# Spherical temperature averaging using Icosahedral grids

I reckon that Spherical triangulation of monthly global temperatures is by far the most honest way to present temperature data, but it has one drawback. You cannot make annual or longterm averages because the triangular mesh is forever changing from one month to another. To make such averages you really need a fixed grid. So does that mean that we have to give up and return to 2D fixed lat,lon grids such as  those used by CRU, NOAA  or NASA, in order to calculate annual distributions? The answer is no because there is a neat method of defining fixed 3D grids that maintains spherical symmetry.  This is based on subdividing an Icosahedron.

Figure 1. Annual average temperature anomalies for 2016 over the pole. Note the equal area triangles of an icosahedron grid.

Figure 1. shows  the annual average temperature anomalies for 2016 using this method, while here  and below is the WebGL (Nick Stokes) 3D interface.

An Icosahedron is a 20 sided object whose sides are equilateral triangles. The mesh is formed by dividing each triangle into 4 new triangles whose vertices are made to lie on a sphere. By repeating this procedure iteratively you generate an equal area mesh over the earth’s surface.

Generating a triangular mesh from an Icosahedron

I was lucky to find an IDL package to do this, and after quite a struggle managed to integrate it with the global temperature data.  I am using a level-4 mesh containing 2562 triangles which seems to be the best fit to the average density of global temperature measurements. Despite this there are still 94 empty triangles lying in remote regions. These have been assigned  zero anomaly and are visible as occasional single light blue triangles.

Another well known way to tesselate a sphere uses hexagons, as on a soccer ball. However it turns out that this is a just a truncated icosahedron!  Check out cutting off each vertex midway. It is just easier to make soccer balls that way.

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### 2 Responses to Spherical temperature averaging using Icosahedral grids

1. Nick Stokes says:

Clive,
The icosahedral meshes are great. Interpolating to them is a challenge, but I see you have that under control. But I don’t see a big problem with annual averages on irregular mesh. You do the averaging locally, without needing the mesh, and then mesh the final result. With missing months, you just decide how many will be allowed before abandoning the station, and interpolate in time to infill within the average.

I did long term trends here, which has the same kind of issue. It’s noisier because fewer stations qualify. Interpolation on a regular mesh has the pros and cons of smoothing generally.

• Clive Best says:

Nick,

You do the averaging locally, without needing the mesh, and then mesh the final result.

Yes I guess that is kind of true, but as the time interval increases so more and more stations will fall out. Hadcrut4 could do the same by taking yearly or even decadal averages of each station before gridding them in each 5 deg cell. Instead they grid them first and then take annual averages, and as far as I know all the rest (including Berkeley) do the same.

Berkeley Earth do a global fit each month onto a lat,lon grid , but (AFAIK) not for each year or each decade.