Global average, integration and webgl.

Another post empowered by the new WebGL system. I've made some additions to it which I'll describe below.

I have written a lot about averaging global temperatures. Sometimes I write as a sampling problem, and sometimes from the point of view of integration.

A brief recap - averaging global temperature at a point in time requires estimating temperatures everywhere based on a sample (what has been measured). You have to estimate everywhere, even if data is sparse. If you try to omit that region, you'll either end up with a worse estimate, or you'll have to specify the subset of the world to which your average applies.

The actual averaging is done by numerical integration, which generally divides the world into sub-regions and estimates those based on local information. The global result always amounts to a weighted average of the station readings for that period (month). It isn't always expressed so, but I find it useful to formulate it so, both conceptually and practically. The weights should represent area.

In TempLS I have used four different methods. In this post I'll display with WebGL, for one month, the weights that each uses. The idea is to see how well each does represent area, and how well they agree with each other. I have added some capabilities to the WebGL system, which I will describe.

I should emphasise that the averaging process is statistical. Errors tend to cancel out, both within the spatial average and when combining averages over time, when calculating trends or just drawing meaningful graphs. So there is no need to focus on local errors as such; the important thing is whether a bias might accumulate. Accurate integration is the best defence against bias.

The methods I have used are:

• Grid cell averaging (eg 5x5 deg). This is where everyone starts. Each cell is estimated as an average of the datapoints within it, and weighted by cell area. The problem is cells that have no data. My TempLS grid method follows HADCRUT in simply leaving these out. The problem is that the remaining areas are effectively infilled with the average of the points measured, which is often inappropriate. I continue to use it because it has often very closely tracked NOAA and HADCRUT. But the problem with empty cells is serious, and is what Cowtan and Way sought to repair.
• My preferred method now is based on irregular triangulation, and standard finite element integration. Each triangle is estimated by the average of its nodes. There are no empty areas.
• I have also sought to repair the grid method by estimating the empty cells based on neighboring cells. This can get a bit complicated, but works well.
• An effective and elegant method is based on spherical harmonics. The nodes are fitted with a set of harmonics, based on least squares regression. Then in integrating this approximation, all except the first go to zero. The integral is just the coefficient of the constant.

The methods are compared numerically in this post. Here I will just display the weights for comparison in WebGL.

The display is below, and I'll discuss below that. The gadget and its operation are described here. You'll see there are new checkboxes with the various WebGL objects beside. You can make them disappear. There is a new facility that I'll describe in a later post that creates new objects suffixed _P and _L for the edge nodes and lines. You can toggle these, and also Grid, which is a 5x5° mesh matching what my grid methods use. It should be switched off when looking at the other methods, since it is not then relevant.

The plot is for Jan 2014, since I can then use production weights that may not be calculated on the same day of the month (recent months might have varying data).I have graphed both the weights (adding to 100) and log weights, since they cover a wide range. I have shown the mesh method first. The weights are much as you might expect - high (red) over Africa, poles and Brazil, and low (blue) over US and Europe, where density is high. The mesh does the best job of reflecting local density.

The plain grid is next, and its faults are obvious. If you look at the Arctic, it is blue, not red. Station weights are a fraction of the cell area only, and can't exceed that. The fraction depends on how many stations in each cell. So a cell representing a big area in Africa can have no more weight than a tropical ocean cell, and an Arctic station is limited to much less, because cells are thin there. You'll see odd patterns in the regular ocean mesh. That is from the interaction of the 4x4 SST grid and the 5x5 integration grid. Mostly there is one node per cell, but where there are two, the weight halves, and four, which is possible, brings the weight down further.

The cell infilling method gets much closer to what is expected in Africa and Brazil. It still somewhat underweights the Arctic, probably through incomplete infilling. There is an an apparent artefact near 0°E ocean, which may reflect a problem with the progoram infilling across this point.

The spherical harmonics weights are interesting. SH gives very similar results to mesh (and infill), butit assigns weights with a much broader brush, by regional density rather than local. The good result suggests that the scale that it is not resolving makes little net contribution to the integral.

I now plot all integration results in the new latest data table; you have to click the TempLS button.