But I think they have a wider use. The general pattern is that a field, temperature t, say, is partitioned:

T=E+A

where T is some kind of expected value, and A is an anomaly. That sounds vague, and there are many ways you could make such a partition. But it becomes clear when associated with the use made of it. The purpose is often to calculate some linear function of T, say L(T). Here L will be some kind of average. But suppose L is an ideal, and we actually have to rely on an approximate version L1 (eg coarse sampling).

So L(T) is approximated by L1(T)=L1(E)+L1(A). Now the objective is to choose E in such a way that

- L1(E) can be replaced by L(E), which is more accurate for that part
- L1(A) loses some of the error that marred L1(T)

I will explain with three examples here, of which the third will be developed in detail, because it bears on my long-term project of more accurately calculating surface temperatures:

- Temperature anomalies for global average (GISS). Here the station temperatures have some kind of mean of past values subtracted. It is usually the mean of some fixed period, eg 1961-90. Here is a typical post in which I explain why absolute temperatures are too inhomogeneous to average with the sample density we have, which the anomalies thus constructed are homogeneous enough, if only because they have approximately the same expected value (0). But in terms of this structure, they don't quite fit. The component E (climatology) that is removed is not calculated with a more accurate estimator, but is discarded, with only the anomaly average reported. But the mode of partition is still important, to be sure that the discarded part does not carry with it variation that should have been attributed to A. Using a consistent base period is one way of ensuring that.
- Another example is one I developed in a post at WUWT. This was about the time samples, for a single station, that go into forming a monthly average. An objection sometimes raised that characterising that by two values per day (max and min) "violates Nyquist". Nyquist doesn't really tell us about this kind of sampling, but I looked at the error introduced in sampling regularly a few times a day (eg every 12 hours). This does create aliasing of some harmonics of the diurnal frequency to zero, which adds spurious amounts to the monthly average. I showed that this is a limited error, but which can be reduced with a T=E+A partition. Now E is an estimate of the regular diurnal cycle, based on fine (eg hourly) sampling of a few reference years. That takes most of the diurnal variation out of A, and so sparse sampling (L1) aliases very little to zero, and causes little error. The fine operator (L) can be applied to the E component.
- The third example, to be developed here, is for the monthly spatial integration of anomalies. I'll describe it first rather generally. Suppose T is a variable to be integrated over any domain, and L1 is an approximate integration operator. Then suppose E is a Fourier fit with the first n orthogonal functions, where the inner product is integration with L1.

Then E contains the low-frequency terms which contribute most to the integral, leaving A with just high frequency fluctuations. These may be of lower amplitude that the whole, but more importantly, they will make little contribution to the low pass integration operator. This is still true for the approximate L1, since the basic reason for the low-pass is cancellation.

In fact, by the Fourier process, L1(A) is exactly 0, since A is orthogonal to the fitting functions, of which the lowest is a constant. So if the improved integration operator L is just exact integration, the varying fitted terms will make zero contribution to the integral, except for the constant term. The improved integral is just the integral of that constant. The improvement comes because the approximate integral of those low frequency components is replaced by the exact.

#### Integration methods for the sphere

I have developed four main methods for averaging temperature anomalies via numerical integration on the sphere, which I have incorporated into my monthly program TempLS, with results regularly reported here (click TempLS button). They are described in greatest detail here, with some further ideas. The methods described are- Simple gridding - just assign to each cell of a lat/lon grid the average of the anomalies in the cell, and then us ethe area-weighted average of the cells with data. This is basically what is done in HADCRUT
- Infilled gridding - where cells without data are assigned a value derived from neighbouring cells in some way, and then the area-weighted average of all cells can be used. With the assignment done by kriging, this is the Cowtan and Way improvement of HADCRUT.
- Finite element integration using an irregular triangular mesh. This is my preferred method, as it assigns to each point on Earth the interpolated value from the three nearest data points for each month.
- Spherical Harmonics. I'll now talk more about that. In its original form, I did a least squares regression of the temperature with a set of spherical harmonics, and took the lowest coefficient as the integral.

