I have been developing different integration methods, with the basic idea is that the agreement between a number of good methods with different basis is a guide to the amount of uncertainty that is due to method. I think it is quite small, and I will show comparative graphs. The key thing is that the different methods within TempLS agree with each other better than the indices agree among themselves.

One of the methods I have been exploring is the use of Spherical Harmonics (SH). This is not so much an integration method as an enhancement, and is so treated in TempLS V4. So the agreement between enhanced but otherwise inferior methods, and the better methods that are not enhanced, is further corroboration of the convergence of integration methodology.

I will illustrate all this with an active graph, of the kind I have been using for the index results themselves. You can, with mouse, change scales, translate axes, choose various subsets to plot, and also smoothe. An added facility here is that you can switch to difference plots in any combination you choose. I will at some stage post a facility like that for WebGL for doing this.

The post will be long, so I start with a table of contents.

#### Table of contents

- Brief summary of methods
- Active plot of results
- Comparison of methods.
- Methods in the context of indices and their variability
- Effect of enhancement
- Conclusion

#### Brief summary of methods

Some recent articles are here, here and here, with links to earlier. The methods I'll list are- No weighting - the simple average of stations reporting in each month is used. This is a very poor method, but of interest here because it becomes quite serviceable after SH enhancement.
- Simple grid. This is the traditional method, where the temperature anomalies within each cell of, say, a lat/lon 2° grid, are averaged, and the global is then the area-weighted average of the cells that have results. Area-weighting relates to the shrinking of area of cells near the poles. It is used by HADCRUT; the paper of Cowtan and Way showed that the effect of accounting for cells without data gave an important correction to trends. I do not now use lat/lon grids, but rather a cubed sphere, or other platonic solid based grid. The alternative is usually icosahedron. In each case I use mappings to make the cells of almost uniform area.
- Grid with infill. This assigns to empty cells a value based on neighbors. Most recently, I do this by solving a Laplace equation, with the known cell values as boundary condition. That sounds complex, but the simple Southwell relaxation method, , which initially guesses the unknown cells, and then replaces with the average of neighbors until convergence, is quite adequate.
- Irregular triangular mesh. This has been my workhorse; it is basically finite element quadrature, with linear interpolation within triangles with stations as vertices. I have thought of it as my best method.
- First order LOESS. This sets up a regular array of nodes (icosahedral), and assigns values to them based on local first order weighted regression with a typically 20 of the nearest stations. The regular array is then simply averaged. I think this is a rival for the best method.