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.
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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.
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Now to gain that 130 meter thickness in 15 year would truly classify as a decadal snowstorm! (and what are we left with, figments, figments everywhere!)
Where do these scientists get their vivid imaginations?
now, this is what global warming is telling us! It will melt every ice particle there is.
Willis,
I said in another thread that the ice pack should be modeled as a wedge. My point was that growing ice extent late in the ice formation season would result in rapid expansion of the thin edge of the wedge, followed by rapid retreat once the melting season started until the thin edge retreated back to thicker ice, whereupon the retreat would slow down. That’s exactly what we are seeing this year in the Arctic. I’ve lived in a winter climate all my life and have seen fresh water lakes and rivers freeze and break up so often it never ocurred to me that there was any question that it was a wedge shape in the first place!
Would like to see someone do an article on ice and Penguins too. The silly bird brains nest as much as 200 miles from their food source and almost die every year going back and forth across the ice trying to get food back and forth to their chicks and mates. 200 miles! And they waddle! So once has to ask how this came to be? They are just barely adapted enough to manage the treck over and over again, so the question arises, why would they have chosen such a breeding location in the first place? Seems to me the more logical explanation is the breeding grounds were locked into their instincts when there wasn’t a whole lot of ice in the first place and the treck to open sea for food was a short one. As the ice grew, they had to slowly adapt to a longer and longer treck, which in turn implies that at some time in the past there was a lot less than 200 miles of ice between the breeding grounds and the open sea. Even bird brains don’t choose their first breeding ground 200 miles from food.
sigh. I meant 1oo miles. A 2 looks just like a 1 when you are so tired that the keyboard starts to get blurry. But Willis posted another article so I had to read it and then a I had to respond to it and then I had this idea about Penguins…. more Ritallin…need more Ritallin….
Modeling the thickness with a rounded cross section, like chopping the top off a sphere, might be more accurate than a cone. The result would probably still be much closer to a cone than to a constant thickness.
In the formula V= 1/3 * A * h, h is height at the apex, not average height.
Area loss: 111,000 km^2 * 273 cm / (100,000 cm/km) = 303 cubic km (34% of total)
——————
Multiplying the area loss by the average thickness seems like such an obvious overestimation but it sails right through climate science peer review.
Funny how these things always make it “worse than we thought”.
davidmhoffer says:
May 30, 2010 at 1:02 am
Would like to see someone do an article on ice and Penguins too. The silly bird brains nest as much as 200 miles from their food source and almost die every year going back and forth across the ice trying to get food back and forth to their chicks and mates. 200 miles!
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A logical deduction is so as to avoid their local predator, the Leopard seal.
How many seals would actually travel that far inland for a meal?
@mindbuilder
I think it is so very, very flat that it makes no difference if it is a cone or a hemisphere.
mb says:
May 30, 2010 at 2:05 am
It seems this is what is being done. As the difference in volume between the two cones is being calculated, the question is moot in any case.
899 says:
May 30, 2010 at 2:23 am
A logical deduction is so as to avoid their local predator, the Leopard seal.
How many seals would actually travel that far inland for a meal>>
Leapord seals don’t hunt on land or ice. They take penguins as prey when the jump off the ice into the sea emaciated, tired and starving from waddling 16o km to the edge of the ice, making them easy prey for the leaport seals. No need to leave your natural habitat for food when they have not choice by to come to yours to eat and are half dead when they get there.
Is climate science calculation a victim of poor imagination, worse than previously imagined?
The shape of the “ice volume” would approximate that of a “contact lens”
Use V = 4/3¶r3 – V = 4/3¶r3 ( first radius is smaller than the second radius)
davidmhoffer says:
May 30, 2010 at 2:34 am
Leopard seals don’t hunt on land or ice. They take penguins as prey when the jump off the ice into the sea emaciated, tired and starving from waddling 160 km to the edge of the ice, making them easy prey for the leopard seals. No need to leave your natural habitat for food when they have not choice by to come to yours to eat and are half dead when they get there.
