In recent posts I have been combining GHCN V3C station data with Hadley HADSST3 Ocean data. The station data are located at specific locations, while HADSST3 data are recorded sporadically within a regular 5×5 degree grid. In addition active station locations and active SST cells vary from month to month. This is particularly true for early years in the temperature series.
I have been using IDL to make a Delauney triangulation in (lat,lon) of all active measurements globally for each month, and then investigate different interpolations of this onto regular grids. However, following a proposal by Nick Stokes, I realised that this is unnecessary because the triangulation itself can produce an almost perfect global average. I believe this is probably the best spatial integration possible because it uses all available data and gives a direct result. There is no need for any interpolation or grid cell averaging. The last two posts showed how such interpolation can introduce biases. This is avoided using triangulation based spatial averaging. Here is the algorithm I used.
- Each triangle contains one measurement at each vertex. Use Heron’s formula to calculate the area of the triangle.
- Calculate the centroid position and assign this an anomaly equal to the average of all 3 vertex values. This centroid value is then used as the average anomaly for the whole triangular area.
- Use a spatial weighting for each triangle in the integration equal to cos(lat)*area. Where Lat is the latitude of the centroid.
- Calculate the global average =
Using this method every possible measurement is used and no extrapolation outside triangular boundaries is needed. Each month is normalised to the surface area covered only by the active triangles. This also means that for areas of dense station data like the US and Europe regional temperature variation can be studied at full resolution. There is no need to average nearby temperature measurements. I think Triangulation is a near perfect perfect solution also to regional and global temperature studies. It is something I will look at next.
To see how all this works in practice we first look at the triangulation results for January 1880 with rather sparse measurements.
In comparison here is January 2015 with dense data in the US and Europe. I am plotting each individual triangle, showing just how dense measurements are in the US. For regional studies it is best to suppress triangular borders and show just the colour coding.
For all anomalies I use the same normalisation period as Hadcrut4 1961-1990. To compare the results to other series I renormalise these also to 1961-1990. The offsets used for each series are shown in the plots. A detailed comparison of recent years is then shown below. The new result neatly lies between those temperature series which use infilling and Hadcrut4 which doesn’t.
A comparison of the monthly values shows a remarkably close agreement with Berkeley Earth.
In conclusion a new method calculating spatial averages of randomly dispersed temperature data has been developed based on triangulation. The original idea for this method is thanks to Nick Stokes. The method has potential benefits also for detailed regional studies.