How do geologists measure green house gass emmissions from large regions
Tracking atmospheric species over the entire planet really is comparatively easy via satellites, compared to plastering entire swathes of land with measurement stations.
You make one infrared image from an imaging satellite at the right wavelength and you got the entire country covered. The right wavelength to choose has to be

originating from the molecule in emission at the right altitude and
be optically thin. If you know your emission is optically thin, the flux in your image is directly proportional to the mass that is emitting this flux. If it's not optically thin, complications apply, but the idea is the same.

An example of the observational fleet that NASA has in orbit can be found here (scroll down the page). There it also becomes clear, that various instruments are needed to cover the entire range of interesting atmospheric species, which necessitates a number of observation platforms/satellites.