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When groundwater is pumped out of the ground by wells, and not replenished fast enough, the groundwater level sinks.

With it, the surface of the area sinks also. Locally, this effect is not apparent, because heights of objects are specified relative to the surface. The change in height stays zero by definition. But relative to the average elevation of surrounding land not connected in terms of ground water, there is a change. What changes is the distance between soil surface and center of gravity of the earth. The subsidence is a change in the shape of the earth, so its center of gravity changes itself, but the magnitude of that change is small enough compared to the surface change to ignore it.

How can the change in surface elevation of a fixed point on the surface of the Earth be measured?

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.

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  • $\begingroup$ To my knowledge the safest/most true and most common method is radar data mapped by satellites. $\endgroup$ – Erik Oct 2 at 10:15
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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.

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Several methods are employed to measure ground subsidence: Spirit leveling, GPS (GNSS), Remote Sensing, Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging to name a few. In the case of modeling subsidence caused by water and oil extraction, the most common measuring methods seem to be Spirit leveling, GPS and Remote Sensing (Interferometric synthetic aperture radar and radar altimetry), with varying degrees of accuracy.

  • Spirit leveling is an efficient and cheap method to determine point elevations. It works well for relatively small areas up to several kilometers, and the surveys can be achieved at less cost. However, when the study area is very large (country or continent-wide), the overall accuracy and consistency of this classical method falls short of space-based techniques: continent-wide classical surveys can have distortions up to a meter or more. Also, it needs good benchmarks, ideally located on a more stable portion of ground around the study area.
  • Continuously operating GNSS stations are another way to measure subsidence by plotting positions regularly and creating time series to detect movement with an accuracy of about a centimeter. This method has the advantage that it is accurate over large areas, however, it is much more expensive, so fewer points can be measured, resulting in a sparser spatial coverage of the study area.
  • Remote sensing with InSAR and satellite altimetry can also provide very dense Digital Elevation Models for little cost, and works well for small to medium sized areas, and usually has a decent accuracy of a few centimeters. They can provide interesting input maps for the study of land subsidence.
  • Other tools such as piezometers, extensometers, can measure compaction and water elevation in aquifers to provide other useful data to better understand subsidence.

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:

Measuring and Monitoring Land Subsidence in California

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In part, this is a geodesy question. I'm not an expert in that area, but there are experts on this site, so perhaps they'll come along. But I want to focus on one thing in the question:

heights of objects are measured relative to the surface

They don't have to be. Heights of objects are measured relative to some vertical datum. That might be the local ground level (but as the question notes, if what you are measuring is local ground level than that's clearly not useful). They might be measured compared to some datum representing sea level. Or they might be measured compared to the geoid, or some reference ellipsoid. The last of these is what is happening when you use GPS measurements.

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  • $\begingroup$ Right, I mean they are specified relative to the surface. Of course I can express a measurement relative to other references, but how do I get it in the first place? $\endgroup$ – Volker Siegel Oct 2 at 19:14
  • $\begingroup$ For that, see the other answer. (D)GPS, or satellite-based radar altimetry.. $\endgroup$ – Semidiurnal Simon Oct 2 at 19:49

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