The best grid for Earth temperature calculation.

Earlier this month, I wrote about the general ideas of gridding, and how the conventional latitude/longitude grids were much inferior to grids that could be derived from various platonic solids. The uses of gridding in calculating temperature (or other field variables) on a sphere are

1. To gather like information into cells of known area
2. To form an area weighted sum or average, representative of the whole sphere
3. a necessary requirement is that it is feasible to work out in which cell an arbitrary location belongs

So a good grid must have cells small enough that variation within them has little effect on the result ("like"), but large enough that they do significant gathering. It is not much use having a grid where most cells are empty of data. This leads to two criteria for cells that balance these needs:

• The cells should be of approximately equal area.
• The cells should be compact, so that cells of a given area can maximise "likeness".

Lat/lon fails because:

• cells near poles are much smaller
• the cells become long and thin, with poor compactness

I showed platonic solid meshes with triangles and squares that are much less distorted, and with more even area distribution, Clive Best, too, has been looking at icosahedra. I have also been looking at ways of improving area uniformity. But I haven't been thinking much about compactness. The ideal there is a circle. Triangles deviate most; rectangles are better, if nearly square. But better still are regular hexagons, and that is my topic here.

With possibly complex grids, practical usability is important. You don't want to keep having to deal with complicated geometry. With the cubed sphere, I posted  here a set of data which enables routine use with just lookup. It includes a set of meshes with resolution increasing by factors of 2. The nodes have been remapped to optimise area uniformity. There is a lookup process so that arbitrary points can be celled. But there is also a list showing in which cell the stations of the inventory that I use are found. So although the stations that report vary each month, there is a simple geometry-free grid average process

• For each month, sort the stations by cell label
• Work out cell averages, then look up cell areas for weighted sum.

I want to do that here for what I think is an optimal grid.

The optimal grid is derived from the triangle grid for icosahedra, although it can also be derived from the dual dodecahedron. If the division allows, the triangles can be gathered into hexagons, except near vertices of the icosahedron, where pentagons will emerge. This works provided the triangles faces are initially trisected, and then further divided. There will be 12 pentagons, and the rest hexagons. I'll describe the mapping for uniform sphere surface area in an appendix.

Lookup

I have realised that the cell finding process can be done simply and generally. Most regular or near-regular meshes are also Voronoi nets relative to their centres. Thatis, a cell includes the points closest to its centre, and not those closer to any other centre. So you can find the cell for a point by simply looking for the closest cell center. For a sphere that is even easier; it is the centre for which the scalar product (cos angle) of 3D coordinates is greatest.

If you have a lot of points to locate, this can still be time-consuming, if mechanical. But it can be sped up. You can look first for the closest face centre (of the icosahedron). Then you can just check the cells within that face. That reduces the time by a factor of about 20.

The grids

Here is a WebGL depiction of the results. I'm using the WebGL facility, V2.1. The sphere is a trackball. You can choose the degree of resolution with the radio buttons on the right; hex122, for example, means a total of 122 cells. They progress with factors of approx 2. The checkboxes at the top let you hide various objects. There are separate objects for red, yellow and green, but if you hide them all, you see just the mesh. The colors are designed to help see the icosahedral pattern. Pentagons are red, surrounded by a ring of yellow.

The grid imperfections now are just a bit of distortion near the pentagons. This is partly because I have forced them to expand to have simiar area to the hexagons. For grid use, the penalty is just a small loss of compactness.

Data

The data is in the form of a R save file, for which I use the suffix .sav. There are two. One here is a minimal set for use. It includes the cell centre locations, areas, and a listing of the cells within each face, for faster search. That is all you need for routine use. There is a data-frame with this information for each of about 8 levels of resolution, with maximum 7682 cells (hex7682). There is a doc string.

The longer data set is here. This has the same levels, but for each there are dataframes for cells, nodes, and the underlying triangular mesh. A dataframe is just R for a matrix that can have columns of various types, suitably labelled. It gives all the nodes of the triangular mesh, with various details. There are pointers from one set to another. there is also a doc string with details.

Appendix - equalising area

As I've occasionally mentioned, I've spent a lot of time on this interesting math problem. The basic mapping from platonic solid to sphere is radial projection. But that distorts areas that were uniform on the solid. Areas near the face centres are projected further (thinking of the solid as within the sphere) and grow. There is also, near the edges, an effect due to the face plane slanting differently to the sphere (like your shadow gets smaller when the sun is high). These distortions get worse when the solid is further from spherical.

I counter this with a mapping which moves the mesh on the face towards the face centre. I initially used various polynomials. But now I find it best to group the nodes by symmetry - subsets that have to move in step. Each has one (if on edge) or two degrees of freedom. Then the areas are also constrained by symmetry, and can be grouped. I use a Newton-Raphson method (actually secant) to move the nodes so that the triangles have area closest to the ideal, which is the appropriate fraction of the sphere. There are fewer degrees of freedom than areas, so it is a kind of regression calculation. It is best least squares, not exact. You can check the variation in areas; it gets down to a few percent.