I’d like to highlight one oddity in the Shakun et al. paper, “Global warming preceded by increasing carbon dioxide concentrations during the last deglaciation” (Shakun2012), which I’ve discussed here and here. They say:
The data were projected onto a 5°x5° grid, linearly interpolated to 100-yr resolution and combined as area-weighted averages.
The oddity I want you to consider is the area-weighting of the temperature data from a mere 80 proxies.
Figure 1. Gridcells of latitude (North/South) and longitude (East/West)
What is area-weighting, and why is it not appropriate for this data?
“Area-weighting” means that you give more weight to some data than others, based on the area of the gridcell where the data was measured. Averaging by gridcell and then area-weighting attempts to solve two problems. The first problem is that we don’t want to overweight an area where there are lots of observations. If some places have 3 observations and others have 30 observations in the same area, that’s a problem if you simply average the data. You will overweight the places with lots of data.
I don’t like the usual solution, which is to use gridcells as shown in Figure 1, and then take a distance-weighted average from the center of the gridcell for each gridcell. This at least attenuates some of the problem of overweighting of neighboring proxies by averaging them together in gridcells … but like many a solution, it introduces a new problem.
The next step, area-averaging, attempts to solve the new problem introduced by gridcell averaging. The problem is that, as you can see from Figure 1, gridcells come in all different sizes. So if you have a value for each gridcell, you can’t just average the gridcell values together. That would over-weight the polar regions, and under-weight the equator.
So instead, after averaging the data into gridcells, the usual method is to do an “area-weighted average”. Each gridcell is weighted by its area, so a big gridcell gets more weight, and a small gridcell gets less weight. This makes perfect sense, and it works fine, if you have data in all of the gridcells. And therein lies the problem.
For the Shakun 2012 gridcell and area-averaging, they’ve divided the world into 36 gridcells from Pole to Pole and 72 gridcells around the Earth. That’s 36 times 72 equals 2592 gridcells … and there are only 80 proxies. This means that most of the proxies will be the only observation in their particular gridcell. In the event, the 80 proxies occupy 69 gridcells, or about 3% of the gridcells. No less than 58 of the gridcells contain only one proxy.
Let me give an example to show why this lack of data is important. To illustrate the issue, suppose for the moment that we had only three proxies, colored red, green, and blue in Figure 2.
Figure 2. Proxies in Greenland, off of Japan, and in the tropical waters near Papua New Guinea (PNG).
Now, suppose we want to average these three proxies. The Greenland proxy (green) is in a tiny gridcell. The PNG proxy (red) is in a very large gridcell. The Japan proxy (blue) has a gridcell size that is somewhere in between.
But should we give the Greenland proxy just a very tiny weight, and weight the PNG proxy heavily, because of the gridcell size? No way. There is no ex ante reason to weight any one of them.
Remember that area weighting is supposed to adjust for the area of the planet represented by that gridcell. But as this example shows, that’s meaningless when data is sparse, because each data point represents a huge area of the surface, much larger than a single gridcell. So area averaging is distorting the results, because with sparse data the gridcell size has nothing to do with the area represented by a given proxy.
And as a result, in Figure 2, we have no reason to think that any one of the three should be weighted more heavily than another.
All of that, to me, is just more evidence that gridcells are a goofy way to do spherical averaging.
In Section 5.2 of the Shakun2012 supplementary information, they authors say that areal weighting changes the shape of the claimed warming, but does not strongly affect the timing. However, they do not show the effect of areal weighting on their claim that the warming proceeds from south to north.
My experiments have shown me that the use of a procedure I call “cluster analysis averaging” gives better results than any gridcell based averaging system I’ve tried. For a sphere, you use the great-circle distance between the various datapoints to define the similarity of any two points. Then you just use simple averaging at each step in the cluster analysis. This avoids both the inside-the-gridcell averaging and the between-gridcell averaging … I suppose I should write that analysis up at some point, but so many projects, so little time …
One final point about the Shakun analysis. The two Greenland proxies show a warming over the transition of ~ 27°C and 33°C. The other 78 proxies show a median warming of about 4°C, with half of them in the range from 3° to 6° of warming. Figure 3 shows the distribution of the proxy results:
Figure 3. Histogram of the 80 Shakun2012 proxy warming since the most recent ice age. Note the two Greenland ice core temperature proxies on the right.
It is not clear why the range of the Greenland ice core proxies should be so far out of line with the others. It seems doubtful that if most of the world is warming by about 3°-6°C, that Greenland would warm by 30°C. If it were my study, I’d likely remove the two Greenland proxies as wild outliers.