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David,
Is it your presumption that seals never take birds from the shore, and that they’ve never done so?
I will think that the birds have a ‘species memory’ regarding that matter of where to not make a home.
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.
mb says:
In the formula V= 1/3 * A * h, h is height at the apex, not average height.
JER0ME says:
It seems this is what is being done. As the difference in volume between the two cones is being calculated, the question is moot in any case.
———–
Actually, it does matter. If the height at the top of a cone is h, the average height of the cone is h/3. So the correct formula for the volume of the cone, using the average height, is
V = 1/3 * A * h = A * (h/3) = A * average height.
In the example, we have two cones. The difference between the volumes of the cones is
V1-V2= A * (average height1 -average height2)
This gives a difference in ice volume which is three times as big as what we are being told, in excellent agreement with Shepherd el al.
The growth of Ice is just as complex as the evaporation and precipitation cycle.
Cold temperatures can thicken the ice from below the surface of water(strength of currents is a factor). The precipitation on the surface in snow has to be packed which then there are different types of snowfall that carry different moisture and mass content. Winds also are a factor in packing snow, moving snow and temps, sunlight, cloud cover, night or day which in turn is planetary rotation.
Why do these climate alarmists love ice so much? Do they have a set amount that they’d be happy with, or would they like to see another ice age? I think we know that in the history of this planet we’ve gone from periods of no ice to much more ice than we currently have. These are the limits of the natural variability. So the loss or gain of a few thousand km2 don’t really concern me. Melt it all for all I care.
What is bad for polar bears is good for penguins.Polar bears supposedly need more ice to hunt and penguins need less to get to the ocean .
In the mining industry we would go out and measure the ice thickness by in situ sampling. Anything else is intellectual waffle and irrelevant.
Willis, you are spot on with your analysis – it’s like drilling one hole into a mineral deposit and extrapolating that result to the rest of the orebody. OK in theory until you actually have to mine it, and then you discover that the statistical tricks used are basically meaningless.
It’s part of the sample volume variance problem, as well as not knowing how to sample a physical object, here a thin sheet of ice of variable thickness.
But who has the cash to measure ice thickness properly? You need to do that to substantiate the various models proposed here.
One would need to do a drilling traverse over the ice sheet to measure the ice thickness. Or maybe a seismic traverse using percussive methods rather than explosives.
That means spending a long time in winter doing field work.
Can you imagine what OSH departments would impose on this activity?
So the next best effort is to wax intellectually using computer modelling, and if the BS is plausible, converted into “fact”.
Well, if Arctic fits on a coaster who would care? Zooming in makes changes obvious:
The region of Jakobshavn Glacier this year (27/05/2009):
http://ice-map.appspot.com/?map=Arc&sat=ter&lvl=7&lat=69.208788&lon=-53.878389&yir=2010&day=147
Same region same day last year (27/05/2009):
http://ice-map.appspot.com/?map=Arc&sat=ter&lvl=7&lat=69.208788&lon=-53.878389&yir=2009&day=147
Looks somewhat different, or not?
Now let’s find the earliest day with comparable circumstances of 2010 in 2009:
http://ice-map.appspot.com/?map=Arc&sat=367&lvl=7&lat=69.208788&lon=-53.878389&yir=2009&day=162
That’s two weeks later (11/06/2009). Let’s assume a melting period of 180 days, then the difference is more than 8 percent.
What’s wrong with this math?
mb says:
May 30, 2010 at 3:11 am
mb, many thanks, I’m glad to have somebody take a hard look at the math. However, its a bit more complicated than that.
First, if you wish to claim that we need to use an apex three times as high as the 273 cm ice thickness used in S2010, we would end up with the thickest ice being about 8 metres thick … which just isn’t happening. So the first thing we can see is that by assuming constant ice thickness out to the edge, they are estimating too large a volume of ice.