The final method, using Spherical Harmonics (SHs), is not like the others, in that it doesn't introduce a subdivision of the area. But I now think it is best seen as not an independent method, but as an improvement applicable to any method. To just write out the regression maths, if L1(xy) is the scalar product, and H is the vector of fitting functions (SH), then the regression coefficients of the fit are

b = L1(H⊗H)

^{-1}L1(H⊗T), where ⊗ is the outer product from the weighted integration

and so b

_{0}L1(H

_{0}) is the average of T

I generally denote L1(H⊗H) as HwH, since L1 is just a weighted sum with weights w (this is the architecture of TempLS). The functions H are orthogonal wrt exact integration L, so L(H⊗H) would be a unit matrix. The approximate version deviates from this, and a figure of merit is the condition number of HwH. This is kept down if the integration method L1 is good.

In addition to the averaging methods listed above, I should add a very primitive one - the simple, unweighted average. This is generally regarded as too unreliable for use, but my original SH method, which actually performed very well, was basically the improvement of this method. You can of course expect better outcomes from improving better methods, and as I shall show, this happens. However, the relative improvement is much less. The reason is that the improvement comes from the improved integration of the SH components, and with, say, the mesh method, the finite element integration was already very good.

#### More on Spherical Harmonics

I described the functions here, with visualisation here, and a movie here. There are two parameters L and M; it is natural to think of M as determining the number of oscillations, with L varying from 0 to M (for fixed M) going from purely latitude oscillation to purely longitudinal. In the latter case (L=M), there are M periods of the sinusoidal variation around the equator. For SHs of order up to M, there are (M+1)^{2}functions, and these are the groupings that I will test below.

#### Tests

I looked at various means of anomalies for each of the months of 2018. I used the anomalies calculated by the mesh method; other integration schemes give slightly different values, but I don't think that changes the issues of integration. For each month and order M of SH, I calculate the mean for each of the four methods- No weighting (simple average)
- Simple grid with no infill
- Infilled grids
- Integrating on irregular triangular mesh.

You can see that the better methods, mesh and infilled grid, start out usually close together, with often the simple average furthest away. But then, as the order M incrases, they all converge for a while. Eventually this gets ragged as the higher order SH's are less well integrated by the integration scheme. The direct cause is ill-conditioning of the HwH matrix. I'll show here a similar plot of log10 of this condition number for each case.

#### Discussion

As you look through the months, the blue mesh line is usually fairly stable, at least up to about M=10. This indicates that the removal of SH components has little effect, because the mesh integration was already good, and replacing with exact integration makes little difference. The other lines start out separated; the black unweighted average often a long way away. But they converge toward the blue line, which says that the SH separation is giving better results.The condition numbers also give a good quantification of the merit of the integration schemes. mesh is markedly better than infilled grid, which is in turn better than simple grid weighting, which in turn is better than no weighting at all. Better conditioning has the practical effect that mesh integration can be used to approximate temperature anomalies with a SH fit (as done monthly here) to higher order, and better resolution, than other methods.

An anomaly in approximation behaviour is February, in which there was convergence toward the blue, but that showed a gradual increase rather than stability after about M=5. I have looked into that; I don't really know what is different about the February mesh that would cause that. The meshes can be seen here. It is true that the February mesh does do a less accurate job with integrating the harmonics, as can be seen from the top row of HwH. I can't track that down to a visible difference in the mesh. The condition number for February rises earlier than it does for March, although not so different from January.

I had thought that, although subtracting fitted SHs did not improve mesh integration here, it might do so if stations were fewer. I tried that in conjunction with my culling process. However, no improvement was found there either. The reason seems to be that when stations are too sparse for SHs to be well integrated, so there is room for improvement, they are also too sparse to allow the fit to be identified. There just aren't enough degrees of freedom.

#### Conclusion

The idea of separating out an "expected" component prior to integration, which can then be integrated with a more accurate operator, has general application. In the case of spatial integration of monthly anomalies on the sphere, it gives a radical improvement of inferior methods. It does not do much for the better mesh integration, but the different behaviour quantifies the merits order of the various schemes.Although the mesh method seems to be at least as good as SH upgrades of grid methods, the latter is faster if meshes would have to be calculated. Simple gridding is fast, as is the SH fitting. However, on my PC even full meshing takes only a few minutes. So I have upgraded the "SH method" in TempLS to be the enhancement of the mesh method, since I can generally re-use meshes anyway.