Regardless of the reason that they are so different from the others, the authors areal-weighting scheme means that the Greenland proxies will be only lightly weighted, removing the problem … but to me that feels like fortuitously offsetting errors, not a real solution.
A good way to conceptualize the issue with gridcells is to imagine that the entire gridding system shown in Figs. 1 & 2 were rotated by 90°, putting the tiny gridcells at the equator. If the area-averaging is appropriate for a given dataset, this should not change the area-averaged result in any significant way.
But in Figure 2, you can see that if the gridcells all came together down by the red dot rather than up by the green dot, we’d get a wildly different answer. If that were the case, we’d weight the PNG proxy (red) very lightly, and the Greenland proxy (green) very heavily. And that would completely change the result.
And for the Shakun2012 study, with only 3% of the gridcells containing proxies, this is a huge problem. In their case, I say area-averaging is an improper procedure.
w.
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As I understand it previous studies compared CO2 as measured from ice cores with temperatures as measured from ice cores. In other words essentially the same samples were used to determine both temperature and CO2. Consequently although absolute dates were inaccurate the relative dating of the CO2 rise and the temperature rise could be measured with much greater precision. By contrast this study attempts compare completely unrelated measures of temperature and CO2. I can’t see how anything like the precision in measurement of relative timing that can be achieved comparing ice cores to ice cores can be obtained.
The only justification offered for rejecting the ice core to ice core comparison method with all its advantages is a poorly explained speculation that the bulk of the world might have warmed somewhat later than the poles. Even if one were to accept that this were so, it still would not admit the conclusion that CO2 caused the warming. That is because you would still have to explain why THE POLES warmed BEFORE CO2 levels rose, a result which this study does nothing to invalidate and which is remains known with much greater certainty (because it is obtained by comparing like with like) than the results in this paper, which as pointed out here, seem to be attempting to paper over a lack of accurate data with dubious statistics.
Mosh
There are numerous scientific studies that suggest there is some sort of temperature relationship between British climate and that in other parts of the world. I thought you liked scientific studies?
By the way, over at the other place I asked for the links to the papers you cited-Julio etc-they’ve become lost in the mass of posts on both threads since you made your comment.
Tonyb
Posted on April 9, 2012 by Willis Eschenbach
Area-weighting” means that you give more weight to some data than others, based on the area of the gridcell where the data was measured…
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Willis, you are probably right about handling the gridcells to calculate “global temperature” and trends, however the whole thing can only be correct, if the termometer data is representative for the gridcells.
To put it in a simple way, if you have a termometer in a certain area, you can only use it for area weighting if you can prove, that the termometer’s data is representative for the whole area. If you can not prove it, then you can not do the weighting.
What we know for sure is, that the termometer data (if collected correctly) is representative for the box containing ther termometer. You can not just draw a gridcell around the box on the map and claim the termometer represents the whole gridcell. But unfortunately this is what some climate scientist do.
So, even if the operations with gridcells are mathematically correct, the result would be a fiction, if there is no scientific proof, that the termoter data is representative for the gridcells.
aaron says:
April 9, 2012 at 11:39 am
Interesting thought … not sure how you’d determine if that were the case, though.
Perhaps the place to look would be the CO2 records from the South Pole. During Antarctic summer (Dec-Jan-Feb) there’s a significant meltback of sea ice. If the melting ice released CO2, we’d expect to see a corresponding variation in the CO2 down there in the south during that time.
… OK, just took a look. Bad news, the annual increase in CO2 at the south pole starts around March and ends in September … and that’s when the ice is forming, not melting as your theory suggests.
Sorry,
w.
xham says:
April 9, 2012 at 11:46 am
You are right, xham. And there are a variety of ways to have equal-area gridcells, not just geodesic. But I still say averaging by any kind of gridcells introduces problems. That’s why I use cluster analysis averaging, because of the errors introduced by any system of gridcells.
These areas just get worse when you have scarce data. You might end up with three adjacent gridcells that contain most of the data, and one gridcell in a distant area. The fact that the gridcells have the same area is meaningless in that situation.
w.
Nice work again Willis!
Off topic Salby Video!
For anyone intrested in Murry Salbys game changing work on the carbon cycle you here have a video where his slides is shown. Its a fantastic lecture!
[ http://www.youtube.com/watch?v=YrI03ts–9I&feature=youtu.be ]
Weighting aprox is in accordance to the latitude area of global area a grid box is.
A 5×5 box at Equator is double area than an 5×5 box around 60N of 60S. At the poles the 5×5 box is not much at at all.
