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.
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.
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:
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.