Sinkhole detection using geophysics

Hmm, I suppose it would depend on the target depth, estimated diameter of the sinkhole, and especially age of the sinkhole. I've seen active sinkholes filled with loose rock, which would be tough to get ERT electrodes in, but ancient sinkholes might just be a depression filled with whatever material deposited on top of them. A lot of methods could potentially work, but there are a lot of limitations to any one of them.

ERT: Probably what I would use if I knew the location of the sinkhole well enough to target it with a surface array, and the surface material has good enough contact resistance to get high quality data from your survey. You can scale an ERT survey to whatever size you need, with larger electrode spacings trading data density for increasing depth of penetration, so you could detect voids of almost any size down to significant depths (you'd need some basic modelling to quantitatively guess what size target you could capture at what depth for any given array). This all depends on there being a resistivity contrast between the void/material that filled the void, and the surrounding material. Any technique is going to require some type of contrast in physical or electromagnetic properties, though.

EM: An EM survey may be quicker to run than an ERT survey, though you won't be able to get as much information in 3D as with ERT. You can get around this restriction to some extent with a multi-frequency or multi-offset EM system, like the EM38-MK2, which can capture four different depths of investigation by having two coil offsets (and then horizontal and vertical dipole orientation for each offset).  Although, that specific system has far too shallow a depth of investigation for detecting sinkholes. In reality, if you want 3D information, ERT is probably easier and more accessible. At any rate, if there is a conductivity contrast between the sinkhole and the surrounding material, you should be able to capture the general size of the sinkhole by the size of the associated EM anomaly.

GPR: I haven't seen much on using GPR to study sinkholes, but I can imagine there would be a relative permittivity contrast at the sides, top, and bottom of the sinkhole. A quick search turned up a paper titled "Characterizing Sagging and Collapse Sinkholes in a Mantled Karst by Means of Ground Penetrating Radar (GPR)" by Rodriguez et al. (2014) in Environmental & Engineering Geoscience. They point out that, at the time of publication, there had not been much research on investigating sinkholes with GPR. They give a detailed description of their process for identifying the boundaries of sinkholes with GPR. One problem they ran into was attenuation of the signal due to the clay/silt material that had filled in the sinkholes they were studying.

Gravity: If you happen to have a field gravimeter, the inside of a sinkhole is likely to have less dense material (be that void space or less compacted soil) than the surrounding subsurface. A line of gravimeter data could determine the extent of the sinkhole, presuming there were no other major density anomalies, such as a fault.

Seismic: There are people much more qualified than myself to explain the pros and cons of a seismic survey for this particular application.

Magnetics: If you expect the sinkhole infill to have significantly different magnetic properties than the surrounding material, perhaps a magnetometer would be useful, but in most cases, a different methodology would be better.