Cos of the latitude. Cos of 0 is 1, Cos of 10 is 0.985, Cos of 20 is 0.94, Cos of 30 is 0.866,Cos of 40 is 0.766, Cos of 50 is 0.643, Cos of 60 is 0.5, Cos of 70 is 0.342, Cos of 80 is 0.174and Cos of 90 is 0.0
Leif Svalgaard says:
April 9, 2012 at 1:22 pm
Steve from Rockwood:
If you have the data as a text file or an Excel file, send it to me leif@leif.org and I’ll graph it.
————————————————-
On its way.
Steven Mosher says (April 9, 2012 at 1:36 pm)
“Its actually BOTH. Added CO2 will warm the earth and the ocean will respond by outgassing more CO2.”
And that’s why the temperature has been going up exponentially since the end of the ice age?
/definite sarc
rgbatduke says:
April 9, 2012 at 12:09 pm
“I don’t think the climate scientists realize how important it is to address and resolve this geometric issue before they spend all of the time that they spend making blind assertions about “global average temperature”. Numerical integration on the surface of the sphere — which is what this is, whether or not they understand that — is difficult business, and one that it is almost impossible to carry out correctly in spherical polar coordinates. There is an entire literature on the subject, complete with alternative tilings and coordinatizations. But even using the Jacobean doesn’t fix spherical polar coordinates when one relies on a sparse grid of samples, let alone a sparse grid of samples biased by their geography.”
Yes, climate scientists have proved time and again that they care little or nothing for the assumptions that underlie their assertions. They are quite happy to take any two temperature readings and treat them as comparable though the two readings exist in geographic areas that bear no resemblance to one another. Thank God that you, Willis, and other serious scientists are willing to “call them out” on this most egregious systemic error.
A 5×5 box from Equator to either 5 N or S is weighted 0.9990. A 5×5 box from the poles to either 85N or S is weighted 0.04362.
That means that the area at the 5×5 grid at Equator is 23 times larger than the 5×5 grid at the poles.
Just for the record, this paper is totally at odds with the recent paper by Lu et al due to be published this month.
Using Ikaite crystals and their heavy Oxygen content he claimed that the Medieval Warm Period was not confined to the Northern Hemisphere but also happened in Antarctica. This suggests a close correlation between Antarctic temperature and Global Temperature.
Steven Mosher says:
April 9, 2012 at 1:27 pm
This is not even close to the problem. Up to 90% of the cells have NO DATA at all. Only a few cells have more than one point. So perhaps Mosher can repeat his averaging discussion with 1 sample in one cell and no data in the surrounding 50 cells.
A quick review of the proxies shows that some are highly auto correlated spatially, but only where they are located close together. This may sound like a stupid observation but localized high correlation and regional correlation are very different. Local correlation only gives you confidence to extrapolate outward to cells that are in the vicinity of the closely correlated points (such as in one area of Greenland). This process only gets you a small amount of grid coverage. Now take that average and move it several thousand kilometers to an area that is physically very different. The extrapolation should be meaningless. If a proxy in central Canada does not correlate well with a proxy in Greenland and there is no data between the two points, how can you say that an average proxy between the two points exists midway distance-wise and is a real estimate of the actual proxy at that location?
W., the seasonality of co2 is due geographic biological factors, not temp. Might look at co2 over a full pdo cycle, unfortunately I don’t think we have that data yet.
For the melt rate, we can look at sea level.
I largely agree with Willis here. I think they probably did as he described and that area weighting by cells is not helpful. If they did something else, I can’t see any reason for invoking 5×5 cells. And if they did want to use cell-based weighting, much larger cells would have been better.
But I also note Chris Colose point that they looked at other methods as well. In fact there is a section (4) in their SI which does, among other things, a Monte Carlo test to see if their average is sensitive to spatial randomness. And the answer seems to be no. That section 4 is headed “Hpw well do the proxy sites represent the globe?”
So while a better weighting scheme could have been used, it would likely make little difference.
I find it hard to believe it really 2012 and we are having this discussion. I could understand it if it was the 13th century and people were just beginning to figure such things out, but now with all our scientific understanding, this strikes me a very elementary. Surely there is standard way of doing this.
Steve from Rockwood is correct.
Not only as he states, but if you regard each of the 80 proxies (I have done so) you will see that very few of the proxies are from high latitudes. Most are from very temperate areas (15-25°C). Of course: that is where the proxy-meisters like to go! Where it is not too cold. SO the underweighting of critical zones has another positive feedback. I call this the “Caribbean Proxy Feedback”.
