One approach would be to use the fact that the surface of a water volume in equilibrium is the same height everywhere, even if the surface is separated. The water level in both ends of a U-shaped pipe is the same. One could use a long pipe that reaches from the measured location to a reference point. Connected bodies of groundwater are large, so that may require pipes of a length in the order of kilometers. Also, with a large distance between the end the air pressure may be different, so there is no equilibrium.
The ground water level itself is a possible reference point, because the ground water can permeate the ground. But we know that it is not in equilibrium, because ground water is removed only in parts of the surface - the wells, and the permeation is slow.
Ground elevation can be measured from satellites, as GPS does. I suspect that the accuracy is too low for many purposes because the relative change in the measured distance is very small. Also, the orbit of a satellite changes with the change of the center of gravity. Because the change is slow, elevation changes in distant regions influencing the satellite create noise in the measurement.
Land that has been heavily disturbed by humans in most developed countries, such as urban and rural regions will have been surveyed for topographical & sub-division purposes.
Such data allows for a digital terrain model (DTM) to be created for such regions. When subsidence occurs, the subsided area can be resurveyed & compared to the original DTM to ascertain the amount of subsidence that has occurred.
Undisturbed or poorly surveyed regions become problematic. The subsidence area & the surrounding area can be surveyed, but then someone has to make a judgement call & decide what the original surface may have looked like to infer an estimate for the degree of subsidence.
The other way is to have satellites periodically measure surface elevations & then to do comparisons between each satellite measurement.
Radar could be used as the measuring system, but LIDAR (& this reference also) can produce more accurate results.
Other forms of subsidence/uplift studies that may require an even higher degree of accuracy, for example, isostatic rebound/subsidence, plate tectonics, etc. are better monitored using a wider mix of other space geodetic techniques, including Very Long Baseline Interferometry, Satellite Laser Ranging, etc. to get down to the millimeter-level. Such techniques use high-end equipment, radio-telescopes, lasers, etc. to help maintain reference frames and model crustal displacements on a global scale.
As a side note, the center of mass can be deduced by observing satellite motions. Satellite laser ranging provides key input as it can measure accurate distances between ground stations and satellites, and VLBI provides Earth orientation and distances between stations. With all those inputs, the origin of the world reference frame (the most recent realization as of 2019 being ITRF2014) is consistent with the center of mass at the centimeter level, and satellite orbits are tracked and corrected to account for Earth dynamics.
The following web page on USGS has interesting information for further reading on this topic: