side by side using figure 2 and perhaps 2009 or 2007 maximum Ice loss extent so we can do an eyeball comparison on your SWAG. Even better the May 2009 thickness vs the September 2009 extent.
For example here is the September 15th 2009 Ice extent: http://arctic.atmos.uiuc.edu/cryosphere/IMAGES/ARCHIVE/20090915.jpg
An eyeball comparison shows all the purple and blue areas on figure 2 are gone. The ice that is left is the green ice. The compacted ice (red) along the shore lines is also gone except that of Greenland’s. Therefore if you were to use a “constant ice thickness” 1.5 or 1.75 meters would be a better guesstimate.
Since 1 centimeter = 0.01 meters the 273 cm thick they have assumed is 2.73 meters.
And yes I am aware of sublimation and melting from the bottom due to “warmer” arctic waters. However the constant still looks about a half to a full meter too high and that would lead to a very high number in the final calculations.
re mb: May 30, 2010 at 3:11 am
and Willis Eschenbach: May 30, 2010 at 4:01 am
mb, as Willis has pointed out, you’re making an elementary mistake in geometry by dealing with ‘averages’. Just calculate the before and after cone volumes directly from V = AH/3 and subtract them. This gives: ΔV = (10,532 – 10,829) = -297km^3, as per Willis’ calculations.
/dr.bill
Ring around the rosie, a pocket full of spheres. Thought you’d outfox me. Well……….
Willis et al
Very interesting, unable to comment on your maths,
Do any of these papers help with your question on ice thickness? [google search ‘defence research thickness of ice’]
http://www.igsoc.org/journal/3/28/igs_journal_vol03_issue028_pg727-732_724-726.pdf
http://www.igsoc.org/journal/12/66/igs_journal_vol12_issue066_pg417-421.pdf
http://www.nasa.gov/topics/earth/features/seaice_skinny.html
sea ice thickness section in wiki does nor provide reference http://en.wikipedia.org/wiki/Measurement_of_sea_ice
the polar bear circle graphic is very confusing
Why do we always have to keep proving to the smug, the-science-is-settled crowd that the earth is not flat? I suppose every few centuries or so it takes a brave, question authority, set of pioneers to rock the boat.
899;
David,
Is it your presumption that seals never take birds from the shore, and that they’ve never done so?>>
My presumption is that an an animal whose main diet is fish, krill, squid and penguins hunts mostly in the ocean and being ungainly on land, rarely pursues prey in that fashion. Brief reading reveals descriptions such as “When hunting penguins, the leopard seal patrols the waters near the edges of the ice, almost completely submerged, waiting for the birds to enter the ocean.” I came across no references regarding hunting of penguins on land or ice.
899
I will think that the birds have a ‘species memory’ regarding that matter of where to not make a home.>>
Think what you want but many of species return repeatedly to the breeding grounds of their birth and this appears to be one of them.
899
Alligators and crocodiles have been known to take prey on dry land, even though they are most fitted to do so in the water, or at water’s edge.>>
Crocodiles in particular are very well fitted to taking prey on land. But they only venture a small distance from water and then only when very hungry. By small distance I’m talking tens of meters not one hundred + kilometers. If the penguins were using the ice expanse to protect their nesting grounds from predation they certainly wouldn’t need 100 + kilometers of it. Or even 10. Or 1. The fact that they can accomplish 100km+ it but only by stretching their physical resources to the absolute limit just to arrive at their food source at maximum vulnerability to their chief predator suggests that it is not natural for them. They evolved with a very narrow ice expanse that was easily crossed and which has built up naturaly over time. I doubt you can propose an evolutionary process that is logical and arrives at the same result.
The reasonable conclusions is that the nesting grounds were established with very narrow ice expanse in place, suggesting that the current ice extent is large by comparison and, given that their nesting grounds are near shore line, the much lower ice extent did not translate to much higher sea levels.
The assumption the ice area’s thickness looks like a cone, being highest in the center and tapering toward the ages, seems to conflict with the animation in Steve Goddard’s May 28 article “The great 2007 ice crunch – it wasn’t just melt.”
This is sort of OT, but I’d like a few comments about mathematical symbols.
Willis wrote:
V = 1/3 * A * h
If I had written this in grade school math class, my teacher would have corrected my use of asterisk (*) instead of the proper times (×). Skipping several years ahead, ASCII-1968 (okay, it was brand new, but that’s okay) doesn’t have a times sign, nor do typewriters, so programming languages used * instead, and we all found it quite natural. Some used uparrows for exponentiation, e.g. x↑ which worked great until ASCII changed it to be carat, ^, but that worked great until C used it for XOR, which was not so great. Keypunches didn’t have that symbol, so several languages used **, e.g. x**2, which works pretty well.
At any rate, here’s my question. How many people see something like x * y and don’t immediately read it as multiplication? For that matter, how many kids are being taught in school that * is the times operator and × is just an antique that only grandparents use?
Then there’s always ¢ which our typewriter did have, but has never made it into common computer usage. Blame that on inflation, just my 2 ¢, now worth $0.25.
Suggestion: Add a vertical line at the ends of the triangle to show the volume (cross-section of the volume) which had to melt to decrease the covered surface area. Maybe color that purple, and color purple the small rectangles which represent the amount of melt needed in the slab model. The difference between the small purple triangles and the end purple rectangles is the difference between the ice retreat in the two models.
Willis…the ice is melting(which tends to happen when temps get above 0C)May 29th,2010…..Inuvik…Northwest Territories…+29C…Edmonton…Alberta…at -2C…28 hours of snow…but is it weather?Hey.At least I can still ski in Banff!