TomT says:
April 9, 2012 at 2:57 pm
Surely there is standard way of doing this.
There is. H.G. Wells wrote about the only sure way. It is going back in time with a thermometer and a cannister of lithium hydroxide.
Leif Svalgaard says: April 9, 2012 at 12:32 pm
……………..
Steven Mosher says:April 9, 2012 at 12:24 pm
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Least flawed way to set the area weighting is triangulation. No output of a single station is used, but each triangle area is given average of the 3 corner stations.
In line with rgbatduke:
From occasional browsing on Google Scholar I’ve found 55 PDFs on spherical geometry, grids, and data collection, a mere dilletante sample.
The Warmistas badly need the classic text “Statistical Analysis of Spherical Data” by Fisher.
My previous aerospace work includes orbital coverage analysis that encountered data gridding problems similar to those of irregularly situated thermometers. One similarity was how the Earth’s rotational velocity makes latitude paramount.
Fisher’s methods are overwhelmingly successful for such applications, but they also provide a metric for data sufficiency, which I gather this 80-point load would fail. Must be why they ignore Fisher.
Steve from Rockwood says:
April 9, 2012 at 2:37 pm
Up to 90% of the cells have NO DATA at all. Only a few cells have more than one point. So perhaps Mosher can repeat his averaging discussion with 1 sample in one cell and no data in the surrounding 50 cells.
Here is the distribution of proxies [from Steve]: http://www.leif.org/research/Shakun-Proxies.png
Looks like none of these guys have ever heard of plotting on an equal area stereonet (or Schmidt net). In geology when you want to know about groupings of data points, you plot on an equal area net, and count the points falling within a circle representing 1% of the total area. In practice you use a plat with a 10cm radius and count the points falling with a 1 cm radius of your major lat/long intersections. There is no need to “correct” for area (ie multiply by sin(latitude) as that is taken care of already.
A Schmidt net of lat/long lines is constructed by calculating the projection of any point on the interior of a half globe into the horizontal plane of the globe *as seen from a point one radius above the centre of the plane*. Somewhere here I have the equations for these which I used long long ago on a galaxy far far away, to do this (in Fortran on an IBM360 (with output to a lineprinter!!) and later on a series of HP calculators.)
Quick course on stereonet usage here: http://folk.uib.no/nglhe/e-modules/Stereo%20module/1%20Stereo%20new.swf
But Willis is essentially correct, that there is no meaningful way to average such paucity of readings over such a large area. A stereonet might be usefully applied to compute an ‘average temperature’ for any particular 1% of the area, but this is totally unhelpful for any area where there ARE no records, and is explicitly based upon a number of assumptions about the homogeneity of the data which is represented. It is one thing to determine the “average pole to the plane” for a couple of hundred individual measurements of strike and dip and use the resultant average as a representative for the plane being measured, and quite another to do so for temperatures. If you are unsure of what I mean here in reference to planes, think of averaging many GPS position readings taken while stationary and using a minimum RMS distance measure to find the “position”. Schmidt nets sorta, kinda do that on paper…sorta, without the RMS Uses the Mark 1 eyeball.
It would of course be beyond hope that any of the warmists would think about talking to geologists or even GIS people about methodologies of plotting data representing 3 dimensional spatial arrangements.
Arrgh. Memory FAIL. A Schmidt net projection is ‘seen’ from a point square root of 2 times the radius above the center of the plane, not the radius distance (that’s for wulff nets).
Mosher: “and that frost fairs in England can reconstruct the temperature in australia”
“During the Great Frost of 1683–84, the worst frost recorded in England”
The first sub zero DJF HADCET was 1684. There have only been 3. 1684, 1740 and 1963.
HADCET: 1684 DJF -1.2
http://www.metoffice.gov.uk/hadobs/hadcet/ssn_HadCET_mean.txt
Does Mosher have thermometer records from Australia in 1684?
From a geostatistical point of the view – my critique of the interpolation technique used is as follows:
a) the grid cells are too small. The cell should ‘match’ the density fo the data so most (if not all) of the cells have a sample point. Very roughly and without revisiting the dataset presented in Willis E’s post, I would say the cells need to be at lest 20x bigger. b) The data is extremely clustered. Kriging (a type of averaging) has the effect of declustering the data and may have proved useful. However as a general comment extrapolating the data beyond the limited sample points is problematic and will mislead. Ask yourself the question, if you had to invest sizeable capital on the bet that a proxy point in Japan accurately predicted the value near Midway, how confident would you be in the proxy sample dataset.