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.

#### Active plot of results

The idea of active plotting is described here, and my regular example is the monthly plot of indices here. The active plot for this post is here, with details below:You can control which data are shown with the checkboxes. The plot can be dragged with mouse, or rescaled by dragging motions behind the relevant axis. Below the plot, the most useful buttons are Smooth, which toggles 12-month moving average, and Data, with produces a new window with a dump of whatever is currently on the plot. The radio buttons on the right of the moveable legend allow you to choose one set to be used as the reference, with the others shown as differences. The top radio restores to the original. You can combine smoothing and differencing. You can click on the color squares on the legend to choose new colors. I'll describe examples below, but encourage you to experiment. All data is set to have anomaly base 1981-2010. This images which follow are just screenshots from settings of this active plot.

#### Comparison of methods

I have shown just a recent period with no smoothing. I have made the no weighting faint, and not attempted to scale so it is within the plot - the idea here is just to show how bad it is. The other methods cluster fairly well, with excursions of simple grid.

Here is the same plot, but with mesh as the reference. Relative accuracy is now clearer. The rather faint infilled now seems closest to mesh, with the blue LOESS a little more deviant, and simple grid a lot more. The better methods are generally within about 0.02°C of mesh.

Here now is the plot of the full period since 1900, with 12-month running average. There is the same relative concordance, but a notable change before about 1960. This reflects the earlier lack of data for Antarctica before the 1957 IGY. There is not only greater variability, but a shift, relative to the mesh base, with grid being higher and LOESS lower.

#### Methods in the context of indices and their variability

Here is a plot of the main method of TempLS with the four major indices. I have used blue colors for the indices. You can see a general clustering of the reddish colors relative to the indices.

And here is the same data plotted relative to mesh. It is easier to quantify the agreement. Again the methods (except for grid) generally agree to within about 0.02°C, with the indices more in the range 0.05-0.1.

Here is the same data, smoothed, going back to 1900. It does not look so smooth. Both methods and indices show the 1957 shift. There is also a similar tendency to drift, with HADCRUT apparently warming relative to TempLS mesh before 1957.

#### Effect of enhancement

Here is a plot of recent years (difference from mesh), contrasting plain methods with SH enhancement with modest order (max frequency) 4. You may notice that earlier plots mentioned orders 6 and 10. I have changed this because 10 is well into the region of instability. One thing to note here is that enhancement brings the no-weighting case from way out to somewhat reasonable. Perhaps more useful is that simple grid goes from rather bad to really quite good. LOESS barely changes. The reason is that the improvement, where it happens, is due to fixing the faulty integration of the low frequency component. LOESS didn't really have a fault toi fix.

Here is a smoothed plot of the whole period. The SH-enhanced plots are particularly affected by the 1957 shift due to Antarctica. The reason is that there are a latitude only modes which are poorly constrained with the gap near the S pole.

Here is a plot of orders 4 and 8 of SH enhancement. Order 4 has 25 functions, and has up to 4 periods around the globe. Order 8 has 81 functions, and 8 periods around the globe. The worst methods show continued improvement, and grid is really quite good at level 8. For LOESS, however, level 8 shows a separation from the results of lower order. It is still quite good, but I think the change represents the start of instability. Level 10 is a good deal worse.

#### Conclusion

I think the important result is that the uncertainty due to integration method is now well quantified. There are methods that agree consistently to within 0.01-0.02°C. This is less than the variation seen between the major indices. That is of course partly due to slightly different data, and also due to method tweaks such as land masking and use of EOFs (NOAA). I don't think the tweaks are a major factor, and I think the indices could actually be improved by using the better integration methods.I have been interested in SH enhancement, being a very distinct method, and I think it helps reinforce the conclusion that there is a narrow band of proper results. But I don't think it is of great practical use, since it rescues bad methods, but may only have downside with good ones.

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