<h2> Area weighted trends for Antarctica</h2>

In an earlier thread I discussed the use of a least-squares program TempLS for calculating regional trends in Antarctica from mixed ground and satellite (AVHRR radiometer) data. It was in the context of recent papers by Eric Steig and colleagues (S09) and Ryan O'Donnell and colleagues(O10), where various statistical methods were used to achieve these objectives. In that analysis the stations were not (down-)weighted by their area density, which has the potential to excessively weight the Antarctic Peninsula warming, and also West Antarctica. I don't think S09 or O10 did this either, and it isn't obvious how to do it. But I've been developing Voronoi tessellation methods, and now I have been able to get them to work for this analysis. I'll also look at some residuals and explained sum of squares (R²).

Area weighting

I've talked in previous posts about the need for area density weighting. The intention is to make the summation used for averaging temperatures etc behave as integral approximations, so that in particular regions won't bias the result just because they have many stations. The clearest way to do this is to allocate a unique region of space around each station, and use that area as the weight in the least squares method. As I've also mentioned, that can lead to a locally noisy distribution of weights, and smoothing is used.

We are processing monthly averages, and so the weights should be adjusted for each month to reflect the stations that report (and not others). That means in our case a new Voronoi tessellation each month. This was done. The area weighting is done only for ground stations. Satellite stations are on a fairly uniform grid, so don't need differential weighting by area. They do, however, need a weighting relative to the ground stations and this is currently a loose parameter.

Area weighting draws attention to the question of what area we are in fact talking about. The original data set included non-Antarctic islands, including Gough Island, which at 40.4 S latitude is less Arctic than Rome. You could only weight these by the amount of ocean they cover, but is coverage of oceans intended? I excluded all the distant islands, even including Scott Island. Of course there is another question there, which is whether the boundaries of Antarctica marked on the map are appropriate for a climate analysis, or whether the boundaries sea ice areas should be included. I've stuck mostly with the map-, as did O10 and S09.

I've used a modified relative weighting factor here, as shown on the maps. It is a ratio of expected total weight for ground stations vs that for AVHRR. A high value means ground stations have more weight - a value of 1 means they are weighted equally.

Varying relative weighting

I'll start with a low weighting factor - 0.2, and using 5 principal components. This means satellite data dominates, so there is not much change from last time. Here is the map:


It looks a bit like this one, though not with warming so concentrated in the peninsula. Overall, though, the trend is a bit higher.
You'll see that I've now quoted an R² reduction. This is the relative change of sums of squares of residuals in going from accounting for local effects (basically, anomalies), to including the spatial trends, It is the ratio explained, and as you'll see, it isn't much. I'll say more about that when we look at individual resuduals. The reason it isn't much is that there is a lot of short-term variation, which dominates the SS residuals. The fact that it is small doesn't mean that it is insignificant.

Now bringing the factor up to about parity:



Now there is a difference. The pattern is somewhat similar to when I upweighted the ground stations previously, but with a more marked cooling spot. The continent trend is down, though still high. There is slightly more intense warming on the peninsula, although West Antarctica is relatively strong. And FWIW, R² is down very slightly.

So now on to letting ground stations dominate, with a factor 5.0:



Now there is even more cooling over E Antarctica, and also stronger warming over the peninsula, and strong over WA. Total warming is well down - now to 0.095 C/decade. And R² is up a bit.

I had thought R² might be used to express a preference for the value of this factor, but the changes are small and inconsistent. Overall, I think what it shows is a marked effect of the downweighting of the denser stations on the Peninsula and around the Ross Sea, and upweighting of the sparser cooling stations in the East.

Numbers of PCs

For the next test involving PCs, I'll set the weighting factor to 2.0, and use 4,5,6,7 PC's:


There are significant differences. With more PC's R² increases, as it must, and also the trend increases significantly, from about 0.1 C/decade to about 0.17 C/decade. I'm not sure what to make of that.I showed the eigenvalue plots for unweighted here; the plots now are too similar to repeat.

Residuals

To get a feel for R² and to see that the scheme is actually working, I plotted some residuals. The black curves are the residuals after subtracting the LS local temperature L. This is in effect the monthly mean. Then the red curves are what you get after taking out the predicted trend.

Clearly the noise is large, and it makes it almost impossible to see the month-to-month effect. So I have plotted also four year moving averages. These do show differences, with an apparent trend in the black residuals, which reduces with the red curves.

For individual stations, the red trend-fitted residuals need not be trendless. The extent to which they are indicates how well the PC's are serving as basis functions.

Next Steps

Now that I have the spatial weighting scheme working, I'll follow Eric's suggestion of just a time series analysis of regions, including West Antarctica. And I'll try to get the validation analysis done. There's another distraction around in that I have the Voronoi tessellation working for the whole globe, so I really want to see what differencethat makes to the indices.