Next, you have the wrong formula for the difference in the volume of the cones. You have neglected the difference in the areas of the bases of the cones. The correct formula is:
V = 1/3 * (A1*h1 – A2*h2)
Even if we use the unphysical value of 8 metres of ice at the apex, this gives us 500 km^3, still well below their value of 851 km^3.
From looking at Figure 2 above, we can see that the thickest ice is at about 350 cm, much smaller than the 8 metres you propose. Using 350 cm as the apex gives us an ice loss of 326 km^3, rather than the 297 km^3 I had estimated before.
And that give us a total global ice loss of 221 km^3, which is still much smaller than their estimate of 746 km^3. Now we can say “Well, we should use 4 metres for the thickest ice”, but it doesn’t make a whole lot of difference, because we are dealing in estimates and approximations. Under any reasonable assumptions, their estimate of global ice loss is off by at least a factor of two, and likely more. (Remember, they have Antarctic ice shelves dropping a quarter kilometre in thickness …)
@hoiv
Yes, so what?
Same dates, but on the other side of Greenland
http://ice-map.appspot.com/?map=Arc&sat=ter&lvl=7&lat=69.208788&lon=-53.878389&yir=2009&day=147
http://ice-map.appspot.com/?map=Arc&sat=ter&lvl=7&lat=69.208788&lon=-53.878389&yir=2010&day=147
how dramatic, the exact reverse effect of your two selections.
So repeat after me, weather is not climate and a local phenomenon is not global.
Sorry, wrong coordinates:
http://ice-map.appspot.com/?map=Arc&sat=ter&lvl=7&lat=67.5&lon=-35.0&yir=2009&day=147
http://ice-map.appspot.com/?map=Arc&sat=ter&lvl=7&lat=67.5&lon=-35.0&yir=2010&day=147
Willis Eschenbach> I don’t claim that the ice is formed like a cone. On your picture above the cone is clearly marked ” how I model the arctic ice pack”. This means that it is your model, not mine. If you find that this model is ridiculously unrealistic, you are complaining about yourself, please leave me out of that fight. I just discussed the calculations you make inside this model, the model which you proposed.
It’s true that in my previous post I forgot that you also change the area of the two cones A1 is not necessarily equal to A2. Fortunately this is easy to repair: The correct formula is
V1-V2=A1*average height1 – A2*average height2
As you see, this does not change the fact that you underestimate the difference between the volumes by a factor of 3.
Your main argument is that in the paper we are discussing the volume is calculated as area * average height, and that this formula might not be valid for the actual shape of the ice. Then you proceded to give an example of a possible shape (the cone) where the formula would not be true. But the example fails, because it is also true for a cone that volume = area * average height.
It seems that now you argue that the ice has some different shape, and that this for some reason leads to different results on its volume. But as a matter of fact, the formula is true in great generality. I dare you to come up with an even distantly possible shape of the ice which does not satisfy that volume = area * average height!
I was intrigued by the (Emperor) penguin puzzle, so I checked out the Wikipedia entry in search of clues. According to this:
1. The penguins begin mating around March (i.e. in late summer), and in March-April walk to their breeding sites between 50 and 120 kilometers from the edge of the pack ice
2. Although not at the edge of the pack ice, most of the breeding sites are still on pack ice rather than land
3. They lay their eggs in May-June. The females go off to feed while the males look after the eggs
4. The eggs hatch in July-August. Both parents take turns to feed them for several months, which necessitates long treks to the sea. The distance to the sea gradually diminishes as the pack ice melts with the approach of summer.
5. Finally, both adults and young (which now have their adult plumage) walk the considerably shorter distance to the sea in December.
It seems evident from this that the main constraint on the breeding cycle is that they must lay their eggs in a location that will not melt into the sea before the young are hatched and able to walk (or swim in icy waters). However, I don’t understand why they have to walk such a long distance from the sea before breeding. Wouldn’t the Antarctic pack ice be close to its minimum extent by April? I wonder if the Wiki article is accurate on this point. It seems inconsistent that the breeding sites would be closer to the sea in December (mid-summer) than in March-April, after a full summer of melting.