Willis Eschenbach says> (Remember, they have Antarctic ice shelves dropping a quarter kilometre in thickness …)
Yeah, you keep saying that. But do you have any real evidence or reason to doubt this? I understand that you are concerned about the Venable ice shelf, a small glacier on the Antarctic coast, and that the error estimate they give is large.
Acording to “Recent Antarctic ice mass loss from radar interferometry and regional
climate modelling” ERIC RIGNOT et al. doi:10.1038/ngeo102, “The strong, widespread
correlation between ice thinning and ice velocity (>50myr−1), for
example, on the Berg, Ferrigno, Venable, Pine Island, Thwaites,
Smith and Getz glaciers, indicates that thinning is caused by the
velocity of glaciers being well above that required to maintain
mass balance, that is, ice stretches longitudinally, which causes
it to thin vertically.”
This makes sense to me, and it would explain extreme thinning. Any comments?
damn, Greenland is still covered in white.
Oh, for cripesakes Gallier2, watch the coordinates, I almost materialized inside of a wall.
Did you know, that there used to be so many grizzlies around here, the natives avoided Butte Creek. And my grandfather, who was born here about 115 years ago, told me the Sacramento River would get so full of salmon during the run seasons, the river would turn to foam.
Okay pardon me if I’ve told this tired story before. Blame it on Anthony for using a favorite catch phrase to head the post. Over a thousand years ago, the Owens River Valley was full of lakes and swamps and rivers and little creeks fed by enormous glaciers that capped the mountains. There was a virtual forest of cottonwoods and aspen and all those water loving trees. Thousands and thousands of people lived there. We saw a popular chipping ground right after a rain – the ground, as far as the eye could see, glittered with obsidian chips from hunters making arrow heads. There’s rock carvings estimated at over 5,000 years old – the Paiute who live there now say the carvings where there when they came to the valley from Nevada thousands of years ago.
Now, of course, everybody knows, water is so scarce in the Owens River Valley, people might shoot you for it. When you look to the east, you can just see the little remnants of the great glaciers, all melted. Those glaciers were formed by the Great Uplift, when water was trapped there, and frozen in the cold. As the years went on, thousands of years, those glaciers melted and formed the lakes and streams. As the glaciers disappeared, the water disappeared. The “old people” (that’s what the Paiute call them) disappeared. The Paiute lived sparingly, watching the trees disappear and the land dry up. Then the white settlers came in and used what was left – compared to what had been there, it was a drop in the bucket.
I’m not saying water transfers are okay, I think it’s wrong to take water miles from it’s source to support badly planned subdivisions. But Mono Lake was only a couple of hundred years from drying up on it’s own, the Angelinos just hastened Nature’s process.
Earth has it’s plan. And remember what Rick says in Casa Blanca – let me paraphrase – the problems of puny people don’t amount to a hill of beans. We hurt ourselves, and we hurt our fellow creatures, but Earth Abides (great book by the way, George Stewart, probably the dustiest book on your local library shelf).
Now, you kids play nice, I’m off to a hockey game.
So repeat after me, weather is not climate and a local phenomenon is not global.
And two years, or ten, does not a trend make.
Willis:
I always thought SWAG 95% CI was +/- 50%
When in doubt, Flip a coin!
Is it safe to assume that melting is uniform over the entire surface of the cone? I’d expect there to be a difference of temperature from edge to center (warmer farther south) so that melting would be more rapid at the edges. Then there is the possible complication of melting from beneath due to water temperature…..
IanM
@juanita
huh?
I do not know about the rest of the world but in my neck of the woods the ice starts forming at the edges of the pond and starts melting towards the center away from the edges. When wind blows it packs ice against a solid object such as the shore.
Although I’m not a climatology buff, it’s fascinating to read these comment threads. Among other things, I see the same tricks tried again and again by those trying to discredit the arguments of the day’s posting. In this case (by noiv), finding some data/figures/whatever that appear to contradict the assertions of the posting, but are often irrelevant. It doesn’t seem to occur to these people that by repeated tricks of this sort, they “give the game away”.
Along these lines, on another comment thread someone, I forget who, linked to a recent/current page on the GISS site. I guess it makes the point (relentless warming), but at the same time there’s something remarkable about the graphs – the winter of 2009-2010 has disappeared! All those record-breaking (low) temperatures in N. A., Europe and elsewhere apparently mean nothing when it comes to the real temperature record.
Let’s look at the big picture for a second –>
http://i50.tinypic.com/35iackn.gif
Global and arctic sea ice are slowly diminishing and there is no evidence that this trend has stopped.
IF the trend continues, then arctic summer sea ice will disappear before the end of this century.
The trend is still intact, and in Wall Street parlance, “the trend is your friend”. (I.e., bet on the trend until it actually stops.)
dr.bill says:
mb, as Willis has pointed out, you’re making an elementary mistake in geometry by dealing with ‘averages’.
———
No, I’m not.
———-
dr.bill futher says:
Just calculate the before and after cone volumes directly from V = AH/3 and subtract them. This gives: ΔV = (10,532 – 10,829) = -297km^3, as per Willis’ calculations.
/dr.bill
———-
The H you are using here is not the average height of the cone. It is the height of the top point. The average height (thickness) of the cone is not H, it is H/3.
When we are dealing with ice sheets, we are measuring the average thickness of the ice sheet, not some abstract, model produced H. If we insist on making the model assumption that the ice forms a circular cone, the height of this cone is three times the average thickness of the cone.
@gallier2
Thanks for pointing to another evidence. Just a hint: the ice is the white stuff. And are you saying, that you need 30 years of ice-free Arctic to agree that things are changing?