To do this, it is necessary to have a space of approximating functions. I'm using spherical harmonics - typically up to 200. So far I've analysed a model in which the global temperature is assumed to consists of a linear trend, spatially varying. But it will be possible to do yearly or monthly plots and, I think, trends and spatial averages for sub-regions, subject to enough data being available. I'd be interested in suggestions and comments, which is my excuse for posting at this fairly early stage.
Below the jump, I'll compare trends over the Earth's surface with plots from the AR4.
These plots used a total of 140 spherical harmonics, which gave reasonable resolution. The calculation is still quite fast, about 100 sec, so a lot more could be used. It is a whole sphere calculation - I've not plotted at the Arctic and Antarctic extremes, because the results then become extrapolations.
First, here are the plots of global trends from the AR4. It's Fig 3.9 from Sec 22.214.171.124.
and the caption says:
Figure 3.9. Linear trend of annual temperatures for 1901 to 2005 (left; °C per century) and 1979 to 2005 (right; °C per decade). Areas in grey have insufficient data to produce reliable trends. The minimum number of years needed to calculate a trend value is 66 years for 1901 to 2005 and 18 years for 1979 to 2005. An annual value is available if there are 10 valid monthly temperature anomaly values. The data set used was produced by NCDC from Smith and Reynolds (2005). Trends significant at the 5% level are indicated by white + marks.
The least squares method doesn't have those minimum requirements, although of course reliability diminishes when the data is sparse. Here are the TempLS plots for the global land/ocean fit: