The spatial integration method used to calculate the global surface temperature has a large impact on the end result. I am using a 3D method based on spherical triangulation of the weather stations combined with SST results from HadSST3. This covers the full surface of the earth and avoids map projection corrections. Here I compare the results I get using CRUTEM4 station data with the traditional results of HadCRUT4.6 both updated for November 2019. First the traditional 5×5 degree method.

Note how the polar regions are sampled by just a few small 5×5 degree bins which are then governed by a cos(lat) projection factor. This problem is avoided in 3D and no projection correction is needed because the triangular surface bins naturally have full coverage. The 3D triangulation method compensates geometrically over areas of sparse coverage.

The net effect of this is to change the spatial integration result for the global surface temperature whenever polar regions are warming faster or slower than elsewhere.

Comparison of recent monthly temperatures with traditional HadCRUT4.6 where CRUT4/HadSST3 uses exactly the same data

Other groups compensate for this in different ways. For example Cowtan & Way use 2D kriging to extrapolate results into unmeasured regions. GIStemp use equal area bins covering the surface but allowing distance weighted contributions from remote stations. Berkeley Earth use a least squares fitting algorithm to also extrapolate into unmeasured regions. These methods more or less agree. I maintain that the most natural surface integration is to treating the surface as a sphere. Spherical triangulation implicitly extrapolates over all triangular elements, yet it remains agnostic to trends.

There is only one other way to avoid both interpolation and spatial bias and that is to use Icosahedral binning. This is the only way I know of to bin data on the surface of a sphere without distortion.

*Related*

Clive,

“There is only one other way to avoid both interpolation and spatial bias and that is to use Icosahedral binning.”The spatial bias, which is the main problem, is due to the inadequate treatment of empty cells, and icosahedral binning won’t fix that. The problem is that just omitting empty cells assigns to them the average value of the remaining cells overall, and for Arctic etc that just isn’t right. The right thing to do is to make a proper estimate of the missing cells using local information. Kriging is one way, but is overkill.

I’ve described how I now do it here and elsewhere. The best estimate of a missing cell is the average of neighboring cells. But they may be empty too. That can be fixed. You can just write the linear equations expressing that relation, and solve them. That is equivalent to solving the Laplace equation, with the known cells as boundary condition. And that is just the 2D equivalent of linear interpolation for the missing cells.

It works as well as the triangle mesh. I have been describing FEM methods which I think are probably better; I’ll post more about that soon. The Laplace usage is a part of that too.

Nick,

Yes, You’re right that if there is an underlying temperature trend in space which is only sparsely sampled using icosahedral binning won’t solve that.

However fitting into unmeasured zones assumes that temperature is a smoothly varying function over the earth’s surface. Doesn’t solving Laplace’s equation say that this function is a sum of spherical harmonics ?

FEM methods in 2D seems to take that one step further, However I am no expert!

Clive

“However fitting into unmeasured zones assumes that temperature is a smoothly varying function over the earth’s surface. “Everything is an unmeasured zone, except where there is a thermometer. The grid cells are artifices. You assume within the cell that the cell average applies, but knowledge does not end at the cell boundary (and was not perfect within).

Spherical harmonics are solutions of the Laplace equation (without boundaries), but so is any real (or Im) part of an analytic function. Including, of course, a plane. As used here, it is the 2D equivalent of linear interpolation (2nd deriv kinda zero).

I’ll be writing more about the FEM method soon, but I think the laplace idea is a separate and useful way if fixing methods. Having written down the equations, you can solve them by standard methods like conjugate gradient, or just by diffusion.