Probably the two most common and simplest are Barnes and Cressman interpolation (see [1] for a modern implementation of Barnes), which use inverse distance weig

Probably the two most common and simplest are Barnes and Cressman interpolation (see [1] for a modern implementation of Barnes), which use inverse distance weighting to combine observations within a neighborhood. An improvement to these techniques frequently used to grid statistical weather forecasts is the BCDG method (see [2]). The core idea of any objective analysis/interpolation scheme is that you model spatial relationships with some sort of function. Kriging exemplifies this - you fit a spatial covariance model and use that to predict values in the far field, using combinations of nearby values. I don't really have a good link or textbook reference at arms reach, but you can quickly re-frame this entire interpolation problem as a data assimilation one where you're attempting to approximate a solution to some 2D or 3D field using sparsely sampled observations; in the weather world, the Real-time Mesoscale Analysis used in the USA is a good example of a system which hybridizes a weather model and observations to create a high-resolution analysis [3]. A problem arises when the field you're analyzing can have shocks or variability much smaller than the scale that you're able to measure from your network. A front is a great example - across a front, temperature and wind direction will change over a very small distance, but most interpolation schemes will smear this out. The problem is that you really do care about that front and its location - that's where the interesting weather happens! > And where does someone get high-resolution priors from? A high-resolution weather forecast model, which uses physics to simulate the atmosphere and produce reference states. 3-km forecasts are readily and easily available in the US. > Are you saying that companies have gone out and done a one-time mapping of temperature etc. at a e.g. 1-mile grid across the US? No, this probably wouldn't be useful. Especially because we get a ~1-km mapping of lower troposphere temperatures from geostationary weather satellites every 10-15 minutes across the entire globe. May not be exactly "surface temperature", but it tells you a whole lot about high-frequency variability in temperature fields. [1]: https://gmd.copernicus.org/preprints/gmd-2022-116/gmd-2022-116.pdf https://gmd.copernicus.org/preprints/gmd-2022-116/gmd-2022-1... [2]: https://journals.ametsoc.org/view/journals/wefo/24/2/2008waf2007080_1.xml https://journals.ametsoc.org/view/journals/wefo/24/2/2008waf... [3]: https://journals.ametsoc.org/view/journals/wefo/26/5/waf-d-10-05037_1.xml https://journals.ametsoc.org/view/journals/wefo/26/5/waf-d-1...